SPECIAL ISSUE PAPER 235 Prediction of losses in belt-type ...

Prediction of losses in belt-type continuously variabletransmission due to sliding between belt and discG Poll, T Kruse, and C Meyer

Department of Machine Elements, Institute for Machine Elements, Engineering Design and Tribology,

University of Hannover, Hannover, Germany

The manuscript was received on 1 September 2005 and was accepted after revision for publication on 10 February 2006.

DOI: 10.1243/13506501JET141

Abstract: The efficiency of belt-type continuously variable transmission (CVT) – apart from thepower consumption of auxiliary systems such as hydraulics – predominantly depends on theenergy dissipated during sliding at the belt-disc contacts. These sliding motions are a conse-quence of elastic deformations and misalignments because of clearances, resulting in a so-called spiral path of the belt elements on the discs and hence tangential and radial slidingmotions.

The performance of such systems can be predicted through an iterative computation bynumerically solving a set of differential equations for the forces and motions coupled with afinite-element computation of the deformations.

A comparison with elaborate measurements of deflections, belt motions, and losses showsthat a relatively simple modified Coulomb type friction model with a steep gradient throughthe origin delivers sufficiently accurate results. The computations reveal the existence of‘locked’ zones with extremely low ‘creeping’ motions.

Thus, the mechanisms of power transmission in belt-type CVT are better understood anddesigners have a validated tool to optimize shaft and disc geometry concerning maximumefficiency.

Keywords: continuously variable transmissions, efficiency, coefficient of friction, sliding motion

1 INTRODUCTION

Belt-type continuously variable transmission (CVT)units are gear units with transmission by tractionforces. They comprise two parallel shafts, each withone fixed disc and one moveable disc. Axial displace-ment of themoveable disc forces the chain to contactthe discs on a different running perimeter. The ratioof transmission changes and the input speed andtorque are transformed (Fig. 1). The subassemblythat comprises both shafts and the chain is referredto as the variator.

In most applications, the normal force required forpower transmission is generated using a hydraulicclamping system. It also provides oil to lubricate the

traction contact between the discs and the chain.The forces applied to the arc of contact and the lim-ited disc stiffness impose elastic deformations onthe discs and shafts and thereby deflections, whichvary along the perimeter (Fig. 2). The necessary clear-ance between the moveable disc and the shaft causesthe moveable disc to tilt. Furthermore, the normalforces compress the chain. All these effects inducesliding movements between the chain and the discs,or what is known as a spiral path, which is largelyresponsible for the loss of power and wear in thevariator (Fig. 3) [1, 2]. Therefore, the performance ofthe variator can be improved by optimizing itsgeometry with respect to deflections.

2 BASES FOR A NUMERICAL MODEL

To design a highly efficient yet compact variator,it is important to understand the key influencing

�Corresponding author: Department of Machine Elements,

Institute for Machine Elements, Engineering Design and

Tribology, Welfengarten 1A, Hannover 30167, Germany. email:

[email protected]

SPECIAL ISSUE PAPER 235

JET141 # IMechE 2006 Proc. IMechE Vol. 220 Part J: J. Engineering Tribology

variables in the arc of contact, such as the slidingvelocity and sliding direction, the longitudinal tensileforce in the chain, and the force components trans-mitted between discs and pins. The loss of powerand the resultant efficiency can be calculated fromthese quantities. A simulation programme has beendeveloped for this purpose. It can be used at a veryearly stage of the development process, i.e. as soonas the first geometric variant of the variator has beendevised. The results are used for determining theideal geometry of the variator and also help to deter-mine the best way to design the other componentsof the transmission unit, e.g. hydraulics. The simu-lation is based on the following assumptions [2, 3].

1. The longitudinal strain and the transverse com-pression are independent of each other. Thisstrictly applies to the rocker pin element chainbecause the functions of the rocker pin elements(axial forces) and links (peripheral forces) are sep-arated, which means that the compression in therocker pin elements does not have any influenceon the longitudinal strain in the links. Likewise,

the longitudinal strain in the links has no effecton the normal forces in the pin elements.

2. The chain is a continual element with even massdistribution and longitudinal and transversestiffness.

3. The wedge-shaped gap between the discs is con-cerned as symmetrical. Both contacts (with themoveable disc and the fixed disc) have the samecontact conditions. The deformations on the driv-ing disc and the moveable disc are of a similarmagnitude. Although the two discs may deflectand tilt differently, the error when assumingequal contact angles on both sides can be con-cerned as negligible.

4. Coulomb’s law of friction applies except for verylow-sliding speed. The friction coefficient has aconstant mean value that is derived from themeasured values. A continuous analytic functionis selected for areas in which sliding speeds arevery small (discussed subsequently).

5. Internal losses in the chains are not taken intoaccount. The rocker pin element chains have avery low level of internal losses. Specific func-tional approaches have to be added to determineinternal losses in push belts [4].

6. The same approaches are used for the driving andthe driven set of discs.

Figure 4 shows the sliding velocity (V sl ¼ V disc 2V chain) and force vectors in the contact between thetapered disc and the chain in a tapered disc incre-ment with a size of dw and a taper angle of b. Thenormal force dN between the chain and the discacts perpendicular to the surface. The varying longi-tudinal stretching of the chain causes sliding move-ments in the direction of the periphery whentensile force is built up or reduced.

In addition there is radial sliding and as a conse-quence, the running radius changes. The slidingangle g, which in the mathematically positive sense

Fig. 1 Functional principle of variable belt

transmission [2] and rocker pin chain [7]

Fig. 2 Causes of sliding movements and losses

Fig. 3 Spiral path [2, 9]

236 G Poll, T Kruse, and C Meyer

Proc. IMechE Vol. 220 Part J: J. Engineering Tribology JET141 # IMechE 2006

is measured against the outward pointing radialbeam, denotes the direction of the sliding move-ment. The effective taper angle bS indicates thespatial direction of the sliding movement. The forceof friction on the chain is directed opposite tothe sliding velocity vector vsl and is the product ofthe normal force and the local friction value m dN.This formally corresponds to Coulomb’s law offriction. However, in reality, there may be differentfriction levels in the contact between the chain andthe discs depending on the position of the arc ofcontact and the peripheral velocity.

The adhering layer of lubricant is first entrained inthe chain’s inlet zone into the wedge-shaped gap

and subsequently gradually squeezed out of the con-tacts. This then results inmixed lubrication and finallyboundary lubrication. Calculating the thickness of thelubricating films, the temperature-related viscosityand the resultant fluid friction and boundary frictionratio are complicated tasks. Therefore, for an initialapproximation, a uniform and location-independentlaw of friction is adopted with which the frictionvalue for low-sliding velocities tends towards zero (1)

m ¼ msl � (1� e�vsl=v0 ) (1)

Figure 5 shows the applied function for low-slidingvelocities (vsl , 1 mm/s) with a sliding coefficient of

Fig. 4 Sliding and force vectors in the contact [3]

Fig. 5 Friction law at low-sliding velocities

Prediction of losses in belt-type CVT 237


frictionmsl ¼ 0.09 and a speed constant v0 ¼ 0.1 mm/s[5]. Thus, the static coefficient of friction is zero;instead of genuine adhesion, creeping takes place atvery low-sliding velocities. This type of behaviourseems physically very plausible, although it has yetto be confirmed in experiments [6].

This approach has the numerical advantage thatadhesion conditions do not have to be checked andthere is always a small sliding velocity with whichthe sliding angle can be determined. The force offriction can be separated into a radial component(m . dN . cos bS cos(p2 g)) and a tangential com-ponent (m dN . cos bS

. sin(p2 g)), which causesthe tensile force to change. Figure 6 shows theradial and tangential components of the force offriction. The equation for dN results after trans-formation, neglecting the change in the radius ofcurvature. It is important to take into account thatthe axial components count twice because of thetwo-pulley chain contacts.

In this way, a system of first-order differentialequations can be established for the calculation(Fig. 7). The motion relationship in the arc of contactcan be described with these equations. As all coupledvariables are known at the outset or can be describedby analytical functions, the calculation can be con-sidered as an initial value problem and solvedaccording to the Runge–Kutta procedure of fourthorder for coupled systems of differential equations[2, 3].

The local taper angle b and widening w of thewedge-shaped gap between the discs are determinedby finite-element calculations. Two-dimensionalfinite-element models for the half-section of the var-iator suffice for these calculations, because theselected element type and rotationally symmetricalcomponents allow to apply stresses which are notaxially symmetrical. Following the finite-elementcalculation, the axial deformation and the actualtapered angle are determined at discrete points onthe running radius and approximated with apolynomial function, delivering the variables as afunction of the arc angle.

3 SOLUTION BY ITERATIVE COMPUTATION

The computation of an operating point is an iterativeprocess, and the calculation is made first for theoutput set of discs. For this purpose, starting valuesare specified for the starting tensile force F0 and thestarting sliding angle g0, and an initial deformationis specified. The nominal axial force S2,targ andthe nominal drive torque T2,targ serve as targetvalues. If no solution is achieved after a first calcu-lation, the starting values (F0, g0) are varied suchthat better agreement with the target values isobtained (Fig. 8).

Fig. 6 Forces acting on the arc element in the axial

section

Fig. 7 Differential equations system [10]



Only the torque is known at the input set of discs. Inthe real gear unit, the clamping force is varied suchthat the nominal transmission ratio is obtained. Asno change of ratio operations can be simulated withthe calculation procedure and the clamping force atthe drive is not known a priori, a different boundarycondition must be found. Both disc sets are inmutual contact only via the chain. After the calculationfor the driven set of discs, the belt tension forces areknown and can thus be used as boundary conditionand target values for the calculation on the drivingside. The tensile force at the output of the driven setof discs is the tensile force at the input of the drivingset of discs. Consequently, the starting tensile forceat the driving set of discs is not available as variationparameter for solving the set of differential equations.However, only one boundary condition, namely, thefinal tensile force F2, still has to be fulfilled. The start-ing sliding angle remains as variation parameter. Forthe driving discs, as already for the driven discs withthe torque, the tensile force is not used for the vari-ation, but instead the tensile force changes as refer-ence variable and controlled variable.

The possibilities of the procedure will now beexplained with the help of an example. The tensileforce F, the local axial force S0, and the local powertransmission losses P0

V,Var are shown in Fig. 9 overthe arc of contact at the two disc sets. The arrows

indicate the direction of the sliding movements (slid-ing angle g), and the arrow length corresponds to themagnitude of the sliding velocity.

The transmitted torque is increased in steps start-ing from zero. Without torque, the tensile forces areidentical in both belt sections. For i ¼ 1, the clamp-ing forces at the driving and driven discs are alsoequal; consequently, identical conditions must existat both disc sets. The sliding movement is directedinwards and backwards at the input. Over the arc ofcontact, the vector changes direction through 1808(purely inward sliding at the orthogonal point). Thesliding velocity becomes smaller.

A well-recognizable region of very small, almostconstant, sliding velocity appears. In this ‘creepzone’, genuine adhesion could appear if the staticcoefficient of friction would not be zero at theorigin. Increased outward drifting appears as theend of the arc of contact is approached. A strongerincrease in the axial force also takes place here, caus-ing high local power transmission losses. The lossesare almost zero in the adhesion region. As thetorque is increased, the axial force peak at thedriven discs flattens and moves in the directiontowards the input. The region of small sliding velo-city becomes smaller and is also displaced in thedirection towards the input. The sliding angle isincreasingly directed backwards.

Fig. 8 Computation scheme for driving and driven discs [2, 10]



The fundamental form of the axial force distri-bution changes hardly at the driven discs, but itsmaximum drifts in the direction towards the outlet.The sliding angle turns into a forward at the inlet. Inthe last section of the figure (131 Nm), the slidingangle is directed forward at the driven disc and back-ward at the driving disc along both contact arcs. Thelarge increase in the power transmission loss indicatesthat the chain is close to the onset of gross slipping.

4 VERIFICATION

Computations were compared with experimentalresults of the authors for verification. Figure 10shows the power transmission efficiency andpower loss from one measurement and twocalculations. The power transmission loss curvesrun approximately parallel with increasing torqueup to a nominal torque of 120 Nm, whereby the

Fig. 9 Calculated forces, power losses, and sliding velocities over the arc of contact on increasing

the drive torque [2, 10]

Fig. 10 Comparison of measurements and calculations for different friction coefficients



calculated values lie slightly below the measuredvalues.

The curve with the friction coefficient of msl ¼ 0.11approaches the measured curve more closely. Asfrom �120 Nm, the calculated power transmissionlosses for a friction coefficient of msl ¼ 0.09 increasestrongly and the power transmission efficiency falls.This indicates that the chain is close to the onset ofgross slipping. In contrast thereto, the powertransmission efficiency maximum of the calculationvariant with a friction coefficient of msl ¼ 0.11 isreached only at �135 Nm. Although the powerdissipated with a friction coefficient of msl ¼ 0.11evidently matches the real system more closely, the

influence of the assumed sliding friction coefficientis surprisingly small in the range of practical interest.Only the threshold torque for the onset of slippingresponds sensitively as expected. The approximationwith a simple friction law already represents realityvery closely.

An explanation for this is that the friction has twoopposite effects that mutually cancel to a largeextent in the system considered here. With smallfriction coefficients, the local friction forces becomesmaller, but the sliding movements increase at thesame time, so that the frictional work hardly changes.

The reasons of the variator deformations describedearlier were investigated by measurements on vari-ous shaft and disc geometries, so that measurementresults are available for various shaft stiffness valuesand tilt play variants. For this purpose, a measuringsystem was integrated in a test gear unit (Fig. 11).

The disc deformations are measured with sixdistance sensors (three each on the fixed disc and onthe moveable disc; right part of the figure), which aremounted on a measuring device. The measuringdevice is pivot mounted on the bearing pots and, asthere are two sensor sets shifted by 1808 with respectto each other, the deformation can be measuredover the entire circumference. A sensor on the leftpart of the figure determines the position of thechain in the region of the arc of contact.

Figure 12 shows an example of the comparisonbetween the measurement (black line) and the calcu-lation (grey line) for the driving and driven discs withtransmission ratio i ¼ 1 and a driving torque of120 Nm. The chain runs at the top at 908 into the disc

Fig. 11 Measuring system for determining the shaft

and disc deformations

Fig. 12 Comparison of measured and calculated deformations [2]



and at the bottom at 2708 out of the disc set again. Thedifferent measured characteristics at the driving anddriven discs are reproduced well by the calculation.This makes it reasonable to conclude that the axialforces and their distribution over the arc of contactcome very close to the real conditions, albeit the calcu-lated curves lie slightly below the measured values.

Power transmission efficiency measurements werecarried out for the same variator execution inaddition to the deformation measurements. Themeasured and calculated results for the power trans-mission efficiency and for the power loss are plottedin Fig. 13 against the drive torque, with a trans-mission ratio of i ¼ 1 and a rotation speed ofn ¼ 1000 r/min. The respective curves match closely.Again, a friction coefficient of msl ¼ 0.11 was used inthe computations, also resulting in a good match forthe axial clamping force required to maintain thetransmission ratio. The model has been validatedwith measurements at three different transmissionratios, several rotational speeds, four differentgeometries of the discs, and two different tiltingclearances of the moveable discs. The accuracymargin of the measurements is 0.11 per cent.

5 SUMMARY

Owing to deformations, sliding movements takeplace in variators of chain type CVT. They are largelyresponsible for the power transmission losses.Measurements and calculations show that theoperational behaviour of continuously variable belttransmissions can be reproduced accurately evenwhen a very simplified friction law is taken as basis.This requires an iterative computation methodbased on a system of differential equations andfinite-element calculations. The distribution of theaxial forces and the deformations of all variator com-ponents can be reproduced realistically therewith,making it a useful tool for variator optimizationalready in the early design stage.

REFERENCES

1 Dittrich, O. Anwendung der Theorie des keilformigenUmschlingungstriebes auf stufenlose Getriebe. BadHomburg, 1992.

2 Sue, A. Betriebsverhalten stufenloser Umschlingungsge-triebe unter Einfluss von Kippspiel und Verformungen.Doctoral Thesis, Hannover University, IMKT 2003.

3 Sattler, H. Stationares Betriebsverhalten stufenlosverstellbarer Metallumschlingungsgetriebe. DoctoralThesis, Hannover University, IMKT 1999.

4 Sladek, W. Moglichkeiten zur Wirkungsgradoptimier-ung von Stufenlosgetrieben. Doctoral Thesis, HannoverUniversity, IMKT 2001.

5 Srnik, J. Dynamik von CVT-Keilkettengetrieben. Doc-toral Thesis, TU Munich, 1998, VDI-Fortschritt-BerichtNo. 372 Reihe 12, (VDI-Publishing, Dusseldorf).

6 Wang, G. Untersuchungen an stufenlos einstellbarenZug- und Schubgliedergetrieben. Doctoral Thesis, TUClausthal, 1991.

7 Faust, H. and Linnenbrugger, A. CVT-Entwicklung beiLuK. 6th LuK Kolloquium, 1998, pp. 159–181 (LuK,Buhl).

8 Sattler, H. Efficiency of metal chain on V-Belt CVT.International Congress on Continuously variablepower transmission, Eindhoven University of Techno-logy, 1999, pp. 99–104.

9 Sue, A. Dittrich, O., and Poll, G. CVT-Wirkungsgrad-berechnung aus der Verformung beim Umschlingung-strieb. CVT 2002 Congress, VDI-reports Vol. 1709,2002, pp. 69–88 (VDI-Publishing, Dusseldorf).

10 Sue, A. and Poll G. Analytisch-iteratives Berechnungs-verfahren fur stufenlose Umschlingungsgetriebe. VDI-reports Vol. 1827, 2004, pp. 691–712 (VDI-publishing,Dusseldorf).

APPENDIX 1

Notation

F0 starting tensile forceF1 force in the unloaded belt sectionF2 force in the loaded belt sectioni ratioP0V,Var local power losses in the variator

q stiffnessR running perimeterS clamping forceS0 local clamping forceT torqueT0 local torquevsl sliding velocity

b tapered disc anglebS effective tapered disc angleg sliding angleg0 starting sliding anglemsl coefficient of frictionv revolution speed

Fig. 13 Comparison between loss and efficiency

measurements and calculations [2]



6 APPENDIX 2

Q ¼ q � E(AB� CD)2 �HJ

F ¼ q2 �AE2

(AB�CD)2 �HJ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(q �AE)2 � F � b �C � (AB�CD)

q

C ¼ F �b2 � (AB�CD) �HJ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(q �AE)2�F �b �C � (AB�CD)

q

K1 ¼ L � (AB� CD)� A � (4q � AH þ BL)þ ADN

K2 ¼ M � (AB� CD)� ABM þ ADO

K3 ¼ �A � (AB� CD) �HJ

EK4 ¼ �N � (AB� CD)þ C � (4q � AH þ BL)

�CDN

K5 ¼ �O � (AB� CD)þ BCM � CDO

K6 ¼ r � r �D(F � b)� 1

� �� (AB� CD) � CGHJ=A

A ¼ sinb� m � cosbS � cos gB ¼ 4 � q � tanbC ¼ cosb� m � sinbS

D ¼ m� � v2 � bE ¼ 2 � R � tanb0 �wG ¼ m � cosbS � sin gH ¼ 1þ tan2 b � cos2 gJ ¼ cos2 bL ¼ �m � sinbS � cos2 gþ cosb �H � JM ¼ m � sinb � cosb � sinbS � sin g � cos gþ G �H � JN ¼ m � cosbS � cos g� sinb � cos2 b �HO ¼ � sinb � cosb � G



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