P e r p m o a

M¢ck. Mack. Tkeorv Vol. 29, No. 6. pp. 865--876, 1994 Copyright ~ 1994 Elsevier Science Lid

0094-114X(93)E0003-O ~nted in Great Britain. All rights re,fred 0094-I 14X/94 $7.00 + 0.00

A N A L Y S I S O F B E L T B E H A V I O R A N D S L I P

C H A R A C T E R I S T I C S F O R A M E T A L V - B E L T C V T

HYUNSO0 KIMt Department of Mechanical Engineering. Sungkyunkwan University. Chunchun-dong 300,

Suwon, 440-746, Korea

JAESHIN LEE Pacific Controls Ltd, Wonchon-dong, Suwon, Korea

(Received 21 August 1991; in revised form 23 August 1993: received for publication I0 Not'ember 1993)

Atetract--The metal V-belt behavior of a V-belt CVT was investigated analytically and experimentally. Numerical results showed that the belt radial displacement increased in the radial inward direction for the driven pulley, while that of the driver increa~.d slightly and decreased with the increasing torque load. Experimental results for the belt radial displacement were in good agreement with the theoretical results. Also, the slip characteristics between the belt and the pulley were studied by using the speed ratio-torque load-axial force relationship derived from the belt behavior analysis, it was found that the gross slip points depend on the torque transmission capacity of the driven side. Experimental results for the slip and axial forces were in the range of the coefficient of friction/~ = 0.08-0.13.

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

The metal V-belt CVT is one of the CVT's that have been developed to the point of automotive applications, it is well known that the metal V-belt CVT is able to achieve more efficient operating levels with respect to drive performance and fuel consumption than conventional multi-ratio gearbox transmissions. The metal V-belt (MVB) transmits power by thrust force between the metal blocks [I]. Figure ! shows the metal V-belt assembly in the driver and driven pulleys and the belt structures. The pulleys are subject to axial forces which can be varied in order to adjust the pitch radii of the belt position on the pulleys. The MVB consists of a number of thin, fiat tension bands and metal V-blocks, which fill the length of the bands completely. The MVB is positioned internally in the V-shaped pulley and operated with variable pitch radius. Continuously variable transmission can be achieved by changing the belt pitch radius via the axial forces on the movable side of the pulley. During the operation of the MVB, a block on the driver side is carried forward by the pulley, and is pressed on the block ahead of it, generating a compression force between the two. As the block travels along the contact arc, this compression force builds up from the leading edge to the trailing edge. On the driven side, the block carries the pulley, so the compression force between the block decreases along the way of the contact arc.

The literature on MVB CVT is very scarce. Gerbert [2] performed some rigorous analysis on the MVB mechanics based on the results of rubber V-belt analysis [3] and obtained a set of nonlinear equations considering the play between the blocks and the angle between the belt and circumfer- ential direction. But his numerical analysis requires some cautions to avoid orthogonal points and to select boundary conditions for the angle. Sun [4] and Katsuya et al. [5] obtained a set of equations to describe the MVB behavior including the effects of friction between the bands. However, they assumed that the friction between the belt and the pulley exists only in the circumferential direction. In actual MVB behavior, the belt moves in radial direction as well as circumferential. The radial motion causes radial component of friction force and makes the belt trajectory noncircular, which complicates the problem enormously. However, it is inevitable to

tTo whom all correspondence should be addressed.

865

866 HYu~o KIM ~md J , ~ I .~

/ ~ r . . ~ " Steel band / / " l ]

/ / ~ Block

Fig. I. Metal V-belt.

Movable flange I

I i

Fixed flange

consider the belt radial motion to obtain an accurate amount of the active arc that is integral part in design of the torque transmission capacity of the MVB CVT.

In this paper, the belt behavior of the MVB c v ' r is investigated analytically and experimentally. Considering the belt radial motion, relatively simple differential equations are derived. Thus, the belt radial displacements are measured and compared with the theoretical results. Also, the slip characteristics between the belt and the pulley are studied by using the speed ratio-torque load-axial force relationship derived from the belt behavior analysis.

For the

METAL V-BELT MECHANICS

analysis of the MVB's behavior, the following assumptions are made:

(I) Metal blocks and bands are treated as a continuous belt. (2) Band tension is constant, i.e. there is no friction between the bands. (3) In an active arc, the belt moves in the radial and circumferential direction and

the compression force changes due to the corresponding kinetic friction. In an inactive arc, there is only a static friction and the compression force remains constant.

(4) The coefficient of friction is constant. (5) The inertia forces are neglected. (6) The bending stiffness of the belt is neglected.

Figure 2 shows a free body diagram of a belt element in the active arc. Representing the belt tension F = T - P, from quasistatic equilibrium requirements, Fig. 2, the following equations are obtained:

dP - d-'O = - p " = F" = 21~Nr cos ~ sin 7, (I)

T - P -- F -- 2Nr(s in / / - /z cos/I cos 3'). (2)

Figure 3 illustrates a radial displacement X by an axial force. The axial component of the force compresses the block in the axial direction and this causes the belt to penetrate in the radial direction as

2 (T - e)(cos ~ - /~ sin//) (3) X = - ~ P: = rEs( s in ~ - I~ cos ~ cos 7 ) '

where Kb is a radial spring constant of the block defined by Gerbert [I] and P: is an axial compressive force.

Now let us consider the relationship between the belt radial and the circumferential displacement. In Fig. 4(a), belt radius R in a state of free tension is represented as a dotted line. When the belt transmits a power, the whole arc of contact between the belt and the pulley is divided into two regions: (I) inactive area and (2) active area. The belt tension changes from the initial tension to Fs in the tight side and F2 in the slack side. The trajectory of belt in this state is shown as a solid

Metal Vobelt ~

P + d P

Nr d I dO

ItNr dO cos "f I ttNr dO cos "f

Fig. 2. Forces on a metal V-belt element in active arc.

867

line in Fig, 4(a). In the inactive arc, the belt radius re is constant since there is no change in the belt tension. In the active arc, the belt tension changes by the kinetic friction and the belt moves in not only circumferential but also radial direction. For the length r d~ of the element

r d~ =. tan y dr. (4)

The length of the belt element aobo at initial state varies into ob and ab' where the arc aob0 is the belt length in free tension state, and ab is in pretension state, and ab' is in loaded state. The relationship between these belt lengths are as follows:

ab - aoboC~ = a b ' - aoboC, (5)

where ce and ~ arc the longitudinal strain of the belt in the inactive and the active arc. Rewriting the above equation with belt radius and angle, equation (5) becomes

re dO - R d0 ~s -- r dO + • d~ - R d~ £. (6)

( a ) ( b ) I I

L IL . . . . :

I

I I i i

X

Fig. 3. Belt deformation by axial forces.

868 HYUN5120 KIM and JAF~qlN LEE

(a) ~ Active (b) a0

Inactive a,2 .b

I

F z = T-P z F l = T-P t

Fig. 4. Belt deformation in radial and longitudinal direction.

Combining equations (4) and (5), and from the geometric relationship X = R - r, the following equation is obtained:

dX l (~bb~f(X--X')+R(F--F')]" d-"O = X' = _ tan 7 (7)

where Cb is a longitudinal spring constant of the belt and related with strain F = Cbc. Now, we have three equations, equations (I), (3) and (7) and three unknowns F, 7, X. Equations

(I) and (7) are first order differential equations for the belt tension F and the radial displacement X, and equation (3) is a constraint for F, X and the sliding angle 7. Solving the equations with respect to the active angle 0 gives the belt tension, radial displacement and the sliding angle.

SPEED R A T I O - T O R Q U E LOAD-AXIAL FORCE RELATIONSHIP

Speed ratio-torque load-axial force relationship can be obtained based on the belt behavior analysis. Axial forces acting on the belt can be calculated by summation of the forces in the active and inactive arc.

f: f' S = S~ + So= P:ds + P:ds (8) in

where S, is an axial force in the inactive area, Sa is an axial force in the active area, $ is the entire arc of contact between the belt and the pulley, 0~, is the magnitude of the inactive arc, 0,, = $ - 0. Contact length ds is expressed as ds = r d0 = (R - X) d0. Substituting P.. from equation (3) into equation (8) gives

K, f[. K. f* X(R _ X)dO S = T XR(R - X,) d0 + -~- :e,,

K,f' = K#2 (RX, - X~)(dp - O) + --~ ~o (RX - X 2) dO. (9)

Equation (9) contains axial force 5". radial displacement X which is related with torque load from equation (3) and the contact arc $ which depends on the speed ratio. Thus. the above equation can be used as a speed ratio-torque load-axial force relationship for the MVB CVT.

NUMERICAL RESULTS

Figure 5 shows the numerical results for the belt radial displacement vs active angle 0. The belt radial displacements were plotted in nondimensional form by introducing the nondimensional

Metal V-belt CVT 869

w

.o x

o z

Driver

, I I [, tO0 200 300

Active angle 0 (degree)

Fig. 5. Nondimensional belt radial displacement.

.I 400

variables as radial displacement X o = X / R , belt tension F o = F / ( R ' K , ) and tension ratio u = Fo/F~ [3]. Since the magnitude of the active arc depends on the torque load, the active angle 0 can be interpreted as a scale of the torque load. As shown in Fig. 5, for the driver pulley, the belt radial displacement remained almost constant for the first 0-80' and decreased, i.e. the belt came out towards the outward radial direction. For the driven pulley, the belt radial displacement increased as the active angle increased. In other words, the belt penetrated in the inward radial direction with the increased active angle.

Figure 6 shows the relationship between the active angle and the traction coefficient 2 for the speed ratio R -- I. The traction coefficient is defined as 2 -. (Ft - F2)/(Ft + F,) and proportional to the torque transmitted. As shown in Fig. 6, as 3. increased, i.e. the torque increased, the active arc increased for both the driver and driven pulley. However, the magnitude of the active arc for

v

tt ,.m

~t <

200

150

100

50

D

river

I J I O.I 0.2 O.3

Traction coe f f i c i ent k

Fig. 6. Active angle vs traction coefficient.

J 0.4

870 HYU.'~oo K~ and J ~ m N lEE

the driven side is always larger than that of the driver. This means that slip occurs in the driven side when the active arc is equal to the whole arc of contact.

E X P E R I M E N T

Figure 7 is an assembly drawing of the MVB CVT test machine. Variable speed a.c. motor (1) drives the driver pulley (2). Torque transmitted to the driven side through MVB (3) is balanced by torque load in powder clutch (6). Various speed ratios can be obtained by adjusting the spring forces in the driver and driven side. Axial forces, speed ratio, torque load and band tension are measured from the test machine in Fig. 7.

Belt radial displacement is measured by a specially designed displacement sensor [5]. Fig. 8 illustrates (a) belt displacement sensor and (b) pulley displacement sensor. The belt displacement sensor is a direct contact type with strain gauges attached on the ring. The pulley displacement sensor is used in order to compensate the belt radial displacement that occurs due to the misalignment of the movable flange of the pulley.

Experiments are performed to investigate (I) the belt behavior and (2) the slip characteristics. In experiment (i), the belt radial displacements of the driver and driven side are measured for the traction coefficient ~ = 0 (no load) and ~. = 0.32 (medium load) for the speed ratio R = I. In experiment (2), the axial force of the driver and driven pulley, band tension, and torque load are measured for the speed ratio R = 3/5. I, 5/3 until gross slip occurs between the belt and the pulley.

G

( I ) Driven spring

(2) Tension transducer

(3) Metal V-belt

(4) Encoder

(5) Tachometer

(6) Powder clutch

(7) Torque transducer

(8) Driver spring

(9) Driver pulley

(10) Variable speed motor

Fig. 7. Assembly drawing of a metal V-belt CVT test machine.

Metal V-beh CVT

(~)

,ll hi I"I Ring

~ Strain gage

Strain gage ~ Pulley Belt --

(b)

(a)

Fig. 8. (a) Belt displacement sensor and (b) pulley displacement sensor.

' Movable flange

871

RESULTS AND DISCUSSION

The following results are based on the material properties and measured dimensions of a commercially available MVB.

Belt radial displacement

in Figs 9 and 10, the theoretical and experimental results of the belt radial displacement were compared for the traction coefficient A = 0. For the driver (Fig. 9) and driven (Fig. 10) side, the theoretical results of the belt displacements remained constant along the entire arc of contact. This is obvious since there is no torque transmitted, thus no active arc exists, which results in constant belt tension in both sides. Experimental results also show almost constant belt radius except the leading edge of the driven side. in leading edge of the driven side, the belt penetrated in the inward radial direction about 0.1 mm and remained almost constant.

0.6

0.4

E E 0.2

i° .. -0.2 m

- 0 , 4

-0.6

Experiment

- ~ - - - ~ ' v = ~ ~ v \ Theory

I I I 0 45 90 |35 180

Contact angle (degree)

Fig. 9. Comparison oftheoretical and expenmental resultsforbeltradial displacement at ~ = 0 in driver pulley.

872 HYUNSOO Km and JJ~z-.N lEE

0.6

0.4

S E 0.2

E 0

,, -11.2 o

-0.4

-0.6

Theory

\ Experiment

| I s

0 45 90 135

Contact angle (degree)

180

Fig. 10. Comparison of theoretical and experimental results for belt radial displacement at 2 = 0 in driven pulley.

In Figs II and 12, the theoretical and experimental results of the belt radial displacement were compared for 2 = 0.32. For 2 -0.32, the experimental running conditions are: total tension, Fj + F2 -- 2456 N and torque load, TL -- 43 Nm. As shown in Fig. 1 I, both theoretical and experimental results of the belt displacement for the driver side remained constant. However. if there is a torque load, the belt tension must change as tight side tension, F, in the leading edge and slack side tension, F2 in the trailing edge. Thus, there should be an active arc where the belt radial displacement changes corresponding to the tension variation. From equations (I), (3) and (7) and the experimental running conditions, the active arc can be obtained as 0 = 123 ̀7 for ~. =0.32 and the theoretical belt radial displacement changes from X=2.81 x 10-~mm to X = 2.59 x 10 ~ mm. So total amount of the belt displacement change is 2.2 x 10 -4 mm, which is too small to be measured. Therefore, in spite of the change in the belt radial displacement, both the theoretical and experimental belt displacement seem to remain almost constant.

in Fig. 12, the theoretical and experimental results of the belt displacement were compared for the driven side at ~. = 0.32. From the numerical analysis, the active arc and the total change of

0.6

0.4 -

,EE 0.2 -

E o 0

,, -0.2 -

-0.4 -

-0.6

Theory

Experiment

I I I 43 90 133 180

Contact angle (degree)

Fig. I I. Comparison of theoretical and experimental results for belt radial displacement at .~. = 0.32 in driver pulley.

Metal V-belt CV'T 873

0.6

A E E

&}

0 .4

0.2

0 -

- 0 . 2 - -

- 0 . 4 - -

- 0 . 6 -

T h e o r y

E x p e r i m e n t

I T T 45 9 0 135

Contact angle (degree)

180

Fig. 12. Comparison of theoretical and experimental results for belt radial displacement at 2 = 0.32 in driven pulley.

the belt displacement were calculated as 0 = 165 ° and X = 9 x 10 -4 mm. Even if the amount of the belt displacement change of the driven side is four times larger than that of the driver, it is still too small to be measured by the transducers used in the experiment. Therefore, the theoretical and experimental curves show almost constant values along the arc of contact except the first 0--30' of the experimental results. As for the sensitivity of the displacement sensors, since the variation of the belt displacement exists in the error range of the transducers due to the vibration, and other factors, it is considered to be difficult to measure such a small amount of the belt displacement even if more accurate sensors with higher sensitivity are used. The rapid decrease of the experimental curve in the leading edge for the driven side (Fig. 12) can be explained by the effect of the belt bending moment. As the belt enters the pulley, the position where the compressive force acts on the metal block changes and the corresponding bending moment makes a sudden change in the belt radius of curvature. Since there is no compressive force on the block that enters the driver pulley, such a change is not observed (Fig. !1).

Slip characteristics

The theoretical and experimental results of the slip characteristics were compared in Figs 13-15. Theoretical curves for the nondimensional axial forces, S/(F~ + F2) were obtained from the speed ratio-torque load-axial force relationship, equation (9) and plotted for the traction coefficient 2. Numerical computations were performed until the magnitude of the driven pulley active arc was equal to the whole arc of contact, i.e. slip occurred between the belt and the pulley. The coefficient of friction between the belt and the pulley,/~ -- 0.08, 0.1 ! and 0.13 were used in the computations in order to compare the experimental results.

Figure 13 shows the theoretical and experimental results of the nondimensional axial forces for the speed ratio R = 3/5. The experimental slip occurred in the driven side at the traction coefficient ). = 0.42. The experimental results of the driven side axial force fits best the theoretical curve of ~u = 0.13. However, the driver side axial force follows the theoretical curve of/.t = 0.08. Since the experiment was performed until the sliding took place in the driven side, the experimental results for the driver were observed outside the sliding point of the/a = 0.08 theoretical curve.

Figure 14 shows the comparison of the theoretical and experimental results of the nondimen- sional axial forces for the speed ratio R = I. As shown in Fig. 14, the slip occurred at 2 = 0.36. The experimental curves for the driver and driven pulley axial forces are in good accordance with the theoretical curve of/a --0.08.

Figure 15 shows the comparison of the theoretical and experimental results of the nondimen- sional axial forces for the speed ratio R = 5/3. The slip occurred at 2 = 0.55. The experimental

874 H Y U N ~ K1M and JAJESHIN LEE

5

I ° O O [] o

0 o

_

~ 2 -~'~'+-----~ ~ + ._ +-¢" + + ++ , + ¢" + :t ++ ++ +

o Z

Experiment O Driver

+ Driven

Theory

I l J 0 0.2 O.4 0.6

Traction coeff icient ~.

Fig. 13. Slip characteristics and axial forces at speed ratio R ~ 3/5.

results for the driver follows the p --0.08 theoretical curve and the experimental results for the driven follows the p =0 .13 theoretical curve. Since the slip depends on the driven side, the experimental results for the driver were observed outside the range of the p = 0.08 theoretical curve.

As shown in Figs 13-15, the slip between the MVB and the pulley depends on the torque transmission capacity of the driven side. The experimental results of the driver pulley axial force are in good accordance with the p = 0.08 theoretical curve, in the driven side, except the R = i case, the experimental results follow the p = 0.13 theoretical curves. Since the slip depends on the driven side, the experimental results for the driver are observed outside the slip range of the

-- 0.08 theoretical curve. The reason why the different coefficient of the friction appears in the

5 --

oo , - [ i a p # o

"~ 3

:6 p=O.08 ~ = O . l l p = O . 1 3 o

m

I - - E x p e r i m e n t 0 Driver

+ Driven

Theory

I Z 0 0.2 0.4

Traction coeff ic ient ).

Fig. 14. Slip characteristics and axial forces at speed ratio R == I.

I 0.6

Metal V-belt CVT 875

5 -

4 -- 0 0

_ ÷ ~ ÷ - -

It = 0.08 p = OAt p = 0.13

Experiment o Driver 1--

+ Driven T h e o r y

1 I 0 0.2 0.4

Traction coefficient k

Fig. 15. Slip characteristics and axial forces at speed ratio R = 5/3.

O z

I 0.6

driver and driven side for R = 3/5 and 5/3 can be explained by different contact states between the belt and the pulley due to the belt misalignments, which takes place inevitably in the MVB CVT drive. However, a more detailed study, including the friction between the band and the block, elasto-hydrodynamic effects of the lubrication oil, would be required to explain the exact slip characteristics of the MVB CVT.

In addition, from the design and operation standpoint, it is useful to obtain the optimum loading curve, in other words, optimum axial force curve, since it is important not to allow slipping and at the same time not to overload the belt for extended life. The optimum loading curve can be plotted by simply connecting the slip points from Figs 13-15. As for the practical design, such a curve should be obtained from extensive experimental data for various speed ratios.

CONCLUSIONS

(I) Numerical results for the MVB CVT belt behavior analysis showed that as the torque load increased, the belt radial displacement increased in the radial inward direction for the driven side while the belt displacement decreased slightly and increased for the driver.

(2) The theoretical results for the belt radial displacement were in good agreement with the experimental results except the leading edge of the pulleys. However, the absolute magnitude of the radial dispalcements was so small that the change in the belt displacement could not be measured in the experimental range.

(3) The slip characteristics between the belt and the pulley were studied by using the speed ratio-torque load-axial force relationship derived from the belt behavior analysis. It was found that the gross slip points depend on the torque transmission capacity of the driven side. Experimental results for the slip and axial forces were in the range of the coefficient of friction

= 0.08-0.13.

REFERENCES

I. S. C. van der Veen. Power, pp. 133-140 (1977). 2. B. G. Gerbert, ASME, 84-DET-22 (1984). 3. B. G. Gerbert, Acta Polytech. Scand. Mech. Eng. Series, No. 67 (1972). 4. D. C. Sun, Trans. ASME I!0, 472-481 (1988). 5. A. Katsuya. T. Sato and K. Kurimoto, Trans. JSAE 44, 71-76 (1990). 6. T. Koyama, Osaka Institute of Technology Report (1987).

876 HYUNSOO KIM and J~,HIN LEE

Zusammeafassug--Das Verhalten vom Treibncmen aus Metall in V-Tn:lbriemen CVT wurde analytisch und ¢xpcrimentell untcrsucht. Die numerische Resultate zeigen, dab der Trcibnemen auf der angetnebenen Rolle sich einw;irts biegt, w~ihrend er auf der antreibenden Rolle sich leicht einw~irts biegt und sich mit steigendem Drehmoment entspaant. Expenmentelle Ergebnisse im Bezung auf die radialen Bewegungen des Treibriemens stimmten gut mit den theorctischen Resultaten. Auch das Rutchverhalten zwischen Riemen und Rolle wurd¢ untersucht anhand der Beziehung zwischcn Geschwindigkeitsverh//lmis, Drehmoment und axiale Belastung, abgeleitet yon der Analyse des Treibrsememsverhaltens. Es ¢rgab sich, dab Momente des krassen Abrutchens abh~ingig yon dcr Drehmoment/ibertragungskapazit',it der angetnebenen Seite sind. Die Versuchsresultate ffir den Abrutchen und die axiale Belastung waren im Bereich, des Reibungskoetfizienten # = 0.08-0.13.