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Mechanical Systems and Signal Processing

Mechanical Systems and Signal Processing 58-59 (2015) 101–114

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n CorrE-m

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Estimation of coefficient of friction for a mechanical systemwith combined rolling–sliding contact usingvibration measurements

Sriram Sundar, Jason T. Dreyer, Rajendra Singh n

Acoustics and Dynamics Laboratory, Smart Vehicle Concepts Center, Department of Mechanical and Aerospace Engineering, The Ohio StateUniversity, Columbus, OH 43210, USA

a r t i c l e i n f o

Article history:Received 10 June 2013Received in revised form14 November 2014Accepted 27 November 2014Available online 13 January 2015

Keywords:Combined rolling–sliding contact systemCam–followerFriction experimentMixed lubrication regimeNonlinear dynamics

x.doi.org/10.1016/j.ymssp.2014.11.01570/& 2014 Elsevier Ltd. All rights reserved.

esponding author. Tel.: þ1 614 292 9044.ail address: singh.3@osu.edu (R. Singh).

a b s t r a c t

A new dynamic experiment is proposed to estimate the coefficient of friction for amechanical system with a combined rolling–sliding contact under a mixed lubricationregime. The experiment is designed and instrumented based on an analogous contactmechanics model, taking into consideration the constraints to ensure no impact and nosliding velocity reversal. The system consists of a cam (rotating with a constant speed)having a point contact with a follower that oscillates about a frictionless pivot, whilemaintaining contact with the cam with the help of a well-designed translational spring.The viscous damping elements for contact are identified for two different lubricants froman impulse test using the half-power bandwidth method. Dynamic responses (with thecam providing an input to the system) are measured in terms of the follower accelerationand the reaction forces at the follower pivot. A frequency domain based signal processingtechnique is proposed to estimate the coefficient of friction using the complex-valuedFourier amplitudes of the measured forces and acceleration. The coefficient of friction isestimated for the mechanical system with different surface roughnesses using twolubricants; these are also compared with similar values for both dry and lubricated casesas reported in the literature. An empirical relationship for the coefficient of friction issuggested based on a prior model under a mixed lubrication regime. Possible sources oferrors in the estimation procedure are identified and quantified.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Friction plays a significant role in the dynamics of mechanical systems under sliding contacts [1–7]. The friction force isoften modeled using the Coulomb formulation, though the analyst must judiciously select the value of the coefficient offriction (μ). In many prior experimental studies as summarized by Persson [8], μ is found from a simple and puretranslational sliding contact (without rolling) system. For instance, Espinosa et al. [9] used a modified Kolsky bar apparatus,while Hoskins et al. [10] used a sliding block of rocks to estimate the normal and friction forces. Furthermore, translationalsliding experiments were employed by Worden et al. [11] to estimate the dependence of friction forces on displacement andvelocity, and then by Schwingshackl et al. [12] to model the non-linear friction interface. Several investigators have also

S. Sundar et al. / Mechanical Systems and Signal Processing 58-59 (2015) 101–114102

conducted friction experiments on rotating systems such as a pin-disk apparatus [13,14], two rotating circular plates [15],and a radially loaded disk-roller system [16,17]. Also, Kang and Kim [18] determined the Coulomb friction insightstabilization equipment using torque and angular displacement characteristics, while Povey and Paniagua [19] estimatedthe bearing friction for a turbo machinery application. Such pure sliding contact experiments cannot be employed toestimate μ for a system with combined rolling–sliding contact since the kinematics is different. Radzimovsky et al. [20]conducted experiments on gears to determine the instantaneous μ over a mesh cycle. However, none of the prior combinedrolling–sliding contact experiments rely on vibration measurements. Furthermore, the conventional or direct approachesgiven in the literature [9–19] focus on estimating the μ for only a pure sliding contact systems (without rolling). Since thekinematics of such systems is generally complicated, new effort is needed to estimate the μ from vibration measurements(measured forces and acceleration) under certain conditions.

Some researchers have experimentally studied cam–follower mechanisms [21,22] from the stability and bifurcationperspective under impacting conditions. In contrast, a cam–follower mechanism with rotational sliding contact (with noimpacts) is used to experimentally determine μ in this study. Since μ cannot be directly measured from vibrationexperiments, an analogous contact mechanics model [23] is developed to aid the process. The goal is to vary the surfaceroughness, lubrication film thickness, contact pressure and velocities at contact (sliding and entrainment). The proposedsystem could then be utilized to simulate the contact conditions seen in drum brakes and geared systems.

2. Problem formulation

Fig. 1 shows the mechanical system with an elliptic cam (with semi-major and minor axes as a and b, respectively).Though the kinematics of combined rolling–sliding contact systems are complex compared to systems with pure slidingcontact, this is one of the simplest systems which one can devise to measure the coefficient of friction in such systems whichwould allow controlled measurements of the reaction forces and system acceleration. The cam is pivoted at E along its majoraxis with a radial run out, e, from its centroid (Gc, with subscript c denoting cam). The angle made by the end point of themajor axis (A) with the horizontal axis (ex) is Θ (t), which is an excitation to the system (where t represents the time). Theequation of the elliptic cam is given by the following, where r is the radial distance from Gc to any point on thecircumference of the cam, and Δ is the polar angle of that point,

r Δ� �¼ abffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a sin Δ� �� �2þ b cos Δ

� �� �2q : ð1Þ

The cam is in a point contact (at Oc) with the follower (at Ob, with subscript b denoting follower), which consists of a thincylindrical dowel pin (of radius rd) attached to a bar (of length lb) of square cross-section (of width wb). The center of gravityof the follower lies at Gb at a distance of lg from the pivot point P (using roller bearings) which is at dy distance above theground. The follower is supported by a linear spring (ks) along the vertical direction (ey), which is at a distance of dx from Pas shown in Fig. 1. The angular motion of the follower is given by α(t) in the clockwise direction from the ex axis; it is alsothe only dynamic degree-of-freedom of the system. The contact mechanics at O between the cam and the follower isrepresented by non-linear contact stiffness (kλ) and viscous damping (cλ) elements. Viscous damping is valid in this studysince the system is designed to not lose the contact at any point of time, and the indentation velocity is low (no impacts),and hence the contact damping force is insignificant (compared to contact stiffness force) regardless the damping

Fig. 1. Example case: a mechanical system with an elliptic cam and follower supported by a lumped spring (ks).

S. Sundar et al. / Mechanical Systems and Signal Processing 58-59 (2015) 101–114 103

mechanisms involved. Damping forces and mechanisms become significant only when the system undergoes impacts,hence the viscous damping assumption is valid for this system under non-impacting conditions. The choice of contactdamping mechanism will not affect the results of the study due to insignificance of the contact damping force. Hence asimple viscous damping model is chosen over a non-linear amplitude dependent damping model. A coordinate system i; j

� �attached to the follower is defined with its origin at Q where i is orthogonal to the follower. The angle subtended by GcOc

!from ex is given by φ(t), which is used in the following equation to calculate the ΔO(t) for the contact point Oc as

ΔOðtÞ ¼ mod φðtÞ�ΘðtÞ;2π� �: ð2Þ

Here, “mod” is the modulus function defined as modðx; yÞ ¼ x�y: floorðx=yÞ, if ya0. The vector QOc!

is represented in the ði; jÞcoordinate system by ψ i iþψ j j. When the instantaneous value of ψi(t) is negative, that would ensure that the cam and thefollower are in contact.

The scope of the current study is restricted to an estimation of μ under a mixed lubrication regime. The key assumptionsin the proposed system are as follows: (i) The bearings at the follower pivot are frictionless and rigid; (ii) the surfaces of thecam and follower have no other irregularities with the exception of random surface roughness; (iii) the sliding frictionbetween cam and the follower can be described by the Coulomb friction model; (iv) the contact force is represented by theHertzian point contact model [23], since it is widely used for point contact in machine elements; and (v) the bendingmoment of the follower is negligible. The Coulomb friction model is used in the study only to model the variation in μ withthe relative velocity like, Benedict and Kelley model [17]. With the variation in friction with relative velocity being low in thesystem under study, it can be assumed to be almost constant. Furthermore, to capture the variation in the amplitude of μwith surface roughness, contact pressure and dynamic viscosity, an empirical model based on the Benedict and Kelley model[17] will be used. Hence this also takes care of applicability of μ under boundary and mixed lubrication conditions. Thespecific objectives of this study are (1) Develop a contact mechanics model for a mechanical system with a combinedrolling-sliding contact to design a suitable experiment and to predict the dynamic response; (2) Design a controlledlaboratory experiment for the cam–follower system to measure dynamic forces and acceleration; (3) Propose a signalprocessing technique to estimate μ using the Fourier amplitudes of measurements and obtain an empirical formula for μ;and (4) Identify the potential sources of errors in the proposed technique.

3. Contact mechanics model

The 0-state of the system (represented by superscript 0) is defined as the state when QOc!¼ 0

!and the major axis is

parallel to the follower (α0 ¼ �Θ0). In the 0-state Q0, Ob0and Oc

0are coincident. From the geometry of the system, α and the

magnitude of POb!

along j (χ) are calculated in the 0-state as

α0 ¼ cos �1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij PE!j4x þj PE!j2x j PE

!j2y�j PE!j2x 0:5wbþ2rdþbð Þ2q

þj PE!jy 0:5wbþ2rdþbð Þj PE!j2x þj PE!j2y

0@ 1A; ð3Þ

χ0 ¼ j PE!jx cos ðα0Þ�j PE!jy sin ðα0Þ�e: ð4ÞHere, j PE!jxand j PE!jy represent the magnitudes of PE

!along ex and ey, respectively. The instantaneous values of the moving

coordinates ψi(t) and ψj(t) are determined from α(t), Θ(t) and the system geometry using the following vector equation:

PE!¼ POb

!þObOc!þOcE

!: ð5Þ

Employing the vector polygon procedure discussed by Sundar et al. [24], the equations for ψi(t) and ψj(t) are obtained as

ψ iðtÞ ¼ χ0þe� �

sin αðtÞ�α0� �þ r ΔoðtÞ

� �þ0:5wbþ2rd� �

cos αðtÞ�α0� �

þr ΔoðtÞ� �

sin αðtÞþφðtÞ� ��e sin αðtÞþΘðtÞ� �� 0:5wbþ2rdð Þ; ð6Þ

ψ jðtÞ ¼ χ0� χ0þe� �

cos αðtÞ�α0� �þ r ΔoðtÞ� �þ0:5wbþ2rd

� �sin αðtÞ�α0� �

�r ΔoðtÞ� �

cos φðtÞþαðtÞ� �þe cos αðtÞþΘðtÞ� �: ð7Þ

Differentiating Eqs. (6) and (7) with respect to time, _ψ iðtÞ and _ψ jðtÞ are obtained as follows:

_ψ iðtÞ ¼ χ0þe� �

cos αðtÞ�α0� �_αðtÞ� r ΔoðtÞ

� �þ0:5wbþ2rd� �

sin αðtÞ�α0� �_αðtÞ

þr ΔoðtÞ� �

cos αðtÞþφðtÞ� �_αðtÞþ _φðtÞ� �þ _r ΔoðtÞ

� �sin αðtÞþφðtÞ� �

�e cos αðtÞþΘðtÞ� �_αðtÞþ _ΘðtÞ� �

; ð8Þ

_ψ jðtÞ ¼ χ0þe� �

sin αðtÞ�α0� �_αðtÞþ r ΔoðtÞ

� �þ0:5wbþ2rd� �

cos αðtÞ�α0� �_αðtÞ

þr ΔoðtÞ� �

sin φðtÞþαðtÞ� �_φðtÞþ _αðtÞ� �� _r ΔoðtÞ

� �cos φðtÞþαðtÞ� ��e sin αðtÞþΘðtÞ� �

_αðtÞþ _ΘðtÞ� �

: ð9Þ

S. Sundar et al. / Mechanical Systems and Signal Processing 58-59 (2015) 101–114104

Here,

_r ΔoðtÞ� �¼ 0:5ab b2�a2

� �sin 2ΔoðtÞ� �

_φðtÞ� _ΘðtÞh i

a sin ΔoðtÞ� �� �2þ b cos ΔoðtÞ

� �� �2h i1:5 : ð10Þ

The angle φ(t) corresponding to the contact point Oc is determined at every instant for a given α(t) and Θ(t) by locating thepoint on the elliptic profile of the cam which is tangential to the follower. Hence the slope of the follower, sb(t)¼tan(�α(t)),should be equal to the slope of the cam at Oc (scOðtÞ) which is calculated as follows:

scOðtÞ ¼ tan ΘðtÞþ tan �1 � b2

a2 tan φðtÞ�ΘðtÞ� �" #" #ð11Þ

Equating sb(t) and scOðtÞ and rearranging, φ(t) is calculated by the following:

φðtÞ ¼ΘðtÞ� tan �1 b2

a2 tan αðtÞ�ΘðtÞ� �" #ð12Þ

The equation of motion of the follower when it is in contact with the cam is derived by balancing the moments(from Fig. 2) about P as

IPb €αðtÞ ¼mbglg cos αðtÞð Þ�FsðtÞdxþFnðtÞχðtÞ�Ff ðtÞ 0:5wbþ2rdð Þ: ð13ÞHere, IPb is the moment of inertia of the follower about P, mb is the mass of the follower, g is the acceleration due to gravity,and χ(t) is the moment arm of the contact force about the pivot P. The elastic force from the spring, Fs(t), is given by thefollowing, where Lus is the original length of the follower spring:

FsðtÞ ¼ ks Lus �dyþdx tan αðtÞð Þþ0:5wbsec αðtÞð Þ� �: ð14Þ

The normal force (Fn(t)) arising from the point contact with the cam is given by

FnðtÞ ¼ �kλ ψ iðtÞ� �

ψ iðtÞ�cλ _ψ iðtÞ: ð15ÞThe non-linear contact stiffness is defined for a point contact based on the Hertzian contact theory [23] as

kλ ψ iðtÞ� �¼ ð4=3ÞYe ρeðtÞ ψ iðtÞ

� �0:5: ð16Þ

Here, Y is Young's modulus (with superscript e denoting equivalent) in accordance with the Hertzian contact theory given bythe following, where ν is Poisson's ratio,

Ye ¼ 1�ν2cYc

þ1�ν2bYb

" #�1

: ð17Þ

The equivalent radius of curvature at the contact (ρe(t)) and the radius of curvature of the elliptical cam at Oc (ρc(Δo(t))) aregiven by

ρeðtÞ ¼ ρc ΔoðtÞ� �� ��1þðrdÞ�1

h i�1; ð18Þ

ρc Δ0ðtÞ� �¼ a sin γ ΔoðtÞ

� �� �� �2þ b cos γ ΔoðtÞ� �� �� �2h i1:5

ab: ð19Þ

Fig. 2. Free-body diagram of the follower; refer to Fig. 1 for the two coordinate systems.

S. Sundar et al. / Mechanical Systems and Signal Processing 58-59 (2015) 101–114 105

The contact damping is modeled as linear viscous damping as the system is designed to not undergo any impact. The viscouscontact damping is given by the following expression, where ζ is the modal damping ratio which will be experimentallyfound under lubricated conditions (as explained later in section 4) and ϑ is the linearized natural frequency of the system,and χn is static equilibrium value of χ(t) (discussed later in this section).

cλ ¼2ζϑIPbχn� �2 : ð20Þ

The friction force is given as

Ff ðtÞ ¼ μFnðtÞsgn vrðtÞð Þ: ð21Þ

The Coulomb friction is used to describe a variation in coefficient of friction with the relative velocity along the lines of theBenedict and Kelley model [17] and Stribeck formulation [32]. When the variation of μ with relative velocity is low, it can beassumed to be almost constant, which is found to be the case with the following system. Here the relative sliding velocity,vr(t) is given by

vrðtÞ ¼ _ψ jðtÞ� r ΔðtÞ� �sin φðtÞþαðtÞ� �þe sin αðtÞþΘðtÞ� �� �

_αðtÞþ _ΘðtÞ� �

: ð22Þ

The static equilibrium point is used as the initial condition while numerically solving Eq. (13). In Eqs. (6), (7) and (13),α(t), ψi(t), and ψj(t) are replaced with their corresponding values at the static equilibrium point (with superscript n), and alltime-derivative terms are set to zero and solved. Using the method of Jacobian matrix as discussed by Sundar et al. [24], ϑ isthen calculated at the static equilibrium point.

4. Experiment for the determination of μ

Since the measured time domain signals are bound to have significant noise, a frequency domain based signal processingtechnique is preferred for the estimation of μ. Accordingly, measured forces and acceleration must not be affected bydiscontinuities and system resonances. Design criteria for the experimental system can be given by the following. First, thefollower must always be in contact with the cam, as a loss in contact would generate impulses in force and accelerationsignals. Second, vr(t) should not change direction during the operation, as that would induce a sudden change in thedirection of the Ff(t), thereby making the measured forces discontinuous. Third, the cam should rotate with a constant speed(Ωc) in order to accurately measure the spectral contents of forces and acceleration. Fourth, at least the first five harmonicsofΩc should lie in the stiffness controlled regime. Fifth, the experiment should permit a mixed lubrication regime. Finally, avariation in the slide-to-roll ratio should be possible in the experiment.

Fig. 3 shows the schematic of a cam–follower experiment having a hollow cylindrical cam of outer radius, a, driven by theoutput shaft of an electric motor. The radial runout between the center of the rotation (axis of the shaft) and the centroid ofthe cam can be easily varied. A point contact is obtained, as the cam and the dowel pin have cylindrical surfaces with theiraxes oriented orthogonal to each other. The contact is continuously lubricated using either a heavy gear oil (AGMA 4EP)[25,26] or a light hydraulic oil (ISO 32) [25,26]. The follower is hinged at one of its ends with two frictionless rolling elementbearings and is supported by a coil spring. A tri-axial force transducer (PCB 260A01 [27]) located at the follower hingemeasures the reaction forces, Nx(t) and Ny(t), along ex and ey, respectively. An accelerometer (PCB 356A15 [28,29]) located atthe end of the follower measures its tangential acceleration. These are dynamic transducers with a very high frequencybandwidth [27,29]. Both force and acceleration signals are simultaneously sampled.

SpringFollower

Frictionless bearings

Cam

Accelerometer

Rigid fixture

Dowel pin

Tri-axial load cell

Output shaft of the electric motor

Housing for electric motor

Lubricated interface

Fig. 3. Mechanical system experiment used to determine the coefficient of friction (μ) at the cam–follower interface.

S. Sundar et al. / Mechanical Systems and Signal Processing 58-59 (2015) 101–114106

5. Identification of system parameters

5.1. Identification of geometrical parameters

The following parameters for the cam–follower system are carefully chosen to satisfy the design constraints stated inSection 4: mb¼0.21 kg, a¼b¼17.5 mm, Ib¼2020 kg mm2, lg¼179 mm, lb¼89 mm, wb¼12.7 mm, rd¼3.2 mm, ks¼2954 N/m,Lus ¼57 mm, dx¼40 mm and dy¼61 mm. The relative positions of the pivot points of the cam and the follower are givenby PE!¼ 86 mm exþ24 mm ey. The averaged surface roughness (R) and root-mean-square roughness (Rrms) of the cam and

follower surfaces are measured using an optical profilometer. For the precision ground surfaces used in the experimentRc¼0.29 μm and Rb¼0.25 μm, while for sand-blasted surfaces Rc¼0.36 μm and Rb¼0.89 μm. The key parameters that dictatea loss of contact between the follower and the cam and the sign reversal in vr(t) are e and Ωc. Inverse kinematics [30] isemployed, as explained below, to predict a range of values for these two parameters over which the system neither has a lossof contact nor a sign reversal in vr(t). For a given value of e and Ωc, the angle of the follower (assuming it is just in contact)with the cam (αk(t)) is kinematically calculated for different values ofΘ(t) in the range [0, 2π] (superscript k represents valuescalculated using the inverse kinematics). By setting ψi¼0 in Eq.(6), αk(t) is calculated using the following equation:

χ0þe� �

sin αkðtÞ�α0� �þ r ΔoðtÞ

� �þ0:5wbþ2rd� �

cos αkðtÞ�α0� �

þrk ΔoðtÞ� �

sin αkðtÞþφkðtÞ� �

�e sin αkðtÞþΘðtÞ� �

� 0:5wbþ2rdð Þ ¼ 0: ð23Þ

Here, rk(Δo(t)) is obtained using Eqs. (1) and (2) as

rk ΔoðtÞ� �¼ abffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a sin φkðtÞ�αkðtÞ� �� �2þ b cos φkðtÞ�αkðtÞ� �� �2q : ð24Þ

Eqs. (23) and (24) are solved along with Eq. (12) after replacing α(t) with αk(t), to get rk(Δo(t)), αk(t) and φk(t). Then,differentiating αk(t) with respect to t, _αkðtÞ and €αkðtÞ are obtained. The normal force is estimated (as stated below) from themoment balance about P and by neglecting the moment due to Ff(t) in comparison with the moment due to Fn(t) because ofsystem geometry,

FknðtÞ ¼IPb €α

kðtÞþFks ðtÞdx�mbglb cos αkðtÞ� �χ0 cos αkðtÞ�α0

� �� bþ0:5wbþ2rdð Þ sin αkðtÞ�α0� ��

þrk ΔoðtÞ� �

cos ðαkðtÞ�φkðtÞÞi: ð25Þ

Here, Fks tð Þ is calculated from Eq. (14) corresponding to αk(t). If the minimum value of FknðtÞ calculated from Eq. (25) is negative, itwould indicate that the follower would lose contact with the cam during the steady-state operation. Similarly, the relative velocity(vkr ðtÞ) is kinematically calculated to check for any sign reversal from Eqs. (9) and (22) by replacing α(t) andφ(t) with αk(t) andφk(t),respectively. The procedure mentioned above is repeated for different values of e andΩc to calculate theΩc–e/a map as shown inFig. 4; the regimes with andwithout loss of contact and reversal in the sliding velocity direction are clearly marked. All experimentsare conducted in the e/a range from 0.05 to 0.15, andΩc is varied only between 10.1 Hz and 11.7 Hz; thus the system is well withinthe contact regime (as shown) with a constant sgn (vr(t))¼�1.With these parameters, the linearized natural frequency of thesystem is found to be 1040 Hz for a steel cam and a steel follower (Yc¼Yb¼200 GPa; νc¼νb¼0.3); thus the first five harmonics ofΩc lie in the stiffness controlled regime. Also, the lubrication regime is identified based on the “lambda ratio” (Λ), which is the ratio

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

10

15

20

25

30

e/a

Ω[H

z]

Fig. 4. Classification of response regimes of the mechanical system with a circular cam in terms of Ωc vs. e/a map with the parameters of Section 5.Key: , Operational range of the experiment.

S. Sundar et al. / Mechanical Systems and Signal Processing 58-59 (2015) 101–114 107

of minimum lubrication film thickness [31] to the composite surface roughness R2rms;cþR2

rms;b

� �0:5� �. With AGMA 4EP oil [25,26]

(with dynamic viscosity, η¼0.034 kgm�1 s�1, pressure viscosity coefficient¼20�10�9 m2/N at 60 1C) 2oΛo5, the system

should lie in the mixed lubrication regime. Also, lower values of Λ are achieved (0.7–1.5) with ISO 32 oil [25,26] (withη¼0.012 kgm�1 s�1, pressure viscosity coefficient¼18�10�9 m2/N at 60 1C). Since the temperature at the contact is higher thanthe ambient (due to continuous sliding), it is assumed that the interfacial oil operates at 60 1C. Furthermore, the slide-to-roll ratiowhich is given by the ratio of vr(t) (as given in Eq. (22)) and entrainment velocity (ve(t) as defined below), varies between 0.75 and1.25 for e/a¼0.116 and Ωc¼11.55 Hz as shown in Fig. 5. Here,

veðtÞ ¼ _ψ jðtÞþ r ΔðtÞ� �sin φðtÞþαðtÞ� �þe sin αðtÞþΘðtÞ� �� �

_αðtÞþ _ΘðtÞ� �

: ð26Þ

The slide-to-roll ratio vrðtÞ = veðtÞ Þ

�could be easily changed by altering the geometry, such as PE

!, a, b and e.

5.2. Identification of the modal damping ratio

The modal damping ratio under lubrication depends on the oil viscosity and the materials in contact; hence it isdetermined experimentally using the half-power bandwidth method with both lubricants. The experimental setup consistsof two masses (m1¼1.4 kg and m2¼1.8 kg) connected by three identical point contacts which are lubricated as shown inFig. 6. These point contacts are obtained by placing three dowel pins (rd¼3.2 mm) attached to m1 in one direction and twomore dowel pins attached tom2 in the orthogonal direction, as shown. The system is placed on a compliant base (foam), andtwo accelerometers are attached to each mass. An impulse excitation is imparted to the system in the vertical direction with

0 0.2 0.4 0.6 0.8 1

0.8

0.9

1

1.1

1.2

1.3

t [s]

Slid

e-to

-rol

l rat

io

Fig. 5. Slide-to-roll ratio for the cam–follower system with e/a¼0.12 and Ωc¼11.55 Hz and other parameters of Section 5.

Dowel pins

Impact hammer

Triaxialaccelerometer

Dowel pins

Fig. 6. Impulse experiment to determine the viscous damping ratio associated with the lubricated contact regime. (a) Experimental setup; (b) top view ofthe dowel pin arrangement showing the three point contacts. Key: , contact point.

S. Sundar et al. / Mechanical Systems and Signal Processing 58-59 (2015) 101–114108

an impact hammer. The response accelerance spectrum of each mass along the vertical direction is then found by averagingsignals from two accelerometers. Impact tests are conducted with two lubricants. Fig. 7 shows the relative accelerancespectra (between m1 and m2), focusing on the system resonance (�1000 Hz). As observed, there is a reduction in theamplitude and the natural frequency with lubrication. For a single point contact, the damping ratio (ζ) with unlubricated,ISO 32 oil and AGMA 4EP oil conditions are found to be 1.8%, 1.9% and 4.1%, respectively. Note that the damping for ISO 32 oilis very close to the dry case. It is assumed that these values of ζ are also valid for the running cam–follower experiment.

6. Signal processing technique to estimate μ

The μ is estimated from measured reaction forces (Nx(t) and Ny(t)) along the ex and eydirections, respectively, and thetangential acceleration (lb €αðtÞ) of the follower at its free end. By dividing the measured tangential acceleration by lb, €αðtÞ isobtained, and then numerically integrating it twice w.r.t. time, the time-varying component of α(t) is computed, while theintegration constant (αd) is obtained from the time-averaged value of αk(t). The instantaneous elastic force Fs(t) is calculatedfrom α(t) using Eq. (14). From Fig. 2, Nx(t) and Ny(t) are evaluated as follows:

NxðtÞ ¼ FnðtÞ sin αðtÞð ÞþFf ðtÞ cos αðtÞð Þ�mblg €αðtÞ sin αðtÞð Þ; ð27Þ

NyðtÞ ¼ FnðtÞ cos αðtÞð Þ�Ff ðtÞ sin αðtÞð Þþmbg�mblg €αðtÞ cos αðtÞð Þ�FsðtÞ: ð28ÞRearrange Eqs. (27) and (28) to yield the friction and normal forces as

Ff ðtÞ ¼NxðtÞ cos αðtÞð Þþ mbg�NyðtÞ�FsðtÞ� �

sin αðtÞð Þ; ð29Þ

FnðtÞ ¼mblg €αðtÞþNxðtÞ sin αðtÞð Þþ NyðtÞþFsðtÞ�mbg� �

cos αðtÞð Þ: ð30ÞSince the dynamic force transducer used does not measure the DC component, a technique to estimate μ is proposed that

utilizes complex-valued Fourier amplitudes while maintaining the phase relationship among the measured signals. First, themeasured Nx(t), Ny(t) and €αðtÞ are converted to the frequency domain using the fast Fourier transform (FFT) algorithm. Then,the harmonic reaction forces are reconstructed (with superscript r) using only their DC components (with superscript d) andthe fundamental harmonic component of Ωc (with superscript 1) as

fNrx ðtÞ ¼Nd

xþfN1x cos Ωctð Þ; fNr

y ðtÞ ¼NdyþfN1

y cos Ωctð Þ: ð31a;bÞ

In the above equation, fN1x and fN1

y (where � represents a complex-valued signal) are known from measurements while Ndx

and Ndy are unknown. Similarly, the following harmonic signals have also been reconstructed as the following where

ςsðtÞ ¼ sin αðtÞð Þ and ςcðtÞ ¼ cos αðtÞð Þ:eςrs ðtÞ ¼ ςds þfς1s cos Ωctð Þ; eςrc ðtÞ ¼ ςdc þfς1c cos Ωctð Þ;fFrn ðtÞ ¼ FdnþfF1n cos Ωctð Þ; eFrf ðtÞ ¼ Fdf þfF1f cos Ωctð Þ;eFrs ðtÞ ¼ Fds þfF1s cos Ωctð Þ: ð32a–eÞ

Since €αðtÞ does not have a DC component, it is written as

f€αr ðtÞ ¼f€α1 cos Ωctð Þ: ð33Þ

Frequency [Hz]

Rel

ativ

e ac

cele

ranc

e[m

/N-s

2 ]

850 900 950 1000 1050 11001

5

10

15

Fig. 7. Relative accelerance spectra in the vicinity of the system resonance. Key: , dry (unlubricated); , lubricated with AGMA 4EP oil; ,lubricated with ISO 32 oil.

S. Sundar et al. / Mechanical Systems and Signal Processing 58-59 (2015) 101–114 109

Substituting these reconstructed harmonic signals in Eqs. (29) and (30) and rearranging, the following DC components andfirst harmonic components of eFrf ðtÞ and fFrn ðtÞ are found as

Fdf ¼Ndxς

dc �Nd

yςds þ �Fdsς

ds þmbgςds þ0:5 fN1

xfς1c �fN1

yfς1s �fF1sfς1s �

; ð34Þ

fF1f ¼Ndxfς1c �Nd

yfς1s þ fN1

x ςdc �fN1

y ςds þmbgfς1s �Fds

fς1s �fF1s ςds �; ð35Þ

Fdn ¼Ndxς

ds þNd

yςdc þ 0:5mblb €α

dþFdsςdc �mbgςdc þ0:5 fN1

xfς1s þfN1

yfς1c þfF1sfς1c �

; ð36Þ

fF1n ¼Ndxfς1s þNd

yfς1c þ fN1

x ςds þfN1

y ςdc �mbgfς1c þFds

fς1c þfF1s ςdc þ0:5mblbf€α1�:

ð37Þ

From Eq. (21) and since sgn (vr(t))¼�1 is a constant, the following relationships can be derived:

Fdf ¼ �μFdn; ð38Þ

fF1f ¼ μ fF1n : ð39Þ

Since eFrf ðtÞ and fFrn ðtÞ are exactly out-of-phase,

∠fF1f ¼ �∠fF1n : ð40ÞSubstituting Eqs. (34)–(37) into Eq. (38) to (40), three non-linear equations with three unknowns (μ, Nd

xandNdy) are obtained

as shown below.

Ndxς

dc �Nd

yςds þ �Fds ς

ds þmbgςds þ0:5 fN1

xfς1c �fN1

yfς1s �fF1sfς1s �

¼ �μ Ndxς

ds þNd

yςdc þ 0:5mblb €α

dþFdsςdc �mbgςdc þ0:5 fN1

xfς1s þfN1

yfς1c þfF1sfς1c �� �

; ð41Þ

Ndxfς1c �Nd

yfς1s þ fN1

x ςdc �fN1

yςds þmbgfς1s �Fds

fς1s �fF1s ςds � ¼ μ Nd

xfς1s þNd

yfς1c þ fN1

x ςds þfN1

yςdc �mbgfς1c þFds

fς1c þfF1s ςdc þ0:5mblbf€α1�j:

ð42Þ

∠ Ndxfς1c �Nd

yfς1s þ fN1

x ςdc �fN1

y ςds þmbgfς1s �Fds

fς1s �fF1s ςds �� �¼ �∠ Nd

xfς1s þNd

yfς1c þ fN1

x ςds þfN1

yςdc �mbgfς1c þFds

fς1c þfF1s ςdc þ0:5mblbf€α1��:

�ð43Þ

These equations are numerically solved to estimate μ, Ndx and Nd

y . In order to computationally validate this technique,predicted forces and acceleration from the contact mechanics model with e/a¼0.3, Ωc¼11.55 Hz and a known μ¼0.3 areused. The signal processing technique (with 9460 Hz sampling frequency and frequency resolution of 1.15 Hz) yields anestimate of μ as 0.302, which is about 99.3% accurate. This method also accurately estimates Nd

xandNdy as �2.26 N and

�12.88 N, respectively, compared to the known values of �2.26 N and �12.7 N, respectively.

7. Experimental results and friction model

Spectral tests are conducted under lubricated conditions with different surface roughness levels at the contact. Care istaken during the experiments to record the steady state force and acceleration measurements only after the initialtransients have sufficiently decayed. Using the measured data and the signal processing technique discussed in Section 6,the_μ (estimated value of μ) is identified for various values of mean surface roughness Rm ¼ 0:5nðRcþRbÞ as shown in Fig. 8.This_μ is compared with the values reported in the literature [13] for dry friction contact. A higher range is observed in thecase of a pure dry friction regime in comparison with_μ for the mixed lubrication regime. Also, observe that_μ with ISO 32lubricant (with a low Λ value) is similar to the dry friction contact case [13].

He et al. [32] used the Benedict–Kelley friction model [17] to develop an empirical relationship between μ and Rm, butthis was specific to a line contact in gears. Hence that relationship is generalized for both point and line contacts as thefollowing, where ⟨ ⟩t is the time-average operator,

μ¼ C1

C2�Rmlog10

pλη vrðtÞ� �

t veðtÞ� �2

t

!: ð44Þ

Table 1Comparison of measurements and predictions (from the contact mechanics model) with μ¼0.51 and e/a¼0.116 at the harmonics of Ωc¼11.55 Hz.

Harmonic of Ωc €α (rad/s2) Nx (N) Ny (N)

Measured Predicted Measured Predicted Measured Predicted

1 122.7 122.6 0.99 0.99 2.48 2.472 3.2 5.5 0.08 0.07 0.07 0.073 0.2 1.3 0.02 0.003 0.02 0.003

Rm [μm]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.2

0.4

0.6

0.8

1

1.2

μ

Fig. 8. Estimated μ for different Rm values and comparison with prior values (including the range) for the dry friction regime [13]. Key: , With AGMA4EP oil; ,With ISO 32 oil; , dry contact – iron pin with steel disk [13]; , dry contact – copper pin with steel disk [13].

S. Sundar et al. / Mechanical Systems and Signal Processing 58-59 (2015) 101–114110

Here, C1 and C2 are the arbitrary constants and pλ is the time-averaged Hertzian contact pressure given by

pλ ¼32π

FnðtÞρeðtÞψ iðtÞ

� �t: ð45Þ

With a non-linear curve-fitting technique, the constants of Eq. (44) are found from the experimental results for eachlubricant: C1¼0.0288 μm and C2¼2.03 μm for AGMA 4EP oil, and C1¼0.0509 μm and C2¼1.6512 μm for ISO 32 oil.

The measured force and acceleration spectra are compared with the contact mechanics model (with estimated _μ) inTable 1 for a typical case with e/a¼0.116, Ωc¼11.55 Hz and_μ¼0.51. The contact mechanics model successfully predicts theforces and acceleration at the first three harmonics of Ωc, which are dominant compared with the higher harmonics.

The normalized coefficient of friction (μ) for the empirical model of Eq. (44) is defined as

μ¼ μ

log10pλ

η vr ðtÞh it veðtÞh i2t

� �: ð46Þ

From Fig. 9 it is observed that μ monotonically increases with Rm. Also μ is lower with AGMA 4EP (with higher Λ) ascompared to ISO 32 oil (with lower Λ). Fig. 9 compares some results of prior friction experiments [16,33–35] in terms ofselected μ values which are calculated based on certain assumptions given a lack of pertinent data. For instance, Shon et al.[16] and Xu and Kahraman [33] conducted experiments under the Elastohydrodynamic lubrication (EHL) regime, and hencetheir μ values are very low. Conversely, Grunberg and Campbell [34] and Furey [35] conducted experiments under poorlylubricated conditions (mixed lubrication regime). It can be easily inferred that μ decreases as Λ increases. There are somedifferences in the μ values from (Eq. (44)) and the ones reported in the literature; these may be attributed to differentlubrication regimes as well as potential sources of error in the μ estimation process which is discussed next.

8. Potential sources of error in the estimation of μ

Some of the common measurement errors which are difficult to minimize include the following. First, a variation in thefrictional load torque on the cam causes small variations inΩc during the experiment. This in turn introduces inaccuracy inthe harmonic contents of the measured forces and acceleration, thereby affecting the estimated μ. Second, a small error inthe angular alignment (κ) of a force transducer could measure Ny(t) cos(κ) instead of the actual Ny(t). From the static analysisit is found that for κ¼51, the μ estimate has only a 0.5% error. Third, if the follower spring is oriented at an angle of σ (from

R [μm]

μ0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0

0.01

0.02

0.03

0.04

0.05

0.06

Fig. 9. Comparison of the modified Benedict–Kelley model from the results of Fig.8 with friction values reported in the literature [16,33–35]. Key: ,Model for AGMA 4EP oil (high Λ); , Model for ISO 32 oil (low Λ); , Shon et al. [16]; , Xu and Kahraman [33]; , Grunberg andCampbell [34]; , Furey [35].

Table 2Error in the estimation of μ for the mechanical system with a circular cam for different values of e at Ωc¼11.55 Hz.

e/a €α (rad/s2) _μ Error (%)¼ _μ–known μknown μ

� 100

At the first harmonic of Ωc At the second harmonic of Ωc

0.05 52.5 1.01 0.284 5.30.10 105.0 4.02 0.286 4.70.15 157.5 9.05 0.288 4.00.20 210.0 16.1 0.292 2.80.25 262.5 25.1 0.296 1.30.3 315.1 36.2 0.302 0.70.35 367.6 49.3 0.31 3.10.4 420.1 64.4 0.32 6.00.45 472.7 81.5 0.33 9.4

S. Sundar et al. / Mechanical Systems and Signal Processing 58-59 (2015) 101–114 111

the vertical in the clockwise direction), the elastic force Fs(t) will be as follows as opposed to the one given by Eq. (14),

FsðtÞ ¼ ks cos αðtÞð Þcos αðtÞ�σð Þ L

us �dyþdx tan αðtÞð Þþ0:5wbsec αðtÞð Þ� �

: ð47Þ

Also the reaction forces will have to be calculated from the following instead of using the expressions of Eqs. (27) and (28),

NxðtÞ ¼ FnðtÞ sin αðtÞð ÞþFf ðtÞ cos αðtÞð Þ�0:5mblb €αðtÞ sin αðtÞð Þ�FsðtÞ sin σð Þ; ð48Þ

NyðtÞ ¼ FnðtÞ cos αðtÞð Þ�Ff ðtÞ sin αðtÞð Þþmbg�0:5mblb €αðtÞ cos αðtÞð Þ�FsðtÞ cos σð Þ: ð49Þ

Based on the static force balance, the error in the estimation of μ is about 9% with only σ¼11, which is very significant.The estimation of μ involves some error-prone numerical methods [36]. For instance, bias errors [37] might be caused in

the computation of the spectral contents of forces and acceleration due to a coarse frequency resolution (constrained by thelength of the measured time domain signal and the usage of the Hanning window). Furthermore, Eqs. (41)–(43) are solvedusing the Levenberg–Marquardt algorithm which has limited accuracy as dictated by its relative and absolute tolerancevalues [38].

The error in μ is simulated for the system with a circular cam for different values of e under a constant Ωc¼11.55 Hz.Using the predicted force and acceleration responses from the contact mechanics model (with known μ¼0.3) in the signalprocessing technique,_μ is calculated, and the results are given in Table 2. For a very low value of e/a, the error in_μ is highbecause the amplitude of €α (and the reaction forces) at the first harmonic of Ωc is very small. As the amplitude of €α at thefundamental harmonic of Ωc increases, the error reduces and reaches a minimum at e/a¼0.3 (error¼0.67%). Beyonde/a¼0.3, the error again starts increasing since the amplitude of €α at the second harmonic of Ωc becomes significantcompared with that of the first. Next the error is calculated for different Ωc with a constant e. The error monotonicallydecreases (as observed from Table 3) with an increase in Ωc; this is because the amplitude of €α (and the reaction forces)at the first harmonic of Ωc increases, while the amplitude ratio of the second harmonic to the fundamental harmonicis a constant.

Table 3Error in the estimation of μ for the mechanical system with circular cam for different cam speeds with e/a¼0.1.

Ωc [Hz] €α [rad/s2] _μ Error (%)¼ _μ–known μknown μ

� 100

At the first harmonic of Ωc At the second harmonic of Ωc

2 3.24 0.135 0.28 6.75 20.26 0.84 0.281 6.38 51.77 2.15 0.283 5.711 97.9 4.1 0.285 5.014 158.5 6.6 0.288 4.017 233.8 9.7 0.292 2.721 356.7 14.9 0.297 1.0

0 0.2 0.4 0.6 0.8 1

8

10

12

14

16

18

20

22

24

Ωc [H

z]

Loss of contact regime

In-contact regime No sign

reversal of v (t)Sign reversal of v (t)

Fig.10. Classification of response regimes of a mechanical system with an elliptic cam in terms of a Ωc–b/a map with e¼0.1a and other parameter valuesgiven in Section 5. Key: , Operational range of simulation.

Table 4Error in the estimation of μ for the mechanical system with an elliptic cam at Ωc¼8.33 Hz and e¼0.1a.

ε €α (rad/s2) _μ Error (%)¼ _μ–known μknown μ

� 100

At the first harmonic of Ωc At the second harmonic of Ωc

0 52.2 2.18 0.283 5.70.31 52.1 52.7 0.281 6.20.44 52.1 104.9 0.281 6.20.53 52.05 156.8 0.284 5.50.6 52.0 208.4 0.288 4.20.66 51.95 259.7 0.294 2.2

S. Sundar et al. / Mechanical Systems and Signal Processing 58-59 (2015) 101–114112

A similar analysis is done for the system with an elliptic cam given e/a¼0.1, for different values of eccentricity

ε¼ 1�ðb2=a2Þ� �0:5� �

with known μ¼0.3 (other parameters remaining the same as in Section 5). Fig. 10 gives a map of

Ωc–b/a, showing different regimes that are obtained using the inverse kinematics procedure of Section 5. Comparison ofFig. 10 with Fig. 4 suggests that ε for an elliptic cam provides a similar motion input as e does for a circular cam. Care is takenso that the system lies in the regime without a loss of contact and no direction reversal of the vr(t). Table 4 shows the_μvalues for an elliptic cam for different ε atΩc¼8.33 Hz. Only a small variation in the error is observed. However, an increase

in the ε increases the acceleration amplitude at the second harmonic of Ωc due to a change in the type of motion input to

the system. Overall, it is inferred that μ can be satisfactorily estimated even for a system with an elliptic cam.

9. Conclusion

The main goal of this article is to estimate μ using both model and experimental measurement and not to validate thecontact mechanics model using the experiment. The major contributions of these analytical and experimental studies are asfollows. First, a new vibration experiment has been designed to estimate μ for a mechanical system with combined rolling-sliding contact under a mixed lubrication regime. This experiment permits the contact pressure, “lambda ratio”, contact

S. Sundar et al. / Mechanical Systems and Signal Processing 58-59 (2015) 101–114 113

velocity (sliding and entrainment), lubrication regime and surface roughness to be changed while satisfying the designconstraints. Thus, the same experiment can be used to estimate μ for similar rolling-sliding contact systems such as gearsand drum brakes. Second, a refined contact mechanics model for a cam–follower system with an elliptic cam is formulatedthat successfully predicts the system responses, as theory and experiment match well. This mathematical model yields abetter understanding of the system dynamics as well as the accuracy of the μ estimation procedure. Third, a new signalprocessing technique is proposed to calculate μ using the complex-valued Fourier amplitudes (without DC component) ofmeasured forces and acceleration. The DC components of the measured signals are also estimated by this method (alongwith μ) by numerically solving a set of nonlinear equations. The validity of the assumptions made is proved by the existenceof a solution to this problem, since a direct solution technique is employed to estimate the coefficient of friction withoutusing any kind of approximation or residue minimization techniques. The chief limitation of this study is related to theangular alignment of the follower spring. Also the error in_μ is controlled by the choice of system geometry and cam speed;in particular the speed should be fairly low in order to avoid impacting conditions.

Acknowledgment

The authors gratefully acknowledge the Vertical Lift Consortium, Inc., Smart Vehicles Concept Center (www.SmartVe-hicleCenter.org) and the National Science Foundation Industry/University Cooperative Research Centers program (www.nsf.gov/eng/iip/iucrc) for partially supporting this research.

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