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THE DERIVATION OF PROFILED SURFACES

FROM DEFLECTED ELASTIC MEMBERS

by

GERALD DRUCE

A thes is submitted fo r the Degree o f Doctor o f Philosophy in the

Faculty o f Engineering o f the U n ive rs ity o f Surrey

Department o f Mechanical Engineering November 1976

SUMMARY

Cam mechanisms have many app lica tions in automatic machinery. Considerations o f dynamic performance require the p ro f i le shape to be defined in d ire c t ly from the fo llo w e r motion, the f l e x ib i l i t y o f tim ing and con figu ra tion in troduc ing a d d itio n a l va riab les . Consequently the design and manufacture o f new master cams using conventional techniques is laborious and expensive. P recis ion is essentia l since the fo llo w e r motion is sen s itive to minor dimensional inaccuracies and d is c o n tin u it ie s o f curvature.

The ob ject o f th is in ve s tig a tio n is to s im p lify both the design and manufacture' o f a new master cam.

The au tho r’ s survey o f previous work showed i t necessary to es tab lish the design sequence before the processes in vo lv in g greatest e f fo r t could be iso la te d . H is associated study o f e x is tin g p ro f i le copying and generating devices found none which s a tis f ie d the c r i te r io n o f extensive a p p lica tio n , but id e n t if ie d weaknesses of previous designs. This thes is considers the adaption o f the copying p r in c ip le to manufacture cam p ro f i le s from the in t r in s ic shape o f a le a f spring deflected to s u it spec ified boundary cond itions, so e lim ina tin g dependence upon co-ord inate data?complex transm issions and specia l templates w h ils t re ly in g upon the re p e a ta b ility o f an instrument in preference to the p o s itio n in g accuracy o f a m achine-tool. P ro file s fo r two d is t in c t fo llo w e r motions can be derived by th is means, th e ir c h a ra c te r is tic s resembling those o f SHM and cyc lo id a l motion re sp e c tive ly .

The c h a ra c te r is tic s are functions o f the displacement equation.A search revealed no comparison nor experimental v e r if ic a t io n o f p rev ious ly published analyses o f the e la s tic a . L im ita tio n s o f the previous analyses led the author to derive new so lu tions using the method o f pe rtu rba tions; these have the advantage o f simple computation and give a d ire c t so lu tio n fo r any interm ediate p o s it io n . Experimental studies showed good c o -re la tio n between th e o re tic a l and measured d e fle c tio n s .

The attachments were designed and manufactured and p ro f i le s cut using th is process.

ACKNOWLEDGEMENTS

I wish to thank my supervisors, Professor J. M. Zarek, Dipl«Ing., B.Sc., Ph.D., C.Eng., M.I.Mech.E., Head of the Department of Mechanical Engineering, University of Surrey, who granted me permission and facilities to undertake this investigation, and Mr. D. F. Nettell, B.Sc., C.Eng., F.I.E.E., F.I.Mech.E., Chief Engineer, Hydraulics Division, Vickers Limited, for their guidance and encouragement.

Acknowledgement is also made to Professor I. M. Allison, Ph.D., C.Eng., M.R.Ae.S., for reading the original draft and making many helpful suggestions, to Mr. B. G. Maudsley of International Computers Limited, for granting permission to continue this investi­gation, to Mr. A. Morel and Mr. J. G. Green for constructing and assisting with the operation of the attachment, to members of the Computing and AVA units of the University of Surrey for assistance and the use of facilities and to Mrs. M. J. Hamilton for typing the manuscript.

ACKNOWLEDGEMENT OF COPYRIGHT

The fo llo w in g fig u res and tab les are reproduced from papers by the author w ith the permission o f the respective e d ito rs :-

Conference Proceedings, In s t i tu t io n o f Mechanical Engineers:-

F igu res :- 3-2, 3-1? 5-5, 5-6, 5-7? 5-9? 5-10,5-12, 5-13, 5-1^•

Tables:- ■ 3-1, 11-1.

Engineering:-

F igu res :- 1-2, 2-1, 2-2, 3-6.

Machine Design and C on tro l: -

F igu res :- 2-3, 8-19, 8-20, 8-21.

The author also acknowledges the permission of the above and the e d ito r o f Control and Instrum entation to copy supporting papers and thanks to the e d ito r o f Machinery and Production Engineering fo r permission to reproduce F ig. 4-4 from Childs (21) and F ig.5.11 from Grodzinslci (47).

-V-

Summary i i

Acknowledgements i i i

Acknowledgement o f copyright iv

L is t o f chapters ’ v i

L is t o f appendices x i

L is t o f p la tes x i i i

L is t o f fig u re s x iv

L is t o f tab les xx

Notation x x i i i

CONTENTS

Page

-vi-

Chapter 1 INTRODUCTION 21.1 Purpose and a pp lica tions o f cam mechanisms 21.2 Configurations o f the cam mechanism 31.3 Advantages o f p ro f i le de riva tio n 41.4 Objects o f the in ve s tig a tio n 51.5 Analysis o f the de flected le a f spring 6

1.6 Summary 7

Chapter 2 ' THE DESIGN OF CAM MECHANISMS 92.1 S pec ifica tio n o f the mechanism 92.2 Timing and cam angles 102.3 Follower motion 102.4 L im itin g fa c to rs 112.5 Pressure angle, curvature and surface stress 12\2.6 Use o f design data 122.7 A n c illa ry d e ta ils 132.8 Continuous path manufacture l42.9 Summary 14

Chapter 3 THE DYNAMIC PERFORMANCE OF CAM MECHANISMS 153.1 The fo llo w e r motion 153.2 S ign ificance o f the cam ch a ra c te ris tic s 173-3 P ra c tica l considerations l 8

3*4 Determination o f the secondary motion causedby p itc h curve e rro rs 19

3.5 Choice o f the cam law 233.6 Summary 24

Contents (cont)Page

PART 1 INTRODUCTION 1

-vii-

PART 2 PROFILE MANUFACTURE 25

Chapter 4 CONVENTIONAL METHODS OF MANUFACTURINGCAM PROFILES - 26

4.1 Q ua lity o f dynamic performance 26

4.2 S pec ifica tion and manufacture 264.3 The p lunge-cu tting process 28

4.4 A pp lica tion o f numerical co n tro l 3°4.5 Accuracy o f manufacture 334.6 Summary 36

Chapter 5 AIDS TO CAM MANUFACTURE AND PROFILE-GENERATING MACHINES 37

5.1 Copying and generation o f p ro f ile s 375.2 Design requirements fo r a cam p ro f i l in g

machine 38

5.3 P ra c tica l considerations 405.4 Aids fo r cam p ro f i le manufacture 4 l5.5 P ro f ile copying machines 425.6 P ro file generation 435.7 Gear driven generating machines 475.8 A pp lica tion o f geneva mechanisms 495.9 Assessment o f previous designs 495.10 Summary 525.11 Conclusion 53

PART 3 PROFILE DERIVATION AND DEFLECTION OFTHE ELASTICA 54

Chapter 6 THE DERIVATION OF CAM PROFILES FROM THESHAPE OF A DEFLECTED LEAF SPRING 55

6.1 Requirements fo r p ro f i le c u ttin g 556.2 Accuracy o f p ro f i le manufacture 556 .3 Cost o f p ro f i le manufacture 576.4 Ease o f use 586*5 F le x ib i l i t y o f p ro f i le c u ttin g 59

Contents (cont)Page

-Vlll-

6.6 Design ob jec tives fo r a cam -pro filingmachine 60

6.7 D eriva tion o f cam p ro f i le s from the shapeo f a deflected le a f spring 6 l

6.8 The manufacturing process fo r derivedp ro f i le s 64

6.9 Summary 65

Chapter 7 ANALYSIS OF THE DEFLECTED LEAF SPRING 67

7.1 A pp lica tion o f the analys is 67

7.2 Requirements and assumptions 67

7.3 The approximate theory o f bending 69

7.4 S ign ificance o f lo n g itu d in a l forces 707.5 L im itin g proportions 737.6 Analysis o f the e la s tic a 737.7 A new so lu tio n o f the bending moment equation

fo r la rge curvature using the method o f pe rtu rba tions 77

7.8 D e flec tion due to shear force 82

7.9 A n tic la s t ic curvature 83

7.10 Conclusions 847.11 Summary 86

Chapter 8 DERIVATION OF THE CAM CHARACTERISTICS ANDASSOCIATED PARAMETERS 89

8.1 Use o f the cam c h a ra c te r is tic s 89

8.2 C ha rac te ris tics fo r Simple Derived Motion 89

8 .3 Performance o f Simple Derived Motion 918.4 Leaf s e ttin g fo r F in ite Pulse Derived Motion 928.5 C ha rac te ris tics fo r F in ite Pulse Derived Motion 948.6 Performance o f F in ite Pulse Derived Motion 958.7 L im itin g parameters 978.8 Pressure angle 978.9 P ro file curvature and surface stress 988.10 D riv ing torque 1018.11 ’’Asymmetric” motion 1028.12 Summary 1°4

Contents (cont)Page

~ix-

Chapter 9 DESIGN OF THE ATTACHMENT 1059.1 D eriva tion o f the p ro f i le 1059.2 The datum in d ic a to r 1079 .3 The measurement o f le a f d e fle c tio n 108

9 -3 * l O ptica l methods 1089-3*2 A pp lica tion o f transducers 1109 . 3 .3 D irec t measurement - the datum

in d ic a to r 1 11

9-3*4 Compensation o f e rro rs 1139.4 Transmission between ro ta ry tab le and s lid e 1149 -5 . The s lid e l l 6

9.6 Accuracy o f le a f s e ttin g 1179-7 Control o f p ro f i le wavyness l l 8

9.8 Summary 119

Chapter 10 DEVELOPMENT OF THE LEAF SPRING MOUNTING 12210.1 S e tting the le a f 12210.2 Requirements fo r the le a f 12310.3 P re lim inary d e fle c tio n te s ts 124

10.3*1 Slope and d e fle c tio n measurements 12510.3*2 Conclusions 127

10.4 Use o f the attachment 12810.4.1 R epea tab ility and p os itio n in g te s ts 12910.4.2 D eflection measurements 12910.4.3 Conclusions 130

10.5 Development o f the attachment 13110.5.1 In ve s tig a tio n o f end e rro rs 13110.5*2 Re-design o f the le a f mounting 13210.5*3 D e flec tion te s ts on leaves mounted

between dowels and knife-edges 135

10.5*4 F ina l version o f the le a f mounting 13410.5-5 D eflec tion te s ts mounting the leaves

between dowels and ro lle rs 15510.5.6 Conclusion 15&

10.6 Proportions o f the deflected le a f 15&10.7 Summary 157

Contents (cont)Page

-x-

Chapter 11 COMMENTARY AND RECOMMENDATIONS FORFURTHER RESEARCH 139

11.1 The improvement o f p ro f i le manufacture 13911.2 Analyses o f the elast-ica l 4 l11.3 Leaf m ate ria ls 14311.4 The le a f mounting l4411.3 Assessment o f p ro f i le de riva tio n 14611.6 Continuous c u ttin g . 14711.7 Assessment o f the derived p ro f i le s 14811.8 Associated inve s tig a tio ns 150

11.8.1 P ro f ile accuracy 15011.8.2 Design procedure 15111.8.3 Angular motion and o ffs e t o f the

r o l le r 15211.8.4 N um erica lly -contro lled p ro f i le

manufacture 15211.8.5 Analysis o f the e la s tica 153

11.9 Conclusion 15311.10 Summary 153

REFERENCES 157

Contents (cont)Page

LIST OF SUPPORTING PAPERS 172

-xi-

APPENDICES

Appendix 1

A l !

A l ! (a)

A l ! (b)

A1.2

A na lys is o f the de flec ted le a f

sp ringA na lys is o f the curve o f d e fle c tio n using the Method o f P ertu rbations to

co rre c t the curvature equation S o lu tio n ne g lec ting the lo n g itu d in a l

end fo rcesS o lu tio n in c lu d in g the lo n g itu d in a l

end fo rcesSeries s o lu tio n fo r the curve o f d e fle c t io n o f a c a n tile v e r in the

e la s t ic a range derived from e l l i p t i c

in te g ra ls

Page

A4

A4

A 7

A l l

A21

Appendix 2

Appendix 3

A na lys is o f the earn c h a ra c te r is tic s

fo r Simple Derived Motion

» *A na lys is o f the cam c h a ra c te r is tic s

fo r F in ite Pulse Derived Motion

A 38

A 47

Appendix 4 P repara tion and examples o f the manu­

fa c tu r in g s p e c if ic a tio n Example 1. To determine the s p e c if i­ca tio n fo r manufacturing the p ro f i le

c o n tro ll in g a r is e movement w ith Simple Derived Motion Example 2. To determine the s p e c if i ­c a tio n fo r m anufacturing the p ro f i le to d rive a r is e movement w ith F in ite Pulse Derived Motion Example 3* To determine the le a f dimensions fo r manufacturing the p r o f i le c o n tro ll in g a r is e movement w ith 1asymmetric 1 Simple Derived

Motion

A6l

A64

A71

A79

\

xii-

Appendix 5

Appendix 6

The manufacture o f cam p ro f i le s fo r

Simple Derived Motion

The manufacture o f cam p ro f i le s fo r

F in ite Pulse Derived Motion

Page

A84

A8?

Appendix 7

A7.1

A7.2

Appendix 8A8.1 A8.2 A8.3 -A8.4 A8.5

A S ,6

A8.7A 8,8

A8.9A8.10

Accuracy o f the machine-tool and

transm issionAccuracy o f the v e r t ic a l knee-and-

column type m il l in g machine

Accuracy o f the spur gear t ra in

Computer programs In tro d u c tio n Common procedures 'LEAFSLOPE' 'LEAFPERTS' 'DEFLECTION' 'SDM-CHAR' 'SDM-PRESANG ’FPDM-PROP' 'FPDM-DISPL * •FINDIF-WAY1

& ' SDM-DISPL1

& . ’ SDM-CURV'

'FPDM-CHAR*’FINDIF-ACCN'

A89

A89

A91

A92A92A92

A93 A94 A95 A96

A9 6

A97 A98

A99

Appendix 9

A9.1A9.2

A9-3A9.4

A9-5

Equipment and m ate ria ls used M ill in g machine Rotary tab le D ia l gauge

»Leaf spring Cam blank

A101

A101A101A101A102A102

PLATES

The p la tes

Plate 1

P late 2

P late 3

Plate 4

P late 5

P late 6

fo llo w the te x t and appendices.

The scalloped surface o f the cam p ro f i le a fte r plunge c u ttin g a t increments o f 0°20’ .

General view showing the E l l io t v e r t ic a l m ill in g machine set up to cut a cam p ro f i le fo r Simple Derived Motion.

Rotary ta b le , transm ission and s lid e mounted on the tab le o f the E l l io t m ill in g machine.

A complete cam p ro f i le a f te r plunge c u tt in g .

The datum in d ic a to r from the rear o f the m ill in g machine.

Leaf spring mounted between dowels and knife-edges, showing sandwich construc tion o f one dowel.

P late 7 Leaf spring mounted between dowels and r o l le r s .

-xiv-

ITGUEES

The fig u res fo llo w the te x t and appendices, those marked * areduplicated in the te x t .

1-1 The basic cam mechanism.1 - 2 * Terminology - disc cam d riv in g a ra d ia l tra n s la t in g fo llo w e r

2-1 Diagram i l lu s t r a t in g the design procedure fo r a new cammechanism.

2-2 Diagram i l lu s t r a t in g the a p p lica tio n o f design data todetermine l im it in g values.

2-3 R o lle r fo llo w e r; pressure angle and th ru s t.

3-1 Invers ion o f the ra d ia l cam mechanism.3-2* L i f t c h a ra c te r is tic s . Comparison o f the i n i t i a l displacements

from re s t fo r d if fe re n t cam laws.3-3* Overlap o f to lerance zones: SHM and Modified Trapezoidal

Acceleration Motions.3-4 Enlarged section o f displacement ch a ra c te r is tic i l lu s t r a t in g

the e ffe c t o f ra d ia l and angular e rro rs .3-5 Graph o f p itc h curve wavyness caused by round-o ff e rro r.

C yclo ida l motion.3-6 Graph o f estimated secondary acce le ra tion o f fo llow e r caused

by round-o ff e rro r . C yclo ida l motion.

4-1 Arrangement o f the machine-tool fo r c u ttin g cam p ro f i le s .4-2 Scalloped surface o f the p ro f i le a fte r the increment c u ttin g

operation.4-3 Influence o f c u tte r path upon the shape o f the p itc h curve:

comparison o f ra d ia l and o s c il la t in g motions.4-4 Curve approximation by p o in t- to -p o in t c u tte r movements w ith in

a to lerance band using numerical c o n tro l.4-5 In te rp o la tio n of a curve by chords using the continuous path

f a c i l i t y o f numerical c o n tro l.4-6 Curve approximation using c irc u la r arc in te rp o la tio n fo r

num erica lly -con tro lle d continuous path manufacture.

-XV-

5-1 Diagram illustrating the design procedure for a cammechanism using a p ro f i le generating or copying device.

5-2 Standard form o f the d e ta il drawing fo r a cam to he cuton a generating or copying device.

5-3 S e tting in s tru c tio n s . Simple Derived Motion.5-4 Functions o f a p ro f i le copying machine.5-5 Diagrammatic arrangement o f a p ro f i le generating mechanism:

SHM.5-6 Graph showing the combination o f lin e a r and sinuso ida l

components o f motion to derive the displacement ch a ra c te r is tic o f cyc lo id a l motion.

5-7 Arrangement o f a double-rack mechanism to generate p ro f i le sfo r cyc lo id a l motion.

5-8 S im p lifie d version o f the double-rack generating mechanism.5-9 Arrangement'o f a double s lid e mechanism combining lin e a r and

sinuso ida l motions to generate p ro f i le s fo r a f in ite -p u ls e motion.

5-10 Graph showing relative movements to generate the profile fora specified lift: double-slide mechanism.

5-11 Graphs o f displacement c h a ra c te r is tic s . P ro file s generatedby an eccentric p lanetary gear mechanism.

5-12 Functions o f a p ro f i le generating device co n tro lle d by aneccentric p lanetary gear mechanism.

5-13 Functions o f a p ro f i le generating device driven throughsynchronised geneva mechanisms.

5-14 D e ta il o f d if fe re n t ia l gear t ra in fo r geneva-driven generatingdevice.

5-15 Diagrammatic displacement c h a ra c te r is t ic : geneva-drivengenerating device.

6-1 Displacement c h a ra c te r is tic fo r an "asymmetric" fo llo w e r motion.6-2* (a) Leaf spring deflected fo r Simple Derived Motion.

(b) Leaf spring deflected fo r F in ite Pulse Derived Motion.6-3 Plan view showing the layout o f equipment on machine-tool to

cut a cam p ro f i le fo r Simple Derived Motion.

Figures (cont)

-xvi-

7-1*

7-2*

7-3*7>

7-5*

Figures

7-6*

7-7

7-87-9

8-1

8-2

8-3*8-4*8-58-6

8-7

8-8

8-9*

8-10*

8-11

8-12

Diagram i l lu s t r a t in g the de riva tions of a lte rn a tiv e analys is o f the e la s tic a .Forces and moments acting on one span o f the deflected le a f spring .C antilever deflected in the e la s tica range.S e tting data. Graph o f l im it in g d e fle c tio n due to bending s tress and contact angle fo r given span o f le a f spring .Graph comparing the d e fle c tio n a t -zj-span due to combined bending and lo n g itu d in a l forces found from a lte rn a tiv e analys is o f the e la s tic a .Analysis o f the e la s tic a using the method o f pe rtu rba tions . Graph comparing the slope a t mid-span found by expanding 1, 2 and 3 terms o f the se ries .Graph comparing d if fe re n t analyses o f the e la s tic a . Angle o f slope a t mid-span fo r vary ing o ffse t-span ra t io .Forces acting upon elements o f a beam subject to pure bending. Forces ac ting to suppress a n t ic la s t ic curvature on a section o f a slender beam subject to a large d e fle c tio n .

Graph o f angle o f slope a t mid-span fo r varying o ffse t-span r a t io . E l l ip t ic in te g ra ls s o lu tio n .Displacement c h a ra c te r is t ic : Simple Derived Motion.V e loc ity c h a ra c te r is tic : Simple Derived Motion.Acce lera tion c h a ra c te r is tic : Simple Derived Motion.Pulse c h a ra c te r is tic : Simple Derived Motion.S e tting data: F in ite Pulse Derived Motion.D eriva tion o f the displacement c h a ra c te r is tic fo r F in ite PulseDerived Motion from the de flected le a f spring .General case o f the transform .Example o f V e loc ity c h a ra c te r is tic : F in ite Pulse DerivedMotion.Example o f Acceleration c h a ra c te r is tic : F in ite Pulse DerivedMotion.S e tting data: ra t io o f d e fle c tio n to o ffs e t a t ^r- and -^-spanfo r varying er .Design data: magnitude o f maximum pressure angle, SimpleDerived Motion.

(cont)

(cont)

Design data: p o s it io n o f maximum pressure angle, SimpleDerived Motion.Design data: minimum convex ra d ii o f curvature, SimpleDerived Motion.Design data: minimum concave r a d ii o f curvature, SimpleDerived Motion.Design data: exte rna l load torque fa c to r, Simple DerivedMotion.Design data: in e r t ia force torque fa c to r, Simple DerivedMotion.Design data: spring torque fa c to r , Simple Derived Motion.Graph comparing externa l load torque fa c to rs fo r d if fe re n t cam laws.Graph comparing in e r t ia force torque fa c to rs fo r d if fe re n t cam laws.Graph comparing compensated spring torque fa c to rs fo r d if fe re n t cam laws.

I l lu s t ra t io n s o f machine-tool po s itions fo r p ro f i le c u tt in g : Simple Derived Motion.P os ition e rro r due to probe t ip rad ius and apparent increase o f le a f th ickness.L im itin g contact angle: F in ite Pulse Derived Motion.S e tting data: graph o f maximum probe t ip rad ius and apparentthickness e rro rs fo r vary ing S ' .S e tting data: graph o f probe th ru s t e rro r fa c to r fo r varying. <T.Example o f components o f e rro r in the measured d e fle c tio n o f the le a f sp ring .Spur gear transm ission between the ro ta ry tab le and the s lid e . A lte rn a tive f le x ib le band transm ission between the ro ta ry tab le and the s lid e .Forces acting upon the s lid e assembly.

-xviii-

10-1 *10-2

10-3

10-4

10-5

10-6

10-7

10.8

10-9

10-10

10-11

10! 2

10 -13

10-14

Figures

10-15

A lte rn a tive means o f d e fle c tin g the le a f spring .Graph comparing th e o re tic a l and measured slopes a t mid-span fo r varying o ffse t-span ra t io s .Graph o f measured d e flec tions a t an^ -J-span fo r varying o ffs e t . Leaves deflected between o ffs e t clamps.Graph comparing th e o re tic a l and measured de flec tion s of le a f spring fo r a complete p r o f i le .Graph comparing f i r s t d iffe rences between de flec tions over increments o f cam angle fo r a complete p ro f i le .Graph comparing th e o re tic a l and measured de flec tion s a t one end o f the span: le a f de flected between o ffs e t clamps.Graph comparing th e o re tic a l and measured de flec tions a t one end o f the span: le a f de flected between o ffs e t clamps.Arrangement o f a le a f mounted between dowels and ro lle rs de fin ing s ig n if ic a n t dimensions.Graph comparing th e o re tic a l and measured d e flec tions a t one end o f the span: le a f mounted between dowels and knife-edgesGraph comparing th e o re tic a l and measured de flec tions a t one end o f the span: le a f mounted between dowels and knife-edgesLeaf deflected by dowels and ro l le r s .Graph comparing th e o re tic a l and measured d e flec tion s a t one end o f the span: le a f de flected between dowels.Graph comparing th e o re tic a l and measured de flec tion s a t one end o f the span: le a f de flected between dowels and ro l le r s .Graph comparing th e o re tic a l and measured de flec tion s a t increments o f cam angle over h a lf the angle o f l i f t : le a fdeflected between dowels.Graph comparing th e o re tic a l and measured de flec tions a t increments o f cam angle over h a lf the angle o f l i f t : le a fdeflected between dowels and r o l le r s .

(cont)

-xix-

Figures re la t in g to Appendices

A3-1 F in ite Pulse Derived Motion. Displacement derived fromf i r s t span.

A3~2 F in ite Pulse Derived Motion. Displacement derived fromsecond span.

A3-3 F in ite Pulse Derived Motion. Displacement derived fromth ird span.

A5-1 C a lib ra tio n o f the datum in d ic a to r . Method o f a lig n in g the probe t ip w ith the c e n tre - lin e o f the dowel.

A6-1 F in ite Pulse Derived Motion. Arrangement o f deflectedle a f on s lid e fo r copying the p ro f i le shape.

Figures (cont)

A8-1* Flow chart fo r computer program 'LEAFPERTS'.

-XX-

TABLES

The tab les fo llo w the te x t, appendices and fig u re s , those marked * are duplicated in the te x t .

3-1 Summary of experimental results (8 2) comparing the performanceo f constant acce le ra tion , simple harmonic and cyc lo id a l motions.

3-2 F ir s t d iffe rences between p itc h curve r a d ii fo r increments o f 0 /J3 - o . l a t s ta r t o f r is e movement. Comparison o f simple harmonic, cyc lo id a l and m odified trapezo ida l acce le ra tion motions.

4-1 Accuracy o f p ro f i le manufacture by various processes.4-2 Accuracy o f machining spherica l surfaces using n-c machine-tools

(104).

6-1 P ro file wavyness fo r varying to lerance. Linear in te rp o la tio n o fc irc u la r arcs fo r n~c continuous path machining.

6-2 Cost ana lys is . Components o f cost fo r d if fe re n t cam manufacturingprocesses.

6-3 R elative costs o f master cam p ro f i le manufacture by d if fe re n tprocesses.

6-4* Histogram: re la t iv e costs o f master cam p ro f i le manufactureusing d if fe re n t processes.

7-1 Comparison o f assumptions invo lved in a lte rn a tiv e analyses o f the elastica*, accuracy and parameters required fo r a numerical s o lu tio n .

7-2 S ign ificance o f lo n g itu d in a l end forces upon the shape o f adeflected le a f spring .(a) Approximate theory of bending.(b) So lution by method o f pe rtu rba tions .

7-3 L im itin g d e fle c tio n o f le a f springs due to maximum bending stressfo r varying th ickness, span and o ffs e t.

7-4 Comparison o f and o ffse t-span ra t io fo r d if fe re n t so lu tions o f the e la s tic a .

7-5 Percentage d iffe rence between the o ffse t-span ra t io s fo r given <Tca lcu la ted from the e l l ip t i c in te g ra ls so lu tio n , the approximate theory o f bending and the method o f pe rtu rba tions .

7-6 V a ria tion w ith cr o f parameters a ffe c tin g the a n t ic la s t iccurvature o f the le a f .

-xxi-

8!

8-2

8-3*

8-4

8-5

8-6

8-7

8-8

8-9

8-10

9-1

10-1

10-2

10-3

10-4

10-5

Tables

10-6

Simple Derived Motion. S ig n ific a n t values o f the dimensionless forms o f the cam c h a ra c te r is tic s .F in ite Pulse Derived Motion. S ig n ific a n t values o f the dimension­less forms o f the cam c h a ra c te r is tic s .F in ite Pulse Derived Motion. Equations o f the displacement c h a ra c te r is t ic .Equations o f the components o f d r iv in g torque and the dimension­less torque fa c to rs .D eriva tion of the spring s t if fn e s s compensating fac to rs fo r d if fe re n t cam laws.Simple Derived Motion. Dimensionless form o f the displacement c h a ra c te r is tic fo r varying ^ .Simple Derived Motion. Dimensionless form o f the v e lo c ity c h a ra c te r is tic fo r varying CTSimple Derived Motion. Dimensionless form o f the acce le ra tion c h a ra c te r is tic fo r varying cf .Simple Derived Motion. Dimensionless form o f the pulse ch a rac te ris ­t i c fo r varying CT*F in ite Pulse Derived Motion. Example o f the s e ttin g data fo r ranges o f machine r a t io , l i f t and angle o f l i f t .

E rro rs a ffe c tin g the le a f s e ttin g , a llow ing fo r accuracy o f s lid e movement, accuracy o f datum in d ic a to r reading, transm ission e rro rs and d e fle c tio n o f the guide r a i l .

S pec ifica tions o f le a f m a te ria ls .Results o f re p e a ta b ility te s ts on the E l l io t MMILM0R,r v e r t ic a l m ill in g machine.Results o f d e fle c tio n measurements a t the -J- and - |-sPan p o s itio n s . Leaf springs deflected between o ffs e t clamps.Table displacements fo r c u tt in g the cam p ro f i le fo r a complete r is e movement, Simple Derived Motion.Results o f d e fle c tio n measurements. Leaf mounted on dowels a t uniform p itc h .Results o f d e fle c tio n measurements. Leaf mounted on dowels and ro lle r s a t uniform p itc h .

(cont)

Tables (cont)

10-7

10-8

10-9*

1 1-1 *

Tables

A7-1

A7-2

F ir s t d iffe rences between d e fle c tio n measurements. Leaf mounted on dowels a t uniform p itc h .F ir s t d iffe rences between d e fle c tio n measurements. Leaf mounted on dowels and ro l le r s a t uniform p itc h .Summary o f the re s u lts o f the d e fle c tio n measurements.Standard devia tions o f the measured de flec tions and f i r s t d iffe rences to compare re s u lts obtained using d if fe re n t designs o f le a f mounting.

Summary o f top ics and ob jects fo r fu r th e r research.

re la t in g to Appendix 7

Methods o f te s tin g the accuracy o f knee-and-column type m ill in g machines. E x trac t from BS4656:Pt3*1971«Analysis o f the p itc h curve rad ius e rro rs re s u lt in g from the maximum perm itted to lerances fo r d if fe re n t classes o f spur gear, BS436:1940.

- XXlll•-

SIGN CONVENTION. All linear vectors are positive when acting radially outwards, away from the centre of cam rotation..

All angles are measured in radians unless otherwise specified.

NOTATION

Abbreviations:-SDM - Simple Derived Motion FPDM - Finite Pulse Derived Motion

k

Ai

A l

act

dS

b

C C ) U

Cl

- dimensionless form o f fo llo w e r acce le ra tion .- dimensionless form o f fo llo w e r acce le ra tion ,FPDM.

- area o f cross-section o f le a f (assumed constan t).

- [H /E I]2

- fo llo w e r acce le ra tion .- fo llo w e r acce le ra tion , FPDM- acce le ra tion ca lcu la ted by f in i t e d iffe rence

techniques.- secondary acce le ra tion o f fo llo w e r.

>

- breadth o f cross-section o f le a f .- ra t io o f cam angles, 'asymmetric' motion.- s u b s titu tio n s fo r d if fe re n t ia t io n aY in te g ra tio n .

- machine ra t io .- constants o f in te g ra tio n .- distance between dowel centres.

- p itc h c ir c le diameter o f spur gear.

. "2vmz

.-1 k -1MMm/s2

in /s 2 m/s2in /s 2 u/s2

J jn ji m/ s2

in MM

fan,

in,

Mtv\

d- thickness of plate. A.n. m

- dowel diameter. X n , m u ,

E - Young's Modulus Pcomplete or incomplete elliptic integral of Legendre's Second Kind.

-xxiv-

■lYiay ~ maximum e c c e n tr ic ity ra t io o f spur gear.

•G - lo n g itu d in a l s tra in in le a f.<1 - fo llo w e r o f fs e t . m ,

Fc

FiFpi.

h

FeFt

I<3

$

I

I

I

- complete or incomplete e l l ip t i c in te g ra l o f Legendre's f i r s t k ind .

- re su lta n t force on fo llo w e r along or tangental to path.

- constant component o f F.~ component o f F due to constant external load.- component o f F due to f r ic t io n .- component o f F due to in e r t ia o f fo llo w e r.- component o f F due to p re-load o f spring .- component o f F due to spring s t if fn e s s .- ra d ia l component o f force on element o f le a f .- te n s ile force on element o f le a f .

- func tio n .

- Modulus o f R ig id ity .

- acce le ra tion due to g ra v ity .

- lo n g itu d in a l force on ends o f le a f spring,F ig . 7-2.

- l i f t o f fo llo w e r.- l i f t o f fo llo w e r, FPDM.- equivalent l i f t s , 'asymmetric' motion.

- second moment o f area o f cross-section o f le a f about p r in c ip a l ax is o f bending.

A f

H

JB.

A

t i -

id.

ujl

A f.

M l,

xV\.A,

hi

MNhi

hi

NMMM

N\bi\jb\

MWUW,m m

UM4

spec ified in te g ra ls .

3,, - su b s titu tio n s in equation (A l„59 ).su b s titu tio n s fo r d if fe re n t ia t io n or in te g ra tio n .

-XXV-

Ky lc - coefficients, equations (7.8), (A1.39)

'k - displacement factor; the linear displacement ,of the slide per degree rotation of the cam blank.

- modulus of elliptic integral, equation (7»l6).

L - free length of leaf spring. /tfh MM

IA - mass of follower assembly A

IA - bending moment. JbJ-ifl,

% - fixing moments. -in, Mm

t i l - M/EI.. -1

A n IA*1

H - speed of cam rotation 'Mil/toth,

t ' A )- substitutions defined in text.

1v - suffix to identify gear wheel; n=l;2;&c.

lv - number of pitches of dowels for mounting leafspring.

- substitution defined in text-.

P - dimensionless form of follower pulse.

p. - diametral pitch of spur gear.• >|

- M,

4 - follower pulse. ■in/p t/sJ

rl - radius of base circle. i n , MW.

f?c - radius of anticlastic curvature. A M , , MtU

Pk - radius from cam centre to pitch curve. AIV, MM

- maximum radius of pitch curve. in, MVM

+ - radius of roller follower An. 'M'M

- probe tip radius. in, 'M'M,

s - spring stiffness (assumed constant) M/m

-xxvi-

s - length o f an a rc . in, ' W U

s - leng th measured along neu tra l ax is spring .

o f le a f in. At 1u.

T - torque to d rive disc cam. Nth

Te - component o f d r iv in g torque due to load.

constant Min

- component o f d r iv in g torque due to fo rc e .

in e r t ia Nr

rs - component o f d r iv in g torque due to spring .

- no. o f tee th on spur gear.

compress Nm

/d - to lerance on l in e a r dimension. A l l , y U f W

~ time fo r disc cam to ro ta te through angle 0 see

/t

kr

lL

^s

U

AL

a t angular v e lo c ity » .- dimensionless torque c o e ff t . corresponding

to Tc.- dimensionless torque c o e ff t . corresponding

t o ' T-j-.- thickness o f le a f .- w a ll th ickness o f r o l le r .- dimensionless torque c o e ff t . corresponding

to Ts .

- s tra in energy.

= x/X .

A A .

An. 'ht'D-c

A l « / A - s u b s titu tio n s defined in te x t . \) z

V - dimensionless form o f fo llo w e r v e lo c ity .N/( - dimensionless form o f fo llo w e r v e lo c ity ,

FPDM.

$ - W/EI. M\2

Ak1\

~ y/Y .- fo llo w e r v e lo c ity .- fo llo w e r v e lo c ity by f in i t e d iffe rence

techniques.

/crx/s N+/sXn/s At/s

-xxvii-

W “ d e fle c tin g fo rce , F ig . 7-2.Wp - probe th ru s t, y -d ire c tio n .

Mhi

A O -

Abp

a

X

ti .

d e fle c tio n in d ire c tio n o f load W, equation A t \ , h H ' H i

(9.U.p ro f i le wavyness. s u b s titu tio n s defined in te x t .

span corresponding to maximum o ffs e t, F igs. A n .

6-2*, 7-2.equivalent spans fo r 'asymmetric* motion, M u ,

F ig .equivalent span along transformed ax is , FPDM, M l , .F ig . 6-2.

0602.X,

VY

Y,

co-ord inate a x is .increment o f span. /tt.increment o f transformed span, FPDM. M l ,

dimensionless form o f fo llo w e r displacement, maximum o ffs e t, F igs. 6-2; 7-2. M t ,

equivalent o ffs e ts fo r 'asymmetric* motion. Al. dimensionless form o f fo llo w e r displacement,FPDM, F ig . 6-2

AvVAM.amm

AM.AM.

n%i

1%ti-

- co-ord inate a x is . j- d e fle c tio n o f le a f a t span x .- perpendicular distance from neu tra l plane to

element o f le a f sp ring .- fo llo w e r displacement a t cam angle ©.- fo llo w e r displacements, FPDM.- d e fle c tio n on transformed ax is a t span x^,

FPDM./tj3 ; - interm ediate fo llo w e r displacements, FPDM.

“ e rro r in p itc h curve rad ius a t successive increment p o s itio n s .[ (Kb + r ) 2 - e2 I 4 (8 . 28) .

xnAM.

Al

M\

AY\

An.

A\

4W/M.

a w

AV\AU

4vMH

-xxviii-

Z - pressure angle, F ig . 2-3-(y1 - dc< / d9

$ - angle o f l i f t (o r re tu rn ) .

r k> ” ebui val en't angles o f l i f t , 'asymmetric' motion. Tad,^ - angle o f l i f t , FPDM.

- equivalent angle o f l i f t on x -a x is , FPDM, T&ot.F ig . 8-7.

^ - contact angle, F igs. 9-2, 9-3«^ - to lerance on . &9 , F ig . 3-4.

o - maximum d e fle c tio n o f le a f sp ring . ' %V\, 1 M M

oib “ R eflection o f dowel due to le a f fo rce . Am . , i M ? A

- d e fle c tio n e rro r due to apparent increase in A f t , / A m

le a f th ickness, F igs. 9-4, 9-6.Sp - d e fle c tio n e rro r due to probe th ru s t, 4YV, / A M \ .

F igs. 9-5, 9-6.

£ - pe rtu rba tion parameter, equation (7 . 26) .£ - p o s itio n in g e rro r o f machine tab le along A M , . / A w ,

x -a x is , equation (10 .3 ).£p - e c c e n tr ic ity o f p itc h c ir c le o f spur gear. A M , / A M .

Q - cam ro ta tio n corresponding to fo llow e r d is -placement y .

0 , - cam ro ta tio n corresponding to fo llo w e r d is -placement y)? FPDM.

- increment o f ro ta t io n . (jT^w@_ - cam ro ta tio n from s ta r t o f event to t ra n s it io n cI j l Q , ( a g / C )

p o in t.- interm ediate cam ro ta tio n s , FPDM corresponding

to yx and y i r0 ^ - equivalent angle o f cam ro ta tio n , FPDM. » V/K2A.)

Mr

f L\

- c o e ff ic ie n t o f f r ic t io n .- 'e q u iva le n t' c o e ff ic ie n t o f f r ic t io n . H rs/tjW.

-xxix-

r - Poisson's Ratio.

- curvature of neutral axis of leaf spring. >vvJ

fa - radius of curvature of’pitch curve. A>a, 4tidU

fa - radius of curvature of profile. '{111 urn

O'

«ao-w

- tensile stress- bending stress- max. working tensile stress

Af /41 m /ubK\i/-ul

3r kj sin (jk o).

4> - angle between x and x axes (transformationfor FPDM), Fig. 8-8.

V - curvature parameter, equation (9-4).

fo- angle of slope of neutral axis.- maximum angle of slope of neutral axis.

c&g.

difl.

(0 - angular velocity of disc cam (assumed con­stant .

^a4js

% - angular velocity of roller about pivot centre, ^ d js- angular acceleration of roller about pivot HYld /s2 centre.

-1-

P A R T 1

INTRODUCTION

- 2 -

1.1 PURPOSE AND APPLICATIONS OF CAM MECHANISMS

The operation of many different types of machinery depends upon the repetition of particular working cycles. During a cycle the working parts make controlled synchronised movements and so must be constrained to trace prescribed paths. A motion may be defined by specifying the path and the times in the working cycle at which • the output member begins and completes a particular movement, together with the requirements governing the displacement, velocity or accelera­tion at any intermediate positions and for any periods of dwell or hesitation. Professor Meyer zur Capellen (80) has recorded the in­fluence of the continual demand for increased quantity of production and for the improvement of the quality of manufactured goods upon the design and performance of machinery, with simultaneous developments intended to reduce the dependence of industry upon repetitive manual effort. These requirements, together with the effects of increasing costs, have provided the incentive for the design and development of mechanisms for the operation and control of automatic machinery.This investigation is concerned with the analysis and development of a simple method of manufacturing accurate profiles of master cams at a cost lower than that of the methods currently in general use and intended for a wide range of applications in ’moderate' and 'medium' speed machinery.

Among the plane mechanisms used for this purpose are linkages, such as the 6-bar chain, and mechanisms such as the slider-crank, the geneva and the cam. In every case it is possible to obtain a complete­ly different motion of the output member to that of the input drive.The cam mechanism provides an efficient means of driving a reciprocat­ing, oscillating or indexing output motion between accurately specified limits. In contrast to other mechanisms the cam does not impose either an inherent motion or a fixed relationship between the times of the outward and return movements of the output member which can also be

Chapter 1

INTRODUCTION

-3-

held absolutely stationary by the cam for any period of dwell. More­over the output motion of certain cam mechanisms can be interrupted for any number of machine cycles, permitting alternative modes of operation of the machine (24). There is a wide choice of suitable motions for the output member and it is possible to meet any reasonable requirement for a particular velocity or acceleration of the output member at an intermediate position or for the proportions of the move­ment allocated to the acceleration of the output member from rest to maximum velocity and to the retardation from this position to rest at the completion of the movement. This flexibility has led to the wide­spread application of the cam mechanism in many different forms.Typical applications include the operation of the valve gear of internal combustion engines (8), the machinery used for the manufacture of boots and shoes (65), electric lamps (4l) and textiles (15), wrapping machines (94) and the peripheral machines of computer installations (24).

1.2 CONFIGURATIONS OF THE CAM MECHANISM

There are many possible versions of the cam mechanism, the basic principle of operation being illustrated by Fig. 1 -1 . The input member, the cam., has a specially contoured working surface and must travel along a prescribed path or rotate about a fixed axis. The output member, the follower, must also be constrained to a single degree of freedom. The position of the follower is determined by ensuring continuous physical contact between the cam profile and the working surface of the follower, so that the resultant motion of the follower is dependent upon the shape of the profile and the motion of the cam. A rotary input drive makes it possible to operate the mechanism continuously for indefinite periods and in this form the cam becomes a profiled disc or cylinder fixed to a rotating shaft. Rothbart (99) has illustrated a selection of possible configurations of the cam and follower, other variants of the mechanism include alternative types of contact between the cam profile and the follower and the means of force closure used to ensure continuous con­tact between these parts.

To avoid elaboration of the text the discussion and analyses will generally refer to the dwell-rise-dwell movement of a radial translating follower driven by a rotating disc cam through a freely pivotted roller

To Fa c e Pa g e 4

T e r m i n o l o g y - h i s c C a m D r i v i n g a

Raua i. Re c i p r o c a t i n g Fo l l o w e r .

and using a spring for the force closure of the mechanism. This version of the mechanism is illustrated in Fig. 1-2 which is also used to define terms relating to the movements and path of the follower. This configuration was chosen to obtain the simplest forms of the equations defining the geometry of the mechanism, the analyses of alternative configurations are given by Rothbart (97)j enabling the results derived in this thesis to be adapted to other forms of the cam mechanism. A dwell-rise-dwell movement of the follower imposes the most severe conditions for motion. The terminology and notation used in this thesis are based on that used by Rothbart (97)*

1.3 ADVANTAGES OF PROFILE DERIVATION

Cam-driven mechanisms are used extensively in the peripheral machines of computer installations. Experience in the design, develop­ment and modification of these machines emphasized the disproportionate effort, and hence cost, involved in the design and manufacture of the master cams from which the production items are copied on a special- purpose machine-tool, a problem also encountered in other industries.The relatively small quantities of batch production encouraged research into simpler methods of manufacture without loss of dynamic performance.

The principal source of this effort arises from the method of specifying the pitch curve by polar co-ordinates at small increments of cam angle. Detail design involves the calculation of these co­ordinates which are either used directly for profile manufacture, as in the plunge-cutting process described by Cheney (20), or to determine the interpolation points and cutter paths using the continuous-path facility of a numerically-controlled machine-tool, typical processes being described by Haringx et. al. (50) and Oshima (93)- Since the quality of follower motion depends upon the continuity of derivatives of the displacement equation any errors caused by rounding-off the radial co-ordinates, machining, or poor blending of successive arcs of the interpolated curve are all significant.

These considerations led to the concept of copying profiles from the intrinsic shape of a leaf spring subject to a largo deflection, so eliminating the dependence upon intermediate co-ordinates between extreme positions of the follower. It is shown that i:he profiles for two distinct

-4-

-5-

follower motions designated Simple Derived Motion (similar to Simple Harmonic Motion) and Finite Pulse Derived Motion (resembling Cycloidal Motion) can be copied using this principle. Elimination of the co­ordinates makes it essential to provide design data for determining the fundamental dimensions of the mechanism, such as the base circle diameter, to suit the limiting conditions of pressure angle, profile curvature and surface stress.

1.4 OBJECTS OF THE INVESTIGATION

The principal requirements for the derivation process were identified from the design sequence for a new cam mechanism and an investigation into the significance of pitch curve errors upon the dynamic performance of the mechanism.

Despite the multitude of publications on cam mechanisms the -author found it necessary to prepare a scheme co-ordinating the design sequence for a new cam mechanism. Parameters suitable for the appli­cation of design data were identified on the flow chart and a separate chart tracing the use of such data to determine the fundamental dimensions was also prepared. The significance of manufacturing accuracy was investigated by estimating the secondary acceleration imposed on the follower by rounding off the radial co-ordinates to the greatest accuracy attainable by conventional manufacturing processes.

A survey of conventional methods of profile manufacture was followed by an assessment of previous designs of profile generating and copying devices to identify the merits and limitations of the proposed method of profile derivation from the shape of the deflected leaf spring. It was found subsequently that Borun (12) had published a brief description of the simpler version of this method of profile derivation. Unlike the design developed by the author it is impossible to derive the dynamically superior finite pulse motion using Borun's method of mounting the leaf.

Associated topics for further research identified during this investigation include:-

- 6 -

(a) a detailed study of the design procedure for a cam mechanism.

(b) the limiting performance of different cam laws.

(c) the accuracy attainable with different methods ofprofile manufacture and the significance of profile errors upon the follower motion.

(d) an analysis of roller motion to estimate limiting condition for slip.

(e) the significance of roller offset.

(f) the large deflection of slender leaves.

1.5 ANALYSIS OF THE DEFLECTED LEAF SPRING

The equation for the curve of the deflected leaf and derivatives of this equation are needed to compare the theoretical and measured deflections, to determine the cam characteristics and to calculate the design data for pressure angle and curvature. It must be possible to obtain the dimensionless forms of the cam characteristics from these equations. In addition allowance must be made for the large offset- span ratio inherent in this application, invalidating the assumption of negligible slope made in the approximate theory of bending. Alter­native solutions for the deflection of flexible members In the elastica range were compared to test the assumptions of pure bending and negligible shear. Further analysis emphasized the significance of longitudinal end forces, and hence the design of the leaf mounting, upon the curve of deflection. The author used the method of pertur­bations to derive a new solution for the large deflection of a leaf spring additionally subject to longitudinal end loads and also designed a leaf mounting suitable for both derived motions with the object of preventing the longitudinal force exceeding an empirical maximum value.

-7-

The cam mechanism provides a means of accurately controlling and synchronising the movements of machine members, the output motion having well-proportioned characteristics, and absolute dwells. It has widespread application in automatic machinery operating in the 'moderate' and 'medium' speed ranges.

From experience in industry the author found that the effort, and consequent cost, of manufacturing master cams hindered the design of new machines and the improvement of existing models.

The principal source of this effort results from the method of specifying the shape of the pitch curve by polar co-ordinates at small increments of cam angle. These co-ordinates become redundant for manu­facturing purposes if the profile is cut by a copying or generating process provided design data ace available to determine the fundamental dimensions. A further advantage results from the elimination of round­off error in the manufacturing process which is shown to be a potential source of poor dynamic performance.

The author found it necessary to prepare a procedure co-ordinating the design sequence for a new cam mechanism, including the application of design data. He also surveyed previous profile generating and copying techniques to assess the proposed adaptation of the copying principle to. derive cam profiles from the intrinsic shape of a leaf spring subject to a large deflection. It is shown that the profiles for two distinct follower motions designated Simple Derived Motion and Finite Pulse Derived Motion are obtainable by this means.

Since the c.am characteristics are functions of the curve of deflection the author also surveyed previous analyses of the large deflection of flexible members. He derived a new solution particularly suited to this application by the method of perturbations.

1.6 SUMMARY

The descriptions and analyses in this thesis generally apply to the dwell-rise-dwell movement of a radial translating follower driven by a disc cam. These results are readily adaptable to other configurations of the mechanism, the relevant analyses being included in a comprehensive study of cam mechanisms by Rothbart (97)-

-8-

- 9 -

Chapter 2

THE DESIGN OF CAM MECHANISMS

2.1 SPECIFICATION OF THE MECHANISM

Surprisingly little has been published on the design procedure for cam mechanisms. The specification depends upon a number of inde­pendent parameters as a result of the flexibility of the mechanism which permits the use of different types of follower, alternative motions of both the cam and the follower and variations in both the shape and position of the path of the follower (Fig. 2-1). The criti­cal feature is the specification of the profile shape in a form immedi­ately suitable for the chosen method of manufacturing the master cam. This shape must produce the required motion of the follower and be compatible with the geometry and the operating requirements of the mechanism as well as the maximum allowable stresses in the component parts caused by the external load and the dynamic forces resulting from high accelerations.

A guide to the selection of suitable combinations of cam and follower types prepared by Rothbart (99) includes alternative forms of cam profile, different movements of the input drive, different paths and means of constraining the follower and different types of contact between the working surfaces of the cam and follower. Consequently every mechanism is to some extent unique and the design must begin from first principles following a sequence-such as that shown diagram- matically in Fig. 2-1. The information needed to begin the design is listed along the top two lines of the diagram and the progress of the work is followed by tracing the lines downwards. These lines also indicate the inter-relationship between the parameters defining the dimensions, geometry, strength and operation of the mechanism.

-10-

The motion and position of the input drive and the path and constraint of the follower is determined by the application, an indica­tion of the space constraints being obtained from the layout' drawing.The logic of the operating of the machine is determined by preparing a timing chart to illustrate the movements of significant parts during a complete operating cycle (75)- Provided it is intended to run the machine at a nominally constant speed it is convenient to relate the time base to the angle of rotation of a mainshaft measured from a speci­fied datum. The chart determines the operating sequence of the machine, the synchronisation of associated movements is checked and the slopes of the lines indicate the severity of accelerations. The information needed to determine the cam angles is taken from this chart, the lift is found from the required displacement of the cam follower.

2.5 FOLLOWER MOTION

At any instant during the operation of the mechanism the posi­tion of the follower is determined by the radius from the centre of cam rotation to the roller centre, and the velocity of the follower is determined by the rate of change of this radius and the angular velocity of the cam. Therefore the shape of the profile and the motion of the follower are interdependent, the selection of either one determining the second. From the manufacturing viewpoint it is easiest to construct the cam profile from blending circular arcs or arcs and tangents, but the resultant follower motion has poor dynamic performance (97)- For this reason it is normal practice to select a suitable follower motion, then the shape of the profile is specified indirectly by calculating the polar co-ordinates of the follower relative to an origin co-incident with the centre of cam rotation at small increments of the angle of lift. The profile is a smooth curve touching the outline of the follower at every increment position in succession. The cam law defines the dis­placement of the follower in terms of the angular rotation of the cam from the start of that movement. The dynamic performance is assessed by examining the derivatives of the displacement equation which define the velocity, acceleration and pulse of the follower during the motion. The factors which influence the choice of a particular cam law are free­dom from shock loads associated with abrupt accelerations and the magni­

2.2 TIMING AND CAM ANGLES

-11-

tude of the peak accelerations and hence of the inertia forces. In theo'ry the first condition is satisfied by using a cam law which satisfies the boundary conditions specified in equation (2.l). (The notation is given on -p, ct-iO), Then the acceleration characteristic for a dwell-rise-dwell-return movement of the follower may include abrupt changes of slope at the limits of the rise and return movements, but since no discontinuities occur the pulse must be finite throughout a complete rotation of the cam. This argument makes no allowance for round-off errors in the radii of the pitch curve or for manufacturing or assembly errors. The effect of round-off errors on the acceleration

o

the follower are analysed in Chapter 3-

e - o •. o • T t =u • dJc

0 ;=

M z

dtf m a x ; d \M . d t z

Q = M A J - A : ^ * 0 ;f t =

U a J . dk c tt% .1)

2.4 LIMITING FACTORS

The procedure for designing a cam mechanism is shown diagram- matically in Fig. 2-1. The starting data is taken from the layout drawing and the timing chart. It will be noted that the specification of the profile can be completed at an early stage of the work hut it is essential that the analysis be completed to ensure that the critical values of pressure angle, curvature and, surface stress are acceptable.The use of design data eliminates the danger- of repeating a considerable amount of this work should any of these values be excessive. The factors influencing the choice of the cam law are cost, the method of manufacture of the master cam, the speed of operation and the working load. The author has shown that comparisons based upon the speed of rotation of the cam disc in isolation are misleading since the acceleration of the follower is also dependent upon the magnitudes of the lift and the angles of lift and return (26). Consequently the published guides of recommended speed ranges for different cam laws tend to be vague (33)

(9 7 ) .

-1 2 -

2.5 PRESSURE ANGLE, CURVATURE AND SURFACE STRESS

Although the displacement equation has the form

(2.2)

the shape of the profile is also dependent upon the radii of the base circle and the roller. All these parameters, together with deriva­tives of equation (2.2) determine the magnitudes of the pressure angle and the curvature of the profile. The pressure angle between the direction of follower movement and the thrust normal to the profile at the point of contact with the follower (Fig. 2-5) determines the load on the guides or pivot of the follower. Large pressure angles are undesirable since the force normal to the profile, and hence the sur­face stress, increases and in extreme cases the follower is liable to jam or to overrun. The limiting value is empirically quoted as 30° (lOO). The profile’may have either concave or convex curvature. Any concave curvature must always exceed that of the roller whilst the minimum radius of curvature of convex surfaces is restricted by the large surface stress induced in the material. The magnitude of this stress is found from the Hertz equation (Chapter 9)- The most effec­tive means of reducing any of these parameters is to increase the diameter of the base circle, assuming that it is not possible to reduce the lift or to increase the angles of lift and return. However this change increases the speed of rotation of the roller (or the sliding velocity of a flat-faced follower), so reducing the load capacity.There is also an increased danger of roller slip occurring (28). E.S.D.U. (32) and Morrison (86) have published data on acceptable levels of surface stress for commonly used cam and gear materials which also assist in the selection of suitable combinations of ma­terials for the cam and the working surface of the follower.

2.6 USE OF DESIGN DATA

If the initial design does not satisfy the limiting conditions for the satisfactory operation of the mechanism it is necessary to revise the starting parameters and repeat the relevant calculations.

-13-

Hence there is a considerable incentive to find a means of testing the dimensions of the initial design to check the limiting values before undertaking the main design. The procedure for this process is illustrated in Fig. 2-2. Two levels of accuracy are involved in this work. The polar co-ordinates specifying the roller position must be calculated for precision manufacture, but the requirements for testing the limiting values are less rigorous. It is adequate to estimate magnitudes from design data presented in the form of graphs or tables. For this purpose the equations for the pressure angle and profile curvature have been re-arranged in dimensionless form to obtain groups of associated factors (34) (68) (69). A par­ticular set of data can apply only to one configuration of the mech­anism and a specific cam law. Consequently the information available is restricted in scope, although the data on surface stress have universal application. It should be noted that the maximum surface stress does not occur under normal running conditions but when the mechanism is running slowly. Then the inertia force is negligible and virtually the entire spring force acts upon the profile. However the accumulation of fatigue cycles at this loading is very gradual. Further consideration is given to surface stress in Chapter 8 (page 5)8),

2.7 ANCILLARY DETAILS

The hub and fastening of the cam, together with the camshaft, are subject to shear and torsional loads. The shear force on the shaft is known from the calculations for the spring design. The author has shown that the three components of the driving torque needed to overcome the load, compress the spring and accelerate the mass of the follower assembly can be reduced to separate dimensionless groups readily presented as design data in graphical form (25). The stiffness of the spring needed for the force closure of the mechanism is determined from the maximum net force acting to separate these parts, allowing a safety margin. A preload is desirable to control the follower at the inner dwell position. Neither the natural fre­quency of the spring nor a harmonic should co-incide with any of the cycle times of the cam at the normal operating speed. The roller follower must be designed according to the load/speed requirements

-14-

for the roller and its pivot. If a form of flat-faced follower is used provision must be made for lubricating the sliding surfaces and the surface finishes must be specified (70).

2.8 CONTINUOUS PATH MANUFACTURE

Computer programs have been written for calculating the co­ordinates specifying the shape of the pitch curve which include pro­cedures for checking the critical values of pressure angle and curva­ture (58) (78). If either of these parameters exceeds the specified limit the input data defining the dimensions of the cam disc is variedaccording to programmed iterative processes until acceptable values areobtained. The program is simplified and computer time used more economically if design data is used to test the input data before it is run. These programmes have also been extended to prepare the con­trol tapes for manufacture of cam profiles using numerically-controlled machine-tools. Alternatively a copying device or a generating mechanism can be used to manufacture the profile for a particular cam law as a continuous cutting operation. This approach eliminates the need to calculate the co-ordinates for the increment positions. The design requirements for cam profile copying and generating devices are con­sidered in Chapter 5 which also includes an examination of previous work in this field.

2.9 SUMMARY

The specification of a disc cam is determined by the configura­tion of the mechanism, the motion, path, mass and type of follower, the external loads, the operating sequence and times and external constraints. The selection of the cam law is influenced by the application, particu­larly the operating sequence, the 'speed1 and the loads.

The cam law^ material and dimensions of the cam and roller are interrelated through empirical limiting values of pressure angle, curva­ture and surface stress which involve functions of the dimensions, the displacement equation and its derivatives. The iteration involved in a direct solution is circumvented by the preparation of specialized design data.

-15-

Chapter 3

THE DYNAMIC PERFORMANCE OF CAM MECHANISMS

3.1 THE FOLLOWER MOTION

The correct operation of a cam mechanism is dependent upon the maintenance of continuous physical contact between the cam pro­file and the working surface of the follower. The position of the follower at any instant is determined by the radius from the centre of .cam rotation to the profile at the point of contact and the motionof the follower is determined by the rate of change of this radiuswith the angular rotation of the cam disc and the instantaneous angu­lar velocity of the cam. As shown in Fig. 3~1 the follower is driven away from the centre of a disc cam by increasing the radius to the profile with rotation and the return movement is achieved by the con­verse process. The magnitudes of the velocity and acceleration of the follower depend upon the lift and the angle of lift in addition to the speed of the cam. The follower displacement along the pathis measured from a datum such as the inner dwell position and linearvectors are considered to be positive when acting radially outwards, away from the centre of cam rotation.

The displacement of the follower from a datum position at the beginning of an event, such as a dwell-rise-dwell movement, is related to the corresponding angle of cam rotation 0 (radians) by the equation for the cam law in the form

which may not be a continuous function throughout one rise or return movement. In the case of a disc cam the velocity of the follower is

where

ti

which, simplifies to

a3~ 100IB

if it may be assumed that the variations in the angular velocity of the cam due to the fluctuating driving torque and changes in the other loads on the same drive are negligible. Then the equations for the velocity, acceleration and pulse of the follower are functions of successive derivatives of the displacement equation (3*l).

Velocity:

Acceleration:

Pulse:

(1 ( e )

0. = , (J I'1

^f(e)

(3 .2 )

A

Equations (3-l) to (3-4) can be rewritten in dimensionless form for the range 0 / a? £ 1 by expressing the displacement as a fractionof the lift, the angle of cam rotation as a fraction of the angle of lift and re-arranging the coefficients.

Displacement: Y =ft

i I ©

( j £ $(3,5

Velocity:

Acceleration:

Pulse:

V

A

P

ai l l

f f

nY

P"1/ 0

(3,6)

(5 7

(VThe follov/er must be accelerated from rest at the start of a

dwell-rise-dwell event to attain maximum velocity at the transition point and is then retarded to rest at the limit of the movement. During this motion the inertia force due to the mass of the follower assembly acts to maintain contact between the cam profile and the follower whilst the velocity is increasing, but the direction of the inertia force

reverses at the transition point. Unless the mechanism runs at a very low speed the inertia force may be expected to exceed the external load, permitting the cam and follower to separate unless a means of force closure is provided, either by using a second cam profile to drive the follower positively in the opposite direction or by including a spring in the mechanism. The spring is designed for a pre-load and stiffness which ensures that the resultant force due to the external load, the inertia force and the spring force always acts on the follower towards the cam. The natural frequency of the spring must not be a harmonic of the speed of rotation of the cam (7 ) •

3.2 SIGNIFICANCE OF THE CAM CHARACTERISTICS

The graphs of the dimensionless forms of the displacement, velocity, acceleration and pulse of the follower are known as the cam characteristics. These curves for a dwell-rise-dwell event (a typical follower movement) provide an effective means of assessing the performance of a particular cam law by identifying the locations and magnitudes of the peak values of acceleration, indicating any abrupt changes of slope or discontinuities in the curves and showing whether the pulse becomes infinite at any position (33)* A study of the operation of the mechanism during this sequence of events esta­blishes conditions for a parameter in terms of the next derivative at significant positions. Unless an abrupt change of velocity occurs (with a corresponding ’infinite’ acceleration) the displacement charac­teristic must be a smooth continuous curve with zero slope at both limits of the rise movements corresponding to the positions of zero velocity. The maximum velocity occurs at an intermediate position.A theoretical analysis by Hrones-(55) has shown that the significant parameter for assessing the dynamic performance of the mechanism is the pulse since the third derivative becomes infinite at any position where the magnitude of the acceleration changes abruptly, imposing a shock load on the system. This source of shock loading is elimina­ted by selecting a cam law in which the acceleration equation satis­fies the boundary conditions specified in equation (2.1 ).

- 18-

3.3 PRACTICAL CONSIDERATIONS

These conclusions are based upon the false assumption that the cam profile is an exact reproduction of the theoretical shape. The only experimental investigation into the dynamic performance of cam mechanisms known to the author is that published by Mitchell (82) in 1950. He compared the acceleration of the same follower driven by three cams having the same nominal lift, cam angles and base circle diameter but having profiles shaped to produce constant acceleration, simple harmonic motion and cycloidal motion of the follower respect­ively. Infinite pulse occurs at the start, and finish of a dwell-rise- dwell event driven with simple harmonic motion and additionally at the transition point in the case of constant acceleration motion, but does not occur with cycloidal motion. A large flywheel was attached to the camshaft which was driven by an electric motor through a long, twisted belt drive. The force on the follower was measured with strain gauges and a recording oscillograph. The speed of rotation of the cams was varied from 20 to 170 rev/min in steps of 10 rev/min.The results, summarised in Table 3-1, are notable for thelow values of the maximum speeds of rotation of the cams and follower accelerations attained and for the marginal superiority of the finite pulse cycloidal motion cam. There is little difference between the acceleration curves for S.H.M. and cycloidal motion at 130 rev/min, above this speed the results were distorted by resonance of the fol­lower and probably by the limitations of the recording equipment. As shown below (p 5) ) this poor performance is explained by the magni­tudes of the profile errors in these cams.

Normal engineering tolerances can be applied to the fundamental dimensions of a disc cam. These are the base circle diameter, the lift and the cam angles. Small variations in these dimensions have negligible effect upon either the limits of follower movement or the actual motion of the follower provided the sum of the increments of cam angle equals the angle of lift. However the conventional approach of tolerancing individual dimensions collapses in the case of the polar co-ordinates used to specify the follower position and, indirect­ly, the shape of the cam profile since the follower motion depends upon the rate of change of the differences between adjacent radii over

FOLLOW

ER USPLLCEM.EMT

l«j)

To F\ce P&ge 1^

0-006.

. 0-005.

g 0 - 0 0 4

Ztuu4 0003

to/3

“ 0 -002.

*o ._ l-J

i2 O OOI.

o UZ!

C a m Mo t io n ./. S im p l e H a r m o n ic ,2. Co n s t a n t Acceleration

3. MoblFIEb TRAPEZOlbAL A CCELERATIO N C U R V E .

4. CYCLO IbA L.5 . 4 - 5 - 6 - 7 P o ly n o m ia l. T M inim um T o le r a n c e o f

±QOOO 2in (SyuX) ONL i f t o f 1,0+n (2 5 ,4 him).

Lift Cha,r k c t ei?is t ic s .

Fig. 3,-2.T

O 0-01 0 0 2 0 0 3 0-04 0-05 0-06 O-07 0-08 0*09 0*10

A n g l e o f C a m R o t a t i o n / 0 '

2X

20 -I

15 -

10 -

5 -

0-

-5 -2-0 *0 0 5 0*010 0-015

CAiNN A N G L E

MOblFlEB T2APEZQlbAl ACCELERATION MOTION

VJAVYNESS OF PITCH CURVE

ANGLE OF L\F T 120Q

l i f t i o **,25*4 fc*.

O* 0 0 0 2 am j decree 5 utvs |d jrec.

S H h \

Fig T - 2) OVEKLkP OF TOLE^iLMCE ZONESSNIA AN > MofclFiE>> TR A?£2QM> AL ACCELERATION MOTIONS,

small increments of cam angle. As shown in Table 3-2 and Figs. 3-2 and 3-3, this difference is smaller than the precision tolerance in the critical regions around the start and finish of the follower movement. (Thus for h=l and =100° the difference between adjacent radii of the pitch cruve spaced at increments of 0.5° does not exceed 0.000 2in until © =5.0^ for cycloidal motion.) Throughout the event the random variation of the error within the tolerance band contain­ing the pitch curve causes a secondary velocity and acceleration to be superimposed upon the motion of the follower. A larger, but constant, error has no effect upon the differences between radii and therefore upon the acceleration of the follower, merely altering the displace­ment by that amount. It is therefore necessary to specify an additional tolerance, the wavyness of the profile, defined as the rate of change of radial error with cam angle.

A Y a % i ~ i am , I daame.- A© 0

3.4 DETERMINATION OF THE SECONDARY MOTION

Since the errors causing profile wavyness may be expected to occur in a random manner there is no reason for supposing that the resultant acceleration characteristic is free from discontinuities with the result that the theoretically superior dynamic performance of a finite-pulse cam law may not be realised in practice. Direct measurement of the profile is misleading since the angular position must be corrected to allow for the displacement of the point of contact from the path of the follower and the actual displacement error is the com­ponent of the radial error along the path. The follower surface will bridge extreme hollows in the profile so the pitch curve should be smoother than the profile. Normally the profile is derived from the pitch curve and one source of error is introduced in the latter at the design stage when the lengths of the radii specifying the positions of the follower at the co-ordinate points are rounded off to sensible values for manufacturing purposes. The maximum round-off error deter­mines the theoretical minimum wavyness of the profile to which must be added an allowance for manufacturing errors. Johnson (63) has shown

To FkCE Page 20

c m (6)ANGLE

FIG 3-4. ENLARGED SECTION OF DISPLACEMENT CHARACTERISTIC TO SHOW E FFE C T OF RADIAL AND ANGULAR ERRORS.

-2.0-

that finite difference techniques can be used to estimate the velocity and acceleration of the follower accurately from the displacements over successive increments of cam angle, making it possible to deter­mine the order of magnitude of the secondary velocity and acceleration of the follower caused by profile wavyness. Since these errors occur in a random manner the accuracy of the calculation is not improved by taking more than three successive increment points. Fig. 3-4 shows an enlarged section of the displacement curve on which the theoretical and actual positions of the follower are plotted at three positions separated by two equal increments of cam angle. ■ Since the actual dis­placements of the follower are used the analysis automatically allows for angular errors in the manufacture of the profile. The velocity of the follower a.t the mid-position 2 is estimated by assuming that the Chord A^ A is parallel to the tangent to the displacement curve through poi-4'

Expressions for the velocity at the intermediate positions C and D are needed before the acceleration can be found.

and the secondary velocity due to profile errors is

Then the acceleration of the follower at the mid-position 2 is

and the maximum secondary acceleration is

Offl - a M - a 2 - ^ | J , -* OJ - V j (W3)

If the tolerances on polar co-ordinates defining the follower position are +t on the radius and + jf on the increment of cam angle then the worse case occurs when ^ * / g “ *“ (or conversely)and both angular increments become ). Then the profile wavyness is

2 AAO' ^ " N • (?.l4)

(Ae - i )and the maximum secondary acceleration is

a.&2 = 2 to2 4

( & Q - 0 2Johnson (63) derived separate expressions for the secondary

accelerations caused by errors in the radial displacement of the follower at the precise angular increment positions and for those caused by errors in cam angle corresponding to the theoretically correct angular displace­ment. He does not justify his conclusion that the resultant secondary acceleration is the algebraic sum of the components due to displacement error and due to angular error. If the displacement error is measured at the precise angular increment positions then effects of angular errors must be included and the total secondary acceleration is determined from the displacement error alone. Consequently equation (3«15) for the secondary acceleration caused by the extreme tolerances on the radial and angular co-ordinates of adjacent points on the pitch curve differs from Johnson's result.

Contrary to established practice, equation (3»12) shows that the magnitude of the secondary acceleration is -inversely proportional to the size of the angular increment and therefore the dynamic performance would appear to be improved by taking larger angular increments. The accuracy of this approach was checked by comparing the acceleration of the follower obtained from the second derivative of the displacement equation with that calculated by the finite-difference method using equation (3»12). For a particular case of cycloidal motion ( = 1 A • I 0 0 j co ~ I / s )and using displacements accurate to 8 places of decimals (inches) it was

-21-

k (3 .1 5 )

-2 2 -

found that the maximum difference was less than 0.1%. Since machining errors may be expected to occur in a random manner there is no theoretical means of estimating the actual acceleration of the follower by this method, but it is possible to determine the wavyness of the profile and the con­sequent secondary acceleration due solely to round-off error in the radii of the pitch curve specified for manufacturing purposes. This was done by repeating the calculations for radii rounded-off to 4 places of decimals. Using 0.5° increments of cam angle the graph (Fig. 3-5) shows that the wavyness of the pitch curve due to this cause alone approaches the empirical maximum of 0.000 2in/degree quoted by Nourse (89). Since machining errors are inevitable the result shows that it is extremely difficult to attain this standard in practice. Beard (9) suggests that the final handsmoothing process of the manual plunge-cutting process of cam manufacture, described by Cheney (20) and discussed in Chapter 4, tends to even out profile irregularities. It may be expected to remove the peaks of the asperities whilst any additional wavyness introduced by the manual operation would have a longer period and so be less significant.

The secondary acceleration of the follower was determined by taking the difference between the acceleration calculated from the second deriva­tive of the displacement equation and that given by the finite-difference method, equation (3*12). Both curves are drawn in Fig. 3-6 to show that both the magnitude and direction of the secondary acceleration fluctuate rapidly, the extreme values equalling the theoretical maximum. The computer programs ’FINDIF-WAV* and 'FINDIF-ACCN' (Appendix 8) were written for this purpose. The graph shows that in particular the gradual start to the motion needed to satisfy the boundary conditions for finite-pulse motion defined by equation (2.1) is lost and, regardless of cam law, the secondary acceleration is a potential source of shock loading. However this analysis makes no allowance for the elasticity of the materials or for the response of the cam copying machine making the production items at low speed. It has been suggested that in use the follower would have a burnishing effect on the profile, this theory could be tested by regular measurement of the master cam. Another means of reducing profile wavyness -would be to use a master cam several times actual size controlling the copying machine through a diminishing linkage and by using an oversize cutter on the copying machine. The latter would involve modifications to the profile of the master cam. The combined effect would introduce a magnifying factor of around 10 which is inadequate to obtain a significant

-23-

improvement (96). Moreover the diminishing linkage introduces a new source of error. A more satisfactory approach is to use a means of manufacture independent of round-off error.

A comparison of the profile errors on the cams used in Mitchell's (82) experiments with the published values obtained from various sources and summarized in Table 4-1 suggests that the poor performance is due to this cause. It should also be noted that the pressure angles of Mitchell's cams are undesirably large (Table 3-1)•

3.5 CHOICE OF THE CAM LAW

For these reasons the choice of the cam law for a new design is not necessarily the obvious one of selecting a finite-pulse charac­teristic. To obtain the correct follower motion .with smooth accelera­tions it is essential to reproduce the differences in radii to a high degree of precision. Other design requirements include the desirability of keeping the absolute values of follower acceleration as low as poss­ible to obtain correspondingly small dynamic and spring forces. These considerations led Kestell (66) to argue

"Cam design and methods of manufacture are inseparable, for the degree of accuracy to which a cam can be pro­duced dictates the degree of sophistication of’ design which it is desirable or even worthwhile to employ."

This view was endorsed by Gardiner in the ensuing discussion and in his own paper Gardiner (4l) expanded the argument, stating that simple harmonic motion cams had been found suitable for machines used in the manufacture of electric lamps which handle delicate glass components. Similarly Molian (85) has concluded

"There is therefore no best generating function for all cases. In the common run of low-speed mechanism design the S.H.M. cam is probably as good as any."

and in a survey of the current state of knowledge of mechanism design and manufacture Meyer zur Capellen (80) wrote

-24-

"The most troublesom problem is manufacturing, which sometimes is not accurate enough and can be very- expensive. Deviations in the curvature caused by a lack of accuracy in manufacturing can generate uncon­trollable dynamic forces or wear problems and the necessity of calculating the elasticity of the whole mechanism; these demonstrate that a suitable kinematic solution does not always yield the best results in practice."

3.6 SUMMARY

The follower motion depends upon the motion and profile shape of the cam. Complete control of the output movements requires con­tinuous physical contact through the higher-pair connexion between these parts.

Established theoretical analysis has shown that the dynamic performance of the mechanism can be assessed by the freedom from shock loading, identified by a finite pulse characteristic.

This analysis is based upon the theoretical shape of the pitch curve. Therefore no allowance is made for manufacturing errors expressed as the wavyness (the rate of change of dimensional error).It is shown that inherent round-off errors in the radii specifying the shape of the pitch curve cause wavyness approaching the empirical limit. The actual acceleration of the follower allowing for round-off error alone shows significant discrepancies from the theoretical curve implying the presence of shock loading in a nominally finite-pulse motion.

Several authorities are quoted to support the contention that profile accuracy rather than cam law governs the performance of cam mechanisms. Therefore a copying or generating process less subject to random error should produce profiles giving superior follower motion.

-25-

a

P A R T 2

PROFILE MANUFACTURE

-26-

CONVENTIONAL METHODS OF MANUFACTURING CAM PROFILES

4.1 QUALITY OF DYNAMIC PERFORMANCE

In the absence of any knowledge of the follower assembly and guides it is only possible to assess the quality of the dynamic per­formance of the mechanism in terms of the higher-pair connexion be­tween the profile and the follower. However the interaction between these parts is vital to the motion and enables criteria for the per­formance and manufacturing accuracy of the profile to be established. The overall accuracy of every movement is determined by the actual lift and cam angles, but the quality of the motion demands close ad­herence to the theoretical characteristics for the displacement, velocity and acceleration of the follower throughout every rise and return event. Therefore the follower must maintain continuous contact with the profile during these movements. The theoretical analysis by Hrones (55) shows that shock loading with consequent vibration, noise and wear results from abrupt changes of acceleration, but practical considerations emphasize the problems associated with the accurate manufacture of cam profiles for the finite-pulse cam laws. The princi­pal problems in cam manufacture are controlling the wavyness of the pitch curve (the locus of the roller centre) and obtaining the veryslight changes in radii at the start and finish of such a motion, a

-6 otypical example being 1 x 10 in over a rotation of 0.5 •

4.2 SPECIFICATION AND MANUFACTURE

The blank of a disc cam is made by conventional manufacturing methods. The profiles of production cams are cut by copying the master on a special-purpose machine-tool, so the basic problem is con­fined to the manufacture of the profile of the one-off master cam.The copying machine can be adapted to scale down the size of the pro­duction item, enabling an oversize master to be used for greater accuracy of the production item, and also to cut the corrected pro­files for cams having different follower paths to that used on the

Chapter 4

-27-

master cam. The importance of controlling the acceleration of the follower makes it necessary to derive the shape of the profile, from the required motion and path of the follower and to match the shape of the working surface of the follower. The locus of the roller centre is specified by polar co-ordinates about an origin co-incident with the centre of cam rotation at small increments of the angles of lift and return. Then the actual cam profile is a smooth curve touch­ing the working surface of the follower at every increment point in succession: it is not expressed as a mathematical expression.

The form of specifying the shape of the profile by means of the pitch curve determines the method of manufacture in which an in­version of the action of the mechanism is reproduced by appropriate movements of the machine-tool. A jit^-borer or a vertical milling machine can be used and the follower is replaced by the cutter which rotates about a fixed axis to generate the working surface of the follower. The cam blank is rotated around the origin and simultaneously displaced along an extension of the path of the follower to produce the same relative motion between these parts as in the actual mechanism. The procedure for cutting the profile of the master cam has been des­cribed in detail by Fromelt (40), Rothbart (lOl), and Cheney (20).The cam blank consists of a turned cylindrical disc, sufficiently large to contain the complete profile. It is mounted upon a rotary table fixed to the table of the machine-tool. For accuracy of loca­tion a special tooling hole may be provided, it is accurately bored at the greatest possible radius from the centre. For mechanisms designed to operate with a roller follower the cutting tool can be an end-inill cutter of the same diameter as the roller to be used on the finished assembly. Fresh co-ordinate positions must be calculated, and the curvature checked, if a cutter having a larger diameter is to be used. The cutter is fixed in the vertical spindle of the machine- tool so that the axes of rotation of the cutter and the cam blank are parallel. The table of the machine-tool is adjusted to obtain the specified radial distance between the centres of the cam and the cutter at a convenient datum position, such as the start of a rise or return event. To obtain the profile shape for a radial translating follower the centre of the cam blank will be displaced along the common centre­line of the cam and the cutter. Therefore this centre-line must co­incide with the direction of one of the feed movements of the table

(Fig. 4-l)« The first cut is made and t-he cutter withdrawn. The rotary table is indexed through one increment of cam rotation to - the next co-ordinate position. The centre distance is adjusted to that specified by the new co-ordinates and another cut made. This procedure is then repeated until the whole sections of the profile controlling the rise and return events have been cut. The sections of the profile at which the follower is required to dwell are made by turning the rotary table cut a circular arc of the correct radius.

■ Cheney (20) recommends two roughing cuts followed by a finishing cut.

4.3 THE PLUNGE-CUTTING PROCESS

The plunge cutting process leaves a scalloped surface of the type shown in Fig. 4.2 to an enlarged scale and Plate 1 is a photo-- graph of an actual profile cut at 0°20f increments. The true profile is obtained by a hand-smoothing operation to remove the areas shaded in the drawing. The smoothing operation begins by filing, then pro­gressively finer grades of emery cloth are used and finally the sur­face is honed. It is evident that the process is a slow one and that the accuracy of the finished surface is dependent upon the skill of the toolmaker. The accuracy of manufacture of cam profiles made by various processes is summarized in Table 4-1. The accuracies quoted by Astrop (5 ) (l-off) and Mitchell (82) (3-off) refer to particular master cams, the remainder are claimed to be repeatable accuracies under production conditions. With the exception of Oshima (93) no information about surface texture or profile wavyness is given, more­over the specified co-ordinates used as the basis of measurement must also be subject to round-off error. The quantity of metal to be removed by the hand-finishing process depends upon the size of the angular increments. Since the accuracy of the manual process is like­ly to deteriorate with the quantity of material to be removed i.t is generally advised to use the smallest practicable angular increments, varying between 0°10' (5 ) and a recommended maximum of 2° (97). How­ever this recommendation is in conflict with the conclusions drawn from the analysis of the dynamic effects of profile wavyness in Chapte:3«

Other configurations of the cam mechanism use an offset trans­lating foliower or an oscillating lever follower driven by a disc cam.

-2 9 -

In the latter case a complication arises in the calculation of the increment positions as the circular arc traced by the roller centre causes it to 'gain' or 'lose' 011 the rotation of the cam in the man­ner shown in Fig. 4-3 and the co-ordinate positions must be corrected accordingly (97). in both cases it may be simpler to make the master cam for the equivalent radial translating follower ?i provided the critical pressure angles and curvatures permit this change. Then the profiles of the production items are corrected upon the copying machine. In the case of the cam for an offset follower it is only necessary to make the required displacement between the path of the cutter and the parallel centre-line through the cam centre but an additional degree of freedom is needed to correct for the circular arc path traced by the roller of an oscillating follower. The rota­tion of the master cam and the production item are synchronised on the copying machine and the master cam controls' the radial displace­ment of the cutter relative to the centre of rotation. The additional degree of freedom is obtained on an additional slide permitting move­ment of the cutter in the direction perpendicular to the radial dis­placement. The additional movement is controlled through a lever having the same length between bearing centres as the oscillating follower (106). Alternatively a plate with a radiussed side can be used provided a means of force closure is included.

The plunge-cutting method of cam manufacture involves con­siderable effort, both in the preparation of the design and the manu­facture of the master cam. It would appear to be an effective applica­tion of numerically-controlled machine-tools although the high cost of these machines makes it more economical to use the copying machines for quantity production. Alternatively the cam profile can be genera­ted as a continuous cutting operation using a special-purpose machine. This approach eliminates much of the design-effort and simplifies the manufacturing process. It is considered in detail in the following Chapter.

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4.4 APPLICATION OF NUMERICAL CONTROL

Cam profiles can be cut by means of the continuous-path facility of numerically-controlled machine-tools using two degrees of freedom. The object of using these machines is to improve the dimensional accuracy of the profile and to eliminate the hand-smooth­ing process. The profile cutting operations which may be programmed for numerical control are:-

(a) a co-ordinate position of the cutter

(b) straight-line cutting parallel to the x- or y- axis

(c) straight-line cutting in the x-y plane at any angle to the x-axis by simultaneous movements parallel to the x- and y- axes at the appropri­ate speed ratio

(d) circular arcs of any radius about a specified centre

(e) parabolic arcs of any size but symmetrical about the focus

(f) lengths of a cubic curve.

The first two of these operations can be performed on a conventional milling machine or jig-borer, but the n-c machine can make much shorter movements.

Various approaches have been developed for manufacturing cam profiles on numerically-controlled machine-tools as a series of extreme­ly short straight line cuts. Childs (21) used tool movements confined within a narrow tolerance band (Fig. 4-4). If the straight-line cuts can be made in any direction the lines become chords of the curve, the length of a particular chord being determined by the maximum accept­able deviation between the true curve and the chord. Therefore the

length of the chord diminishes with the radius of curvature and there is a minimum radius determined by the response of the machine (Fig. 4-5) (92). Buhayar (l6) calculated a large number of increment positions in order to use the numerically controlled machine-tool for the conventional plunge-cutting process. By making eight cuts per degree rotation it was possible to eliminate the hand-smoothing operation but the surface finish was 125/^in which compares unfavour­ably with a possible surface finish at 20 to 4yUin obtainable by hon­ing (4). Oshima (93) approximated the cam profile by short chords subtending 1° or 2° at the centre of rotation. In an extreme case the lift could equal the base circle radius of the cam disc, so that the length of an arc subtended by a given angle at the outer dwell position would be double the length of an arc subtended by the same angle at the inner dwell position. Therefore the deviation from the true shape of the profile increases significantly with the radius of the cam using this method of manufacture. By working from the pro­file in preference to the pitch curve Oshima claimed that it is poss­ible to produce the control tape to suit any diameter of cutter. In practice precautions must be taken if there are any concave portions of the profile to ensure that undercutting does not occur. The con­tour was cut by making step movements of lO^am (0.000 4in) parallel to the x- or y- axes as needed to obtain the specified direction and length of chord. Since the cutter radius is at least 500 times the step length the finished profile is much smoother than the stepped path of the cutter, the shape of the pitch curve depends upon the relative diameters of the cutter and the roller. Therefore the cut­ter should be as large as possible and it is also an advantage to use an enlarged master cam.

Oshima used a special-purpose instrument to measure the radius of the profile through a radial translating knife-edge follower. The displacement of the follower was measured by Moir£ fringe techniques and an accuracy in the order of lytm was' claimed. However the accuracy of 0.1° quoted for the measurement of the cam angle may be questioned since an optical dividing head can be accurate to 0°0'10M (120). The worst error on the profile radius was found to be 20ytm (0.000 8in) which is inferior to that claimed by several sources for the hand- finishing process in Table 4-1. The acceleration of the follower was determined from the measured displacement data for cycloidal motion

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for cams cut with chords subtending angles of 1° and 2° at the cam centre. The curves resemble Fig. 3-6 with serious deviations front the shape of the theoretical acceleration curve. The fluctuating maxima and minima have peak values approximately three times that of the theoretical maximum and it is improbable that the theoretical finite-pulse motion would be realized in practice.

More sophisticated methods of interpolating curves for con- tinuous-path machining by numerical control involve the use of blend­ing circular arcs or parabolic arcs which are symmetrical about the focus (50) (92)* As in the case of straight line chord interpolation the positions of the blending points and the lengths of the arcs are determined by the maximum acceptable deviation from the true curve. Fewest interpolation points are needed for parabolic interpolation but the complexity of the calculations demands the use of a computer to prepare the control tapes. These processes were developed for general engineering purposes with the object of producing smooth con­tinuous curves having common tangents to every pair of arcs through the blending point. The particular case of cam profile manufacture imposes the additional requirement that the curvatures of the arcs be equal at the blending points. As shown in Fig. 4-6 this is imposs­ible for circular arc interpolation and it is not claimed for para­bolic arc interpolation. The radius of curvature of the pitch curve

From the conditions established for blending the arcs it follows that a discontinuity in the radius of curvature must be due to an abrupt change in the second derivative. Therefore an abrupt change of fol­lower acceleration and consequent shock loading of the mechanism must occur at every blending point. By approximating the cam profile to a succession of blending cubic curves using the 2C,L program and the TABCYL routine described by Wilkinson (ll6) the worst discontinuity in tho acceleration of the follower at a blending point was found to

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be 2.5$ of the change of acceleration over that section of the profileCUL-btc (Xc.cefara.kionin a particular case, a ;consfam4 cam, showing that the profile

had been accurately manufactured in the critical regions around the start and finish of the event, but since the changes are abrupt shock loading must still be induced.

4.5 ACCURACY OF MANUFACTURE

The accuracy of continuous-path machining using numerical con­trol was investigated at the National Engineering Laboratory with the object of comparing products made on two machine-tools of different design using different programming languages. The results published by Sim (104) are summarized in Table 4-2. Spherical surfaces were manufactured for ease and accuracy of measurement, consequently the experiment did not test the ability to cut shapes involving variations in curvature or points of inflexion which are features of cam profiles. No information was given about wavyness. The maximum errors would be acceptable for many cam profiles provided these result from a gradual accumulation of error, however a comparison of the maximum and rmsvalues is not conclusive in this respect. The round off error in the-.5dimensions specifying the curves did not exceed 5 x 10 inch, but the analysis by Rogers and Schaffer (96) shows that errors of this magni­tude can cause large secondary accelerations of the follower.

The use of numerical control eliminates much of the draughting effort, moreover the program is available for subsequent designs. Wilkinson (ll6) claimed that manual adjustment of the machine-tool occupied 90$ of the time needed to manufacture the master cam, but acknowledged that quantity production using numerical-control is un­economic. All the methods outlined above for manufacturing the cam profile by the continuous-path facility of a numerically-controlled machine-tool have disadvantages associated with the quality of the surface texture and the dynamic performance of the mechanism. Dis­appointing performance of cams made on numerically-controlled machine- tools have been reported to the author by several independent sources. After careful consideration the conventional plunge-cutting process was preferred and .successfully used for the manufacture of high accu­racy cams for the zoom lenses of television cameras (5) which requires

profile accuracy and the least possible wavyness, although dynamic considerations do not apply in this particular case. Many of the - benefits claimed for numerical-control can also be obtained from mechanical methods of copying or generating the profile and are con­sidered in the following Chapter.

Manufacturing errors liable to occur during the conventional plunge-cutting process of cutting the profile of the master cam are:-

(a) an offset error between the line of centres of the cam and cutter relative to the path of the follower.

(b) inaccuracies in the manual setting of either the radial or the angular co-ordinate at a particular position.

(c) an eccentricity between the centre of rotation of cam blank and the origin of the polar co-ordin­ates used to determine the cam positions.

(d) an angular misalignment between the axes of rota­tion of the cam blank and the cutter.

(e) the deflection of the transmission and structure of the machine-tool due to the cutting load.

(f) manufacturing errors in the gears and lead screws of the machine-tool.

(g) inaccuracies in the manual smoothing process.

All these errors except the second and the last cam also occur with numerical-control manufacture and with copying and generating machines, including those used for making the production items. Generating devices are particularly susceptible to the deflection of component parts which transmit the cutting forces.

Translating followers may be offset deliberately to reduce the‘pressure angle for a particular lift, angle of lift and base circle diameter and also to eliminate the reversal of side thrust between the follower and guides on the rise and return movements.Fenton (38) has shown that offset errors can affect the intermediate motion of the follower significantly. However this error is expressed by a continuous function and so does not cause abrupt changes of velocity or acceleration. It can arise at either stage of manufacture or on assembly of the mechanism.

Research into the dynamic performance and the accuracy of manu­facture of gear cutting machines by Welhourn and Smith (ill) showed that normal cutting forces caused deflections between the hob and work­piece which exceed the specified tolerance for the finished gear. In the addition to the elastic deflection of the transmission and structure the load/deflection curve showed an hysterysis movement of bolted joints in the structure. Applying these findings to the machine-tools used for cam manufacture, jthe structure should he welded and well-damped since the magnitude of the cutting force varies with the depth of cut and direction of the resultant force varies with the pressure apjgle. Cheney's recommendation to use two roughing and a finishing cut is confirmed (20 ) and one design of copying machine (ll8)' adjusts the synchronised speed of rotation of the master cam and the blank according to the pressure angle. The transmission of the profile-cutting machine should be short and direct and should incorporate precision gears. The cutter must be accurately gound to ensure concentricity and the specified radius of cut. The workpiece and supporting members should be as stiff as poss­ible, this requirement conflicting with the need to keep the static load on the table as low as possible to ensure uniform, free movements. The material of the cam blank should be homogeneous and free from inclu­sions, Welbourn and Smith observed random errors in the order of 0.000 3 to 0.000 4in in gear teeth profiles manufactured from 3°/° nickel-chromium cold drawn steel to B.S. 970: 1955; En36 and recommend further research into the surface of gear materials and the deformations which occur dur­ing the machining process. Measurements made on the Philips contour milling machine whilst taking light finishing cuts showed that the maxi­mum error in the position of the cutter was 2>tm (0.000 08in) which was magnified to 5/iro (0.000 2in) on the profile of the finished cam (50).

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4 .6 SUMMARY

The method of manufacturing the master cam profile is influenced by the indirect specification in terms of the polar co-ordinates of the pitch curve and the shape of the working surface of the follower. The equations of these curves are unknown. The profile is manufactured by reproducing an inverted form of the motion of the mechanism, replacing the follower with the cutter.

The procedure and accuracy of profile manufacture by the conven­tional plunge-cutting process and various applications of the continuous- path facility using numerical control were investigated to identify the sources of error and to establish the requirements to be satisfied by an alternative method. Manufacturing errors result from inaccurate mount­ing of the cam blank or cutter, setting errors due to the operator, faults in the transmission or the elastic deflection of these members due to the cutting force, deflection of the structure of the machine-tool due to the cutting force, wear, and inaccuracies in the slides.

Inspection of Table 4-1 shows no significant advantage for numeri­cal control. Abrupt changes in the curvatures at the blending points of the interpolating curves used for continuous path manufacture cause shock loading. Conventional plunge cutting was preferred for an application in which profile accuracy and smoothness, but neither cost nor dynamic per­formance were controlling factors.

t

It is particularly important to reduce to a minimum the magnitude of random errors causing profile wavyness. Larger errors expressed as continuous functions have negligible effect upon the profile shape and therefore are far less critical.

A significant development in the performance of cam mechanisms may be expected to result from the development of a manufacturing process which eliminates the need to blend interpolating curves and the source of round-off error whilst requiring the simplest possible setting of the machine-tool with minimum dependence upon the accuracy of lead screws and gear transmissions. The positioning device must not be subject to the manufacturing forces.

Chapter 5

AIDS TO CAM MANUFACTURE AND PROFILE-GENERATING MACHINES

5 .I COPYING AND GENERATION OF PROFILES

The design and manufacturing effort involved in the production of a new master cam using the conventional plunge-cutting process des­cribed by Cheney (20) has provided a strong incentive to develop aids to cam manufacture and means of copying or generating the cam profile which eliminate the need to calculate the increment positions specify­ing the shape of the profile, enable the profile to be cut as a con­tinuous process and require no manual finishing operation.

Previous publications have not made a clear distinction between copying devices and generating mechanisms so the following definitions are proposed

A Copying Device produces a section of the cam profile . between two specified cam angles by copying a previously prepared template. Different values of lift and angles of lift or return are obtained by adjusting the velocity ratio between the displacement of the follower tracing the profile of the template and the movement of the cut­ter and by changing the ratio of the directly proportional relationship between the travel of the template and the rotation of the cam blank. The same device is used for cutting profiles to produce different follower motions by changing the template. Such a machine is more sophisti­cated than the copying machine used to manufacture the production items from the master cam.

A Profile Generating Mechanism produces the rise or return section of the cam profile by a continuous cutting action along a path determined by t-he output movement of a con­strained mechanism from one or more input drives of known path, direction and displacement. ?ih.e output motion of

this mechanism determines the instantaneous position of the cutter relative to the centre of cam rotation and the input drive(s) must be synchronised with the rotation of the cam blank. The lift and the angles of lift and return can be varied by adjusting the proportions of the mechanism and/or by changing the displacement of the input drive.

5.2 DESIGN REQUIREMENTS FOR A CAM PROFILING MACHINE

The merits of various devices developed for this -purpose can be evaluated by preparing the design requirements to he satisfied by a'cam profiling machine intended to have the widest possible application.

(a) The device should be universally applicable to suit any size of cam within the dimensional constraints of the device itself and those of the machine-tool to which it is fitted. The dimensions affected directly by this requirement are the base circle diameter, the lift and possi­bly the angles of lift and return which may be influenced by the availability of change gear ratios.

(b) The device should he capable of generating pro­files to suit any cam law. For this purpose it ' should be possible to make profiles to match any type of follower, any path of the follower and any displacement relative to the centre of cam rotation, including any requirement for particu­lar magnitudes of velocity or acceleration of the follower at particular positions or for an "asymmetric" motion in which the transition point is displaced from the mid-position of the motion.

(c) The various sections of the cam profile controlling consecutive follower movements, such as a dwell- rise-dwell event, must blend smoothly. It must be

possible to identify the conclusion of the cutting action for a particular section of the profile positively and with a high degree of precision. After one complete revolution of the cam blank the cutter must return to the starting position.

<?(d) The generating mechanism or means of determin­

ing the instantaneous position of the cutter relative to the centre of cam rotation mustbe isolated from the cutting forces to elimin­ate errors due to the elastic deformation of the component parts and any movement of bolted joints (ill). This precaution should also result in a compact design requiring a small operating force. It should be driven posi­tively and must synchronise the displacement accurately with the corresponding rotation of the cam blank.

(e) The effects of backlash, bearing clearances and stick-slip friction must have minimal effect upon the accuracy of the output motion.

As shown in the previous Chapter it is generally possible to make the master cam for a radial translating follower. Then the corrections to make the profile of the production cams suit the type and path of the follower can be made on the copying machine (106).

In addition it is desirable that the device meets additional requirements associated with the cost and operation of the special- purpose equipment.

(f) The device must assemble easily upon a conventional machine-tool suitable for cam manufacture without modification to that machine.

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(g) It must be simple to understand and to use, not requiring specialized knowledge or skill from the operator.

(h) The use of the profile cutting machine should result in improved accuracy and surface texture of the product and a signi­ficant reduction in both the design and manufacturing effort in comparison with the conventional plunge-cutting method of cam manufacture.

The effectiveness with which the deflected leaf spring method of deriving cam profiles meets these requirements is discussed in Chapter 11.

The design of a new cam mechanism to be manufactured by a copying or generating process would begin by making a preliminary check of the limiting parameters to determine the principal dimen­sions following the procedure set out in Fig. 2-2 and would continue with the scheme shown in Fig. 5-1* As in Fig. 2-1 the progress of the work is traced by following the lines downward from the starting dimensions at the top of the diagram. Although Fig. 5-1 does not appear to be simpler than Fig. 2-1 which represents the design proce­dure for a cam to be made by the plunge-cutting process, the actual work is considerably reduced. There is no need to calculate the incre ment co-ordinates and to present this data in the form needed for manu facturing purposes. It is envisaged that a standard form of the type shown in Fig. 5-2 would be adequate for many purposes.

5.3 PRACTICAL CONSIDERATIONS

A study of profile generating and copying mechanisms shows that many designs rely upon the transmission of cutting forces through moving parts and that some depend upon friction drives. Profile inaccuracies will inevitably result from the elastic deflections of these parts and the vital synchronisation is lost if slip occurs in the transmission. The operation of some mechanisms is dependent upon a reversal of motion during the generation of one section of the pro-

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-4l~

file although this should not result in any take-up of backlash or clearance since the direction of the component of the cu.tting force along the path of the follower does not change* For these reasons severe restrictions may be placed upon the depth and rate of cutting and the choice of materials which can be used for the master cam may be limited (47) (6l).

The output motion derived from any constrained mechanism must be a continuous function determined by the input motion or motions and by the proportions of members of the mechanism. For this reason it is inflexible, a particular mechanism is only capable of generating one family of curves. However this limitation is not a crucial one provided the resulting follower motion can be shown to produce good dynamic performance, and it is compensated by the feasibility of com­piling comprehensive design data identified in Fig. 2-2 to assist in the design of the new cam mechanism.

5.4 AIDS FOR CAM PROFILE MANUFACTURE

Various aids have been developed to assist the toolmaker in the manufacture of the master cam by the plunge-cutting process. These aids are intended to simplify the operation of re-setting the machine- tool at every increment position and hence to reduce the danger of human error. In some cases the aid overcomes the restriction that the increment cuts can only be made at the co-ordinate positions specified on the drawing.

The task of setting the machine for increment cutting is assisted by the provision of a digital readout display giving the angular posi­tion of the rotary table and cam blank with the co-ordinate position of the machine-tool table related to the specified datum (122). The accuracy can be improved by using Moire" fringe methods of linear mea­surement. The toolmaker is provided with tabulated co-ordinate data calculated to the same order of accuracy as the digital display, but plunge cuts can only be made at the specified co-ordinate positions and the profile must be finished by the manual smoothing operation.

Jellig (60) describes a method of deriving cam profiles direct­ly from the drawings. The machine is set manually by aligning a pointer onto a large scale drawing of the geometric development of the cam profile. It is difficult to accept the enlargement of 500:1 claimed in this paper on account of size, the practical difficulties involved in the preparation of such a large drawing would outweigh the benefit of improved accuracy resulting from the diminution in scale of the finished product. The feed of the paper relative to the pointer must be synchronised with the rotation of the cam blank, so the stretch of the paper and the effects of varying atmospheric humi­dity would also affect the accuracy of manufacture. The method is intended for increment cutting but the toolmaker is not restricted to the calculated co-ordinate positions. Due to- the additional work of preparing the drawing there is a significant increase in the design effort involved and the cost of the product.

A more sophisticated approach enables the entire cam profile to be cut as a continuous operation by tracing the outline of a true drawing of the shape of the profile to an enlarged scale of 5 :1 pre­pared with an electrically conductive ink (48). The drawing is secured to a second rotary table synchronised to turn with the cam blank. A stylus free to move radially over the drawing is positioned to maintain a constant spark gap between the tip of the stylus and the outside edge of the line. Hale claims that the drawing can he prepared to an accuracy of 0.005in which reduces to + O.OOlin for the position of the cutter. No information about the accuracy of the machine-tool'was given, judging from the results published by Cheney (20) and Haringx (50) the profile would be accurate to+0.003 - 0.005in. This arrange­ment may be compared with a photo-electric sensing device used to con­trol a flame-cutter from an ordinary drawing. The flame-cutter is not subject to cutting forces but may operate at a higher speed. It is understood that the inertia forces were sufficiently large to cause a tendency to over-run, the overall accuracy of this machine being quoted as + O.OlOin (62).

5.5 PROFILE COPYING MACHINES

A machine for making the profiles of master cams by copying prepared templates has been manufactured by Sylvania Electric Products

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Inc. (U.S.A.) (121). Only one template is needed to cut any size of cam to a particular cam law as the relative movements of the template, cutter and cam blank are controlled through change gears. The arrange­ment of this machine is shown diagramatically in Fig. 5-H Tn this instance the mechanism is not inverted for manufacturing purposes. The clutch is needed to engage the template at the cam angles corresponding to the start of a lift or a return movement. The proportionality of the displacement of the cam blank relative to the cutter and that of the follower tracing the profile of the template is determined by an adjustable linkage.

5.6 PROFILE GENERATION

All the devices described above are capable of manufacturing cam profiles to suit any follower motion but none satisfies the defini­tion for generation of the profile shape. Mechanisms capable of generat­ing cam profiles have been developed, but since such a device must have a unique output motion for a given lift and angle of lift of the cam it is restricted to one cam law. The generating action can produce pro­files for one of the common cam laws or it may be a special motion hav­ing acceptable dynamic performance.

The cam profile to obtain constant velocity follower motion is readily generated on a milling machine with the use of a dividing head,driven to rotate the cam blank at constant angular velocity. As in the plunge-cutting process the mechanism is inverted so that the axis of the cutter remains stationary. The rotation of the cam is synchronised with the feed of the table of the milling machine at uniform velocity in the direction parallel to the path of the follower. Whilst simple to generate this profile gives extremely poor dynamic performance be­cause of the ’infinite' acceleration at the start and finish of every follower movement. It is used on some automatic lathes.

Simple Harmonic Motion profiles can be generated by a scotch yoke mechanism or by an end-cam (6l). The radius of the driving crank of the scotch yoke must be equal to exactly half the lift of the cam and the crank must make exactly half of one revolution whilst the cam rotates through the angle of lift or return for the event being generated.

Therefore the length of the crank between centres must be adjustable or else it must be made specially to suit the cam being cut. The working surface of the end cam shown diagrammatically in Fig. 5-5 is cut at an angle to the plane perpendicular to the axis of rota­tion, so that the difference between the heights of the cylinder measured parallel to the axis of rotation at the radius of the fixed follower equals the lift of the cam. The axial movement with simple harmonic motion is generated by forcing the inclined face against the fixed follower whilst rotating the end cam. The lift can be ad­justed by varying the radius to the fixed follower. The requirements for the angular rotation 'of the end cam are the same as those for the scotch yoke. In both cases the details of the construction can be varied but the cutting force must he transmitted through the compo­nents. The lift and the accuracy of blending successive portions of the profile are dependent upon the accuracy of setting or manufactur­ing the driving radius of the mechanism and of determining the rota­tion of the input member. |

The displacement curve for the more complex' function defining cycloidal motion is constructed geometrically by taking the algebraic sum of the linear and sinusoidal components in the manner illustrated in Fig. 5-6. The cam profile can be generated in the same manner. Jensen (6l) illustrates a device (Fig. 5-7) in which the motions of two racks are combined to obtain the parallel displacement of a third member which carries the cutter. The cam blank turns with the rotary table T which is connected through the gears Gl and G2 to the rack Rl. The gear G3 meshes with both Rl and a second rack R2 constrained to slide in the direction perpendicular to the travel of rack Rl. The displacement of rack R2 is directly proportional to the angle of rota­tion of the cam blank. A third rack R3. is parallel to rack R2 and is driven with simple harmonic motion through a- scotch yoke mechanism.The driving crank E of the scotch yoke is integral with the gear G3. The gear G4 meshes with racks R2 and R3- ft is freely pivoted to a slide S free to move in the direction parallel to the travel of R2 and R3- The cutter is mounted on this slide. The mechanism can be analysed as an epicyclic gear train of infinite diameter to show that the displacement of the slide S is the mean of the displacements of the racks. Therefore the drive through gears Gl and G2 must cause displacements of racks Rl and R2 equal to twice the lift for rotations

of the cam blank through the angle of lift or return. During this rotation of the cam blank the crank driving the scotch yoke must make exactly one revolution. Consequently the radius of the crank becomes the awkward dimension of h/ -Tf . It is also essential that the line of centres of the driving crank be parallel to the path of the rack R3 at the start of the generating action.

For these reasons the mechanism illustrated in Fig. 5-7 requires precision manufacture and setting and it is sensitive to the transmis­sion of cutting forces through components of the mechanism.Jensen (6l) claims that -'excellent' results were obtained from this mechanism when cutting a cam from 'Pexiglass'. This mechanism can be simplified and cheapened by combining the actions of racks R1 and R2. The simplified version shown in Fig. 5-8 also imposes less restriction upon the size of the gear G3 carrying the driving crank for the scotch yoke.

Another means of superimposing a (sinusoidal motion upon a linear displacement was designed by Wildt (112) and is described and illustrated by Jensen (6l). A revised version of this illustration appears in Fig. 5-9* The operation of this device depends upon the relative motions between two moving slides and a fixed frame. One slide is made to complete one oscillation with simple harmonic motion relative to the machine frame whilst a second slides moves with linear motion relative to the first slide in a direction inclined to the amplitude of the sinusoidal motion. The cam blank is mounted 'on the second slide and rotated through an angle directly proportional to the linear displacement of that slide from a specific datum. Gears are eliminated by transmitting the drive through flexible steel bands. The cutter rotates about a fixed axis.

The input drive is applied to the leadscrew L-L (Fig* 5-9) to move the slide S2 in the direction Q parallel to the centre line U-U. The cam blank is mounted upon a rotary table R supported by bearings in the slide S2. Two flexible steel bands B3 and B4 aro secured at F to a disc fixed to tho rotary table R. These bands are wound around the disc in opposite directions and their outer ends are secured to the meachine frame. By making the straight line portions of the bands parallel to the centre-line U-U the angle of rotation of the cam blank

is directly proportional to the displacement of slide S2. Another pair of flexible bands, BI and B2, are fixed to the slide S2 atipositions J and M respectively. These bands are secured to spindle A at K and used in the same manner as bands B3 and B4 to obtain a positive drive in both directions of movement. Spindle A is mounted on bearings in the slide SI. The eccentric E fixed to the spindle runs in the bore of the trunnion block T which is free to slide in guides fixed to the machine frame. Then the linear displacement of slide S2 in direction Q imparts a sinusoidal oscillation to slide51 in the direction P-P.

Jensen's drawing implies that spindle A is mounted in bearings on slide S2 but this arrangement would prevent relative movement be­tween the slides.

The movements of slides SI and S2 are shown separately inmovesFig. 5-10• The centre of cam rotation on slide Sl^along the straight line from UI to U2 through the lift h. Simultaneously the eccentric E completes one revolution to give slide SI simple harmonic motion in direction P-P. The ordinate of the displacement characteristic is parallel to the centre line U-U and the equation of the sinusoidal component of motion must be such that the displacement curve has zero slope at the start and finish of the event being generated. Therefore the eccentricity must be determined for the cam being cut. Comparison of Figs. 5-6 and 5-10 shows that this device does not generate cycloi­dal motion. Moreover the axis of rotation of the cutter is displaced from the centre-line U-U by the sinusoidal action, further complicating the analysis of the follower motion derived from this profile. For the same reason it would be necessary to drive the lead screw through a flexible member.

This design is structurally stiffer than the rack driven mechan­ism shown in Fig. 5-7 since the cutting force is transferred from slide52 to the lead screw L. However the offset of the cutter axis from the centre-line U-U and the effect of the pressure angle combine to applya couple to the cam blank which must be reacted through the flexible bands B3 and B4. The operation is sensitive to manufacturing and assem­bly conditions such as the accuracy of the eccentricity between the

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centres of shaft A and disc E. The error due to elastic deflectionof the steel bands is a function having zero magnitude atboth limits of motion provided the cutter centre lies on a radius of

5.7 GEAR DRIVEN GENERATING MACHINES

An alternative approach to the generation of cam profiles is to show that the output motion of a particular mechanism generates cam profiles which impart a follower motion with good dynamic perfor­mance although it does not conform to a familiar cam law. Grodzinski (44) showed that eccentrically mounted spur gears can be used to con­vert an imput'drive at constant angular velocity into a fluctuating output motion in which a periodic angular acceleration and retardation is superimposed upon the input motion. (Other speed variations caused by manufacturing and assembly errors of the gear train were ignored in the analysis.) Correct depth of mesh was maintained by using a connecting rod to join the centres of gear rotation. The bearing for the output gear ran in slides and the motion was transmitted through an Oldham coupling.

The design is simplified by exploiting the properties of the involute tooth form to accept a varying depth of mesh. Then both gears can rotate about fixed centres provided the eccentricities are not so large that the gear teeth fail to engage. This arrangement has the disadvantages of reduced load capacity and an increase in the velocity of sliding between the tooth profiles. Grodzinski (44) (45) quotes different values for the maximum possible eccentricity ratio

both gears. In fact this ratio is determined by the extremes ' of maintaining engagement at the maximum separation of the true centres of the gears and jamming of the teeth at the smallest centre distance. Therefore the ratio is dependent upon the dimensions of the teeth which are proportions of the pitch but independent of both the eccen­tricity and the pitch circle diameter. A sensible relationship would be the ratio of the whole depth of the tooth to the eccentricity.

coa6(Vvu,ou.s

the cam at these positions. Therefore this error is insignificant.

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(\;lThe use of fixed centres of rotation made it possible to ex­tend the application of eccentrically mounted gears to form planetary gear trains (46). A fluctuating output motion is obtained through the driven sun gear by driving both the input sun gear and the planet carrier independently with uniform angular velocities. Particular combinations of the relative magnitudes and directions of these in­put drives result in a periodic output motion which includes distinct hesitations, typical characteristics are shown in Fig. 5-H*Grodzinski recognised that the smooth output motion with blending hesitations was suitable for cam profile generation since adjoining sections of the profile could be joined by stopping the generating action during a^hesitation (45). The operation and functions of this profile-generating machine are shown diagrammatically in Fig. 5-12.For manufacturing purposes the cam mechanism is inverted in the same manner as in the plunge-cutting process. The rotary table carrying the cam blank is fixed to a slide driven through a rack and pinion from the output of the planetary gear train. The drive from the electric motor is taken through four 'infinitely-variable' velocity ratio drives to various parts o,f the mechanism to obtain the maximum flexibility of cam size and cam angles.

The use of 'infinitely-variable'' velocity ratio transmissions implies the use of friction drives. The cutting force is taken through components of the mechanism and it is essential to ensure that no slip occurs. The use of the eccentrically-mounted gears in the planetary gear train results in variations of the length of the path of contact, the pressure angle, and the backlash. These gears must be designed for the worst possible condition. Theoretically it would be possible to increase the load capacity of the gear train by using several planet gears, but this would involve precision manufacture and assembly. The theoretical analysis of the generating mechanism makes no allowance for manufacturing or assembly tolerances causing differences from the nominal dimensions. In a particular machine the independent settings of the variable velocity ratio drives could be calibrated to reduce the significance of some errors, such as the tolerance on the eccen­tricity of the pitch circle. It must be possible to identify the start and finish of a particular event accurately to blend adjoining sections of the cam profile. The application of this mechanism was restricted to cutting soft materials such as perspex (4?)*

5*8 APPLICATION OF GENEVA MECHANISMS

One difficulty experienced with the use of profile generating mechanisms is that of blending adjoining sections of the finished profile. Wildt has shown that this problem can be overcome by com­bining the output motions of two geneva mechanisms through a bevel differential gear to enable the entire profile to be cut in one con­tinuous operation (115)* The arrangement of the mechanism is shown diagrammatically in Fig. 5~13- The geneva mechanism has the advantage of a positive lock during dwells and the start and finish of an in­dexing movement can be readily identified. The input drive synchro­nises the rotation of the cam blank with both input cranks of the geneva wheels. The output movements are taken from the geneva wheels to the gear G1 and the planet carrier C of the differential gear train respectively (Fig. 5-14). The output gear G2 of the differential is converted into the linear displacement of the cam blank. Both geneva wheels must rotate gear G2 through equal angles in opposite directions, these movements must not be superimposed. The number of stations on the geneva wheels must be matched to the velocity ratio of the differential gear train. Different angles of lift and return are ob­tained by changing the velocity ratio of the drive to the cam during a dwell and the cam angles for the starts of the rise and return eventsare determined by the relative positions of the driving cranks of the

fgeneva mechanisms. Then the complete profile is cut from one indexing movement of each geneva wheel to obtain a displacement characteristic of the form shown diagrammatically in Fig. 5-15* The cam characteris­tics for this profile would include an infinite pulse due to the abrupt accelerations associated with the start and finish of a crank-driven geneva wheel. Hunt et al (57) have shown that the crank can be re­placed by a linkage to obtain a finite-pulse indexing movement.

5.9 ASSESSMENT OF PREVIOUS DESIGNS

All the copying and generating mechanisms included in this sur­vey incorporate a mechanism to control the relative motion between the cam blank and the cutter and to transmit the cutting forces. The accuracy of the finished profile depends upon the following factors (not all of which apply to a particular design):-

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the dimensional accuracy of the components.

the synchronisation of the rotation of the cam blank and the displacement of the cutter.

backlash in the meehansim.

the deflection of the components due to the machining forces.

the transmission of manufacturing forces through friction drives.

the fluctuation in the instantaneous velo­city ratio of a gear transmission.

the accurate identification of the limit of the motion corresponding to the section of the profile to be generated.

The simplest manufacturing aid uses digital displays of the co-ordinate positions of the table. These are easier to read than dials and possibly more accurate. This approach can be considered an intermediate step between manual plunge-cutting and numerical-con- trol.

The problems associated with manually-prepared drawings or developments of the cam profile to an enlarged scale are identified on pages 42 & 43 . The accuracies quoted for these methods are inferior to the plunge-cutting and manual smoothing process (Table 4-1) and additional drawing-office time is required.

Profile copying by any method transfers the manufacturing pro­blem to the template, although manufacturing errors may be reduced by including a diminishing factor in copying mechanism. A mechanical linkage requires at least one infinitely-adjustable member if earns are to be cut for any magnitude of lift and all the members are subject to direct or bending loads due to the transmission of the cutting force.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

For this reason an hydraulic control of the copying action is to be preferred (67), but this solution requires a suitable machine-tool and increases the cost of the equipment required.

The generating mechanisms (except that driven through the geneva wheels) are only capable of producing the section of the pro­file controlling a rise or return movement of the follower at one setting of the device. The magnitude of the lift depends upon the radius of a crank or an eccentric, or the position of the reaction member in the case of a face cam (Fig. 5-5) • The operation of the differential rack mechanism (Fig. 5-7) and the double slide mechanism (Fig. 5-9) rely on the precise radius to obtain the correct combination of displacements, imposing a serious practical problem since this dimension is a function of 'Tj . A similar requirement applies to the geneva-driven mechanism (Fig. 5-15) as any discrepancy between the lift movements will be reproduced in the cam profile at the blend­ing point. Less stringent tolerances are acceptable for the scotch yoke mechanism and the inclined face cam mechanism since a correspond­ing allowance of, say, + 0.003in on the lift is generally acceptable provided the actual profile is complete and conforms to the displace­ment equation. The inclined face cam is easy to manufacture and any magnitude of lift can be set by adjusting the radius to the reaction member. To prevent inherent slip the contact between these parts must be made through a suitably tapered roller. This mechanism is inherently simple and robust.

Emphasis has been placed on the importance of manufacturing the correct shape of the profile in the regions close to the start and finish of every follower movement. The output of generating mech­anisms consists of a continuous repetition of the motion used to cut the profile, separated by hesitations and dwells in the cases of the eccentric gear mechanism (Fig. 5-12) and the double geneva mechanism (Fig. 5-13) respectively. Except in the case of the last it is vital that these mechanisms be accurately set at the starting position and that the driving member be given the correct- displacement. The same requirements apply to the use of a template for a copying process.

No allowance is made in any of the gear driven mechanisms for potential manufacturing and assembly errors. These are identified in

Chapter 9 (pages 114 - 5 ) and Appendix 7. These errors areparticularly significant in the case of the, eccentric gear mechanism and other applications which involve partial revolutions of gear wheels.

The double geneva mechanism (Fig. 5-13) has the considerable advantage that the complete profile including dwells can be manufactured at one setting of the mechanism. The gears rotate in opposite directions whilst the rise and return sections of the profile are being cut, but as the reaction force from the continuous cutting action does not reverse the backlash must always be taken up in the same direction. However some clearance must be provided between the driving member and the slot and between the locking faces to permit relative motion and to ensure that the motion cannot be jammed by the locking action. In this applica­tion these clearances must be taken up at the changeover positions, implying a corresponding discontinuity in the profile.

Since these mechanisms operate at very low speeds only the cut­ting force has any significance. The components of this force acting parallel and perpendicular to the path of the follower vary with the depth of cut and the pressure angle, hut retain the same direction whilst manufacturing one section of the profile. Consequently the reversal of the direction of a drive is not significant except in the case of the geneva mechanism. Attempts to reduce the elastic deflection by enlarg­ing the components or by magnifying displacements aggravate the problem of accommodating the device on the machine-tool and increase the weight. Similarly friction drives need to be as large as possible to transmit the greatest possible torque for a given coefficient of friction.

For these reasons the rate.and depth of cutting and the choice of material may be restricted. It has been conceded that some generat­ing mechanisms are only capable of cutting soft materials such as pers­pex (47) (6l).

5.10 SUMMARY

Aids to master cam manufacture are intended to reduce design effort, simplify manufacture and reduce the cost of the product. A clear distinction is made between copying and generating devices and

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the design requirements are established for an ’ideal* profile cut­ting machine. The practical restriction to a particular follower motion is not considered serious provided the conditions for good dynamic performance are satisfied.

The survey of previous work shows that the aids to cam manu­facture and the use of copying machines do not overcome the basic problem of manufacturing an accurate profile. None of the generat­ing devices meets the design requirements completely, in many instances correct operation depends upon the precise manufacture of certain dimensions, exact setting and exact displacement of the input member. Elastic deflection results from the transmission of cutting forces through the members of the mechanism and in one case this force is taken through friction drives which synchronise the generating action.

This assessment is supported by Cram who concludes that none of these devices proved sufficiently'successful to gain widespread application (23)-

5.11 CONCLUSION

To be successful a direct method of manufacturing cam profiles must he designed to isolate the components of the device from the cut­ting forces(, synchronise the rotary and linear movements in a simple and direct manner, operate correctly with components manufactured and assembled to practical tolerances and identify the limits of every section of the profile to a high degree of precision.

PROFILE DERIVATION.AND DEFLECTION OF THE ELASTICA

Chapter 6

THE DERIVATION OF CAM PROFILES FROM THE SHAPE OF A DEFLECTED LEAF SPRING

6.1 REQUIREMENTS FOR PROFILE CUTTING

The methods of manufacturing master cam profiles surveyed in Chapters 4 and 5 are assessed in terms of

(a) accuracy(b) cost

■(c) ease of use(d) flexibility

To gain acceptance any new process must have comparable accuracy and cost with methods in current use.

6.2 ACCURACY OF PROFILE MANUFACTURE

As shown in Chapter 3i follower'acceleration is extremelysensitive to profile errors. It is impractical to specify the cutter

-4position for manufacture to a greater accuracy than 1 x 10 in. Con­sequently round-off error is an inherent source of wavyness which alone can approach the empirical maximum of 2 x 10~^ in/degree quoted by Nourse (89). For this reason jig-borers are used for the manual plunge- cutting process. Alternatively the continuous-path facility of a numerically-controlled machine-tool must trace a curve of continually varying curvature -including points of inflexion. Linear interpolation,Fig. 4-5, requires a specified maximum tolerance between the chords and the true curve. The number of interpolation points is reduced by using bilateral tolerances, Fig. 4-5(b), the chord length and the angle subtended at the centre of curvature being functions of the radius and tolerance. Typical values, including the profile wavyness due solely to the tolerance are quoted in Table 6-1. Assuming constant curvature the angle between adjacent chords equals that subtended at the centre. Analysis showed the

effect of round-off errors on chord length and subtended angle to benegligible, except for the combination of largest tolerance ( 5 x 10 in )and smallest radius ( 0.5 in ) when the angular error was 3*5$* For a

-5 -5tolerance of 5 x 10 in the errors did not exceed 5 x 10 %. Theseresults emphasize the need to use the smallest possible tolerance,especially with large radii. Even so the angles between adjacent chordsare significant and the author considers it essential to smooth themachined profile by honing. Then linear interpolation becomes preferableto circular arc or parabolic interpolation since neither of these can besmoothed to eliminate the abrupt changes of curvature at the blendingpoints, Fig. 4-6. This conclusion concurs with Beard (9) who consideredthe final honing operating of the plunge-cutting process to have thevalue of smoothing profile irregularities and is supported by the decisionof Rank-Taylor-Hobson (5) to use the manual plunge-cutting and honingprocess for cutting precision master cam profiles', achieving an accuracy

-4 oof + 3 x 10 in on a barrel cam track exceeding 1000 rotation.

The copying and generating devices surveyed in Chapter 5 are intended to manufacture master cam profiles by blending the portions of the profile controlling successive follower movements or dwells. These machines must be reset at every blending point. Production items are copied from the master cam on a special-purpose machine. The accuracy of master cams cut in this way is subject to positioning errors at both limits of every section whilst both master and production cams are liable to errors resulting from the accumulation of tolerances, backlash and elastic deflection of machine members. Loads result from the cutting, inertia and friction forces. Loss of synchronisation between displace­ment and cam angle results from transmission errors, ^uch as those identified in Appendix 7* Copying devices must he capable of reproducing profiles having continuously varying curvature including maxima, minima and points of inflexion. Additional errors result from the threshold (the maximum displacement of the tracer without response of output) and frictional forces. Koenigsberger (72) has shown that these effects com­bine to produce a hysterysis effect upon reversal of tracer movement.Due to the inertia forces accuracy is related inversely to the cutting speed (67). The significance of these copying errors is demonstrated by comparing the positioning accuracy of the SIG n-c profile cutting machine (+0.000 4in) with the copying accuracy (relative to the master

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cam) of the very similar hydraulic copying machine (+0.000 8in) (125)• However it does not follow that the pitch curve wavyness of cams cut on the copying machine is inferior to that of the master cam.

These limitations may be overcome by using the copying principle to determine the pitch curve radii for an increment cutting process.Then every machine movement can have the same direction with sufficient travel to take up backlash and threshold. Since these machine settings are independent of the positioning accuracy and the tolerances on the lift and angles of lift and return may reasonably be ten times those for increment positions it becomes possible to use a milling machine instead of a jig-borer (relative capital cost 1:4 (126) ).

6.3 COST OF PROFILE MANUFACTURE<

Few publications quote the time or cost of cam design, cutting master cams or copying production cams. A comparison of the manufacturing methods of master cams is complicated by the significance of certain dimensions. For example the angle of lift (and hence the number of increment positions) largely determines the time needed to complete a plunge-cutting process. The estimates of the relative cost of cam manu­facture by different processes summarized in Table 6-3 are based on data quoted by Cheney (20), Wilkinson (ll6), Sylvania Corporation (121), SIG (125) and PERA (126), supplemented by the author's experience. The estimates make no allowance for the preparatory design needed to deter­mine the configuration and basic dimensions of the mechanism, checking the limiting dimensions or for raw material and turning the blank, all of which can be considered independent of the production process.

The estimates of the components of the cost of specifying and cutting the master cam profile using different processes are derived in Table 6-2 and compared in Table 6-3. The times quoted.in Table 6-2 com­prise the total time required to prepare the manufacturing instructions including any computer time expressed as the equivalent cost in man-hours, plus the time required for all the machining operations. It was assumed that all.the machine-tools have the same utilisation factor and rate of depreciation whilst the maintenance cost is proportional to the capital cost. Differences in power consumption and floor area occupied were

considered negligible. Then the unit cost of the machine-tool is the product of the time required and the relative capital cost, taking a vertical milling machine as 1. The numerically-controlled machine-tools considered are a 3-axis machine with a linear interpolation continuous path facility (relative capital cost 6, circular arc interpolation increases the capital cost by 10$) and a special-purpose cam cutting machine using co-ordinate data (relative capital cost 9 to 11 -).

The cost of drawing office and tool room labour is approximately equal, making it possible to express the machine-tool costs in terms .of man-hours. Taking the cost of labour as £3000 per annum-(1974), a vertical mill as £5 per hour (126) and allowing for overheads by doubling the labour cost the unit machine-hour is equivalent to 1 - man-hours.

The cost indices compared in column A of Table 6-3 range from 1 for both numerically-controlled machining and profile generation to 6 for completely manual plunge-cutting, the last reducing to 4.9 if the co­ordinate data is prepared on a computer. This comparison makes no allow­ance for profile accuracy. The results quoted in Table 6-1 show that a

-4 -5reduction of the tolerance from 5 x 10 to 5 x 10 trebles the number of interpolation points for continuous path cutting with linear inter­polation, so the proportion of the computer time required for calculating the co-ordinates and punching the control tape would be correspondingly increased. Processes considered to have inadequate accuracy are elimin­ated from column B of Table 6-3. This index is based on the lower estimate for numerically-controlled linear interpolation followed by honing which equals that for the special-purpose cam milling machine.The-author's method of profile derivation from the shape of a deflected leaf spring introduced in Section 6.7 is also at the low end of this index.

6.4 EASE OF USE

The ease of using a particular process depends upon the setting procedure and special-purpose equipment needed, the number of independent control movements to be co-ordinated manually during the machining pro­cess and the number of repetitive operations. Numerically-controlled machines, especially the special-purpose cam-profiling machines, have

considerable advantage in this respect. In contrast the plunge-cutting process is repetitive, but relies upon familiar techniques whilst copying and generating processes may require specialised skills. It is antici­pated' that such devices require a longer time to set up, therefore cutting time must be correspondingly reduced.

It is important that the operation of any copying or generating process be immediately apparent, not requiring special training or skill.

6.5 FLEXIBILITY OF PROFILE CUTTING

To obtain maximum utilisation the machine-tool must be capable of other work. Most of the copying and generating devices surveyed in Chapter 5 satisfy this requirement since they form attachments to conven­tional milling machines or jig-borers. These attachments are cheap in comparison with the capital cost of the machine-tool. However generating devices are only capable of producing profiles for one output motion whereas cams are manufactured to a variety of cam laws according to the operating requirements and accuracy attainable. A further variation is introduced if it is necessary to displace the transition point (the position of maximum follower velocity) from the mid-point of the motion to reduce the maximum surface stress, permit a lower spring force or for operational reasons. Ratios of cam angle controlling follower acceleration up to 6:1 are quoted (34). Such motions are termed asymmetric in contrast to symmetrical motion in which the transition point occurs at mid-displacement of the follower and the magnitudes of the peak acceleration and retardation are equal. As shown in Fig. 6-1 the displacement character­istic for asymmetric motion consists of blending sections of two symmetrical motions. It is essential that the slopes be equal at the blending point.

Profiles of any reasonable shape can be cut immediately by numerical control or the plunge-cutting process, but generating and copying devices require to be reset at the transition point. It is argued that a restriction on the proportions of asymmetric motion (e.g. to ratios of integers) is not serious.

6.6 DESIGN OBJECTIVES FOR A CAM-PROFILING MACHINE

To gain acceptance, a new method of profile manufacture must he simple to prepare and use whilst producing acceptable follower performance with a choice of output motions including asymmetric proportions. The cost must be comparable. Ideally the process shouldnot be responsible for inherent wavyness of the pitch-curve. Means ofattaining these objectives are to:-

(a) avoid dependence upon calculated co-ordinates subject to round-off error.

(b) simplify the setting procedure by introducing a positiverelationship between the angle of cam rotation and cutterposition.

(c) depend upon the repeatability of measuring instruments in preference to movements liable to positioning error.

(d) transmit the cutting forces entirely through the structure of the machine-tool.

By eliminating dependence upon co-ordinate data to specify the profile shape the design cost and probability of error are reduced. Since there is no round-off error a source of profile wavyness is removed. A positive connexion between the cam angle and displacement reduces the danger of random error and the significance of positioning error provided the dis­placement is set correctly for the actual angular position of the cam. Transmission errors due to such causes as gear eccentricity are continuous functions (Si) and so cause negligible secondary acceleration (Chapter 3)-

Compared with plunge-cutting, the proposal to use an independent instrument to determine the radius in preference to the manual feed provides superior accuracy, especially if the operation requires the reading to be restored to a specified graduation for every cut, because the repeatability is approximately 50% of the positioning error (l4). Moreover the operator is not required to read and remember a random sequence of awkward dimensions, so the rate of production should be increased whilst fatigue and the probability of error are reduced. The

-61-

concept of restoring an indicator to a datum position for every cut provides a basis for the application of servo-mechanisms to control the process. Since profile accuracy is now independent of positioning accuracy it becomes possible to use a vertical milling machine instead of a jig-borer provided a tolerance of +0.003 in on the lift is accept­able and that the ends of a rise or return event can be identified with sufficient accuracy to manufacture correct profiles in these critical regions.

Therefore superior accuracy to the plunge-cutting process appears to be possible with reduced manufacturing time on a cheaper machine-tool.

6.7 DERIVATION OF CAM PROFILES FROM THE SHAPE OF A DEFLECTEDLEAF SPRING

A device for this purpose must be sufficiently small and light to be mounted on the table of a conventional milling machine without struc­tural alteration or impeding the feed movements for cam manufacture. The considerations of the previous section in conjunction with the survey in Chapter 5 show that the limitations of copying processes result from reliance upon an existing template. (For this purpose a template is defined as a substantial plate having a specially contoured edge reproduc­ing the follower motion. This edge must be manufactured to precision accuracy since every error will be reproduced in every profile. The cam profile is copied through appropriate velocity ratios to suit the required lift and angle of lift. Therefore the use of a template presupposes accurate profile manufacture). Dependence upon a prepared template is eliminated by deriving the shape of the pitch curve from that of a leaf spring deflected with parallel ends offset by the lift, Fig. 6-2, to form an analogue of the displacement characteristic. The span must be propor­tional to the angle of lift. The means of tracing the shape of the leaf must not impose such force that the deflection is altered significantly, although the maximum radial error can exceed that causing wavyness since it is a continuous function. Because the leaf can be deflected to suit any span and offset within the elastic range of the material it is possible to use coarse increments of velocity ratio in the transmission whilst retaining infinite variation of both lift and angle of lift.

To Face Page 62

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The boundary conditions for a leaf spring deflected into the shape shown in Fig. 6-2 (a) are

X = O ; ^ = 0;If - = o.

X; , - Y ; f t • O.

and are related to the cam dimensions by

)( = ]x = 'k © r (6.2)

Y = 4 Jwhere the angleq © and ^ are measured in degrees and -Y. is the displacement factor, the scale distance on the abscissa corresponding to a cam rotation of 1°. The magnitude of the displacement factor depends upon the velocity ratio of the transmission ( section 6.6 (b) ) It follows that

d & doc (6.3)

and h = 4 H .dk (Ul

(6.4)

provided the assumption that the cam rotates at uniform angular velocity is valid. This motion is symmetrical with zero velocity at both limits of a dwell-rise-dwell event, satisfying the basic requirement for a cam mechanism.

Analysis of the deflected leaf spring using'the approximate theoryof bending shows that the maximum bending moments, and therefore the

2 2maximum values of d y/dx , occur at the ends. Consequently the follower acceleration must be finite and the pulse infinite at these positions. Similarly, the acceleration is zero at mid-span where the slope and velocity attain maxima, so there is no discontinuity of pulse during the motion. The follower motion derived from this profile has similarities to simple harmonic motion which is widely used for cams in "slow" and "moderate" speed applications (33) (85) (97)• This motion is designated

the Simple Derived Motion. An application of this principle to derive the infinite pulse motion was subsequently found to have been published in a brief paper by Borun (12). However he omitted the influence of longitudinal end forces on the curve of deflection considered in Chapter 7 and experimental work reported in Chapter 10 showed it essential to develop a superior means of mounting the leaf. The author's extension of the principle to a finite pulse motion having theoretically superior dynamic characteristics also relies upon his own method of mounting the leaf.

The author's study of the deflected leaf spring showed that a finite-pulse follower motion having superior dynamic performance, similar to that of cycloidal motion, can be derived in the same manner by rotating the leaf through the maximum angle of slope relative to the direction of tracing the radial displacement ( Fig. 6-2 (b) ); The shape of this pro­file is derived from three successive spans. The new origin 0^ is a point of inflexion, the transformed and y ^ axes being tangeptal and normal respectively to the neutral axis through this point. The limit of the working length is the next point of inflexion having the same slope. The angle of maximum slope relative to the original x-y axes is

enabling the span and offset to be found through the relationship between

(6.5)

Appendix 1. The boundary conditionsfor this configuration are

(6.6)

The second derivative must be zero because the curvature is zero at the points of inflexion. The boundary conditions are related to the cam dimensions by

-64-

DC, = -4 0 ,

X, - -0,X - <

Since the mid-position of the lift co-incides with the intermediate point of inflexion the acceleration is zero at the transition point.

Therefore the pulse is finite throughout a dwell-rise-dwell event and this motion resembles the cycloidal and 3-4-5 polynomial motions recommended for high-speed applications (33) (85) (87) (97) •This motion is named the Finite Pulse Derived Motion.

The author has extended the application of this principle beyond the infinite-pulse motion identified by Borun. In contrast to generating processes it is possible to derive the shapes of cam profiles producing two radically different motions using the same attachment.

6.8 THE MANUFACTURING PROCESS FOR DERIVED PROFILES

As in the case of the plunge-cutting process (Chapter 4) the derived profile is cut by replacing the contact surface of the follower with the cutter which must generate the same surface. Since the cutter axis is stationary the mechanism must be inverted to reproduce the same relative motion between cam and follower. The displacement of the cam centre is to be determined by the deflection of the leaf spring at the proportion of span corresponding the ratio of the cam angle to the angle of lift, so a transmission is needed to convert the rotation of the cam into a proportional linear motion. For the Simple Derived Motion,Fig. 6-2 (a), this displacement must be parallel to the x-axis, the deflection being measured in the perpendicular direction. So it is simplest to make the master cam for a radial or offset translating fol­lower, the correction for an oscillating follower being made on the copying machine for production items (106).

The repeatability of an instrument for measuring displacement is superior to the calibration (BS2795;197l)- Therefore the pitch curve radius at every increment position must be determined by restoring the

T o Face Page 65

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-65-

surface of the leaf to a specified datum at the corresponding propor­tion -of the span. This arrangement has the additional advantage that the actuating movement does not require accurate alignment with the feed. The conversion from rotary to linear motion implies a relatively long sliding member to obtain stability and to suit a rotary table of 10 to 12 in diameter. It is logical to mount the leaf spring on this member. The instrument must be stationary relative to the machine-tool structure. This layout has the merit that only the offset of the leaf, not the travel of the probe'tracing the leaf, has to be aligned accurately with the path of the cam centre. It also enables the span of the leaf, a function of Tf ? to be determined by turning the rotary iable through the angle of lift. It would be difficult to set the span by this means in the converse arrangement since the connexion is lost.

The cam profile is a smooth curve continuously touching the contact surface of the follower. In the case of a roller follower, the shape of the pitch curve is derived from that of the leaf spring, so the replacement of the roller by a cutter of the same diameter automatically compensates for the pressure angle.

The outline scheme for a device to manufacture the Simple Derived Motion profile using this principle is shown diagrammatically in Fig. 6-3- The simple version cuts the profile by plunge-cutting, it must then be smoothed and honed.

6.9 SUMMARY

Estimates for specifying and cutting cam profiles using different processes (Tables 6-3 and 6-4) show that numerical control with linear interpolation and generating devices cost approximately the same amount, but manual plunge-cutting is five times more expensive.

The wavyness of numerically-controlled linear interpolation deteriorates as the radius increases, requiring the smallest possible tolerance (Table 6-1). The author recommends honing these profiles which should then be superior to profiles cut with n-c circular or parabolic arc interpolation because the discontinuities in the second derivative at -the blending points cannot be eliminated in this way. Comparable accuracies are attainable with the manual plunge-cutting process.

-66-

The design objectives for profile cutting are met by a copying process in^which a deflected leaf spring replaces the conventional template. It is shown that this principle can be applied to derive profiles for two follower motions designated Simple Derived Motion and Finite Pulse Derived Motion similar to Simple Harmonic and cycloidal motions respectively.

Compared with the plunge-cutting process this method of profile derivation is expected to improve the profile accuracy because round-off error is eliminated and the transmission synchronises the cam angle and follower displacements. Then positioning error becomes irrelevant as the pitch curve radius is automatically set for the actual angular position of the1 cam, but the limits of the angle of lift must be identi­fied accurately. The accuracy of radii depends upon the repeatability of the linear measuring instrument which is 50$ of the positioning error. Since no reliance is placed upon the accuracy of lead screws a cheaper machine-tool can be used.

Savings result from the elimination of co-ordinate data, reduced machining time because the simpler setting procedure is simpler and the use of a vertical milling machine instead of a jig-borer. The estimated cost (Table 6-3) is at the low end of the index, comparable with n-c machining plus honing.

It was found subsequently that an application of this principle to derive cam profiles for an infinite pulse motion had been published in a brief paper by Borun (12). His analysis omitted the longitudinal and clamping forces acting upon the leaf.' The experimental work reported in Chapter 10 showed that an improved means of mounting the leaft was essential. Borun made no reference to the derivation of a finite-pulse motion, such as that identified by the author.

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qjLX

-67-

Chapter 7

ANALYSIS OF THE DEFLECTED LEAF SPRING

7.1 APPLICATION OF THE ANALYSIS

The theoretical performance of cam mechanisms using the derived profiles will be assessed by comparing the characteristics of the

manufacturing process eliminates the need to specify any intermediate co-ordinates the equations for the follower velocity and acceleration contain derivatives of the deflection equation which is itself needed for comparing the theoretical and measured deflections and determining design data. For this application the leaf must be deflected to form one of the curves illustrated in Fig. 6-2, the deflection-span ratio being significantly greater than that encountered in general engineering practice. Consequently a solution based upon the approximate theory of bending is liable to error and previous solutions for the deflection of slender leaves in the elastica range were investigated. To assess the significance of longitudinal end forces the author derived further solutions using the method of perturbations. The derivations of these solutions are traced in Fig. 7-1? and the factors influencing the selection of the preferred solution are summarized in Table 7-1 •

7.2 REQUIREMENTS AND ASSUMPTIONS

The shape of the deflected leaf is determined by the motion to be used, the lift, angle of lift and machine ratio, equation (A2.1). The boundary conditions defining the deflection of the leaf into the required configuration, Fig. 6-2, were established in the previous chapter. From equations (6.1) and (6.2)

follower motion with those of cam laws in common use. Although the

0 .

(7.1)

-68-

2where h and aj (radians) are the lift and angle of lift, respectively for Simple Derived Motion and are given by equations(6.5) and (6.7 ) for Finite Pulse Derived Motion. For the analysis it was found convenient to express the relative proportions of the span X and offset Y by the associated parameter

where Yo anSFe maximum slope.

The equations for the cam characteristics require the curve of deflection to be expressed in terms of rectangular co-ordinates. Neither the deflecting forces nor the free length of a span are known, therefore the solution must be independent of these parameters. A general solution, suitable for preparing design data, must also he independent of parameters defining the elastic properties and cross- section of the leaf, but provision must be made to ensure that the maximum stress is confined to the elastic range in every application.

For the analysis it was assumed that:-

(a) the material obeys Hooke’s Law, is isotropic andhas the same value of Young's Modulus in tensionand compression.

(b) the stresses are confined to the elastic range.

(c) the applied loads act along a principal plane ofthe cross-section.

(d) plane cross-sections perpendicular to the neutral axis of the unloaded beam remain plane and perpen­dicular to the neutral axis after loading.

(e) the leaf was initially straight.

(f) the cross-section is uniform.

(7.2)

Then the Euler-Bernoulli LawT i

1 _ dxl Me d s E I (7.3)

derived from the pure bending of a beam may be used to solve a problem in which the bending moment varies along the span. This approximation implies that the neutral axis intersects the centroid of the cross- section and (d) that the deformation due to shear is negligible.

7.3 THE APPROXIMATE THEORY OF BENDING

In general engineering components are designed for strength and stiffness. Thbn the maximum slope of a loaded beam is small and the free length can be equated to the span. Consequently the approximation

can be substituted into the bending moment equation (7*3) to obtain

which can be solved readily with sufficient accuracy for most purposes (19) (107). For small curvatures it follows that the bending moment due to longitudinal forces acting on the ends of a beam is negligible, but this argument is invalid for the large deflections involved in

bending. Since the loading conditions are indeterminate it is necessary to investigate the significance of longitudinal end forces upon the curve of deflection. Further considerations involve the effects of anticlastic curvature and the practical requirement that the profile shape has to be derived from an external surface of the leaf instead of the neutral axis.

1 + /

2(7.5)

this application, nor is it compatible with the assumption of pure

T o Face Page 7 0

Fo r c e s & m o m e n t s A c t i n g u p o n

ONE SPNN OF kKEFLECTEL LEKF SPRING,

F i g , 7 - 2

Cantilever. Le flecte l in the

E l a s t i c a R a n g e .

f- 1 <3. "7 -3

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Euler (37) used the exact form of the bending moment equation

d (p

d s

N\El

to analyse the deflection of leaf springs under different conditions of loading and constraint, so these solutions involve the length of the neutral axis and the slope. For general engineering purposes it is more convenient to use rectangular co-ordinates in conjunction with the approximate theory of bending. No allowance is made for the difference between the span and the free length whilst the variation in shear force due to the changing slope is negligible. Thai equation(7 .5 ) is integrated successively to obtain equations for the slope and deflection at any position along the span, the constants of integration being determined by the boundary conditions, equations (7-1)» to obtain the deflection equation for the cubic spline

which satisfies the requirements for a general solution. The cubic is anticipated by the assumption of constant shear force.

7.4 SIGNIFICANCE OF LONGITUDINAL FORCES

The effect of longitudinal forces upon the curve of deflection was investigated using the approximate theory of bending for the con­figuration illustrated in Fig. 7-2. The bending moment equation

(7-6)

(7.7)

was solved by putting

xj = < r4 k 2 + i q * . + 1 >

-71-

The solution is found in Appendix Al.l(b) by equating coefficients and substituting the boundary conditions to obtain the equations for the slope and span of the leaf,

OCX.4a. fz)(M- WV-e TV/dye z w CL (7.9)

*'V

CL« w - (7.10)

where CL = fj-j j £ i j 2 (7.11)

and N\ •= W (■£■ ~ -C -2_) (7.12)

«■ _ * “* )

This solution is extended in Section 7.7 below using the method of perturbations to improve the accuracy by allowing for the large curva­ture.

Since the forces W and H are independent it is impossible to solve equations (7*9) and (7.10) for a given span and offset. Frisch- Fay (39) overcame this problem in a similar analysis by making the usual assumption for beams subject to small curvature that the free length equals the span, so the difference between the lengths of the neutral axis in the deflected condition and the span equals the elastic deflection due to the longitudinal force. The restriction to small curvature permits another assumption that the longitudinal force is constant, enabling a second relationship to be established from the Young’s Modulus equation.

"DIFF

EREN

CE

Ox

IOS)

To Face Page 72

0

-3

-4

- 5 -

0-1

DEFLECTION! DY APPRO/, THEORC / 'M.IDWWG FOR LONGITUDINAL FORCE Q.ocr Due TO FRICTION AT SUPPORTS EQUATION (7.10),

0-2 0'5_CT

deflection dy methodNX - ''F PERTURBATIONS X ALLOWING FOR

' \FRICTION AT v\ SUPPORTS

EQUATION (7.-30).

EQUIV. COEFFT. OF FRICTION,

5 1 = SLA (l(6) '4o = wax. OMa.de.

-®f sTope.

deflection dy -numericalINTEGRATION OF SLOPE FROM ELLIPTIC INTEGRALS SOLUTION EQUATION (7.19), , = 0,SPAN= T O Varying o ffset ,EL= 2-16

->s AT 4-SPAN. 'V DEFL'N by approx, theory

^s-DEFL'kI by specified solution '\

- 6

FlJ. 7-5. Xiffekence Between Reflections at / span sue to Longitudinal Loads usia|£ mternative solutions.

-72-

The error becomes significant for offset-span ratios exceeding 0.05, far smaller than the proportions required for deriving cam profiles.This limitation was overcome by expressing the longitudinal force H as the product of the equivalent "coefficient of friction" andthe deflecting force W. Then the effect of varying longitudinal forces upon the curve of deflection and the deflecting force was investigated for a leaf of unit span having a modulus of rigidity

E l - Z A b M j - L 2

The results are plotted in Fig. 7-5 and summarized in Table 7-2(a). Comparing the extreme practical case in which the longitudinal force results from friction at a support

yfa/'X ~ 0.25 and (T = 0 . 6 (nominal)

with the equivalent leaf subject to the deflecting force W only'.thelongitudinal force causes the angle of maximum slope to diminish by0.5$ whilst the deflection at -zjr-span increases by 1%. The tensilestress due to the longitudinal force at each limit of span is lessthan 0.4$ of the maximum bending stress, so the neutral axis effectivelyintersects the centroid of the cross-section. The corresponding strain

-5is 2 x 10

Therefore the assumption of pure bending is valid and the longitudinal strain due to the frictional forces H at the end supports (Fig. 7-2) is negligible.

According to the approximate theory of bending the deflecting force varies inversely as the square of the span and the moment due to . the longitudinal force is inversely proportional to the span for a given offset-span ratio. Therefore the error in the deflection at T^-span varies inversely with the span and the results quoted are representative. The magnitude of the longitudinal force is unpredictable, but Fig. 7-5 shows that it can affect the curve of deflection significantly.

-4 .A maximum change of deflection set empirically at 2 x 10 m can be ensured by designing the mount for the leaf to prevent the maximum longitudinal force exceeding 10$ of the deflecting force.

-73-

The limiting proportions of a deflected leaf are determined by the elastic limit of the material. Some margin is essential, but a design factor is considered unnecessary. Since the tensile stress due to the longitudinal force H is negligible in comparison with the maximum bending stress the limiting proportions may be estimated by substituting the maximum permitted bending stress O3 in equation (Al.85), Appendix A1.2, page A20, to obtain

7.5 LIMITING PROPORTIONS

Design data calculated from equation (7*13) fo*1 hardened and tempered steel leaves of representative thicknesses are presented for a maximum

quoting the corresponding angles of lift for varying machine ratios.

These results show that leaves made of high-tensile material can be subjected to the large offset-span ratios required for this appli­cation without exceeding the yield stress.

7.6 ANALYSIS OF THE ELASTICA

Reference to Table 7-3 shows that the leaf can be deflected until the parameter CT exceeds 0.6 without the maximum bending stress exceeding the yield point. This corresponds to an offset-span ratio of 0.47, the maximum angle of slope being 37°. Then the denominator of the curvature equation becomes

(7!3)

obending stress of 80 tonf/in in Fig. 7-4 and Table 7-3? the former

instead of the approximation to 1 used to derive equation (7.5)* Con­sequently the bending moment calculated by the approximate theory is too large, introducing errors in the deflecting force, bending stress

and the integrals of equation (7.5) defining the slope and deflection of the leaf. Further error results if the span is equated to the free length of the leaf.

Therefore the deflection of the leaf must be analysed by more rigorous methods.

The derivation of more accurate solutions for the deflection of a leaf spring in the elastica range using, or approximating more closely, the correct equation for curvature and in some cases including longitu­dinal end forces are traced in Fig. 7-1- With the exception of the finite element analysis by Seames and Conway (103) all the previous, solutions are derived from the Euler-Bernoulli Law and all are based on assumptions (a) to (e) in Section 7-2 above. Consequently the axial component of the deflecting force resulting from large curvature is also neglected.

\Provided the longitudinal forces are negligible the shape of the

deflected leaf, Fig. 6-2, corresponds to a series of identical cantilevers having the fixed end at a maximum or minimum and deflected by equal and opposite concentrated loads at the free ends, co-incident with the.points of inflexion, Fig. 7-3 • None of the previous solutions for the deflection of a cantilever in the elastica range satisfies the requirements for a general solution of the cam characteristics completely. The approach used by Euler (37) of expressing the curvature of the neutral axis in terms of j was followed by Barten (6), Bisshopp and Drucker (ll) and Hummel and Morton (56). All these solutions involve elliptic inte­grals and are confined to a definition of the co-ordinates of the free end in terms of the free length, flexural rigidity and deflecting force. Mitchell (84) derived similar expressions-suitable for wider application in terms of rectangular co-ordinates, including an equation for the deflection at any intermediate span. In his introduction Mitchell commented:-

-75-

"The main problem encountered in the theory of non-linear bending (i.e. not using the approximate equation for the curvature of the deflected member) is that in which, the free shape of the rod and load distribution being given, the loaded, or deflected, shape, is sought. In its most general form this problem is one of considerable mathematical difficulty and it is only in special cases that a solution can be found by other than numerical methods."

Bisshopp and Drucker (11) allowed for the foreshortening- of the projected length of the deflected leaf on the x-axis but assumed pure bending (Fig. 7-1). Then the equation for the Euler-Bernoulli Law became

These authors derived separate expressions relating the x- and y- co-ordinates of the free end of the cantilever to the angle of maximum slope in terms of complete and incomplete elliptic integrals of Legendre's first and second kinds. The parameters which are unknown in this appli­cation can be eliminated by re-arranging the solution to obtain the equation for the offset-span ratio. Using the notation of Jahnke and Emde (59)

Y = F(-$ - F d ^ - g f E « ) - E <©,)]

e i (7 .1 5 )

(7.16)

\ - (I +

6, = Sth.1 ( 1 / ( 1 J ? ) )

Equation (7.15) also formed the starting point of an analysis by Professor Arscott (3) included in Appendix A1.2 with extensions by the author. This solution relates the offset-span ratio to the angle of maximum slope by the series expansion

Y 2 - 6X

v. — 0 / —— t + — q r T3 3 5 V ' (7'17)

compared with the expression

— = - / t a n ((&)y 3 ■ (7 .18)

given by the approximate theory of bending. The deflection at inter­mediate positions along the x-axis can be determined by numerical inte­gration of the slope, equation (Al.89) page A20.

• X t « ( 1 - f ) ____________

derived from the correct equation for curvature.

A finite elements technique for analysing the curve of deflection published by.Seames and Conway (103) is significant because the solution corrects for the variation in bending moment along the length of the cantilever. The solution is derived by approximating the curve of the neutral axis to a series of blending circular arcs. These arcs can be made sufficiently small that the approximate theory of bending may be applied without significant error to determine the radii of the arcs. Starting from the (known) slope at the free end the span and deflection are found from the sums of the projected lengths of the arcs on the x- and y- axes, continuing until the magnitude of the slope equals that specified for the free end, intermediate values being obtained in the process. The accuracy of the slope determined by this method depends upon the size of the arcs.' The second derivative can only be found by numerical methods.

-77-

Because it requires known values of the slope at the free end and the parameter W/EI whilst involving iteration to match a speci­fied offset-span ratio the finite differences method is unsuitable for determining the cam characteristics. However it enables the error resulting from the assumption of pure bending in alternative solutions to be assessed. Since the difference would he greatest for large curva­tures the deflections were calculated at increments of span for identical conditions using the finite elements technique and numerical integrationof equation (7*19) for CT = 0.5 and (T =0.6. It was found that the maxi-

-5 ■mum discrepancy was smaller than 2 x 10 m .

Therefore the assumption of pure bending does not introduce signi­ficant errors In the solution for the deflection.

7.7 A NEW SOLUTION OF THE BENDING MOMENT EQUATION FOR LARGE CURVATURE USING THE METHOD OF PERTURBATIONS

The author used the method of perturbations to derive a new solu­tion (Appendix l.l) for the large deflection of a leaf subject to both deflecting and longitudinal forces, Fig. 7-2, which has significantly greater accuracy than that given by the approximate theory of bending.It has the advantage over previous solutions of the elastica that the deflection can be calculated immediately for any intermediate position.An iterative procedure is necessary if the offset and span are specified in preference to the deflecting forces and the computer program 'LEAFPERTS1 was written to solve this problem. The series solution is expressed directly in terms of rectangular co-ordinates and is based upon the same assumptions as the previous analyses of the elastica surveyed in Section 7-2 above.

The method of perturbations was used to obtain a series solution of the non-linear differential equation given by the Euler-Bernoulli Law

2El(7-20)

by expressing the boundary conditions, equations (7-1) in dimension­less form.

02,Putting =: (7.21)

and AT = 4 = (7-22)Y

the derivatives become

= X = Y iX )< Ar (7.23)

z X2 • { 7m2h )

Substituting, in the bending moment equation (7-20)

E l/ I )I k = N\ ~ V/x. -f

n + « z l 7z = K " w > u 7 (7-25)

y \where the perturbation parameter £ - (7 .26)

The solution is obtained by expanding the perturbed term as a series and putting

/Or e? a -+ *+■ ^ A^z "T" (7.27)

where v^; v^ — etc., can all be differentiated twice.

Substituting (7-27) in the expanded form of (7»25) gives

-79-iia T g y i N\ - W X u , r H L 1 g , + <£+■1 + £ w / ^

j 1 + 1 £ ( / ) / 3g2 + ( 0 +

7 . 2 2

which is solved by equating the coefficients of powers of the pertur­bation parameter. So the accuracy depends upon the number of terms in the series.

Neglecting the longitudinal forces H ( Appendix 1.1(a) ) the deflection is

tf x.' (o')

+ _ U _ — /s~ id/'t A-xi + j_£_2 4 o //2 n<&

~ J £ . Y { z4 Q - t- ?5~ [ S n ! eP!^ IT&o /2go

H*where £0/ = FA/S’!

(2)

(7.29)

and 0 = W / E l

Part (0), identical to the solution given by the approximate theory of bending, is the solution for the unperturbed condition obtained by equating the coefficients of C . Parts '(l) and (2) are the corrections obtained by equating the coefficients of £ and £ X respectively.

The general solution including the longitudinal forces is derived in Appendix 1.1(b) for two terms of the series expansion.

n

2H cN\ + V/ \

CL I2 v / x - 'ZfAi

+ '3a-1-Z.BH3 V V •2 ^

f m 1 v,

clx _a.x JL - 2 j- jl

•~A-XL 0-X•+*+ Zol (M + w Y /!

I7.5K1 I *-/ V

« y l *

-'ZroO

OlX | -

J [WVM+y\4 H 3 W v

12 ri a / v

+ E e-

BH3 (

3/:/w yf>«3 M - l

iyjzJh. „ i / k - v/j^ M + y

i («••*• - £) + 2 -e

tA + ) - 4 ( K -^ ) ( K+ £ )*

j iI 3.TC- — Q.OC / I

*£ A A JL I ^ 7

t - (.7 J o j

Unlike the solution derived by Frisch-Fay (39) this result applies to large deflections. .

percentage

DIFFERENCE

DETWEEN

000

obtained

PROWL

PERTURBATIONS

& ELLI

PTIC

INTEGRAL

SOLU

TION

ST o Face Page 81

m e t h o l o f p e r t u r b a t i o n sCoi'AP^lSON OF SLOPE AT M l k - S r A N 2>t EKPM'VfclMG 1, ?, & S TERMS OF SERIES

PERT

URBA

TION

S T

o

-81-

Equation (7.29) can be compared directly with the previous solutions for the deflection of the elastica, equations (7 -1 6) and (7.17) since all apply to the same conditions. The values of the offset-span ratio ( Y/X ) calculated from these results for increments of the maximum angle of slope expressed as the parameter cjr were used to determine the percentage difference.

where suffix S identifies the solution, equation (7-17) or (7-29)

and E the value given by equation (7-16).

The corresponding curves drawn in Fig.’7-6 show that the corrections introduced by expanding the series to contain all products of the per- turbation parameter up to £ and £ respectively maintain the percentage difference, equation (7 -3 1)? within 0.1$ for values of <r smaller than 0.27 and 0.43 respectively. These solutions should not be used for larger slopes because the percentage difference then in­creases rapidly, although the curves for CT superimposed upon Fig.7-4 show that this restriction is over-ridden by practical considerations for spans shorter than 2in since the material would yield.

Equation (7-30) was solved for a span of l.Oin, varying both the offset and the ratio of longitudinal to deflecting force, using the pro­gram TjEAFPERTS1. The results given in Table 7-2(b) confirm the conclusions deduced from the approximate solution but showed that for large slopes the maximum bending moment calculated by the approximate theory is excessive. Therefore the bending stress is up to 10$ lower than the quoted value for the curves drawn in Fig. 7-4.

Neglecting the longitudinal forces, the series expansion of the elliptic integrals solution derived by Professor Arscott (3) has similar accuracy to the result given by the method of perturbations from the

1 0 0

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powers of £° and £ . The curves relating the offset-span ratios for these solutions to are drawn in Fig. 7-7 •

Within these limits of accuracy the method of perturbations provides an analytical solution to the problem posed by Mitchell (page 75)-

Only the elliptic integrals solution, equation (7*16), and the finite elements method are valid for any offset-span ratio. The former has continuous functions for the slope and the second derivative of the deflection.

7.8 DEFLECTION DUE TO SHEAR FORCE

Every solution considered in this chapter .neglects the deflection due to shear force. Timoshenko (107) determined the displacement of the free end of a cantilever, Fig. 7-3? from the strain energy due to the bending moment and shear force. Provided there is no distortion of the cross-section at the fixed end (implied by the symmetry of this application, Fig. 6-2(a) ) the deflection of the free end is

Where X is the length of one span and Y the offset

The first term within the brackets is identical to that determined by the approximate theory of bending, the additional deflection due to shear is given by the second term. Substitution of the values from Table 7-3 shows the deflection due to shear stress alone is 0.02$ of the offset.

Therefore the error due to the neglect of the deflection due to shear is insignificant. Differences in scale are immaterial since the

-83-

shear term is constant for a given offset-span ratio, material and bending stress.

7.9 ANTICLASTIC CURVATURE

The longitudinal strain due to pure bending on an element distant y from the neutral axis (Fig. 7-8) is

Case (19) analysed the anticlastic curvature of an initially straight beam of uniform cross-section, showing that straight lines which are parallel to the principal axis of bending on a section of the unloaded

the application of a bending moment. The radius of anticlastic curvature through the centroid of the section

predicts significant distortions of slender leaves subject to large deflections. However experiments have shown that this does not occur,

not exhibit anticlastic curvature. The work by Case shows the apparent contradiction results from the neglect of the radial force F^ acting upon the element, Fig. 7-8. Assuming pure bending, the tensile force acting upon each end of the element is

R (7 .33)

so a complementary strain - V must exist in the transverse section.

member become (to a close approximation) arcs of concentric circles upon

(7.3*0 (19)

according to Ashwell (2) it is generally assumed that slender members do

Ft - Ey FA / K

Fr = 2E* j Fa J-fe / K

(7.35)

(7-36)

-84-

the resultant radial forces on the tensile and compressive sides of the neutral axis being opposed. Consequently the effect of these forces is negligible in the case of a stiff beam subject to small curvature and the radius of anticlastic curvature is given accurately by equation (7*34).

However it is shown in Fig. 7-9 that the corresponding anti­clastic curvature of a slender leaf subject to large deflection would destroy the equilibrium of the radial forces. According to Case (19)the moment about the centroid 0 of the cross-section of arc ACBcis sufficient to suppress the anticlastic curvature if

4// < < > 6 ' (7.37)

A more rigorous analysis by Ashwell (2) supported by experiment confirmed this result, but showed that close agreement with equation (7*34) is con­fined to the range

To j ( R-bij Z 1 (7 .38)

The calculations summarized in Table 7-6 show that large deflections of the slender leaf needed for this application meet the requirement for the suppression of anticlastic curvature. A consequence is a slight reduction in the stiffness of the leaf (l9)» emphasizing the importance of deriving equations for the cam characteristics which are independent of the elastic properties of the leaf material.

7.10 CONCLUSIONS

The anticipated range of the offset-span ratio corresponding to 0.25 < O' < 0.50 and the need to calculate the intermediate deflections as accurately as possible determine the choice of the method used to analyse the deflection of the leaf. It has been shown that the assumptions involved in the application of the Euler-Bernoulli Law are valid provided the solution is derived from the correct expression for curvature and precautions taken to prevent the longitudinal force exceeding 10$ of the deflecting force. The significant features of the various solutions

-85-

summarized in Table 7-1 show that the offset-span ratio may exceed the accuracy range of the solutions derived by the series expansion of elliptic integrals and by the method of perturbations unless the expansions of the latter are extended to higher powers of the pertur­bation parameter. The finite elements method depends upon the parameter

W/EI and the slope at mid-span, both of which are unknown in this application.

The elliptic integrals solution equation (7.16), derived by Bisshopp and Druclcer (11) was considered most suitable for this appli­cation since it is valid for any offset-span ratio. ■ '

Since equation (7-16) is unsuitable for calculating the maximum angle of slope corresponding to a given offset-span ratio the inverse operation was performed for increments of the parameter CT . The accuracy of published tables was found inadequate for this purpose so the computer program 'LEAFSLOPE' (Appendix 8) incorporating procedures for calculating elliptic integrals by Hoffsommer and Van der Reit (53) was written and run. The program was extended to compare the results given by the approximate theory of bending, equation (7-6), and the series expansion of elliptic integrals, equation (7 .1 7 ). Tbe results included in Table 7-4 were used to plot Figs. 7-6 and 7-7•

Then the intermediate deflections of a particular setting of the leaf determined by G2 and the span X can only be found by numerical integration of the slope, equation (7.19)? using a Simpson's Rule pro­cedure (71) incorporated in the computer program 'DEFLECTION' (Appendix 8). Comparison between the deflections calculated with this program and the approximate theory of bending for the same offset-span ratio shows the maximum difference occurs in the regions of 0.3 and 0.7 of the span.(This difference can exceed 6$ for = 0.5*) The second derivative needed to calculate the follower acceleration and curvature of the pitch curve is readily obtained from equation (7-19)-

For a given offset and span the deflecting force increases with the longitudinal force (Table 7-2), so the restriction

-86-

has the additional advantage of minimizing the deflection of the supports. However it imposes a problem in the design of the mounting since the coefficient of friction between two dry steel surfaces is 0.3. The principles of alternative methods of deflecting the leaf for this purpose are illustrated in Fig. 10-1.

7.11 SUMMARY

The analysis of the curve of the deflected leaf spring is required:-

(a) to derive the equations for the cam characteristics.

(b) to determine the intermediate deflections.

(c) to obtain design data for checking limiting conditions.

The cam characteristics must be independent of the dimensions and elastic properties of the leaf spring and of the deflecting force.

The method of deriving the profile shape from that of the deflected leaf spring has the advantage of eliminating the requirement to specify the co-ordinates of the pitch curve at small increments of cam angle which is essential for any direct machining process, For the purposes of this investigation the theoretical co-ordinates are needed for comparison with measured values.

the material to yield. Then the denominator of the curvature equation

Slender leaves of high-tensile steel can be subjected to large deflections, the maximum angle of slope exceeding 30°» without causing

differs significantly from the assumption of unity used to derive the bending moment equation for the approximate theory of bending. Therefore a more rigorous solution is needed for this application.

Previous solutions for the deflection of a cantilever in the elastica range were derived from the Euler-Bernoulli Law, equation (7.3)» neglecting deflections due to shear and longitudinal forces.The solution obtained from the correct expression for curvature relates the offset-span ratio to the angle of maximum slope as a function of elliptic integrals, equation (7 .1 6 ).

The author used the method of perturbations (Appendix l.l) to solve the bending moment equation for the large deflection of a leaf additionally subject to a longitudinal force, Fig. 7-3* The accuracy of this solution depends upon the highest power of the perturbation parameter included in the series expansion. With zero longitudinal force the offset-span ratio is within 0.1$ of that calculated from the elliptic integrals solution, equation (7-l6), for values of the parameter CT smaller than 0.09 (approximate theory of bending), 0.27 ( 6 ) and 0.43 ( £2 ).

Evaluations of the solutions for varying magnitudes of longtu- dinal force expressed as a ratio of the deflecting force showed that longitudinal forces larger than 10$ of the deflecting force can have considerable effect upon the shape of a leaf deflected with the range of offset-span ratio envisaged for this application. Then the maximum

-4change of deflection due to the longitudinal force is under 3 x 10 in. (0.4$). The tensile stress due to the longitudinal force is less than 0.2$ of the maximum bending stress, justifying the assumption of pure bending.

Previous work showed the deflection due to shear to be about 0.02$ of the offset and that slender leaves subject to large curvature do not exhibit anticlastic curvature.

Therefore the Euler-Bernoulli Law provides a valid basis for analysing the large deflection of a leaf spring, Table 7-1* It is essential to derive the solution from the correct expression for curva­ture, but the deflections due to shear force and longitudinal forces smaller than 10$ of the deflecting force can be neglected. Subject to these conditions the relationship between the offset-span ratio and the maximum angle of slope expressed in terms of elliptic integrals, equation

-88-

(7-16), is valid for the complete range of Cf possible in this application. It provides the datum for determining the accuracy of alternative solutions, but the deflection at intermediate positions can only be found by numerical integration of the slope, equation(7!9).

-89-

Chapter 8

DERIVATION OF THE CAM CHARACTERISTICS AND ASSOCIATED PARAMETERS

8.1 USE OF THE CAM CHARACTERISTICS

For the method to be a practical proposition it is necessary to show that the dynamic performance of cam mechanisms using either the Simple Derived Motion or the Finite Pulse Derived Motion is acceptable. This can be done by showing that the equations of motion of the follower satisfy equations (6.1) and, in the case of a finite pulse motion, the additional requirements specified in equations (2.1) must also be met. The curves of the characteristics and the magnitudes of the dimensionless forms of the velocity and acceleration of the follower must be similar to those of proven cam laws. These values are needed for design purposes to determine the forces acting upon the mechanism and the operation of associated parts. The established pro­cedure used to derive the equations for the velocity, acceleration and pulse of the follower is based upon the assumption that the fluctua­tions of angular velocity of the cam disc are negligible. Then the relationship

is used to derive the equations of the cam characteristics, (3«2),(3-3) and (3.4).

8.2 CHARACTERISTICS FOR SIMPLE DERIVED MOTION

The shape of the working span of the deflected leaf spring used to manufacture the shape of the cam profile for Simple Derived Motion must satisfy the boundary conditions specified by equations (6.1) and has the form illustrated in Fig. 6-2 (a). The span of the leaf is measured along the x-axis which is tangental to the neutral axis at one limit of the working length. The length of the span is directly pro­portional to the angle of lift and is determined by the displacementfactor

The x-axis is parallel to the transverse feed of the mill table (Fig. 6-5). It is most convenient to determine the magnitudes of the cam characteristics for integer increments of cam angle. Hence it was decided to relate the offset-span ratio of the deflected leaf to the angle of maximum slope by equation (7 .1 6) and to calculate co-ordinate positions on the curve of the neutral axis by numerical integration of the slope equation (7 .1 9 ) at equal intervals of span using the program ’DEFLECTION' (Appendix 8). Since other un­certainties are introduced by associated parameters it is unnecessary to determine the magnitudes of the velocity and acceleration to the same degree of precision needed for the deflection.

for the Simple Derived Motion and the Finite Pulse Derived Motion.This results from the introduction of an additional factor, the machine ratio defined by equation (6.2 (a)) which relates the displacement of the slide to the angle of rotation of the cam blank. Therefore it is possible to derive different shapes of profile for the same lift and angle of lift by changing the machine ratio and altering the span to correspond. For this reason the curves of the characteristics are drawn for a range of CT between 0,1 and 0.6- (where CT is the sine of the maximum angle of slope of the deflected leaf). The relationship be­tween CT and the offset-span ratio calculated from equation (7 .1 6 ) is shown by the graph (Fig. 8-1) and the displacement characteristic for the Simple Derived Motion is drawn in Fig. 8-2. The velocity of the follower is obtained from the slope equation (7*19) through the relationship

There is one major difference between the familiar cam charac­teristics based upon trigonometric or polynomial functions and those

whereCJL&

from equation (6.2 (b)) and is given by equation (8.1) to obtainthe equation for the dimensionless group (Appendix 2).

T o Face Page 31

Velocity Characteristic

Acceleration CharacteristicSitApLE liERWEh KOTION.

Angle of Cam Rotation 1 / F*/£ 8 ~A

•91-

V B 4<r © ©

(8,4)

The velocity characteristics for the same range of CT are drawn in Fig. 8-3* ^he corresponding equations for the acceleration and pulse of the follower are

t f ( f e T f e r 7d a d \cLSJ vrfxCL l%,5

and

a x s U 9/ 1 d-hj

respectively. The. derivation of the corresponding dimensionless groups is given in Appendix 2 to obtain the equations

J

and1- w - i r r a ? )

P= 4>J3 ( l - I X ' - z S I ' & f O -

4 G o r M 4 % ! f ) r k (si)The curves of the acceleration characteristic are drawn in Fig. 8-4 and those for the pulse in Fig. 8-5. The significant values of the dimensionless groups are summarized in Table 8.1 and the computer pro­gram ’SDM-CHAR’ is included in Appendix 8.

8.3 PERFORMANCE OF SIMPLE DERIVED MOTION

Examination of the cam characteristics for the Simple Derived Motion, Figs. 8-2, 8-3, 8-4 and 8-5, shows that the basic boundary conditions for a dwell-rise-dwell movement of the follower defined by equations (6.1) are satisfied, but there is finite acceleration, and therefore shock loading of the mechanism, at both the start and finish of the motion. However the transition from acceleration to retardation

at the point of maximum velocity of the follower is smooth. The magnitude of the maximum velocity increases with (T , but the curva­ture of the acceleration characteristic increases with 0" with the result that the peak acceleration is smaller for large values of (T . This is another reason for using the largest possible offset-span ratio- .compatible with the limiting bending stress of the material of the leaf. The maximum acceleration is 10-15$ greater than that ob­tained with Simple Harmonic Motion for the same lift and timing. As may be expected from a study of equation (7.6) the acceleration characteristic for small angles of slope approaches the straight line of the Cubic No. 2 cam law (97) which has the equation

A - f o f l - 2( | ) J (8.3)

to give peak values of 6 and -6 for the dimensionless group at the positions = O and - ( respectively. Therefore the advantage of the ease of manufacture must be balanced against the shock loading inherent in this motion and the application of the Simple Derived Motion profile must be confined to "moderate speed" applications such as those for which the Simple Harmonic Motion profile would be used. The pres­sure angle, curvature of the pitch curve and the torque needed to drive the mechanism are considered below. It' is necessary to know the off­set and span accurately for the precision setting of the leaf, but for design and manufacturing purposes it is not essential to know the shape of the curve. The setting can be checked by measuring the deflection at = k and ~ \ - The sum of these measurements should equalthe lift and the magnitude of each can be checked against the values read from Fig. 8-11. The accuracy of tabulated or graphical data for the velocity and acceleration of the follower should be adequate for design purposes. Therefore the effort involved in the design of the mechanism is reduced considerably.

8.4 LEAF SETTING FOR FINITE PULSE DERIVED MOTION

The theoretical superiority of finite-pulse cam laws is demon­strated in Chapter 5 and the means of obtaining the shape of the cam profile for the Finite Pulse Derived Motion from that of the deflected leaf spring is identified in Chapter 6 and illustrated by Fig. 6-2 (b).

-92- . ,

-93-

The procedure for setting up the leaf for this purpose and the method of manufacturing the cam profile is described in Appendix 6. To set the leaf and analyse the follower motion it is necessary to find thespan and offset relative to the original x~ and y-axes. Since thelift and the angle of lift are known the angle of maximum slope of theleaf relative to the original x-axis can be calculated from equation (6.4)for a given machine ratio . This enables <T to be found from equation (6.5) and the offset-span ratio to be calculated from equation (7*16) . The span is found independently from

X - { fa see « (8. 10)so the offset can be calculated from the known offset-span ratio. The equivalent angle of lift (the angle of lift for Simple Derived Motion with the same span and machine ratio) is given by

/ x - X . / M (8.(1)

The optimum proportions for the deflected leaf are obtained indepen­dently of the cam dimensions by varying the machine ratio. The computer program ’FPDM-PROP' (Appendix 8) was written to calculate the values of

the angle of maximum slope of the leaf ( $£>).CT = sin (%).the offset-span ratio for this value of <T from

equation (7.16) using Table 7-4 (Y/X).the span of the deflected leaf (X). the offset of the deflected leaf (Y). the equivalent angle of lift (I).

over ranges of the lift (h- ), angle of lift (>) and the machine ratio. Sample data calculated by means of the program 'FPDM-PROP1 (Appendix 8) is given in Table 8-10 and was used to construct the graphs, Fig. 8-6. Reference to the graphs enables the designer to select the most suit­able machine ratio for the profile to be cut.

T o Face Page 34-

T a b l e 8 % . F inite P u l s e t e i v E B M o t i o n

Equations of the L isplkcement Characteristic .for the Configuration shown ih Fig. 8-f.THE COMPLETE ANALYSIS IS GIVEN IN APPENDIX 3>HALF -SPAN Oa-P,

^ =" (I*' y ~ ^ ^ i os ^ ^

e’ " ( ( | ' x - ( r Y ^ %}) x (2T )

J.lh -k d T Jic-fo-/, - q / . < o ( f e . 2 )

AT p,= l^X'7 *' I/i -I y) toS$>

© I = X cos fa■+ i y s A % ) j-k

(8.20) (6.24

SPAN P, P2

Q, « 0t T

AT p,

( |.X - (y -5 CoS

^ ^ x + X SaX (fo t X losfo= 0 X + (Xte-i/o - Y s v u f 0) / 4

(8 .23)

(l3,24)

(8.25)

H M F SPf\M ~ Q z

q = H i + ( f / M ' o - h) ^ ^

©l - &TT + f ®X X $*C% - I £x X 7 % - / ) f o l h ' i * U / f lR

(U£>)

IB. 2^

8.5 CHARACTERISTICS FOR FINITE PULSE DERIVED MOTION

The motion of the follower is analysed in Appendix 3* For this purpose it was found necessary to derive the equations for the cam characteristics independently for each span of the working length of the deflected leaf along the original x-axis. Referring to Fig. 8-7 the length is divided into three sections

(a) the half-span from the point of inflextion 0^ to the maximum at P^.

(b) the complete span from the maximum at P^ to the minimum at P .

(c) the half-span from the minimum at P^ to the point of inflexion at Q^.

The slope of the leaf is the same at 0^ and the displacement of the cam follower is equal to the ordinate of the corresponding point on the curve of deflection relative to the ^ “^i axes* computerprogram ’DEFLECTION' can be used directly to find the deflection (y) of the leaf for equal increments of the equivalent angle of lift The analysis given in Appendix 3 enables the corresponding displacement of the follower and angle of cam rotation to be found. (It should be noted that the increments of the actual angle of cam rotation are not equal.) The equations of the cam characteristics are summarized in Table 8-3, and the 'DEFLECTION' program was modified to calculate the follower displacement and the corresponding angle of cam rotation ('FPDM-DISP').

The slope of the leaf relative to the original x-y axes is given in terms of the equivalent angle of lift by equation (7*19) • Sincethe angle between the x- and x^ axes is known from equation (6.4) it is possible to derive the transform for the equation of the slope of the neutral axis relative to the new x^-y^ axes* general configura­tion for any point on the curve is illustrated by Fig. 8-8 . The rela­tionship between the angles of slope of the same point and the angle between the x- and x^- axes is

y = > - 0 (B.fZ)

-94-

0/ = t ia f a ( g . f t j

-95-where the angles are defined in Fig. 8-8.

and(f> - ztmC1 Oho (8.14

substituting in (8.12)

detx

and in this case 0 — •

(6. ft)

Then the dimensionless form of the velocity of the follower is given by substituting from equations (7.19) and (8.2) to obtain

n / \

V = = tot (%) Jc&X\j (•- f )w A, /1

1

| r>—-/Cio

f I

The acceleration of the follower is obtained by differentiating equation (8.1 5 ) and substituting this result in (8.5 ) to obtain the dimensionless form

A »■ XT' = 8cf-totac [Am I^ - f0 M M

(O'•A \CioL. cA a

%

Vo

where dy/dx is given by equation (7 .1 9 ) .

8.6 PERFORMANCE OF FINITE PULSE DERIVED MOTION

The values of the dimensionless groups of the cam characteris­tics at significant positions during a dwell-rise-dwell event are summarized in Table 8-2. Since both the velocity and the acceleration

T o F /\C £ PAGE S t

OI

cQ<7

-96-of the follower are functions of dy/dx, equation (7-19) , the valueswere calculated for increments of the equivalent angle of lift ( )and the actual angle of cam rotation calculated from the appropriate equation of (8.19), (8.23) or (8.27). The computer program 'FPDM-CHAR1 (Appendix 8) was used to calculate the velocity and acceleration of the follower for unit lift and an angle of lift of 120° over a range of machine ratios. These characteristics were drawn for Figs. 8-9 and 8-10 to show that the Finite Pulse Derived Motion satisfies the bound­ary conditions specified in equations (2.1). Therefore the follower should not be subject to any shock loads provided the manufactured pro­file of the cam conforms exactly to the theoretical shape. In compari­son with the conventional plunge-cutting method of manufacturing the profile which depends upon the specified polar co-ordinates to define the shape of the pitch curve (106) this process has the advantages that the theoretical shape of the pitch curve is not subject to round-off error and that the displacement should always be' set correctly for any angle of cam rotation. There is an abrupt change of slope in the acceleration characteristic corresponding to a finite discontinuity in the pulse curve, a feature confirmed by Nutbourne et al (90). To this extent the theoretical performance of the Finite Pulse Derived Motion is inferior in severely loaded applications to that of a cam law having a smooth continuous acceleration characteristic (such as cycloidal motion) since there must be a sudden, but finite, change in the rate of application of force at these positions. In practice it is also neces­sary to allow for the effects of the secondary accelerations induced by round-off errors in the polar co-ordinates and random manufacturing errors, discussed in Chapter 3« Practical requirements restrict the selection of the machine ratio and the span and offset of the leaf which are related by equation (A3.l). In this case the maximum abso­lute value of follower acceleration varies directly with the offset-span ratio of the leaf whilst the limiting value is determined by the angle of thrust between- the surface of the leaf and the path of the probe (Chapter 9). Ihe minimum offset-span ratio is determined by the lift and dimensions of the attachment. The commentary on the accuracy of the setting dimensions and of the design data applies equally to both motions derived from the shape of the deflected leaf spring.

-97-8.7 LIMITING PARAMETERS

The procedure for designing a cam mechanism from first princi­ples is; outlined in Chapter 2 and follows the flow chart shown in Fig. 2-1 and modified in Fig. 5-1 for the special case of a cam to be manufactured by a copying or generating process. Inspection of these flow charts shows that the limiting values of such parameters as the maximum pressure angle and the minimum convex and any concave radii of curvature are functions of several independent dimensions and of derivatives of the displacement equation. As a result it is necessary to follow an iterative process, repeating sections of the calculations until acceptable values are obtained. The iteration is. circumvented by using prepared design data to enable suitable starting values to be chosen immediately (Fig. 2.2). For ease of use this data should be presented in graphical or tabular form and has universal application if the variables can be formed into dimensionless groups (110). A particular set of data is confined to one cam law and in some instances, such as pressure angle, to one configuration of the mechanism. Conse­quently the preparation of comprehensive design data is only an economic proposition if it is intended to use the same cam laws for most new designs.

8.8 PRESSURE ANGLE

The general equation for the pressure angle between the direction of the thrust acting through the centre of rotation of the roller and the path of the centre at that position (Fig. 2-3) is ( 9?)

= /tan1

where 0-r is the offset of the path of the translating follower and

Equation (8.28) simplifies to

-98-

in the special, but frequently encountered, case of a radial trans­lating follower. Equation (8.29) can be re-arranged using dimension­less ratios

$ XvL = t — -£-yz — — (B ,3o )

The scheme used by the Engineering Sciences Data Unit (34) for thepresentation of design data for the magnitude of the maximum pressureangle and the corresponding angle of cam rotation from the start of arise movement of the follower involves the use of two graphs. Thesedata are restricted to one cam law and a particular configuration ofthe mechanism. One graph relates the maximum value of theparameter A+ (degrees) to the corresponding value of theratio (R /h). The angle of cam rotation at which the pressure angleattains the maximum is read from the second graph which relates(©Al ly to (R./h). This scheme has the advantage that the effect qmax . A °of varying factor <T' can be shown simultaneously by drawing separate curves for different values of 0“ and was used by Martin (77) to obtain the curves shown in Figs. 8-12 and 8-13 the Simple Derived Motion. For the simpler cases of Simple Harmonic and Cycloidal Motions (the latter was used in the ESDU publication cited above, for varying ratios of acceleration and retardation periods) Warriner (110) has shown that the complete data can be included in one graph by constructing curves of tan K (degrees) against a base of (e/j3) for varying values of (R^/h).

8.9 PROFILE CURVATURE AND SURFACE STRESS

Martin continued by determining critical radii of curvature of the pitch curve for the disc cam driving a radial translating follower with the Simple Derived Motion using a solution derived from Equation (4.1) Substitution of the magnitudes of the displacement and of derivatives of the displacement equation also involves the factor <7 and the value CT = 0.5 was used to calculate the values needed to plot Figs. 8-l4 and 8.15. The radius of curvature of the pitch curve is expressed in terms of the ratio C^/h) for different values of (R^/h) and a range of angles of lift between 30° and l8o°. The minimum radius of convex curvature

-99-at the nose of the cam is read from Fig. 8-l4 and minimum radius of convex curvature of the section of the pitch corresponding to the

it is smaller than the radius of curvature at the nose. The proportions of some cams are such that concave curvature of the pitch curve also occurs. The minimum value of the radius of concave curvature was read from the tabulated results to plot Fig. 8-15. The radius of convex curvature of the profile is obtained by subtracting the radius of the roller from that of the pitch curve

This value of is needed to determine the surface stress in the material of the cam, in this case the published data (33) (86) have universal application. In those instances where concave curvature of the profile exists the radius of curvature of the cam profile is the sum of the radii of the roller and the pitch curve

In the limiting case this radius must always exceed that of the roller, more significantly this value determines the maximum diameter of cutter which can be used to produce this profile. (If the diameter of the cutter is different from that of the roller follower the polar co-ordin­ates specifying the pitch curve of the cutter must be ammended to suit both the new radius of the cutter and to allow for the resultant cor­rection to the pressure angle.)

The same principles are used to derive the corresponding data for the Finite Pulse Derived Motion.

The thrust acting normal to the cam profile and the roller through the "point" of contact causes an elastic deformation of both curved surfaces. The local surface stress in the material of the cam disc can be estimated from the Hertz equation, but this solution is based upon the false assumptions of static conditions, perfect align­ment and truly cylindrical surfaces. Both Juvinall (65) and Morrison (86) quote the work of Smith and Lui (105) into the significance of

acceleration period of the follower (rise movement) is included where

(8.30)

(8.31)

-100-

relative motion between the cylinders upon the magnitude of the sur­face stress. This analysis shows that even a tendency for relative sliding to occur between the surfaces causes a shear loading at the contact surfaces and considerable increases in the magnitudes of both the direct and the shear stresses, including reversals of direction of these stresses in the case of rotating cylinders. According to Juvinall a combined rolling and sliding action between the surfaces is a source of high local temperatures in the materials, causing differential thermal expansion which further increases the stress in the material. Therefore the calculated value of the Hertz stress is not a valid criterion of loading. Errors in the rigorous solution will result from imperfect alignment of the mechanism which may be expected to vary during any movement of the follower-as a result of the fluctuation in both the thrust and the pressure angle. A valid criterion for the loading is obtained by using the estimated Hertz stress in conjunction with experimentally determined values of the limiting load for the same combination of materials under specified conditions of rolling and/or slip. Some data has been published by Morrison, but acknowledgement is made in the ESDU publication (32) to the need for further research into this subject (see below, Chapter 11).

Therefore the ability to estimate the limiting condition for roller slip in a new mechanism would form a valuable extension of the design procedure. A theoretical investigation supervised by the author (28) into the angular motion of the roller about the pivot centre for a radial translating follower during the rise and return movements has shown that the angular velocity is given by

L *

- A ( m )

and the expression for the angular acceleration of the roller is

7 = 7 ( Y )1 ~J [ cLb. J

Both equations (8.32) and (8.33) ca-n He arranged in dimensionless form for the preparation of design data and proposals for the extension of this analysis to derive expressions for estimating the limiting condi­tion for roller slip are made in Chapter 11, (page IS'2. ).

-101-

It is also possible to derive design data to determine the torque needed to drive the cam disc during a follower movement. The author (2 5) has shown that this torque is given by

T - F ( fC’g /Lj. ) JuQM. fe) - (8,34)

8.10 DRIVING TORQUE

where F is the force acting along the path of a radial translating follower (Fig. 2-3). Substituting the magnitude of the pressure angle for this configuration of the mechanism from equation (8.29) simpli­fies equation (8.34)

T - ' F Jj (8.35)

The force F acting upon the follower is the algebraic sum of:-

(a) the external load.(b) the frictional resistance to motion.(c). the appropriate component of the weight of the

follower assembly.(d) the force required to accelerate the mass of

the follower assembly.(e) the spring force.

The spring force can be subdivided into the preload and the product ofthe stiffness of the spring and the displacement of the follower.Then the force equation can be expressed as

F = F + Ma + S (8.36)c ywhere F = F + F + F + Mg (8.37)c ex f pi

In equation (8.37) it is assumed that the external load is constant andthat the path of the follower is vertical* Then equation (8.35) can be expanded into the components of the force

T = (F + Ma + S )c y dx (8.38)

-102-

du yKSubstituting — 6~ ~ ///ct&

and separating the terms

T - 5 £ + fYJr + s .YYW <0 (8.39)

Dimensionless torque factors can be derived by separating the terms of equation (8.39) and substituting from equations (3*5)j (3*6) and (3*7) for the dimensionless forms of the displacement, velocity and acceleration of the follower respectively to obtain equations (8.40) to (8.45) given in Table 8-4. These equations were used to plot the curves for the three components of torque drawn in Figs. 8-l6, 8-17 and 8-l8. The actual values are read from these graphs for an actual mechanism, making it possible to determine the torque required to drive an actual mechanism.It is also possible to compare the driving torques for different cam laws under standard conditions of lift, angle of lift and speed of rotation of the cam disc, assuming the same mass of the follower assembly, constant external load and pre-load of the spring. Under these conditions a valid comparison is dependent upon the correction of the spring stiffness to allow for the different magnitudes of the maximum retardation of the follower during a rise movement and the least displacement from the start of the motion at which this retardation is attained. Then the compensated spring stiffness torque factor is |-Am|t h/y where t is obtained from Sequation (8.45). The torque curves for the Simple Derived Motion are compared with those of cam laws in common use in Figs. 8-19, 8-20 and 8-21. Since the follower acceleration is also a function of the factor "cT the compensated spring torque is also used in Fig, 8-l8. The magnitudes of the spring stiffness compensating forces needed to plot Figs. 8-l6 to8-21 are given in Table 8-5.

8.11 'ASYMMETRIC1 MOTION

In common with most cam laws the characteristics for both derived motions obtained from one setting of the leaf are 'symmetrical', the maximum velocity occurring at

X = i • X i .

-103-

a

and having equal absolute values of peak acceleration

occurring at the positions 9^ and ©^ respectively where 0 ^ “ — 0^.

It is theoretically possible to manufacture profiles for 'asymmetric* motions (page 59) using the derived motions. For this purpose the leaf must be deflected with different spans and offsets, but it is essential that the slope (and hence the follower velocity) at the transition point be identical for both configurations. The position of the transition point is determined from the proportions of the cam rotation corresponding to the follower acceleration and retardation, such that

(8.4?)

e T * + J 2 (8.48)

Then the boundary conditions for the 'symmetrical' motions forming the special motion (Fig. 6-1) are

4 = 20r = <8.49)Xk={<1 - 2-khX(8.50)

and

J g - 2 ( 1 - 0 [j = 2 ( 1 - (8 .51)

= 2X(i-l)X (8.52)

whilst the offsets are found from the cumulative lift and by equating the offset-span ratios.

VK • + Y * . - 21 ' . (8.53)

^ (8.54)

The design data determined for the appropriate symmetrical motion applies to this case provided each section is analysed separately by substituting

-104-

the equivalent values of the lift and angle of lift, and modifying the base circle radius to the notional value for the .outer portion of the profile (see Example 3* page ^73).

8.12 SUMMARY

The theoretical performance of cam mechanisms using both pro­files derived from the shape of the deflected leaf spring is shown tobe similar to those of proven cam laws.

The dimensionless forms of the characteristics are derived fromthe expression defining the curve of the leaf spring.

iThe acceleration characteristics for Simple Derived Motion

(Fig. 8.4) is continuous throughout a dwell-rise-dwell event but the finite values at both limits indicate shock loading at these positions.The curve is similar to that of simple- harmonic motion, showing this profile to be suitable for "moderate" speed applications. The absolute value of the peak acceleration varies inversely with the offset-span ratio, concurring with the recommendation to use the largest possible offset-span ratio.

The Finite Pulse Derived Motion satisfies the requirements established in Chapter 2 completely. The acceleration curve is similar to that of a cubic polynomial. The relationship between the machine ratio and the span and offset of the deflected leaf is presented graphi­cally. In this case the absolute value of the peak acceleration varies directly with the offset-span ratio of the leaf. Practical considerations restrict both the maximum and minimum values of the offset-span ratio for this application.

The specialized design data needed for setting the leaf spring are derived and examples of the data for determining the limiting values of pressure angle, curvature, driving torque and the angular motion of the roller are quoted.

No allowance is made for the effects of manufacturing errors on the motion.

-105-

Chapter 9

DESIGN OF THE ATTACHMENT

9.1 DERIVATION OF THE PROFILE

The basic arrangement of the attachment to be assembled on a milling machine or jig-borer for deriving the shape of a cam profile from that of a deflected leaf spring was established in Chapter 6 and illustrated diagrammatically in Fig. 6-3* In addition to the design requirements for a copying device identified in Chapter 5 the attach­ment should:-

(a) incorporate standard equipment whenever possible.

(b) require the fewest possible precision tolerances(smaller than _+ 0. 002 in) compatible with theproduction of accurate profiles.

The analysis of follower motion in Chapter 3 shows the criterion formanufacturing accuracy is pitch curve wavyness due to random errors suchas positioning and round-off. An empirical maximum of 0.0002 in/degree is quoted by Nourse (89) for conventional manufacturing processes, although reference to Table 4-1 shows this is difficult to achieve.Since wavyness defines the rate of change of error it is argued that larger tolerances are permissible on the lift and.cam angles, say _+ 0.003 in and + 0.5° respectively.

Either increment or continuous cutting could be used to derive master cam profiles from the shape of a deflected leaf spring. Experi­mental comparison of theoretical and measured deflections requires readings at increments of span and, since this investigation is primarily concerned with establishing the principle of this method of cam manu­facture, increment cutting was chosen for cutting profiles by this method on a conventional milling machine. Consequently a final smoothing

-106-

operation is required, but it is recommended in Chapter 6 that profiles

interpolation also be given a honed finish.

The method of cam manufacture is based upon the inversion of the disc cam mechanism described in Chapter 4. The roller follower is replaced by the cutter rotating about a fixed axis to generate a cylinder of the same nominal diameter. Therefore the cam blank must rotate about a parallel axis and simultaneously trace a path in the plane of rotation to reproduce the relative motion between cam and follower. The cam blank is mounted on a rotary table fixed to the table of the machine-tool. The rotation of the blank controls the linear displacement of a slide through a mechanical transmission. The deflected leaf is mounted on the slide so that the displacement of the cam blank relative to the cutter is determined by the deflection of the leaf, so it is simplest to align this movement with the transverse feed of the machine-tool (y-direction, Fig. 6-3).This arrangement produces the profile for a radial or offset translating follower. Master cams for mechanisms having oscillating lever followers can also be made in this way, the profiles of the production cams being corrected on the copying machine (106).

Manufacture of the cam profile for Simple Derived Motion (Appendix 5) requires the slide travel to be aligned with the longitudinal feed (x - axis). As shown in Fig. 9-1 (a) the starting radius for cutting the profile controlling a return movement is given by the maximum ordinate

and the ordinate at any intermediate cam angle 0^ during' the movement is

manufactured on numerically-controlled machine tools using linear

(9!)

(9.2)

where y. is the deflection at

(9-3)

-107-

The centre-distance between cam and cutter is set by adjusting the transverse feed until the Datum Indicator registers the return of the surface of the leaf to a pre-determined position. Therefore the common centre-line through the cam and cutter axes must be aligned with the transverse feed (y-axis) for both derived motions.

9.2 THE DATUM INDICATOR

In contrast with both the manual plunge-cutting process (20) and numerically-controlled profile manufacture (93) this method of setting the machine-tool is intended to eliminate the requirement for movements through a sequence of specified dimensions. Instead, the displacement at any cam angle is determined by restoring the Datum Indicator to a specific graduation which is readily identified. The sole function of this device is to indicate the return of the leaf surface at the incre­ment of span corresponding to that angular position of the cam to a specified datum. The- indicator must satisfy the following design require ments:-

(a) it must be a precision instrument having good repeatability: it is not required to measure displacements.

(b) the indication must be clear and give warning of the approach to the datum.

(c) ideally it should impose no thrust upon the leaf.

(d) the accuracy must not be impaired by the curvature of the leaf.

(e) it must operate correctly in workshop conditions.

(f) it must be insensitive to machine vibration.

(g) it must be capable of identifying the limits of every span.

(h) it must be capable of locating the leaf at the supports.

-108-

Duncan (29) claims to have obtained experimental results verifying Love's theoretical analysis (based upon the approximate theory of bending) (76) showing the deflection of the outer surfaces of thin plates to be very little different from that of the unstressed neutral plane•

Therefore it is reasonable to base the analysis of the cam characteristics on the shape of the neutral axis although the profile shape is derived from the surface of the leaf.

9.3 THE MEASUREMENT OF LEAF DEFLECTION

The measurements needed to compare the theoretical and observed deflections of the leaf could be taken either in position on the machine- tool or using special apparatus. Whilst the latter is a satisfactory test of the theoretical analysis the application of the principle depends upon the development of an effective means of determining the datum position on the machine-tool under factory conditions. The solution was also influenced by the requirement for low cost.

9.3.1 OPTICAL METHODS

Optical methods of measuring deflection or curvature have the advantage that no additional force is imposed on the leaf. Interferometry methods involving optically flat, polished specimens and special-purpose equipment (29) were rejected as impractical. Durelli and Parks (3 1) describe the use of a perspex model to measure the deflection of a beam using Moire'" fringe techniques. Duncan and Sabin (30) measured the large radius of curvature of flexed thin plates by a method combining the principles used by Martinelli to determine the radius of curvature of a soap film with Ronchi's method for testing the accuracy of optical mirrors. The curvature parallel to a specified plane was defined by the partial curvature parameter

ijj = iko- E d ?

fa fa W I 2 ( f - r) (9A)

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where d is the plate thickness andw the deflection due to a load W acting in that direction.

An experiment on a cantilever deflected by a concentrated load is reported. Observations made at intervals along perpendicular directions of traverse determined the components of curvature of the plate in those directions. The theoretical magnitude of the parameter / varied between +8.0 and -3*4. Some discrepancies between the observed and

Measurement of curvature also forms the basis of -Lightenberg's method (74) intended to determine bending moments from the deflection of small models. A camera is focussed upon the image of a large grating reflected on the polished surface of the model. The film is exposed

of equal curvature across the surface of the model. The magnitude of the curvature is found by counting the number of fringes from the known conditions at a reference point to the section of interest. The focal length must be accurately related to the (larger) radius of curvature of the grating to ensure that the errors due to the deflection of the model are negligible. This method measures the change of slope between the un­loaded and loaded shapes of the model. The analysis of the results is based upon the approximate theory of bending.

The return of the surface of the leaf to a specified datum can be determined by the position of the reflection of a narrow beam of light from the surface. Because of the curvature the plane of this beam con­taining the angles of incidence and reflection must be accurately aligned with the z-axis (Fig. 6-3).

None of these methods was considered suitable. Only the reflection method met the requirements for industrial application, but the beam of light is inevitably obstructed by some supports of the leaf. To be visible to the machine operator the equipment must be located at the rear of the machine-tool, so restricting the range and the magnitudes of the angles of incidence and reflection. Durelli and Parks (31) commented that it is generally unnecessary to use a sensitive instrument for measuring the deflection of a perspex model since the deflections are large, permitting the use of course gratings. These authors quote the example of a simply

calculated values exceed 10$, which must relate to the term. The standard deviation is 1.7 x 10 X .

before and after the model is loaded to obtain Moire" fringes of contours

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support ed perspex beam 12 x 1 x | inch. The grating had a density of 300 lines/inch, so 1 fringe represents a deflection of 1/300 in.Lightenberg (74) claimed the support of Landwehr and Grabert (73) in his criticism of interferometry methods and the difficulty of measuring second derivatives easily.

"As second derivatives are not measured very easily it might be thought that they could be obtained from measurement of the deflections w themselves. In that case we should have to use interferometry measurements with all the inherent difficulties to obtain sufficient accuracy."l

Lightenburg claimed superior accuracy for his method, the standard devia­tion between the theoretical and observed values of the bending moment is6.5 x 10 2 over the range 0.5 to 3*5 kgf-cm (0.4 to 3*1 Ibf-in) of the theoretical values. The maximum difference is less than 0.5$ of the theoretical value. However Lightenburg warns that errors occur in the regions of loads and built-in supports. It is uncertain whether the clamps secure the model with zero slope and the deflection is affected by the clamping action. Other practical problems involve the mounting of the leaf and controlling the slope of the unloaded leaf. Similar problems may be anticipated with the metal leaves used for profile derivation.

These considerations led to the decision to test the practibility of measuring the deflection directly.

9.3.2 APPLICATION OF TRANSDUCERS

Various types of transducer are used to measure linear displacement' through changes of capacitance, inductance, potential difference, resis­tance or electromotive force, or by means of the piezo-electric effect (88). The last two have dynamic effects, making them unsuitable for this appli­cation. All types except change of capacitance depend upon a mechanical connexion for the operating force and so would increase the deflection of the leaf. The largest force is required by those types of variable resistance and potentiometric transducers actuated by wiping contacts which give an undesirable step-function output. Capacitance transducers

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in which the leaf itself acts as one plate have the merit of requiring no operating load, but since the areas of the plates are significant accuracy is liable to be adversely affected by the curvature of the leaf. This error could be balanced by using two transducers on opposite sides, but one would be obstructed by the supports which could also inter­fere with the measurement. In every case additional amplifying and measuring equipment is needed.

For these reasons it was decided to begin by testing one of the simplest and cheapest solutions, a dial gauge actuated by a linkage.

9.3.3 DIRECT MEASUREMENT - THE DATUM INDICATOR

The displacement can be measured directly on the machine-tool by transmitting the transverse movement of the leaf- through a probe to actuate the linkage. The setting for an increment cut is determined by returning the linkage to a specific position. Measurement of distance is not involved and warning of approach to that position is given. It is difficult to accept Borun's claim (1 2 ) that the thrust of 0 .5 to 1 kgf (l to 2 .2 lbf) needed to actuate an hydraulic tracer for continuous copying can be imposed on a leaf of cross-section 0 .3 x 25 nim ( 0.012 x 0.984 in.) without disturbing the curve of deflection unless the leaf is externally re-inforced. SIG (125) recommend a minimum thrust of 7 lbf. for this purpose using a conventional template, moreover calcu­lations based on the analyses in Appendix 1 show that much smaller forces cause significant diflections.

Therefore the possibility of restricting the thrust on a freely mounted leaf within, say, 0 .1 lbf. was investigated.

The position of a probe constrained to trace a prescribed path can be determined with a high standard of repeatability using a dial gauge, but the conventional plunger stylus design requires an operating force of 0.2 lbf (BS907:1965) so the type recommended for testing the accuracy of machine-tools (BS4656:197l) having a jewelled movement actuated by a lever stylus was preferred. It requires an operating force of about 0.06 lbf and the specified repeatability is better than 0.0001 in (BS2795:197l)• Since the application does not involve the accurate

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linear measurement the relationship between probe displacement and dial reading is immaterial although any significant diminution is undesirable because the approach to the cutting position would appear more rapid. The use of this principle simplifies the plunge-cutting process since the machine settings require no co-ordinate data. Every setting is identified by returning a needle to the same graduation, eliminating the dependence upon the positioning accuracy of the machine- tool (Appendix 7) and round-off error in the radii which is a significant source of pitch curve wavyness (Chapter j>). This permits the use of a milling machine instead of a jig-borer and cuts at smaller increments of cam rotation.

The simplest solution of replacing the stylus with a cranked spindle proved unsatisfactory as it dug into the leaf, making fine adjust­ment impossible. Free movement was obtained by .extending the coupler of a parallel crank mechanism to form the probe, Plate 5- Tb® cranks were duplicated, so the minature ball bearings used for the pivots could be preloaded to counteract axial and diametral slackness. The frame could be positioned anywhere on the support spindle which formed part of a cranked frame clamped to the column face of the milling machine. Fine adjustment is obtained by turning the face of the dial gauge. Normal operation requires the tip of the probe to be returned to a specific position, but the use of the instrument to calibrate the span (Appendices 5 and 6) requires accurate alignment of the probe travel with the y-axis.

The linkage proved easy to use, it was found possible to identify the datum position whilst making a cut, although it is inadvisable to apply the principle in this way with manual control due to the danger of accidental overtravel, but settings could be adjusted without stopping the machine. According to Judge (64) small amplitude, high-frequency vibration of a dial gauge has the beneficial, effect of improving the repeatability.

Alternatively the probe could be mounted on a pair of flexure pivot bearings (35) (117). This suspension permits free movement through the operating displacement in the y-direction (Fig. 6-3), but would be very stiff in the x-direction to react the side thrust due to leaf curvature without slackness.

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The working edge of the probe must be aligned accurately parallel with the axis of bending to prevent errors in increment positions resulting from leaf curvature. The thrust along the path of the probe, including the force needed to actuate the dial gauge, was estimated to be about 0 .1 lbf, depending upon the travel. As shown in Figs. 9-2 and 9-4 the slope causes an apparent increase in leaf thickness measured along the path of the probe; for the same reason the nominal line of contact between the radiussed tip of the probe is displaced from the path. Both these errors reduce the trans­verse movement needed to restore the datum indicator to zero, opposing the change in deflection caused by the probe thrust. As a close approximation the cumulative error acting to increase the table move­ment is

= ~ ^p) ^ “ •) (9-5)

9-3-4 COMPENSATION OF ERRORS

where cb-f is the probe thrust errorand is the contact angle between the directions of probe travel

and the normal to the leaf, Fig. 9-2.

The probe thrust error depends upon the parameter Wp/EI, the span and the offset, Wp being the probe thrust along the y-axis. The maximum error at mid-span for varying <T was derived from a version of the program ’LEAFSLOPE* to prepare Fig. 9-5-

All these errors are continuous functions, small in comparison with the lift. The apparent thickness and probe tip radius errors are zero at both limits of a dwell-rise-dwell event for either derived motion, but the thrust error is always a maximum at mid-span (and therefore has the same finite magnitude at 0^=0 and 0 ^ = ^ for Finite Pulse Derived Motion). The components of deflection error plotted for a particular case of Simple Derived Motion in Fig. 9-6 show that zero resultant error at mid-span may be obtained by selecting the leaf thickness from Fig. 7-4 and 9-5, using the appropriate probe tip radius, Fig. 9-4. This design data shows that the thickest possible leaf compatible with the limiting

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bending stress should be used to obtain maximum stiffness. Even so, reference to Examples 1-3 (page 4M-) shows that a resultant error may be unavoidable, but the secondary acceleration from this cause is negligible.

As shown in Example 3 these errors complicate the remounting of the leaf during the manufacture of an ’asymmetric' profile, Fig. 6 -1

and page 59 * Since it is essential that the follower velocity (and therefore the slope of the neutral axis) be identical at the transition point for both settings it is necessary to use different leaf thicknesses to equate the resultant slopes (not the magnitudes of the probe thrust errors) at this position. For a given offset and leaf thickness the minimum machine ratio is determined from the limiting bending stress on the shorter span using Fig. 7-4. The probe thrust error for the shorter span is found from Fig. 9-5 enabling the leaf thickness for the longer span to be found indirectly by equating the resultant offset- span ratios to calculate the probe thrust error for the longer span.Since the profile is derived from the leaf surface the zero' of the datum Indicator must be restored. The difference between the apparent thick­ness errors is small in comparison with the lift.

9.4 TRANSMISSION BETWEEN ROTARY TABLE AND SLIDE

The cam blank was mounted on a mandrel fitted to a manually operated rotary table, a tray being provided to contain the swarf (Plate 4). The travel of the slide carrying the deflected leaf spring must be directly proportional to the angle of cam rotation with provision for a range of displacement factors, equation (6.2), to suit various cam angles and lifts. It is also necessary to engage and disengage this drive at any position for cutting dwells or to change the setting of theleaf. The arrangement of the compound gear train driving a rack forthis purpose is shown in Fig. 9-7• A friction cone clutch is incorporated in the layshaft and locks must be fitted to both the rotary table and theslide. The machine ratio defined as the linear travel of the rack perradian rotation of the rotary table specifies the drive in a convenient numerical (but dimensional) form (Appendix 2, page A38).

I ZP3

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The machine ratio is altered by using different numbers of teeth for wheels and on the change-gear principle. Since the transmission not subject to machining forces the objections made in Chapter 5 to frictional drives in generating devices does not apply. The backlash is permanently taken up in one direction by connecting a bias weight to the slide. The torque, estimated at 50 lbf-in, required an axial clamping force of about 500 Ibf between the members of the clutch. The concentricity problem was overcome by connecting the cone to the lay- shaft through an Oldham coupling which has a 1:1 velocity ratio in all positions (50).

The sources of manufacturing and assembly errors in the spur gear train are summarized in Appendix 7» According to Tuplin (109) the criterion for random pitch error is the resultant dynamic load, but this is irrelevant 'to this application. The gears used in the transmission are excessively strong to obtain the required centre- distances with correctly proportioned wheels, so a relatively fine pitch can be used to ensure the simultaneous mesh of several pairs of teeth. Since BS436:1940 specifies the maximum material condition as the true profile shape the effect of random pitch and tooth thickness errors on slide travel is counteracted.- The accumulative pitch error remains significant, especially as a rack is involved. The procedure for setting the span (Appendices 4 and 5) synchronises the cam and slide positions at both limits of travel within the repeatibility of the rotary table. Therefore the worst combination of gear errors produces maximum asynchronisation at mid-position. An analysis based on this assumption to determine the maximum errors in cam angle and pitch curve radius for the various classes of gear (BS436:194o) are summarized in Table A7-2 of Appendix 7*2. Accordingly class A2 precision cut gears were specified for the transmission.

The small torque and incomplete rotation of the layshaft make it feasible to use a flexible band drive of the type used in the double slide generating mechanism, Fig. 5-9, in place of the gears. The arrangement shown in Fig. 9-8 allows increased dimensional freedom since the centre distance between cam and layshaft can exceed the sum of the disc radii. Sufficient overlap must be provided to permit the maximum possible slide travel, the bands must be duplicated to obtain a positive drive in both directions of travel. Tensioning straightens the bands

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and aligns them with the direction of slide travel* The discs are easier to make than gears. Pitch and tooth errors are eliminated and the effect of disc eccentricity is reduced by using the largest possible diameter. The effect of elastic deflection of the bands is analagous to accumulative pitch error, the accuracy of slide displace­ment should be similar to that for the precision gear transmission. A range of machine ratios is obtained by permanently assembling a set of disc and band units, one operating whilst the remainder idle.

9.5 THE SLIDE

The slide carrying the deflected leaf spring must trace a straight- line path, the direction being determined by the motion specified for the cam profile. The assembly shown in Plates 3 and 5 was constructed as a single unit on a substantial rolled steel angle. It was secured to the machine table, overlapping on the column side. This arrangement was chosen for stiffness, but has the disadvantage of restricting the trans­verse feed and hence the maximum pitch circle radius to 6-J in on the Elliott milling machine. A universal mill must be used to cut profiles for Finite Pulse Derived Motion. At that time the type and size of leaf spring had not been determined, so the slide and slide travel were designed to suit the largest size. Due to the position of the rack the driving force is applied eccentrically. To prevent the mechanism jamming the travel is constrained entirely by the circular-section guide rail nearer the rack (42). The static forces acting on the slide, Fig. 9-9 , were analysed to determine the slope of the slide relative to the theoretical path and the change of deflection opposite the datum indi­cator due to the horizontal forces acting on the guide rail during move­ment. The bearing reactions were represented by concentrated loads and it was assumed that the entire load was taken by the guide rail. The changes of slope and deflection opposite the probe were calculated to be 0.006° and 0.00016 in respectively for a slide movement of 6 in along a in diameter guide rod. Neither deflection reversed direction, the bias weight preventing any reversal of the resultant forces. Increasing the rod diameter to 1 .0 in reduced the changes of slope and deflection to 0.002° and 0.00005 in respectively. This layout has the additional advantage that the rails are required to be parallel within 0.0005 in, in the horizontal plane only.

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Increment cutting requires repeated slide, movements of 0.010

to 0.070 in. Bell and Burdekin (10) have shown that'stick-slip action during low-speed movements along plain metallic slideways is minimised by using PTFE bearings. Standard Glacier "DU" (83) journal bearings were suitable for the guide rail, but split housings assembled with shims were needed to take up the +0.004 in tolerance on the bearing pads used for the square-section rails. These pads were secured with an expoy resin adhesive to avoid the damage to the plastics surface caused by the conventional countersunk screw fastening. As the load on these bearings is under 0.5$ of the maximum rating it was safe to specify the maximum metal condition for both the journal bearing housing and the guide rail to minimise the bearing clearance. These bearings proved entirely satisfactory.

9.6 ACCURACY OF LEAF SETTING

The span and offset of the leaf should be set up on the slide to align the span with the direction of travel within the accuracy of machining (Appendix 7*l)* Then the span is set directly from the angle of lift by turning the rotary table. Errors in the span and offset can be assessed in terms of the nominal values of the parameter Cf and the lift, the span being read from Table 7-4. The results of this analysis for CT = 0.5 are summarized in Table 9-1*

Errors in the span result from transmission inaccuracies and positioning error of the rotary table. The former can he estimated from the analysis of permitted gear tolerances in Table A7-2, but assuming the worst cumulative error over the complete span instead of the half-span condition used in Appendix 7*2. So, for this case, no gear can make more than half a revolution. This error varies with the span, but the positioning error is constant. A tolerance of +0.002 in on the alignment of the slide travel over 6 in movement has negligible effect on the span. Since the transmission error exceeds the positioning error the accuracy of the span is improved by using the shortest possible span compatible with the limiting bending stress and maximum contact angle. For the proportions quoted in Table 9-1 the tolerance on the angle of lift is within +P°6 , much less than the suggested value of +0°301.

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The offset is subject to errors caused by:-

(a) misalignment of the slide travel.

(b) changes of guide rail ’deflection.

(c) repeatability of the dial gauge.

The maximum cumulative errors for settings having the same nominal value of CT but different offsets are summarized in Table 9-1• The error of offset is within the suggested tolerance of +0.003 in.

No allowance was made for bearing clearances, but the force analysis showed these must always be taken up in the same direction.The change in the actual value of cf due to these errors is small, sothe consequent variation in follower velocity and acceleration isnegligible.

9 .7 CONTROL OF PROFILE WAVYNESS

Independent setting of the cam angle and pitch curve is subjectto error in both the increment of span and the difference in radii, i.e.the measured deflections of the leaf at the two positions. The effectsof transmission error on increments of span corresponding to 1° of cam

_5rotation is 7 x 10 in for a machine ratio of 2:1. This causes an error_5of 2 x 10 in between deflections at mid-span, diminishing with slope.

Misalignment of the slide travel has negligible effect upon the measured differences over increments of span between 0.010 and 0.070 in. As before, it is assumed that the bearing clearances are always taken up in the same direction. Errors in the zero reading of the datum indicator due to tip radius and apparent increase in leaf thickness counteract that due to probe thrust. For the graphical results given in Fig. 9-6 the maximum change between deflections from these causes is 3 x 10 in/ degree. This maximum does not co-incide with maximum slope. Random errors in the measured deflections result from the repeatability of the dial gauge (+0.0001 in at any position) and the positioning error of the rotary table (+0°1X at any position). The latter corresponds to an error in the difference between deflections of about +0.0004 in at maximum

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slope, less elsewhere. Then, depending upon the offset and maximum slope, the maximum cumulative error in the difference between deflec­tions over 1 ° increments of cam rotation is in the order of 0.0006 to0.0008 in, compatible with the values quoted for the manual plunge- cutting process in Table 4-1. Individual measurements are also subject to round-off error. This magnitude of error applies only to the measured deflections over increments of cam rotation needed for com­parison with the theoretical curve. A significant improvementdn pitch curve accuracy should result from the application of the profile deriva­tion process (Appendices 5 and 6) since the transmission ensures that the pitch curve radius is determined for the actual cam angle, eliminat­ing the effect of positioning error on this dimension. Then the worst pitch curve wavyness is reduced to 0.0002 in between any two increment positions, due entirely to the repeatability of the dial gauge.

Any increment cutting process must compromise between pitch curvewavyness and the quantity of material to be removed during the final

0*1smoothing operation, but angular increments as small as 0 20 appearreasonable (Plate l).

9.8 SUMMARY

Experimental comparison between the shape of the deflected leaf and the theoretical curve requires measurements at intervals of span.Since the primary object of the investigation is to establish this principle of profile manufacture the cams were manufactured by increment cutting to permit the use of a conventional milling machine.

Previous work on the deflection of thin plates showed very little difference between the deflections of the outer surface and the unstressed neutral plane, so the analysis of follower motion based on the curve of the neutral axis can be applied to profiles derived from the surface of the deflected leaf.

The linear travel of the leaf relative to the datum indicator is proportional to the cam rotation from the start of the event. The datum indicator registers the return of the leaf surface aligned with the probe to a prescribed position. Therefore the adjustment of the machine table

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is always determined by restoring the dial gauge to the same graduation, eliminating the need to read co-ordinate data. Improved accuracy com­pared with plunge-cutting results from the absence of any round-off error and reliance upon the repeatability of the instrument instead of the positioning accuracy of two independent feeds.

A lightly-loaded parallel crank mechanism was used for the datura indicator in preference to transducers or optical methods. The specified repeatability of the dial gauge is better than 0.0001 in and the maximum thrust along the path of the coupler is estimated to be about O.i lbf.The leaf deflection caused by this thrust opposes errors due to the apparent increase in leaf thickness with slope and the tip radius of the probe. The tip radius can be calculated to neutralise the error at mid­span.

A compound gear train connects the rotary table to a rack on the slide carrying the leaf. Analysis of the cumulative gear errors showed that precision cut gears to BS436:1940: Class A2 are required. Since no cutting force is imposed on the transmission a friction clutch can be incorporated in the layshaft.

The criterion for profile accuracy determined by follower motion is pitch curve wavyness. It is impractical to specify a maximum wavyness less than 0.0002 in/degree for conventional means of profile manufacture, but tolerances on the lift and angles of lift and return depend upon the application. Values of +0.003 in and 0° 30^ are suggested.

Errors due to the transmission and the deflection of the guide rail are continuous functions causing negligible secondary acceleration of the follower. Therefore pitch curve wavyness during the copying process depends upon the repeatability of the dial gauge. The maximum difference between adjacent radii is 0.0002in, the wavyness depending upon the size of the angular increment. Taking the standard deviation at one third of this value the lower limit for angular increments is 0° 20'*',

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If cam angle and leaf deflection are measured separately for . comparison with the theoretical curve of deflection,, then the position­ing error of the rotary table becomes significant. The maximum wavyness at mid-span is in the order of 0.0006 to 0.0008 in/degree for 1 ° incre­ments, reducing with the slope. These readings are also subject to measurement and round-off errors.

I

1

122

Chapter 10

DEVELOPMENT OF THE LEAF SPRING MOUNTING

10.1 SETTING THE LEAF

Accurate setting of the leaf to conform with the theoretical curve of deflection is vital for the success of the process. One span of the deflected leaf, Fig. 6-l(a), must satisfy the boundary conditions.

derived from equations (6.1 ) for infinitely varying magnitudes of both span and offset within the steps of the machine ratio and the capacity of the machine-tool.

The critical tolerance on the cam profile is that governing pitch curve wavyness. The means of eliminating or reducing the significance of the principal components of wavyness resulting from random round-off and positioning errors were discussed in the previous chapter. Pitchcurve errors attributable to the leaf result from the apparent increasein leaf thickness with slope, Fig. 9-2, and the deflection due to probe thrust. As both are continuous functions the secondary acceleration corresponding to an error of even 0.010 in at mid-span is negligible.

Smooth blending of successive portions of the profile depends uponthe accuracy of aligning the datum indicator with the limits of span andthe positioning accuracy of the rotary table. The maximum cumulativeerror from these causes for a machine ratio of 1 .0 , a span of 2.0 in and

= 0 .5 results in a slide displacement of 0.004 in from the limit of_5span. The corresponding deflection of the leaf is less than 1 x 10 m ,

reducing with larger spans and machine ratios. This error is an order ofmagnitude smaller than the least tolerance for machining the profile,

-3although the slope of the leaf at that position is 4.4 x 10 . Whilst it

(10.1)

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increases with slope this error cannot affect follower acceleration since the differences over increments of cam angle are unaltered.

10.2 REQUIREMENTS FOR THE LEAF

The deflected leaf must reproduce the theoretical curve of deflection accurately and consistently. The analysis in Chapter 7 and Appendix 1.1 showed that longitudinal forces acting at the limits of

Jspan significantly affect the shape of the deflected leaf. Since these forces are indeterminate the mounting must be designed to ensure that the longitudinal forces cannot exceed 10$ of the deflecting force to satisfy the latter requirement. The analysis of the curve of deflection derived from the Euler-Bernoulli Law was based upon the assumptions and approximations made in Sections 7«2 and 7«3* Consideration of the forces and stresses in the same Chapter amplifies and extends these requirements:-

(a) it must be possible to assume that the material is homo­geneous, isotropic, obeys Hooke's Law and has the same value of Young's Modulus in tension and compression.

(b) the material must have high values of Young's Modulus2and yield stress, say 80 tonf/in .

(c) the leaf must be initially straight in the unloaded condition.

(d) precision tolerances must apply to the dimensions of the cross-section throughout the length of one leaf.

(e) the surfaces must have a smooth texture.

(f) the leaves should be readily available as standard stock items.

A recommendation to use a ductile material having a fine grain structure to obtain the best co-relation with the theoretical curve proved illfounded. Of these materials 40/60 brass was only available in coiled form. Attempts to straighten it by cold rolling hardened the metal

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and caused irregularities in the thickness. Annealed l8/8 chromium steel was available as flat lengths but damaged very easily. The elastic range of both materials was incompatible with large offset-span ratios.

Two materials which met the requirements well are gauge plate and feeler stock, Table 10-1. Gauge plate had to be specially hardened and tempered to Rockwell C58. These operations required considerable care, the furnace must be large enough to prevent significant temperature gradients occurring along the length and the leaves must be quenched by dipping endwise into an oil bath. The chromium-tungsten-vanadium alloy has the merit of hardening to form a fine grain structure which is ductile with deep hardening properties, but is liable to distort. The feeler stock was commercially available already hardened and tempered to Rockwell C50-52.

Neither Borun (12) nor any of the authors of analyses of the elastica cited in Chapter 7 gave any details of materials.

The tests reported in this Chapter showed the feeler stock to be most suitable for this application. The quoted tolerance on thickness is^O.OOO 25 in. Measurement of ten samples of different batches showed that the tolerance was maintained with a tendency towards the upper limit on both dimensions of the cross-section. The maximum variation in thick­ness along the 12 in length of one leaf was 0.000 2 in, equivalent to a 5 % difference in the second moment of area about the axis of bending. There was a similar variation in width, but the linear relationship and larger dimension reduced the difference to 0.15$. Since a working span is only 10 to 30$ of the length the significant difference should be smaller.

10.3 PRELIMINARY DEFLECTION TESTS

The objects of these tests were:-

(a) to identify suitable materials for the leaves.

(b) to find suitable dimensions for the cross-section of theleaf.

To Face Page 125

I

Fig. 1

0-1.

M

eans

of

'r

efl

ec

tin

g

the

-Lea

f S

pr

ing

.

-125-

(c) to find the optimum range for the offset-span ratio.

(d) to compare the theoretical and measured deflections of the leaf spring.

(e) to gain experience in deflecting the leaf under these conditions.

These tests were conducted before the attachment described in the previous Chapter was constructed.

In practice the maximum offset-span ratio is restricted by the yield stress of the material, the deflecting force (in magnitude or the consequent deflection) and the limiting contact angle, Figs. 9-2 and 9-3- The force and strength considerations, together with desirability of compatible probe thrust and combined tip radius and apparent thickness errors (Fig. 9-6) implies a high stiffness hut also involves the dimensions of the cross-section.

10.3! SLOPE AND DEFLECTION MEASUREMENTS

The leaf was deflected between parallel offset clamps of the type used by Borun (12) to take the shape shown in Fig. 10-l(a). The measured offset, span and slope at mid-span were compared with the theoretical relationship given by the series solution for the elastica (3 )

~r “ | sinC/i) +■ T s (7*17)

chosen for ease of calculation. The result is within 0.1$ of that given by the elliptic integrals solution, equation (7 -1 6 ) for angles of maximum slope smaller than 16° (Fig. 7-6). In addition the symmetry of the curve was checked by comparing the measured deflections at and span.

These measurements were made by mounting the leaf between machine vices secured to the table of a recently-overhauled milling machine. The accuracy and repeatability of the lead screws used for the measurements

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were checked using slip gauges and a plunger-type dial gauge (Appendix 7*2). The maximum discrepancies found were 0.001 5 in/in on the longi­tudinal feed (x-axis, Fig. 4-1) and 0.000 8 in/in on the transverse feed (y-axis). The standard deviation on the repeatability in both directions was 0.000 5 in. No tendency to yaw was identified. Specially- made jaws were fitted to the machine vices to suit the datum indicator.The fixed jaws were aligned individually parallel to the longitudinal

o ifeed within 0 1 and the offset obtained by inserting a spacer betweenone fixed jaw and the leaf. The offset was measured from the lead screw graduations in conjunction with the datum indicator and the span found with an internal micrometer. It was unnecessary to set specific dimen­sions provided the fixed jaws were accurately parallel.

Trials with different materials showed the annealed materials were unsuitable. The gauge plate had to be hardened and tempered specially, then satisfactory results were obtained using a 0.016 x 1.000 in section but thicker leaves required a clamping force beyond the capacity of the machine vices. Feeler stock of 0.018 x 0.500 in section also gave satisfactory results. Subsequent work revealed inadequacies in the use of clamps to mount the leaf. A thinner section of feeler stock,0.012 x.0.500 in, was found best for the final design, Fig. 10-l(c). No corresponding size of gauge plate is available, so feeler stock is recom­mended despite the superior composition of gauge plate. Feeler stock has the additional advantages of being readily available and suitable for immediate use. The leaf spring used by Borun (12) had a rectangular cross-section 0.3 x 25 mm (0 .0 12 x 0.984 in), but the offset-span ratio of 0 .5 quoted in his paper ( cf = 0.633) implies a minimum span of about3.8 in for a limiting bending stress of 80 tonf/in . The corresponding probe thrust error over such a long span of a freely mounted leaf would be unacceptably large.

The bearing edge of the original datum indicator was sharpened to a nominal knife-edge and the origin for longitudinal measurements found by aligning the knife-edge with edge of one vice jaw. Since the curvature is zero at mid-span the slope was estimated by measuring deflections at small increments of span each side of the mid-position. By calculating the slope from the difference between the deflections at these positions symmetrically arranged about mid-span the result is independent of the

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probe thrust, tip radius and apparent thickness errors. The results compared graphically in Fig. 10-2 are within the tolerance band corresponding to the cumulative errors equal to the standard deviation of 0.000 5 in on the two measurements. Although the theoretical deflections at intermediate positions along the span had yet to be calculated the symmetry of bending was investigated through the relationships

The measurements were repeated after reversing the leaves in the clamps to test for errors due to initial curvature. In these measurements the probe thrust error should be equal and opposite, the tip radius and apparent thickness errors being identical for both positions. These

drawing attention to the difficulty of aligning the probe with the edge of the vice jaw, but not any initial curvature of the leaf. A displace­ment £ of the origin along with x-axis causes a discrepancy in the measured deflections

The significance of this error increases with the slope.

Finally the shape of the deflected leaf was copied onto the edge of a steel block by making plunge cuts at increments of span. No difficulties were experienced in maintaining contact between the probe of the datum indicator and the leaf or in reading the dial gauge whilst the machine-tool was running.

10.3.2 CONCLUSIONS

The tests in which the leaves were deflected between parallel offset clamps showed that both gauge plate and feeler stock are suitable materials. Both materials must be hardened and tempered.

(10.2)

results were less consistent than those for the angle of maximum slope,

where y is the sngle of slope at these positions.

i

-128-

The following features of the design required particular attention

(a) the alignment of the probe of the datum indicatorwith the side of the clamp.

(b) the means of setting the span accurately.

(c) the accurate location of the clamps and spacers.

(d) the clamp assembly must be as stiff as possible.

(e) the manufacture of sharp edges on the jaws andspacers to identify the limits of span accurately.

10.4 USE OF THE ATTACHMENT

Manufacture of cam profiles using the attachment illustrated diagrammatically in Fig. 6-3 requires an infinitely adjustable span within the limits of machine dimensions and one step of the machine ratio to suit the specified angle of lift. The original design illustrated in Plates 2-5 followed the arrangement published by Borun (12) which used two. clamps to secure the leaf. One clamp was fastened permanently to the slide, the second one being adjustable in the direction of slide travel. The span was set by aligning the fastened clamp with the probe of the datum indicator, turning the rotary table through the angle of lift and setting the movable clamp in the same way. The offset was obtained by inserting spacers having a tongue-and-groove location between the leaf and the fixed jaw of the appropriate clamp.The attachment was mounted on the table of an Elliott "Milmor" verticalmilling machine of the knee and column type for a series of tests com­mencing with the repeatability tests summarized in Table 10-3 and cul­minating in the manufacture of cam profiles. The deflection of the leaf was found by measuring the displacement of the table in the y-direction from an origin co-incident with one limit of span, Fig. 10-l(a), after the transverse feed movement had restored the dial reading of the datum indicator to zero. A depth micrometer mounted on a substantial block was used for these measurements. The block was clamped to the slideway

-129-

machine d in the knee of the milling machine so that the feeds of the micrometer and table were accurately aligned. The observations were checked by reading the graduations of the lead screw.

10.4.1 REPEATABILITY AND POSITIONING TESTS

The results of the repeatability and positioning tests are summarized in Table 10-2. These measurements were used to determine the repeatability of the transverse feed readings needed to compare the measured and theoretical deflections and the positioning accuracy involving both the operation of the rotary table and transverse feed movements. The standard deviation of the repeatability tests 1 and 2

on the transverse feed are compatible with the recommendations of BS2795 for the dial gauge. A positioning error of the rotary table in tests 3 » 4 and 5 is transmitted through the drive to cause a proportionate error in slide position and hence in the measured deflection of the leaf, although this error is insignificant in use since the action of the trans­mission automatically compensates for the original error. Therefore pitch curve wavyness cannot result. The larger standard deviation for test 5 emphasizes the accuracy and care needed in mounting the leaf and calibrating the zero of the datum indicator when using clamps to mount the leaf.

10.4.2 DEFLECTION MEASUREMENTS

The first series of deflection measurements tested the symmetry of the deflected leaf using equations (1 0 .2) at © (ffi 4 . and& f ~ 4 . The measured deflections were compared with thetheoretical values calculated by numerical integration of the slope .equation

4 • 4 cr ~x ( 1 4 j r )

d B - I6(cr)z [1+ f Y (7 .1 9 )

using the computer program 'DEFLECTION' (Appendix 8). Both the specially hardened and tempered gauge plate 0 .0 16 in thick and standard feeler stock, 0.018 in thick, were used. The results quoted in Table 10-3 and

plotted in Fig. 10-3 show a considerable improvement upon the prelimi­nary tests, the standard deviation for offset-span ratios under 0.3 being 0.001 1 in. (See also Table 10-9.)

This form of the attachment was then used to cut the complete profile for Simple Derived Motion using the procedure described in Appendix 5- The transverse displacement of the mill table, equal to the rise at that cam angle, was measured at 1° increments, the plunge cuts being made at 0° 20X increments. The readings are given in Table 10-4 and plotted in Fig. 10-4. The first differences between successive radii in Table 10-4 are plotted separately in Fig. 10-5 to provide a measure of the dynamic performance. The standard deviation between the theoretical and measured displacements for the portion of the span

complete span (Table 10-9), the major discrepancies occurring in the critical regions close to the limits of span. Consequently the start and finish of the lift would not conform to the theoretical motion.

Although the technician and the author were unfamiliar with the process the only serious difficulty encountered in using this form of the attachment occurred when mounting the leaves. Development of the mounting to eliminate this problem was essential in view of the associated end errors. In addition design data, Fig. 8-11, was prepared to enable check measurements at /- and span to be made before cutting a profile.

10.4.3 CONCLUSIONS

These experiments showed that this principle of profile manufacture is a practical proposition provided the source of error at the limits of span could be identified and eliminated in conjunction with an improved means of mounting the leaf.

between and was 63% of that for the

Unlike the conventional plunge-cutting process described by Cheney (20) only one operator was required. In addition the time needed to cut the profile was shorter and a cheaper machine-tool was used.

-131-

The development was concerned principally with improving the leaf mounting.

10.5.1 INVESTIGATION OF THE END ERRORS

Following the conclusion by Horvay (54) that the end error resulting from the means of applying the bending moment is confined to a length approximately equal to the thickness of the leaf the effects of the clamping force were ignored in the analysis of the curve of deflection (Appendix l). The results of further experiments plotted in Figs. 10-6 and 10-7 confirmed the previous readings. Since the co-relation between the theoretical and measured deflections was best in the central portion of the span where maximum slope occurs these errors could not result from a calibration error. Nor were these readings consistent with the theoretical shape of a deflected leaf subject to large longitudinal forces, Table 7-2.

Therefore the error must be an inherent fault of this method of mounting the leaf.

The forces due to the tightening action take up the clearanceson the moving jaw of the clamp, causing the rotation observed in thepreliminary experiments, page 126. A torque of approximately 20 lbf-fthad to be applied to a ■§■ BSF screw to eliminate the clearance at theinner edges of the jaws. This corresponds to a compressive force of about 3000 lbf (36), vastly in excess of the deflecting force. The distribution of the compressive force is indeterminate, but it must cause the corners of the jaws to deflect and produce complementary stresses in the plane of bending of the leaf. In addition the fric­tional force is sufficient to apply longitudinal forces up to, say,800 lbf corresponding to a direct stress of over 3° tonf/in and a strain of 2.4 x 10*" .

These results concur with Lightenburg's observations (74), page 110, and show that this error must result from the compression of the leaf between the jaws of the clamps and not from a longitudinal force

10.5 DEVELOPMENT OF THE' ATTACHMENT

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because the solution derived by the method of perturbations, equation (7 -30)> shows that the maximum error from this cause occurs around the ?}:- and span positions where co-relation improved.

Therefore the method of deflecting the leaf used by Borun (12) is unsuitable for this application.

10.5.2 RE-DESIGN OF THE LEAF MOUNTING

It had been recognised that Borun's method of deflecting the leaf between offset clamps confines the manufacturing process to profiles having an infinite-pulse characteristic. The improved method of mounting the leaf was also required to suit the configuration needed for the manufacture of profiles for the Finite Pulse Derived Motion, Fig. 8-7« Then the mounting must:-

(a) incorporate several spans of identical length and offset, successive spans having slopes of opposite sign.

(b) enable the limits of every span to be identified accurately.

(c) ensure zero slope relative to the x-axis, Fig. 8-7, at thelimits of every span.

(d) impose the least possible forces upon the leaf.

(e) permit unobstructed access of the probe over successivespans.

These conditions are met by deflecting the leaf in the manner shown diagrammatically in Fig. 10-1(b). It must extend beyond the span AB needed to manufacture the profile controlling a rise movement with Simple Derived Motion to apply additional forces at the outer mid-span positions C and D. All these nominal 'line' contacts must be located accurately

at the specified x-y co-ordinates, parallel to the z-axis, Fig. 4-1. Cylindrical dowels are suitable for positions A and B provided the minimum radius of curvature of the leaf exceeds that of the dowel. Since one dowel would obstruct the probe of the datum indicator it is necessary

-133-

to adopt the sandwich construction shown in Plate 6. This arrangement is possible because the maximum deflecting force has been reduced from 3000 lbf to about 20 lbf.

The deflection of the neutral axis at the outer supports is half the offset. The positioning problem was simplified by using knife-edges in preference to dowels, the load being about 2$ of the rated capacity (95)* This arrangement is' illustrated in Plate 6.

IO.5 .3 DEFLECTION-TESTS ON LEAVES MOUNTED BETWEEN DOWELS AND KNIFE-EDGES

A rectangular steel plate was secured to the slide in place of theclamps. The dowel holes were machined after assembly on the verticalmilling machine to align the span with the slide travel as accurately as possible. Then the centre-distances could be set by turning the rotary table through the angle of lift. The offset of the dowel centres,

is set through the transverse feed. The sides of the plate were machined in position to be accurately parallel to the y-axis and square with the upper and lower surfaces. The knife-edges were attached to these sides using spacers and shims to set the distances CA and BD, Fig. 10-l(b). These dimensions were checked with slip gauges. The correction for half the apparent thickness of the leaf on the y-axis was calculated from the angle of slope and a bar gound to locate the knife-edges relative to the dowels for the required deflection.

It was found that the datum-indicator could be calibrated easily by aligning the probe with the centre-line of a dowel using the procedure described in Appendix 5- The measurements were concentrated on the criti­cal region near the limits of span. Several sets of readings were taken for two offset-span ratios, the leaf being re-positioned or a new leaf inserted every time. The results plotted in Figs. 10-9 and 10-10 show good co-relation between the theoretical and measured values, the standard

-4 -4deviation being 2 .6 x 10 in reducing to 1 . 1 x 10 in between theexperimental results. The tendency towards the high readings suggests the presence of a longitudinal force.

(10.4)

The improved results show that the discrepancies found in the original tests can be attributed to the clamps.

Although care was taken to prevent the leaf rubbing against the knife-edges wear occurred rapidly. Difficulty was experienced in locating the ordinate of the knife-edges more accurately than +0.002 in, although this error did not appear to influence the shape of the central span. Rectangular blocks, hardened and ground before assembly would have been easier to manufacture and locate with improved wear resistance.

10.5.4 FINAL VERSION OF THE LEAF MOUNTING

The manufacture of cam profiles for Finite Pulse Derived Motion requires three successive working spans of the leaf to be deflected with the same span and offset, Fig. 8-7. The method of securing the support plate to the slide and machining the dowel holes in situ has the merits of eliminating assembly errors and ensuring the best possible alignment of the span with the slide travel. It was found easy to set the span by turning the rotary table through the angle of lift which compensates automatically for transmission errors within the repeatability of the rotary table.

The problems associated with the knife-edges were overcome by mounting the leaf between dowels located at the limits of span, providing at least one 'idle1 span at each end. Zero end slope was ensured by inserting ground spacers between locating blocks secured to the mounting plate and the leaf, pressing it against the outermost dowels. This wedging action does not require much force and is immaterial since the preliminary tests showed that the resultant errors are confined to the outer ends of the ’idle' spans. Only one edge of the support plate requires accurate machining to be parallel t.o the slide travel. The machining is further simplified by positioning every dowel along a common centre-line, the offset being obtained by fitting rollers of the appro­priate wall thickness over the dowels, Fig. 10-11 and Plate 7. Free rotation is obtained by making the bore of the rollers exceed the dowel diameter, so the critical dimensions of the rollers are the wall thick­ness and concentricity.

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-135-

This arrangement has the merit of restricting the magnitude of any longitudinal force acting on the leaf to the maximum frictional force possible between parts of the mounting. The reaction force is just that required to deflect the leaf and the limiting condition applies between the bore of the rollers and the dowels. Then the maximum longitudinal force which can act on the leaf is (Fig. 10-8).

which can be made a minimum by lubricating the sliding surfaces. This is essential if the analysis of the curve of deflection,' Appendix 1, is to apply to the mounted leaf.

This method of mounting the leaf is equally suitable for manu­facturing both forms of the derived profile. It was found very easy to use, the reverse bending action making it possible to machine the com­plete profile at one setting if the angles of lift and return are equal.

10.5.5 DEFLECTION TESTS MOUNTING THE LEAVES BETWEEN DOWELS AND ROLLERS

The deflection tests were repeated,' using the same method. The results given in Tables 10-5 and 10-6 are compared with the theoretical curves in the large-scale graphs, Figs. 10-12 and 10-13, for the critical regions close to the end of a span and for a half-span in Figs. 10~l4 and 10-15. The standard deviations for the complete series of tests are given in Table 10-9. Different offsets were obtained by mounting the leaves between plain dowels and by fitting rollers over the dowels. The superior results obtained from the latter test emphasize the importance of ensuring' that the least possible longitudinal force he applied to the leaves.The accuracies are compatible with those obtained from the repeatability test on the machine-tool, Table 10-2, line 3- The standard deviation of the first differences between pitch curve radii over 1° increments are particularly consistent, showing that good control of this source of pitch curve wavyness had been achieved.

10.5.6 CONCLUSION

The dowel and roller method of mounting the leaf imposes the least possible deflecting forces and prevents any significant longitu­dinal forces being applied.

The assembly was easy to set up and use. The setting accuracy was compatible with that of the machine-tool and with the published values for other methods of cam profile manufacture, Table 4-1.

10.6 PROPORTIONS OF THE DEFLECTED LEAF

The proportions of the deflected leaf are defined by the offset- span ratio and are restricted by the limiting bending stress, Fig. 7-4.The least probe thrust error is obtained by using the shortest possible span. This configuration also gives the least possible peak accelerations for Simple Derived Motion, but the lift is restricted by the need to prevent yielding. Consequently longer spans and thicker leaves may be necessary to obtain the required lift. It is impossible to equate the maximum probe thrust and apparent thickness errors for longer spans, but a resultant error at mid-span is not considered serious since the resul­tant secondary acceleration is a continuous function and has a small magnitude. The leaf thickness and machine ratio needed to determine the setting data are found from Figs. 7-4 and 8-11 using the procedure illustrated in Example 1, page

Later experiment showed that the excessive deflection of longer spans due to probe thrust could be prevented by pulling a hard rubber strip backed by a steel plate against the upper surface of the leaf.Square location is assisted by the corrugated shape of the leaf. The limiting frictional force was adequate to prevent a thrust of 2 lbf applied at mid-span causing a measurable deflection on a dial gauge.With this arrangement a small probe tip radius should be used for least tip radius and apparent thickness errors which are not counteracted with this arrangement (Fig. 9-2).

As shown in Example 2, page A71, different conditions apply to Finite Pulse Derived Motion because the peak accelerations are direct

-137-

functions of the maximum angle of slope, Table 8-2, and the contact angle must be confined within an empirical maximum of,' say, 50°-

Standard feeler stock 0.500 in deep by 12 in long having a thickness of at least 0.012 in hardened and tempered to Rockwell C50-52 was found most suitable for the leaves, Table 10-1.

10.7 SUMMARY

It is essential that the leaf be set accurately with the speci­fied span and offset, using a mounting certain to give close agreement with the theoretical curve of deflection. This requirement implies that the loading should equal the deflecting force and that any longitudinal force acting on the leaf does not exceed 10$ of the deflecting force.

The method of mounting the leaf used in the previously published method of manufacturing profiles for the infinite-pulse Simple Derived Motion by Borun (12) was found unsuitable. Experiments showed that the curve of a single span deflected between offset clamps differed signifi­cantly from the theoretical shape in the regions close to the clamps.Since the solution for the deflection of the leaf derived by the author using the method of perturbations showed that the maximum error due to a longitudinal force occurred around the 7$ and span positions this error is attributed to the excessive compressive force of about 3000 lbf. needed to close the clamps completely.

The author obtained good co-relation between the measured and-4.theoretical curves of deflection (standard deviation 2.6 x 10 in) by

mounting a longer leaf between dowels and knife-edges, Fig. 10-l(b), despite inaccurate location and wear of the knife-edges. He developed a dual-purpose mounting suitable for manufacturing both derived profiles by deflecting several spans of equal length between a row of dowels and rollers, controlling the slope at limits of the 'idle1 end spans,Fig, 10-11. Then the radial force between the rollers and leaf along the active span(s) is merely that needed to obtain the required deflection and longitudinal forces are 'controlled within the limiting value by lubricating all sliding surfaces and further reduced by leverage of the rollers.

The leaf thickness is determined from the limiting bending stress, the required offset and the acceptable probe' thrust error, the shortest possible span being used to obtain maximum stiffness, and aiming to equalize the sum of the apparent thickness and probe tip redius errors to the probe thrust error at mid-span. Alternatively the probe thrust can be reacted by re-inforcing the leaf.artificially, clamping it between the support plate and a hard rubber strip backed by a steel plate.

-139-

COMMENTARY AND RECOMMENDATIONS FOR FURTHER RESEARCH

11.1 THE IMPROVEMENT OF PROFILE MANUFACTURE

The quality of the specification and manufacture of the master cam profile is vital for the successful operation of a new cam mechanism. The effort, and consequent cost, involved in profile design and manu­facture using conventional processes deters both experimentation and the modification of existing machines. Considerable, incentive exists for the development of a simpler profile manufacturing process independent of closely spaced co-ordinates or the preparation of a new control tape and having inherent control of pitch curve wavyness.

The survey of generating mechanisms in Chapter 5 showed that none satisfies the design requirements adequately because these devices are sensitive to manufacturing tolerances and difficult to set accurately.In all these designs members are subject to the cutting forces, some even transmitting this load through friction drives. This conclusion is supported by the absence of commercial applications of profile generating mechanisms except the special case of the constant velocity cam for con­trolling automatic lathes. A copying mechanism using a conventional template merely transfers the problem of profile manufacture from the master cam to the template whilst introducing the requirement for step- lessly variable velocity ratios in. both the linear and angular drives.Again no copying device for master cam manufacture satisfying the criterion for continued application was found during the literature search or dis­cussions between the author and manufacturers.

The survey of relative costs in Tables 6-3 and 6-4 shows marginal ' difference between numerical control and a viable copying process, these indices being under half that for conventional plunge-cutting. Whilst

Chapter 11

-140-

superior positioning accuracy is obtained and operator's errors are eliminated with numerical control the profile machined by circular or parabolic arc interpolation is suspect because abrupt changes of curvature, and hence of follower acceleration, occur at every blending point, Fig. 4-6. Of no consequence in most applications of this facility these discontinuities are unacceptable in the special case of cam profiles. The author considers numerical control with linear inter­polation, used by Childs (22) and Oshima (93) j to be superior provided the master cam profile is subsequently honed to remove the edges left by the linear movements of the cutter although no previous author mentions any smoothing operation. The need to finish the profile makes this form of numerical control unsuitable for quantity production, more­over it is cheaper to make the production items on a special-purpose copying machine once the master cam has been made. The author found several cam manufacturers in agreement with Astrop (5) who rejected numerical control in favour of the manual plunge-cutting and honing process, following the disappointing performance of cams cut on numeri­cally-controlled machine-tools.

The copying principle was considered to provide the best basis for developing an improved method of profile manufacture because the template and transmission need only be used to determine the machine setting, the cutting forces being reacted entirely through the structure of the machine-tool. Then it is feasible to replace the template with a deflected flexible member, using the intrinsic curve of deflection instead of the manufactured profile. The shape must be predictable and repeatable, satisfying the boundary conditions, Fig. 6-2(a),

01 = °» 1 - 0; i t = °' 7* - X ; M - Y j % m o , I ^

requiring maximum stiffness of the flexible member and least possible reaction force between it and the tracer. Since the span and offset are set to suit the application coarse steps of velocity ratio in the transmission are acceptable. The author showed that profiles for Finite Pulse Derived Motion (resembling cycloidal motion) can be manufactured by this process in addition to those for Simple Derived Motion (similar to SHM) described by Borun (12).

-Infi­

The author found no previous comparison of the various analyses of the deflection of flexible members in the elastica range, nor any experimental verification of the theoretical solutions apart from the approximate measurements reported by Hummell and Morton (56). Conse­quently the previous papers contain no recommendations for suitable materials. With the exceptions of Conway (22), Mitchell (82) and some work by Frisch-Fay (39) every contemporary author analysed a cantilever, neglecting longitudinal forces. Only Mitchell included a direct solution for the deflection at any intermediate position along the span (involving functions of elliptic integrals).

The techniques used to derive previous solutions and the associated assumptions are summarized in Fig. 7-1 and Table 7-1, Numerical solution of equation (7 .1 0 ) based upon the approximate theory of bending showed that longitudinal forces exceeding 10$ of the deflecting force have a significant effect upon the theoretical curve of the neutral axis.Other investigations showed the assumption of pure bending and the neglect of deflection due to shear to be valid.

To make a more rigorous analysis-of the significance of longitu­dinal forces and to obtain a result independent of transcendental functions the author adopted a fresh approach, using the method of perturbations to derive new solutions for the slope and deflection of a leaf spring subject to large curvature from the Euler-Bernoulli Law, equation (7-3). Separate solutions for the configuration illustrated in Fig. 7-2 were derived in Appendix 1 . 1 to obtain equations (7*29) and (7-30) by respectively neg­lecting and including the logitudinal end forces. Whilst these equations contain many terms, the limit of accuracy depending upon the highest power of the perturbation parameter included, in the series expansion, the computation is much simpler than that involved in the solution of elliptic integrals. Consequently these calculations, including the iteration to find the maximum angle of slope corresponding to a given offset-span ratio, are within the capacity of a programmable calculator.The deflection at any intermediate position can be calculated directly for given loads.

ll.2 ANALYSES OF THE ELASTICA

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The specification of the deflected leaf needed to derive general equations of the cam characteristics is the converse of that used in most engineering applications. In this special case the forces, flexural rigidity and leaf dimensions must be replaced by functions of the lift, angle of lift and machine ratio. Provided the longitudinal forces are negligible these requirements are met by

F W - F l ^ e , ) -2]i e c m - f M . ) ]i[2 J Vh

where

X , x

T" = StK

V ft 0 +g=)P

(7 .1 6 )

derived by Bisshopp and Drucker (ll). This relationship between the offset-span ratio and the angle of maximum slope was chosen as the basis for the cam characteristics because it is valid for any value of . The disadvantages associated with this solution are:-

(a) the intermediate co-ordinates can only be calculated by numerical integration of the slope, equation (7.19). This is insignificant for the general application of the principle as graphical design data has been pre­pared to check any leaf setting for either motion,Fig. 8-6, from Table 8-6.

(b) the equations for the cam characteristics and limiting values are functions of S' . Equation (7.16) is un­suitable for calculating this angle from a given offset- span ratio and tables of elliptic integrals were found to have inadequate accuracy. Therefore the computer program ’LEAFSLOPE’ incorporating procedures for cal­culating these functions (53) was used to find the offset-span ratio for small increments of the parameter

In practice the maximum offset and leaf thickness for a given span are limited by the yield stress, Fig. 7-4. The curves of slope vs offset-span ratio drawn in Fig. 7-6 compare the author’s solution

derived by the method of perturbations for zero longitudinal force, equation (7.29), with the elliptic integrals solution, equation (7 .1 6 )

which covers most of the practical range. Therefore the cam character­istics could also be expressed as functions and derivatives of equation (7*29). Solutions valid for larger values of <T can be obtained by extending the series expansion to the next power of the perturbation parameter. The corresponding curve for the series expansion of the elliptic integrals solution (3)

is less accurate than the solution for the first power of the pertur­bation parameter, Fig. 7-6. The merit of simplicity is insignificant in computing so this solution was rejected.

The geometric derivation of the cam characteristics for Finite Pulse Derived Motion (Appendix 3 ) is incompatible with the elliptic integrals solution for the deflected leaf spring, equation (7»l6), making it impossible to determine the follower displacement at a speci­fied cam angle by this method. Consequently the indirect approach of calculating both the cam angle and the displacement from increments of the equivalent range of lift , Fig. 8-7, was used in the programs •FPDM-DISPL* and ’FPDM-CHAR' (Appendix 8). The simpler computations associated with the solutions for the deflection derived by the method of perturbations (Appendix l.l) have the further advantage that an i terative solution to the problem of calculating the follower displace­ments, and hence the velocity and acceleration, at regular increments of cam angle becomes possible for the finite-pulse motion.

11.3 LEAF MATERIALS

The choice of leaf material is restricted by the requirements for accurate dimensions of cross-section, a high yield stress and commercial availability. The first is met by gauge plate and feeler stock. Only feeler stock is commercially available in the hardened and tempered

to show close agreement for values of 0“ smaller

(7.17)

-144-

condition, the author experienced considerable difficulty in obtaining gauge plate which had been heat treated to the manufacturer's specifi­cation. Feeler stock has the advantage of being available in fine increments of thickness, although the maximum thickness of 0.025 in may be inadequate for long spans. Careful distinction between the compo­sitions and heat treatments of feeler stock supplied by different manu­facturers is advised, the specification for the material found most satisfactory by the author is given in Table 10-1.

11.4 THE LEAF MOUNTING

The theoretical characteristics for Simple Derived Motion are obtained from the curve of one complete span, Fig. 6-2(a), by deflecting the leaf to make the offset equal the lift and the span directly propor­tional to the angle of lift. The leaf mounting must satisfy the boundary conditions, equations (6.1), and ensure that the longitudinal force is within the limiting value. The leaf setting for Finite Pulse Derived Motion, Fig. 6-2(b),is similar. The only previous publication referring to this method of profile manufacture known to the author is a brief paper by Borun (12) describing the derivation of Simple Derived Motion from a leaf deflected between offset clamps. The author found this arrangement unsatisfactory, moreover it-cannot be used for the finite- pulse motion. Although the offset could be obtained easily by inserting ground spacers it was difficult to set the span accurately, to calibrate the datum indicator and to mount the leaf in the clamps. The force required to close the jaws completely was many times the theoretical deflecting force, making it impossible to confine the longitudinal force within the prescribed limit. However the largest discrepancies between the measured and theoretical deflections occurred in the critical regionsclose to the ends of the span, not around the and span positions

-4 .predicted by the analysis. The standard deviations were 11.9 x 10 m-4for the complete span and 6.9 x 10 in for the central portion, Table

10-9* The error was similar at both ends, which is incompatible with an angular misalignment. It is attributed to the large compressive force on the leaf and corners of the jaws.

The author overcame the problems associated with the leaf mounting by developing the assembly shown in Fig. 10-11 and Plate 7 which is equally .suitable for both derived motions. The superior correlation between the theoretical and measured deflections using the improved mounting is shown by the reduction in the standard

. -4deviation over the complete span to 4.7 x 10 in. The difference was larger around the -zj:- span position for leaves mounted betweenIfAbfia- (Q-'S1 Ikjxxi d-oo0-s2$plain dowels}{and rollers, Table 10-6, demonstrating the importance of limiting the longitudinal force by fitting rollers over the dowels and lubricating all sliding surfaces. The critical tolerance of0.0002 in/degree specified by Nourse (89) was met by 79% of the first differences between deflections, Tables 10-7 and 10-8, although these readings, unlike the machine settings for profile derivation, are subject to both round-off errors and the positioning error of the rotary table. Therefore the actual machine settings should have superior accuracy*-

The thickness can be selected for short spans to equalize the additional deflection of a freely mounted leaf due to probe thrust with the opposing cumulative error caused by the apparent increase in leaf thickness and the probe tip radius, Fig. 9-6, but the limiting bending stress prevents this approach being used for lifts exceeding0.5 in (Fig. 7-4). This is a serious restriction preventing the general application of the principle, so it is advised that the leaf be externally reinforced by holding it between the mounting plate and another plate secured to the dowels. The limiting friction force caused by a force of 3 to 5 lbf acting perpendicular to the plane of bending along the edges of the leaf would be adequate to react a probe thrust of 1 lbf. Precautions to prevent this restraint affecting the curve of deflection or impeding the threading action are essential.With this arrangement the apparent thickness and probe tip radius errors would be superimposed upon the theoretical displacement characteristic.It is argued that these errors are insignificant because both are con­tinuous functions of the contact angle equation (9.5)• Therefore they are not sources of random secondary accelerations. The maximum displace­ment error at mid-span is about 0*003 in, so the difference between the maximum acceleration and retardation is negligible.

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The unit costs of the different methods of profile manufacture estimated in Table 6-2 compare profile derivation favourably with alternative processes because the time required for detail design and manufacture is 30$ less than that needed for conventional plunge- cutting and a cheaper machine-tool can be used without loss of accuracy. It is recommended that the cuts he made at nominally equal angular increments for ease of operation, although precision setting is no longer essential because the pitch curve radius is determined auto­matically from the angle of cam rotation. Unlike the other manual processes the operator is not required to read and set a series of random dimensions, instead the transverse feed is adjusted until the dial gauge reading is restored to the same graduation. Therefore the speed of cutting is increased and the probability of error reduced. Profile derivation has the additional advantage of inherently superior accuracy because the critical differences between adjacent radii depend upon the repeatability of one instrument and not the positioning accuracies of two independent feeds. The process only involves the spindle and transverse table feeds so a vertical milling machine (a 'universal' machine is essential for Finite Pulse Derived Motion) can be used instead of a jig-borer (relative capital cost 1:4). Assuming the machine-tool and rotary table exist the capital cost is confined to the manufacture of the attachment and the preparation of design data.

In comparison with both conventional plunge-cutting and numerical control profile derivation requires a longer time to prepare the machine tool.

Apart from replacing the clamps with the dowel and roller assembly the original design of the attachment, Plates 2 to 5, proved satisfactory. The gear transmission is essential to obtain the long slide travel associated with the dowel and roller mounting. This arrangement has the advantage of requiring less height than the alternative flexible band transmission, Fig. 9-8, because it is only necessary to assemble the gears to suit the specified machine ratio(s). The use of plastics bearings to suppress stick-slip action during low-speed indexing movements

11.5 ASSESSMENT OF PROFILE DERIVATION

of the slide proved successful. It is recommended that these bearings be used also for the layshaft.

No modification to the machine-tool was necessary.

The span determined by turning the rotary table through the angle of lift is subject to the transmission errors and positioning accuracy of the rotary table, but the end positions can be restored within the repeatability of the rotary table. The maximum cumulative error from these causes, plus the misalignment of the datum indicator, is estimated at 0° 10X compared with a tolerance of, say, +0° 301 on the angle of lift. Besides meeting this tolerance it is essential that the requirements for zero slope be satisfied. The repeatability of the end positions is particularly important since it affects the pitch curve radius directly, but in practice the slope of the leaf is so small that a slide position error of +0.002 in is insignificant. Random transmis­sion errors co-incident with the transition point have greatest signifi­cance because of the large slope. Hence the error in the difference between adjacent radii due to slide position error can exceed 0.001 in at this position, Table A7-2. For maximum accuracy class A2 precision cut gears having a relatively fine pitch were specified to ensure the simultaneous mesh of several pairs of teeth, the bias weight maintaining contact between the same faces. The slide travel was checked against the angular position of the rotary table.

11.6 CONTINUOUS CUTTING

The obvious extension of the present investigation is the replace­ment of the increment cutting process by a continuous profiling operation which would also eliminate the honing process. (A similar machine could be developed for profile grinding.) The force of 7 kgf (15-5 lbf) required to actuate the tracer on the modern SIG hydraulically-controlled cam copying machine (125) would be excessive even with additional restraint for the leaf, Borun (12) quotes the more reasonable range of 0.5 to 1.0 kgf. (l to 2.2 lbf) for a similar application. The control system must be capable of accurately reproducing a continuously varying curvature, includ­ing points of contraflexure, with the least possible wavyness, so continuous

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cutting requires additional expensive equipment and a more sophisti­cated machine-tool. A rolling element is essential to ensure free movement of the tracer along the leaf, the radius must be as small as possible to minimize the inevitable asymmetry of the displacement derived by this means.

The reduction in machining time, but not setting time, must be balanced against the increased capital cost. The continuous cutting process would be most suitable for the non-repetitive manufacture of small batches since no master cam. would be needed.

11.7 ASSESSMENT OF THE DERIVED MOTIONS

It is easiest to set the attachment for machining the profile for 'symmetrical' Simple Derived Motion because the slide travel is aligned with the longitudinal feed and the shape is copied from one span of the deflected leaf. This motion has characteristics similar to Simple Harmonic Motion, there is little difference between the maximum velocities (dimensionless forms), but the peak accelerations of the derived motion are about 10$ greater. Both are functions of the parameter (f , it is evident from the graph of the acceleration characteristic (Fig. 8-4), that the machine ratio should be chosen for maximum CT compatible with the yield stress to obtain the least acceleration for a specified lift and angle of lift. The pulse becomes infinite at both ends of a dwell-rise-dwell event, so Simple Derived Motion is subject to the same restrictions of 'moderate' load and speed recommended for Simple Harmonic Motion by Molian (85) and Rothbart (97)5 although the author has argued that these limits are difficult to quantify (26). Both these motions have the advantages associated with small pressure angles. ■

There are two possible methods for deriving 'asymmetric' profiles, Fig. 6-1, although the ratio of cam angles for acceleration and retar­dation is restricted in both cases. The method followed in Example 2 (page A 71 ) requires the leaf to be reset when the tracer reaches the transition point. It is essential that the value of cr be 'identical' for both settings to blend the slope smoothly, therefore the probe thrust errors must be equal, effectively limiting the ratio of span

lengths to about 3 ;1. If different leaf thicknesses are used the difference in apparent thickness errors is reproduced in the lift, corresponding to about 15 - 30$ of the tolerance, unless both set­tings are compensated to allow for this predictable error. A more satisfactory solution would be to react the probe thrust by re-inforcing the leaf. Then the same leaf could be used for both settings, equalising the apparent thickness and probe tip radius errors at the transition point. The ratio of the cam angles of accelerating and retarding the follower must be an integer to suit the arrangement of dowels along a common centre-line.

External restraint for the leaf also permits the use of a heavier probe.’ Then the shape of the leaf can be traced through a roller, exaggerating the probe tip radius error to derive an 'asymmetric' profile without resetting the leaf. The roller radius cannot exceed the least radius of leaf curvature, limiting the pro­portions of the motion to about 2:1, but without any other restriction.

The setting instructions and procedure for Finite Pulse Derived Motion are more involved and a ’universal’ milling machine is essential, combining to increase the cost of the process. The velocity and acceleration characteristics are drawn for a special case in Figs. 8-9 and 8-10 because the span can neither be eliminated nor replaced with a parameter determined from the offset-span ratio in equations (8.19), (8.23) and (8.27), Table 8-3. These curves resemble those for cycloidal motion, the simpler expressions for the dimensionless forms of the maximum velocity and acceleration, Table 8-2, give values approximately 20$ and f>0% greater for the derived motion. Abrupt changes of slope occur at the maximum and minimum of the acceleration characteristic,Fig. 8-10, indicating finite discontinuities of pulse at these posi­tions. Such discontinuities occur only at positions of zero accelera­tion with the commonly-used cycloidal and modified trapezoidal accelera­tion motions, but do happen at the peak accelerations of trapezoidal motion considered "excellent for high speeds" by both ESDU (33) and Rothbart (97).

Ramps cannot be produced by this process. According to Molian (85) these are associated mostly with the specialized requirements of automotive cams whereas the derivation process was conceived primarily

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for machinery cams maintaining continuous contact between the profile and follower. -

Assessment of the maximum accelerations should he made in con­junction with the improvements in pitch curve accuracy inherent in the derivation process.

11.8 ASSOCIATED INVESTIGATIONS

Subjects for further research identified during the progress of this investigation were discussed in two conference papers presented by the author (26) (27). The objects of these investigations and appli­cations of the results are summarized in Table 1 1 -1 .

11.8.1 PROFILE ACCURACY

Theoretical analyses of mathematically perfect profiles and mechanisms demonstrate the significance of finite pulse characteristics to ensure good dynamic performance at high speed. In addition to impact, with the consequent vibration and wear, the author has shown that infinite pulse at the transition point causes inevitable roller slip (28) and an abrupt reversal of inertia torque (25). For these reasons, ignoring practical considerations, finite-pulse cam laws are generally recommended for high-speed operation.

The difficulty of achieving the required quality of motion inpractice is demonstrated by comparing the empirical tolerance on pitchcurve wavyness (2 x 10 4 in/degree) quoted by Nourse (89) with the minutedifferences between radii at increments of 5r° or 1 ° in the critical

-5regions of low follower velocity, e.g. 1 .x 10 in for cycloidal motion (Table 3“2). Consequently the tolerance zones for finite and infinite pulse motions overlap, Fig. 3-3* Negligible advantage results from en­larging the master cam to, say, four times full size. Again, the absolute limit of positioning accuracy of a jig-borer is an order of magnitude larger than these differences, so the pitch curve wavyness due to round-off error alone can approach the tolerance. Assuming rigid components the finite differences analysis showed this wavyness to be a source of large secondary accelerations, Fig. 3-6.

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Therefore the disparity between the finite pulse and some infinite pulse cam laws is not as distinct as the theoretical analyses suggest. In practice the elasticity of the copying machine used to make the production items and the components of the mechanism itself reduce the severity of the secondary accelerations, Beard (9) has suggested that the honing operation used to finish plunge-cut profiles has the same effect. The load applied to the master cam during the copying process is inadequate to burnish the profile through repeated use, but this process could be another means of smoothing the profile in addition to, or replacing, the honing operation.

The author considers that further investigations into the accu­racy attainable by different manufacturing processes, the significance of profile errors and a quantitative assessment of different cam laws are particularly important for the improvement of cam performance. The fundamental criterion is follower acceleration, so this study must involve the whole mechanism. During the discussion on the first of the author's papers on this subject (26) Professor Meyer recommended the investigation be extended to include follower mass and the spring characteristics as additional variables.

11.8.2 DESIGN PROCEDURE

The literature survey revealed a multitude of papers concerned with particular aspects of cam design, but little attempt to co-ordinate the work even in the preparation of design data. A flow diagram illus­trating the design sequence for a new cam mechanism prepared by the author (27) has been developed into the forms shown in Figs. 2-1 and 5-1. The procedure for using design data to avoid major iterations is shown similarly in Fig. 2-2, making feed-back loops redundant in the main dia­grams. An extended study of the design sequence for the complete mechanism and associated components is an essential preparation for the development of comprehensive computer programs capable of determining the basic dimensions and pitch curve co-ordinates to suit a given con­figuration, cam law and limiting conditions.

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Associated investigations have been concerned with the analysis of the angular motion of roller followers and the significance of.off­setting reciprocating followers.

General equations have been established for the angular velocity, acceleration and pulse of the roller (28), It is intended to extend this analysis in conjunction with the author's study of driving torque (2 5) to estimate the limiting condition for roller slip.

Fenton (38) and Kloomok & Muffley (68) have shown that the maxi­mum pressure angle during an event, and hence the force normal to the profile, can be reduced by offsetting a reciprocating follower, but the effect upon the remainder of the motion has been neglected. Nor is any analysis of the forces and torques for this configuration known to the author•

11.8.4 NUMERICALLY-CONTROLLED PROFILE MANUFACTURE

Profile cutting on a numerically-controlled machine-tool using the continuous path facility with parabolic arc interpolation requires the least number of blending points per unit length for a given maximum tolerance between the interpolated and true curves (92). The arcs are symmetrical about the foci, having equal slopes at the blending points but abrupt changes of curvature. Improving this facility to suit the special requirements of cam manufacture involves the development of the computer program used for preparing the control tape to blend higher derivatives. Equalizing the curvature eliminates the instantaneous changes of follower acceleration at the blending points, but abrupt changes of pulse would remain. The ideal program would ensure that both the curvature and rate of change of curvature of the interpolated pitch curve are continuous throughout the event.

It is doubtful whether master cams are manufactured in sufficient quantity to justify this work commerically, the literature survey showed a definite preference for the simpler linear interpolation.

•11.8.3 ANGULAR MOTION AND OFFSET OF THE ROLLER

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The solution for the deflection of the leaf spring with large curvature and different systems of loading was derived readily for this configuration using the method of perturbations. Reference to Fig. 7-6 suggests that the accuracy of the results obtained in Appendix 1.1 would be improved by including higher powers of the per­turbation parameter in the series expansions. Similar expressions can be derived for other configurations and systems of loading. The ease of solving these equations using a programmable calculator facilitates more extensive experimentation to test these theoretical results.

11.9 CONCLUSION

The technique of deriving cam profiles from the intrinsic shape of a deflected leaf spring satisfies the purpose of this investigation by simplifying both the design and manufacturing processes. It is applied to a conventional machine-tool and has inherent control of pitch curve wavyness. Both the technician and the author found the ■ process easy to use.

11.10 SUMMARY

The objects of developing an improved method of master cam pro­file manufacture are to reduce the cost and time required by simplifying both the design procedure and the machine-tool operation whilst improv­ing the inherent accuracy of the process.

Neither generating nor copying mechanisms have met the criterion of continued commercial application due to:-

(a) difficulties in setting the device.

(b) sensitivity to manufacturing tolerances.

(c) unsuitability for transmitting the cutting force.

Conventional copying machines transfer the problem of profile manufacture from the master cam to the template. It was overcome in this invesgitation

11.8.5 ANALYSIS OF THE ELASTICA

by altering the method of operation, only using the template to deter­mine the setting. Then the tracer thrust can be reduced significantly, permitting the replacement of the profiled template by the intrinsic shape of a deflected leaf spring. Mounting the leaf to suit the lift and cam angle simplifies the transmission.

The author found no previous comparison nor a rigorous experi­mental verification of the theoretical analyses of the elastica. He showed good correlation between the solutions obtained by evaluation of transcendental functions and a finite-element technique supporting the assumption of pure bending. After demonstrating the significance of longitudinal forces upon the curve of the neutral axis he used the method of perturbations to derive a new solution for the deflection and slope of a leaf spring subject to large curvature with deflecting and longitudinal loads. Unlike previous solutions the deflection at any intermediate position can be found directly.

The cam characteristics were determined from the elliptic integrals solution derived by Bisshopp and Drucker (ll) because it is valid for any offset-span ratio although intermediate deflections can only be found by numerical integration of the slope. The author's solution has the merit of simplicity, but is inaccurate for maximum slopes exceeding 24°. An accurate solution for the practical range determined by the yield stress can be obtained by extending the series expansion to the third power of the perturbation parameter.

The author identified the means of deriving the cam profile for a finite-pulse motion. He knows of only one previous publication mentioning this principle in which Borun (12) refers only to the simpler infinite-pulse motion derived from a single span.

Borun's method of deflecting the leaf cannot be used for the dynamically superior finite-pulse motion. The author found it required an excessive clamping force and encountered difficulties in setting the span and calibrating the datum indicator. He overcame these faults by developing the dowel and roller assembly which also restricts the longi­tudinal force. It is subject to apparent thickness and probe tip radius errors which counteract the probe thrust error. None has significant

-155-

effect upon the follower motion because they are continuous functions, but -the large thrust error of a freely mounted leaf for offsets exceed- ing ^ in presents a serious restriction. Therefore it is advised that the leaf be re-inforced by pulling a plate against the top surface.

The tolerance for pitch curve wavyness was satisfied by 79$ of the first differences between the measured deflections, Tables 10-7 and 10-8. Unlike the machine settings for profile derivation the measured deflections are subject to'both round-off and positioning errors. There­fore the pitch curve should have superior accuracy. The results emphasized the importance of limiting the longitudinal force and lubri­cating the surfaces.

Simple Derived Motion is similar to Simple Harmonic Motion. It is restricted to "moderate" speeds and loads because the pulse is infinite at positions of zero follower velocity. The peak accelerations of the derived motion are about 10$ greater. Asymmetric motions can be derived either- by resetting the leaf at the transition point or by trac­ing the leaf through a roller. Practical restraints limit the ratio of cam angles in both cases.

Finite Pulse Derived Motion resembles cycloidal motion, but abrupt changes of pulse occur at the maxima and minima of the follower acceleration. This is also a feature of trapezoidal' acceleration motion considered "excellent for high speeds" (35) (97)-

The derived profiles cannot include ramps.

Although subject to transmission, calibration and alignment errors the process gives inherently superior control of profile wavyness com­pared with conventional increment cutting.

-156-

Associated research investigations identified during the investi­gations are:-

(d) to develop the derivation principle into continuous profile cutting.

(e) to.investigate the accuracy of different profile manu­facturing processes, the significance of profile errors and the limiting performance of different cam laws.

(f) to develop a systematic design procedure for the com­plete cam mechanism.

(g) to estimate the limiting condition for roller slip.

(h) to determine the significance of follower offset uponthe complete motion.

(j) to extend the numerically-controlled continuous pathfacility to blend the curvatures of adjoining parabolic arcs.

(k) to apply the method of perturbations to other systemsof loading and configurations of the leaf spring, andto test these results experimentally.

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1 ANDERSON, D. G.

2 ASHWELL, D.'G.

3 ARSCOTT, F. M.

4 A.S.M.E.

5 ASTROP, A. W.

6 BARTEN, H. J.

7 BAUMGARTEN, J. R.

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67 KHAIMOVICH, E. M.

68 KLOOMOK, M.MUFFLEY, R. V.

69 KLOOMOK, M.MUFFLEY, R. V.

"Precision machines assure cam accuracy." Iron Agev!73, n!5, 15 Apr. 1954, p.l40!42.

"Cam design and manufacture."(Industrial Press, N.Y.), 1965-

Canadian Westinghouse Co. Ltd.Personal communication to the author.

"How profile errors affect cam dynamics." Machine Designv.29, n.3, 7 Feb. 1957, p ! 05-108.

"Engineering precision measurements." (Chapman and Hall), 1957-

"Engineering considerations of stress, strain and strength."(McGraw-Hill), 1967-

"Evaluation and design of machinery primarily used in the manufacture of boots and shoes."Proc.I.Mech.E.v ! 7 8, pt!, n.24, 1963-64, p.625-683.

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"Plate cam design. Pressure angle analysis."Product Engineeringv.26, n.5, Hay 1955, p.155-160.

"Plate cam design. Radius of Curvature."Product Engineeringv.26, n.9, Sept. 1955, p.186-192.

KLOOMOK, M. MUFFLEY, R. V.

KNIGHT, K. R.

KOENIGSBERGER, F.

LANDWEHR, R. GRABERT, G.

LIGHTENBERG, F. K.

LONGSTREET, J. R.

LOVE, A. E. H.

MARTIN, D. J. L.

MAW, N. SMITH, M. R, WALDRON, P.

"Plate can design. Design of cam followers.".Product Engineeringv.27, n.9, Sept. 1956, p.19 7-20 1.

Procedure No. DPI (Simpson's Rule). Dept, of Physics, University of Surrey.

"Design principles of metal-cutting machine-tools."(Pergamon), 1964.

"Interferenzoptische Versuche zur Plattenhiegung."Ingenieurarchiv■ v.XVIII, 1950, heft.1.

"The Moire method - a new experimental method for the determination of moments in small slab models."Proc.Society for Expt'I Stress Analysis v.1 2 , n.2, 1955, p.83-98.

"Systematic corelation of motion." Machine Designv.25, n.12, Dec. 1953, p.215~220l

"Mathematical theory of elasticity." ’(Cambridge), 1892.

"The deflected leaf spring cam law." University of Surrey Final Year Project 1968.

"Register of digital computer programs on mechanisms."Advances in Mechanism

MERRITT, H. E. "Gear engineering.” (Pitman), 1971.

MEYER ZUR CAPELLEN, W.

MICHALEC, G. W.

MITCHELL, D. B.

MITCHELL, D. C. BURKE, A. E.

MITCHELL, T. P.

MOLIAN, S.

MORRISON, R. A.

"Kinematics - a survey in retrospect and prospect."J.Mechanismsv.1 , n.3-4, 1966, p.2 1 1-228.

"Precision gearing."(Wiley), 1966.

"Tests on dynamic response of cam-follower systems."Mechanical Engineeringv.72, June 1950, p.467-471

Dec. 1950, p.1009-1011.

"P.T.F.E. impregnated dry bearings."Engineers 1 Digestv.l6 , n.2, Feb. 1955, p.53-58.

"The non-linear bending of thin rods." Trans.A.S.M.E.Ser.E, v.8l, 1959, p.40-43.

"The design of cam mechanisms and link­ages."(Constable), 1968.

"Load/life curves for gear and cam materials."Machine Design,v.40, n.l8, 1 Aug. 1968, p.102-108.

NEKLUTIN, C. N. "Springs for cam followers." Machine Designv.33, n.25, 7 Dec. 1961, p.195-200.

\

! 67-

88 NORTON, H. N.

89 NOURSE, J. H.

90 NUTBOURNE, A. W. MORRIS, R. B. HOLLINS, C. M.

91 NUTBOURNE, A. W.

92 OLESTEN, N.O.

93 ' OSHIMA, T.

94 REEVE, J. E.

95 ROARK, R. J.

96 ROGERS, M. D.SCHAFFER, R. R.

"Handbook of tranducers for electronic measuring systems."(Prentice-Hall), 1969-

"Recent developments in cam profile measurement and evaluation."S.A.E. Paper No. 964A, 1965-

"A cubic spline package: Part 1.Users’ Guide."Computer-Aided Design •v.4, n.5, Oct. 1972, p.228-238.

"A cubic spline package: Part 2.The Mathematics."Computer-Aided Designv.5, n.l, Jan. 1973? P-7-13*

"Numerical control."(Wiley), 1970.

"Cam profile machining by. n-c milling machine^"Rev. Electrical Communication Lab. Tokyo v.l8, n.3-4, Mar-Apr. 1970, p.226-234.

"Cam indexing mechanisms."OEM Designv.l, n.l, Oct. 1971, p.62-66.

"Formulas for stress and strain." (McGraw-Hill), 19 6 5.

"Kinematic effects of cam profile errors." A.S.M.E. Paper No. 63-WA-269, 1963-

97 ROTHBART, H. A. "Cams-design, dynamics and accuracy." (Wiley), 1956.

-168-

98 ROTHBART, H. A.

99 ROTHBART, H. A.

100 ROTHBART, H. A.

101 ROTHBART, H. A.

102 SCHACHTE, J. J.

103 SEAMES, A. E. CONWAY, M. D.

104 SIM, R. M.

105 SMITH, J. 0. LIU, C. K.

"Cam dynamics of high-speed systems." Machine Designv.28, n.5, 8 Mar. 1956, p.100-107-

"Basic cam systems."Machine Designv.28, n.ll, 31 May 1956, p.133-136-

"Limitations on cam pressure angle."Product Engineeringv.28, n.l, Jan. 1957, p.193-195-

"Which way to make a cam?"Product Engineeringv.29, n.9, 3 Mar. 1958, p.68-71.

"Accuracy of machine-tools."American Machinistv.lll, n.7, 27 Mar. 19 6 7? p.163-170.

"A: numerical procedure for calculating the large deflections of straight and curved beams."Trans.A.S.M.E.v.79, Ser.E, 1957, p-289-294.

"A comparison of two surface-fitting programs for numerically-controlled machine-tools."N.E.L. Report No. 211, 1966-

"Stresses due to tangental and normal loads on an elastic solid with applica­tion to some surface stress problems." J.Appl.Mechanics v.20, n.2, June 1953, p.157-166.

-169-

107 TIMOSHENKO, S.

108 TOBIAS, S. A.

109 TUPLIN, W. A.

110 WARRINER, D.

111 WELBOURN, D. B. SMITH, J. D.

112 WHITTAKER, E. T. WATSON, G. N.

113 WIGAN, E. R.

114 WILDT, P.

115 WILDT, P.

.106 STANTON, R. F. V. "Making cams for automatic cigar and cigarette machinery."Machinery(L)v.79, n.2016, 5 July 1951, p.15-19.

"Strength of materials."(Van Nostrand), 1958.

"Design of small isolator units for the supression of low-frequency vibration."J.Mech.Eng.Sci.v.l, n.3, Dec. 1959, p.280-292.

"Gear Design."(Machinery Pub. Co.), 1962.

"The presentation of design data with particular reference to the angular acceleration of cam rollers."University of Surrey, Final Year Project, 1973.

"Aspects of shaping, hobbing and shav­ing machines for medium-sized gears."I.Mech.E. Conference on Gearing, 1970, p.1-9.

"A course of modern analysis."(Cambridge U.P.), 1927.

"The sine spring."S.R.D.E. Report No. 1029, 1949*

"Triebkurve."German Patent No. DP 637 037, 1934.

"Zwanglaufige Triebkurvenherstellung."VDI - Tagungshaft 1, 1953, p.11-20.

-170-

116 WILKINSON, D. G. "Cam manufacture using 2C,L."(LESLIE, W. H. P. (Ed)) "Numerical control-programming

languages."(North Holland Pub. Co.) 1970 p.23-52.

117 WITTRICK, W. H. "The theory of symmetrical crossedflexure pivots."Australian J. of Sci. Research Ser.A, v!, n.2, 1948, p!21-134.

118 "Electronically-controlled cam mill-i ing machine."

(Research Engineers Ltd.)Machinery(L)v.84, n.2l49, 22 Jan. 1954, p!93~196.

119 "Producing high-accuracy cams."(Lawrence Scott Co. Ltd.)Machinery(L)v .89, n.2278, July 13, 1956, p.100-112

'n.2279, July 20, 1956, p .156-162.

120 "Metrological instruments."(Carl Zeiss, Jena), 1962.

121 "Cutting cams as they are used."American Machinistv!07, n.9,, 29 Apr. 1963, p.47-49.

122 "Digital readout aids cam milling."(Schleght L. G. & Son. Inc.)Tooling and Productionv .34, n.5, Aug. 1968, p.91-92.

123 ' "Kopp type EKF 35 N/C cam millingmachine."Machinery(L)v.ll4, n.2953, l8 June 1969, p.976.

-171-

124 . "Kollmann n-c cam milling machine."Machinery(L)v.115, n.2980, 24 Dec. 1969, p.1014.

125 S.I.G. Swiss Industrial Co. (U.K.) Ltd,Machine Tool Division. •Private Communication to the author17 Dec. 1973.

SUPPORTING PAPERS

DRUCE, G. "The electro-mechanical control of mechanisms." Controlv.12, n.115, Jan. 1968, p.27-4l.

DRUCE, G, "The design and performance of cam mechanisms." Paper read to the Borough Polytechnic Conference on Mechanisms, London, 1 July 19&9•

DRUCE, G. "Cams - the case for the triple harmonic pro­file."Machine Design and Control v.7» n.6, June 1969, p.36-39-

DRUCE, G, "Cam torques compared." Machine Design and Control v„8, n„3, Mar. 1970, p.22-25-

DRUCE, G. "Speed categories for cam mechanisms."Paper read to the Symposia on Mechanisms Cranfield Institute of Technology, l4 July 1970 published in Advances in Mechanism v.l, 1970.

DRUCE, G. "Research in cam mechanisms."Paper read to I.Mech.E. Discussion Meeting on 'Mechanisms", London, 27 May 1971-

DRUCE, G. STRIDE, F.

"A survey of devices for the generation of cam profiles." ■Paper read to I.Mech.E. Conference on 'Cams & cam mechanisms', Liverpool,10-11 Sept. 1974.

DRUCE,- G. HALTON, R. WARRINER, D.

"The rotary motion of roller cam followers." Ibid

DRUCE, G.

-173-

Technical File No. 25- "Cams and followers,11 Engineeringv .2 1 6 , n.l, Jan. 19 7 6 , pplO.

A P P E N D I C E S

-A2-

CONTENTS

Appendix 1

Al.l

Al.l (a)

Al.l (b)

A1.2

Appendix 2

Appendix 3

Appendix 4

Analysis of the deflected leaf springAnalysis of the curve of deflection using the Method of Perturbations to correct the curvature equation Solution neglecting the longitudinal end forcesSolution including the longitudinal end forcesSeries solution for the curve of deflection of a cantilever in the elastica range derived from elliptic integralsAnalysis of the cam characteristics for Simple Derived Motion

Analysis of the cam characteristics for Finite Pulse Derived Motion

Preparation and examples of the manu­facturing specification Example 1. To determine the specifi­cation for manufacturing the profile controlling a rise movement with Simple Derived Motion Example 2. To determine the specifi­cation for manufacturing the profile to drive a rise movement with Finite Pulse Derived Motion Example 3- To determine the leaf dimensions for manufacturing the profile controlling a rise movement with ’asymmetric' Simple Derived Motion

PageA4

A4

A7

All

A21

A38

A47

A61

A64

A71

A79

Appendix 5 The manufacture of cam profiles for A84Simple Derived Motion

Appendix 6 The manufacture of cam profiles for A87

Finite Pulse Derived Motion

Appendix 7 Accuracy of the machine-tool and A89transmission

A 7 ! Accuracy of the vertical knee-and- A89

column type milling machine A7.2 Accuracy of the spur gear train A91

Appendix 8 Computer programs A92A8.1 Introduction A92A8.2 Common procedures A92A8.3 'LEAFSLOPE' ‘ A93A8.4 'LEAFPERTS' A94A8.5 ’DEFLECTION' & 'SDM-DISPL' A95

A8.6 * SDM-CHAR' A96

A8.7 ’SDM-PRESANG' & 'SDM-CURV' A96

A8.8 'FPDM-PROP* A97A8.9 ’FPDM-DISPL’ & 'FPDM-CHAR' A98

A8.10 'FINDIF-WAY' & ’FINDIF-ACCN’ A99

Appendix 9 Equipment and material used A101A9.1 Milling machine A101A9.2 Rotary table A101A9»3 Dial gauge A101A9.4 Leaf spring A102A9-5 Oam blank A102

-A3-

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-A6l-

PREPARATION AND EXAMPLES OF THE MANUFACTURING SPECIFICATION

The procedure for preparing the manufacturing specification of the cam together with the associated parts of the follower is illustrated diagrammatically in Fig. 5-1* The method of using design data to make a preliminary check of the basic dimensions is shown similarly in Fig. 2-2. This explanation is based upon the design of a disc cam to drive a radial roller follower, appropriate modifications should be made to suit other configurations of the cam mechanism (97)* Examples of standard forms of drawing for the cam and setting the leaf spring are given in Figs. 5-2 and 5-3 respectively. The design proce­dure is the same for both Simple Derived Motion and Finite Pulse Derived Motion, details depending upon the application. Examples are included in this Appendix.

The sequence and times of each follower movement and the speed of cam shaft rotation are known from the' timing chart (Chapter 2). This information identifies the portions of the profile controlling the rise and. return movements with any dwell periods, so determining the cam angles. The mass of the follower assembly and the operating force(s) are estimated and used in conjunction with the design data on follower acceleration to calculate the forces acting on the profile and follower at critical positions, the spring force, spring rate and the driving torque. The design data is also used to check the maximum pressure angle, the curvature of the pitch curve and so the surface stress in the cam disc, providing information for material .selection and determining the facewidth. If the magnitude of any of these parameters is unaccept­able the diameters of the base circle or roller and/or the facewidth must be adjusted or, in the case of excessive stress, the calculations repeated for alternative materials. Usually it is most effective to increase the base circle diameter if space permits since this change reduces the maximum pressure angle (and so the maximum force normal to

APPENDIX k

-A62-

the profile) whilst increasing the radii of curvature of the pitch curve (to reduce the surface stress) without weakening the roller or pivot. The disadvantages are reduced load capacity of the roller (due to increased speed) and hence increased danger of roller slip (28).

The cam shaft is subject to varying torsion, bending and direct shear loads with rotation. Having determined the shaft diameter the hub of the cam disc and the fastening can be designed.

For a "symmetrical" follower movement with Simple Derived Motion, Example 1 (page A6*f), the offset of the leaf equals the lift whilst the leaf thickness and the span (hence the machine ratio and the gears for the transmission) are found from Fig. to suit the limiting bendingstress and to minimize the resultant error caused by probe thrust, the apparent increase in leaf thickness and probe tip radius* Normally the shortest possible span should be used for maximum stiffness. The magni­tudes of these errors estimated from Figs. 9-ti and 9-5 were used in Example 1 to equalize these errors at mid-span with closely-spaped dowels.A small resultant error at this position, inevitable with longer spans unless the leaf is stiffened artificially, does not affect the follower motion adversely and so is acceptable.

The smaller of the angles of lift and return determines the criti­cal dimensions of the cam, but equal attention must be given to both leaf settings to ensure that both have the optimum leaf thickness and span.In the case of an "asymmetric" motion, Fig. 6-1 and Example 3 (page A79), it is essential that both leaf settings have the same slope at the tran­sition point, consequently the offset-span ratios allowing for probe thrust error must be matched at this position. Then the dial setting of the datum indicator requires adjustment to allow for the difference between the leaf thicknesses. The resultant error in the lift caused by the difference between apparent leaf thicknesses at the transition point is negligible compared with the tolerance on the lift.

In the case of Finite Pulse Derived Motion, Example 2 (page A7l)j the limiting contact angle over-rides the requirement that the curve must not be re-entrant as an additional criterion regulating the maximum offset- span ratio. This is checked from the tabulation of the program 'LEAFSLOPE*

against the offset and span found from the setting data, Fig. 8-6.This configuration has the advantage over Simple Derived Motion that the offset is smaller than the lift and it is shown in Example 2 that an adjustment of span to simplify the machining of the mounting plate by rounding off the equivalent angle of lift has negligible effect upon the lift. The apparent leaf thickness and probe tip radius errors are zero at mid-span with this motion, but no allowance is made in Example 2 for the probe thrust error at these positions. The procedure used in Example 3 could be followed, but it would be preferable to stiffen the leaf artificially, permitting the use of longer spans.

Since the span is proportional to the product of the cam angle and displacement factor, equation (A2.3)j the optimum length can be used within narrow limits. The span is set from the rotary table, the centre- distance of the dowels should also be specified for inspection. The arrangement of the dowels, rollers and leaf is shown to an enlarged scale in Fig. 10-8. The offset is

where the symbols are defined in Fig. 10-8. Hence the dowel diameter and wall thickness of the rollers are the critical dimensions.

This method of profile derivation described in Appendices 5 and 6 eliminates the need to specify any pitch curve co-ordinates, but design data, Fig. 8-11, has been prepared in dimensionless form to check any setting relative to the x-y axes by measuring the deflections at x/X = % and x/X = -jj.

(A^.l)

EXAMPLE 1.

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-a8C-

THE MANUFACTURE OF CAM PROFILES FOR SIMPLE DERIVED MOTION

The procedure for determining the leaf setting for a 'symmetrical* follower motion is given in Example 1, page A64. If the angles of lift and return differ the span and possibly the leaf thickness and/or the machine ratio must be altered to suit. The master cam is cut to drive a radial or offset translating follower, the correction for an oscillat­ing follower being made on the copying machine.

The support angle assembly illustrated in Fig. 6-3 and plates 2 to 5 is mounted on the table of a vertical milling machine, aligning the slide travel and the datum edge of the mounting plate parallel to the longitudinal feed (x-axis) within 0.001 in per 12 in travel of the slide. The run-outs of the machine-tool spindle and the mandril inserted in the rotary table should not exceed 0.001 in, for a radial translating follower their common centre-line should be within 0.002 in per 12 in travel of the transverse feed (y-axis). The gears G1 and G2 forming the transmission, Fig. 9-7? are selected to suit the machine ratio.Since the torque transmitted is small in ..comparison with the load capa­city of the gears and layshaft assembly a large tolerance on the position of the layshaft centre is acceptable to suit varying centre-distances between gears G1 and G2. Once the support angle and spindles are aligned all feeds except those controlling the transverse table movement (y-axis) and the rotary table are locked for the remainder of the process.

The frame supporting the datum indicator, Plate 2, is clamped to the slideway on the column face of the milling machine, the probe travel being aligned with the transverse feed for calibrating the span limits with the cam angles. The zero reading should require a probe travel of approximately 0.1 in from the equilibrium position, the final 0.010 in actuating the dial gauge.

APPENDIX 5

The dowel holes are machined for a transition fit at a uniform pitch equalling the span along a common centre line parallel to the slide travel. The datum edge of the mounting plate can be skimmed after assembly. The position of the first dowel hole is not critical on either axis, the conditions being met by turning the rotary table through the angle of lift to index the slide through one span to pitch the dowel holes. If the cam angles differ it is preferable to make parallel rows of dowel holes, aligning the dowels for change-over on the y-axis.The slide positions corresponding to 0 = 0 and 9 = aP are calibrated by the datum indicator. The edge of the probe is aligned visually with the centre of the dowel, moving the slide from the rotary table. The transverse feed positions corresponding to the zero reading are read from the lead screw dial, or by setting up a depth micrometer, at +L0 of this position and the angular positions corrected in the manner shown in Fig. A5-1 to converge on the limit of span. The slide is then locked, the cluch released and the rotary table set to the required cam angle.

The rollers are made from a tube by grinding the outside diameter to obtain the required wall thickness. On assembly the sliding surfaces are lubricated with a thin oil and the leaf threaded through. Zero slope is obtained at the outer ends of the idle spans by wedging spacers and shims between the spacing blocks and the leaf opposite the outermost dowels, Fig. 10-11. The special split dowels must be used where necessary to allow the probe uninterrupted access to the leaf.

The cam blank, complete except for the profile, is mounted on the mandril fitted into the centre of the rotary table and checked for square­ness. The centre-distance to the cutter spindle equals the pitch curve radius at 9 = 0 allowing, say, 0 .0 1 5 in excess for a finishing cut and honing. The datum indicator is set to zero, fine•adjustment being made by rotating the dial. The cutter diameter equals that of the roller follower. The first plunge cut is made by lowering the spindle.

The machine table is withdrawn to separate the probe and leaf and the rotary table indexed through one angular increment, the slide being moved through the proportional distance through the transmission. The machine table is then advanced using the transverse feed until the dial gauge reads zero. Then the difference between the pitch curve radii

-A 86-

equals that between the leaf deflections at the corresponding positions along the span, Fig. 9-1• The next plunge cut is made and the process continued until the whole profile has been machined.

Dwells are cut as a continuous operation. The slide and rotary table must both be locked before the clutch is released. The centre- distance remains that of the last plunge-cut and the rotary table is turned through the angle of dwell whilst the slide remains stationary.The transmission must then be re-engaged to plunge-cut the succeeding section of the profile.

One or two roughing cuts should he followed by a finishing cut, completing every section of the profile in turn. The heights of the scallops, Fig. b-2 and Plate 1, are within the limit for honing spheroidal graphite cast iron or unhardened steel hut initial filing to remove the sharp edges is quicker and reduces wear of the honing tool.

The method of calculating the leaf settings for an 'asymmetric' profile, Fig. 6-1, is given in Example page A79- Tbe machine ratio must be unaltered throughout a rise or return event, the setting proce­dure is simplified if the larger equivalent angle of lift is an integer product of the smaller one as different spacings of the dowels can be arranged along one line. The different offsets are obtained by changing the- rollers. The portion of the profile between one limit of the event and the transition point is plunge-cut, then the machine table is lowered to clear the datum indicator for changing the leaf and rollers without disturbing either the transverse feed or the rotary table. The table is raised, the dial gauge reset to zero and the profile completed.

APPENDIX 6

THE MANUFACTURE OF CAM PROFILES FOR FINITE PULSE DERIVED MOTION

The principle for deriving the shape of a profile for a ’symmetrical’ finite pulse motion is identified in Chapter 6 and illustrated in Fig. 6-2(b). The method of determining the leaf setting is shown in Example 2, page A71.

For this profile the slide travel must be inclined by the angle of maximum leaf slope to the transverse feed of the machine table (x-axis) so a universal milling machine is essential. Theprocedure described in Appendix 5 for mounting the support plateassembly and datum indicator on the machine-tool is followed, at this stage the slide travel is aligned perpendicular to the transverse feed. There are two possible methods of machining the dowel holes.If the equivalent angle of lift, equation (A3.2), can be read from the graduations of the rotary table without estimating divisions or round-off error then the holes can be machined by the process used for Simple Derived Motion, Appendix 5« The datum edge of the mounting plate is skimmed after which the machine table is rotated through the angle of maximum leaf slope.

The alternative method is illustrated in Fig. A6-1. After machining the first dowel hole the rotary table is turned through halfthe angle of lift ( i^q ) and the transverse feed moved through halfthe lift ( ^ hq ) to locate the centre for the second hole. This pro­cedure is repeated for every dowel hole. Then the machine table is swung through the angle of maximum leaf slope and the datum edge of the mounting plate machined parallel to the common centre-line. (It should be rough machined and relieved to reduce the amount of metal to be removed at this stage since cutting force is restricted by the stiffness of the guide rails.) This method is independent of the equivalent angle of lift.

-A88-

The remainder of the process is identical to that used for Simple Derived Motion, except that the starting and finishing posi­tions are at mid-span.

APPENDIX ?

A7.1 ACCURACY OF THE VERTICAL KNEE-AND-COLUMN TYPE MILLING MACHINE

The recommendations for the accuracy of a new machine-tool of this type specified in BSA6 3 6:Pt3 :1 9 7 1 relevant to this application are summarized in Table A7-1.

As shown in Appendices 5 and 6 the only movements of the machine- tool required for the derivation of a cam profile using this principle are the transverse feed of the table (y-axis, Figs. A-l and 6-3) and the vertical feed of the cutter spindle (z-axis). BS4636 recommends the use of lever stylus dial gauges to BS2795:Pt2:1971j straight edges and squares for these accuracy tests.

According to Schachte (102) the accuracy of a surface cut on a machine-tool depends upon:-

(a) the geometric accuracy of the machine-tool.

(b) the measuring accuracy.

(c) the positioning accuracy.

(d) the deflection of the transmission and structure of themachine-tool under load.

The geometric accuracy specified in BSA636 is determined by the accuracy of alignment of the slideways, surfaces of the machine table and rotary table, and the axes of rotation of the rotary table and cutter spindle with the co-ordinate axes defined in Fig. A-l. In addition to angular misalignment between any pair of these members the accuracy of manufacture is affected by curvature of a slideway and slackness or wear between sliding surfaces. These faults may be localized.

ACCURACY OF THE MACHINE-TOOL AND TRANSMISSION

-A90-

The measuring accuracy depends upon the accuracy of the pitch of the lead screws and graduations of the dials whilst positioning accuracy depends upon the ability of the operator to align the gradua­tion and reference mark and is also affected by backlash, the effect of reversing the direction of friction forces between sliding surfaces and the deflection of both sliding members and their supports under static and cutting forces. The last three have directional effects.

Repeatability is the measure of the variation in the machine-tool setting corresponding to a nominal position on the dial, expressed as a standard deviation. The directional effect can be counteracted by ensuring that final adjustments of position are always made in the same direction and positioning accuracy can be improved by using a dividing head attachment on the rotary table in preference to visually aligning the dial graduation and reference mark.

The deflection of the structure and transmission of the machine- tool is caused by the static load of the fixtures and workpiece and by the machining forces. Because the pressure angle acts in different quadrants during the rise and return actions the component of cutting force along the x-axis, Fig. 4-1, also reverses direction, tending to cause relative movement between the rotary table and machine table.

In this application the accuracy of the profile is independent of the positioning accuracy and repeatability of the machine-tool, the results summarized in Table 10-9 being based upon the zero reading of the dial gauge and not lead screw dial readings. According to Welbourn and Smith (ill) the stiffness of the machine-tool between cutter and workpiece is in the order of 10^1bf/in.

J -A91-

A7.2 ACCURACY OF THE SPUR GEAR TRAIN

For a machine ratio of 2 the spur gears forming the transmission, Fig. 9-7? are:-

The displacement factor, equation (6.2), is 0.03^9 in/degree.

The rack is 13 in long, has 38 teeth and meshes with gear G3 .The action of the bias weight (Chapter 9) takes up the backlash in one direction, irrespective of the direction of slide motion although some backlash remains in the worm gear of the rotary table.

The setting procedure, Appendices 3 and 6, ensures that the span of the leaf equals the displacement of the slide corresponding to the rotation of the cam blank through the angle of lift or return. There­fore the boundary conditions, equations (6.1) and (6.6), should be satisfied despite the errors in the transmission.

The error in pitch curve radius has been calculated to obtain the values quoted in Table A7-2 for a cam having unit lift and the worst possible.case in which the largest pitch error of every gear is a maximum in the same direction for the classes of gear specified in BSA3 6:19 0.The nominal rotation of gear Gl corresponding to an advance of 1 tooth is 2T(/96 radiansand the nominal advance of the rack is

DIAMETRAL PITCHNO. OF TEETHGEAR Gl G2 G3

9 6 9 6 A 8

12 12 12

The results show that precision cut gears to Class A2 are required for this application.

-A92-

APPENDIX 8

COMPUTER PROGRAMS

A8.1 INTRODUCTION

All the programs were written in ICL Algol 60 and run on the ICL 1905F computer at the University of Surrey using 80-column punched cards for input and line printer output. Some programs having output exceeding the capacity of one line printer are divided into several channels.

A8.2 COMMON PROCEDURES

The additional standard function ARCSIN (X) was available as an external procedure.

The following procedures are included only in the first program in which they occur

SIMPS The numerical integration of a given function for aspecified value of a variable between given limits and to a specified accuracy (7l)»('DEFLECTION' statements 5-35)

FUNC The equation for the slope of the neutral axis of thedeflected leaf (7*19)1 Fig. 7-5•('DEFLECTION' statements 36-03)

K(A) The evaluation of the complete elliptic integral ofLegendre's first kind for a specified modulus A (53)•('LEAFSLOPE' statements 9-50)

G(A) The evaluation of the complete elliptic integral ofLegendre's second kind for a specified modulus A (53). ('LEAFSLOPE' statements 31-57)

F(R,P) The evaluation of the incomplete elliptic integral ofLegendre’s first kind for specified modulus R and ampli­tude P (53).('LEAFSLOPE' statements 58-100)

E(R,P) The evaluation of the incomplete elliptic integral ofLegendre's second kind for specified modulus R and ampli­tude P (53).('LEAFSLOPE' statements IOI-I6 3)

A8.3 'LEAFSLOPE'

The analysis of the curve of the neutral axis of a deflected leaf spring having a uniform cross-section, Fig. 7-5? gives a relationship' between the offset-span ratio and the angle of maximum slope expressed as a function of elliptic integrals, equation (7 »1 6 ), or as a series expansion of the elliptic integrals solution, equation (7.17). The former also provides a relationship between the- free length and the span (ll).

The sine of the maximum angle of slope ( tt ) occurs in every equation for the cam characteristics of both Simple Derived Motion (Appendix 2) and Finite Pulse Derived Motion (Appendix 3 ) derived from this analysis (Appendix 1.2), but inspection of equations (7.16) and (7.17) shows that <T cannot be calculated for a given offset-span ratio. Since an accurate value is required the data was prepared by calculating Y/X for small increments of <3~ using the procedures K(A), G(A), F(R,P) and E(R,P) to evaluate the elliptic integrals involved in equation (7 .1 6 ) and also from equation (7*17) for comparison with the result for the cubic spline obtained from equations (7 .6 ) and (7 .1 8 ) using the approximate theory of bending. The ratio of free length to span was also calculated for every increment.

A separate program, 'LEAFPERTS', was written to solve the results obtained from the method of perturbations (Appendix l.l). The results are summarized in Table 7-^*

A8.*f ’LEAFPERTS*

The solution derived by the method of perturbations for the curve of the neutral axis of a leaf spring deflected into the con­figuration shown in Fig. 7-2 (Appendix l.l) provides a direct means of calculating the deflection at any intermediate position provided the deflecting and longitudinal forces are known, equation (7-3°)• Since the span and offset, but not the forces, are known in this application the solution was obtained by following the iterative pro­cess defined in the flowchart, Fig. A8-1. The significance of the longitudinal force upon the shape of the curve was evaluated by using the relationship

over the range 0.03</<^ 1.0. To save computer time and to co-relate the results with those obtained from the program ’LEAFSLOPE1 (Appendix 8.3) the offsets and corresponding values of QT obtained from the latter were included in the input data for ’LEAFPERTS’.

The starting values for the iterations were calculated in terms of the parameters

using the approximate theory of bending, modifying them empirically to compensate for the foreshortened span (statements 36-39)-

W (A8.1)

where i-s the equivalent ’coefficient of friction'

(Al.ll)

and

E l

W ( A i . 12 )

£ l

Having determined the functions of the deflecting force (statements A9-8 5) the bending moment was iterated until the increment reached a specified minimum to converge on the boundary conditions

-Ab­

using equation (Al.7 6), (statements 86-97)* The offset calculated from equation (7 *30) was expressed as a ratio of the-required value to iterate the deflecting force in a similar manner to converge on the boundary conditions.

(statements 98-1 1 0 ), the bending moment being revised for every iteration of the deflecting force. The conditional statements 93 and 106 checked the boundary conditions to stop the iteration immediately the values fell below specified limits.

Finally the deflections at l/A- and l/20- span were calculated and the errors on the slope and offset tabulated.

A8.5 'DEFLECTION''SDM-DISPL'

The follower displacement for Simple Derived Motion equals the deflection at the corresponding proportion of the span

but the preferred solution, equation (7 *1 6 ), is confined to the relation­ship between the offset-span ratio and the angle of maximum slope. There­fore the intermediate positions had to be found by numerical integration of the slope, equation (7*19)» using the Simpson's Rule procedure 'SIMPS' (statements 5-35 of 'DEFLECTION'). Consequently the complete process must be followed to obtain one intermediate value..

The program 'DEFLECTION' was written for calculating the deflection at increments of cam angle for a given offset and (f , simultaneously tabulating the dimensionless form y/h. The conditional statement 76

checks the maximum bending stress against the limiting value included with the input data, a warning is printed should this value be exceeded.

X

A/ © (6 .2 )

The dimensionless lift characteristics for the range 0 .1 cr < 0.6 in 0 .1 steps were calculated at 1 ° increments for angles of lift of l80°

and 200° to accord with the ESDU publication (3*0 using the program 'SDM-DISPL' to obtain the data given in Table 8-6.

A8.6 'SDM-CHAR'

This program was written to prepare the design data for the dimensionless forms of the velocity, acceleration and pulse characteris­tics for a radial translating follower, driven with Simple Derived Motion which are specified by equations (8.A), (8.7) and (8.8) respectively.The values are calculated for 1° increments of the cam angle for anglesof lift of l80° and 200° over the range 0 < 0/aP < l/2 for 0.1 < CT ^0.6in 0.1 steps.

Sample results are summarized in Tables 8-7? 8-8 and 8-9. Graphsof the characteristics are drawn in Figs. 8-3? 8-4 and 8-5.

A8.7 ’SDM-PRESANG f•SDM-CURV1

The use of prepared design data, Fig. 2-2, requires analyses of the limiting conditions imposed by the pressure angle and curvature of the pitch curve. The pressure angle equation

rflI

docUSA) (97)

was rewritten in dimensionless form

Vat ft') = j? ~

X +(nb, this parameter is independent of speed)

(A8.5)

to calculate the parameter $ td 0*3) (where $ is in degrees) for increments of cam angle expressed as ©//vf for values of between0.1 and 10.0 and the usual range 0.1 4 0.6 in 0.1 steps. The resultswere used to prepare the graphical design data, Figs. 8-12 and 8-13 (77)•

-A97-

In this program the procedures 'SIMPS' and 'FUNC' were used to determine the dimensionless form of the displacement (as in the program 'DISPLACEMENT') and the first derivative from equation (7*19) .

This program was amended to calculate the radius of curvature of the pitch curve to locate the magnitudes and positions-of minima. Equation (.1) was rewritten in dimensionless form

and the results obtained in a similar manner except that separate tabluations were needed for every angle of lift between 3°° < < l80°in 30° steps. The results were used to prepare the graphical design* data, Figs. 8-lA and 8-15 (77)•

A8.8 'FPDM-PROP'

The principle involved in the derivation of cam profiles having finite-pulse characteristics is identified in Chapter 6. The equations for the follower displacement, velocity and acceleration, Table 8-3, equations (8.l6) and (8.17) respectively, derived in Appendix 3 are functions of the span and offset of the leaf (Fig. 8-7).

As shown in Example 2 it is necessary to select compatible starting values for the machine ratio and the parameter J using the graphical data presented in Fig. 8-6. This program was written to calculate this data using the symbols

s <f sin ( )

PSI-0 % maximum angle of slope at mid-span

Y/X offset-span ratio

X span on x-axis (Fig. 8-7)(coat)

-A98-

Y

BX

BX/BU

V/U offset-span ratio on the transformedXl“yl axes

The offset-span ratio (Y/X) was calculated-by equation (7-16) using the procedures 'K(A)f, 'G(A)', *F(R,P)', and 'E(R,P)f (statements 68-80).

Sample results are summarized in Table 8-10 and the curves plotted in Fig. 8-6.

A8.9 'FPDM-DISPL''FPDM-CHAR*

These programs were written to calculate the cam characteristics for Finite Pulse Derived Motion, information taken from the tabulation of 'FPDM-PROP' (Table 8-10 or Fig. 8-6) forming part of the input data to avoid repetition of the elliptic integrals procedures in these programs. These calculations, based upon the displacement equations, Table 8-3j- and equations (8.l6) and (8.17) for the follower velocity and acceleration • respectively are complicated by the additional-relationship between the parameter CT and the offset-span ratio over successive spans of the deflected leaf. Consequently these programs were written to define the characteristics for a given mechanism, leaf setting and machine ratio defined in the input data and lack the general application of the dimen­sionless forms used in 'SDM-DISPL' and ''SDM-CHAR'.

In 'FPDM-DISPL' the follower displacement is calculated from the leaf deflection found by numerical integration of the slope, equation (7 *1 9 ) for equal increments of the equivalent angle of lift ( ) usingthe procedures ’SIMPS’ and 'FUNC. Consequently the increments of the true angle of lift are unequal and liable to round-off error (which does not affect the derived profile). For convenience of calculation the

&

/ s

equivalent angle of lift (deg) corresponding to span X

ratio of equivalent to actual angle of lift

offset on the y-axis

-A99-

deflection is found for complete spans of the leaf, commencing and finishing at mid-span. For this reason the results are tabulated in three columns. In addition to the displacement and true cam angle the number of increments and the leaf deflection are recorded.

The program 'FPDM-CHAR' calculates the dimensionless forms of the follower velocity and acceleration for equal increments of the equivalent angle of lift (M/ ). The procedures VV(T) and VW(T) (statements 5-23) calculate the velocity from equation (8.1 6 ) whilst AA(T) and AB(T) (statements 23-51) determine the acceleration from equation (8.17). Two versions are needed to allow for the reversed direction of slope over successive spans. The actual cam angle is found by cross-referencing the counter number with the value read from' 'FPDM-DISPL'. The results were used to plot the characteristics drawn in Figs. 8-9 and 8-10.

More general results could be obtained by treating the dimension­less parameter lift/machine ratio as a single variable in the mannershown in Fig. 8-6. Then the characteristics could be evaluated for unit lift over a range of angles of lift as the magnitude of isunaltered provided the equivalent angle of lift ( ) and the ratio h/cremain constant.

A8.10 'FINDIF-WAV1'FINDIF-ACCN'

These programs were written to investigate the significance of the random round-off error in the pitch curve co-ordinates upon the follower acceleration. Since this error is non-existant in the derived profiles equations (3*12) and (3»13) were used to calculate the round­off error, profile wavyness and consequent secondary acceleration for SHM and Cycloidal Motion, chosen for their similarity to the derived motions.

'FINDIF-WAV1 calculates the true follower displacement for equalincrements of cam angle over half the angle of lift, both specified inthe input data. The radius is rounded-off by multiplying the true dis-

n+1 n /placement separately by 10 and 10 (where n is the numbeb of

-A100-

significant figures in the rounded-off value). The function 'ENTIER' is used to determine the rounded-off value, the conditional statements 02 and 57 being used to determine the last figure. This result is then multiplied by 10”n to obtain the rounded-off value. (Statements 38-46 and 52-60.) These results were used to construct Eig. 5-5.

The similar program rFINDIF-ACCN’ estimates the order of magni­tude of the secondary follower acceleration from the difference between the 'true' value calculated from the second derivative of the displace­ment equation and that calculated by finite difference techniques using equation (5*13) from the rounded-off radii. The results are plotted in Fig. 3-6 .

AJOl

APPENDIX 9

EQUIPMENT AND MATERIALS USED

A9.1 MILLING MACHINE

Elliot "MILMOR" vertical milling machine,

Table working surface 50 x 10J inlongitudinal traverse 28 intransverse traverse 14 invertical traverse 20 in

traverse 5 incentre-line to column face - maximum 29j in

- minimum 8J indistance to table - maximum 18 inspeed - maximum 63 rev/min

- minimum 4880 rev/minpower of motor 2i hp

A9.2 ROTARY TABLE

A. A. Jones & Shipman type 4012 with indexing attachement 4014-002.

Table - diameter height

Diameter of centre bush hole

A90 DIAL GAUGE

Verdict Gauges type T6 dial indicator

Dial diameterGraduationsRange

10 in 4^/8 in

1 in

2 in 0.0005 in 0.060 in

A102

A9.4 LEAF SPRING

Starrett feeler strip type 6 6 7.0 .0 12 x 0 .50 0 x 12 in.Material:- BS970:Pt5:060A96 Hardened and tempered to 5&0 VN.

A9 .5 CAM BLANK

Material:- free-cutting 5-5$ copper-aluminium alloy, BS1474:1972:FC1 .

P L A T E S

T o Fa c e Pa g e

PLATE X

The scalloped surface of the cam profile after the first increment cutting operation. The cuts were made at nominal increments of 0° 20', the radius being set from the surface of the deflected leaf spring.

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T erminology - Disc Cam Driving a

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Fig. 2-1. Design Procedure for

a Cam Mechanism.

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I N V E R S I O N O F R A D I A L

CANA M E C H A N I S M

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/. S i m p l e Ha r m o n i c .

2. C o n s t a n t A cceleration

3. MoblFIEb TRKPEZOILftL A c c e l e r a t i o n C u r v e .

4. Cy c l o id u.5 . 4 - 5 - 6 - 7 P o l y n o m i a l .

T J A l N l t A U f A T o l e r a n c e o f

± Q O O O 2 in ( 5 /« r« ) O N

L i f t o f I.Oih ( 2 5 , 4 * * )

Lift C h a r a c t e r i s t i c s .

t r* i 1-------- 1-------- r

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A n c l e o f C a m R o t a t i o n / 0 '

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SPEED OFrotationOF CAMFACE* FOLL*R BASE EXTERNAL MASS PATH TYPE RISE OUTER RETURN INNERPROPERTIES

rnSHAFT CAM FOLLT*

HUB FASTENING SHAFT

CURVATURE OF PROFILE <L> (D)

SURFACE STRESS (L) ID)

PRESSURE ANGLE (L) (D)

MAXIMUMFOLLOWERACCELERATION

C E E ftuso FIGS. t-\ & 2-2

Fig. 5 - 1 Design Procedure fo r

Generated Profiles.

F23

S p e c i f y P a t h o f F o l l o w £ R

F ig. 5-2 St a n d a r d Fo r m o fD e t a i l D r a w in g o f Ca m .

DIMENSIONS CF DLANK L.L?L*fc,

Vr

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Profile irAEMSioNs /Ease Circle 'hiN.LiftRo l l e r ^ in.

CAM JANGLES, QFRoh To

IMMER WELL RISEOuter hWELt. 'RETURN.

G1 G2'ROTARYi /yb»l e

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Fig. 5-1 S etting In s t r u c t i o n s

CHANGE GEARS Gear Ho . of P i t c h

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a- OUTER WELL

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The diagrammatic arrangement of a mechanism which generates Simple Harmonic Motion. The amplitude of the axial displacement is changed by varying the radius•

Fit3. 5-6

The combination of linear and sinusoidal components to derive the displacement for cycloidal motion.

F26

ft <5. 5

The diagrammatic arrangement o f a mechanism to generate cyc lo ida l motion. The p itc h c irc le s o f gears and the p itch lin es o f racks are represented b.y the ou tlines of the respective pa rts . (R ef.8).

displacement: y - 2 ( ^ 2 +

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^ T - ro ta ry tab le

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oR o t a t i o n o f In p u t S h a f t ,

U- 27

R o t a t i o n o f In p u t s h a f t

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G e a r M e c h a n i s m . .

(g r o l z i n s k i ( 4 ? ) ),

F 30

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Arrangement of a generating mechanism driven by synchronised geneva mechanisms.

Fig. 5-14, Arrangement of input drives from the geneva mechanisms to the bevel

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1- CUTTER (FlYEft AX IS). ■2- CAN BLANK RQUNTEX OH R O T A R Y TABLE.3- ROTARY TABLE W I T H hSVIhING HEAD A T T A C H M E N T

£ CONCENTRIC 6 EAR W H E E L .4- LAYSHAFT CARRYING C O M P O U N D <3EAR %

INCORPORATING A C O N E CLUTCH.5- SLLAE - MOUNTED ON TABLE TT - TRIVEW THROUGH

RACK IN DIRECTION!'X"X'.6 - EFFECTIVE LENGTH OF MFLECTOd LEAF SPRING ?-TABLE OF MACHINE-TOOL

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T able 3-1.Comparison of Follower Motion for Constant FcceLERATioN, S.H.hA. & Cycloidal Mqt/ons.S u m m a r y o f E x p e r i m e n t s R e s u l t s ( M i t c h e l l(ty),

Base Circle Bi*. 8-SI2 Lift z-ooo a ,

A n g l e s o f Lift 2 70°Re t u r n

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Wht. Pressure Ancle (dig). 37 30 37

IAAX. SpEEB OF (/cv /t>un ).Rotation of Disc Can\

170 170 /SO

M at. Linear Acc'n of Racial Follower

4*4 5-4 6-9

A c c u r a c y - m e a n d e v i a t i o n o f r a d i u sFROM T H E O R E T IC A L VALUE LESS THAN + O - O 0 I #w. MAX. DEVIATION - 0.0015/*,

TABLE V2, D i f f e r e n c e s B e t w e e n P a d i i o f P i t c h C u r v e

for Increments of C m Angle at Start of LiftL I F T v l . O : A N G L E O F L I F T 0 4 % * > \

C A M L A W

C A M S U N C Y C L O iB A L MOTION M O B . T R A P A C C 'NANGLE A I S P ’L ' B I F F . A i s p - l B i f f B l S P 'L B IFF .

e / f V - f t m

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0 0 5 0 - 0 0 6 15 0 - 0 0 0 81 0 - 0 0 1 2 60 - 0 0 2 7 0 0 - 0 0 0 6 0 0 - 0 0 0 8 3

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MACHINE MAX.ERROR

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MAXIMUM ROUND-OFF ERROR FOR BOTH MACHINE-TOOLS - 5 x IQ"S

T a b l e 6 - 1Linear I n t e r p o l a t io n of Circular Arcs.

NukuRicmly- Controlled Continuous- Path Profiling,Length of Chor/ci, Angle Subten%e% at Centre& PROFILE WNYYNfc'SS,

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3-02

I A B L E 6 ~ / (.coht),Li n e a r . I n t e r p o l a t i o n o f C i r c u l a r A r c s .N u m e r i t , a y - C o n t r o l l e d C o n t i n u o u s - P a t h P r o f i l i n g ,

L e n g t h o r C h o r d , A n g l e S u b t e n d e d k t C e n t r e -- & P r o f i l e W d v y n e s s .

TOLERANCE

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T a b l e 6 - 1 Udncl).Li n e a r I n t e r p o l a t i o n o f C i r c u l a r A r c s .N u m e r i c a l l y - C o n t r o l l e d C o n t i n u o u s - P a t h P r o f i l i n g ,

L e n g t h o f C h o r ^ , A n g l e S u b t e n l >e l > a t . C e n t k ?& P R O F I L E W L n V Y N E S S .

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H M r ; S P A N 0 , . - P ,

®*. x„- I x Y u & 2

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R I O

T u rle 8 - 4 ,■Equations fo r Thriving Torque & Dimension less Torque Factors. (2s)

COMPONENT OF TORQUE TORQUE DHAENS\ONLESS> TORQUE FACTORCONSTANTLOAriINERTIAFORCESPRING‘STIFFNESS

Tc * FVl (8-40) T . MAftX yi ((Vdi)y £\ =■ §jyy (e-«)

to “ A „ V (6-4$)U

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<r-5-9T9 5-380*1 1*0

0*2 -5-319 1-0 5-32

0*3 1*0 5-820*4 -S688' ho 5-630*5 -5-526 1*0 T530*6 -5-541 1*0 5-34

TABL

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to O O' A- to to X ft to ft ft ft

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CTLU—Joft

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IIh-

to o (V to X X f t o o G to to O' to X. X X- to ftto o to O' X- X X O' <*“ G X V— X X X to o c «—ft o o C\, ft ft -c ft O' X O' CC-to to N X to N'tftto (V o o to ft X V— X to ft O' to o tt-to c to •X G fta (M o to O' <7 ft o- iN, G ft ft to ft c O' ft X ft to

X G- to o X" to ft C\!ft ft' i— X to to o X ft T— X K— toNT G G o G ft to y ft-to < X ft X X o O*— *— G G o o o o O G O o o G G o o o oto G o O O G U*‘o o o lZ>o C5 c’o o o o

to o i— X ft X ft ft -o to X o IO X X O' o O'X o to­r* v— v~ X X ft •X o X o ft fs.to ft to to<} G ft to O' ft o G ft G O' G ft ft O' o C?'ft XT— £> O to 00 ft'o CO to to to ft O' to to O ft O' to oa X Ud to O' ft ft G to <r-to ft ft ft O' X ■c— IN. X toG X o *— to ft ft X ft V" X to to o ft ft c— X ,— to•x G G G C: \— ft y to ft to X X ft X X O' oo G O O O o o O O . 0 o o o G o O o G- oo O O o o o o o O o o o o O* o o o o

UJSI «J <c oO 7 2 ft

c o o o o o o o o o o o o o o o o oT~f\JtofttoXftX>0O«— <\ltofttoXftT~ <f~ s~ <c—- '— ‘i— ■!'— 18

0 0.

0982

587?

0.

0982

7606

0.

0983

0570

0.

0^83

4883

0.

098^

0747

0.

0984

8496

190

0.09

9746

14

0.09

9748

80

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335

0.09

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94

0.09

9768

85

0.09

9780

5520

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8063

0.

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2251

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2573

0.

0998

3040

0.

0^98

3660

0.

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4198

TABL

E 8-

7.

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DERI

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CH

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000

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(DEO

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200

Fff2.

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O ♦ O O O O O C- C O C *— <e" <t“ v" ■ r- v- v— «- c- v- e-OCO

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NT O' tu O <Nj TO TO TO N TO O' O <N, f t f t TO TO TO TO TO. TO

O * OCj O O O O O O O 'c- i— -v~«— <r- v- v -c-c >

TOTO O O ft TO C- TO v- TO TO- TO f- TO O' <TO <NJ TO O TO ft O' CN O TO TO v~ O TO TO O' TO ft O TO TO < TO O TO N O N v-O C CV <\| v*- O TO ft O tfT O «- O TO- TO O' N O NT O' N N<N C TO <c- v- ft O' CC O TO v- CV TO O CC. OC v~ TO TO ft TO Oft to TO CO ft TO ft TO ft TO Cy O TO «- TO v~ TO N O C Orv ft O CV TO ft TO N CC O C v- v- Cv ft ft. TO TO TO TO ft ftC * O © O O O O © © O <e*» -s— v v- v v* v*-v- v- v t -

ftTO- O TO n.’ o ft <T- TO ft ft O O N fN> ft ft ft O' *- v- TOTO C ft N N O'- fNJ TO v- TO O' O' "M" fNi O N v~ O xj- o 00 No c N- o o- to nC nt n to «- rv o «• x c to D- rv rv ft ft,00 ■ O ftTO TO‘ 0CTOftTOCCftTOftOTOftN<c—0CN0CfNiTO C TO OC ft ft TO TO ft TO fN. O TO 't~ TO C TO TO- 0C O Oc- TO- CO **- (V TO ft TO N CC CN C- v- t~ tNi ft. ft, TO TO TO TO TO ft

O • O O © © © © © © © c- v~ y* tr c- <™ t— v- »r- c~ v~c.

N.X TO «CLL £L 3C CQ<21 Z ft TO ■>—i £X (/> ft ft! ft UJ tu -J o ft •TO /*x fttt, t i l UJ it. TO O O ft w

UJCO

C ft. O ft o ft O ft O ft O ft O ft O ft O ft o ft o O fN J ftN O tV ftN O fN iftN O tN J ftN O ru ftN O O O C> C L: v~ «- V- V- CV (N. (NJ (N, NT NT NT NT TO TO TO TO ft.O O O O O O' C- o o o o o c- o o o o o c- c o

O ft o ft O ft O ft O ft C L ' ft' O ft- C ft O ft' OO'O'OOoONNTOTOftftTOxJNTNinjiNJt- C’vi v— v— v v v sr- v- t~ <r* t*- v— «*■ r* v-

C’ ft O

O f t O f t O f t iO f t O f t O f t O f t O t n o u N O f t o r- v- (Nl OJ I'O ft) TO TO ft ft TO TO N N OO CO O' O' O

8LE

8-8

SIMPLE

DERIVED

MOTION

- DIMFNSTONLFSS

ACCELERATION

CHARACT6RTST

LIFT

=1.000

: ANC

LE OF

I. TFT

SETA

(DFG)

= 200

F m

(a 'O

tfoNOftoft-'Cfto

s J O O N N N o C & ' f V ' N ' C ' i C C <> Mi r r N r C tf <> tf O Ox- tf j" r ir C. N V u' x- cc tf ft. x— m C- O' O O' f t (> f t sC K ® - C C i ( ' o C ' N ' C < ! ' C O ^ O Cv- oc ft> it- u N 10 rv. U' N <- N c:; in uo tf- t f c tf. t f oM<*. V'OCVr-O-N'JO'Y.W'r-sIN', ICiO-'CO'CC x- O tf- U* X*- tf- x- C O- N- *C xl- tf O ft tf «-• ft O tf C-in ft ft ft ft ft ft tf ?n tf tf tf tf tf tf x*~ x- o o o

Cacx~otf tf tf ft) • tf,o .o

ft tf tf (> <1 tf- V" X— O- T~ ft) ft tf • tf tf tf OC tf tf- ft C'o- ft tf cc tf tf o tf tf O' tf tf', C- tf ft X tf ft1 tf tf- otftfC.-tfO'tf<t~x~OCCCtfO cc v~ O' tf -3 tf Cv tf- o ft tf o ft tf ft tf ft tf OC Ct tf ft tf tf tf tf. O o 0C-- oft O' ft ft tf O v—- tf i ft- tf’• \— tf t— (V x™ tf xr— tf tf, <— Oft X- O' tf to tf tf O CC; ft) ft X- o ft tf O'- ft' tf CO ft Oin tr, ft ftftftftfttftftftf, tftftfv-x-s-ooo

o

otf ft CO • tf

ftOjtfft\fti(Vx“ ft. ftoctfo tf tfc o m ft tfs tftf tf tf tf, C tfft ft ft tf o ftft CO tf' O' ft ft c— Ofttfie— OCftft tf o

ftCCO'O tf OC Cftx—ft t f f t ft f t ft ft- O O'O' x- ft tf N O tf X- oc- O ft. X- ft. OC O r otf ft tf 0- tf ft c:

ft O tf tf X— OX" ft CC tf O' oX> O' ft ft tf ft•o < X C ft C-tf tf cc tf . ft) fttf ft c: tf tf oft ft. ft ft ft ft ft tf tf tf tf) (V tf (V tf x- i-

tfftfttf o* tf

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m o tf c tf c tf c in c. in o m o in o m o in oO' O' X; CO tf tf tf ND tf tf ft ft tfi tf (VI tf x**- xr“ o o— x- x- r x~ V" x~ x~ x' x— xr~ v X" «■* x*” v* x~ <- xr- «—LU fttr, oia. ft ao u

o tf o m o tf o tf o tf o tf o tf o tf o tf o m oX— x“ (M(\jtftfftfttftf.'0'0tftfaC00O0'C

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COMPARISON OF THEORETICAL ANA MEASURE*, REFLECTIONS

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0 . 5 2 7 6 0 . 0 8 0 5 0 . 0 8 0 8 0 .08030 . 5 2 8 2 0 . 0 8 0 6 0 . 0 8 2 2 0 . 0 8 2 0

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0. 62 I 5 0 . 0 9 4 7 0 . 0 9 5 6 0 . 0 9 3 30 . 6 2 2 9 0 . 0 9 5 0 0 . 0 9 3 8 0 . 0 9 4 40 . 6 2 3 6 0 . 0 9 5 0 0 . 0 9 3 8 0 . 0 9 3 40 . 6 2 4 0 0 . 0 9 5 1 0 . 0 9 2 5 0 . 0 9 4 00 . 6 2 4 6 0 . 0 9 5 2 0 .0953 0 . 0 9 4 90 . 6 2 4 8 0 . 0 9 5 2 0 . 0 9 4 7 0 . 0 9 5 10 . 6 2 5 0 0 . 0 9 5 3 0 . 0 9 6 0 0 . 0 9 5 80 . 6 2 6 2 0 . 0 9 5 4 0 . 0 9 2 4 0 . 0 9 3 4

0 . 684-3 0 . 1 0 2 7 0. 1 0 2 0 0 . 0 9 7 00 . 6 8 5 3 0 . 1 0 2 8 0. 1060 0 . 1 0 2 20 . 6 8 6 2 0 . 1 0 3 0 0 . 1 0 0 2 0. 1000

0 . 7 4 8 5 0 . 1 1 2 3 0. 1114 0. 10750 . 7 4 8 8 0 . II23 0 . 1 1 1 3 0. 10860 . 7 4 9 0 0. 1 123 0. 1 I 04 0. 1 145

0 . 8 0 8 0 0.1 191 0. 1 190 0. 11600 . 8 0 8 8 0. 1192 0. 1208 0 . I |400 . 8 1 |3 0 . 1 / 9 6 0. 1195 0. 1(27

0 . 8 4 2 6 0. 1236 0 . 1 2 7 1 0. 1286

L E A F ;b £ fL .£ C T E h 'RETW EEW O F F S E T CLMAPS

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THEORETICAL hi S PLACEMENT

U & FI RSTD IFFERENCE

MEASURED DISPLACEMENT OF MILLING MACHINE TABLE

(*) 0*«e< 32*11'FIRSTDIFFERENCE0‘«0<32*ll' (l-l)32TI’< © < 64*22'

FIRSTDIFFERENCE

32*ll'< 6< 64*22'or 64*22' 0.0000I* 63*22' 32‘ 62* 22’ 1 43* 61* 22' 304* 6ff22' 535* 5 9*2 2’ 0.00824* 58* 22' 1 1 7T 57 22' 1588* 56* 22' 2059* 5 5*22 256Iff 54* 22' 0.0313II* 53* 22' 37512* 52*2 2' 44213* 51* 2 2' 51314* 50* 22' 58915* 4ff 22' 0,066916' 4 8*2 2’ 753IT 47 22 - 84118' 4 6*22' 932Iff 4 5*22' 1 02720* 44* 22' 0 . I 12521* 4 3*22’ 122622* 42* 22' 133023* 41* 22’ 143624* 4ff 22' 15442 5* 39* 22' 0.16552 b* 3 8*22' 176727- 37*22 188 12 8* 3 6*2 2 1 99629’ 3 5* 22’ 21123CT 34* 22' 0 . '222931* 33* 22' 23463 2* 32* 22' 246432 II' 0.2486

0

0.0003 I I I 6 23 230.0035 4 I 47 51 570.0062 67 71 76 80 0084 88 91 95 98

0101 I 04 106 108 I I I 0112 114I I SII 6 I I 7

0.0 117 I I 8

0

0. 0000 7 27 48 750.0105 141 I 83 226 278 0.0332 392 455 525 599 0.0675 757 843 932 I 025 0.1122

1220 1322 1426 1534 0. 1644 17 57 1872 1983 2101

0.2220 2340 245 5 0. 2478

0.0007 20 21 27 300. 00364243 52 540.0060 63 70 74 76 0.0082 86 89 93 970.0098 I 02 104 108 110

0.0113I I 5II I 118 I I 9

0.0 I 20 I I 5

0.0000 I I 26 47 72 0.0100 I 40 179 224 274 0.0328 388 453 52 I 594 0.0674 757 842 933 I 027 0.1124 I 225 1326 (434 I 542 0.1653 1765 1880 I 997 2113 0.2235 2355 2472 0.2495

0. 001 I I 5 21 25380.004039 45. 50 54 0.0060 65 68 73 800.0083 85 91 94 97

0.0101 101 108 108 I I I

0.0112 I I 5 1 17 I I 6 I 22

0.0120 I I 7

ALL DISPLACEMENTS IN INCHES

I. £ AF 'SD£frL£CT£ii BETV/EgM off SET C LA KPSNVEASlifcEfe DISPLACE ME MTS OF MILLING N\ACHlNE 'T/ABLE.SEE FIGS.(0-4 & io-s,

F122

TKEIE 10-5.R E F L E C T I O N OF L E N F S P R I N G . '

Leaf. M o u n t e d u p o n D o w e ls a t C o n s t a n t P itch see figs. 10-12 & 10.14MOORE & WRIGHT FEELER STRIP- O-OIOx O-SOO* 12 (tn)SPAN (y) - 1-255Un) OFFSET (Y) -.0-2 60 0 (tn) EQUIVALENT MNSLE OF LIFT - ?>t>-O"OFFSET/SPkN Rf\TlO~ 0*207111 £16 H(\ (sn tyQ) - 0-500971

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F12S

TABLE 10-6.D E F L E C T IO N OF LEAF S P R IN G .Leaf M o u n t et* u pon Towels & Rollers at C o n s t a n t PitchS T A R R S T T PEELER STRIP- 0-012* 0-500* IZ(<h)s p a h ( x ’) - l a s s L a ) o f f s e t ( V ) - 0 - 4 . I 3 NEQUIVALENT A M S LE o f l i f t - 3 6 ' O 0OFFSET/SPAN RATIO- 0-330? SISMA(SnY°)~0'4603IS

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