291

Characterisation of friction reduction with

tangential ultrasonic vibrations using a SDOF

model

Shravan Bharadwaj and Marcelo J. Dapino*

Department of Mechanical and Aerospace Engineering,Smart Vehicle Concepts Center,The Ohio State University,201 W. 19th Ave., Columbus, OH 43210, USAE-mail: dapino.1@osu.edu*Corresponding author

Abstract: Active control of friction between sliding surfaces is offundamental and practical interest in automotive applications. It hasbeen shown that the friction force between sliding surfaces decreaseswhen ultrasonic vibration is superimposed on the sliding motion. Thisprinciple can be applied to systems in which solid state lubricationor friction modulation is advantageous. The ultrasonic vibration maybe applied longitudinally or normal to the direction of motion. Anumber of friction models have been considered in order to analysethis phenomenon. The degree of friction reduction has been shown todepend on the ratio of the sliding velocity to the vibration velocity. Sincefriction is a system response, it is necessary to include system dynamicsin the analysis of ultrasonic lubrication. A nonlinear single-degree-of-freedom-model is formulated and numerically approximated to quantifythe effect on friction reduction of control force, intrinsic coefficient offriction, mass load, tangential contact stiffness at the sliding interface, andsystem stiffness. Model results are in close agreement with experimentalmeasurements.

Keywords: piezoelectric actuators; system dynamics; ultrasonic lubricationvehicle design.

Reference to this paper should be made as follows: Bharadwaj, S.and Dapino, M.J. (2013) ‘Characterisation of friction reduction withtangential ultrasonic vibrations using a SDOF model’, Int. J. Vehicle

Biographical notes: Shravan Bharadwaj received the MS degree inmechanical engineering from the Ohio State University, with MarceloDapino as advisor, and the B. Tech degree from the NationalInstitute of Technology, Trichy, India. His MS research focused onsystem dynamics, experimental methods, and mechanical design of smartmaterial components for automotive applications. Prior to pursuing theMS degree, he worked in various automotive companies with focus ondesign and manufacturing of vehicle engine and chassis systems. Hiscurrent affiliation is with Apple Inc., Cupertino, California, where he isa manufacturing design engineer.

Copyright © 2013 Inderscience Enterprises Ltd.

Design, Vol. 63, Nos. 2/3, pp.291–304.

Int. J. Vehicle Design, Vol. 63, Nos. 2/3, 2013/

292 S. Bharadwaj and M.J. Dapino

Marcelo Dapino is the Honda R&D Americas Designated Chair inEngineering at the Ohio State University, where he is a Professor inthe Department of Mechanical and Aerospace Engineering. At OhioState University, he serves as Associate Director for Research ofthe Smart Vehicle Concepts Centre, a National Science FoundationIndustry/University Cooperative Research Centre, and is a Senior Fellowof the Centre for Automotive Research. His research interests focus onresearch, development, and manufacture of smart material systems. Heserves on the Executive Committee of the ASME Aerospace Division andhas led the organisation of major ASME and SPIE conferences.

1 Introduction

Friction is the resistance to motion that occurs when two objects in contact slide orroll relative to each other. Depending on the nature of the system, friction can arisedue to dry contact, fluid shear, or internal forces. The tangential force required toinitiate motion (static friction force) is greater than the tangential force requiredto maintain relative motion (kinetic friction force) (Bhushan, 2002). Friction is asystem response rather than a material property. Hesjedal et al. (2002) proposedlaws of friction for macroscopic bodies in contact, which state that friction forceis directly proportional to the applied load and is independent of the apparentcontact area. According to Coulomb’s law of friction, the kinetic friction force isindependent of the sliding velocity.

Friction reduction due to ultrasonic oscillations has been reported in theliterature (Littmann et al., 2001a,b; Storck et al., 2002; Bharadwaj and Dapino,2009). The degree of friction reduction is often quantified through the frictionratio, defined as friction force with ultrasonic vibrations over friction force withoutultrasonic vibrations. The friction ratio depends on the velocity ratio, defined as thebase sliding velocity over the velocity of the superimposed ultrasonic vibration. Theeffectiveness of ultrasonic lubrication is higher at low velocity ratios, and accordingto the Dahl and Coulomb friction models (Leus and Gutowski, 2008; Tsai andTseng, 2005), it decreases to zero for a velocity ratio of 1.

As described in Bharadwaj and Dapino (2009), at velocity ratios lower than 1,the oscillation creates a pulsating tangential force since this force follows the signumfunction. The average of this pulse is lower than the average without the oscillatingcomponent, hence frictional force reduction is observed. In the limit when thevelocity ratio is zero or unity, the pulse signal becomes asymmetric either withzero average or average equal to the maximum value of the pulse, correspondingrespectively to maximum friction reduction or zero friction reduction. Thenonlinearity associated with the signum function poses computational challengessince very small time steps are required in order to maintain accuracy, thus resultingin numerically stiff equations and longer simulation times. Leus and Gutowski(2008) argued that that the friction force can be reduced in one vibration cyclewithout instantaneous change of the friction force’s direction.

Ultrasonic lubrication can be utilised in various vehicle systems, e.g., suspensionjoints, steering components, seat rails, brakes, powertrain components, etc.

Tangential ultrasonic vibrations using a SDOF model 293

Superposition of high-frequency vibrations on low-frequency disturbances or dithercontrol has been proposed for suppressing squeal in automotive brakes (Cunefareand Graf, 2002; Badertscher et al., 2007; Hagerdorn et al., 2005; Jearsiripongkuland Hochnelert, 2006; Hagedorn and von Wagner, 2004). Active or smart padshave been developed (Michaux et al., 2007) which include embedded piezoelectricelements between the brake pad material and the metallic backing plate togenerate a normal harmonic force. This force provides a small variation about themean clamping force. It has been reported that tangential dither excitation canstabilise an unstable system having self-excited oscillations. The effect of variousdither waveforms on stability of a system has also been studied (Hagedorn andvon Wagner, 2004). The application of high frequency dither to eliminate self-excited stick slip oscillations and active friction reduction using ultrasonics arerelated phenomena since the governing equations in both cases are the same. Theformer has been studied using Stribeck friction and an averaging technique (Doet al., 2007). Using this approach, approximate relations involving the amplitudeand frequency of the excitation needed to suppress friction induced oscillationshave been developed. Bharadwaj and Dapino investigated the use of ultrasoniclubrication in active seat belts (Bharadwaj and Dapino, 2010). By modulating theeffective friction coefficient between the D ring and webbing, precise control of thechest force was proposed. This approach is advantageous as it precludes the useof load limiters in the seat belt retractor, which tend to be massive and complex.Experiments conducted with the ultrasonic energy applied normal to the slidingvelocity, as is the case in the seat belt application, show a 60% reduction in frictionforce between a seat belt webbing and an ultrasonic waveguide.

The focus of this paper is to understand how system dynamics affect theeffectiveness of ultrasonic lubrication and to develop guidelines for designingdynamic systems with ultrasonic lubrication. A lumped parameter model for anultrasonically driven system is presented in Section 2. The system consists ofa single degree-of-freedom (SDOF) resonator sliding in dry friction conditionsand subjected to an ultrasonic excitation force. Results of the model simulationsare presented in Section 4, including time and frequency domain calculations,energy and power calculations, and a sensitivity study considering the effects ofultrasonic control force, coefficient of friction, mass loading, tangential contactstiffness at the sliding interface, and system stiffness. The SDOF resonatorprovides a representative platform for the design of generic dynamic systemsfor automotive applications. Although the development of practical ultrasonicallycontrolled systems may require analysis at the multiple DOF or continuos level,such analyses are heavily application dependent and cannot be easily conceptualisedthe way a SDOF resonator can. The design guidelines and methods presented inthis article can be extrapolated to practical systems.

2 Dynamic model for ultrasonically lubricated system

2.1 Friction models

In Littmann et al. (2001a,b) and Storck et al. (2002), the Coulomb model has beenused to define the friction force, and analytical expressions relating the friction

294 S. Bharadwaj and M.J. Dapino

reduction ratio with the velocity ratio have been developed. Friction reduction asdescribed by the Coulomb model solely depends on the velocity ratio, thus it isindependent of system parameters such as mass, stiffness, coefficient of friction,tangential contact stiffness, and frequency of ultrasonic excitation. However, it hasbeen shown that the contact stiffness at the interface has an effect on ultrasoniclubrication (Leus and Gutowski, 2008; Tsai and Tseng, 2005). Using numericalanalysis (Leus and Gutowski, 2008), it was shown that the Dahl friction model,which accounts for tangential contact deformability, provides a more completedescription of the system compared with the Coulomb model.

Representation of the discontinuity at zero slip velocity has been a concern,since physical friction processes are continuous (Bharadwaj and Dapino, 2010).To address this issue, more involved and complex models have been proposed.A micro-slip approach such as the Iwan model considers the Coulomb frictionelement connected to a spring. When the force in the spring reaches a certainmagnitude, it reverses direction. However, such models have not been able todifferentiate between static and kinetic friction coefficients. More sophisticatedfriction models classified as state variable friction laws use differential equations todescribe the friction force. These models are hysteretic in nature and have internaldynamics. Dahl friction is one such example. In the Dahl model, the contactasperities are modelled as micro-springs. The effective contact stiffness is taken intoconsideration, and the product of the stiffness and the elastic displacement of theasperities gives rise to the friction force.

2.2 System model

We model the ultrasonically driven system as a single degree-of-freedom resonatorwith Dahl friction acting at the sliding interface and subjected to a harmonicexcitation, as shown in Figure 1. Here, Vt is the macroscopic velocity of the systemand x is the vibrational amplitude relative to the static equilibrium position of thespring. Viscous friction and the Stribeck effect are neglected.

Figure 1 Single degree-of-freedom model of an ultrasonically driven system consideringfriction (see online version for colours)

The governing equation of motion for this system is

mx + Ksx = Fe cos ωt − FD, (1)

Tangential ultrasonic vibrations using a SDOF model 295

where m is the effective mass of the sliding body, Ks is the system stiffness, Fe

is the control force generated by the piezoelectric transducer, ω is the excitationfrequency, and FD is the Dahl friction force. This force is defined by a differentialequation (see, for example, Storck et al., 2002)

FD = KtV

{1 − sgn(V )

µmgFD

}, (2)

where Kt is the tangential contact stiffness and µ is the coefficient of friction. Therelative velocity V is equal to Vt − x.

3 Simulation results

Characterisation of the friction reduction is accomplished by plotting the frictionratio and power dissipation ratio as a function of the velocity ratio. The powerdissipation ratio is the power dissipated due to friction in the presence of ultrasonicvibrations over the power dissipated due to friction in the absence of ultrasonicvibrations. Values for the system parameters used in these simulations are shownin Table 1.

Table 1 Simulation parameters

Parameter Value

Mass, m 0.02 kgSystem stiffness, Ks 133 × 106 N/mExcitation frequency, ω 2π60 × 103 rad/sPiezo control force, Fe 0 to 1920NTangential stiffness of bristles, Kt 0.056 × 106 N/mCoefficient of friction, µ 0.1

Figure 2 shows a comparison of friction ratio versus velocity ratio utilising both theCoulomb and Dahl models. The Coulomb model predicts a higher degree of frictionreduction than the Dahl model. In addition to the velocity ratio, the percentagereduction according to the Dahl model depends on the mass, coefficient of frictionand tangential stiffness. In Coulomb’s model, the instantaneous friction force isdiscontinuous and exhibits a pulsating waveform. In the case of Dahl’s model, theforce is a continuous oscillatory function (Figure 3).

3.1 Energy considerations

The friction ratio provides information on the effectiveness of the friction reductionas it relates to the velocity ratio. In this section we quantify the energy flow in thesystem including input, stored, and dissipated energy. Considering first the systemwithout ultrasonic vibrations (x = 0), energy must be supplied to the system as itslides with velocity Vt,

Ei = FiVtT, (3)

296 S. Bharadwaj and M.J. Dapino

Figure 2 Comparison of Coulomb and Dahl models (see online version for colours)

Figure 3 Instantaneous friction force vs. time according to (a) Coulomb model and(b) Dahl model (see online version for colours)

where Fi is the applied force and T is the period of harmonic motion. Part of theenergy is converted to kinetic energy and part of it is dissipated due to friction,thus

FiVtT =12mVt

2 + FnVtT, (4)

where Fn is the intrinsic friction force.In the presence of ultrasonics, in addition to the input energy to slide the mass,

energy needs to be supplied to the system for generating ultrasonic vibrations. Thepiezoelectric control force Fe(t) vibrates the mass. The total input energy into thesystem is

Einput = FiVtT +∫ T

0Fe(t)x(t)dt

=12mVt

2 + FnVtT +∫ T

0Fe(t)x(t)dt. (5)

This energy is converted partly to kinetic energy, partly to spring potential energy,and the remaining is dissipated due to friction. The average kinetic energy over oneperiod is

Eku =1T

∫ T

0

12m(Vtot(t))

2dt =

1T

∫ T

0

12m(Vt + x(t))2dt. (6)

Tangential ultrasonic vibrations using a SDOF model 297

The average energy stored over one period is

Epu =1T

∫ T

0

12KS(x(t))2dt, (7)

and the frictional energy dissipated during one period is

Efu =∫ T

0Ff (t)(Vt + x(t))dt. (8)

Energy balance gives

12mVt

2 + FnVtT +∫ T

0Fe(t)x(t)dt

=1T

∫ T

0

12m(Vt + x(t))2dt +

1T

∫ T

0

12KS(x(t))2dt

+∫ T

0Ff (t)(Vt + x(t))dt, (9)

or equivalently,

FnVtT +∫ T

0Fe(t)x(t)dt

=1T

∫ T

0

12m(x(t))2dt +

1T

∫ T

0mVtx(t)dt

+1T

∫ T

0

12KS(x(t))2dt +

∫ T

0Ff (t)(Vt + x(t))dt. (10)

With the definitions

Esu =1T

∫ T

0

12m(x(t))2dt +

1T

∫ T

0mVtx(t)dt +

1T

∫ T

0

12KS(x(t))2dt, (11)

Eiu =∫ T

0Fe(t)x(t)dt, (12)

division of (10) by Ef = FnVtT gives

1 +Eiu

Ef=

Esu

Ef+

Efu

Ef, (13)

or equivalently,

Efu

Ef= 1 +

Eiu

Ef− Esu

Ef. (14)

Here, Efu is the energy dissipated due to friction in the presence of ultrasonics,Ef is the energy dissipated due to friction in the absence of ultrasonics, Eiu is the

298 S. Bharadwaj and M.J. Dapino

ultrasonic energy input into the system, and Esu is the system’s internal energy(kinetic and potential). Expression (14) can be converted to a power balance,

Pfu

Pf= 1 +

Piu

Pf− Psu

Pf, (15)

where P denotes power and the subscripts denote the same quantities as before.Thus, the power dissipation ratio is given by

φfu = 1 + φiu − φsu, (16)

where φiu is the ratio of the ultrasonic input power to the friction power dissipatedwith no ultrasonic vibrations and φsu is the ratio of the power used by thesystem to the friction power dissipated with no ultrasonic vibrations. Physically,φiu represents the input power supplied to the system and φsu represents the powerconsumed by the system in the form of potential and kinetic energy.

3.2 Time and frequency domain responses

Simulations were performed at a velocity ratio of 0.5, piezo control force of384N and excitation frequency of 60kHz. Time domain plots of displacement,velocity, instantaneous friction force and instantaneous power dissipated are shownin Figures 4 and 5.

Figure 4 (a) Displacement vs. time and (b) velocity vs. time (see online versionfor colours)

Figure 5 (a) Instantaneous friction force vs. time and (b) instantaneous power dissipatedvs. time (see online version for colours)

Figures 4(a) and 4(b) show that the displacement and velocity are not sinusoidal,despite the excitation being sinusoidal. As evidenced by the Fast Fourier Transform

Tangential ultrasonic vibrations using a SDOF model 299

(FFT) of the velocity, Figure 6(a), there are two frequency components, onecorresponding to the excitation frequency and the other to the natural frequencyof the system,

√ks/m. Analytical models based on Coulomb or Dahl theory,

which model the vibration velocity as a single sinusoid, do not incorporate theeffect of the system’s natural frequency. In Figure 6(b), it is observed that theFFT of friction force shows a number of frequency components, which arisefrom individual harmonics and harmonics of the sum and difference of the twofrequencies.

Figure 6 Fast Fourier Transform of (a) velocity and (b) friction force (see online versionfor colours)

3.3 Effect of dynamic system parameters

The effect of load (Bharadwaj and Dapino, 2009), velocity ratio, and tangentialstiffness (Leus and Gutowski, 2008) have been analysed using Coulomb and Dahlequations. In this analysis we incorporate the effect of system dynamics throughthe parameters mass, system stiffness, contact stiffness, piezo control force, relativevelocity and coefficient of friction. The velocity, displacement, friction force andpower are not harmonic functions as proposed by the Coulomb and Dahl modelsdue to the effect of the natural frequency of the system. The simulation resultspresented here correspond to a chosen set of parameter values (Table 1) which arebased on existing experimental and computational data in the literature (Littmannet al., 2001a,b; Storck et al., 2002; Bharadwaj and Dapino, 2009; Leus andGutowski, 2008; Tsai and Tseng, 2005).

3.3.1 Effect of control force

In a typical ultrasonic system, the control force is generated by piezoelectricelements when excited with a voltage. The relationship between the force generatedand the applied voltage is assumed linear. This control force acts as an externalexcitation to the SDOF model at a given ultrasonic frequency. Figure 7(a) showsthat an increase in control force decreases the friction ratio. With higher controlforce, the velocity of vibration increases, which in turns decreases the velocity ratio.The power dissipation ratio correlates with the amount of input power required toobtain a desired velocity ratio and corresponding friction ratio. At velocity ratiosat or above 1, the ultrasonic vibrations have no effect relative to the case when noultrasonic vibrations are applied (Figure 7(b)). At low velocity ratios, high frictionpower is dissipated with ultrasonic vibrations on. The higher the control force, the

300 S. Bharadwaj and M.J. Dapino

higher the dissipated power will be. As the control force is increased, the term Piu

in (15) increases. Thus, the power dissipation ratio increases with increasing controlforce.

Figure 7 Effect of control force on (a) friction ratio vs. velocity ratio and (b) powerdissipation ratio vs. velocity ratio (see online version for colours)

3.3.2 Effect of coefficient of friction

The coefficient of friction determines the resistance offered to the sliding andvibratory motion of the mass. When the intrinsic coefficient of friction is low,friction reduction is achieved more easily than when it is high. Figure 8(a) showsthat the friction reduction is less effective as the friction coefficient increases. PowerPf in (15) is affected by the value of the friction coefficient. A high coefficient offriction implies a lower φiu and φsu terms in (16). As a result, the power dissipationratio decreases with an increase in friction coefficient as shown in Figure 8(b).

Figure 8 Effect of coefficient of friction on (a) friction ratio vs. velocity ratio and(b) power dissipation ratio vs. velocity ratio (see online version for colours)

3.3.3 Effect of mass load

An increase in mass load makes the friction reduction system less effective.Figure 9(a) shows that the friction ratio curve begins to flatten as load increases.This implies that for a given control force, the maximum possible friction reductionis limited by the mass of the sliding body. An increase in normal load increasesthe friction force and leads to higher required ultrasonic energy. In Figure 9(b),the power dissipation ratio decreases as mass loading increases. The system energyincreases, implying that at low velocity ratios, the power dissipated with ultrasonic

Tangential ultrasonic vibrations using a SDOF model 301

vibrations is very low. The potential energy in the spring is much smaller inmagnitude compared to the input energy. The effect of mass is significant in φiu,which decreases with increasing load.

Figure 9 Effect of mass load on (a) friction ratio vs. velocity ratio and (b) powerdissipation ratio vs. velocity ratio (see online version for colours)

3.3.4 Effect of contact stiffness

The friction force in the case of Dahl’s model is the product of the tangentialcontact stiffness and the elastic displacement of the asperities. Thus, a high contactstiffness implies higher friction acting on the system. The system is studied inthe absence of an excitation force for different values of the contact stiffness.In Figure 10(a), as the tangential contact stiffness is increased, friction reductionis more pronounced. From a design stand point, surfaces in contact should beas stiff as possible for friction reduction to be effective. In Figure 10(b), thepower dissipation is always less than unity since φiu is zero. The ratio dropswith increasing contact stiffness. By increasing contact stiffness, the system energyincreases and hence the power dissipation ratio decreases.

Figure 10 Effect of friction induced oscillations on (a) friction ratio vs. velocity ratio and(b) power dissipation ratio vs. velocity ratio (see online version for colours)

3.3.5 Effect of system stiffness

The Coulomb and Dahl models do not take the system stiffness into consideration.These analyses are based on assuming an infinitely stiff spring to which a massis connected. In practice, the amount of friction reduction depends on systemcompliance. When the excitation frequency matches the system natural frequency,

302 S. Bharadwaj and M.J. Dapino

the displacement and hence the velocity of vibration increase. The velocity ratiotherefore decreases, which in turn lowers the friction ratio (Figure 11(a)). At highervibration velocities, the power dissipation due to friction is also high. For lowervalues of KS , the power dissipation ratio is not significantly affected (Figure 11(b)).Thus, it is desirable to operate systems at their natural frequency to maximise theeffect of friction reduction.

Figure 11 Effect of system stiffness on (a) friction ratio vs. velocity ratio and (b) powerdissipation ratio vs. velocity ratio (see online version for colours)

3.3.6 Correlation with experimental data

The system model described above is applied for correlation with experimental datafrom the literature (Storck et al., 2002). Model parameters are shown in Table 2.Simulations are performed based on specified values for mass, tangential stiffnessand coefficient of friction (Littmann et al., 2001a,b; Storck et al., 2002; Leus andGutowski, 2008; Tsai and Tseng, 2005). Figure 12 shows that the system modelbetter describes the experiments relative to the Coulomb and Dahl models. Atlow velocity ratios, experiments convey a higher friction ratio compared to thetheoretical model prediction. Due to inertial effects of load, coefficient of frictionand friction induced oscillations, it is not possible to attain a friction ratio ofzero, as predicted by the Coulomb and Dahl models. As a result, higher ultrasonicenergy will need to be supplied into the system. The dynamic model selected foranalysis is a simple framework capable of describing most aspects of active frictionreduction.

Table 2 Simulation parameters

Parameter Value

Mass, m 0.02 kgSystem stiffness, Ks 133 × 106 N/mExcitation frequency, ω 2π × 60, 000 rad/sPiezo control force, Fe 0 to 1920NTangential stiffness of bristles, Kt 0.056 × 106 N/mCoefficient of friction, µ 0.1Solver Fourth order Runge–Kutta methodTime step size 1.667 × 10−8 s (Fixed)

Tangential ultrasonic vibrations using a SDOF model 303

Figure 12 Comparison of models with experimental data (see online version for colours)

Source: Storck et al. (2002)

4 Concluding remarks

Active friction control is a function of system parameters mass, system stiffness,contact stiffness, coefficient of friction, friction induced oscillations and externalexcitation. The displacement and velocity of the mass have two frequencycomponents which correspond to the natural and applied frequencies. Frictionforce, however, has a number of frequency components and shows a decreaseupon application of ultrasonic excitation. Numerical analysis elucidates the effectof various parameters on friction reduction in presence of ultrasonic vibrations.The inertial effects of load in addition to coefficient of friction and friction inducedoscillations prevents the system from achieving a zero friction state. For practicalapplications involving friction control, design of ultrasonic systems can utilise theseresults and be tuned for optimum performance. A parametric study shows thesensitivity of friction reduction and power dissipation ratio to system parameters.This information can be used for design of automotive components with adaptivefriction properties.

Acknowledgement

This work was supported by the member organisations of the Smart VehicleConcepts Center (www.SmartVehicleCenter.org), a National Science FoundationIndustry/University Cooperative Research Center.

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