Forces between an abrasive belt and pulley

Mech. Mach. Theory Vol. 22, No. 1, pp. 97-103, 1987 Printed in Great Britain All rights reserved

0094-114X/87 $3.00+0.00 Copyright © 1987 Pergamon Journals Ltd

FORCES BETWEEN A N ABRASIVE BELT A N D PULLEY H. KIM and K. MARSHEK~" Department of Mechanical Engineering, University of Texas at Austin, Austin, TX 78712, U.S.A.

M. NAJI Department of Mechanical Engineering, Wichita State University, Wichita, KS 67208, U.S.A.

Almtraet--The friction characteristics between an abrasive belt and a driver and driven pulley are studied. The normal and tangential forces are measured by separate force transducers. The experimental results for forces on the belt show close agreement with those predicted using an equation developed from theoretical considerations.

NOMENCLATURE

In this paper, an equation relating coefficient of friction and belt tension is described for an abrasive grinding belt on a pulley. Analytical results from the derived equation are compared with experimental data obtained at very low belt speeds.

T, 7",, T2= belt tensions at 0, 0 ffi a, 0 ffi 0 p = average normal pressure between belt and pulley P2 = average normal pressure at slack side end of belt R = radius of pulley 0 = angle of contact = active angle z = average shear stress A, ~ffireal contact area A •ffi apparent contact area /4 =fficoefficient of friction

THEORY

Normal and tangential static equilibrium for a belt element of length RdO in the so-called active arc of contact, as shown in Fig. 1, requires that

C, a, B ~ constants.

T -pR = 0 d T - xR dO -~-O.

INTRODUCTION

(2)

Since the friction force per unit area ~ is proportional to the actual contact area[2],

The classical Euler equation for a fiat belt power transmission assumes a constant coefficient of friction between the belt and pulley; i.e. the coefficient of friction is independent of normal belt pressure. However, Bartenev[1, 2] showed that the coefficient of friction between a rubber structure and a hard surface depends upon the normal pressure, material constants and shear stress, and Belofsky[3] concluded that the coefficient of friction changed with creep based on the experimental measurements by Grimmer and Thormann[4]. Firbank[5] suggested that substantial amounts of power are transmitted by static friction as well as kinetic friction in a flat belt drive. Firbank[6] measured the distribution of normal and tangential forces in the arc of contact for two different flat belts. Childs[7] investigated the real contact areas between a flat belt and a transparent persl~x pulley by the technique of frustrated total internal reflection. He showed that real areas of contact were less than one-third of the apparent area and varied with load, elastic modulus and surface roughness of the belt.

ffi e ( A J A )

(3)

where c is the friction constant or limiting shear stress and A J A is the ratio of the actual contact area to the apparent contact area. The ratio A./A can be written

÷ To whom all correspondence should be addressed. MMr :: I--G

(!)

Fig. 1. Diagram of abrasive belt on a driven pulley. 97

98

H, KIM et aL

as ,4./a

(4)

= ap ~

where a, and n are constants. Substituting equations (1), (3) and (4) into equation (2) gives @ -

sp" dO =

0

(5)

where B is a constant ( B = c a ) which depends on the belt material and the belt pulley surface characteristics. Applying the boundary condition p = p~ at 0 = 0 yields p = [p~-" + (1 - n)BO] Ill -~ p=p2exp(B0)

for

n=l.

for

n #: 1

(6a) (6b)

A friction force ~ and a coefficient of friction/z can be obtained by combining equations (3) and (4) and by using ~ = t t p as = Bp"

(7)

= Bp"-'.

(8)

The constant B for the three values of n from Chiids'[7] surface model can be determined using equation (6) and the experimental data for load and active angle 0. The values of a and n from Childs are based upon the work of Greenwood and Williamson[8] who studied the contact of nominally flat elastic surfaces. Table 2 shows the theoretical estimates of n for different surface models[7]. Childs'[7] model generally requires that, constant a be determined by measurement. Childs mentions that this measurement in practice is difficult and inaccurate. In this paper, instead of measuring a directly, shear force per unit area ~ as well as normal pressure p are measured along the active angle of contact, and these values are used indirectly to determine B. EXPERIMENT

Figure 2(a) is an assembly drawing of the low speed abrasive belt test machine. Figure 2(b) is a photograph of the machine. In Table 1, there is a list of standard machine components as well as custom machined pans. In order to measure the tangential and normal belt forces acting on a pulley, two types of transducers were designed. Figure 3(a) shows a normal force transducer which is designed like a curved beam. The beam has a slot at the middle point to prevent a buckling effect. Two strain gauges are attached beneath one of the cantilever beams. Another two gauges are attached to the pulley disk, and form a half bridge. A tangential force transducer which also functions like a cantilever is shown in Fig. 3(b). Because of the geometric characteristics of the transducers, strain values of the first and last 10-13 degrees are inaccurate. Both 6 in. dia aluminum transducer disks are used as driven (idle) pulleys (2). A ten turn poten-

Table 1. Parts list for the abrasive belt test machine 1~ 2. 3, 4. 5. 6. 7. 8. 9. I0. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

6 in. ~ aluminum driving pulley Driven (transducer) pulley I in. ~ steel shaft I in. ~ steel shaft 5in. ~ torque disk Stand 10-turn potentiometer 30:1 gear reducer Handle Carriage plate---aluminum Carriage plate---aluminum ~in. ~b stainless steel rail Stopping block Pillow block--I in. bore Abrasive Bett--I in. width, 60 in. length, Grade 100 Cord Roller Weight (tension) Weight (torque) Linear bearing----~in. bore Table

Table 2. Theoretical estimates of n for different surface models Surface model Identical spherical asperities Identical cylindrical asperities Identical wedge asperities Spherical asperities Gaussian height distribution

n 2/3 I/2 I !

tiometer (7) is connected to the end of the driven pulley shaft (3). Output from the strain gauges as well as the potentiometer are transmitted to a computer data acquisition system and plotted. The input shaft of a 30:1 gear reducer (8) is rotated by a hand wheel (9). This rotation causes rotation of the driving pulley (1) which is connected to the gear reducer (8) through a shaft (4). The driving pulley causes the driven pulley to be turned through an abrasive belt (15). The 1 in. dia steel shafts (3, 4) are each supported by two pillow blocks. Each pulley is mounted on a transverse sliding plate (10, 11) which is bolted to linear bearings (20). The bearings run on two 5/8 in. dia stainless steel rails (12). The two rails are bolted to the top of table (21). Initial belt tension is applied to the belt with a deadweight (18) which is attached through a cord (16) and over a roller (17) to the driven pulley plate. Since the driving pulley carriage is fixed by a stopping block (13), the applied deadweight is transmitted directly to produce a belt tension. The torque load is applied through a torque disk (5) with deadweights (19). The torque disk converts a given weight to a torque. A variety of torque loads can be obtained simply by adding or subtracting deadweights. Abrasive belts 1 in. wide and 60 in. long of grade 100 were used for the experimental study. In the

Forces between an abrasive belt and pulley

99

(a)

Fig. 2. (a) Assembly drawing of experimental apparatus. (b) Experimental machine.

study, total tension (T~ + 7"2) was held constant at 40 lbf and the belt was run at a pulley speed of 4 rpm. At this low speed, effects of centrifugal force on the force transducers are negligible. RESULTS Figure 4(a) and (b) show the trend in which the forces on the force transducers change with increased values of transmitted torque. With zero torque load, normal force remains constant. The small amount of tangential force indicated is due to the initial deflection of the tangential force transducer. When the tangential transducer enters the belt-pulley con-

tact area, one side of the two contact wings on the pulley load cell [Fig. 3(b)] contacts the belt. The resulting normal pressure causes an initial strain in the transducer. Even if the pressure on the wing can be balanced when both wings contact the belt, there still remains a small initial unbalanced strain. This strain cannot be recovered because of the static friction. Consequently, a small value of the tangential force is indicated along the entire arc of adhesion (inactive arc) [Fig. 4(b)]. As the torque load increases, the driving force on the belt at the exit of the belt pulley contact area increases and extends backwards to match the applied torque load. Peak belt tension exists at the

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- - - - -

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- - - - - -

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Angular position(degrees) Fig. 4. (a) Normal force vs angular position for various transmitted torque conditions for driven pulley. (b) Tangential force vs angular position for various transmitted torque conditions for driven pulley, lO0

Forces between an abrasive belt and pulley

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pulley exit and decreases along the active arc. At the same time, tangential force is distributed according to the well-known creep theory. At low and medium torque load, tangential force increases with increasing normal pressure for the driven pulley. However, the tangential force curves in Fig. 4 do not show the fully-developed slip motion for the entire active angle. At heavy load, the tangential force increases almost linearly in the first half of the active angle and slip occurs in the second half of the curve (approx. 90-180 °) where the tangential shear force can be computed according to the z = #p relationship. With increasing torque load, the tangential force (traction) on the belt (i) increases up to a peak value and (ii) extends backwards towards the entry point. Eventually the whole arc of contact is involved as an

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I01

(T, - T : ) / ( T I + T2).

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AnguLar position (degrees)

Fig. 6. Comparison of theoretical equation with experimental results. (a) Normal force vs angular position. (b) Tangential force vs angular position.

102

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Fig. 7. (a) Normal force vs angular position for various transmitted torque conditions for driving pulley. (b) Tangential force vs angular position for various transmitted torque conditions for driving pulley.

active arc and then the belt-pulley contact experiences a gross slip. Figure 5 shows the dependence of :t/Tt on ( T 1 - T2)/(TI + T2). As torque load increases, the active angle increases and reaches a maximum value where the belt experiences gross slip. The constant B is calculated with equation (6) from active angle data in Fig. 5 and from values of n by Greenwood and Williamson[8].

Calculated forces for three theoretical models are plotted along with the experimental results for maximum torque load in Fig. 6. In order to show clearly the differences between models, the tangential force scale is enlarged. Figure 7(a) and (b) show the experimental normal and tangential belt forces for a driving pulley. Comparing with the experimental results for the driven pulley [Fig. 4(a) and (b)], the curves for normal and

tangential belt forces for the driving pulley are different from those for the driven pulley forces. For the maximum torque load case, the curves appear identical when one is rotated 180 ° about the force axis. The measured active angles for the driving pulley are in close agreement with the values for the driven pulley. DISCUSSION

Comparing experimental with theoretical results, the n = ~ model is a better model than the n = ½and n = 1 models. F o r the model with n = 3, the calculated constant B is 0.398 + 0.012. There is only a small difference between the theoretical model and the experimental results. The small differences in force magnitudes and active angles between experimental and theoretical results are due to the use of

Forces between an abrasive belt and pulley two force transducers with slightly different surface characteristics. The trend for the tangential force follows that predicted by the theoretical model

[Fig. 6(b)]. As shown in Fig. 4(a) and (b), the normal and tangential forces change almost linearly rather than in an exponential manner. The n = 1 case yields the classical Euler equation which predicts an exponential change in the normal and tangential forces along the active angle. A pseudo linear result is obtained with the n = ] model. For the n = 1 theoretical model, the coefficient of friction # is a constant. The coefficient of friction/~ in the n = ~ model changes in proportion to p-i/3. This, in theory, explains why the coefficient of friction drops with increasing belt tension. As Archard[9] explained, the n = ] model gives the contact characteristic of an abrasive belt: an increase in load does not create new contact areas only increases the size of existing contact areas. At low and medium torque load, substantial amounts of power are transmitted by static friction as observed by Firbank [Fig. 4(b)]. Although Firbank mentioned the possibility of power transmission by static friction in the inactive angle, a high proportion of the power is transmitted by static friction in the active angle at low and medium torque load as shown in Fig. 4(b). This can be explained by partial stick-slip motion of the belt surface. Microscopically, the surface of the belt backing material can be considered as a conglomeration of asperities. At low torque load, only a portion of the asperities experience slip, and the remaining parts stick. As the belt pressure increases, the portion experiencing slip increases. However, at low or medium torque load, the belt never experiences a fully-developed slip. At heavy load, the portion of slip increases rapidly until all the contact asperities slip. At this point, the tangential force curve follows the expected theoretical values. In the gross slip region shown in Fig. 4(a) and (b), the coefficient of friction for the Euler equation is calculated using experimental results as ~ ffi 0.22. For the n = ~ model, the friction coefficient /z changes from 0.198 to 0.250. The Euler equation could be used with a conservative value of/z, a value which is smaller than the apparent coefficient of friction, to estimate the transmitted torque.

103 CONCLUSIONS

A theoretical equation has been developed for the belt-pulley coefficient of friction and for the normal and tangential forces on an abrasive belt. From experiments, values for the constants n and B have been determined as n = ~ and B = 0.398 _+ 0.012. The coefficient of friction changed in proportion to p-i/3. For the gross slip condition, the calculated coefficient of friction changed from 0.198 to 0.250 as compared to a constant value of 0.22 for the Euler theory. F o r low to medium torque load, power is transmitted by partial stick-slip motion of the belt surface, and the belt does not experience a fully-developed slip. Normal and tangential forces have been measured using two different belt-pulley interface force transducers. These forces were found to change almost linearly rather than in an exponential manner as predicted by the classical Euler equation. The theoretical equation developed in this paper shows a good agreement with the trends of the experimental results. Acknowledgement--The authors wish to thank the Norton Company, Coated Abrasive Division, for sponsorship of this work.

REFERENCES 1, G. M. Bartenev, Rubber structure and coefficient of friction. Rubber Chem. Technal. 35, 371-378 (1962). 2. G. M. Bartenev, V. V. Lavrentjev and N. A. Konstantinova, The actual contact area and friction properties of elastomers under frictional contact with solid surfao~s. Wear 18, 439-448 (1971). 3. H. Belofsky, On the theory of power transmission by a flat, elastic belt. Wear 25, 73-84 (1973). 4. K. J. Grimmer and D. Thormann, Focrdern u Heben 15, 845--852 (1965). 5. T. C. Firbank, Mechanics of the fiat belt drive. ASME Mechanisms Conference and International Symposium on Gearing and Transmissions, Calif., 72-PTG-21 (1972). 6. T. C. Firbank, On the forces between the belt and driving pulley of a fiat belt drive. ASME Design Engineers Technical Conference, Ill., 77-DET-161 (1977). 7. T. H. C. Childs, The contact and friction between fiat belts and pulleys. Int. J. Mech. Sci. 22, 117-126 (1980). 8. J. A. Greenwood and J. B. P. Williamson, Contact of nominally fiat surfaces. Proc. R. Soc. A 29'5, 300-319 (1966). 9. J. F. Archard, Elastic deformation and the laws of friction. Proc. R. Soc. ,4 243, 190-205 (1957).

LES FORCES ENTRE UNE COURROIE ABRASIVE ET LA POULIE Rtmmt--Ce travail est une etude thtorique et experimentale des caract~ristiques du frottement entre une courroie abrasive et une poulie menante ou mente. Les forces normales et tangetielles sont mesurtes s~pareraent avec des dispositifs particuliers. Les rtsultats exptrimentaux concordent avec ceux predits par une equation baste sur des considtrations th~oriques et developpte par les auteurs.