Int. J . mech. 8c/. Pergamon Press. 1970. Vol. 12, pp. 1053-1063. Printed in Great Britain

MECHANICS OF THE BELT DRIVE

T. C. FIRBANK School of Mechanical Engineering, University of Bradford

(Received 25 April 1970, and in revised form 20 Ju/y 1970)

Summary--The mechanics of the belt drive is considered when the belt possesses a soft pliable envelope to grip the pulley and strong tension members to transmit the power. I t is concluded that shear strains in the belt envelope are a large factor in determining drive behaviour. This is in contrast to the Elastic Creep Theory which explains the traditional belt drive in terms of longitudinal strains.

N O T A T I O N

C belt creep ffd belt creep on driven pulley ffD belt creep on driving pulley E 1 tight-side belt extension E l slack-side belt extension F traction on pulley surface G shear modulus P power

R~ pulley radius Rz radius of tension member when belt wraps on pulley S distance measured along "arc of adhesion" t thickness of belt envelope measured from surface to tension member

U speed of pulley surface V speed of tension member

tight-side belt speed slack-side belt speed

W belt width Y tension modulus k constant depending on "speed differential"

angular measure of "active" are AT change in tension

shear strain ~ coefficient of kinetic friction p, limiting value of static friction coefficient

coefficient of friction w~ angular velocity of pulley

I N T R O D U C T I O N

DUR~G recent years belts bo th for power transmission and conveyor work have been developed having a flexible load-carrying member made of high tensile fibres or steel cords, enclosed in an envelope made of some resilient material such as rubber. The envelope, which is firmly bonded to the tension member , provides the belt wi th the necessary frictional and shock-absorbing qualities, and t ransmits the load from the pulley surface to the tension member . The la t ter is in tended to have such a high extension modulus as to render the

7o 1053

1054 T . C . FIRBANK

be l t v i r t u a l l y i n e x t e n s i b l e d u r i n g ope ra t ion , a n d in these c i r c u m s t a n c e s i t is o f

i n t e r e s t to e x a m i n e afresh the m e c h a n i c s of be l t power t r a n s m i s s i o n . The n e e d

for th i s e x a m i n a t i o n is clear once i t is rea l ized t h a t t he e s t ab l i shed Creep T h e o r y

of b e l t - d r i v e mechan ic s , o r ig ina l ly p u t f o rwa r d b y R e y n o l d s 1 a n d s u b s e q u e n t l y

deve loped b y Swif t s is based on t he idea t h a t be l t b e h a v i o u r is g o v e r n e d b y t he

e las t ic e x t e n s i o n or c o n t r a c t i o n of t he be l t a r i s ing f rom t e n s i o n v a r i a t i o n s w i t h i n

it . Creep T h e o r y is n o t well k n o w n , however , so a b r i e f s t a t e m e n t of i t is m a d e

here, t oge the r w i t h ce r t a in resu l t s wh ich m a y be d e d u c e d f rom it.

C R E E P T H E O R Y

The theory assumes that the belt is flexible and extensible, and is sufficiently thin to make shear and bending strains negligible. Then whenever a change in belt tension occurs due to frictional forces between the belt and the pulley, the belt will extend or contract elastically and move relative to the rigid pulley surface. This motion is called "elastic creep" ~nd is associated with sliding friction as opposed to static f r ic t ion--an important point which has bearing on later discussion. Thus for a driving pulley, throughout that part of the angle of contact which is effective in transmitt ing power, the belt and pulley surfaces will be in sliding contact and the surface speed of the pulley will be greater than that of the belt. That part of the angle of contact c~ effective in t ransmit t ing power is termed the angle of creep or the "effective arc" whilst the remainder is called the "idle arc" (Fig. 1).

A

v,

V2

T2 C

FIG. 1. Dr iv ing drum. A B is the idle arc. BC is the ef fect ive arc.

The mechanism of power transmission by a driving pulley may be described as follows. The belt runs onto the pulley with tight-side tension T x and speed V 1 which matches the surface speed V 1 of the pulley. Both speed and tension remain constant as contact con- tinues through the "idle arc". Thereafter, sliding contact occurs and frictional forces are developed to match changes in belt tension. Finally, the belt leaves the pulley with slack- side tension T 2 and slower speed V~. I f now the overall elastic creep is defined by

c = v ~ - v ~ v~

then it may be shown that C -- E 1 - E I where E x and E~ are the fractional extensions of the belt at entry and exit to the pulley. I f Hooke's Law is assumed for the belt material then, in the absence of belt slip, the relationship

P ocC

may be deduced, where P is driving-pulley shaft power. Fig. 2 shows this relationship for a woven cotton belt.

I t will be clear already from the foregoing brief and incomplete account that Creep Theory cannot be expected to explain the behaviour of the inextensible type of belt described in the Introduction, and that the new circumstances require a fresh examination of the principles involved.

300

2 0 0

~3

8 .~ I 00

Mechanics of the belt drive

L, (200) I /./- T - 2 0 0 Ibs

T - 150 Ibs

T - IOO Ibs

I I I I I I 0 I 2 3

1 0 5 5

Slip, %

FIO. 2. Characteristic curves. Woven cotton belt 2 in. wide. T = mean tension, l, = deduced from load-stretch relation.

A S S U M P T I O N S U N D E R L Y I N G F R E S H A N A L Y S I S

(a) The load carrying member is thin, inextensible and flexible; (b) in the absence of tangential frictional forces transverse plane sections of the belt remain transverse planes; (e) the belt adheres to the pulley surface at the running on point; (d) the coefficient of kinetic friction between the belt and the pulley has a constant value/z~ and the static friction coefficient has a fixed limiting value ft,; (e) the speeds are such that inertia forces may be ignored.

A oonsequence of (a) is that the linear speed of the load-carrying member is constant throughout its length. Assumption (b) implies that substantial shear deformation of the belt does not occur unti l it enters onto the pulley. Assumption (e) implies that the surface speed of the belt adjusts to that of the pulley, the body of the belt making the necessary elastic adaptations. A speed differential other than that due to bending deformation may thus occur between the belt surface and the tension member, its magnitude depending on the degree of shear in the envelope. Concerning (d) the experimental evidence is that/z~ is not constant and depends on such factors as rubbing speed, presence of moisture and dust, surface temperature, etc. Moreover, in the circumstances, the limiting value of the static friction coefficient/~, is probably dependent on amount of vibration present in the drive. However, assumptions of this sort are customary and have the virtue of mathe- matical simplicity.

Analys is of belt mechanics at the driving pulley

The belt transmits power from the input pulley to its own load carrying member. If, for the moment, the transfer is assumed to take place without losses, then

where F is the total traction on the pulley. Equilibrium considerations give

-~R~ -~- ( T 1 - T2) R l, (2)

Combining equations (1) and (2) gives R~ w~ = V implying that a short element of the load-carrying member has an angular velocity about pulley centre O equal to that of the pulley. However, since power losses due to friction and hysteresis are bound to occur, the load-carrying element will have an angular velocity about 0 less than oJ~, in order to satisfy FR~ (%> ( T 1 -- Ts) V.

1056 T . C . FIRBANK

Now consider some poin t in con tac t wi th the dr iving pul ley and, i f possible, m o v i n g a t exac t ly the same speed. Then a s ta te of shear strain will develop in the bel t envelope which will increase as the pul ley ro ta t ion continues. This shear s t ra in wi th its associated shear stress p rov ided by s ta t ic f r ic t ion forces will cont inue to increase unt i l the avai lable fr ict ion forces are exceeded. Moreover , the fr ic t ion forces br ing abou t a fall in tension in the load-earrying m e m b e r which in t u r n results in a drop in the normal pressure be tween the bel t and the pulley. A t the same t ime the fr ict ional forces mus t increase to provide the increasing shear strains in the bel t envelope. Accordingly the fr ic t ion force mus t inc rease - -as it m a y if the fr ict ion is s t a t i c - - u n t i l a l imi t ing va lue is reached. Beyond this poin t the bel t mus t slip on the pulley. I t would appear t hen t h a t the arc of con tac t p robab ly comprises two dis t inct zones ; one in which sl ipping occurs, ex tend ing backwards f rom where the bel t leaves the pul ley up to the poin t of l imit ing fr ict ion ment ioned above, and a zone of adhesion over the remainder of the arc of contact .

A somewhat similar s ta te of affiairs is visual ized in Creep Theory, bu t the ex ten t of t he two zones is control led by a different l imi t ing factor. According to Creep Theory, the ex t en t of the arc of slip is de te rmined by ~k. I n the case of the inextensible bel t bo th Pk and ~, appear to be de termining factors.

Shear stresae~ and s trains in the arc of adhesion

The speed differential be tween the bel t surface and the tension m e m b e r inferred in t he previous section causes the progressive g rowth of shear s t ra in in the bel t envelope unt i l t h e suppor t ing frict ion forces a t the pul ley surface are insufficient to p reven t slip. I t follows

Tension member D ~, ,V

, . / / \ y ~ / / ' ' point

~ 1 Pulley _'~_~centre

FIe . 3. Arc of adhesion. A U = arc of adhesion.

t h a t the arc of adhesion is no t " id le" as in t he case of the extensible bel t and t h a t i t t r ansmi t s t r ac t ive effort b y means of s ta t ic friction. The remainder of the t rac t ive effort is of course t r ansmi t t ed by kinet ic fr ict ion be tween the pul ley and the bel t in the arc of slip.

Since the speeds of the pul ley and the tension m e m b e r are constant , the shear s t ra in in the arc of adhesion will develop in a l inear manne r such tha t y -- ke, where s is the d is tance measured f rom the en t ry poin t to the pul ley (Fig. 5) and k is a cons tant dependent on the speed differential be tween the bel t surface and the tension member .

Mechanics of t he bel t dr ive

I ÷

0"5 I

O.4

0"3

i 0'2:

0"1

1 I T, I 0 0.01 0 , 0 2 0 ' 0 3 0 " 0 4

Creep

FIG. 4. Creep character is t ics : (a) calcula ted for dr iv ing pulley, (b) calculated for d r iven pulley. (T 1 + T s = 150 lbf.)

1057

• Tensio~ member T I ~ elope

o,.o,.,o° i

FIQ. 5. Effect of e las t ic i ty of tens ion member .

M A T H E M A T I C A L F O R M U L A T I O N

BoU

Assuming t h a t t he be l t th ickness is small in re la t ion to pu l ley radius and p~ a n d / z , a re constants . T h e n for t he arc o f slip T = T 2 ea~ a.

I f angle ~ is the ex t en t of the are of slip, t h e n the tens ion change over this arc is

T,(e.~=- 1).

1058 T. C. FIRBANK

The tension change over the arc of adhesion

-~ sum of t rac t ion forces over the arc

average shear force per uni t length × Or - ~) × R~

= ~ ~ ~ , ( ~ - ~) R,*

Hence,

and

The result T, =

T 1 - T s _-- e ~ [ 1 + (1r--~)/2 p,] -- 1 T~ + T~ e r e [ I + 0r-- ~)/2 p,] + 1

follows (~ > 0). W h e n ~ -- 0, i.e. the whole t r ac t ive effort is p rovided by s ta t ic friction, then

(, F o r smaller values of the rat io TIlT s we have

where O~p<~p, and the fr ic t ion is s tat ic friction.

Belt creep

The impor tance of bel t creep in re la t ion to the design of mul t id r ive sys tem in coll iery conveyor p lan t has been poin ted ou t by Sande r s ) The defini t ion of overal l bel t creep s ta ted in the sect ion headed " B e l t Creep" does no t apply to the inextensible bel t , bu t an al ter- na t ive definition, involv ing the idea of speed loss is as follows:

Overal l creep CD U - V(R~/Ri) A B -- U = ~-~ (see Fig. 3).

V is t he speed of the tension m e m b e r and U is the speed of the pul ley surfaee, This defini- t ion provides a measure of the speed loss be tween the dr iv ing pul ley and the bel t tension member .

F r o m Fig. 3, A B (T 2 e~ /Rp) p, t 1

e,k~ p, t( Tl + Tz) 1 e,,=<<{ 1 + [( , , -- <~)/2] ~,) + 1 WG(,.,-- <~) R~

Hence, C D e~L~ p, t 1

(ml + T,) = e ~ { 1 + [(~ - ~)/2] ~,) + 1 Wa(~, - ~) R;"

Thus creep defined in this way is propor t ional to t and inversely propor t ional to G. I t is inversely propor t ional to the square of the pul ley radius and is dependen t on bo th p t and /~,. A poin t to no te is t ha t creep becomes infinite when a -- ~, i.e. when sl ipping fr ic t ion ex tends th roughou t the ent i re arc of contact . Fig. 4 shows a curve re la t ing (T 1 - TI) and CD based on the formulae of this section.

* Since the normal force per uni t length a t the poin t separat ing the arc of adhesion f rom the arc of slip is T~ eu~/R~.

Mechanics of the belt drive 1059

Influence of elasticity of tension member: Fig. 5

The dotted lines show the shear distortion of the envelope due to the speed differential between the pulley surface and the tension member only. The full lines show the actual shear distortion due to progressive contraction of the tension member as the tension in it diminishes.

Then for any point distant s from the entry point of the pulley y = shear strain due to speed differential + shear strain due to

contraction of tension member

ks + l ~ ['" AT Jo T ds

where AT is the tension drop along arc s and/¢ is a constant depending on the magnitude of the speed differential.

Now

AT = W I n G ds .Jo"

SO

GW ?" 1"" y = ~ + - ~ j o J 0 ~ .

The solution of this integral equation gives the distribution of y, i.e.

k sirOa 4 ( G W I Y O a.

Y = 4 ( a W I r O

I f now the belt is a ~ u m e d to be thin, i.e, W# is large as in Creep Theory, a distribution of shear strain as shown in Fig. 6(c) is obtained.

"~ ~ '~ . Pulley ~,. en?ry

i Arc of adhesion I

l FIG. 6. Distribution of shc~r strain: (a) no exte]~ion of tension member, (b) small extension of tension member, (c) large extension of tension

member.

Curve (a) shows the corresponding shear strain distribution when the shear is due to speed differential only and the tension member is inelastic. The area under the curves is a measure of the traction in the are of adhesion and it is clear tha t for the elastic belt the arc is virtually " idle" as Creep Theory supposes. On the other hand, in the case of the inexten- sible belt, appreciable power is t ransmitted by the arc of adhesion.

The driven pulley

In the case of a conveyor system there is no driven pulley to consider. However, the experimental work~escribed later involved the use of a belt-testing machine which applies

1060 T .C. FIRBANK

load to the belt under test by means of a driven pulley connected to a d.c. generator. The machine measures the combined creep on both driving and driven pulleys, so consideration must be given to the mechanics of belt action at the driven pulley.

The subject may be approached on lines similar to those adopted for the driving pulley. The results are:

T,--T== 1 -- e- ,~={ 1 - - [ ( , r -- ot) /2] kt,} o~>0. Tz+T, 1 + e-vk={1 - - [ ( , r - - tx) /2] p . } '

When there is no arc of slip

T,- T, ( , , /~) t , T,-T, = 2 - ( , , / 2 ) t , ' o < t , - < t , , , o, = o.

Creep is given by

Ca e-~k = po t 1 Tx + T, = e-,k={1 - [ ( , r - =)/2] p J + ] WO(,r - =) R~' ~ >~ O.

Otherwise ~. t (T , - T.)

Oa = OW~ = R~

when there is no arc of slip. Fig. 4 shows a curve relating T a - T t and Oa baaed on the formulae of this section.

Total creep

The combined creep on both driving and driven pulleys may be obtained by adding the values of Ca and Ca corresponding to the same value of T , - T= for each pulley.

The influo~e of the elasticity of the towion memb,r

The conclusions relating to the driving pulley also apply to the driven pulley. The argument is the same except that y now equals the shear due to the speed differential + the shear due to extension of the tension member. Also AT is the tension rise in the arc of adhesion instead of the tension drop. When the extension is large the arc of adhesion becomes relatively "idle" and transmits a much smaller proportion of the total traction.

E X P E R I M E N T

The belt consists of a single layer of flexible steel cords embedded in a rubber envelope. The rubber thickness is 0.2 in. on both sides of the belt and the shear modulus for the rubber was experimentally determined at 80 lbf/in I. A specially designed tensile test on a strip of belt 1 in. wide gave an extension of 0.2 per cent for an applied load of 250 lbf.

The schematic layout of the belt . testing machine is shown in Fig. 7. This machine has been used for experimental purposes in the Universi ty of Bradford Mechanical En~dn- eering laboratories for a number of years. I t consists of two trunnion mounted d.c. motors. One, which is fixed in position, is used as a driving motor and drives the other as a generator by means of the test belt. The generator, which is r~ounted on a trolley, supplies d.c. to

Driven pulley

r~

Ti

F ie . 7. Constant mean tension.

Driving pulley

Mechanics of the belt drive 1061

banks of load resistors which are used to vary the generator load. An electrical control panel incorporated in the machine serves to keep the driving motor speed constant a t all loads and also includes a "lost revs" counter which indicates the accumulated difference between the driving-pulley and driven-pulley revolutions during a test run.

The torque on each motor is measured by a system of balance weights and a spring balance. The total tension in both strands of belt is predetermined by weights supported by the trolley.

P R O C E D U R E

A weight of 150 lbf was applied to the tensioning device and the motor speed adjusted to 900 r.p.m. With the speed kept constant, a series of load increments was applied to the drive and the corresponding "lost revs" and motor torque noted. To ensure repeatability, this was performed on several different occasions and the belt surface was examined regularly for signs of wear and deterioration.

R E S U L T S

The experimental results are represented in Fig. 8. Corresponding results (c) as

I +

0,5'

i i

0 4 (c)

0 ' 3

0"2

O.I

FIG. 8.

(b)

I I T I 1 0 0'01 0"02 0 ' 0 3 0 " 0 4

Creep Creep chazaeteristics: (a) experimenta/remdts, (b) calculated results,

(c) elastic creep theory results.

predicted by Elastic Creep Theory are shown for comparison. I f the value

T,-- T, = 0.9,

be taken to give s rough indication of the onset of slip, then the result

= 1 + ~rps T, 2

gives p, = 0.32.

1062 T.C. FmBANK

Again if the value (T, - T I / T , + T2) = 0.45 be taken to indicate slip over the whole a r c

of contact, then the result

TI- Tl e . ~ ' - 1 T, + T s = e . k ~ + 1

gives p ~ = 0.32. Hence it would appear that the proposed arc of adhesion is terminated when p, = pk

and there is no sudden change in the tractive force at the point. I f this surmise is presumed to be correct and the value p, =/~k = 0.3 substituted in the proposed formulae for deter- mining total creep loss between the driving and driven pulleys, then the curve (b) shown in Fig. 8 is obtained. This curve approximates to the experimental curve (a) moderately well.

Calculations based on the proposed analysis suggest that for a power output up to about 40 per cent of the working maximum for the drive,--ruling out overall slip---static friction in the arc of adhesion provides the means of power transmission. Beyond that point an arc of slip develops, first on the driving pulley and then on the driven pulley. In general, the calculated creep on the driving pulley is seen to be somewhat greater than that on the driven pulley (Fig. 4).

D I S C U S S I O N

I t would be of interest to test the belt under a wider range of tensions and speeds. Unfortunately the testing machine was designed to study the performance of belts made from relatively elastic materials and will not accept a tensioning force of more than 150 lbf. I f this is exceeded, misalignment between the pulley shafts causes tracking difficulties. There seems to be little point in testing the belt at lower mean tensions.

The graphs of Fig. 8 suggest strongly that speed loss between the driving and driven pulleys is mainly due to creep arising from shear strains in the belt envelope, and that effects arising from belt extension are negligible. I f this is the case then the possibility of power transmission by static friction forces must be regarded seriously. At the same time it is admitted that the test provides only indirect evidence on this point. Better agreement between the experimental and theoretical curves could be obtained by a more judicious choice of values for p and G, bearing in mind that both quantities show a farily wide range of variation. The dynamic modulus G, for example, may differ from the static modulus by as much as 30 per cent. Exercises of this.kind, however, serve no real practical purpose. The most that can be expected from performance tests of this nature is to identify the over-riding influences in the situation.* The fact tha t the limiting value p, does not appear to differ significantly f r o m / ~ may be ascribed to the effects of vibration and the inherent unsteadiness of the driven pulley mounting which is not rigidly fixed but rests on rollers.

C O N C L U S I O N S

In t he absence of elastic extension, shear strains in the belt envelope or cover are a controlling factor in the mechanics of belt action and performance. Formulae based on this idea are in reasonable agreement with results obtained on a re l iab le t e s t i n g m a c h i n e . I n all p r o b a b i l i t y s u b s t a n t i a l a m o u n t s of power

are t r a n s m i t t e d b y s t a t i c as opposed to k i ne t i c f r i c t ion forces. E x p e r i m e n t a l

r e su l t s cove r ing a wider r a n g e of m e a n t e n s i o n a n d be l t speed t h a n those a t t h e

d i sposa l o f t he a u t h o r are neces sa ry to p r ov i de r e p r e s e n t a t i v e va lues of t h e

p h y s i c a l c o n s t a n t s s u i t a b l e for t he p roposed theo re t i ca l fo rmulae .

Acbnow~t4em~nt~--The author wishes to thank Mr. D. Stott and Mr. J. L. Lancaster of BTR Industries Ltd. (Leyland) for their help and advice during the execution of this project.

* See Addendum.

Mechanics of the belt drive

R E F E R E N C E S

1. O. REYNOLDS, The E ~ r t e e r 38, 396 (1847). 2. H. W . SWIFT, Proc. Inst. mech. Engrs 2, 659 (1928). 3. F. N. SANDERS, E ~ i n e e r 219, 1090 (1965).

1063

Addendum-- I t is to be expected that at the joining, slight changes in cross-section and a change in belt strength will have some bearing on the amount of belt creep.