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Indian JSciRes1(2) 583-586 2014

1Corresponding author

SQUARE HOLE CREATED BY

SHAHRAMHOSSEINI CHALE

Producing square holes in the industry is very

complexity of manufactureUsing a drill ith

production A mathematical model for drill mech

cam is designed y using the mathematical mode

where it is not fixed and moves in a non-circular

made by drill to cam and bit Examples of non-co

was to examine the universal couplings

KEYWORDS Reuleaux Polygon Square Hole Uni

Square holes are used widely in industry Exa

applications are used in some of the couplisquare shape of the hole and shaft will

involvement of couplings accordingly prevent f

There are a variety of methods for creating squ

of these methods is making a square hole by

Using CNC machine despite the high accur

much In this method the hole plan on workpiec

CAD softwares and then after becoming t

transferred to CNC machine (Alan Overby 1

method is to use an Electrical Discharge Ma

EDM is a process that uses electrical disch

electrode to erode an electrically conductive

Fleming 2005) regardless of the mechanical p

as strength and hardness In this machiningelectrode is constructed to the shape of desire

creating electrical discharge between tool

workpiece the spark is generated and the shap

on workpiece caused (KY Song et al 2013)

noticed that no physical contact between the too

workpiece is created in the EDM method

Broaching machining method is another meth

used to create square holes on workpieces

commonly used for machining of internal or ext

profiles that are difficult to generate by ot

processes such as milling and turning Origin

was developed for non-circular internal pro

square hole and keyways (U Kokturk and ESquare hole drilling mechanism including a tri

that can have rotational motion within a squ

investigate the mathematical model for the ca

called Curves of Constant Width is propos

constant width provide a great example of ve

also allow us to gain access to a large amount

In particular Reuleaux polygons a special se

constant width give unexpected results to pract

such as drilling a square hole (Peter Fumich et

existence of non-circular curves of constant

standard Euclidean plane has been known s

ofEuler eg the Reuleaux triangle was presente

983113983123983123983118983098983090983090983093983088983085983088983089983091983096983080983119983150983148

RILLING DOWN BY USING UNIVERSA

COUPLINGS

SHTORI1a

EHSAN ZAMANIb AND AHMADR

aUniversity of Kashan bShahrekordUniversityc

Shahrekord University

ABSTRACT

common and useful and at the same time along with pr

on-coaxial couplings in addition to easier manufacturi

anism has been proposed in the first part of this paper T

l The main limitation of the square hole drilling mechan

ath There is thus a need to non-coaxial couplings in order

axial couplings are Richard Schmidt Oldham and Univers

ersal Couplings Edm Machining Curves Of Constant Width

ples of these

gs Since thelead to full

eewheeling

are holes One

NC machine

cy costs too

e designed by

o G-Code it

986) Another

chine (EDM)

rges from an

material (Ben

roperties such

method thed hole and by

electrode and

e of electrode

It should be

electrode and

d that can be

Broaching is

ernal complex

er machining

lly broaching

files such as

Budak 2004)ngle-like cam

are frame To

a discussion

d Curves of

tor space and

f information

of curves of

ical questions

al 1987) The

width in the

ince the time

d by Reuleaux

to Hornblower the inventor of

(Horst Martini and ZokhrabMustmathematical properties of the

some very important applicatio

width can be freely rotated in

contact to all four sides of the sq

be used for drilling holes

squaresReuleaux polygons are fo

non-uniform circular arcs to ea

Martini H 2000)Some Reuleau

The curves of constant width

constant width in all directionsT

vertical distance between the t

curve

Figure 1 Curves of

The square frame which the cam

body of the drillAs long as the c

whole inner surface of square

rotation inside the square fram

designed on the cam it could b

surface and do milling in a squa

transmit torque between two o

Richard Schmidt coupling (fig 2

and Universal coupling or Hook

The inventor of the coupling

Alabama said that a similar link

to some German engineers for ye

discouraged from applying

erroneously assumed that the cen

its own bearing Actually Schmi

is free to assume its own center

983148983145983150983141983081

983113983123983123983118 983098 983088983097983095983094983085983090983096983095983094 983080983120983154983145983150983156983081

NON-COAXIAL

ZA ASHRAFIc

blems such as high cost and

g also reduces the costs o

e mechanism of the drill and

ism is cam center of rotation

to transmit rotational motion

al couplings which this paper

the compound steam-engine

afaev 2008) Inrecent yearseuleaux triangle have led to

s Since a curveof constan

square always maintaining

uare a Reuleauxtriangle can

of maximum area into

rmed by connections through

ch other (Kupitz YS and

x curves are shown in fig 1

ave a plane shape with a

e proposed term width is a

o parallel lines tangent to

constant width

otates inside it is fixed to the

m is rotated 360 degrees the

frame is covered by cam

e Accordingly if a bit is

able to cover entire square

re shape Off-axis couplings

f-axis shafts For instances

) Oldham coupling (fig 3)

coupling (fig 4) are noted

ichard Schmidt of madison

rrangement had been known

rs But those engineers were

he theory because they

er disk had to be retained by

dt found that the center disk

of rotation In operation al



Indian JSciRes1(2) 583-586 2014 584

three disks rotate with equal velocity (Neil Sclater and

Nicholas P Chironis 2007)

In Oldham coupling it should be noticed that it can be proved

all three disks rotate with equal rotational velocity and thus

the rotational velocity of driver shaft and driven shaft will be

equal (Yutaka Nishiyama 2009)

Figure 2 Richard Schmidt coupling

Figure 3 Oldham coupling

Figure 4 Universal coupling

A mathematical model used for cam mechanism is provided in

the first part of this paper In the second part the universal

coupling used for transmitting torque from drill to cam has

been studied

MATHEMATICAL MODEL FOR THE CAM

Equations have been derived based on the simplest curves of

constant width the circle According to fig 5 width of curve

calculated by the equation

)1(k t ht h =++ )()( π

Where h shows perpendicular line to drawn tangent to the

curve t is the angle between h line and horizontal axis and k is

a constant Family of tangents is obtained from equation

2()(sincos)( t ht yt xt y x F minus+=

Figure 5circle simplest curve of constant width

To find envelope of this family we use the equation (Andrew

David Irving 2006)

(30=part

part=

t

F F

x and y are measured in terms of the parameter t using

equations 2 and 3

(4t t ht t ht y

t t ht t ht x

cos)(sin)()(

sin)(cos)()(

prime+=

primeminus=

By choosing h(t) to different forms variety curves of constan

width equations are obtained It should be noted that h(t)

should satisfy equation 1 For example by choosing h(t)=1

equation of the simplest curves of constant width the circle is

obtainedBy choosing bt

at h += )2

3(cos)( 2

curve width

would be equal toa+2b(Andrew David Irving 2006) (Stanley

Rabinowitz 1997) If a=2 and b=8 then

(5t t t t t y

t t t t t x

cos3sin3sin)93(cos)(

sin3sin3cos)93(cos)(

minus+=

++=

The diagram of equation 5 is plotted in fig 6 using the

software maple which has shown the profile of cam curve



Indian JSciRes1(2) 583-586 2014 585

Figure 6diagram of cam curve

1Universal couplings for torque transmission

In square hole drilling mechanism the cam cen

is not fixed and is displayed in a non-circular pa

a non-coaxial coupling should be used for torqu

The universal coupling is one of the most widel

coaxial couplings which has the ability to t

between parallel and non-parallel shafts and is

two universal joints The joint was used

mechanism for the first time at 1352 AD Figu

universal joint which is designed in the 16 th

maintained in Deutsches museum in munich (

Thoss et al 2006)

Figure7Initial figure of universal joint in th

According to fig 8 the relation between the r

of the driver shaft ( 1ϕ ) and the rotational angl

shafts ( 2ϕ ) obtained as equation 6

β

φ φ

cos

tantan 1

2 =

From equation 6 the velocity ratio between t

and driven shaft is calculated into equation 7

ratio diagram has also been plotted in parameter

2

11

2

)sin(sin1

cos

φ β

β

ω

ω

minus=

ter of rotation

h Due to this

transmission

y used of non-

ansmit torque

composed of

on the clock

e 7 shows the

century and is

Chr Seherr-

16th

century

tational angle

of the driven

6

e driver shaft

The velocity

1 in fig 9

7

Figure 8 Universal joint a

Figure 9 The ω2 fraslω1r

The driven shaft acceleration is al

1

22

2

2

12)sinsin1(

2sinsincos

φ β

β β ω α

minus=

In the case shown in fig 10 if tw

doubled mode the Universal c

capable to transmit torque betwee

Figure 10 The universal cou

connection of two

In Universal coupling the relatio

driver shaftand rotational angle

in equation

2

11

3cos

tancostan

β

φ β φ =

In case which driver and driven

therefor the rotational velocity o

equal

gles in schematic form

tio diagram in φ1

so calculated from equation

82

1

o Universal joints are used in

upling is caused which is

n off-axis shafts

ling made of successive

niversal joints

between rotational angle of

of driven shaft is expressed

9

shafts are off-axis then and

f driver and driven axes are



Indian JSciRes1(2) 583-586 2014 586

CONCLUSION

Parametric equations of cam curve are obt

follows

t t t t t y

t t t t t x



minus+=

++=

Triangular cam profile is obtained from equatcan be rotated within a square frame So if the

on the cam it can make a square hole (fiq 11)

Figure 11 The cam and the bit mount

Because the cam center of rotation is not fi

couplings are used to transmit torqueUnlike

joints which rotation velocity of driver and dri

not equal in universal couplings the drive

velocities are equal And therefor these kinds o

more appropriate to transmit torque from drill to

REFERENCES

[1]

Alan Overby 1986 Cnc machini

McGraw-Hill

[2]

Ben Fleming 2005 The EDM How To

publications

[3]

KY Song DK Chung MS Park C

ldquoDevelopment of strip EDMrdquo Procedia CIRP 6

[4]

U Kokturk E Budak 2004 ldquoOp

broaching tool designrdquo intelligent Co

Manufacturing Engineering ndash 4 Faculty of E

Natural Sciences Sabanci University Orhanli

Istanbul Turkey[5]

Peter Fumich Candice Qui

FreelanPaulSharaba Josh Orzel 1987 ldquoCurv

Width and Reuleaux Polygonsrdquo Mathemati

60(3) Jun 131-140

[6]

Horst Martini ZokhrabMustafaev 2

construction of curves of constant widthrdquo Co

Geometric Design 25(2008) 25(9) December

[7] Kupitz YS Martini H 2000 ldquoOn th

inequalities for Reuleaux polygonsrdquo J Geom

[8] Neil Sclater Nicholas P Chironis 200

and Mechanical Devices Sourcebook McGraw-

ined in t as

10

ion 10 which bit is mounted

d on it

xed universal

the universal

ven shafts are

r and driven

couplings are

cam

g handbook

ook Fleming

Chu2013

53-57

timization of

putation in

gineering and

Tuzla 34956

nn David

es of Constant

cs magazine

008 ldquoA new

mputer Aided

51-755

isoperimetric

8 171-191

Mechanisms

ill

[9]

Yutaka Nishiyama 2009

Air Conditionersrdquo Osaka Keida

77-84

[10]

Andrew David Irving 2006

amp Centre Symmetry Setsrdquo MSc

45

[11] Stanley Rabinowitz1997

Constant Widthrdquo MathPro Press

[12]

HChr Seherr-Thoss F S

Universal joints and driveshaf

Heidelberg

ldquoFrom Oldhams Coupling to

iRonshu September 60(3)

ldquoCurves of Constant Width

mini-dissertation Jun pp 1

ldquoA Polynomial Curve o

pp 23 ndash 27

hmelz E Aucktor 2006

s Springer Verlag Berlin



Indian JSciRes1(2) 583-586 2014 584

three disks rotate with equal velocity (Neil Sclater and

Nicholas P Chironis 2007)

In Oldham coupling it should be noticed that it can be proved

all three disks rotate with equal rotational velocity and thus

the rotational velocity of driver shaft and driven shaft will be

equal (Yutaka Nishiyama 2009)

Figure 2 Richard Schmidt coupling

Figure 3 Oldham coupling

Figure 4 Universal coupling

A mathematical model used for cam mechanism is provided in

the first part of this paper In the second part the universal

coupling used for transmitting torque from drill to cam has

been studied

MATHEMATICAL MODEL FOR THE CAM

Equations have been derived based on the simplest curves of

constant width the circle According to fig 5 width of curve

calculated by the equation

)1(k t ht h =++ )()( π

Where h shows perpendicular line to drawn tangent to the

curve t is the angle between h line and horizontal axis and k is

a constant Family of tangents is obtained from equation

2()(sincos)( t ht yt xt y x F minus+=

Figure 5circle simplest curve of constant width

To find envelope of this family we use the equation (Andrew

David Irving 2006)

(30=part

part=

t

F F

x and y are measured in terms of the parameter t using

equations 2 and 3

(4t t ht t ht y

t t ht t ht x

cos)(sin)()(

sin)(cos)()(

prime+=

primeminus=

By choosing h(t) to different forms variety curves of constan

width equations are obtained It should be noted that h(t)

should satisfy equation 1 For example by choosing h(t)=1

equation of the simplest curves of constant width the circle is

obtainedBy choosing bt

at h += )2

3(cos)( 2

curve width

would be equal toa+2b(Andrew David Irving 2006) (Stanley

Rabinowitz 1997) If a=2 and b=8 then

(5t t t t t y

t t t t t x



minus+=

++=

The diagram of equation 5 is plotted in fig 6 using the

software maple which has shown the profile of cam curve



Indian JSciRes1(2) 583-586 2014 585













Thoss et al 2006)





β

φ φ

cos

tantan 1

2 =




2

11

2

)sin(sin1

cos

φ β

β

ω

ω

minus=

ter of rotation

h Due to this

transmission

y used of non-

ansmit torque

composed of

on the clock

e 7 shows the

century and is

Chr Seherr-

16th

century

tational angle

of the driven

6

e driver shaft

The velocity

1 in fig 9

7




1

22

2

2

12)sinsin1(

2sinsincos

φ β

β β ω α

minus=





connection of two



in equation

2

11

3cos

tancostan

β

φ β φ =



equal


tio diagram in φ1


82

1



n off-axis shafts


niversal joints



9





Indian JSciRes1(2) 583-586 2014 586

CONCLUSION


follows

t t t t t y

t t t t t x



minus+=

++=










REFERENCES

[1]


McGraw-Hill

[2]


publications

[3]



[4]





Istanbul Turkey[5]




60(3) Jun 131-140

[6]








ined in t as

10


d on it

xed universal

the universal

ven shafts are

r and driven

couplings are

cam

g handbook

ook Fleming

Chu2013

53-57

timization of

putation in

gineering and

Tuzla 34956

nn David

es of Constant

cs magazine

008 ldquoA new

mputer Aided

51-755

isoperimetric

8 171-191

Mechanisms

ill

[9]



77-84

[10]



45



[12]



Heidelberg






pp 23 ndash 27





Indian JSciRes1(2) 583-586 2014 585













Thoss et al 2006)





β

φ φ

cos

tantan 1

2 =




2

11

2

)sin(sin1

cos

φ β

β

ω

ω

minus=

ter of rotation

h Due to this

transmission

y used of non-

ansmit torque

composed of

on the clock

e 7 shows the

century and is

Chr Seherr-

16th

century

tational angle

of the driven

6

e driver shaft

The velocity

1 in fig 9

7




1

22

2

2

12)sinsin1(

2sinsincos

φ β

β β ω α

minus=





connection of two



in equation

2

11

3cos

tancostan

β

φ β φ =



equal


tio diagram in φ1


82

1



n off-axis shafts


niversal joints



9





Indian JSciRes1(2) 583-586 2014 586

CONCLUSION


follows

t t t t t y

t t t t t x



minus+=

++=










REFERENCES

[1]


McGraw-Hill

[2]


publications

[3]



[4]





Istanbul Turkey[5]




60(3) Jun 131-140

[6]








ined in t as

10


d on it

xed universal

the universal

ven shafts are

r and driven

couplings are

cam

g handbook

ook Fleming

Chu2013

53-57

timization of

putation in

gineering and

Tuzla 34956

nn David

es of Constant

cs magazine

008 ldquoA new

mputer Aided

51-755

isoperimetric

8 171-191

Mechanisms

ill

[9]



77-84

[10]



45



[12]



Heidelberg






pp 23 ndash 27





Indian JSciRes1(2) 583-586 2014 586

CONCLUSION


follows

t t t t t y

t t t t t x



minus+=

++=










REFERENCES

[1]


McGraw-Hill

[2]


publications

[3]



[4]





Istanbul Turkey[5]




60(3) Jun 131-140

[6]








ined in t as

10


d on it

xed universal

the universal

ven shafts are

r and driven

couplings are

cam

g handbook

ook Fleming

Chu2013

53-57

timization of

putation in

gineering and

Tuzla 34956

nn David

es of Constant

cs magazine

008 ldquoA new

mputer Aided

51-755

isoperimetric

8 171-191

Mechanisms

ill

[9]



77-84

[10]



45



[12]



Heidelberg






pp 23 ndash 27

