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Mechanics of the belt drive
Int. J . mech. 8c/. Pergamon Press. 1970. Vol. 12, pp. 1053-1063. Printed in Great Britain
MECHANICS OF T H E BELT D R I V E 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. NOTATION C ffd ffD E1 El F G P R~ Rz S t U V W Y k AT ~ p, w~
belt creep belt creep on driven pulley belt creep on driving pulley tight-side belt extension slack-side belt extension traction on pulley surface shear modulus power pulley radius radius of tension member when belt wraps on pulley distance measured along "arc of adhesion" thickness of belt envelope measured from surface to tension member speed of pulley surface speed of tension member tight-side belt speed slack-side belt speed belt width tension modulus constant depending on "speed differential" angular measure of "active" are change in tension shear strain coefficient of kinetic friction limiting value of static friction coefficient coefficient of friction angular velocity of pulley INTRODUCTION
D U R ~ G recent years belts b o t h for power transmission a n d c o n v e y o r w o r k h a v e been developed h a v i n g a flexible l o a d - c a r r y i n g m e m b e r m a d e o f high tensile fibres or steel cords, enclosed in an envelope m a d e o f some resilient material such as rubber. The envelope, which is firmly b o n d e d to the tension m e m b e r , provides the belt w i t h the necessary frictional a n d shock-absorbing qualities, a n d t r a n s m i t s t h e load from the pulley surface to the tension m e m b e r . The l a t t e r is i n t e n d e d to h a v e such a high extension m o d u l u s as to render t h e 7o 1053
1054
T . C . FIRBANK
b e 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 o p e r a t i o n , a n d i n t h e s e c i r c u m s t a n c e s i t is o f interest to e x a m i n e afresh the mechanics of belt power transmission. The need for t h i s e x a m i n a t i o n is clear o n c e i t is r e a l i z e d t h a t t h e e s t a b l i s h e d Creep T h e o r y of belt-drive mechanics, originally put forward by Reynolds 1 and subsequently d e v e l o p e d b y S w i f t s is b a s e d o n t h e i d e a t h a t b e l t b e h a v i o u r is g o v e r n e d b y t h e e l a s t i c e x t e n s i o n or c o n t r a c t i o n o f t h e b e l t a r i s i n g f r o m 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 , h o w e v e r , so a b r i e f s t a t e m e n t o f i t is m a d e here, t o g e t h e r w i t h c e r t a i n r e s u l t s w h i c h m a y be d e d u c e d f r o m it. CREEP
THEORY
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 friction--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 transmitting power, the belt and pulley surfaces will be in sliding contact and the surface speed of the pulley will be greater t h a n that of the belt. That part of the angle of contact c~ effective in transmitting 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. D r i v i n g drum. A B is t h e idle arc. BC is t h e effective arc.
The mechanism of power transmission by a driving pulley m a y be described as follows. The belt runs onto the pulley with tight-side tension T x and speed V1 which matches the surface speed V1 of the pulley. Both speed and tension remain constant as contact continues 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 slackside 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.
1055
Mechanics of the belt drive L, (200)
I
/./-
300
T -
200
Ibs
T - 150 Ibs
200
T - IOO Ibs
~3
8 .~
I00
I 0
I
I
I
I
I
2
Slip,
I 3
%
FIO. 2. Characteristic curves. Woven cotton belt 2 in. wide. T = mean tension, l, = deduced from load-stretch relation. ASSUMPTIONS
UNDERLYING
FRESH
ANALYSIS
(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 r u n n i n g 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 m a y 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 until 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 t h a n that due to bending deformation m a y 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 a m o u n t of vibration present in the drive. However, assumptions of this sort are customary and have the virtue of mathematical simplicity. A n a l y s i s of belt mechanics at the driving pulley
The belt transmits power from the i n p u t 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 t h a n oJ~, in order to satisfy F R ~ (%> ( T 1 -- Ts) V.
1056
T . C . FIRBANK
N o w consider some p o i n t in c o n t a c t w i t h the driving pulley and, if possible, m o v i n g a t e x a c t l y t h e s a m e speed. T h e n a state of shear strain will develop in t h e belt envelope which will increase as t h e pulley r o t a t i o n continues. This shear strain w i t h its associated shear stress p r o v i d e d b y static friction forces will continue to increase until the available friction forces are exceeded. Moreover, t h e friction forces bring a b o u t a fall in tension in t h e load-earrying m e m b e r which in t u r n results in a drop in t h e n o r m a l pressure b e t w e e n t h e belt and t h e pulley. A t t h e s a m e t i m e t h e frictional forces m u s t increase to p r o v i d e the increasing shear strains in t h e belt envelope. Accordingly the friction force m u s t i n c r e a s e - - a s it m a y if t h e friction is s t a t i c - - u n t i l a limiting v a l u e is reached. B e y o n d this p o i n t the belt m u s t slip on t h e pulley. I t would appear t h e n t h a t t h e arc of c o n t a c t p r o b a b l y comprises two distinct zones ; one in which slipping occurs, e x t e n d i n g b a c k w a r d s f r o m where t h e belt leaves t h e pulley up to t h e p o i n t of limiting friction m e n t i o n e d above, a n d a zone of adhesion o v e r t h e r e m a i n d e r of the arc of contact. A s o m e w h a t similar s t a t e of affiairs is visualized in Creep Theory, b u t the e x t e n t of t h e two zones is controlled b y a different limiting factor. A c c o r d i n g to Creep Theory, t h e e x t e n t of the arc of slip is d e t e r m i n e d b y ~k. I n t h e case of t h e inextensible belt b o t h Pk a n d ~, a p p e a r to be d e t e r m i n i n g factors. S h e a r stresae~ a n d strains i n the arc of adhesion
The speed differential b e t w e e n t h e belt surface and t h e tension m e m b e r inferred in t h e previous section causes the progressive g r o w t h of shear strain in t h e belt envelope until t h e supporting friction forces a t the pulley surface are insufficient to p r e v e n t slip. I t follows
Tension
,.
/
member
D~,
,V
/\ y
~
/
point
/' '
~1
Pulley
_'~_~centre
F I e . 3. Arc of adhesion. A U = arc of adhesion.
t h a t t h e arc of adhesion is n o t " i d l e " as in t h e case of t h e extensible belt a n d t h a t it t r a n s m i t s t r a c t i v e effort b y m e a n s of static friction. The r e m a i n d e r of the t r a c t i v e effort is of course t r a n s m i t t e d b y kinetic friction b e t w e e n t h e pulley and t h e belt in t h e arc of slip. Since t h e speeds of t h e pulley and t h e tension m e m b e r are constant, t h e shear strain in t h e arc of adhesion will develop in a linear m a n n e r such t h a t y -- ke, where s is t h e distance m e a s u r e d from t h e e n t r y p o i n t to t h e pulley (Fig. 5) a n d k is a c o n s t a n t d e p e n d e n t on t h e speed differential between t h e belt surface and the tension m e m b e r .
Mechanics of t h e belt drive
1057
0"5I O.4 0"3 I
÷
i 0'2: 0"1
0
1
I
T,
I
0.01
0,02
0'03
0"04
Creep FIG. 4. Creep characteristics: (a) calculated for driving pulley, (b) calculated for d r i v e n pulley. (T 1 + T s = 150 lbf.)
~
•
Tensio~ member
TI
elope
o,.o,.,o° i FIQ. 5. Effect of elasticity of tension m e m b e r .
MATHEMATICAL
FORMULATION
BoU A s s u m i n g t h a t t h e b e l t thickness is small in relation to p u l l e y radius a n d p~ a n d / z , are constants. T h e n for t h e arc o f slip T = T 2 ea~a. I f angle ~ is t h e e x t e n t of t h e are of slip, t h e n t h e tension change o v e r this arc is
T,(e.~=- 1).
1058
T. C. FIRBANK
The tension change o v e r t h e arc of adhesion -~ s u m of t r a c t i o n forces o v e r t h e arc a v e r a g e shear force per u n i t length × Or - ~) × R~
= ~~
~ , ( ~ - ~) R,*
Hence,
and
T, = T h e result 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. t h e whole t r a c t i v e effort is p r o v i d e d b y static friction, t h e n
(, F o r smaller values of t h e ratio TIlTs we h a v e
where O~p<~p, a n d t h e friction is static friction.
Belt creep T h e i m p o r t a n c e of belt creep in relation to t h e design of m u l t i d r i v e s y s t e m in colliery c o n v e y o r p l a n t has b e e n p o i n t e d o u t b y S a n d e r s ) The definition of overall belt creep s t a t e d in t h e section h e a d e d " B e l t Creep" does n o t apply to t h e inextensible belt, b u t an altern a t i v e definition, i n v o l v i n g t h e idea of speed loss is as follows: Overall creep
CD -- U - V(R~/Ri) B U = A ~-~
(see Fig. 3).
V is t h e speed of t h e tension m e m b e r a n d U is t h e speed of t h e pulley surfaee, This definition provides a measure of t h e speed loss b e t w e e n t h e driving pulley a n d t h e belt tension member. F r o m Fig. 3, AB (T2 e ~ / R p ) p, t 1
e,k~ p, t( Tl + Tz)
1
e,,=<<{1 + [(,,-- <~)/2] ~,) + 1 WG(,.,-- <~)R~ Hence,
CD e~L~p, t 1 (ml + T,) = e ~ { 1 + [(~ - ~)/2] ~,) + 1 Wa(~, - ~) R;" T h u s creep defined in this w a y is proportional to t a n d inversely proportional to G. I t is inversely proportional to the square of t h e pulley radius and is d e p e n d e n t on b o t h p t and /~,. A p o i n t to n o t e is t h a t creep becomes infinite w h e n a -- ~, i.e. w h e n slipping friction e x t e n d s t h r o u g h o u t the entire arc of contact. Fig. 4 shows a c u r v e relating (T1 - TI) and CD based on t h e formulae of this section. * Since t h e n o r m a l force per u n i t length at t h e p o i n t separating t h e arc of adhesion f r o m t h e 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 e n t r y 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
A T = W I n G ds .Jo" SO
GW ?" 1""
y = ~+-~joJ0 ~. The solution of this integral equation gives the distribution of y, i.e. k Y = 4(aWIrO
sirOa 4 ( G W I Y O a.
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 t h a t for the elastic belt the arc is virtually "idle" as Creep Theory supposes. On the other hand, in the case of the inextensible belt, appreciable power is transmitted by the arc of adhesion.
The driven pulley I n 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 m a y be approached on lines similar to those adopted for the driving pulley. The results are:
T,--T== 1 -- e-,~={ 1 -- [(,r -- ot)/2] kt,} Tz+T,
o~>0.
1 + e-vk={1 -- [ ( , r - - tx)/2] p . } '
When there is no arc of slip
T,- T, (,,/~) t, T,-T, = 2 - ( , , / 2 ) t , '
o
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 Ta - Tt and Oa baaed on the formulae of this section.
Total creep The combined creep on both driving and driven pulleys m a y 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 t h a t 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. EXPERIMENT 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 University of Bradford Mechanical En~dneering 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 b y means of the test belt. The generator, which is r~ounted on a trolley, supplies d.c. to Driven pulley
r~
Ti
F i e . 7. Constant mean tension.
Driving pulley
Mechanics of the belt drive
1061
banks of load resistors which are used to vary the generator load. A n 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 b y a system of balance weights and a spring balance. The total tension in both strands of belt is predetermined b y weights supported by the trolley. PROCEDURE A weight of 150 lbf was applied to the tensioning device a n d the motor speed adjusted to 900 r.p.m. W i t h the speed kept constant, a series of load increments was applied to the drive and the corresponding "lost revs" a n d motor torque noted. To ensure repeatability, this was performed on several different occasions a n d the belt surface was examined regularly for signs of wear and deterioration. RESULTS The experimental results are represented in Fig. 8. Corresponding results (c) as
0,5'
i i
(b)
(c)
04
0'3
I
+
0"2
O.I
0
I
I
T
0'01
0"02
I
0'03
1
0"04
Creep FIG. 8. Creep chazaeteristics: (a) experimenta/remdts, (b) calculated results, (c) elastic creep theory results.
predicted b y 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
T, gives p, = 0.32.
= 1 + ~rps
2
1062
T . C . FmBANK
Again if the value (T, - T I / T , + T2) = 0.45 be taken to indicate slip over the whole of contact, then the result
TI- Tl
arc
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 determining 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 m a x i m u m 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. I n general, the calculated creep on the driving pulley is seen to be somewhat greater t h a n that on the driven pulley (Fig. 4). DISCUSSION 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 n o t 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 b y 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, m a y differ from the static modulus b y 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 t h a t the limiting value p, does not appear to differ significantly f r o m / ~ m a y 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. CONCLUSIONS
In t h e 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 o n a r e l i a b l e 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 o f p o w e r are t r a n s m i t t e d b y s t a t i c as o p p o s e d t o k i n e t i c f r i c t i o n forces. E x p e r i m e n t a l results covering a wider range of m e a n tension a n d belt speed t h a n those a t t h e disposal of the a u t h o r are necessary to provide r e p r e s e n t a t i v e values of the 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 h e p r o p o s e d t h e o r e t i c a l f o r m u l a e . Acbnow~t4em~nt~--The author wishes to t h a n k 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
1063
REFERENCES 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).
A d d e n d u m - - I t is to be expected t h a t at the joining, slight changes in cross-section and a change in belt strength will have some bearing on the amount of belt creep.
MECHANICS OF T H E BELT D R I V E 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. NOTATION C ffd ffD E1 El F G P R~ Rz S t U V W Y k AT ~ p, w~
belt creep belt creep on driven pulley belt creep on driving pulley tight-side belt extension slack-side belt extension traction on pulley surface shear modulus power pulley radius radius of tension member when belt wraps on pulley distance measured along "arc of adhesion" thickness of belt envelope measured from surface to tension member speed of pulley surface speed of tension member tight-side belt speed slack-side belt speed belt width tension modulus constant depending on "speed differential" angular measure of "active" are change in tension shear strain coefficient of kinetic friction limiting value of static friction coefficient coefficient of friction angular velocity of pulley INTRODUCTION
D U R ~ G recent years belts b o t h for power transmission a n d c o n v e y o r w o r k h a v e been developed h a v i n g a flexible l o a d - c a r r y i n g m e m b e r m a d e o f high tensile fibres or steel cords, enclosed in an envelope m a d e o f some resilient material such as rubber. The envelope, which is firmly b o n d e d to the tension m e m b e r , provides the belt w i t h the necessary frictional a n d shock-absorbing qualities, a n d t r a n s m i t s t h e load from the pulley surface to the tension m e m b e r . The l a t t e r is i n t e n d e d to h a v e such a high extension m o d u l u s as to render t h e 7o 1053
1054
T . C . FIRBANK
b e 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 o p e r a t i o n , a n d i n t h e s e c i r c u m s t a n c e s i t is o f interest to e x a m i n e afresh the mechanics of belt power transmission. The need for t h i s e x a m i n a t i o n is clear o n c e i t is r e a l i z e d t h a t t h e e s t a b l i s h e d Creep T h e o r y of belt-drive mechanics, originally put forward by Reynolds 1 and subsequently d e v e l o p e d b y S w i f t s is b a s e d o n t h e i d e a t h a t b e l t b e h a v i o u r is g o v e r n e d b y t h e e l a s t i c e x t e n s i o n or c o n t r a c t i o n o f t h e b e l t a r i s i n g f r o m 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 , h o w e v e r , so a b r i e f s t a t e m e n t o f i t is m a d e here, t o g e t h e r w i t h c e r t a i n r e s u l t s w h i c h m a y be d e d u c e d f r o m it. CREEP
THEORY
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 friction--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 transmitting power, the belt and pulley surfaces will be in sliding contact and the surface speed of the pulley will be greater t h a n that of the belt. That part of the angle of contact c~ effective in transmitting 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. D r i v i n g drum. A B is t h e idle arc. BC is t h e effective arc.
The mechanism of power transmission by a driving pulley m a y be described as follows. The belt runs onto the pulley with tight-side tension T x and speed V1 which matches the surface speed V1 of the pulley. Both speed and tension remain constant as contact continues 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 slackside 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.
1055
Mechanics of the belt drive L, (200)
I
/./-
300
T -
200
Ibs
T - 150 Ibs
200
T - IOO Ibs
~3
8 .~
I00
I 0
I
I
I
I
I
2
Slip,
I 3
%
FIO. 2. Characteristic curves. Woven cotton belt 2 in. wide. T = mean tension, l, = deduced from load-stretch relation. ASSUMPTIONS
UNDERLYING
FRESH
ANALYSIS
(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 r u n n i n g 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 m a y 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 until 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 t h a n that due to bending deformation m a y 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 a m o u n t of vibration present in the drive. However, assumptions of this sort are customary and have the virtue of mathematical simplicity. A n a l y s i s of belt mechanics at the driving pulley
The belt transmits power from the i n p u t 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 t h a n oJ~, in order to satisfy F R ~ (%> ( T 1 -- Ts) V.
1056
T . C . FIRBANK
N o w consider some p o i n t in c o n t a c t w i t h the driving pulley and, if possible, m o v i n g a t e x a c t l y t h e s a m e speed. T h e n a state of shear strain will develop in t h e belt envelope which will increase as t h e pulley r o t a t i o n continues. This shear strain w i t h its associated shear stress p r o v i d e d b y static friction forces will continue to increase until the available friction forces are exceeded. Moreover, t h e friction forces bring a b o u t a fall in tension in t h e load-earrying m e m b e r which in t u r n results in a drop in t h e n o r m a l pressure b e t w e e n t h e belt and t h e pulley. A t t h e s a m e t i m e t h e frictional forces m u s t increase to p r o v i d e the increasing shear strains in t h e belt envelope. Accordingly the friction force m u s t i n c r e a s e - - a s it m a y if t h e friction is s t a t i c - - u n t i l a limiting v a l u e is reached. B e y o n d this p o i n t the belt m u s t slip on t h e pulley. I t would appear t h e n t h a t t h e arc of c o n t a c t p r o b a b l y comprises two distinct zones ; one in which slipping occurs, e x t e n d i n g b a c k w a r d s f r o m where t h e belt leaves t h e pulley up to t h e p o i n t of limiting friction m e n t i o n e d above, a n d a zone of adhesion o v e r t h e r e m a i n d e r of the arc of contact. A s o m e w h a t similar s t a t e of affiairs is visualized in Creep Theory, b u t the e x t e n t of t h e two zones is controlled b y a different limiting factor. A c c o r d i n g to Creep Theory, t h e e x t e n t of the arc of slip is d e t e r m i n e d b y ~k. I n t h e case of t h e inextensible belt b o t h Pk a n d ~, a p p e a r to be d e t e r m i n i n g factors. S h e a r stresae~ a n d strains i n the arc of adhesion
The speed differential b e t w e e n t h e belt surface and t h e tension m e m b e r inferred in t h e previous section causes the progressive g r o w t h of shear strain in t h e belt envelope until t h e supporting friction forces a t the pulley surface are insufficient to p r e v e n t slip. I t follows
Tension
,.
/
member
D~,
,V
/\ y
~
/
point
/' '
~1
Pulley
_'~_~centre
F I e . 3. Arc of adhesion. A U = arc of adhesion.
t h a t t h e arc of adhesion is n o t " i d l e " as in t h e case of t h e extensible belt a n d t h a t it t r a n s m i t s t r a c t i v e effort b y m e a n s of static friction. The r e m a i n d e r of the t r a c t i v e effort is of course t r a n s m i t t e d b y kinetic friction b e t w e e n t h e pulley and t h e belt in t h e arc of slip. Since t h e speeds of t h e pulley and t h e tension m e m b e r are constant, t h e shear strain in t h e arc of adhesion will develop in a linear m a n n e r such t h a t y -- ke, where s is t h e distance m e a s u r e d from t h e e n t r y p o i n t to t h e pulley (Fig. 5) a n d k is a c o n s t a n t d e p e n d e n t on t h e speed differential between t h e belt surface and the tension m e m b e r .
Mechanics of t h e belt drive
1057
0"5I O.4 0"3 I
÷
i 0'2: 0"1
0
1
I
T,
I
0.01
0,02
0'03
0"04
Creep FIG. 4. Creep characteristics: (a) calculated for driving pulley, (b) calculated for d r i v e n pulley. (T 1 + T s = 150 lbf.)
~
•
Tensio~ member
TI
elope
o,.o,.,o° i FIQ. 5. Effect of elasticity of tension m e m b e r .
MATHEMATICAL
FORMULATION
BoU A s s u m i n g t h a t t h e b e l t thickness is small in relation to p u l l e y radius a n d p~ a n d / z , are constants. T h e n for t h e arc o f slip T = T 2 ea~a. I f angle ~ is t h e e x t e n t of t h e are of slip, t h e n t h e tension change o v e r this arc is
T,(e.~=- 1).
1058
T. C. FIRBANK
The tension change o v e r t h e arc of adhesion -~ s u m of t r a c t i o n forces o v e r t h e arc a v e r a g e shear force per u n i t length × Or - ~) × R~
= ~~
~ , ( ~ - ~) R,*
Hence,
and
T, = T h e result 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. t h e whole t r a c t i v e effort is p r o v i d e d b y static friction, t h e n
(, F o r smaller values of t h e ratio TIlTs we h a v e
where O~p<~p, a n d t h e friction is static friction.
Belt creep T h e i m p o r t a n c e of belt creep in relation to t h e design of m u l t i d r i v e s y s t e m in colliery c o n v e y o r p l a n t has b e e n p o i n t e d o u t b y S a n d e r s ) The definition of overall belt creep s t a t e d in t h e section h e a d e d " B e l t Creep" does n o t apply to t h e inextensible belt, b u t an altern a t i v e definition, i n v o l v i n g t h e idea of speed loss is as follows: Overall creep
CD -- U - V(R~/Ri) B U = A ~-~
(see Fig. 3).
V is t h e speed of t h e tension m e m b e r a n d U is t h e speed of t h e pulley surfaee, This definition provides a measure of t h e speed loss b e t w e e n t h e driving pulley a n d t h e belt tension member. F r o m Fig. 3, AB (T2 e ~ / R p ) p, t 1
e,k~ p, t( Tl + Tz)
1
e,,=<<{1 + [(,,-- <~)/2] ~,) + 1 WG(,.,-- <~)R~ Hence,
CD e~L~p, t 1 (ml + T,) = e ~ { 1 + [(~ - ~)/2] ~,) + 1 Wa(~, - ~) R;" T h u s creep defined in this w a y is proportional to t a n d inversely proportional to G. I t is inversely proportional to the square of t h e pulley radius and is d e p e n d e n t on b o t h p t and /~,. A p o i n t to n o t e is t h a t creep becomes infinite w h e n a -- ~, i.e. w h e n slipping friction e x t e n d s t h r o u g h o u t the entire arc of contact. Fig. 4 shows a c u r v e relating (T1 - TI) and CD based on t h e formulae of this section. * Since t h e n o r m a l force per u n i t length at t h e p o i n t separating t h e arc of adhesion f r o m t h e 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 e n t r y 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
A T = W I n G ds .Jo" SO
GW ?" 1""
y = ~+-~joJ0 ~. The solution of this integral equation gives the distribution of y, i.e. k Y = 4(aWIrO
sirOa 4 ( G W I Y O a.
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 t h a t for the elastic belt the arc is virtually "idle" as Creep Theory supposes. On the other hand, in the case of the inextensible belt, appreciable power is transmitted by the arc of adhesion.
The driven pulley I n 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 m a y be approached on lines similar to those adopted for the driving pulley. The results are:
T,--T== 1 -- e-,~={ 1 -- [(,r -- ot)/2] kt,} Tz+T,
o~>0.
1 + e-vk={1 -- [ ( , r - - tx)/2] p . } '
When there is no arc of slip
T,- T, (,,/~) t, T,-T, = 2 - ( , , / 2 ) t , '
o
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 Ta - Tt and Oa baaed on the formulae of this section.
Total creep The combined creep on both driving and driven pulleys m a y 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 t h a t 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. EXPERIMENT 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 University of Bradford Mechanical En~dneering 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 b y means of the test belt. The generator, which is r~ounted on a trolley, supplies d.c. to Driven pulley
r~
Ti
F i e . 7. Constant mean tension.
Driving pulley
Mechanics of the belt drive
1061
banks of load resistors which are used to vary the generator load. A n 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 b y a system of balance weights and a spring balance. The total tension in both strands of belt is predetermined b y weights supported by the trolley. PROCEDURE A weight of 150 lbf was applied to the tensioning device a n d the motor speed adjusted to 900 r.p.m. W i t h the speed kept constant, a series of load increments was applied to the drive and the corresponding "lost revs" a n d motor torque noted. To ensure repeatability, this was performed on several different occasions a n d the belt surface was examined regularly for signs of wear and deterioration. RESULTS The experimental results are represented in Fig. 8. Corresponding results (c) as
0,5'
i i
(b)
(c)
04
0'3
I
+
0"2
O.I
0
I
I
T
0'01
0"02
I
0'03
1
0"04
Creep FIG. 8. Creep chazaeteristics: (a) experimenta/remdts, (b) calculated results, (c) elastic creep theory results.
predicted b y 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
T, gives p, = 0.32.
= 1 + ~rps
2
1062
T . C . FmBANK
Again if the value (T, - T I / T , + T2) = 0.45 be taken to indicate slip over the whole of contact, then the result
TI- Tl
arc
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 determining 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 m a x i m u m 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. I n general, the calculated creep on the driving pulley is seen to be somewhat greater t h a n that on the driven pulley (Fig. 4). DISCUSSION 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 n o t 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 b y 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, m a y differ from the static modulus b y 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 t h a t the limiting value p, does not appear to differ significantly f r o m / ~ m a y 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. CONCLUSIONS
In t h e 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 o n a r e l i a b l e 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 o f p o w e r are t r a n s m i t t e d b y s t a t i c as o p p o s e d t o k i n e t i c f r i c t i o n forces. E x p e r i m e n t a l results covering a wider range of m e a n tension a n d belt speed t h a n those a t t h e disposal of the a u t h o r are necessary to provide r e p r e s e n t a t i v e values of the 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 h e p r o p o s e d t h e o r e t i c a l f o r m u l a e . Acbnow~t4em~nt~--The author wishes to t h a n k 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
1063
REFERENCES 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).
A d d e n d u m - - I t is to be expected t h a t at the joining, slight changes in cross-section and a change in belt strength will have some bearing on the amount of belt creep.