brake groan

21
Identification and quantification of stick-slip induced brake groan events using experimental and analytical investigations Ashley R. Crowther a) and Rajendra Singh b) (Received: 07 September 2007; Revised: 10 June 2008; Accepted: 12 June 2008) To describe the brake creep-groan phenomenon and several types of stick-slip motions, we propose both analytical and experimental investigations for an automatic transmission equipped vehicle.A lumped torsional model is employed to approximate the dynamics of the real mechanical system.This model assigns inertia to the drive, brake rotor, brake caliper and tire/vehicle, and by appropriate algorithms the simulation time histories include the effects of the friction non-linearity coupling brake and rotor.We consider how to force this particular system, from what physical state, and finding appropriate parameters and solutions. Important computational issues, as related to the stick-slip or slip-stick transitions, are addressed with the algorithms. Driving forces are assigned with two functions, one appropriate for comparison with the test and the other to study stick-slip orbits, which have been found to have various types of motion depending on the controlling brake actuation parameters. Since the groan features a discontinuous friction force and finite- repeating motions some comparisons are made in the frequency and time- frequency domains. These demonstrate the expected stick-slip frequency and multiple orders. © 2008 Institute of Noise Control Engineering. Primary subject classification: 62.1; Primary subject classification: 13.7.1 1 INTRODUCTION The brake creep groan type noise and vibration problem may occur in vehicles when the driver slowly moves some small distance and stops (e.g. in stop-go traffic, at traffic lights, in garage maneuvers). In automatic transmission (AT) vehicles, the torque converter impeller applies small torque on the station- ary turbine and hence the rest of the powertrain. With slow pedal release the brake friction torque will be overcome by the driving torque (impeller) and the brake rotor will start to slip against the brake pad. This initial slip of the rotor excites vibration in the driveline and brake subsystems, allowing a condition where the rotor and pad may stick again. In practice this stick-slip motion may repeat until the rotor and pad reach some steady-sliding state, or stick and even halt the vehicle. The friction-induced motion is highly dependent on pedal actuation. Groan may also occur on brake appli- cation as reported by Brecht et al. 1 ; it typically can be a long event for brake release on takeoff and a short event for vehicle braking to stop. Jang et al. 2 suggest approaches to minimize the vibration by increasing damping or inertia of the rotating bodies and by reduc- ing system compliance. Kim et al. 3 examined the effect of particle sizes of pad constituents and concluded that groan was more severe with smaller abrasive particles with respect to filler or lubricant particle sizes (such as rubber and graphite). Further studies of the material compositional effects on groan, such as the static to kinetic ratio and velocity dependence of the friction coefficient are given in Ref. 2. By using a brake test bench they made correlations between a reduction in amplitude of friction force oscillations with reduced friction ratio and proposed new pad material formula- tions to give these characteristics. Bettella et al. 4 examined the path to receiver and the influence of panels, windscreen and acoustic cavity. By examining the transmission losses in the airborne path they concluded that the groan noise is structure-borne. Their sound pressure and acceleration data show a fundamen- tal frequency at 60 Hz (stick-slip) and multiple orders, due to the discontinuity in the friction force; these modes should be separated from those of the acoustic cavity. Donley and Riesland 5 examined the significant a) Acoustics and Dynamics Laboratory, Mechanical Engi- neering Department, The Ohio State University, 201 West 19th Ave., Columbus, OH 43210 USA; email: [email protected]. b) Acoustics and Dynamics Laboratory, Mechanical Engi- neering Department,The Ohio State University, 201 West 19th Ave., Columbus, OH 43210 USA Noise Control Eng. J. 56 (4), July-Aug 2008 235

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Page 1: Brake Groan

Identification and quantification of stick-slip induced brake groan eventsusing experimental and analytical investigations

Ashley R. Crowthera) and Rajendra Singhb)

(Received: 07 September 2007; Revised: 10 June 2008; Accepted: 12 June 2008)

To describe the brake creep-groan phenomenon and several types of stick-slipmotions, we propose both analytical and experimental investigations for anautomatic transmission equipped vehicle.A lumped torsional model is employedto approximate the dynamics of the real mechanical system. This model assignsinertia to the drive, brake rotor, brake caliper and tire/vehicle, and byappropriate algorithms the simulation time histories include the effects of thefriction non-linearity coupling brake and rotor. We consider how to force thisparticular system, from what physical state, and finding appropriateparameters and solutions. Important computational issues, as related to thestick-slip or slip-stick transitions, are addressed with the algorithms. Drivingforces are assigned with two functions, one appropriate for comparison with thetest and the other to study stick-slip orbits, which have been found to havevarious types of motion depending on the controlling brake actuationparameters. Since the groan features a discontinuous friction force and finite-repeating motions some comparisons are made in the frequency and time-frequency domains. These demonstrate the expected stick-slip frequency andmultiple orders. © 2008 Institute of Noise Control Engineering.

Primary subject classification: 62.1; Primary subject classification: 13.7.1

1 INTRODUCTION

The brake creep groan type noise and vibrationproblem may occur in vehicles when the driver slowlymoves some small distance and stops (e.g. in stop-gotraffic, at traffic lights, in garage maneuvers). Inautomatic transmission (AT) vehicles, the torqueconverter impeller applies small torque on the station-ary turbine and hence the rest of the powertrain. Withslow pedal release the brake friction torque will beovercome by the driving torque (impeller) and thebrake rotor will start to slip against the brake pad. Thisinitial slip of the rotor excites vibration in the drivelineand brake subsystems, allowing a condition where therotor and pad may stick again. In practice this stick-slipmotion may repeat until the rotor and pad reach somesteady-sliding state, or stick and even halt the vehicle.The friction-induced motion is highly dependent onpedal actuation. Groan may also occur on brake appli-

a) Acoustics and Dynamics Laboratory, Mechanical Engi-neering Department, The Ohio State University, 201 West19th Ave., Columbus, OH 43210 USA; email:[email protected].

b) Acoustics and Dynamics Laboratory, Mechanical Engi-neering Department, The Ohio State University, 201 West19th Ave., Columbus, OH 43210 USA

Noise Control Eng. J. 56 (4), July-Aug 2008

cation as reported by Brecht et al.1; it typically can bea long event for brake release on takeoff and a shortevent for vehicle braking to stop. Jang et al.2 suggestapproaches to minimize the vibration by increasingdamping or inertia of the rotating bodies and by reduc-ing system compliance. Kim et al.3 examined the effectof particle sizes of pad constituents and concluded thatgroan was more severe with smaller abrasive particleswith respect to filler or lubricant particle sizes (such asrubber and graphite). Further studies of the materialcompositional effects on groan, such as the static tokinetic ratio and velocity dependence of the frictioncoefficient are given in Ref. 2. By using a brake testbench they made correlations between a reduction inamplitude of friction force oscillations with reducedfriction ratio and proposed new pad material formula-tions to give these characteristics. Bettella et al.4

examined the path to receiver and the influence ofpanels, windscreen and acoustic cavity. By examiningthe transmission losses in the airborne path theyconcluded that the groan noise is structure-borne. Theirsound pressure and acceleration data show a fundamen-tal frequency at 60 Hz (stick-slip) and multiple orders,due to the discontinuity in the friction force; thesemodes should be separated from those of the acousticcavity. Donley and Riesland5 examined the significant

235

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flexible bodies of a McPherson strut system by using amulti-body dynamics software package. They calcu-lated frequency response functions for relativedisplacement between the rotor and the pad (with unitforce applied at the rotor/pad interface); several modesin the range of 30–100 Hz are evident in the compli-ance function. This is of relevance to our estimate ofbrake subsystem modes. Our approach to understand-ing brake groan draws inspiration from many otherfriction-induced problems such as those reviewed byMartins et al.6.

Brake groan remains a complex engineeringproblem that has yet to be fully understood or analyzed.To aid this void in the literature, we propose an analyti-cal investigation of four-degree-of-freedom torsionalmodel (as shown in Fig. 1). It includes driveline andbrake torsional subsystems, with friction interfacethrough the brake rotor and pad. This model shouldconceptually describe the brake creep-groan phenom-enon and several types of stick-slip motions, and yet itcould be easily extended. Note that two dynamicsubsystems are coupled by a friction interface, an

Fig. 1—Torsional model utilized for brake groan anand (c) Rotor/tire side view. System parame=0.189 and Jd=73.1 kg m2; Stiffness, kd=8=1.15, cb=4.53, ct=13, dd=23.9 and for Cdt=200 N msrad–1.

236 Noise Control Eng. J. 56 (4), July-Aug 2008

important aspect of the non-linear model (as illustratedby Figs. 1 and 2. The frequency ratio for the subsystemmodes dominating the stick-slip motions is not nearzero (or infinite), hence neither the brake or powertrainsubsystems may be considered as a rigid body. Manyparameters and conditions affect stick-slip orbit(s)including, but not limited to, system modes, frictioncharacteristics, driving torque magnitude, rate andmagnitude of brake release, rate and magnitude ofbrake reapplication and hydraulic system dynamics.

is: (a) Driveline subsystem, (b) Brake subsystemare as follows: Inertia, Jd=9.54, Jr=0.652, Jb, kb=68000, kt=22000 N mrad–1; Damping, cdI brake force dt=600, for Case II,

Fig. 2—Brake friction force as a function of slid-ing velocity.

alysters790

ase

Page 3: Brake Groan

2 PROBLEM FORMULATION

We establish the following objectives for an ATequipped vehicle: a) Obtain and analyse an example setof real system data; b) Develop a groan model in termsof two dynamic subsystems coupled with friction inter-face and briefly identify relevant system parameters; c)Examine the computational issues associated withstick-slip motion simulation and obtain solutions to thecreep-groan response; and d) Show a preliminarycorrelation between theory and experiment. Thelumped torsional model is employed to approximatethe dynamics of the real mechanical system. The modelassigns inertia to the drive, brake rotor, brake caliperand tire/vehicle, and by appropriate algorithms thesimulation time histories may include effects of thefriction non-linearity (Fig. 2) coupling brake and rotor.We consider how to force this system, from what physi-cal state, and finding appropriate parameters andsolutions. In doing so the driveline subsystem (Fig.1(a)) has parameters determined from a sensible reduc-tion and the brake subsystem (Fig. 1(b)) parameters areestimated from modal testing. For the initial valueproblems there are important computational issues, asrelated to the stick-slip or slip-stick transitions, whichmust be addressed before programming solutionalgorithms. To relate solutions to the vehicle test weattempt to reproduce roughly a function for brakepressure as controlled by the driver (Fig. 3(a)) andcompare the time domain characteristics of thetransient stick-slip motions to those measured. Next weseek an analytical insight by implementing a brakerelease function (Fig. 3(b)), that leads to steady statestick-slip orbits, which have been found to have varioustypes of motion depending on the controlling brakeactuation parameters7. That the groan features a discon-tinuous friction force and finite-repeating motionsallows for comparisons in the frequency domain for theexpected stick-slip frequency and multiple orders. Thusthe frequency content is examined for simulated andtest signals including the transient ebb and flow of realgroan signals via both time and frequency domainanalyses.

3 MEASURED BRAKE GROAN EVENTS

3.1 Vehicle Experiment

A typical medium sized passenger vehicle is used toduplicate the driver behavior of creeping forward inalmost stationary traffic or at traffic lights. The vehicleis stationary on flat ground, the engine is idling and theapplied brake is resisting a small amount of torquefrom the torque converter. The driver releases the brakevery slowly until groan occurs and with careful actua-

Noise Control Eng. J. 56 (4), July-Aug 2008

tion of the brake pedal the recording of sustained groanevents is possible. Consider the slow speed nature ofthe creep-groan event; the wheel rotates no more than30° over several seconds. Sound pressure levels aremeasured at the driver’s right ear position and atop thewheel arch 100 mm away from the vehicle body. Accel-erometers are mounted on the brake caliper andsuspension strut (Fig. 4). The ‘brake caliper accelerom-eter’ measures motion near tangential to the padcontact area on the brake rotor, the ‘suspension strutaccelerometer’ measures fore-aft motions. Data aresampled at 48 kHz. Tests are roughly 60 s in length

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

800

1000

1200

1400

1600

F n(N)

Time (s)

τ

γβFn∧ατ

1 2 3 4L

(a)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 21000

1020

1040

1060

1080∧Fn

FmF n(N)

Time (s)

α1← →

α2← →

α3← →βγ1↑

↓βγ2↑

βγ3↑

(b)

Fig. 3—Functions for transient brake normalforce, Fn�t�: a) Case I with variousprofiles—1. �=300 N, �=0.05 s,�=0.1 s, 2. �=240 N, �=0.1 s,�=0.15 s, 3. �=180 N, �=0.15 s, �=0.2 s and 4. �=100 N, �=0.2 s, �=0.25 s; (b) Case II with �=1, �=30, �1=0.95, �2=0.90, �3=0.995. Note: Fn�t�=Fm−� tanh �t, �=−1/� ln��1−�2��1+�2�−1� and �=−1/2� ln��1−���1+��−1�, see Ref. 7.

Fig. 4—Accelerometer locations for brake groanvehicle test.

237

Page 4: Brake Groan

with the driver carefully inducing roughly 15 groanevents, each of around 0.5–2 s duration. The brake isfully applied between groan events.

Multiple intermittent brake groan events are shownfor measured accelerations in Fig. 5. Events A, B, E andG are a short creak sound, rather than sustained groan;here the driver has released the brake only slightly pastthe point of initial slip before reapplication. See Fig. 6for a close up look of event B; evident is an impulse, ortwo, followed by a decay transient with rich spectra.Events C, D, F, H and I have a more continuous repeti-tion of this motion over a finite duration. Event F isquite sustained, Fig. 7 provides the first 0.8 s�19.8–20.6 s�; here the groan is more transient untilsettling to a fairly steady state for 1 s �20.6–21.6 s�, asshown by Fig. 8. The impulses are repeating at close to75 Hz until the groan dies away fairly abruptly ataround 22.1 s.

3.2 Quantification of Groan Events

Groan severity could be quantified from timedomain data in various ways, such as the length of the

12 14 16

-0.50

0.51

1.5

Acceleration(g)

12 14 16

-2

0

2

4Acceleration(g)

AB C D

AB C D

Fig. 5—Vehicle test acceleration time histories forto I: a) Suspension strut fore-aft accelerom

12.698 12.7 12.702 12.704 12.706-0.5

0

0.5

Acceleration(g)

12.698 12.7 12.702 12.704 12.706-2

0

2

Acceleration(g)

Fig. 6—Acceleration time histories for final impulserometer; b) Caliper tangential accelerome

238 Noise Control Eng. J. 56 (4), July-Aug 2008

groan event, peak to peak range and mean-squarequantities. As follows are a set of metrics (given by Qwith subscript i) illustrated via the test data. First weexamine the “steady-state” response given by Figs. 8and 9 and compare to a “no-groan” record of the samelength, T (though such a figure is not shown). For theno-groan case the vehicle is idling and stationary andthe accelerometers and microphones pick up vibrationsfrom engine firing and background noise. The peak topeak range is Q1=�max−�min, for all T andmean-square values (a ‘power’ like quantity from theperspective of signal processing) are defined as Q2

= 1T�ta

tb�2 �t�dt, with T= tb− ta. Assignments formeasured signals are, acceleration, �= x and soundpressure, �=P. The metric Q1 (Table 1) shows accel-erations for “steady-state” groan are at least an order ofmagnitude greater than for no groan. For soundpressure the results are not as distinct. Values of Q2,give a similar finding, accelerative “power” is distinctlygreater for the groan case and sound “power” (mean-square pressure) gives mixed results. This gives an

20 22 24 26

20 22 24 26e (s)

F G H I

F G H I

(a)

(b)

mittent brake groan with groan events labeled Ab) Caliper tangential accelerometer.

08 12.71 12.712 12.714 12.716 12.718

08 12.71 12.712 12.714 12.716 12.718(s)

(b)

(a)

groan event B: a) Suspension strut fore-aft accel-

18

18Tim

E

E

intereter;

12.7

12.7Time

e ofter.

Page 5: Brake Groan

erom

understanding of which time domain measurementswill give a clear representation for groan events. Theaccelerometers (say at the suspension strut which is apractical location) give groan readings clearly distinctfrom engine vibrations. Metric 3 is the length of agroan event, Q3= tb− ta, from the first significant accel-

19.8 19.9 20 20.1

-0.5

0

0.5

1

Acceleration(g)

19.8 19.9 20 20.1-1

0

1

2

3Acceleration(g)

Fig. 7—Acceleration time histories for initial non-sstrut fore-aft accelerometer; b) Caliper tan

20.6 20.7 20.8 20.9 21

-0.5

0

0.5

1

Acceleration(g)

20.6 20.7 20.8 20.9 21-1

0

1

2

Acceleration(g)

Fig. 8—Acceleration time histories for steady stateaccelerometer; b) Caliper tangential accel

20.6 20.7 20.8 20.9 21-0.5

0

0.5

Sound

Pressure(Pa)

20.6 20.7 20.8 20.9 21-0.4-0.20

0.20.40.60.8

Sound

Pressure(Pa)

Fig. 9—Sound pressure time histories for steady staphone; b) Driver’s right ear position micro

Noise Control Eng. J. 56 (4), July-Aug 2008

eration spike, at ta, (quite clear in Fig. 7 at approxi-mately 19.82 s) to the last acceleration spike at tb.Note, to “quantify” what is a significant accelerationspike we have taken approximately 12.5% of the peakto peak range given in Table 1 as the cutoff magnitude,

.2 20.3 20.4 20.5 20.6

.2 20.3 20.4 20.5 20.6(s)

(a)

(b)

y state section of groan event F: a) Suspensionial accelerometer.

.1 21.2 21.3 21.4 21.5 21.6

.1 21.2 21.3 21.4 21.5 21.6(s)

(a)

(b)

ion of groan event F: a) Suspension strut fore-afteter.

1.1 21.2 21.3 21.4 21.5 21.6

1.1 21.2 21.3 21.4 21.5 21.6e (s)

(b)

(a)

ortion of groan event F: a) Wheel rim micro-e.

20

20Time

teadgent

21

21Time

port

2

2Tim

te pphon

239

Page 6: Brake Groan

i.e. for suspension strut, xcut� ±0.25 g and for caliper,xcut� ±0.5 g. Assigning such limits allows for easierdecision making in data processing. Metric 4 is an‘energy’ like quantity that is defined as: Q4

=�ta

tb�2 �t�dt. Unlike the “steady-state” sample of Fig.8, most transient groan events grow and decay and haveoutlier minimum and maximum values, hence metricsQ1 and Q2 may be less appropriate measures. In Fig.10, Q3 and �Q4 are rank ordered by �Q4 for Events A-I.Event F is the worst case, in terms of Q3 and �Q4,followed by Event D. Next is Event C which has higher“energy” than Event H yet was shorter in length; alonger event does not necessarily equate to more severegroan in terms of vibration energy, however, thereceiver perception (driver) could be studied. Compar-ing with �Q4 �g s0.5� rather than Q4 �g2 s� gives unitsscaled better to Q3 (s).

3.3 Identification of Groan in FrequencyDomain

The example data of Figs. 8 and 9 are chosen forfrequency domain analysis given the steady-state

Table 1—‘Steady-State’ groan and no groan compMean-square values.

Event

Suspension Strut Caliper

Q1 (g) Q2 �g2� Q1 (g) QNo Groan 0.101 4.65e−4 0.127 6.‘Steady-

State’Groan

2.038 0.200 3.801 0

RatioGroan/No-

Groan

20.1 430.0 29.9 1

0.1

0.2

0.3

0.4

0.5

0.6

Q40.5(gs0.5)

A E G B I H C D FGroan Event

0

0.4

0.8

1.2

1.6

2

2.4

Q3(s)

Fig. 10—Groan event comparison with metrics Q3

(thin lines) and �Q4 (thick line) for sus-pension strut (solid), Caliper (dotted).Events are rank ordered from ‘Best’ to‘Worst’ based on Q4 for suspension strut.Note Q3 lines are overlaid at this magni-fication.

240 Noise Control Eng. J. 56 (4), July-Aug 2008

nature and Figs. 11 and 12 give corresponding spectraup to 300 Hz for the superimposed with the no-groancase. The engine idling contributes significantly toamplitudes under 50 Hz. The acceleration signals givethe clearest account of groan with the highest peak at76 Hz and near multiples at 152, 226, 303, 378 and454 Hz, i.e. up to the first six orders of the dominantharmonic, and discernible peaks up to the 11th order(frequencies above 300 Hz are not shown in the plots).As suggested by earlier authors1–5, brake groan is astick-slip event and here the stick-slip frequency wouldbe 76 Hz, with multiple orders due to the impulsivebehavior shown in accelerations at stick-slip transi-tions.

This data can be further processed to establish themost significant frequency band for groan. Herewe use a frequency domain averaging methodto find the band-limited mean-square quantities,

defined as �ms�n�= 1fb

��n−1�fbnfb �2�f�df, where n

=1,2 , . . . , �fs / �2m�f��. The band limit, fb, is selected asfb=m�f, and we select �m�f�=10 Hz. For �f=0.183 Hz, m=55 and fb=10.07 Hz. This could be putas follows: with these parameters, all 55 frequency binsfrom the mean-square values of Figs. 11 and 12 areaveraged into one single bin at fb /2 and n multiplesthereof. The resulting band-limited spectral compari-sons are given in Figs. 13 and 14 for both groan andno-groan cases and these yield several insights. First, itcan be seen that the caliper and strut have similarresponses, with main peaks at the above mentionedstick-slip frequency and multiple orders, the impulse atevery stick and slip is also exciting the system with abroadband effect. The noise is minimal at the wheel rimmicrophone compared to the driver right ear position.This matches the perception of those present for thetest, i.e. for the human ear, brake groan seems louder ormore perceptible within the vehicle than standing nextto the wheel. The driver’s right ear microphonesuggests that the receiver should perceive the transient

n metric Q1: Peak to peak range and metric Q2:

Wheel Arch Drivers Right Ear

� Q1 (Pa) Q2 �Pa2� Q1 (Pa) Q2 �Pa2�4 1.113 0.199 0.511 0.034

1.569 0.044 1.264 0.094

1.4 0.2 2.5 2.8

ariso

2 �g2

94e−.113

62.8

Page 7: Brake Groan

groan events mostly in the 50 to 250 Hz bandwidth.The results show the noise is likely to be structureborne and is exciting the acoustic cavity. A frequencydomain metric could be defined as the area under thefrequency curve over 50–250 Hz (or some other span)and used for comparative evaluations. Further the timedomain metrics could be applied after signals are bandpass filtered with suitable cutoffs to reduce extraneousaffects from engine and background noise sources. Forsound pressure measurements this would improvecalculations of Q1–4, but thought should be spared forthe effect of filtering in the case of transient signals.

The complex values from short time FFTs may beused to construct waterfall plots (Fig. 15) showing the

0 50 10010

-5

10-4

10-3

10-2

10-1

100

Acceleration

Magnitude(g)

0 50 10010

-5

10-4

10-3

10-2

10-1

100

Acceleration

Magnitude(g)

F

Fig. 11—Acceleration spectra for steady state porti(thin line): a) Suspension strut fore-aft acRecord length, T=1 s with a Hanning winfor a frequency resolution of �f=0.183 H

0 50 10020

40

60

80

Sound

Pressure(dB)

0 50 10020

40

60

80

Sound

Pressure(dB)

Fr

Fig. 12—Sound pressure spectra for steady state posample (thin line): a) Wheel rim micropholength, T=1 s with a Hanning window. Saquency resolution of �f=0.183 Hz.

Noise Control Eng. J. 56 (4), July-Aug 2008

time-frequency content. The transient signals do bringto the spectral calculation process an averaging effecton frequencies and their amplitudes; yet the results arequite illustrative. To improve clarity for the figures, thelower cutoff frequency is selected as 50 Hz, removinghigh amplitude content from engine idling. Thetime-frequency characteristics of the five most majorgroan events C, D, F, H and I are captured. For eachevent, the variation in brake pressure leading to groanwould be different, depending on how the driveractuated the pressure, also the length and mean squarequantities are different (Table 1 and Fig. 10). Note that

150 200 250 300

150 200 250 300ncy (Hz)

(a)

(b)

f groan Event F (thick line) and no groan sampleometer; b) Caliper tangential accelerometer.. Sampled at 48 kHz, zero padded to 218 points

50 200 250 300

50 200 250 300ncy (Hz)

(b)

(a)

n of groan Event F (thick line) and no groanb) Driver’s right ear position microphone. Recorded at 48 kHz, zero padded to 218 points for a fre-

reque

on ocelerdowz.

1

1eque

rtione;mpl

241

Page 8: Brake Groan

the dominant spectral energy stays within a fairlynarrow band around 76 Hz during the transient andsteady state motions.

4 DEVELOPING A MODEL ANDIDENTIFYING BRAKE SYSTEMPARAMETERS

4.1 Dynamic Model Reduction

Careful reduction of the powertrain is important forproper selection of parameters for the reduced model.For clarity, the process is illustrated in steps as shownin Fig. 16. The powertrain is initially discretised withthe most significant inertias and compliant elements(Fig. 16(a)). Here we have Je, the engine inertia, Jtc, thetorque converter inertia and Jtr1, the inertia of the trans-

10-910-810-710-610-510-410-310-210-1

BandLimited

Power(g2 )

0 50 100 150 20010

-910-810-710-610-510-410-310-210-1

BandLimited

Power(g2 )

Fig. 13—Band limited mean-square acceleration foand no-groan steady state sample (solid lCaliper tangential accelerometer. Record48 kHz, zero padded to 218 points for a fr=m�f=10.07 Hz, where �m�f�=10 Hz wi

10-7

10-6

10-5

10-4

10-3

10-2

10-1

BandLimited

Power(Pa2)

0 50 100 150 20010

-810-710-610-510-410-310-210-1

BandLimited

Power(Pa2)

Fig. 14—Band limited mean-square sound pressureline) and no-groan steady state sample (sear position microphone. Record length, Tzero padded to 218 points for a frequency=10.07 Hz, where �m�f�=10 Hz with m=

242 Noise Control Eng. J. 56 (4), July-Aug 2008

mission input components, these three inertias rotate atengine speed (ignoring the torque converter dynamics).The reduced order model of the driveline is given bythree inertias, Jd, representing the combined inertia ofthe powertrain elements from the engine to the finaldrive ring gear, Jr, the inertias of the brake rotor, wheelhubs and rims and Jt, the tire inertia and equivalentinertia of the vehicle mass. These inertias areconnected via lumped stiffness elements, kd, an equiva-lent drivetrain stiffness and kt, the tire stiffness. Brakegroan occurs at low speeds where the first gear isengaged, hence the transmission gears down with speedratio, ntr, the inertias of transmission output sidecomponents, Jtr2 and the final drive pinion, Jfd1, rotateat the corresponding speed (here the driveshaft shaft

250 300 350 400 450 500ency Hz

(a)

(b)

ady state portion of groan Event F (dotted line)a) Suspension strut fore-aft accelerometer; b)th, T=1 s with a Hanning window. Sampled atncy resolution of �f=0.183 Hz. Band limit, fb=55.

250 300 350 400 450 500ency Hz

(b)

(a)

steady state portion of groan Event F (dottedline): a) Wheel rim microphone; b) Driver’s rights with a Hanning window. Sampled at 48 kHz,lution of �f=0.183 Hz. Band limit, fb=m�f

Frequ

r steine):lengequeth m

Frequ

forolid

=1reso55.

Page 9: Brake Groan

inertia can be split and lumped evenly to Jtr2 and Jfd1).Then the final drive unit gears down with speed rationfd, to the axle speed, where there are the rotatingelements, Jfd2, the final drive ring gear, Jrl and Jrr, theleft and right brake rotors and wheel rims and Jtl andJtr, the left and right tires combined with an equivalentvehicle inertia, Jv. Note that Jtl=Jtr=Jtire+0.5Jv=Jtire

Fig. 15—Waterfall plot showing ‘averaged’ spectravia suspension strut fore-aft accelerometewindow. Sampled at 48 kHz, zero padded�f=0.732 Hz.

.

.

.bT ( , t)δ

δ

δ

k tdk tJJrT (t)v

(d)

(a)

T (t)d

dJ

nbr

bl

v

v

en

½T (t)

T ( ,

T ( ,T (t)

½T (t)

Jtr

Jtl

Jrl

rrJ

fd2JJfd1

k tl

kal

k tr

arkkps

k is tr

Jtr2

tr1J

+ tcJ fdJe osk

Fig. 16—Model reduction process for the driveline=0.2, Jtc=0.1, Jtr1=0.02, Jtr2=0.01, Jfd1=Jtl=Jtr=73 kg m2 (inc. vehicle mass); Stif=10000 and k =k =22000 N mrad–1.

tl tr

Noise Control Eng. J. 56 (4), July-Aug 2008

+0.5mvrt2, where Jtire is the inertia of one tire, mv the

vehicle mass and rt the tire radius. Two wheel brakingis assumed for simplicity in later transient simulations,thus the engine drives and the brakes slow the wholevehicle mass via one pair of wheels. The stiffnessconnections are kis, transmission input shaft, kos, trans-

tents of transient groan Events A-I as measuredort time record length, T=0.2 s with a Hanning16 points for a frequency resolution of

.

.

a

δbT ( , t)

T ( , t)b δ

nfdJfd1

t

vT (t)

J

rJk t

akfd2J

(c)

(b)

trnJeq1 keq1

fdn

eT (t)

2trn

r( )

2

koseJ Jtc+

J +tr1 tr2Jtr

isk pskka

tk

fd1JJfd2

Jr

tJ

(t)

T (t)

n

v

ystem. System parameters for (a) are: Inertia, Je5, Jfd2=0.05; Jrl=Jrr=0.65 ands, kis=20000, kos=35000, kps=30000, kal=kar

l conr. Shto 2

trn

nt2

Te

t)

t)

subs0.00fnes

243

Page 10: Brake Groan

mission output shaft, kps, driveshaft, kal and kar, left andright axles and ktl and ktr, left and right tires. Thedriving torques include engine torque, Te�t�, braketorque Tb�t� and vehicle resistance torque Tv�t� (seeSec. 5.3).

The first reduction step (Fig. 16(b)), modifies thecomponents rotating at engine speed to equivalentparameter values at transmission output speed. Onlybraking of the right wheel is considered, hence the leftaxle and tire subsystem are now omitted, giving Jr

=Jrr=Jrr1+Jrr2+Jrr3, Jt=Jtr and ka=kar, kt=ktr, note Jrr

is composed of brake rotor, Jrr1, wheel hub, Jrr2 andwheel rim, Jrr3 inertias. The second reduction step,shown as Fig. 16(c), sums in series the spring elementsbetween engine and final drive pinion and lumpsengine, torque converter and transmission inertiastogether, giving:

Jeq1 = ntr2 �Je + Jtc + Jtr1� + Jtr2, �1a�

keq1 = � 1

ntr2 kis

+1

kos+

1

kps−1

�1b�

The final reduction step, shown as Fig. 16(d) andcorresponding to Fig. 1(a), modifies the componentsnow rotating at driveshaft speed to equivalent param-eter values at the final drive ring gear speed. Theseparameters are then divided by two, considering thatthe model is for only the right axle, brake and tiresubsystems:

Jd = 0.5�nfd2 �Jeq1 + Jfd1� + Jfd2� = 0.5�ntr

2 nfd2 �Je + Jtc

+ Jtr1� + nfd2 �Jtr2 + Jfd1� + Jfd2� �2a�

kd = � 2

nfd2 keq1

+1

ka−1

, �2b�

Td = 0.5ntrnfdTe �2c�

Note the brake subsystem includes Jb, an approxi-mated inertia for the caliper and knuckle and kb, thetorsional stiffness of the suspension arms. Given thecomplexity of the suspension/brake knuckle assemblyand for the sake of analysis regarding the stated objec-tives, these parameters will be defined via �b=�Jb /kb,the natural frequency of the first torsional mode of thecaliper/knuckle/suspension system (see Sec. 4.2.2).

The reduced brake groan model is designed so as tobe applicable to both transient and stick-slip periodicanalyses. Since brake groan is induced by the stick-slipphenomenon the equations of motion are given by thefollowing set of differential equations where J, C andK represent inertia, viscous damping and stiffnessmatrices respectively, T is the external torque vectorand � is the angular displacement vector.

244 Noise Control Eng. J. 56 (4), July-Aug 2008

J� + C� + K� = T�,t� . �3�

Note that the friction torque would introduce apiecewise non-linearity. Under the slipping conditionthe governing system is of dimension four and is givenby �= d r b t�T and

J = diag�Jd Jr Jb Jt�; �4a�

K = �kd − kd 0 0

− kd kd + kt 0 − kt

0 0 kb 0

0 − kt 0 kt

; �4b�

C = �cd + dd − cd 0 0

− cd cd + ct 0 − ct

0 0 cb 0

0 − ct 0 ct + dt

; �4c�

T = �Td

− Tb

Tb

− Tv

�4d�

The viscous damping is included via c terms alongwith stiffness elements and inertial damping terms, dd

and dt, that are applied on the powertrain equivalentinertia and tire/vehicle inertia. For the sticking condi-tion the dimension is reduced by one, giving, �= d r/b t�T, where r/b indicates brake and rotorcoordinates, and

J = diag�Jd Jr + Jb Jt�; �5a�

K = � kd − kd 0

− kd kd + kt + kb − kt

0 − kt kt ; �5b�

C = �cd + dd − cd 0

− cd cd + ct + cb − ct

0 − ct ct + dt ; �5c�

T = � Td

kb

− Tv

. �5d�

The brake and rotor travel together when stickingand will become offset for slipping motions between

sticking points. Defining =r−b as the relative

displacement offset gives a torque offset kb in Vector(5d). The vehicle displacement may be determined asx =r .

v t t
Page 11: Brake Groan

4.2 Identification of System Parameters andEigensolutions

4.2.1 Driveline subsystem

The powertrain system parameters are given similarvalues to those provided for models in Ref. 8. Theseparameters are given with Fig. 16. The vehicle mass ismv=1500 kg while transmission and final drive speedratios are ntr=2.4 and nfd=3.2, respectively. The tireradius is rt=0.31 m. Using the reduction method thedriveline parameters are obtained as given with Fig. 1.

4.2.2 Brake subsystem

A floating caliper type brake system (Fig. 17) of acommon vehicle is used for obtaining approximateparameters for our model. Note that it is not the samebrake as for the experiment of Sec. 3 but it is a similarlysized vehicle. With experimental modal analysis thebrake system is found to have several modes, in theregion of 20 to 150 Hz. Distinctive motions includebending around a longitudinal axis (fore-aft vehicle),bending around a vertical axis and twisting around atransverse axis (though the vehicle). Naturally thebrake and suspension system is complex to model andthus it is desirable to develop a small lumped systemapplicable for non-linear analyses. Given the momentarm transmitting the brake friction force as torque onthe suspension arms, it is suspected that the twistingmotion of the caliper and knuckle around some centreof rotation (on a transverse axis) would be excitedsignificantly by stick-slip dynamics at the rotor/padinterface. With this in mind, four particular results fromthe modal tests are considered (Fig. 18). The acceler-ometer mounting positions are shown in Fig. 17:a) Location A: mounted in the vertical plane atop thefixed part of the brake caliper; b) Location B: mountedin the transverse plane on the upper control arm. Animpact hammer is used to excite the system in the verti-cal plane (by Location A) and the accelerance typefrequency response functions are obtained with adigital signal analyzer.

(a) (b)

Fig. 17—Typical Floating-Caliper Brake Systemwith Acceleration Measurement Loca-tions A and B.

Noise Control Eng. J. 56 (4), July-Aug 2008

First examine Fig. 18(a) for the result of Test I, herethe floating part of the caliper is removed (as shown inFig. 17) and the measurement is at Location A. Twomain response frequencies we consider are around 42and 120 Hz, believed to be bending about a longitudi-nal axis and twisting about a transverse axis, respec-tively (also evident are several other modes such as at56, 110 and 134 Hz). We can identify that the mode at120 Hz (as well as 110, 114 and 134 Hz) is associatedwith the twisting motion; examine Fig. 18(b) for Test II,with measurement at Location B in the transversedirection. Notice that the modes below 100 Hz are nolonger present and thereby must be associated onlywith vertical motions. Now two tests (III and IV) are

0.0E+002.0E-014.0E-016.0E-018.0E-011.0E+001.2E+001.4E+001.6E+00

20 45 70 95 120 145

Frequency (Hz)

Accelerance(V/V)

Test III

Test I

(a)

4.0E-01

6.0E-01

8.0E-01

1.0E+00

20 45 70 95 120 145Frequency (Hz)

Coherence Test I

(b)

4.0E-01

6.0E-01

8.0E-01

1.0E+00

20 45 70 95 120 145Frequency (Hz)

Coherence Test III

(c)

0.0E+00

5.0E-01

1.0E+00

1.5E+00

2.0E+00

2.5E+00

20 45 70 95 120 145Frequency (Hz)

Accelerance(V/V)

Test II

Test IV

(d)

Fig. 18—Experimental modal analysis for typicalfloating-caliper brake system: a) Ac-celerance with impact and measurementat Location A for Test I (Brake released)and Test III (Brake applied); b) Coher-ence for Test I; c) Coherence for Test III;d) Accelerance with impact at LocationA and measurement at Location B forTest II (Brake released) and Test IV(Brake applied).

245

Page 12: Brake Groan

conducted with the fixed part of the caliper reattachedand the brake firmly applied. Test III (Fig. 18(a)) is formeasurement at Location A. Comparing with Test I wesee that the peak at 40 Hz is much smaller and the120 Hz mode (or a nearby mode) has shifted signifi-cantly down to about 90 Hz. This illustrates an impor-tant point for the stick-slip dynamic analysis: somemodes will change significantly during stick-slip orslip-stick transitions. Similarly Test IV, with measure-ment at Location B, also shows this mode shifting to90 Hz. The coherence plots show expected character-istics; Test I close to the impact point has better coher-ence than Test III, and coherence is good at resonancesand poor at anti-resonances. Test II and IV show thesame coherence characteristics (though the plots arenot shown). The shift could be attributed to one or moreof the following modal couplings: a) increased inertiaand mass when the fixed part of the caliper isreattached; b) an addition to the rotor inertia due to thetorsional motion of the brake subsystem via the brakeclamping force; and c) an addition to the driveshaftstiffness coupling the brake subsystem to the essen-tially grounded driveline gearing (by large inertia andstatic friction).

We can use this understanding to estimate the effec-tive kb as follows. First, modal measurements are madeof the brake system to approximate inertia values(given the lack of availability of solid models). Thedimensions of the floating and fixed parts of the caliperare assumed (Table 2a) giving a total mass of about5.5 kg for the caliper; similar the to manufacturer’sfigure. Inertia about the centroid, Jyy, is calculated viaJ =m�h2+w2� /12 (rectangular block), where m, h and

Table 2—Estimated brake subsystem geometry andknuckle (original and adjusted).

ComponentHeight

(m)Width

(m)Depth(m)

Mas(kg

Caliper(fixed part)

0.14 0.03 0.065 2.20

Caliper(floating part)

0.14 0.05 0.065 3.27

Component Radius (m)

Wheel hub �Jrr2� 0.06Brake rotor �Jrr1� 0.14 (outer) 0.07 (inner)

Knuckle 0.12Knuckle (adjusted) —

yy

246 Noise Control Eng. J. 56 (4), July-Aug 2008

w are mass, height and width respectively. Then inertiaabout the tire rolling axis is obtained with an approxi-mated radius of rotation and by applying the parallelaxis theorem. The brake rotor and hub are assumed tobe simple discs, with measurements, masses andinertias as given in Table 2b. Finally, the brake knuckle(the most complex shape) is also approximated as adisc, providing a mass of about 4 kg, again similar tomanufacturer’s figure.

Under the condition of Test I, the slipping equationfor the brake, as extracted from Eqn. (3), gives thenatural frequency for slipping brake subsystem as fb1

= 12� �kb /Jb�0.5. The combined inertia is that of the fixed

part of the caliper plus knuckle, Jb=0.03 kg m2, hencegiven the test result �fb1=120 Hz�, an approximatedstiffness is found as kb=17000 N mrad−1. Test III hasthe caliper reattached and the brake applied, so now thecombined inertia consists of the whole caliper, knuckle,wheel hub and rotor, Jb+Jr=0.184 kg m2; note thewheel rim inertia, Jrr3, is removed for the test. Thiscondition is considered to be governed by the stickingsystem equations; however the tire is removed andhence kt=0. Also, Jd� �Jb+Jr�, so for the purpose oflinear modal analysis, the drive inertia may be consid-

ered grounded, giving �Jr+Jb�r/b+ �kd+kb�r/b=0.Now the resulting natural frequency of the coupledbrake-axle torsional system is fb2= 1

2� ��kb+kd� / �Jb

+Jr��0.5=60 Hz. This illustrates the mode shift asshown in Fig. 18(a), however, it is significantly far froman observed shift to 90 Hz as found in Test III. Withsuch approximations we obviously do not expect thelinear system theory to closely match the modal tests

ia parameters: a) Caliper, b) Wheel hub, rotor and

Inertia aboutCentroid�kg m2�

Radius ofRotation

(m)

Inertia about TireRolling Axis

�kg m2�0.004 0.1 0.023

0.006 0.13 0.061

Depth (m) Mass(kg)

Inertia about Centroid/TireRolling Axis

�kg m2�0.025 2.20 0.0040.02 7.20 0.088

0.045 3.97 0.007— 3.97 0.096

inert

(a)

s)5

6

(b)

Page 13: Brake Groan

and an improvement must be made. Looking at theassumptions made in geometry, the most inaccuratelydescribed is likely the knuckle inertia, given itscomplex shape. This inertia may be understated, givenasymmetry and since a proportion of the suspensionlinkage inertias (if known) must be added. Increasingto a value similar to the brake rotor (the adjusted valuein Table 2b) gives Jb=0.119 kg m2 (fixed part of caliperand adjusted knuckle) for Test I and with fb1=120 Hz,we now estimate a more reasonable kb

=68000 N mrad−1 (perhaps a more likely value for thetorsional rigidity of the suspension linkages). For TestIII, Jb+Jr=0.272 kg m2 (whole caliper, adjustedknuckle, wheel hub and rotor) and the torsional modein the sticking condition is fb2=85 Hz; closer to the90 Hz measured. These calculations of brakesubsystem parameters are quite approximate and servethe purpose of providing order of magnitude valuesrather than just “guessing” some generic values. Asignificant challenge would be to add additionaldegrees of freedom to the brake subsystem of Fig. 1and thus one must describe only those modes that havemost significance to the stick-slip dynamics.

4.2.3 Eigensolutions

Eigensolutions yield the natural frequencies,damping ratios and mode shapes under the slipping andsticking system conditions. These are given in Table 3;for further discussion on these modes see Ref. 7.

Table 3—Damped frequencies, damping ratios, mo=200 Nmsrad−1�: a) Under the slipping c

fdiSL (Hz)

iSL

0n/a

Coordinate �0SL �0

SL �

Drive �d� 1.00 n/a 1.Rotor �r� 1.00 n/a 0.Brake �b� 0.00 n/a 0.Tire �t� 1.00 n/a 0.

fdiST (Hz)

iST

2.389.45%

Coordinate �0ST �

Drive �d� 0.33 −3Rotor/Brake �r/b� 0.25 −3

Tire �t� 1.00 −3

Noise Control Eng. J. 56 (4), July-Aug 2008

5 NON-LINEAR FRICTION ANDTRANSIENT EXCITATION MODELS

5.1 Overview

Two braking cases are applied for the transient force.Case I (Fig. 3(a)) is an attempt to approximate thedriver behavior and will see a finite end to motions.Case II (Fig. 3(b)) will see either steady-state stick-sliporbits or steady sliding solutions. Case I simulationsmay be used to examine trends with time domainmetrics for various brake force profiles and some quali-tative comparisons with experimental results are made.Case II simulations demonstrate the effectiveness ofour computational methods and bring an understandingof steady-state stick-slip orbits7. Friction laws, initialforcing functions, initial conditions, and stick to sliptransitions are discussed next.

5.2 Friction Laws

The brake torque is a function of the average radiusof contact, rb, between the brake pad and rotor and

brake friction force, Fb� , t�, as shown in Fig. 1(c). Akey simplifying assumption is that the dry frictionforce is described with Coulomb’s law with theadditional consideration of a static friction coefficient,µs, different from kinetic, µk. Figure 2 illustrates thefriction law for an instantaneous or constant value ofthe brake normal force, Fn�t�. With consideration of thedirection of sliding and the sticking regime, the brakefriction force or torque9 may be expressed with a piece-wise function defined in terms of the slipping velocity,

= − :

nd phase (rad) vectors for the model of Fig. 1 �dttion, b) Under the sticking condition.

.33%

34.85%

95.42%

�1SL �2

SL �2SL �3

SL �3SL

−3.09 0.02 0.11 0.00 n/a−3.07 1.00 −3.09 0.00 n/a

n/a 0.00 n/a 1.00 00.04 0.01 0.01 0.00 n/a

4.634.52%

54.63.25%

�1ST �1

ST �2ST �2

ST

1.00 −3.09 0.01 0.050.08 −3.09 1.00 −3.110.04 0.02 0.00 −0.11

dal aondi

(a)45

1SL

00190013(b)

0ST

.06

.05

.04

r b

247

Page 14: Brake Groan

Fb =�µkFn � 0

µsFn = 0+

FbST

� 0

− µsFn = 0−

− µkFn � 0

�; �6a�

Tb = rbFb�,t� . �6b�

Henceforth = r− b is termed as the relative velocityand its integral/derivative as the relative displacement/acceleration. Further definitions include, Fb

SL= ±µkFn,the slipping friction force, Fb

B= ±µsFn, the breakawayfriction force (or threshold) and Fb

ST, the actual stickingforce or the shear force between sticking elementsduring sticking motions. In terms of the friction torque,given by Eqn. (6b), we have corresponding definitions,Tb

SL, TbB and Tb

ST, for slipping, breakaway and stickingfriction torques, respectively. The slipping condition is

valid for all t when ���0 and the sticking condition is

for all t when �0. The stick to slip transition occursat the instant when the sticking friction force exceedsthe breakaway force (for subsequent analysis we alsodefine here the torque expressions which provideidentical conditions):

�FbST� � �Fb

B� = µsFn; �7a�

�TbST� � �Tb

B� = rbµsFn. �7b�

At some instantaneous time, t, the motions may be atthe stick-slip boundary, which is defined by the follow-

ing, where =0±:

�FbST� = �Fb

B� = µsFn; �8a�

�TbST� = �Tb

B� = rbµsFn. �8b�

Crossing the boundary from the sticking regimerequires only that Eqns. (7a) and (7b) be satisfied.Crossing the boundary from the slipping regimerequires both:

= 0± and �9a�

�FbST� � �Fb

B� = µsFn, �9b�

�TbST� � �Tb

B� = rbµsFn �9c�

Note that FbST is multi-valued when sticking, consider-

ing Eqn. (7a), i.e., FbST� −µsFn�0�µsFn�, and may

lie anywhere on the vertical axis between these points

(Fig. 2). Finding the exact intersection point, =0, forthe above Eqn. (9a), is impractical, whether solving via

248 Noise Control Eng. J. 56 (4), July-Aug 2008

numerical integration or when assembling piecewiseanalytical solutions. Various approaches such as bisec-tion methods10 require a bracket within which a valueclose to zero may be taken as zero. We illustrate thisbracket in Fig. 2, which is referred to as the zero veloc-ity tolerance:

= 0 where � − � � 0 � �� . �10�

In our simulations we apply a time-varying tolerance asa function of the slope of relative velocity.

5.3 Engine Load and Brake ActuationFunctions

The engine torque is assumed as Te=20 N m, whichfor the reduced model (Fig. 1) gives a drive torque,Td=0.5ntrnfdTe=76.8 N m. Vehicle resistance is simpli-

fied with dt, providing a drag torque of −dtt Nm, asshown in Eqns. (4c) and (5c) and as Tv in Fig. 1. Thissimplifies complicating friction laws governing thedynamic interactions between the tire and road surfaceand between axle and wheel bearings. It is consideredthat initially there is sufficient brake torque to hold thevehicle stationary and to aid in formulating brake

forces we introduce Fn=Td /rbµs as the brake slip line.Its intersection with brake force is the time of initialslip.

Brake Force Case I: Two hyperbolic tangentfunctions are assembled together with a delay region.Three parameters, �, � and � respectively control theamplitude, slope, and delay between functions; a fewexamples are illustrated in Fig. 3(a). The first function,

Fn1�t�= Fn−� tanh ��t−��, crosses Fn at its mean aftertime, �, which is related to slope by, �=−0.5�−1 ln��1−���1+��−1�. Once the solution reaches t�2� the

delay function is Fn2�t�= Fn−�� till t reaches 2�+�where a second hyperbolic function is applied as

Fn3�t�= Fn+�−2� tanh���3�+�− t��. This is Fn1

inverted, offset and scaled in amplitude. It brings afairly abrupt end toward the groan events in simula-tions by crossing well above the slip line (consider toothat µk�µs and a brake stick line is less determinable).The abrupt end is similar to the vehicle test experience;the driver could ease the pedal into a groan event butfor the pedal retractions it is harder to control themechanism. The three functions give the brake forcealgorithm as:

Fn�t� = �Fn1�t� 0 � t � 2�

Fn2�t� 2� � t � 2� + �

Fn3�t� t � 2� + �� �11�

Note that to truly control the “length” of the groanevent, the time between crossings of the brake slip line

Page 15: Brake Groan

is a useful quantity to fix or vary, and depends on slopeand delay parameters:

L = 2� + � − 0.5�−1 log��1 + 0.5��1 − 0.5�−1� ,

�12�

The function in Eqn. (11) has limits in smoothness andif values are assigned sufficiently small for � (or suffi-cient large for �) then there will a significant disconti-nuity to the brake force.

Brake Force Case II: Here the force (Fig. 3(b))crosses the brake slip line near the maximum of thehyperbolic function. Parameters � and � still controlamplitude and slope, however, Case I and Case IIprofiles are quite different for the same � and �. It is

intended that the function is applied as 0�Fn�t�� Fn

+� for all t and Fn�t� approaches a finite value as t→� so as to lead to steady state solutions of some type.Specific details are provided in our recent article7.

5.4 Initial Conditions

Initial conditions are determined so that at t=0 s thesystem has no motion and is under the stick state, i.e.with matrices Eqns. (5a)–(5d) and coordinate vector�= d r/b t�T. Given the above mentioned forcingfunctions the initial torque vector is T�0�= �Td 0 0�T,yielding initial conditions,

��0� = K−1T�0� , �13a�

��0� = 0. �13b�

6 COMPUTATIONAL ISSUES

6.1 State Vectors and Stick-Slip Transitions

An algorithm for stick-slip transitions is proposednext. Two Runge-Kutta solvers with adaptive step sizecontrol are used. These are designated in this paper as“RK23”11 and the higher order “RK45”12, which utilize2/3 and 4/5 orders respectively. The governing secondorder differential equations are reduced to the

first order form as, U=BU, where U

= d r b t d r b t�T. Matrix B is not calculatedexplicitly, rather each equation is determinedseparately. On each successful time step, tn, the solver

returns values of U�tn�, then velocities within U are set

as Ui�tn�=Ui+4�tn�, i=1 to 4. Under the slipping condi-

tion, accelerations, Ui�tn�, i=5 to 8, are determinedwith the system described by Eqn. (3) with Eqns.(4a)–(4d). For the sticking condition they are deter-mined in similar fashion using the expanded form of

Eqn. (3) with Eqns. (5a)–(5d), for d and t. So as tonegate the need for applying the brake/rotor displace-

Noise Control Eng. J. 56 (4), July-Aug 2008

ment offset (and associated programming), i.e., �t�=0 for all t, the brake and rotor coordinates are keptdistinct, yielding:

r = b = �Jr + Jb�−1�cd�d − r� − ct�r − t� − cbb

+ kd�d − r� − kt�r − t� − kbb� �14�

Since the rotor and brake have the same acceleration,

b= r and velocity, b= r, during the sticking motions

these two components of U are equal. However, theirdisplacement solutions remain distinct and thisapproach allows the fixed dimension of U for stickingand slipping states. During sticking motions thesedisplacements will change at the same rate and thus

=r−b holds. Programming the solution in such away may slightly increase the computational time ofthe solver, but significantly decreases the programmingcomplexity; the solver does not need to stop and start ateach crossing of the stick-slip boundary to change thedimension of U. Additionally, the small computationalcost associated with writing and storing a separateoverall solution vector and in initializing the solver ateach transition is avoided.

6.2 Algorithm for Stick to Slip Transitions

Assume the system in stick at t0=0 s and record itsstate as V�t0�=0. First, solve the sticking systemequations until the sticking torque is greater than thebreakaway torque, i.e., Condition given by Eqn. (7b).Then switch to the slipping system equations. Knowing

=0 during sticking allows the sticking torque to bedetermined via subtraction of slipping equations forrotor and brake as:

= 0 = r − b =1

Jr�cd�d − r� − ct�r − t� + kd�d

− r� − kt�r − t� − TbST� +

1

Jb�kbb + cbb − Tb

ST�

�15�

Introduce an inertia ratio parameter, �=Jr / �Jr+Jb�,and rewrite Eqn. (15) as:

TbST = �1 − ���cd�d − r� − ct�r − t� + kd�d − r�

− kt�r − t�� + ��kbb + cbb� �16�

As stepping time is reduced, �t→0, the accuracyimproves for predicting the crossing of threshold givenby Eqn. (7b). Thus the employment of a fine time stepis essential. A measure of error for how far the predic-tion exceeds the threshold is specified as ESL�r�=1− �Tb

ST�trSL−�t� /Tb

B�trSL−�t��, where tn= tr

SL are the firstslip time(s) after the transition. Note that the state is not

249

Page 16: Brake Groan

changed until the step after crossing the threshold; thisis discussed in Sec. 6.4. We define an acceptable boundwithin which this error must lie, −�SL�ESL�r���SL�and instruct the solver to either stop or leave a flag ifESL falls out of the bound.

Once the system reaches the stick-slip transition,several considerations must be made. First, simulations

with �0�=0 will have the sign of the sliding frictiontorque undetermined for the initial slip step, as

sgn��0��=0. However, from the condition given byEqn. (7b) we know which direction the solution hastaken through the stick-slip boundary, i.e. Tb

ST�trSL

−�t��TbB�tr

SL−�t�=rbµsFn for the positive boundaryand Tb

ST�trSL−�t��−Tb

B�trSL−�t�=−rbµsFn, for the

negative boundary. Hence the first slip step after transi-tion requires the brake friction torque to be assigned as:

Tb�trSL� = sign�Tb

ST�trSL − �t��rbµkFn �17�

For later stick-slip transitions, the relative velocity isconstant before slipping, but will not be zero, as at thelast instant of slip (before the prior stick) it fell within

the tolerance −����. This residual value may bemisleading to the actual direction of transition throughthe boundary at the first instant of slip, hence Eqn. (17)is still applied in the same manner. Residual values arebest left unchanged, rather than set to zero; any changewill add/subtract small energy to the system throughthe damping parameters. The next consideration is thatthe zero velocity tolerance must be set as �=0, foreven though the next step will give relative velocity,

�trSL+�t��0, this value may still lie within the

bracket, −���tn����. This would satisfy the firstitem of the slip to stick transition, condition given byEqn. (9a), and if Eqn. (9c) is also satisfied then thesystem will immediately return to the sticking state.This results in an unsatisfactory stick-slip “chatter,”repeating at the period, �t, until clear sticking orslipping motions are achieved. This problem is resolvedby temporarily setting the velocity tolerance as �=0.At some time after transition, the tolerance needs to bereturned to a non-zero value, otherwise the solution

will never find the approximate =0 intersection. If therelative velocity increases/decreases after breakaway, itnaturally must decrease/increase again before reaching

=0. Hence as t progresses we determine a switching

value, S�tn�=sgn��tn−1��tn��. A value S�tn�=1

indicates is increasing/decreasing and some at tn, the

switch becomes S�tn�=−1, indicating the step where changes direction. The zero velocity tolerance is thenreset as ��0 at this step and the switch is not queriedagain until after the next stick-slip transition.

250 Noise Control Eng. J. 56 (4), July-Aug 2008

6.3 Algorithm for Slip to Stick Transitions

Once the system is clearly slipping the stateis recorded as V�tn�=1, i.e. slip. Considering that� is switched back on, we must assign a value.One method is to choose a fixed value, run solutions,

check the intersection =0 is not overshot, reduce�t if necessary and repeat until acceptable solutionsare reached; this is a time consumingapproach. Instead, we define a time-varying tolerance,

��tn�=0.5�tn−1��tn− tn−1�, representing approximately

half the change in between previous and currenttime steps. The solution now cannot cross over

=0 without falling within the bracket,

−0.5�tn−1��tn− tn−1���0.5�tn−1��tn− tn−1��. Initialstick times are defined as tn= tm

ST, the residual values are

defined at these times as ST�m�.

6.4 Critical Computational Considerations

A critical consideration is to record prior states,

V�tn−1�, switches and values; �tn−1� and tn−1, for priorsuccessful steps and but not for mid-interval or unsuc-cessful steps. The methodology given is only appropri-ate to single step solvers, multi-step methods will

require V and values for additional times such as tn−2,tn−1, tn and tn+1, bringing additional complications.Further, improvements can be made to results andspeed by careful judging of timing for state transitions.For transitions in state found during evaluations ofmid-interval steps, it is best to force that evaluation andthose subsequent to be for the original state. The transi-tion is recorded and then applied after the step has beensuccessful. This is most easily explained with anexample using the classical second-order Runge-Kutta(midpoint) method,10 where h is the interval, k theestimation and O�hn+1� the error term:

k1 = hf�xn,yn� , �18a�

k = hf�x + 0.5h,y + 0.5k � , �18b�

Fig. 19—Undesirable (a) and desirable (b) Transi-tions in state.

2 n n 1

Page 17: Brake Groan

yn+1 = yn + k2 + O�h3� �18c�

With reference to Fig. 19(a), examine a transition fromslipping to sticking (note that the equivalent parameters

for Eqn. (3) would be y= and h=�t). The first estima-tion, k1=hf�xn ,yn� over the whole interval, h, is for theslipping state, with a certain slope of the derivative, asindicated in Fig. 19(a) by the arrow at �xn ,yn�. Theestimation is applied to a trial step in the midpoint ofthe interval, yn+0.5k1. This trial step now falls withinthe zero velocity tolerance and the midpoint derivativeis evaluated for sticking as f�xn+0.5h ,yn+0.5k1�=0(assuming a sticking is satisfied). This yields�xn+1 ,yn+1�= �xn+h ,yn�, consequently the stickingtransition has been found without actually having a

0.0177 0.0178 0.0179 0.018

72

72.5

73

0 0.01 0.02 0.03 0.04 0.05

40

50

60

70

80

Torque(Nm)

TbB

TbB

TbST

TbSL

slipstick

TbSL Tb

ST

TbST(RK45)

TbST(RK23)

(RK23,45)

last stick times

Torque(Nm)

Time (s)

(a)

(c)

Fig. 20—Comparison of transition points predictedTb

B�t�, brake friction torque, Tb�t�, compos

torque, TbSL�t�; (b) Relative Velocity, and

by 50); (c, d) Enlargement for critical regtion is for Case II brake rorcing with �=3RK45, solid—RK23).

Fig. 21—Prediction of state transition points for 0�suring overshoot of breakaway torque thr

ues �m� at initial stick times, tST.

ST m

Noise Control Eng. J. 56 (4), July-Aug 2008

component of y fall within the tolerance. Additionallythe two derivatives are non-smooth over the interval h.In practice, the Runge-Kutta routines (in Matlab) useadaptive step size control and if mid-interval statetransitions occur then the solutions most often fail tomeet solver error tolerances. Then the step size getssmaller, yet the same set of circumstances can repeat;the result is the step size becoming unnecessarily smallbefore the solution is accepted, hence wasting compu-tational effort. Figure 19(b) illustrates how this isavoided. Here the midpoint derivative is evaluated forthe original state, but the transition is held off till thenext step. The solution is smooth over the interval h,giving �xn+1 ,yn+1�= �xn+h ,yn+k2� and then stickingapplies to all evaluations for yn+2. The solution is still

0 0.01 0.02 0.03 0.04 0.050

0.2

0.4

0.6

Velocity(rad/s)

0.02625 0.0263 0.026350

1

2

3

4

5x 10

-3

Time (s)

δ.

δ.(RK45) (RK23)

εδ.

δε.

RK45osses tolerance

RK23crosses tolerance

δ.(RK23,45)

ε .δ x 50

Velocity(rad/s)

(RK23)(RK45)

slip slipstick

(b)

(d)

unge-Kutta methods: a) Breakaway torque,f sticking torque, Tb

ST�t� and slipping friction

-varying zero velocity tolerance, � (multipliedin (a, b), here � is not multiplied by 50. Simula-, �=0.4 s, µk=0.4 and µs=0.6. (Key: dotted—

tf: a) Error estimation parameter, �EST�r��, mea-ld at last stick times, tr

ST−�t and b) Residual val-

Angular

cr

Angular

by Red o

timeions0 N

tn�esho

251

Page 18: Brake Groan

non-smooth over two-intervals of h and the new stepwill likely fail to meet tolerances of error, but the solverwill carry on after only a few reductions in h, as themid-interval transitions are avoided. Additionalcomputational switches need to be implemented forthis forcing of stick or slip state.

6.5 Adaptation to Various Initial Conditions

The above discussion has been framed around thegiven initial conditions and application to alternateconditions or transitions in time may be made. For thisstudy, commencing in the stick state the initial condi-tions yield a solution with a known characteristic to notimmediately slip. There are no special considerations,however by setting V�t0�=0 the original state is

recorded. In the case of initial conditions where =0

0 0.5 1 1.5 2 2.50

0.2

0.4

Time (s)

δ(rad/s)

.

(a)

0 0.05 0.1 0.15 0.2 0.250.250

0.15

0.30.3

1.9 2 2.10

0.15

0.30.3

Time (s)Time (s)0 0.05 0.1 0.15 0.2 0.250.250

0.15

0.30.3

. δ(rad/s)

δ(rad/s)

.

(b) (c)

0 0.05 0.1 0.15 0.2 0.250

50

100

Time (s)

Torque(Nm)

1.9 2 2.10

50

100

Time (s)

Torque(Nm)

TbB

TbB

Tb Tb

(d) (e)

Fig. 22—Sample simulation for Case I brake forc-ing, �=300 N, �=50 and L=2 s: a)Sticking and slipping velocity; b) Initialtransient of sticking and slipping veloc-ity on brake release; c) Final transient ofsticking and slipping velocity on brakerise; d) Initial brake force,Tb (composedof sticking and slipping torques) vs.breakaway force, Tb

B; e) Final brakeforce, Tb (composed of sticking and slip-ping torques) vs. breakaway force, Tb

B.Note the central region of (a) featuressteady-state stick-slip orbits for a finitetime.

252 Noise Control Eng. J. 56 (4), July-Aug 2008

with immediate slip, the knowledge of state V�t0� at t1

is important to ensure �=0 until the time-varyingtolerance is switched on. For initial conditions startingin the slip state, V�t0�=1 and an initial value for zero

tolerance must be set as ��t1���0�. After the firststep � is switched back to the time-varying tolerance.

6.6 Comparison of Lower and Higher OrderRunge-Kutta Routines

Here we compare the stick-slip transition pointsusing the lower and higher order Runge-Kutta routines.Solutions are run until steady state orbits for stick-slipmotion are found. Both solutions use a maximum timestep of �tmax=2−15 s; which although Matlabdocumentation13 recommends using as a last resortwith their solvers, is helpful for accurate predictions ofstate transitions. All results in this paper use this �tmax

and while variable stepping is applied by the solvers wespecify output times for later frequency domain analy-sis at �t=�tmax. For an example case (Fig. 20) theRK23 ran nearly 2.1 s and RK45 nearly 4 s. Fromvisual inspection the solution could have been consid-ered steady-state after 1.6 s. The higher order solvertook much longer to meet the strict conditions forsteady-state stick-slip.7 To compare typical orbits weshift the start of the second last orbit back to t=0 s andprovide overlays of the results for the last two orbits.Figure 20(a) provides Tb�t� and the enlargement (Fig.20(c)), shows Tb

ST�t� (in stick) approaching the break-away threshold, which at the point exceeding thethreshold drops to Tb

SL�t�. Both lower and higher ordermethods find the threshold crossing with reasonableaccuracy. Figure 21(a) provides the error values, EST�r�,plotted versus solution time, at times immediatelybefore slipping, tn= tr

SL−�t. The achievement of EST

�0.41% and mean values, EST=0.186% (RK23), EST

=0.206% (RK45), shows reasonable bounds for error.

Figure 20(b) provides �t� along with ��t��50 (to beon the same scale). The enlargement (Fig. 20(d)) illus-

0 0.5 1 1.5 2 2.5

-0.5

0

0.5

θ r(g)

0 0.5 1 1.5 2 2.5-2

0

2

θ b(g)

Time (s)

........

Fig. 23—Sample simulation for Case I brake forc-ing, �=300 N, �=50 and L=2 s: Accel-eration signature showing entire groanevent. Note the similarity in shape toEvent F of Fig. 5.

Page 19: Brake Groan

trates the relative velocity approaching the zero cross-

ing and the last few variations in �. When fallswithin the bracket it flattens to the residual value,

ST�m� and �t�= ST�m� for tmST� t� tr

SL. The ST�m�values are given in Fig. 21(b) for all m (all slip-sticktransitions). Notice the maximums have significant

difference, ST�m�=1.57�10−3 (RK23) and ST�m�=2.18�10−3 rad/s (RK45), although either one isacceptable considering the orders of magnitude differ-

ence to peak relative velocity, �0.55 rad/s, of the

orbit. The mean values are found as ��m�=5.55

�10−4 (RK23) and ��m�=6.31�10−4 rad/s (RK45). Itis clear that the two methods yield similar results, withlittle divergence over the entire solution. The lowerorder �2/3� method is indeed faster and as it performsbetter for all of the above indices is used for furtherstudies.

7 CONCLUSION

Brake Force Case I (Fig. 3(a)) is designed tosimulate the driver input in the experiment while allow-ing for parameters to describe amplitude of brakerelease, rate and the delay before brake reapplication.The actual input has not been measured; this function isan approximation. Responses for a sample simulationare shown in Figs. 22 and 23, where �=300, �=50 andL=2. Figure 22(a) shows the relative velocity betweenbrake and rotor (slipping velocity) for the entire groanevent. Stick-slip responses are evident. Figures22(b)–22(e) plot together the initial and final transientsof velocity and brake torque. With reference to condi-tional statements Eqns. (7)–(9), the brake is initially

sticking, =0, with sticking torque, Tb=TbST�Tb

B, thenat t�0.05 s the actuating brake force is reduced to the

point where TbST�Tb

B giving an initial slip, �0 andnow Tb=Tb

SL. Accordingly the torsional responses atslipping frequencies (an initial value problem) bring areturn of slipping velocity at t�0.06 s to within toler-

0 0.5 1 1.5 2 2.5-0.5

0

0.5

θ r(g)

..

0 0.5 1 1.5 2 2.5

-101

θ b(g)

Time (s)....

..

0 0.5

-0.5

0

0.5

θ r(g)

0 0.5-2

0

2

θ b(g)

....

(a)

Fig. 24—Acceleration signatures for sample simula=50 and L=2 s; b) �=225 N, �=50 and

Noise Control Eng. J. 56 (4), July-Aug 2008

ance of zero velocity, i.e. −����. The stickingtorque is within the breakaway value, Tb

ST�TbB so the

bodies stick again and now =0 and Tb=TbST. This first

stick is short, however, the motions fall into a steadyorbit by t�1 s, with sticking period, �ST=0.0196 s,slipping period �SL=0.0076 s and stick-slip period �SS

=0.0272 s. The motions are periodic where at eachtransition there is an initial value problem for responsewith a certain homogeneous and particular solution.These solutions are governed by Eqn. (3) with the

values and at transition points and the shape of theresponse dependent on the modal properties of slippinggiven by Eqns. (4a)–(4c) and sticking given by(5a)–(5c) matrices and Te, Tb and Tv. Referring Figs.22(c) and 22(e) the rise in brake force (on reapplicationby the driver) sees an abrupt end to the orbit and the

brake then sticks for all t and now =0 and Tb=TbST.

Figure 23 is for qualitative comparison to Event F (andsized as such). The overall shape compares well but thisis not to say a direct comparison may be made as thesystem parameters and forcing conditions are notmatched. The simulated stick-slip frequency is 37 Hzcompared to the dominant 76 Hz inferred from experi-ment. Yet, the forcing conditions can lead to differentstick-slip frequencies7. Other researchers have reportedstick-slip frequencies in this order �30–60 Hz�4,5 fortest and simulation. Some simulation results showmore of a transient tail (than an abrupt end), forexample see Fig. 24 with three variations of �, � and L.This was also seen in some of experimental data (e.g.Event I of Fig. 5). Influences on this shape include bothdamping and the timing of the stick-slip response withrespect to the brake force rise (see Fig. 22(e) and notethat the brake nearly slipped again at t�2.06 s)

Figure 25 shows the acceleration spectra for therotor and brake, calculated from a steady-state portionof the signal with 19 stick-slip cycles for a samplelength ts�0.5. The brake groan, in both simulation andexperiment (see Figs. 11 and 13 for experiment) exhib-

1.5 2 2.5

1.5 2 2.5(s)

0 0.5 1

-0.50

0.5

θ r(g)

0 0.5 1-2

0

2

θ b(g)

Time (s)

....

(c)

s for Case I brake forcing with a) �=550 N, �s and c) �=550 N, �=10 and L=0.45 s.

1

1Time(b)

tionL=2

253

Page 20: Brake Groan

its a primary stick-slip frequency and then multipleharmonics. For the simulation we know the harmonicsare due to discontinuities in calculated accelerationschanging at stick to slip and slip to stick points. We caninfer that the same behavior is occurring in the experi-ment.

Brake Force Case II has Fn�t� approaching a finitevalue as t→� (see Fig. 3(b)), hence solutions fall intoa steady-state motion; either stick-slip orbits or steadysliding. Four different motion types are illustrated (Fig.26), each has different slipping, sticking and stick-slipperiods, which lie within defined ranges and may varywith � and � (see Ref. 7). The stick-slip frequenciesrange from 25–95 Hz. In these simulations, for a given�, the slope, governed by �, was shown to be able toinfluence which orbit was found due to variation in thetransient solution flow leading to the orbit. The timedomain result of Figs. 22 and 23 (of Brake Force CaseI) features a finite portion of Type I Motions as illus-trated by Fig. 26(a). For the experiment, thetime-frequency content (Fig. 15) of the measured groanevents A-I do not indicate any significant variation inthe dominant harmonic (stick-slip frequency), it couldbe construed that the real system is operating within thebounds of a condition where there is only one motiontype. This however is for the successful inducement ofgroan in a test; if the driver releases the pedal enough asteady-sliding (Fig. 26(d)) motion is quickly obtained.Hence at least two motion types are easily identified inthis one experiment. In driving several automatic andmanual transmission vehicles (in at least two conti-nents) we have observed (from our own audible percep-tion) that most vehicles seem to have one tone thoughmultiple tones could be heard depending on the actua-tion (pedal) pressure. It would be of interest to see iffuture test data reveals multiple motion types. Adetailed nonlinear analysis of the manual transmission

0 36.7 50 100

10-4

10-3

10-2

10-1

100

Acceleration

Magnitude(g)

0 36.7 50 100

10-4

10-3

10-2

10-1

100

Acceleration

Magnitude(g)

F

Fig. 25—Acceleration spectra for stick-slip cycles fwith �=300 N, �=50 and L=2 s): a) Ro

254 Noise Control Eng. J. 56 (4), July-Aug 2008

equipped vehicle is also needed; here investigation iswarranted for brake groan driven by mass, particularlyin slow braking while reversing down a steep slope.

150 200 250

150 200 250ncy (Hz)

(a)

(b)

in a simulated groan event (brake forcing Case Icceleration; b) Brake acceleration.

-1 -0.5 0 0.5 1

x 10-3

0

0.1

0.2

0.3

0.4

0.5

δ - θt (rad)

δ(rad/s)

.

slipstick

-1 0 1 2

x 10-3

0

0.1

0.2

0.3

0.4

. δ(rad/s)

δ - θt (rad)

stick

slip

-1 0 1 2

x 10-3

0

0.1

0.2

0.3

0.4

0.5

. δ(rad/s)

δ - θt (rad)slip

stick

-1 0 1 2

x 10-3

0

0.1

0.2

0.3

0.4

0.5

. δ(rad/s)

δ - θt (rad)

(a) (b)

(c) (d)

Fig. 26—Phase planes for transient simulationsleading to stick-slip orbits with Case IIbrake forcing and �=0.4 s, µk=0.4 andµs=0.6: a) Type I motion with �=30 N;b) Type II motion with �=80 N; c) TypeIII motion with �=200 N; d) Type IVmotion (steady sliding) with �=300 N.Key: X marks initial conditions; Mediumthick line for 0� t� t2

ST; Thin line for t2ST

� t� tST ; Thick line for tST � t� t .

reque

oundtor a

f−1 f−1 f

Page 21: Brake Groan

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3. J-K. Kim, C-K. Lee and S-S. Kim, “Investigation on Correla-tion between Groan Noise and Coefficient of Friction with Fric-tion Material’s Difference”, SAE Technical Paper 2004–01–0827, (2004).

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5. M. Donley and D. Riesland, “Brake Groan Simulation for aMcPherson Strut Type Suspension”, SAE Technical Paper2003–01–1627, (2003).

6. J. A. C. Martins, J. T. Oden and F. M. F. Simoes, “Recent Ad-vances in Engineering Science—A Study of Static and Kinetic

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Friction”, Int. J. Eng. Sci., 28(1), 29–92, (1990).7. A. R. Crowther and R. Singh, “Analytical Investigation of Stick-

Slip Motions in Coupled Brake-Driveline System”, NonlinearDyn., 50(3), 463–481, (2007).

8. A. R. Crowther, R. Singh, N. Zhang and C. Chapman, “Impul-sive Response of an Automatic Transmission System with Mul-tiple Clearances: Formulation, Simulation and Experiment”, J.Sound Vib., 306, 444–466, (2007).

9. S. W. Shaw, “On Dynamic Response of a System with Dry Fric-tion”, J. Sound Vib., 108(2), 305–325, (1986).

10. W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flan-nery, Numerical Recipes in Fortran, Cambridge UniversityPress, (1992).

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12. J. R. Dormand and P. J. Prince, “A Family of Embedded Runge-Kutta Formulae”, J. Comput. Appl. Math., 6, 19–26, (1980).

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255