+++experimental identification of the contact parameters between a v-ribbed belt and a pulley

Mechanism and Machine Theory 45 (2010) 1424–1433

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Mechanism and Machine Theory

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Experimental identification of the contact parameters between a V-ribbedbelt and a pulley

Gregor Čepon a, Lionel Manin b, Miha Boltežar a,⁎a University of Ljubljana, Faculty of Mechanical Engineering, Aškerčeva 6, 1000 Ljubljana, SI, Sloveniab Université de Lyon, CNRS, INSA de Lyon, LaMCoS UMR 5259, Villeurbanne F-69621, France

a r t i c l e i n f o

⁎ Corresponding author. Tel.: +386 1 4771 608 (diE-mail address: [email protected] (M. Bolt

0094-114X/$ – see front matter © 2010 Elsevier Ltd.doi:10.1016/j.mechmachtheory.2010.05.006

a b s t r a c t

Article history:Received 1 June 2009Received in revised form 3 May 2010Accepted 8 May 2010Available online 30 June 2010

The aim of this paper is to identify the contact parameters between a belt and a pulley that canbe used in a two-dimensional multibody belt-drive model. Two experimental setups areproposed in order to identify the contact stiffness and the friction coefficient between theV-ribbed belt and the pulley. The friction coefficient is identified at various initial belt tensions andrelative velocities between the belt and the pulley. The measurement procedure and the contactformulation are verified with a numerical experiment.

© 2010 Elsevier Ltd. All rights reserved.
Keywords:V-ribbed beltFrictionContact stiffnessANCFMultibody dynamics
1. Introduction

V-ribbed belt-drive systems have become increasingly important to the automotive industry since their introduction in the late1970s. Usually, V-ribbed belts in automotive engines drive multiple-accessory pulleys, leading to compactness, smaller pulleydiameters and a longer belt life. To ensure stable working conditions the dynamic responses of such systems have been studiedextensively. A review of the literature [1] identifies two well-defined groups of studies. The first group deals with the transversebelt span response [2,3] and the rotational response [4] of the pulleys in the belt-drive. The second group deals with describing thebelt-pulley contact formulation. Most of the contact models are based on classical creep theory [5] or the shear theory [6].However, the two groups suffer an unsatisfactory connection with each other: the first does not take into account the belt-pulleycontact behavior and the second neglects the vibration due to the transmission.

Leamy and Wasfy [7,8] attemped to bridge this gap between the above-mentioned groups of studies, developing a general,dynamic finite-element model of a belt-drive system, including a detailed frictional contact. This finite-element model was able topredict the belt creep over the pulleys and the belt-drive vibrations. The contact between the belt and the pulley was modeledusing a well-known penalty method, together with a Coulomb-like tri-linear creep-rate-dependent friction law. Using theabsolute nodal coordinate formulation (ANCF), originally proposed by Shabana [9], the authors in [10] developed a more generalplanarmodel of the belt-drive. Most recently, Čepon and Boltežar [11] presented a belt-drivemodel using the ANCFwith a detailedcontact formulation between the belt and the pulley. The belt-pulley contact was formulated as a linear complementarity problem(LCP), using the discontinuous Coulomb friction law to model the frictional forces.

All the above-mentioned studies [7,8,10,11] describe belt-drive numerical models; however, no work regarding theidentification of the belt material and the contact parameters has been presented. For any reliable simulation of a belt-drive thematerial and the contact parameters should be obtained from experiments. In Ref. [12], Čepon, Manin and Boltežar presented

rect line); fax: +386 1 2518 567.ežar).

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1425G. Čepon et al. / Mechanism and Machine Theory 45 (2010) 1424–1433

methods for identifying the stiffness and damping of V-ribbed belts. Some researchers have experimentally studied the contactbetween a grooved pulley and V or V-ribbed belts [13,14]. The traction capacity of V-ribbed belts depends greatly on thedistribution of the contact pressure over the flank regions of the belt and the roughness of the contacting surfaces. It wasexperimentally demonstrated in [13] that with an increasing belt tension the contact area between the belt and the pulleyincreases. For an accurate description of the contact between a V-ribbed belt and a pulley it is necessary to build a three-dimensional finite-element model [14]. However, these models are not appropriate for modeling the belt-drive dynamics overlonger time scales as they are computationally inefficient.

The aim of this paper is to present methods for the experimental identification of the contact parameters between the pulleyand V-ribbed belts. The identified contact parameters will be suitable for incorporation into the planar multibody, belt-drive modelspresented in Refs. [7,8,10,11]. The procedure includes an experimental measurement of the contact-penalty parameters as well as ameasurement of the friction coefficient. As planar belt-drivemodels are not able to model the belt-rib and pulley-groove geometries,the measured contact stiffness and friction coefficient will include the cumulative contribution of the belt-rubber material as well asthe contact geometry. The identified friction characteristics will be used to validate the contact model presented in Ref. [11].

2. Experimental identification of the contact parameters

In this section we will present a method for identifying the contact stiffness and the friction coefficient between a pulley and aV-ribbed belt. The contact parameters will be identified for a V-ribbed belt with five ribs and a K-rib section (5PK), as shown in Fig. 1.

2.1. Identification of the contact stiffness

The penalty-contact model is used in Refs. [7,8,10,11] to compute the normal contact forces between the belt and the pulley. Itis implied in Ref. [11], that due to the mainly resting contact between the belt and the pulley (except when the belt enters thepulley), the value of the normal contact forces is almost independent of the value of the stiffness coefficient. However, it does havean influence on the amount of deformation between the belt and the pulley. It is shown in Ref. [6] that the radial deformation of thebelt in the pulley grooves influences the angular-speed loss between the belt and the pulley. Thus, an accurate identification of thecontact stiffness is necessary in order to precisely predict the angular-speed loss between the driver and the driven pulley.

In general, the penal normal contact force includes both elastic and dissipative components. The elastic component Fel is a functionof the penetration, which in general represents the deformation of the bodies in contact. The dissipative component Fdiss is usually afunction of the relative normal contact velocity. The penal force in the normal contact direction can thus be written as:

where

FN = Fel gNð Þ + H gNð ÞFdiss gNð Þ; ð1Þ

gN is the amount of penetration, ġN is the relative normal velocity in the contact and H gNð Þ is a step function:

H xð Þ = 1; x≤ 00; x N0 :

�ð2Þ

When the belt and pulley come into contact the normal relative velocity is negative ġN≤0 and the pulley penetrates a certaindistance gN≤0 into the belt.When the contact between the belt and the pulley is established, the belt-rib deforms, which results inan increased contact area between the belt and the pulley, see Fig. 2. As reported in [13], the contact area also depends on thesurface roughness in correlation with the normal contact force.

Thus, it is expected that the contact stiffness will be a function of the rubber-belt material, the belt-rib and pulley-groovegeometries as well as the roughness of the contacting surfaces. In order to measure the cumulative influence of these parameterson the contact stiffness the experimental setup in Fig. 3 is proposed. The 5PK belt segment of length 4 cm is pressed against theplate, which has identical grooves to the pulley for the K-section belts. The compression of the belt segment and the grooved platewas achieved with the universal Zwick/Roell Z050 testing machine in the temperature chamber. The tests were conducted at fourdifferent temperatures, which enabled us to determine the influence of the temperature on the mechanical properties of the belt-

Fig. 1. Belt segment (5PK).

wherecoefficvalue

Fig. 2. Contact between the belt rib and the pulley-groove. ( ) low normal contact force; ( ) high normal contact force.

Fig. 3. Contact stiffness measurement.

Fig. 4. Deformation of 4-cm-long belt segment versus the compression force. ( ) T=22 °C, (—–) T=40 °C, ( ) T=60 °C, ( ) T=80 °C.

1426 G. Čepon et al. / Mechanism and Machine Theory 45 (2010) 1424–1433

rubber. In Fig. 4 the compression force versus the deformation of the belt segment is presented. It can be assumed that the maindeformation of the belt segment is due to the deformation of the soft friction-rubber layer. Temperatures up to 80 °C have practicallyno influence on the mechanical properties of the rubber layer, and hysteresis can be observed during the loading and unloadingphases. The measurements indicate a nonlinear relation between the force and the deformation. If the curve at 22 °C is taken as areference, the loading and unloading phases can be approximated with a quadratic polynomial, see Fig. 5. The presented curve isnormalized to 1 cm of the belt segment length. The elastic component of the penal normal contact force can be written as:

Fel gNð Þ = 3025:6⋅106Lcontg2N + 6:5⋅105LcontgN N½ �; ð3Þ

Lcont[cm] is the length of the belt segment in centimeters. The value of penetration variable gN is given in [m] and the
whererelative normal velocity is given in [m/s]. Eq. (3) enables us to compute the normal contact forces between the belt and the pulley,depending on the deformation (penetration) between the belt and the pulley. The dissipative component Fdiss of the penalty forceis usually a function of the relative normal velocity and is computed with the equation:
Fdiss = CP gN ; ð4Þ

CP is the damping coefficient. Due to the mainly resting contact between belt and pulley (ġN≈0) the influence of dampingient on normal contact forces is usually negligible [11]. In order to avoid numerical problems and unsurprised vibrations theof the damping coefficient is chosen to be CP=300 Ns/m.

Fig. 5. Approximation of the measured contact stiffness at temperature T=22 °C. ( ) experimentally obtained contact stiffness normalized to 1 cm of bellength, ( ) approximation of the contact stiffness with a quadratic function.

Fig. 6. Experimental setup for measuring the friction coefficient.


t

2.2. Identification of the friction coefficient

The torque transmission between the pulley and the flat or V-ribbed belts is mainly governed by the friction. By assuming thatthe belt is sliding against the pulley and that the friction is fully developed along the groove, we can write the Euler–Eytelweinequation for the tension ratio:

TdTj

= eμβ: ð5Þ

The variables Td and Tj are the tensions in the tight and slack belt spans and β represents the contact arc between the belt andthe pulley. From the tensions in both belt spans the friction coefficient can be obtained as:

μ =1βln

TdTj

!; ð6Þ

μ is the global friction coefficient, including the influence of the belt material as well as the belt and pulley-rib profiles. The
wheretest setup consists of a specially developed test rig, as shown in Fig. 6. In order to measure the tension in the belt spans two loadtransducers were used. A total of three arcs of contact were studied (β1=40°, β2=68°, β3=82°), see Fig. 7. For each contact arc,the tests were performed with eight different initial tensions T0 and five pulley rotational velocities ω. The range of the pulley'srotational velocities was deduced from the experiment shown in Fig. 8. The angular-speed loss was measured between the driver

wheredriven

Fig. 7. Schematic view of the experimental setup for measuring the friction coefficient.

Fig. 8. Experimental setup for measurement of an angular-speed loss.


and the driven pulley. From themeasured angular-speed loss themaximum sliding velocity between the belt and the pulley can beobtained using the equation:

gT;max = ωlossR; ð7Þ

R is the pulley radius (in our experimental setup R=0.05 m) and ωloss is the measured angular-speed loss between theand the driver pulley. The measurement of the maximum relative tangential velocity between the belt and the pulley for

different operational conditions is presented in Fig. 9.
threeIn order to avoid the influence of temperature on the friction coefficient the belt segmentwas tempered to the given temperature
before each measurement.A typical force measurement in the tight and slack span is shown in Fig. 10(a) (β=82°, T0=270 N). The corresponding friction

coefficient, computed using Eq. (6), is shown in Fig. 10(b). It is evident that the coefficient of the sticking friction is slightly higher thanthe coefficient of the sliding friction. However, as this difference is not significant the friction coefficient during slidingwill be used asthe reference friction coefficient μref=μsliding.

If the reference frictioncoefficient is constant thenormal contact forceperunit lengthcanbeobtainedusing theequation [5]:

fN =1RTde

−μrefϕ; 0≤ϕ≤β: ð8Þ

In Fig. 11 the analytically computed normal contact force, using Eq. (8), is presented. The mean value of the contact force isassigned to themeasured reference friction coefficient. Using this procedurewe can identify the influence of the relative tangentialvelocity and the normal contact force on the friction coefficient. The values of the friction coefficients obtained for three differentcontact arcs between the belt and the pulley are presented in Fig. 12. The friction characteristics differ slightly from the value of thecontact arc between the belt and the pulley. However, in all three cases the shape of the friction surfaces is almost identical. Clearly,the friction coefficient depends on the normal contact force and the relative tangential velocity, if the dependence of the friction

Fig. 9. Maximum relative tangential velocity between the belt and the pulley versus the torque on the driven pulley: (– □ –) rotational velocity=22 rad/sT0=310 N, ( ) rotational velocity=62.8 rad/s, T0=310 N, ( ) rotational velocity=76.8 rad/s, T0=411 N.

Fig. 10. Measured belt span axial forces and computed friction coefficient at β=82°, T0=270 N and ω=0.21 rad/s: (a) Axial force in belt spans; (b) Computedfriction coefficient with Eq. (6). (—–) axial force in tight belt span, (⋯) axial force in slack belt span.


,

coefficient on the normal contact force was expected, the observed strong influence of the relative tangential velocity on thefriction coefficient was not.

In Fig. 13(a) the averaged values of all three friction surfaces are presented. The values between the measured points can beinterpolated using bicubic splines. The friction characteristics can also be presented in an XY plane (see Fig. 13(b)), where thedependence of the friction coefficient on the normal contact force and the relative tangential velocity is more clearly seen.

3. Numerical belt-drive model

The numerical belt-drive model is based on multibody system dynamics together with the absolute nodal coordinate systemANCF [11]. A viscoelastic belt material that obeys the Kelvin–Voigtmodel is described by the following constitution relation:

σlðx; tÞ = Eεl x; tð Þ + c∂εl x; tð Þ

∂t ; ð9Þ

σl is the longitudinal distributed stress, E is the Young's modulus, c is the viscoelastic damping factor and εl is the Lagrangian
wherestrain. The system of the equations of motion, including all the beam elements in the belt and the constraint equations describingthe connectivity constraints, can be written as:
MB CTeB

CeB 0

" #e::

λB

� �= Q f + Q eB

Q dB

� �; ð10Þ

where

whereand th

Fig. 11. Normal contact force between belt and pulley.

Fig. 12. Measured values of the friction coefficient versus the normal contact force and the relative tangential velocity. ( ) contact arc β=82°, (– o –)contact arc β=68°, ( ) contact arc β=40°.


MB is the constant mass matrix of the belt [9], CeB is the Jacobian of the constraint equations, λB is the vector of the Lagrangeliers.
multip
In a belt-drive, the belt is constrained to move over the surface of the pulley. Both the normal reaction force FN and thetangential friction force FT are generated when the belt element contacts the pulley's surface.

Each belt element has five possible contact points, which are equally spaced along the length of the element, see Fig. 14. Thenormal reaction forces of each contact point are computed using the penalty method, Eq. (1). In the tangential contact directionthe Coulomb's friction law on the acceleration level is used:

sticking : jFiT jbμ0FiN ⇒ g::iT = 0

sliding : jFiT j = μ0FiN ⇒ g

::iT N 0

g; i∈IH; ð11Þ

μ0 is the coefficient of friction, the set IH contains all possible sticking contacts and g::iT is the relative acceleration in
where
tangential direction. In order to compute the possible sticking forces the contact problem in the tangential direction has to beformulated as a linear complementarity problem. The equation of motion, including the contact forces between the belt and thepulley, can be written as:

q::

r = HF WNλN + WTλTð Þ + h; ð12Þ

λN and λT are the contact forces in the normal and tangential directions. For a detailed description of the belt-drive modele contact formulation between the belt and the pulley the interested reader is referred to [11].

Fig. 13. Interpolation of the friction surface with bicubic splines: (a) Presentation in space; (b) Presentation in xy plane. ( ) averaged experimentallymeasured friction coefficients, ( ) interpolation of friction characteristics with bicubic splines.

Fig. 14. Belt-pulley contact formulation.


4. Incorporation of the measured friction characteristic into the numerical model

Themeasured contact parameters can be included into the flexible multibody belt-drivemodel presented in [11]. For each timestep of the numerical simulation, depending on the value of the normal contact force and the relative tangential velocity, thefriction coefficient is deduced from Fig. 13(a). If the contact parameters (the relative velocity and the normal contact force) are notwithin the measured friction characteristics the nearest measured value is taken.

The numerical modeling of the experiment (Fig. 15) is performed in order to validate the belt-drive model and the contactformulation presented in Ref. [11]. However, with the numerical model of the experiment we simulate the actual experiment formeasuring the friction coefficient. In the contact between the belt and the pulley the measured friction characteristics, presentedin Fig. 13(a), are applied. The numerical experiment is simulated at the same initial belt tensions and angular velocities of thepulley as in the real experiment. During the simulation, the forces in supports A and B are computed. Finally, using Eq. (6) thefriction coefficient, based on the numerical simulation, can be obtained. This friction coefficient can be compared with the oneobtained using the actual experiment. The agreement between both friction coefficients can indicate the validity of the numericalmodel and the friction-measuring procedure.

The belt model is based on two-dimensional beam elements, and ten beam elements are used tomodel the belt segment. In thecontact region between the belt and the pulley a finer discretization is applied in order to precisely compute the contact forces.During the numerical simulation for each contact point between the belt segment and the pulley the normal contact force and therelative tangential velocity are computed. Using these data the friction coefficient from Fig. 13(a) can be deduced. The frictioncoefficients obtained using the numerical simulations are presented in Fig. 16. The values obtained for the different contact arcsbetween the pulley and the belt are practically the same.

In Fig. 17 a comparison of the averaged friction coefficients, obtained with the actual and numerical models of the experiment,is presented. Some differences, especially at low and high normal contact forces, can be observed. These differences between theexperiment and the numerical model are due to the simplified expression for the computation of the reference normal force. The

Fig. 16. Computed values of the friction coefficient with the numerical model versus the normal contact force and the relative tangential velocity. ( )contact arc β=82°, (– o –) contact arc β=68°, ( ) contact arc β=40°.

Fig. 15. Numerical experiment for validation of the planar contact model.

Fig. 17. Obtained friction coefficient with numerical and actual experiment. ( ) actual experiment, (– o –) numerical simulation of experiment.


used analytical expression from Eq. (8) neglects the effect of the bending stiffness. This bending stiffness causes the appearance ofpeaks in the normal forces at the entrance and exit sections of the belt-pulley contact. This means that the reference normal forcethat is appointed to the measured friction coefficient is not computed with total accuracy.


However, we can conclude that with the numerical model and the measured friction characteristics we are able to predict thefrictional forces to a sufficient degree of accuracy. Here it is necessary to emphasize that any accurate prediction of the frictionalforces relies strongly on the experimentally obtained friction characteristic. From our experience the value of the contact stiffnessis not directly proportional to the number of belt-ribs. For instance, from the experimentally obtained contact stiffness for the K-section belt with five ribs, it is not possible to accurately predict the values of the contact stiffness for belts that have more or lessribs. Moreover, the values of the contact parameters of the same belt type may also depend on the manufacturer. So, for precisesimulations of the belt-drive dynamics using two-dimensional belt-drive models [7,8,10,11], the presented identification of thecontact parameters in this paper should be performed.

5. Conclusion

In this paper, methods for identifying the contact parameters between a V-ribbed belt and a pulley are presented. Twoexperimental setups were proposed in order to determine the contact stiffness and the friction coefficient between the belt andthe pulley. A nonlinear relation between the normal contact force and the deformation was observed. A quadratic approximationfunction was used in order to relate the normal contact force with the deformation. The measured friction coefficient dependedupon the normal contact forces and the relative tangential velocities. The friction characteristics were interpolated with bicubicsplines, which enabled the incorporation of the measured friction characteristics into the numerical model. Finally, the two-dimensional belt model and the measured friction procedure were verified with a numerical model of the experiment. Goodagreement between the friction characteristics obtained from the actual and numerical experiments was obtained.

Acknowledgments

This work was sponsored by the Slovenian Research Agency, under contract 3311-04-831674, and the French Rhone-AlpesRegion.

References

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[11] G. Čepon, M. Boltežar, Dynamics of a belt-drive system using a linear complementarity problem for the belt-pulley contact description, Journal of Sound andVibration 319 (2009) 1019–1035.

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+++experimental identification of the contact parameters between a v-ribbed belt and a pulley

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