Dynamic Modeling and Stability Analysis of Flat Belt Drives ...

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Dooroo KimGraduate Student

Research Assistant

Michael J. Leamy1

e-mail: [email protected]

Aldo A. Ferri

Department of Mechanical Engineering,Georgia Institute of Technology,

Atlanta, GA 30332

Dynamic Modeling and StabilityAnalysis of Flat Belt Drives Usingan Elastic/Perfectly PlasticFriction LawThis paper presents an analysis of a nonlinear (piecewise linear) dynamical model gov-erning steady operation of a flat belt drive using a physically motivated elastic/perfectlyplastic (EPP) friction law. The EPP law models frictional contact as an elastic spring inseries with an ideal Coulomb damper. As such, the friction magnitude depends on thestretch of the elastic belt and is integral to the solution approach. Application of theextended Hamilton’s principle, accounting for nonconservative work due to friction andmass transport at the boundaries, yields a set of piecewise linear equations of motion andaccompanying boundary conditions. Equilibrium solutions to the gyroscopic boundaryvalue problem are determined in closed form together with an expression for the mini-mum value of the EPP spring constant needed to transmit a given torque. Unlike equi-librium solutions obtained from a strict Coulomb law, these solutions omit adhesionzones. This finding may be important for interpreting belt drive test-stand results and theexperimentally determined friction coefficients obtained from them. A local stabilityanalysis demonstrates that the nonlinear equilibrium solutions found are stable to localperturbations. The steady dynamical operation of the drive is also studied using anin-house corotational finite element code. Comparisons of the finite-element solutionswith those obtained analytically show excellent agreement. �DOI: 10.1115/1.4003796�

IntroductionBelt drives are used in numerous applications to transmit power

etween machine elements. Some common applications includeehicle accessory drives; household appliances, such as vacuumleaners and washing machines; elevators and hoisting equipment;nd consumer devices powered by electric motors. In each appli-ation, the frictional interaction occurring between the belt and theulleys is critical for proper operation of the drive, motivating theeed for a thorough understanding of its behavior.

The effects of friction on flat belt drives have been investigatedy numerous researchers resulting in thorough reviews �1,2�. Ofhe various friction models explored, the most frequently used isoulomb’s Law, as depicted in Fig. 1�a�. The Coulomb law as-

erts that the friction force is proportional to the coefficient ofriction � and the normal force n. The Coulomb friction model, assed in traditional classical creep analysis, results in a frictionorce dependent only on the sign of the relative motion betweenhe belt and the pulley surfaces. The classic creep theory is re-iewed by Johnson �3�, while more recently, Bechtel et al. �4�xtended this theory to include inertial effects due to stretching.ubin �5� derived a closed-form solution for the mechanics of anxtensible belt.

The numerical difficulties posed by the discontinuity in Cou-omb’s Law has motivated the use of “smooth” or “regularized”pproximate models. One such friction model is the creep-rate-ependent friction model �6,7�, as depicted in Fig. 1�b�. Thereep-rate-dependent friction model exhibits dependency on bothhe direction and magnitude of the relative velocity between theelt and pulley, vrel. A closed-form solution for steady operation ofbelt drive using the trilinear creep-rate law was developed by

1Corresponding author.Contributed by the Dynamic Systems, Measurement and Control Division of

SME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CON-

ROL. Manuscript received May 14, 2010; final manuscript received January 18,
011; published online April 11, 2011. Assoc. Editor: Michael A. Demetriou.
ournal of Dynamic Systems, Measurement, and ControlCopyright © 20

https://dynamicsystems.asmedigitalcollection.asme.org on 06/29/2019 Terms of

Leamy and Wasfy �8�. Although the creep-rate-dependent frictionmodel alleviates numerical difficulties, its main drawback is that itdoes not correlate easily with microscopically informed frictionstudies.

In this paper, friction is modeled as an elastic spring, withspring constant, kf, in series with an ideal Coulomb damper, asdepicted in Fig. 2. In elastic-plastic deformation studies, this fric-tion model is referred to as the elastic/perfectly plastic �EPP� fric-tion model �9,10� and arises from microscopic friction studies. Itsfriction force is dependent on the displacement, as seen in Fig.1�c�. The situation depicted in Fig. 1�c� assumes unidirectionalmicro/macro slip as can be expected in steady belt drive opera-tion. The situation with velocity reversals is more complicated,with a hysteresis effect �9�.

The EPP friction force per unit length is described by

f�s� = �kfu�s� , u�s� ��n�s�

kf

�n�s� , u�s� ��n�s�

kf

� �1�

where s denotes the arc length, kf denotes the EPP spring stiffness,� denotes the coefficient of friction, n�s� denotes the normal forceper unit length, and u�s� denotes the tangential belt displacement.At a certain location on the pulley, the belt displacement exceeds�n�s� /kf, and the friction changes from elastic to fully developedsliding �Coulomb-like� friction. This location is termed the tran-sition point and is denoted by s=s�.

The EPP friction model is employed in this study to provide amore accurate description of the frictional contact between thebelt and pulley. In addition, the inclusion of friction elasticity isexpected to help alleviate high parameter sensitivities encounteredin previous belt drive studies. This may be particularly beneficialin the study and design of more complex systems, such as V-belts
for continuously variable transmissions �CVTs�. It is notable that
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revious V-belt studies employing the Coulomb friction law re-uired extremely precise initial parameter values �upwards of 16igits� to achieve solution convergence �11,12�.

The EPP friction law as applied to a belt drive has many simi-arities to shear creep models used in other belt drive studies; see,or example, Refs. �13–18�. In shear creep analysis, the tension-arrying cords rest on a thick belt carcass in contact with theulley, where this carcass shears due to a speed difference be-ween the cords and the pulley surface. The speed of the cordspon initial contact with the pulley is an unknown variable, whichust be found concurrent with the solution, which greatly com-

licates the solution procedure �16�. In addition, it is not clear inhese models what happens to the accumulated belt shear once theelt exits the pulley; for example, it is assumed the belt entersach pulley with zero shear, regardless of the exit shear at therior pulley. In the EPP analysis presented herein, the thickness ofhe belt carcass is assumed small and therefore the belt does nothear. Instead, the elastic spring can be likened to an asperity onhe pulley surface. A further difference is that the belt drive prob-em is posed with an unknown initial spring strain �u /�s, whichnters as a boundary condition, instead of an unknown velocity.

This paper first describes an axially moving, flat belt systemnd details the belt kinematics. In particular, a new expression iseveloped for the belt velocity as a function of strain, whichvoids assuming the initial belt velocity equal to the surface pul-ey speed. It is the first study known to the authors to employamilton’s principle to derive the belt drive nonlinear equationsf motion and accompanying boundary conditions, which requiressing the extended principle with virtual mass transport. An ad-antageous feature of this approach is its ability to recover allcceleration terms in a straightforward manner, including those

Fig. 1 Friction laws employed in flat belt drive studies

ig. 2 Elastic/perfectly plastic spring and sliding damper rep-
esentation of friction
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due to tangential stretching, which have only recently been uncov-ered using more traditional Newtonian approaches �4�. Equilib-rium solutions satisfying these equations are found correspondingto steady drive operation. A stability analysis demonstrates thatthe nonlinear equilibrium solution is stable under local perturba-tions. The effect the EPP spring stiffness has on belt drive opera-tion is explored using an example drive. The analytical results arealso compared with results generated by an in-house, dynamicfinite element simulator and excellent agreement is noted.

2 Flat Belt ModelFigure 3 depicts a two-pulley, flat belt drive where the driver

and driven pulleys have equal radii, R. The analysis presentedhere can be extended to a general belt drive with different pulleyradii and any number of pulleys, but we examine the equal-sizetwo-pulley drive to streamline the discussion. The angular coordi-nate, �i, is referenced to each pulley and is measured from the beltentry points. The subscripts, i=1,2, denote the driven and driverpulleys, respectively. An Eulerian formulation is employed suchthat arc length coordinate, si=R�i, is fixed in space. A uniform,axially moving flat belt wraps around the rotating pulleys andexperiences tangential displacement ui�s , t�. When the belt firstenters the pulley at si=0, the EPP spring is unstretched. The beltloses contact with the pulley at si=sf =R�.

To streamline the development, the remainder of this sectionwill discuss the derivations for the driven pulley, and the 1 and 2subscripts will be dropped. However, the analysis presented caneasily be extended to the driver pulley. For the driven pulley, thebelt enters the pulley with low tension TL and velocity vL. It thenleaves the pulley with high tension TH and corresponding velocityvH.

2.1 Belt Tension. A linear, constitutive law relates the belttension T to the belt strain � by

T = EA� �2�

where E is Young’s modulus and A is the cross-sectional area ofthe belt. Assuming the belt to be linearly elastic, the tension dis-tribution can be decomposed approximately into two components

T = To + EA� �3�

where � denotes the incremental strain of the belt element and Todenotes a reference initial tension at the pulley inlet. Similarly, thestrain can also be decomposed approximately into

� = �o + � �4�

where �o denotes the reference strain, and the additional strain ofˆ

Fig. 3 Two pulley belt drive system with system variables

the belt element � is related to the displacement by

Transactions of the ASME


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� =�u

�s�5�

he reference tension To is given by

To = TL − EA�u

�s�0,t� �6�

t will be shown below that no solutions can be found such thato=TL.

2.2 Belt Velocity. For a nonrotating observer attached to theulley, the � coordinate is fixed. For an observer rotating with theulley, the belt makes small displacements as time progresses. Asresult, the belt’s velocity v�s� is different from the entering

elocity of the belt vL due to the strain difference �u /�s�s , t��u /�s�0, t�. This difference in speed is accounted for by theaterial derivative

Du

Dt= � �u

�s�s,t� −

�u

�s�0,t�� s

�t+ � �u

�t�s,t� −

�u

�t�0,t��

= � �u

�s�s,t� −

�u

�s�0,t��vL + � �u

�t�s,t� −

�u

�t�0,t�� 7�

s a result, the velocity at each location s is given by

v�s,t� = vL +Du

Dt= vo + vL

�u

�s+

�u

�t�8�

here vo corresponds to the velocity at To,

vo = vL�1 −�u

�s�0,t�� −

�u

�t�0,t� �9�

ote that if the belt adheres to the pulley at the inlet, �u /�s�0, t�ust be zero by definition and then v�0, t�=R�p, where �p de-

otes the angular velocity of the driven pulley. This together withq. �8� yields an expression for vo

vo = R�p −�u

�t�0,t� �10�

inally, Eqs. �9� and �10� yield an expression for the inlet velocityn terms of the initial spring strain,

vL =R�p

1 −�u

�s�0,t�

�11�

Equations of MotionUsing the mechanics associated with Coulomb’s friction law, a

teadily rotating belt drive develops a single adhesion and slipone on each driver and driven pulley. Together, the adhesion andlip zones span the entire belt-pulley contact region �3�. For mod-rate dynamic �unsteady� operation, the same friction state haslso been predicted to hold �19�. It is therefore reasonable to ex-ect that the EPP friction model would admit a similar solution forhe same drive �see Fig. 4� in which there exist an initial adhesionone, followed by an elastic creep zone, and then a final fullyeveloped slip zone. The elastic creep zone would see the EPPpring stretching up until a location s=s�. The fully developedreep zone would be defined between s=s� and s=R�. Here, thePP spring would be stretched to its full capacity and the friction

orce would be fully developed. Note that under this scenario, theelt would need to enter the pulley with zero incremental strain

ˆ�0, t�=0, To would equal TL, and a zone of constant tension couldxist. However, it will be shown that in order to find solutions tohe resulting boundary value problem, the initial incrementaltrain cannot be equal to zero and therefore To�TL. As a result,
he elastic creep zone begins at so=0.
ournal of Dynamic Systems, Measurement, and Control


3.1 Extended Hamilton’s Principle. Hamilton’s principle isused to derive the equations of motion and the appropriate bound-ary conditions. The kinetic energy, potential energy, external vir-tual work, and virtual mass transport are derived for the belt in thetwo subdomains governed by the two functions in the piecewiseEPP friction model. In the first subdomain, denoted by the sub-script e, the friction acting on the belt is in the elastic creep zone.In the second subdomain, denoted by the subscript c, the frictionis in the fully developed slip zone. The two subdomains meet atthe transition angle s�. Using the velocity equation in Eq. �8�, thekinetic energies of the flat belt system are given by

Te�s,t� =so

s�

1

2m�vo + vLue� + ue�2ds �12�

Tc�s,t� =s�

sf 1

2m�vo + vLuc� + uc�2ds �13�

where m denotes the mass per unit length and u=�u /�t and u�=�u /�s.

The potential energy of the belt in the elastic creep zone is dueto the axial stretching and extension of the EPP spring such that

Ve�s,t� =so

s� �Toue� +1

2EA�ue��

2� +1

2kf�ue�2�ds , �14�

where an assumption of tensile strain �i.e., no compression� isassumed. The potential energy of the belt in the slip zone is onlydue to the axial stretching,

Vc�s,t� =s�

sf �Touc� +1

2EA�uc��

2�ds �15�

The nonconservative work due to friction �in the fully developedslip zone� and tension acting at the boundaries is given by

�WNC = −s�

sf

�n�s��ucds − TL�ue�s0,t� + TH�uc�sf,t� �16�

where n�s� represents the normal force per unit length. Taking thefirst variation of Eqs. �12�–�15� yields

�Te�s,t� =so

s�

m�vo + vLue� + ue��vL�ue� + �ue�ds �17�

�Tc�s,t� =�

sf

m�vo + vLuc� + uc��vL�uc� + �uc�ds �18�

Fig. 4 Friction zones for the flat belt drive: adhesion, elasticcreep, and fully developed slip

s

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�Ve�s,t� =so

s�

��To + EAue��ue� + kfue�ue ds �19�

�Vc�s,t� =s�

sf

�To + EAuc��uc�ds �20�

variation capturing momentum transport at the boundaries mustlso be included in order to accurately formulate Hamilton’s prin-iple for the fixed volume considered �20–23�. Here, this term isimply

�M = G�v�so,t��ue�so,t� − v�sf,t��uc�sf,t�� 21�

here G=mvL represents the mass flow rate. A constraint mustlso be enforced to keep the displacements, ue and uc, continuoust the transition point s�. This holonomic constraint equation re-ating the generalized coordinates is given by

g = ue�s�,t� − uc�s�,t� = 0 �22�

ith the first variation

�g = �ue�s�,t� − �uc�s�,t� = 0 �23�

etting denote a Lagrange multiplier such that

t1

t2

�gdt =t1

t2

��ue�s�,t� − �uc�s�,t��dt = 0 �24�

he accumulated variations can then be substituted into Hamilton’srinciple

t1

t2

��L + �WNC + �M − �g�dt = 0 �25�

here �L=�T−�V, �T=�Te+�Tc, and �V=�Ve+�Vc. Using Eqs.16�–�24� and integrating by parts in the usual manner, Hamilton’srinciple yields a set of equations of motion

mue + 2mvLue� + �mvL2 − EA�ue� + kfue = 0, so � s � s� �26�

muc + 2mvLuc� + �mvL2 − EA�uc� + �n = 0, s� � s � sf �27�

ubject to boundary expressions

��TL − �To + EAue�� ue��so= 0 �28�

��mvL�vo + vLue� + ue� − �To + EAue�� − �ue��s� = 0 �29�

��mvL�vo + vLuc� + uc� − �To + EAuc�� − �uc��s� = 0 �30�

��TH − �To + EAuc�� uc��sf= 0 �31�

amilton’s principle as formulated does not provide the normalorce per unit length n since it is a workless constraint force.owever, an approach using balance of momentum in the normalirection does �19�

n =T�s� − Gv�s�

R�32�

here the tension T�s� and the velocity v�s� can be expressed inerms of the displacement fields, see Eqs. �3� and �8�.

Because ue�s�� and uc�s�� can be varied, the natural boundaryonditions at s� are taken in Eqs. �29� and �30� such that

= mvL�vo + vLue��s�� + ue�s�� − �To + EAue��s

�� 33�

= mvL�vo + vLuc��s�� + uc�s�� − �To + EAuc��s

�� 34�

etting the Lagrange multiplier equal for both conditions yields

41009-4 / Vol. 133, JULY 2011


mvL�vo + vLue��s�� + ue�s�� − �To + EAue��s

�� = mvL�vo + vLuc��s��

+ uc�s�� − �To + EAuc��s�� 35�

Since Eq. �22� enforces the displacements at s� to be equal for alltime, the velocities are also equal �i.e., ue�s��= uc�s��. Cancelinglike terms and simplifying, the boundary condition in Eq. �35�simplifies to

ue��s�� = uc��s

�� 36�Note that from Eq. �3�, the tension distribution is linearly relatedto the strain. Therefore, Eq. �36� can be equally expressed as

Te�s�� = Tc�s�� 37�

For the boundary conditions at s=so and s=sf, the natural bound-ary conditions are chosen such that

TL = To + EAue� = Te�so� �38�

TH = To + EAuc� = Tc�sf� �39�

3.2 Linearization. For use in evaluating stability of equilib-rium solutions, the equations of motion in Eqs. �26� and �27� arelinearized around a steady-state �i.e., equilibrium� solution by in-troducing

u�s,t� = u�s� + u�s,t� �40�

where u�s , t� denotes small perturbations about the equilibriumsolution u�s�. In order to facilitate a solution, it is necessary �andreasonable� to assume that the transition point s� is unaffected bythe small perturbations in displacement. Suppressing the tilde �� notation, the linearized equations of motion about the equilib-rium solution for the two subdomains are

mue + 2mvLue� + �mvL2 − EA�ue� + kfue = 0, 0 � s � s� �41�

muc + 2mvLuc� + �mvL2 − EA�uc� +

�

R�EAuc� − G�vLuc� + uc��

= 0, s� � s � sf �42�

Since s� is known in the perturbed problem, only four boundaryconditions are required to establish a well-posed problem. TheEPP spring must still have zero displacement at so. Additionally,the tension at sf is chosen to remain constant. Finally, the dis-placement and tensions must be matched at s�. These consider-ations lead to the following boundary conditions consistent withEqs. �28�–�31�:

ue�so,t� = 0 �43�

�uc

�s�sf,t� = 0 �44�

ue�s�,t� − uc�s�,t� = 0 �45�

�ue

�s�s�,t� −

�uc

�s�s�,t� = 0 �46�

4 Equilibrium Solutions: Steadily Rotating Drive

The equilibrium displacement solutions, ue and uc, are gov-erned by

�mvL2 − EA�ue� + kfue = 0, so � s � s� �47�

�mvL2 − EA�uc� + �n = 0, s� � s � sf �48�

These equations account for all tangential accelerations, includingdue to stretching, that have only recently been recovered �4� usingmomentum balance techniques; more details can be found in Ref.
�24�. The boundary conditions governing the equilibrium solu-


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ions obtained previously from Hamilton’s Principle are repeatedere as

Te�so� = TL �49�

ue�s�� = uc�s�� 50�

Te�s�� = Tc�s�� 51�

Tc�sf� = TH �52�o fully pose the equilibrium problem, two other boundary con-itions are enforced, which arise from consideration of the EPPriction law. These are zero spring displacement at so and conti-uity of the friction force at the transition point

ue�so� = 0 �53�

kfue�s�� =�

R�Tc�s�� − Gvc�s�� 54�

Using Eqs. �47� and �48� with the boundary conditions in Eqs.49�–�54�, closed-form solutions for ue�s�, uc�s�, and s� are pos-ible using symbolic manipulators, such as MAPLE

®. These solu-ions are lengthy functions of the belt and system parameters, asell as the low and high tensions TL and TH, and as such are notiven here. Instead, numerically evaluated solutions will be pre-ented for typical drives in Sec. 5. However, the closed-form so-utions do reveal that du /ds�0� is not zero, and thus that To�TL.

hen solutions are sought enforcing the incremental strain to beero at s=0, at least one of the boundary conditions �49�–�54�ust be violated, further reinforcing the necessity of the nonzero

u /ds�0� result.An analysis for the driver pulley follows similarly with the

ension boundary conditions at the inlet and exit reversed suchhat

Te�so� = TH �55�

Tc�sf� = TL �56�ote that the coefficient of friction for the driver pulley is treated

s a negative value, consistent with the EPP friction law depictedn Fig. 1�c�.

Finally, it is of interest to predict the minimum kf required toransmit a given tension difference across the pulley. When kfkf ,min, then s�=R� and thus the belt undergoes elastic creep

hroughout the entire contact arc. If kf is smaller than kf ,min, theystem cannot undergo the prescribed tension change using theiven contact arc. This minimum value of kf can be determined byolving the equilibrium equations with the following boundaryonditions:

Te�0� = TL �57�

Te�R�� = TH �58�

ue�0� = 0 �59�

kf ,minue�R�� =�

R�TH − GvH� �60�

4.1 Stability Analysis. Because the system considered hereins nonlinear, it is of interest to determine the stability of the equi-ibrium solution found. The linearized equations will be used toetermine its local stability. Solutions are sought of the form

ue�s,t� = �s�ei�t, so � s � s� �61�

uc�s,t� = ��s�ei�t, s� � s � sf �62�

here strictly real values for � indicate stable oscillatory solu-
ions. Complex values of � are either asymptotically stable �posi-


tive imaginary part� or unstable �negative imaginary part�.Substituting Eqs. �61� and �62� into the equations of motion in

Eqs. �41� and �42� and using the boundary conditions �43�–�46�lead to the homogeneous boundary value problem:

�mvL2 − EA�

d2

ds2 + 2mvLi�d

ds+ �kf − m�2��s� = 0,

so � s � s� �63�

�mvL2 − EA�

d2�

ds2 + �2mvLi� +�

R�EA − GvL��d�

ds

+ �− m�2 −�G

Ri��s� = 0, s� � s � sf �64�

with boundary conditions

�so� = 0 �65�

d�

ds�sf� = 0 �66�

�s�� − ��s�� = 0 �67�

d

ds�s�� −

d�

ds�s�� = 0 �68�

Using Eqs. �63�–�68�, the eigenfunctions, n�s� and �n�s�, andeigenvalues, �n, can be easily found using a symbolic manipula-tor. Because axially moving systems are inherently gyroscopic,the eigenvalues and the corresponding eigenfunctions occur incomplex conjugate pairs �25,26�. Unlike nongyroscopic un-damped linear systems, these complex eigenfunctions do not passthrough equilibrium simultaneously. The eigenfunction for the en-tire arc length is a piecewise defined function given by

�n�s� = n�s� , s � s�

�n�s� , s � s�� 69�

Finally, the displacements of the axially moving flat belt are

ue�s,t� = n�s�ei�nt + n�s�e−i�nt, so � s � s� �70�

uc�s,t� = �n�s�ei�nt + �n�s�e−i�nt, s� � s � R� �71�

where ue and uc are real-valued functions. Assuming that thelower eigenvalues are more likely to have negative imaginaryparts, the higher eigenvalues are neglected when determining sta-bility. As a rule of thumb, inspection of the first three eigenvaluesis assumed to be sufficient.

Similar to other studies of axially moving media �27�, when thebelt velocity vL is less than the critical transport speed given by

vcr = �EA

m�1/2

�72�

the system �63�–�68� is positive definite and the eigenvalues �ncontain positive imaginary parts, thereby determining the equilib-rium solutions to be stable. For illustrative purposes, the first threeeigenfunctions, �1�s�, �2�s�, and �3�s�, for a typical drive areplotted in Fig. 5 as a function of arc length.

5 Numerical Results for an Example Belt DriveIn this section, the character of the EPP belt drive equilibrium

solutions is explored and compared with analogous solutions formodels employing Coulomb’s friction law. Numerical results forboth friction laws were generated using example belt drive param-eters. The low tension is considered constant at 200 N and thehigh tension is varied from 400 N to 4000 N, effectively varying
the extent of the elastic and slip zones. Other parameters em-
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loyed are given in Table 1. The solutions used for a flat belt drivemploying Coulomb’s law are well known and can be reviewed inefs. �3,8�.The friction forces on the driven and driver pulleys for both the

oulomb and EPP friction models are depicted in Figs. 6 and 7.oth models exhibit slope discontinuities at the transition points

eparating friction zones. For the results with Coulomb’s law, thelope discontinuity occurs at the transition point between the ad-esion and slip zones and is accompanied by a jump in the frictionorce. For the results with the EPP friction model, the slope dis-ontinuity occurs between the elastic creep and the fully devel-ped slip zones and no friction jump is exhibited. The behaviorith increasing high tension is similar for the two friction laws in

hat the slip zone increases with increasing high tension. For the

ig. 5 Real part of the first three eigenfunctions for a typicalrive

Fig. 6 Friction force

Fig. 7 Friction force

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Coulomb friction model, the adhesion arc approaches zero as hightension increases; ultimately, only the slip zone exists on the pul-ley arc; this behavior limits the maximum torque transfer of thepulley. For the EPP friction model, at a small-enough tension dif-ference �not shown�, there exists only an elastic creep zone. As thehigh tension increases, the elastic creep zone on each pulleyshrinks �but the maximum friction force in the zone grows�, andthe fully developed slip zone grows until the maximum torquetransfer is achieved. As expected, similar trends are exhibited inthe belt tension results, see Figs. 8 and 9. The tension distributionsfor both friction models are continuous, but with a discontinuousslope for the Coulomb friction model and a smoothly varyingslope for the EPP friction model. The elimination of these tensionslope discontinuities could potentially alleviate parameter sensi-tivities reported earlier in continuously variable transmission stud-ies employing Coulomb friction modeling �11,12�.

As noted in the Figs. 7 and 9, the elastic creep and fully devel-oped slip zones vary with the tension difference TH−TL. To studythis behavior further, the transition angle �� and the slip angleextent �s were calculated for both the driven and driver pulleys asa function of the high tension TH, see Fig. 10. It is evident in Fig.10 that the transition angle �� and the slip angle extent �s are notidentical for both pulleys. At each value of high tension TH, theelastic creep arc is larger on the driven pulley than on the driver

Table 1 Physical properties used in the example flat belt drivestudy

Parameter Assigned value

1036 kg /m2

A 0.0001 m2

E 800,680 kN /m2

�2 120 rad/s� 1.2R 0.08125 mkf 8.1�107 N /m2

ith Coulomb friction
w
with EPP friction



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ulley. Note also that the transition angle consumes the entireulley arc �i.e., 180 deg� at high tensions of 380 N and below forhe driven pulley, while for the driver pulley, the transition angles 180 deg at high tensions of 310 N and below. However, becausehe minimum high tension for the driven pulley was higher than10 N, the driver pulley cannot achieve this value in the equaladii example considered, and thus always has a fully developedlip zone. Note finally that at a high tension of 8674 N, the fullyeveloped slip arc for both pulleys is 180 deg such that the elasticreep zones no longer exist.

The effect of the EPP spring stiffness value on the initial ten-ion To and initial incremental strain du /ds�0� is explored next.hese metrics are plotted as a function of the EPP spring stiffness

n Figs. 11 and 12. As kf increases, and thus friction approaches aoulomb-like description, the initial tension converges to the low

ension TL and the initial incremental strain approaches, but nevereaches, zero. This latter point is evident in Fig. 13, where aagnified view of large kf clearly shows the initial incremental

train being nonzero.

Fig. 9 Tension

Fig. 8 Tension w

ig. 10 Transition and slip angle as a function of high tension
H


6 Verification via Finite Element SimulationsNext, to verify the predicted equilibrium solutions, simulation

results were generated using an in-house, dynamic finite elementsimulator customized for belt drives. The finite element model isan extended version of the one developed in Ref. �28� and em-ploys an explicit predictor-corrector time marching scheme and apenalty formulation for contact. The pulleys are modeled as rigid,circular constraints and the belt is modeled with two-noded trusselements. The discretization consists of 100 truss elements perhalf pulley. For this study, the belt nodes were modified to store anadditional state variable, which tracks the relative circumferentialbelt displacement, and thus allows the EPP spring stretch to becomputed. Each time the node loses contact with the pulley, thisstate variable resets to zero displacement. The dynamic model issimulated to steady-state by slowly ramping-up the imposeddriver pulley angular velocity �2 until a constant value of 120rad/s is achieved; the driven pulley is loaded by a constant oppos-ing torque M of 45 N m, which establishes a tension difference in

th EPP friction

Coulomb friction

Fig. 11 Initial tension To as a function of the EPP spring con-

wi

ith

stant kf

JULY 2011, Vol. 133 / 041009-7


FE f

Ff f

Fig. 14 Analytical and finite element predicted co

041009-8 / Vol. 133, JULY 2011

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the belt spans. The same parameters in Table 1 were used for thefinite element study, with additional necessary parameters pro-vided in Table 2. These include the penalty spring stiffness k andthe span length lo. Results were generated for three different val-ues of the EPP spring constant: kf =2.8�107, kf =2�108, and kf=8.1�109 N /m2.

The resulting tensions, TL and TH, as well as the angular veloc-ity of the driven pulley, �1, were calculated as outputs from thefinite element model. The values of TL, TH, and �1 are summa-rized in Table 2 for each of the three values of kf. These valueswere then used as input parameters to the analytical model so thatdirect comparisons could be made. A comparison of the contactforces per unit length predicted by the analytical model and thefinite element model for the three values of kf is provided in Figs.14–16. The friction force and normal force per unit length areplotted for both the driven and driver pulleys in one plot using anangle � measuring location counter clockwise from the vertical�see also Fig. 3�. As such, � starts at 0 deg for the inlet of thedriven pulley and 180 deg for the inlet of the driver pulley.

Figure 14 provides the analytical and finite element results forkf =2.8�107 N /m2. This is the minimum EPP spring value �com-puted from the analytical solution using boundary conditions�57�–�60�� such that the system can transmit a torque of 45 N m.The nonexistence of a fully developed slip zone can be verifiedfrom the finite element solution by noting that a slope discontinu-ity is not evident in the friction force. Figure 15 provides theresults for an intermediate value of kf =2�108 N /m2 leading to aslip angle extent of approximately 0.98 rad �56 deg�. As the EPPspring stiffness kf increases further, the spring can be approxi-mated as rigid. This Coulomb-like solution is provided in Fig. 16�kf =81�108 N /m2� where the slip zone begins at ��=1.93 rad�110.6 deg� and the slip angle extent is �s=1.21 rad �69.4 deg�.This slip angle is close to the approximated value of 1.24 radpredicted using the Coulomb solution provided in Ref. �3�. Simi-lar to Coulomb’s law, the solution in Fig. 16 displays a sharpchange in slope at the transition angle.

One objective of this investigation of the EPP friction model

s for the example flat belt drive study

kf =2.8�107 kf =2.0�108 kf =8.1�109

156.2 N 155.1 N 158.3 N712.2 N 716.7 N 707.1 N

119.1 rad/s 119.1 rad/s 119.1 rad/s3.14 rad 2.16 rad 1.93 rad

0 rad 0.98 rad 1.21 rad2.34 rad 2.08 rad 1.93 rad0.80 rad 1.06 rad 1.21 rad146.4 N 155.1 N 158.3 N

1.2�104 2.5�10−7 �0

7 2

ig. 12 Initial incremental strain du /ds„0… as a function of thePP spring constant k

ig. 13 Detail of the initial incremental strain du /ds„0… as aunction of the EPP spring constant k

Table 2 Additional properties and result

Parameter Assigned value Parameter

M 45 N m TL

�2 120 rad/s TH

lo 2�R=0.5105 m �1

k 150 kN /m2 �1�

�1s�2

�

�2sTo

du

ds�0�

ntact forces per unit length, kf=2.8Ã10 N/m



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as to smooth sharp changes in model behavior. If the EPP springonstant is chosen to be too large, the benefits of the EPP elastic-ty are lost. Therefore, the EPP spring stiffness should be chosenppropriately such that the belt can undergo a prescribed tensionhange but also avoid sharp changes in system behavior near tran-ition points. Finally, note that Figs. 14–16 document very goodgreement between the analytical equilibrium solutions and theteady-state behavior predicted with the finite element simulatorver a range of EPP stiffnesses. This comparison serves to verifyhe model development presented herein.

ConclusionsA model and accompanying analysis have been detailed, which

escribe the dynamical operation of a flat belt drive subject tolastic/perfectly plastic friction. This friction law was chosen forts connection to physically observed friction behavior and for itsegular representation. Nonlinear �piecewise linear� equations ofotion were developed using an extended version of Hamilton’s

rinciple, which captures nonconservative work due to frictionnd virtual momentum transport. All accelerations due to stretch-ng and slipping are recovered. At steady state, the solution to theoverning equations is generally composed of an elastic creepone followed by a fully developed slipping zone. An adhesionone is not admissible, which is surprising considering Coulombtudies exhibit one. For small transmitted torques, it is possible forhe solution to exhibit only an elastic creep zone. It was shownhat, for belt velocities below the critical speed, this steady-stateolution is stable with respect to small perturbations. For an ex-mple, belt dive, comparison of analytically predicted steady-stateontact forces to those predicted by an in-house dynamic finitelement simulator shows excellent agreement. It is believed thathe elastic/perfectly plastic friction law could be useful inollow-on studies, which seek more representative belt drive mod-ls �flat, V-belt, and others� exhibiting enhanced numerical stabil-

Fig. 16 Analytical and finite element predicted

Fig. 15 Analytical and finite element predicte

ty.



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