SECTION 1 CHAPTER 1 BELT DRIVES.pdf

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SECTION I : CHAPTER 1

1.1 Introduction 2

1.2 Ratio of bet tentions 2

1.3 Power transmtted 21.4 Maximum power transmtted 2

1.5 Effect of centrifuga tension 2

1.6 Intitia tension in bet 3

1.7 Length of bet.i.; . 3

1.8 Flat bets. . 34

.. 5Example IExample 2

Example 3 6Exercise I 8

1.9 V-bet drives 9

1.9.1 L inked V-bets 91.9.2 Timng bet 9

Example 4 9

Exercise 11 10

Example 5 II

Example 6 13

!I~

II ,

i~

It~~l!====::!J ~

r . , J 'V'- A <if(f ~j · lf; . ~."~~ • • • ... • li ,~,

. , . t i ; 9 .,ill r

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chapter 1

BELT DRIVES

MECHAN IC A L ENGINEER ING : BE LT DR I VES

...-----41 PREFACE It------.l!:::=:======:J

The emphasis of this study materia is to provide the user not only with materia that will

prepare him/her for the engineering examnation, but aso to depict reaistic situations encoun-

tered in engineering practice.

The book contans the deveopment and explanation of specific topics, illustrative examples

and exercises designed to test the user's ability to apply the theory.

The emphasis isplaced on deveoping the student's ability to anayze problems - amost impor-

tant skill for any engineer.

Notice must be taken that some of the chapters are only applicable to the mning industry.

Please refer to the chapter's index for guidance regarding the above.

Iapologize for any errors which may have escaped the scrutiny of the author. Ihould be glad

to receve any corrections or constructive criticism

D.J. BotesAuthor

AUGUST 2007lst Edition

1OBJECTIVE

1

With completion of this chapter the user will be ableto:

• Understand the basic working principles of bet drives.

• Solve practica problems on a l types of bet drives.

• Know the advantages and disadvantages.of the different types of bet drives .- -



1.1 Introduction

Bet( drives are used to transmt power between par-

a~1 shafts. The shafts must be separated a certan.1 inimum distance, which isdependant upon the type01' bet used, in order to work most efficiently. Bets

'icve the following characteristics:

r -~ "

I

Ii~

II

I~ - - - - - - - - ~ - - - - - - - - - - - - ~.

They may beused for long centerdistancesBecause of the slip and creep of bets,theangular - veocity ratio between thetwoshafts isnether constant nor exactlyequa to the ratio of the pulley dia-

meters.When using flat bets, clutch action may

beobtaned by shifting the bet fromaloose toa tight pulley.When V-bets are used, some variationin the angular - veocity ratio may beobtaned by employing a smal pulley

withspring-loaded sides. The diameterof the pulley isthen a function of the

bet tension and may bevaried bychang-ingthe centre distance.

Byemploying step pulleys, an econom-ca means of changing the veocity ratiomay beobtaned,

', change inbet tension due 10friction forcesbetween!h,. bet and pulley will cause the bet to eongate or,0;"1 tract and move reative tothe surface of the pulley.

Ths motion iscaused by eastic creep and is associ-ated with sliding friction as opposed tostatic friction.':'h=••tion at the driving pulley, through that position:n' tlle angle of contact that is actualy transmttingr',wer, is such that the bet moves more slowly thanIhe surface speed of the pulley because of theeastic

~ECHANICAL ENGtNEERING : BEI.T DRIVES

creep. For the driving pulley the bet first contracts

thepulleywith the tight-side tension T, andaveocityv,which isthe same as thesurface veocity of thepul-ley. Then creep or slidingcontact begins, and thebet

tension changes inaccordance with the friction forces.

At theendof theeffectivearc thebet leaves thepulley

with aslack-side tension T,.

1.2 Ratio of belt tensions

Consider a bet partly wound round a pulley so thatthe angle of wrap ise, and letT, and T, be tensionsin the bet when it is about 10 slip in the direction

shown.

R

d O2

dO

T+dT

,/ - .-,,,,,,,,,

,i-

TI

,.,.,.,.,,

.,

··•T 2

If the tensions at the ends of an eement subtendingan angle de at the centre areT and T +dT, and thereaction between the bet and the pulley is R, then,resolving forces radialy:-

If neglecting the:secondorder of smal quantities:

Resolving forces tangentialy:

( T +ar)- T = 1 1 R

dT=lR (2)

And substituting R in (I) into (2):

1.3 Power transmitted

If the bet isused to transmt power between twopul-

leys, with T, and T, the tight and slacksidetensionsin Newton's respectivey, then with abet speed V inm/s:

1.4 Maximum power transmitted

If thebet veocity can bevariedwhile themaximumbet tension T, remans fixed, the power transmtted

willbemaximum when:

1.5 Effect of centrifugal tension

For abet, of mass mper unit length, and pulley ofradius R and speedof bet V m/s.

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II,~

I

F

d O d O

IIL ~

For the centrifuga tension Tclet F be the centrifuga

h)rCl!acting on an eement of the bet subtending ananglede at the centre, then resolving forces radialy:

d eF=2T -

c 2

ri1isis the tension caused by centrifuga force on the

bet and is additiona to the tension due to the trans-mssion of power.

------_._-----NECHANICAL ENG INEERING: BELT DRlVES

1.6 Initial tension inbelt

The bet is assembled with an initia tension, T;.

Where power is beng transmtted, the tension in thetightside increases fromT, toT, and onthe slack side

decreases from T; toT,.

Now, if thebet isassumedtoobey Hookes lawand its

length to reman constant, then the increase in lengthof the tight side isequa to the decrease in length of

theslack side. Hence, sincethe lengths and cross-sec-

tiona areas of thebet are the same on each side:

T\ -~ =~ -T2-

T\ +T2Ti = 2

1. 7 Lengthof belt

Open bet drive:

d

c

a.) Length of bet

D

b.) Angleof lap:

e D-dcos- =

2 2c

Crossed bet drive:

r ,:J'",

D

c

a.) Length of bet:

+ 2e

b.) Angleof lap:

d

1.8 FlatBelts

It is commonly made of oak-tanned leather or of a

fabric which has been impregnated with rubber.

Themodern flatbet consists of astrong eastic core,

such asstee or nylon cords.

To take the tension and transmt thepower,combinedwith a flexible enveope to provide friction between

thebet andpulley.

Advantages:

Veryefficient for highspeeds

They arequietCan transmt largeamounts of

power over longcenter distances.They don't require largepulleysCan transmt power around corners orbetween pulleysatright anglestoeachother.

Are particularly useful in group-drive

instalations because of theclutching

action thatcan beobtaned.An driveefficiency of about98%Flatbet drives absorb morehorizonta

vibrations fromthe systemthanether

gears orV-bets.

DDDDDDDDD

POTCHEFSTROON COLLEGE: DAWIE BaTES (copyrighted)



NECHANICAL ENGINEERING: BELT DRIVES

~-_._--_ ._--_.-----_._-----------_._--------------------

I~::-.----------4~==E=x=am===p=e===::!.J

An open bet drive is required to transmt 25kW at 200 r/mn. The diameters of the driving

pulley and the driven pulley are200 mmand ISOmm respectivey.

Theangleof contact of the driven pulley is 1760 and thecentre distance between the twopara-

le shafts is 1,5m. Take thecoefficient of frictionas0,37 and determne:

(a) The tight and slack side tensions.(b) The angle of contact of the two pulleys if acrossed bet drive isused.

(c) The length of bet for an open drive.(d) The length of bet foracrossed drive.

SOLUTION:

(a) The bet speed:

1tDN

V=601tX 0,2X200

= 60= 2,094rnls .......)

The ratio of bet tensions:

T ,_I =e"e

ITz

1760,37 x 57,3

j T=Tzxe

~ __----------------=--3-,1-1-6-T,-Z---.......)--------------------------~

Tensionon the slack side:

P= (T\-Tz)v

25X 103 = (3,116Tz - T2)X2,094

T2= 5,644kN .......)

Tension in tightside:

TI= 3,116Tz

= 3,116x5,644= 17,583kN .......)

(b) Angle of contact forcrossed bet drive:

r ::.;-__" • . . .• . . • . . • ....----:' .• . ... • .

- - - - - - - - - - -

1,5

cl > R+rcos-=-2 1,5

cl > = 2 cos? (0,1~~075)

=166,60 .......)

POTCHEFSTROOM COLLEGE: DAWlE BOTES (copyrighted)



':~'-i:')11 ~tklphr•-- ----- ----------- -----------------------------------------------~ECHANlCAL ENGINEERING: BELT DRlVES

r-I

Il!1,III

I

III

e = 360°-<1>

= 360°-166°

= 193,4° ~

(c) Length of bet for open drive:

L=~D+d)+ (D~J +2e2 4e

=3,55m ~

(d) Length of bet forcrossed bet.

L= 1t (n-a)» (D+dJ +2e2 4c

1t ( ) (O.2*Q,ISY="2 0,2+0,15 + 4(1,5) +lt1,5)

= 3,57m ~

000000000

.~ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ~

r1t-------------t Example 2. ~ = = = = = = = = = = = = = = = = = = ~

A flat bet 5mm thick is to transmt 37,3kW fromapulley of 500mmdiameter running at 15rls to another pulley. The angleof bet contact at the driving pulley is 1750and the

, ~ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ~

coefficient of friction is0,4. Thedriveistobedesigned sothat the tensile stressinduced inthe

bet will not exceed2,76MPa.

Caculate:

(a) A suitablewidth for thebet.(b) The tota pull exerted on the shaft bearings if the pulley ismounted cenrra ly between

them and thetota weght of shaft andpulley is350N.

SOLUTION:

(a) For thewidth of the bet:

0)

r;"

MaJXimum tension in belt /0

thickness x safe tensile stress

The ratio betweentensions:

04)(~

= e' S7,3

= 3,39

T)

T 2 =339,

= 0,295 T ) ~

Thebet speed:

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-------~-----------------r .1 EC HAN lC A L ENG IN EER IN G: B E LT D R IV E S

1 -I '

7t(n +t)NV = 60

=7t X(0,5+0,005) x 15

=23,8 rols -+

Substitute into power equation:

P = (T)-T2)v

37,3 X 103 = (T) - 0,295T2) X 23,8

T) = 2,223 kN -+

Width of bet:

i!IjI,

ItIII

I,

-~)

t X 0

2223

5 X 10-3 X 2,76 X 106

= 0,161 ID -+

(b) Tota pull on shaft bearings:

Total pull = (Tt+T2)+WSHAFT +WPUI.J.E(

= [2,223+0,295 {2,223}]+ 0,35

= 3,229 kN -+

L_,_-------I

:1Example 3

11

A line shaft with pulleys has a mass of 3,5 tonnes and a radius of gyration of 225 mm

It isdriven by a 600 mm diameter pulley and flat bet from an eectric motor.

The friction between the bet and the pulley is 0,25 and the bet forms a contact angle of 200°

around the pulley.

The tension in the bet when the shaft isstationary is 1,34 kN and the torque as a result of fric-

tion in the bearings can be assumed as 40 Nm

Caculate:

(a) The ratio of the two tensions inthe bet.

(b) The effective tension in the bet.

(c) The torque that is avalable to acceerate the line shaft from a stationary condition.(d) The time needed to drive the system from a stationary condition to a speed of 240 r/rnin.

SOLUTION:

(a) The ratio of tension:

2000,25 x

= e 57,3

=2,39 ~

(b) For the effective jension in the bet:

P O TC HE F ST R OO M C OL LE GE : D AW IE B OT E S (copyrighted)



- , ..-~~---.--------------~~ECHANlCAL ENGINEERING: BELT DRIVES

rT1+T2

Intal tension=--2-

T +T_) __ 2 =1340

2T)+T2=2680N ---------- (1)

and

T ) =2 39T2 '

T) =2,39T2 ---------- (2)

Substitute (2) into (I):

(2,39T2 +T2)= 2680

2680T2= 339,

=790,56N ~

and

T)=2,39T2

= 2,39X

790,56=1889,44N ~

Then. effective tension:

T)-T2=1889,44 - 790,56=1098,88N

=1,099kN ~

(c) For the torque avalable to acceerate the lineshaft:

Torqueexerted by the bet:

T =(T) -T2)X R

=1099 X 0,62

=329,66Nm

Effectivetorque =Torque exerted bybet- friction torque.

TA=T-Tp

= 329,66-40

=289,66Nm ~

d) For thetime taken todrive thesystem

Angular acceeration:

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,.,"rxn IctMP!~: ~-.-.----------.--------------------------------------_._.-

t.1ECHANlCAL ENGINEERING: BELT DRIVES

TA=M.0..k2

289,66=(3.5 X 103)x a. x (0,225)2

0.= 1,635rad/s2 ~

For stationary 00 =rad/s, and the fina angular veocity:

2nN0)2=60

2n X 240

60

=25,13 rad/s ~

and

0)2= 00\ +0. t

25,13-0

1,635

= 15,37sec ~

000000000

?f~~ 1 I Exercise I I f~\ J/ 1t===1====::::.J

L A flat bet 200mmwide and 12mmthick transmts 15kW The driving pulley of

358 mm diameter runs at 576 r/min and rHl: angle of bet contact is 1650while the

coefficient of friction is 0,4.

Caculate:

(a) The effective pull in the bet.

(b) The factor of safety used inthe design of the bet. Ultimate tensile stress for

bet materia is 15,4 MPa.[1,344 kN 18,8)

2. A crossed-bet drive isto transmt 7,5 kW at 1440 rlmn of the smal pulley. The smal

pulley has a diameter of 250 mm the veocity ratio is 2 and the center distance between the

two pulleys is 1,5 m It isdesired to use a flat bet 6 mm thick with a coefficient of friction

0,3. If the maximum alowable stress in the bet is 1,7MN/m' and the bet has a mass of

0,97 kg/drn", determne the width of the bet. Assumng no slip.

(Centrifuga tensile stress =mY', with m in kg/m)

[78mm)

3. A pulley that ismounted on a line shaft is driven from an eectric motor by means

of a flat bet. The tota mass of the rotating system is3,5 tonnes while the radius of

gyration is230 mm The pulley on the line shaft has adiameter of 600 mm friction

coefficient between the bet and the pulley is0, 25 and the contact angle is 2000 on the

pulley. The initia tension in the bet is 1,35 kN. Torque as a result of friction in the

bearings is 52 Nm

Caculate:

(a) The torque avalable to acceerate the line system for stationary.

(b) The speed the line shaft will rotate if it isdriven from stationary for aperiod of 21

seconds.[280,13 Nm; 303,4 r/mn]

POTCHEFSTROOM COLLEGE: DAWIE BOTES (copyrighted)



------------ -------_._------------------------------------~ECHANIC AL ENGINEERING: BELT DRIVES

1.8 V-belt drives

II .,

I

L -----1

A V-bet is made of fabric and cord, usualy cotton.

rayon, or nylon, and impregnated with rubber. Incr nrrast to flat bets, V-bets are used with smalersheaves and at shorter center distances. V-bets are

~;ghtly less efficient than flat bets, but a number of

rherncan beused on a single sheave, thus making a

multiple drive. They areendless, which eimnated the

joint used in flat bets. Efficiency of 70 - 96%.

1.0.1 Linked V-belS:

In composedof a large number of rubberized-fabric

:;nksjoined by suitable meta fasteners.

Tl.is typeof bet may bedisassembled at any pointandzdjusted to length by removing some of the links.

This eimnate the necessity for adjustable centersandsirnnlifies the tension for maximum efficiency and

aso reduces the inventory of bet sizes which would

usualy bestocked. Normaly used inpresses and roll-

ers, etc.

1.8.2 Timng bet:

It ismade of rubberized fabric and stee, having teethwhich fitinto grooves cut out on the periphery of the

pulleys.

The timng bet does not stretch or slip and conse-quently transmts power at aconstant angular veocityratio The factthat thebet is toothed provides severa

advantages over ordinary beting.

Advantages:

No initia tension isnecessary, so that

fixed center drives may beused.The eimnation of the restrictionof speeds, theteeth make it possible

torun at nearly any speed.

Disadvantages:

Initia cost of the betThe necessityof grooving the pulleys.

The attendant dynamc fluctuations

caused at thebet-tooth meshingfrequency.

Example 4

000000000

A V-bet isto transmt 23kW fromapulley of 254mmmean diameter rotatingat 1800r/rnin

toa second pulley of 915mmmean diameter. Thecenter distance between thetwopulleys is

1meter, included V-grooveangle of the pulley is 40°, coefficient of frictionbetween the betand pulley is 0,2, tension in the tight-side of the bet is887,5 N, massof bet is0,755kgper

meter length.

Caculate the number of V-betsrequired forthedrive.

SOLUTION:

Betspeed:

1tDN

V=601tx 0,254 x 1800

=60

=23,94 rnls ~

With the mass per meter given, thecentrifuga tensionT,is:

c=m

d=2S4cm D=91Smm

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ME CHAN ICAL ENGINEER ING: BELT DRtvES IQ

Totalpowern=

Beltpower23000

T =mv2c

= 0,755 X 23,942

= 432,7N --- )0

=8316

= 2,77

say, n = 3 belts --- )0

Contact angle:

e D-deos-=---2 2e

0,915-0,254

2 x 1

=0,331

e =141,40 --- )0

000000000

~ ~ / i ~ I = = = E = x = e r = c i = s e = n = = ~ 1 1o determne theslack-side tension:

1.The following information regarding aV-betdriveisgiven:~ - Tc =e"Ocoecc+

T 2 - T c

887,5 - 432,7 0,2• 14 ,~ COIcc20·- - - . . .:- - - ~ = e 57,3

T 2 - 432,7

T2=540,15N --- )0

Safebet tension per betDensity of betmateria

Sectiona area of betCoefficient of friction between betandpulley

Angle of contact onsmal pulleyDriving pulley diameter

Drivingspeed

Number of V-bets

6,5MPa1,15Mg/m750mm

0,25

1650

845mm920rlmin

2he power foreach bet:

Grooveangle 440

P=(T1-T1)v

=(887,5 - 540,15) x 23,94

=8,316kW ~

Caculate:

(a) The power transmtted by thedrive(b) The power transmtted if 4%bet slip takes place(c) Efficiency of thedriveif theeectric motor deivers275kWand slipisneglected.

Thus, the number of bets required: [240kW; 230,26 kW; 83,73%]L ~-----------------

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~ECHANICAL ENG INEERING: BELT DRIVES

I~ . 2. The power transmssion to apump mustbeachieved at themost economca bet veocity

viaa V-betdrive from a 2900rlmn motor. The maximum permssion tension inabet is600Nand itsmass is0,3 kgper meter length. Thegrooveangle is 36° and thecoefficient

of friction between the bet and the V -groove sides is0,3. If the distance between two

shafts is 1,2 m:

Caculate:

(a) The size of the V-bet pulleys totransmt thepower tothepump

(b) The least angle of wrap(c) The power transmtted per bet

[329mm; 170mm; 172,4°; 9,8 kW)

000000000

~- - - - - - - - - - - - - - - - - ~=====E =x =a =r n =p =l e ==5 ====~- - - - - - - - - - - - - - - - ~

A flat bet is tobedesigned totransmt 1ID kW at abet speed of 25 ms between two pulleysof diameters 250 mm and 400 mm, having apulleycenter distance of I meter. The alowable

bet stress is 8,5 MPa, and bets are avalable having a thickness-to-width ratio of 0,1 and amateria density of 1100 kg/m. Given that thecoefficient of friction between the bet and

pulleys is0,3,

Determne:

(a) The mnimum required bet width.(b) The necessary instalation forcebetween thepulley bearings.(c) The force between the pulley bearings when the full power is beng transmtted.

I~-.--~-----------------------....ISOLUTION:

II

------------------

From the sketch the angle of wrap is:

a R-r

cos-= --2 1000

200-125

1000

e = 171,40 -:- 1801t

= 2,99 radians --- )0

Themassof themateria:

m=pxV

=1100X 00 xO,l x l

= (11000 2) kg --- )0

The centrifuga force:

POTCHEFSTROO~1 COLLEGE: DAWIE BOTES (copyrighted)



-.--------.----~---->---------------.~ECHANICAL. ENGINEERING: BELT DR IVES

I?

--------------------_._--------------------------------

T = mv 'c

= 110002 X 252

=(68,7500 2 X 103) N ~

The maximum tension for thebet:

7;=ooxtxcr

=00X 0,100X 8,5 X 106

=(8500 2 X 104)N ~

Use the tension ratio forthe slack side tension:

T\ - Tc =e"eTz -Tc

(85 - 6,875)00 2 X 104

=eO,3 2,99

T2 - (6,875 X 002 X 104)

T2 ={38,72400 2 X 104)N ~

(a) The bet width when thpower todrive thebet is:.

i,I P = (T\-T2)v,I 110 X 103 = (85 - 38,724)00

2X 10

4X 25

I 00= 0,0975 m

1 . , =_9_7,_5_mm_ ~ _ _ _ I

POTCHEFSTROOM COLLEGE: DAWIE BaTES (copyrighted)

(b) For the instalation forcebetween thepulleybearings:

Tt =85co2

X 104

=85 X 0,09752 X 104

= 8,08 kN ~

T2 = 38,724 co2 X 104

= 38,n~X 0,09752

X 104

=3,681 kN ~

The initia tension of thebet:

2~ = r;+T28,08 +3,681

T ; = -=---2---- '---- '-

= 5,88 kN ~

The inclination of thebet tothecenter lineof the driveis:



r-I EC HAN ICA L ENGINEER IN G: BE l.T D R IV ES

r - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ,

I

Then, initia forcebetween bearings:

F; =2 T I cos Q >

=2 X 5,88 cos 4,30

=11,727 kN ~

(c) Force between bearing when full power is transmtted:

F = (TI - T2)cos Q >

= (8,08- 3,681) cos 4,30

= 4,387 N ~

II

L000000000

Example 6

A bet drive is fitted with a gravity idler. The driverotates anti-clockwise at 360 rpm and thecoefficient of friction between bet and pulley is0,3. If any sag inthebet canbeneglected,

• Determne:

I (a) The initia bet tension

I . ~_)_T_h_e_po_w_e_r_tranm_l_·tt_e_d_b_Y_th_e_b_e_lt_. ~

SOLUTION:

Il

Vertica

p

Driver

<l300mm

4()()mm

180N

Takemoments about the leverpivot:

CM=ACM

P x 300= 180 x 400

p= 240N ~

Buttheresultant of the initia bettensions of theidlerwillbeat anangle of 45° tothehorizonta.

cp = 45° - 30°

=15°~

POT CHEFSTROO~ COL LE GE : D AW IE BOTE S (copyrighted)



HE CHA rllCAL ENG [NEERING : B EL T D R I VES•..:.~~_~~~~_'.';_~ _

R

pR=-

cos 15°

240=--cos 15°

= 249 N~

T2=Rcos 45°

= 249 x 0,707

=176,04 N ~

Due to theaction of the idler, theslacksidetension remans

constant at 176,04N.

The angle of wrap will be:

IL ~

14

e = (180+90) x 1;0

= 3,66 rad ~

For T, the tension ratio is:

T . = e'S

T2

T . = e0,3 x 3,66

176,04

Tl = 528,1 N ~

Thebet speed:

1t ONV=-

60

1t x 0,3 X 360=

60

= 5,655 rnls ~

Then, thepower transmtted:

P=(T1-T2)v

= (528,1 - 176,04) x 5,655

= 1,99kW ~

POTCHEFSTROOM COllEGE: DAWIE BOTES (copyrighted)

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