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Research ArticleSensing and Quantifying a New Mechanism forVehicle Brake Creep Groan

Dejian Meng 1 Lijun Zhang 1 Xiaotian Xu2 Yousef Sardahi2 and Gang S Chen 2

1School of Automotive Studies Tongji University Shanghai 201804 China2College of IT amp and Engineering Marshall University Huntington 25755 WV USA

Correspondence should be addressed to Lijun Zhang tjedu_zhanglijuntongjieducn

Received 29 August 2018 Revised 12 January 2019 Accepted 7 February 2019 Published 26 February 2019

Academic Editor Adam Glowacz

Copyright copy 2019 Dejian Meng et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

is paper investigates the creep groan of a vehiclersquos brake experimentally analytically and numerically Experimentally the effects ofacceleration on caliper and strut noise brake pressure and tension are measured e results show that the measured signals andtheir relevant spectra broadly capture the complex vibrations of creep groan is includes the simple stick-slip severe stick-slipvibrationsresonances multiple harmonics half-order harmonics stick-slip-induced impulsive vibrations steadyunstable vibra-tions and their transitions Analytically a new mathematical model is presented to capture the unique features of half-orderharmonics and the connections to fundamental stick-slipresonant frequency and multiple harmonics e analytical solution andthe experimental results show that the vibro-impact of the brake pad-disc system can be triggered by severe stick-slip vibrations and isassociated with instable impulsive stick-slip vibration with wideband e induced stick-slip vibro-impact can evolve into a steadyand strong state with half-order stick-slip fundamental and multiple-order components is new mechanism is different from allpreviously proposedmechanisms of creep groan in that we also view some type of creep groan as a stick-slip vibration-induced vibro-impact phenomenon in addition to conventional stick-slip phenomenae newmechanism comprehensively explains the complexexperimental phenomena reported in the literature Numerically the salient features of phase diagrams of instable stick-slip andvibro-impact are examined by using a seven-degree-of-freedom brake system model which shows that the phase diagrams of thedynamics of creep groan with and without vibro-impact are substantially different e phase diagram of the dynamics with vibro-impact is closer to the experimental results In contrast to existing mechanisms the proposed new mechanism encompasses theinstable stick-slip nature of creep groan and elaborates the inherent connections and transition of the spectrogram e newknowledge can be used to attain critical improvements to brake noise and vibration analysis and design By applying the proposednew model in addition to existing models all experimental phenomena in creep groan are elaborated and quantified

1 Introduction

Noise vibration and harshness (NVH) are important fac-tors for customersrsquo rating of vehicles In vehicle brake NVHproblems creep groan is a specific brake noise that usuallyoccurs at low wheel speed and low brake pressure [1ndash5] erange of the vibratory frequency of the creep groan is usually50 to 500Hz Due to the dramatic increase in the number ofcomplaints about brake creep groan much attention hasrecently been paid to this problem in the automotive in-dustry To understand the mechanism and find the coun-termeasures of brake creep groan much research has beenconducted on vehicles [3ndash5] brake dynamometers [6ndash8]

tribometers [9ndash12] and chassis dynamometers [13] Creepgroan-related vibrations from the caliperknuckle could betransferred to the lower arm and then to the frame front andthe rear MTG to body or be transferred to the strut and thento the body giving rise to an uncomfortable structurallyborne noise In addition to the brake system both thesuspension system and driveline system have effects on theoccurrences and features of creep groan [4 8 9 13]

Experimentally researchers have studied varied creepgroan phenomena and the effects of structural parameters oncreep groan such as sliding stick-slip [7 12 14] torsionalstick-slip of torque [15] tangential or torsional stiffness[9 10] and variations of caliper suspension chassis and

HindawiShock and VibrationVolume 2019 Article ID 1843205 10 pageshttpsdoiorg10115520191843205

drivelines [12 13] e tribological parameters of pad-discinterface are also characterized in terms of varied frictionmodels such as LuGre and the bristle friction law in additionto the Coulomb and Stribeck model with varied slopesmagnitudes and gap of static and dynamic friction co-efficients [16ndash23]

Analytically and numerically different models havebeen developed to characterize creep groan Conven-tionally friction oscillator models with a single degree offreedom (DOF) and two DOFs have been used as a calipermodel with a contact sliding interface to quantify vibra-tions [17ndash19] Usually two major subsystems are includedin groan analytical models e nonrotating substructureincludes the brake mounting brackets and suspensionsystems e rotating structure is the drive train sideincluding the rotor wheel and tire In general allsuspension-type modes could exhibit resonances in thecreep groan range By considering the components of thebrake and driveline and even the tire and vehicle massthree four five and seven-DOFs models have been used toinvestigate varied aspects of brake friction vibrations[20ndash22 24ndash26] e multibody dynamic models of thechassis corner have also been used [27] Moreover finiteelement (FE) models of the disc brake for creep groananalysis have been proposed to quantify creep groan[28ndash30] FEA models have advantages in the calculation ofthe interface contact as well as multiple elastic subsystemsincluding pad caliper suspension and driveline systemsHowever FE models of creep groan have not been suc-cessfully used to elaborate the complex mechanisms ofcreep groan except for evaluating the effects of structuraldesign change on vibrations In fact for this kind ofproblem of elastic bulky components connected by rela-tively compliant parts such as springs finite elementanalysis may not be the most suitable tool for simulatingproblems due to their weakness dealing with ldquoweak con-nectionsrdquo among different components Possible ill con-ditioning due to large differences in the stiffness of variedcomponents and interconnections can degrade the accu-racy of the results Moreover the modeling of interactingcomponents in low-frequency events in an elastic modelwith huge numbers of elements and nodes is far fromefficient and the data interpretation is very difficult Assuch theoretical models with varied DOFs have been usedto classify varied mechanisms and resources of low-frequency vibrations [31ndash33]

e sensing and evaluation of ldquocreep groanrdquo noise hasbeen a challenge for the NVH community e creep groanis usually not a purely tonal sound although ultimately itcould generate a tonal subjective perception of users [34]e vibrations resulting in creep groan noise have con-ventionally been considered friction-induced vibrations ofthe stick-slip mode characterized by simple phase diagramsof velocity and displacement of caliper motions Howeverthe recorded vibration usually does not exhibit simple stick-slip periodic vibration but rather exhibits complex prop-erties with multiple frequencies time-varying frequenciesand their transitions e real creep groan has been found toexhibit complex motions in addition to typical stick-slip

motions which include single stick-slip continuous stick-slip resonance varied complex motions and transitions[7 9 11 15 20 30] Basically stick-slip is a sustainedldquoattach-detachrdquo vibration process that arises via specificfriction properties specific system elastic properties andrelative motion [31 35] In particular negative slopes offriction-velocity curves yield equivalently negative dampingto system dynamic motion and could cause system in-stability and a self-excited vibration which are likely lockedin certain natural frequencies or resonances e spectralcontent of the creep groan usually shows a response atmultiples of a certain fundamental frequency of stick-slip orits induced resonance e stick-slip vibrations may occur atthe 1st 2nd or 3rd order of the fundamental frequencycorresponding to the sustained ldquostick-sliprdquo cycle e stick-slip motions mainly depend on the properties of the me-chanical system and are also related to different charac-teristics of the interface friction [36ndash38] However existingtheories and models are unable to explain these phenomenacomprehensively and consistently For example in the eventof creep groan there could be half-order components inaddition to fundamental components corresponding to thestick-slip repetition frequency and the multiple components[16 21 24 30]e root causes of the half-order componentsand their transitions with other components have not beenexplained e inherent connections among fundamentalstick-slip multiple harmonics half-order component res-onance impulsive components of wideband and transitionshave not been explained consistently

In this study a vehicle road test of creep groan under thedownhill condition is conducted and the varied complicatedtypes of vibrations in creep groans are comprehensivelyrecorded which consist of simple stick-slip motioncontinuousdiscontinuous stick-slip motions and slidingimpulsive motions with varied spectrum signatures fromhalf-order fundamental order multiple orders time-varying spectrum transition and wideband componentsA new mechanism of severe stick-slip motion-inducedvibro-impact is proposed to interpret the occurrence andthe transition of the half-order vibrations and a theoreticalvibro-impact model with transient pad-disc separation isapplied for the analysis

Conventionally the phase diagram of brake vibrationwith limit cycles has been used as experimental evidence ofthe stick-slip mechanism in creep groan In this study thephase diagram of experimentally recorded brake vibrationsis found to exhibit a complex unsymmetrical pattern Tocharacterize this a seven-DOFs model with and withoutvibro-impact effect were used for simulations to derivephase diagrams which showed that only the model with thevibro-impact effect gives rise to an unsymmetrical phasediagram

Creep groan has been investigated widely and manymechanisms have been proposed in the last two decades Incontrast to all of these existing mechanisms the proposed newmechanism can encompass the transient vibro-impact natureof a braking process which is different from conventionalexcitation resources such as periodic torque stick-slip neg-ative friction-velocity gradient and resonance e severe

2 Shock and Vibration

stick-slip vibration-induced interface separation gives rise tounique vibrational features of vibro-impact such as half-ordercomponents is mechanism is different from all previouslyproposed mechanisms in that we also view severe creep groanas a stick-slip-caused vibro-impact phenomenon in addition toconventional stick-slip phenomena is research offers someinsights to guide the analysis of brake creep groan

2 Vehicle Brake System and Experiments

21 Experimental Test Setup Based on our existing un-derstanding when creep groan occurs the vehicle-drivingforce does not necessarily come from the engine is maycome from the inertia force when the vehicle is downhill Inmost existing research the vehicle tests were conducted onhorizontal road when the engine was running e enginecould induce some extra vibration in the suspension whichaggravates the complexity of creep groan motion To reducethe effect of the engine in this study a vehicle road test ofcreep groan was conducted under a downhill vehicle con-dition In the experiment an A-class car is used which isequipped with six-speed automatic transmission and hasMcPherson suspension in front and torsion beam suspen-sion in the rear e total mass of the vehicle is 1431 kgincluding two passengers in the front seats

e vehicle is put on a ramp road with a slope of 10edriver presses the brake pedal and starts the engine releases thehandbrake and hangs in the N gear and finally releases brakepedal slowly until the vehicle starts to moveWhen creep groanoccurs the brake pressure is maintained to prevent creep groanfrom occurring if possible

22 Measurement Instrumentation Two triaxial acceler-ometers are mounted at the piston side of the caliper andsuspension strut respectively To induce the effect of addedmass by vibration transducer and its attachment the mass ofone vibration transducer is less than 5 g and each vibrationtransducer is stuck to the surface of the caliper and strut usinga 454 instant adhesive e X Y and Z directions of theaccelerometer on the caliper are aligned to be the tangentialradial and axial directions of the disc respectively An oilpressure sensor connecting the brake tube with a hose is usedto measure the brake pressure ese sensors are arranged inthe front left chassis corner of the car as shown in Figure 1e sampling frequency of the vibration signal is 5120Hz

23 Experimental Results Existing research has found thatthe specific stick-slip vibrations of brake calipers are theindex of the occurrence of groan and have a good corre-lation with the subjective rating of creep groan noise In thisstudy all kinds of caliper vibrations associated with creepgroan noise are recorded in the test and analyzed usingadvanced methods Figure 2 shows the time history oftypical oil pressure and the vibration accelerations andspectrograms of calipers recorded while severe creepgroan occurs Before severe creep groan occurs simple stick-slip motions were recorded which are quasi-harmonic vi-brations e accelerations in all directions are consistent

except that acceleration in the X direction is larger than thatin the other two directions e result of short-time Fourier-transform or spectrogram of clipper acceleration in the Xdirection is shown in Figure 2(e) Because the interior noiseof creep groan occurs within 500Hz the vibration frequencyis analyzed up to 500Hz in Figure 2(e) As shown in Figure 2when the brake pressure drops to 8 bar creep groan lasts forapproximately 18 s According to the characteristics of ac-celeration in the X direction the whole process is dividedinto seven stages which are respectively named A to G inFigures 2(b)ndash2(d) As seen the characteristics of caliperacceleration in amplitude and frequency evolve complexlywith time when creep groan occurs

e accelerations in stages B and F have smaller am-plitudes than the other stages (A C D and E) and exhibitsimple and clear harmonics that correspond to fundamentalperiodic stick-slip motions (approximately 90Hz) andmultiple harmonic components is type of regular stick-slip has been widely studied and reported in existing re-search is kind of severe vibration is likely to be a systemsʼresonance in which the stick-slip frequency is close or equalto the specific natural frequency of the system

e accelerations in stages A C and E have relativelylarger amplitudes and are half-order of the fundamentalperiodic stick-slip motions (approximately 45Hz) widebandcomponents and transitions in addition to the fundamentalcomponents (approximately 90Hz) and multiple compo-nents is indicates that stick-slip motions have evolved intounstable motions and triggered impulsive effects

e accelerations in stage D have the largest amplitudesand clearly have half-order fundamental periodic stick-slipmotions (approximately 45Hz) fundamental components(approximately 90Hz) and their multiple components issuggests that steady strong and periodic impulsive vibra-tions were formed by the severe stick-slip motion

Table 1 shows the recorded tension in the experimentsWhen creep groan occurs the tension varies obviously emaximum and minimum of the tension variation respectivelycorrespond to the maximum static friction force and dynamicfriction force Based on the radius of the piston equivalentbrake radius level arm and recorded tension the frictioncoefficients are calculated and shown in Figure 3

3 New Model of Vibro-Impact forMechanism of Instable Stick-Slip inCreep Groan

Various stick-slip vibration models of brake creep groanhave been developed and these models can generally beclassified into two categories one is the tangential vibra-tion model and the other is the torsional vibration modelSome research has also considered both the tangential andtorsional vibrations of the brake system in creep groaniswork quantified the influence of system parameters such asspeed and friction properties on tangential and torsionalstick-slip vibrations e existing models for characterizingcreep groan have been focused on the sliding motion be-tween the pad and disc and the associated vibrations of

Shock and Vibration 3

0 2 4 6 8 10 12 14 16 18Time (s)

0

4

8

12

16

Brak

e pre

ssur

e (ba

r)

(a)

Time (s)0 2 4 6 8 10 12 14 16 18

ndash30ndash20ndash10

0102030

X-ac

cele

ratio

n (m

s2 )

A B DC FE G

(b)

0 2 4 6 8 10 12 14 16 18Time (s)


0102030

Y-ac

cele

ratio

n (m

s2 ) A B E F GDC

(c)

0 2 4 6 8 10 12 14 16 18Time (s)


0102030

Z-ac

cele

ratio

n (m

s2 ) A B E F GDC

(d)

0 2 4 6 8 10 12 14 16 18Time (s)

400

5005060

403020100

300

Freq

uenc

y (H

z)

200

100

0

(e)

Figure 2 Test results (a) time history of oil pressure (b) X-acceleration of caliper (c) Y-acceleration of caliper (d) Z-acceleration of caliper(e) spectrogram of caliper X-acceleration

(a) (b)

Tension sensor

(c)

Figure 1 Sensors used to test brake creep groan (a) accelerometer at strut and oil pressure sensor (b) accelerometer at caliper (c) tensionsensor


suspension and driveline systems [20 21 30] However theinstable stick-slips and complex features such as half-orderharmonics observed in the last decade have not beenexplained using existing models [16 21 24 30] To inter-pret the system complex motions in creep groan recordedin stages A C D and E in Figure 2 we propose the fol-lowing mechanism noting that during the whole creepgroan process the oil pressure and speed remain constantor change slightly In stages B and F the strong systemstick-slip vibrations and possibly the resonance of somesuspension subsystems are steadily established and arecharacterized by the fundamental frequency approximately90Hz and their multiple harmonics (eg approximately180Hz or 270Hz) e similar phenomena of severe stick-slip vibrationsresonance have been reported in manyexisting reports is critical state of both severe vibrationsand low oil pressure on the pad is likely to further cause thetransient separation of the pad-disc interface which leadsto vibro-impact vibrations in stage C (similarly A and E)e triggered vibro-impacts exaggerate the vibrations andpossess a half-order component of 45Hz and multiplecomponents At the same time the interface separationclose and vibro-impact cause unstable impulsive excitationof the system leading to the vibrations with a widebandspectrum

is composite eect allows vibrations in stage C that arestronger than those in stage B e systemrsquos stick-slip vi-brations and the induced instable vibro-impact graduallyevolve into a steady state denoted as stage D which hasclearer periodic signature and larger amplitudes Since thewhole process is unstable the vibration instability or slightchanges in oil pressure or speed could allow the system tofurther evolve into stage E (similar to stage C) and nallychange to stage F (similar to stage B)

To further quantify this mechanism we propose thefollowing analytical model Figure 4 shows the schematic

model of the pad-disc interface of an idealized brake Incontrast to all existing studies focusing on sliding andtorsional vibrations we consider the critical vibrationsvertical to the sliding surface which are likely triggeredwhen the system enters the severe stick-slip vibrationsresonance with specic fundamental stick-slipresonancefrequency ω (stage B ω 90 Hz)

Consider the idealized vertical vibration equation of thepaddisk for the contactseparation event or vibro-impacteect due to severe stick-slip vibration with specic fun-damental stick-slipresonant frequency ω

meuroy + k + kc( )y F cosωt yltyc

meuroy + ky F cosωt ygeyc(1)

in which ω could be specic stick-slip fundamental fre-quency or some stick-slip-induced resonant frequency ofsubsystems such as suspension y is the pad displacement inthe direction perpendicular to the disc yc could be assumedto be the root mean square of the roughness of interface k isthe stiness of the pad-caliper system kc is the contactstiness of the pad-disk interface which consists of solidbody stiness and surface roughness stiness characterizedby elasticity and the Hertz contact model and m is the massof the pad e vibro-impact could occur after severe stick-slip is established e simple and severe stick-slips in thebrake system have been widely characterized in existingresearch and will not be proceeded here e stick-slipfundamental frequency of the system is the specic exci-tation frequency to vibro-impact motion Equation (1) is a

Table 1 Friction coecient under 8 bar oil pressure

Times 1 2 3 4 5 6 Average Standard deviationOil pressure (bar) 767 767 765 766 765 765 766 00095Maximum of tension (N) 38829 39104 38242 39320 38795 39853 39024 545Minimum of tension (N) 35141 35603 35930 36047 35413 35850 35664 344Static friction coecient 03987 04014 03936 04041 03991 04102 04012 00056Dynamic friction coecient 03608 03654 03698 03705 03643 03690 03666 00038

0 5 10 15 20 25 30 35 40 45 500365

0370375

0380385

0390395

040405

Relative velocity (mms)

Fric

tion

coef

ficie

nt

Figure 3 Fitted friction coecient vs relative velocity curve

k

yx

m

kc

yc

Figure 4 Schematic model of pad-disk interface of an idealizedbrake


strong nonlinear equation and the solutions can be derivedas follows

(a) Primary resonant solutions

y 2F

ω22 minusω2

1( 1113857m(2Δ + 1)+

3F

4ω2m(2Δ + 1)21113890 1113891sinωt

minusπF

2ω2m(2Δ + 1)cosωt +

F

πω2m(2Δ + 1)

middot 1113944infin

n135

cos(nminus 1)ωtminus cos(n + 1)ωt

nminus

F(ωtminus π)

πω2m(2Δ + 1)

middot 1113944infin

n135

sin(nminus 1)ωt + sin(n + 1)ωt

n

(2)

inwhichω1 km

radicω2

(k + kc)m

1113968ω0 2ω1ω2

(ω1 + ω2) α0 πω0(2ω1) β0 π(1minus (ω2ω1))2and Δ (ω2

1 minusω2)(ω22 minusω2

1)(b) Superharmonic solutions

When the system natural frequency is close to themultiple times for the specific excitation frequencymultiple harmonics are excited

y F 2n2ω2 minusω2 minusω2

1( 1113857

m n2 minus 1( )2ω4

sinωt

+2F ω2

2 minusω21( 1113857

πm n2 minus 1( )2 ω2

1 minus n2ω2( 1113857ω2cosωt

minus 1113944

infin

n234

2F ω22 minusω2

1( 1113857


1 minus n2ω2( 1113857ω2cos nωt

(3)

is could be attributed to the periodic effect ofstick-slip excitation which in principle can be de-veloped into multiple harmonics by using theFourier series concept

(c) Subharmonic solutionsWhen a certain system natural frequency is close tohalf the specific excitation frequency the half-ordercomponent (frequency is half of the specific exci-tation frequency) can be excited

y 4lF

π2mω1 l2 minus 1( ) ωminus lω1( 1113857

middot 1113944infin

n135

cos[((n + 1)ωt)l]minus cos[((nminus 1)ωt)l]n

l 2 4 6

(4)

is solution consists of a dominant component whichis a half-order component with frequency equal to half of thespecific stick-slip fundamental frequency ω2 Notably this

kind of specific response also contains multiple harmonics ofthe half-order

e above analysis suggests that if systemsrsquo verticalnatural frequency is close to the specific excitation frequency(from stick-slip motion) or close to the multiple order or halforder of the specific excitation frequency the system willexperience larger vibrations which correspond to primarysuperharmonic or subharmonic resonance Notably allexisting stick-slip models of creep groan including the mostrecently developed ones with a varied friction model areunable to predict this half-order feature

is analytical result also explains the generation andtransition between half order and fundamental order andmultiple orderse vibrations associated with the half orderin vibro-impact exaggerate the severe stick-slip vibrationsWhen the severe stick-slips are high enough and close to acertain threshold the vibro-impacts could be triggeredmaking the vibrations more severe e triggered vibro-impact could evolve from an unstable state into a steadystate or deteriorate back to pure-type stick-slip motionwithout interface separation

Moreover the unstable vibro-impact tends to impartrandom impulsive excitations to the system whichallows system vibrations to exhibit wideband spectrumfeatures

By integrating the above theoretical observations in-cluding stick-slip vibrationsystem resonance the inducedvibro-impact vibrations and the relevant impulsive effectsthe major spectral signatures and transitions in Figure 2(c)can be fully understood and explained

4 Characterization of Phase Diagrams ofCreep Groan with and withoutVibro-Impact

Creep groan is generally caused by stick-slip motionbetween the disc and pads as a self-excited vibration of thebrake assembly e phase diagrams are widely used tocharacterize creep groan e phase diagram of caliperacceleration with a limit cycle has been considered the firstexperimental evidence of creep groan and stick-slipfriction-induced vibrations is section will furtherquantify the phase diagrams of creep groan dynamicsbased on experiments and numerical analysis of a systemmodel with and without vibro-impact To this end amultiple-DOFs model is used and real measured frictiondata are used in the simulation to minimize the effectsfrom other factors

Creep groan is generally caused by the stick-slip motionbetween discs e phase diagrams with the limit cycles ofbrake calipers have been used as the first experimental ev-idence of the stick-slip mechanism of creep groan In thissection we employ the experimental data for friction tosimulate a seven-DOFmodel as shown in Figure 5 with andwithout vibro-impacts

e parameters are summarized in Table 2 which isslightly different from the previous seven-DOF model [26]


k1 k1dx k2dxc1 c1dx c2dxk2 k1cy k2cy

c2 c1cy c2cy

k3 k1cx k2cxc3 c1x c2xk4 kdxc4 cdxk5 kcxc5 ccxc6 ccy

F 1000N

Ff1 Ff2 μsF

(5)

In the following analysis a nonlinear vibro-impact orthe separation eect between the disc and pad in a brake isconsidered In the dynamic model a spring and a damperare modeled between the mass of the disc and two padsWhen each part in the brake model starts to oscillate there isa severe condition in which the surfaces of disc and pads losecontact and separate e motion equation of the system isgiven as follows

mc euroxc minusccx _xc + c2cx _x2 minus _xc( )minus c1cx _xc minus _x1( )minus kcxxc+ k2cx x2 minusxc( )minus k1cx xc minusx1( )

mc euroyc minusccy _yc + c2cy _y2 minus _yc( )minus c1cy _y1 minusyc( )minus kcyyc+ k2cy y2 minusyc( )minus k1cy y1 minusyc( )

md euroxd minuscd _xd + S δx2( )c2dx _x2 minus _xd( )minus S δx1( )c1dx _xd minus _x1( )minus kdxd + S δx2( )k2dx x2 minusxd( )minus S δx1( )k1dx xd minusx1( )

m1 eurox1 minusF + S δx1( )c1dx _xd minus _x1( ) + c1cx _xc minus _x1( )+ S δx1( )k1dx xd minusx1( ) + k1cx xc minusx1( )

m1 euroy1 minusS δx1( )Ff1 minus c1cy _y1 minus _yc( )minus k1cy y1 minusyc( )m2 eurox2 Fminus S δx2( )c2dx _x2 minus _xd( )minus c2cx _x2 minus _xc( )

minus S δx2( )k2dx x2 minusxd( )minus k2cx x2 minusxc( )m2 euroy2 S δx2( )Ff2 minus c2cy _y2 minus _yc( )minus k2cy y2 minusyc( )

(6)

in which S(δxi)(i 1 2) is a piecewise linear function ispiecewise linear function denes the contact loss betweenthe brake disc and pad as follows

S δxi( ) 1 δxi ltLi

0 δxi gtLi (7)

in which Li represents the free length of the equilibriumspring between disc and pad Since S(δxi) is a discontinuousfunction we need a smooth function to approximate itperfectly us we employ a continuous function with asmooth factor σ

S δxi( ) 12minus tanh

σ δxi∣∣∣∣∣∣∣∣minus Li( )( )2

(8)

where σ is often a large positive number Figure 6 shows theeects of a smooth function with dierent values of σ

As seen in Figure 6 the approximation tends to convergewhen σ 1000 Because the results are sensitive to the in-terface separation approximation we use a value of σ 106for the simulation

Figure 7 shows the phase diagrams from simulations andexperiments Figure 7(a) shows simulation results for the

Knuckle

Caliper

Piston

Disc

Brake pad

Brake fluid

(a)

mc

c2cx c1cxc1dx

c2cy c1cy

ccy

ccx

c2dx

cdx

k2cx k1cxk2dx

kdx

k1dx

k2cy k1cy

kcy

kcx

m2 m1md

v0

F F

Y

X

(b)

Figure 5 Schematic of xed-caliper disc brake (a) and seven-DOF model (b) [26]

Table 2 Parameters used in system model of xed-caliper discbrake

Mass Value (kg) Stiness Value(Nmiddotmminus1) Damping Value

(Nmiddotsmiddotmminus1)m1 035 k1 15times 105 c1 65m2 035 k2 75times 105 c2 30mc 556 k3 75times 105 c3 195md 704 k4 10times 106 c4 40

k5 15times 106 c5 50k6 15times 106 c6 50


brake model without contact loss Figure 7(b) shows thesimulation results for the brake model with contact lossFigure 7(c) is the phase diagram evaluated from measuredvibration data in stage C in Figure 2(b)

Phase diagrams have been used to judge whether thebehaviors of the dynamical system are cyclic Generally ifthe phase plane trajectory is a limit cycle the system is cyclicif the phase plane trajectory forms a ring belt area the system

σ = 1000σ = 100σ = 10

S (δx i)

|δxi| ndash Li040302010 05 06 07 08 09 1

0

02

04

06

08

1

(a)

σ = 10e6σ = 10e5

σ = 10e4σ = 1000

|δxi| ndash Li

S (δx i)

0

02

04

06

08

1

ndash15 ndash1 ndash05 0 05 1 15times10ndash3

(b)

Figure 6 Approximating eects for various values of σ

ndash2 0 2 4 6 8 10 12yc (m)

ndash3

ndash2

ndash1

0

1

2

3

4times10ndash4

times10ndash4

y c (m

middotsndash1)

(a)

ndash4 ndash2 0 2 4 6 8yc (m)

10

ndash4ndash5

ndash3ndash2ndash1

01234

times10ndash4

times10ndash4

y c (m

middotsndash1)

(b)

ndash3 ndash2 ndash1 0 1 2 3

dycd

t (m

middotsndash1)

times10ndash5yc (m)

ndash003

ndash002

ndash001

0

001

002

003

(c)

Figure 7 Phase diagrams (a) simulation results for brake model without contact loss (b) simulation results for brake model with contactloss (c) phase diagram evaluated from measured vibration data in stage C in Figure 2(b)


is quasi-periodic and if the phase diagram is distributedwithin an area of the phase plane and not clear the systembehaviors could be random or even chaotic

As seen in Figure 7 the simulation results of the brakemodel with interface separation or vibro-impact (Figure 7(b))exhibit a much more irregular trend than the case withoutvibro-impact (Figure 7(a)) is suggests that the irregularityis increased by vibro-impact effects Moreover comparedwith Figures 7(a) and 7(b) more closely resembles the ex-perimental results shown in Figure 7(c) which exhibits theunsymmetrical ldquofingerprintsrdquo in the phase diagram

e phase diagram of the system with vibro-impact(Figure 7(b)) exhibits random close loops with multiplesharp corners and exhibits transition from the half order tointeger order with an unsymmetrical signature which arecloser to the measured results (Figure 7(c)) Pure stick-slipvibrations in contrast exhibit a fixed phase diagram withrelatively fixed signature

5 Conclusions

rough experimental analytical and numerical in-vestigations of creep groan vibrations of vehicle brake wecan draw the following conclusions

(1) e experimental results comprehensively exhibitvaried vibrations in creep groan phenomena whichinclude simple stick-slip stick-slip resonanceinstableslidingimpulsive stick-slip stick-slip-induced vibro-impacts and their back-and-forthtransitions

(2) In contrast to conventional theories of creep groanwhich comprehensively characterized the simplestick-slip sliding stick-slip stick-slip-induced reso-nances of suspension systems we develop a newmechanism and analytical model of creep groan andstick-slip-induced vibro-impacts which quantita-tively explain the existence of half-order harmonicsand its transitions to a stick-slip fundamentalfrequency

(3) e severe stick-slip vibration-induced vibro-impactvibrations are unstable and could be triggered whenstable stick-slip vibration is strong enough and coulddeteriorate back to pure stable stick-slip vibrationswithout vibro-impact e transitions have inherentconnections with clear trend patterns of increase ordecrease in specific frequencies in the spectrogramwhich are also accompanied by a wideband spectrumdue to the random impulsive effect in the unstabletransition e varied vibrations associated withcreep groan are not irrelevant but rather have reg-ularity and internal relationships

(4) Based on numerical simulations the phase diagramsof the dynamics of brake creep groan have differentpatterns for cases with and without vibro-impactse irregularity of the phase diagram is increased bythe vibro-impact effect which is more similar to theexperimental phase diagram

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (grant numbers 51575395 51705366and U1564207)

References

[1] M K Abdelhamid ldquoCreep groan of disc brakesrdquo SAETechnical Paper Series 1995

[2] M Gouya and M Nishiwaki ldquoStudy on disc brake groanrdquoSAE Technical Paper Series 1990

[3] J Brecht W Hoffrichter and A Dohle ldquoMechanisms ofbrake creep groanrdquo SAE Technical Paper Series 1997

[4] V Vadari and M Jackson ldquoAn experimental investigation ofdisk brake creep-groan in vehicles and brake dynamometercorrelationrdquo SAE Technical Paper Series 1999

[5] M Bettella M F Harrison and R S Sharp ldquoInvestigation ofautomotive creep groan noise with a distributed-source ex-citation techniquerdquo Journal of Sound and Vibration vol 255no 3 pp 531ndash547 2002

[6] H Jang J S Lee and J W Fash ldquoCompositional effects of thebrake friction material on creep groan phenomenardquo Wearvol 251 no 1ndash12 pp 1477ndash1483 2001

[7] X Zhao N Grabner and U Von Wagner ldquoeoretical andexperimental investigations of the bifurcation behavior ofcreep groan of automotive disk brakesrdquo Journal of 9eoreticaland Applied Mechanics vol 56 no 2 pp 351ndash364 2018

[8] M Gonzalez R Dıez and E Canibano ldquoComparativeanalysis of vented brake discs in groan noise test on dyna-mometerrdquo SAE Technical Paper Series 2004

[9] Z Fuadi K Adachi H Ikeda H Naito and K Kato ldquoEffect ofcontact stiffness on creep-groan occurrence on a simplecaliper-slider experimental modelrdquo Tribology Letters vol 33no 3 pp 169ndash178 2009

[10] S W Yoon M W Shin W G Lee and H Jang ldquoEffect ofsurface contact conditions on the stick-slip behavior of brakefriction materialrdquo Wear vol 294-295 pp 305ndash312 2012

[11] P D Neis N F Ferreira J C Poletto L T Matozo andD Masotti ldquoQuantification of brake creep groan in vehicletests and its relation with stick-slip obtained in laboratorytestsrdquo Journal of Sound and Vibration vol 369 pp 63ndash762016

[12] X Zhao N Grabner and U von Wagner ldquoAvoiding creepgroan investigation on active suppression of stick-slip limitcycle vibrations in an automotive disk brake via piezoceramicactuatorsrdquo Journal of Sound and Vibration vol 441pp 174ndash186 2019

[13] K Joo H Jeon W Sung and M H Cho ldquoTransfer pathanalysis of brake creep noiserdquo SAE International Journal ofPassenger Cars-Mechanical Systems vol 6 no 3 pp 1408ndash1417 2013


[14] C Cantoni R Cesarini G Mastinu G Rocca andR Sicigliano ldquoBrake comfort-a reviewrdquo Vehicle System Dy-namics vol 47 no 8 pp 901ndash947 2009

[15] N Ashraf D Bryant and J D Fieldhouse ldquoInvestigation ofstick-slip vibration in a commercial vehicle brake assemblyrdquoInternational Journal of Acoustics and Vibration vol 22 no 3pp 326ndash333 2017

[16] Z Fuadi S Maegawa K Nakano and K AdachildquoMap of low-frequency stick-slip of a creep groanrdquo Pro-ceedings of the Institution of Mechanical Engineers Part JJournal of Engineering Tribology vol 224 no 12pp 1235ndash1246 2010

[17] J Brecht and K Schiffner ldquoInfluence of friction law on brakecreep-groanrdquo SAE Technical Paper Series 2001

[18] H Hetzler D Schwarzer and W Seemann ldquoAnalytical in-vestigation of steady-state stability and hopf-bifurcationsoccurring in sliding friction oscillators with application tolow-frequency disc brake noiserdquo Communications in Non-linear Science and Numerical Simulation vol 12 no 1pp 83ndash99 2007

[19] H Hetzler D Schwarzer and W Seemann ldquoSteady-statestability and bifurcations of friction oscillators due to velocity-dependent friction characteristicsrdquo Proceedings of the In-stitution of Mechanical Engineers Part K Journal of Multi-body Dynamics vol 221 no 3 pp 401ndash412 2007

[20] A R Crowther and R Singh ldquoAnalytical investigation ofstick-slip motions in coupled brake-driveline systemsrdquoNonlinear Dynamics vol 50 no 3 pp 463ndash481 2007

[21] A R Crowther and R Singh ldquoIdentification and quantifi-cation of stick-slip induced brake groan events using ex-perimental and analytical investigationsrdquo Noise ControlEngineering Journal vol 56 no 4 pp 235ndash255 2008

[22] D Wei J Ruan W Zhu and Z Kang ldquoProperties ofstability bifurcation and chaos of the tangential motiondisk brakerdquo Journal of Sound and Vibration vol 375pp 353ndash365 2016

[23] X Zhao N Grabner and U von Wagner ldquoExperimental andtheoretical investigation of creep groan of brakes throughminimal modelsrdquo Pamm vol 16 no 1 pp 295-296 2016

[24] Z Fuadi K Adachi H Ikeda H Naito and K Kato ldquoEx-perimental model for creep groan analysisrdquo LubricationScience vol 21 no 1 pp 27ndash40 2009

[25] L Zhang P Zhang and D Meng ldquoTests and theoreticalanalysis of creep groan in vehicle disc brakerdquo AutomotiveEngineering vol 38 no 9 pp 1132ndash1139 2016

[26] S Hu and Y Liu ldquoDisc brake vibration model based onStribeck effect and its characteristics under different brakingconditionsrdquoMathematical Problems in Engineering vol 2017pp 1ndash13 2017

[27] M Donley and D Riesland ldquoBrake groan simulation for amcpherson strut type suspensionrdquo SAE Technical Paper Se-ries 2003

[28] K Uchiyama and Y Shishido ldquoStudy of creep groan simu-lation by implicit dynamic analysis method of FEA (Part 2)rdquoSAE Technical Paper Series 2014


[30] D Meng L Zhang J Xu and Z Yu ldquoA transient dynamicmodel of brake corner and subsystems for brake creep groananalysisrdquo Shock and Vibration vol 2017 Article ID 802079718 pages 2017

[31] Z Li H Ouyang and Z Guan ldquoFriction-induced vibrationof an elastic disc and a moving slider with separation and

reattachmentrdquo Nonlinear Dynamics vol 87 no 2 pp 1045ndash1067 2017

[32] M Stender M Tiedemann N Hoffmann and S OberstldquoImpact of an irregular friction formulation on dynamics of aminimal model for brake squealrdquo Mechanical Systems andSignal Processing vol 107 pp 439ndash451 2018

[33] A Mercier L Jezequel S Besset A Hamdi and J-F DieboldldquoNonlinear analysis of the friction-induced vibrations of arotor-stator systemrdquo Journal of Sound and Vibration vol 443pp 483ndash501 2019

[34] M Abdelhamid and W Bray ldquoBraking systems creep groannoise detection and evaluationrdquo SAE Technical Paper Series2009

[35] X Sui and Q Ding ldquoInstability and stochastic analyses of apad-on-disc frictional system in moving interactionsrdquo Non-linear Dynamics vol 93 no 3 pp 1619ndash1634 2018

[36] N Gaus C Proppe and C Zaccardi ldquoModeling of dynamicalsystems with friction between randomly rough surfacesrdquoProbabilistic Engineering Mechanics vol 54 pp 82ndash86 2018

[37] D Tonazzi F Massi L Baillet J Brunetti and Y BerthierldquoInteraction between contact behaviour and vibrational re-sponse for dry contact systemrdquo Mechanical Systems andSignal Processing vol 110 pp 110ndash121 2018

[38] G Lacerra M Di Bartolomeo S Milana L BailletE Chatelet and F Massi ldquoValidation of a new frictional lawfor simulating friction-induced vibrations of rough surfacesrdquoTribology International vol 121 pp 468ndash480 2018


International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018


Active and Passive Electronic Components

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of



Journal ofEngineeringVolume 2018

SensorsJournal of



RotatingMachinery


Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation


Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

drivelines [12 13] e tribological parameters of pad-discinterface are also characterized in terms of varied frictionmodels such as LuGre and the bristle friction law in additionto the Coulomb and Stribeck model with varied slopesmagnitudes and gap of static and dynamic friction co-efficients [16ndash23]

Analytically and numerically different models havebeen developed to characterize creep groan Conven-tionally friction oscillator models with a single degree offreedom (DOF) and two DOFs have been used as a calipermodel with a contact sliding interface to quantify vibra-tions [17ndash19] Usually two major subsystems are includedin groan analytical models e nonrotating substructureincludes the brake mounting brackets and suspensionsystems e rotating structure is the drive train sideincluding the rotor wheel and tire In general allsuspension-type modes could exhibit resonances in thecreep groan range By considering the components of thebrake and driveline and even the tire and vehicle massthree four five and seven-DOFs models have been used toinvestigate varied aspects of brake friction vibrations[20ndash22 24ndash26] e multibody dynamic models of thechassis corner have also been used [27] Moreover finiteelement (FE) models of the disc brake for creep groananalysis have been proposed to quantify creep groan[28ndash30] FEA models have advantages in the calculation ofthe interface contact as well as multiple elastic subsystemsincluding pad caliper suspension and driveline systemsHowever FE models of creep groan have not been suc-cessfully used to elaborate the complex mechanisms ofcreep groan except for evaluating the effects of structuraldesign change on vibrations In fact for this kind ofproblem of elastic bulky components connected by rela-tively compliant parts such as springs finite elementanalysis may not be the most suitable tool for simulatingproblems due to their weakness dealing with ldquoweak con-nectionsrdquo among different components Possible ill con-ditioning due to large differences in the stiffness of variedcomponents and interconnections can degrade the accu-racy of the results Moreover the modeling of interactingcomponents in low-frequency events in an elastic modelwith huge numbers of elements and nodes is far fromefficient and the data interpretation is very difficult Assuch theoretical models with varied DOFs have been usedto classify varied mechanisms and resources of low-frequency vibrations [31ndash33]

e sensing and evaluation of ldquocreep groanrdquo noise hasbeen a challenge for the NVH community e creep groanis usually not a purely tonal sound although ultimately itcould generate a tonal subjective perception of users [34]e vibrations resulting in creep groan noise have con-ventionally been considered friction-induced vibrations ofthe stick-slip mode characterized by simple phase diagramsof velocity and displacement of caliper motions Howeverthe recorded vibration usually does not exhibit simple stick-slip periodic vibration but rather exhibits complex prop-erties with multiple frequencies time-varying frequenciesand their transitions e real creep groan has been found toexhibit complex motions in addition to typical stick-slip

motions which include single stick-slip continuous stick-slip resonance varied complex motions and transitions[7 9 11 15 20 30] Basically stick-slip is a sustainedldquoattach-detachrdquo vibration process that arises via specificfriction properties specific system elastic properties andrelative motion [31 35] In particular negative slopes offriction-velocity curves yield equivalently negative dampingto system dynamic motion and could cause system in-stability and a self-excited vibration which are likely lockedin certain natural frequencies or resonances e spectralcontent of the creep groan usually shows a response atmultiples of a certain fundamental frequency of stick-slip orits induced resonance e stick-slip vibrations may occur atthe 1st 2nd or 3rd order of the fundamental frequencycorresponding to the sustained ldquostick-sliprdquo cycle e stick-slip motions mainly depend on the properties of the me-chanical system and are also related to different charac-teristics of the interface friction [36ndash38] However existingtheories and models are unable to explain these phenomenacomprehensively and consistently For example in the eventof creep groan there could be half-order components inaddition to fundamental components corresponding to thestick-slip repetition frequency and the multiple components[16 21 24 30]e root causes of the half-order componentsand their transitions with other components have not beenexplained e inherent connections among fundamentalstick-slip multiple harmonics half-order component res-onance impulsive components of wideband and transitionshave not been explained consistently

In this study a vehicle road test of creep groan under thedownhill condition is conducted and the varied complicatedtypes of vibrations in creep groans are comprehensivelyrecorded which consist of simple stick-slip motioncontinuousdiscontinuous stick-slip motions and slidingimpulsive motions with varied spectrum signatures fromhalf-order fundamental order multiple orders time-varying spectrum transition and wideband componentsA new mechanism of severe stick-slip motion-inducedvibro-impact is proposed to interpret the occurrence andthe transition of the half-order vibrations and a theoreticalvibro-impact model with transient pad-disc separation isapplied for the analysis

Conventionally the phase diagram of brake vibrationwith limit cycles has been used as experimental evidence ofthe stick-slip mechanism in creep groan In this study thephase diagram of experimentally recorded brake vibrationsis found to exhibit a complex unsymmetrical pattern Tocharacterize this a seven-DOFs model with and withoutvibro-impact effect were used for simulations to derivephase diagrams which showed that only the model with thevibro-impact effect gives rise to an unsymmetrical phasediagram

Creep groan has been investigated widely and manymechanisms have been proposed in the last two decades Incontrast to all of these existing mechanisms the proposed newmechanism can encompass the transient vibro-impact natureof a braking process which is different from conventionalexcitation resources such as periodic torque stick-slip neg-ative friction-velocity gradient and resonance e severe
















0 2 4 6 8 10 12 14 16 18Time (s)

0

4

8

12

16

Brak

e pre

ssur

e (ba

r)

(a)

Time (s)0 2 4 6 8 10 12 14 16 18


0102030

X-ac

cele

ratio

n (m

s2 )

A B DC FE G

(b)

0 2 4 6 8 10 12 14 16 18Time (s)


0102030

Y-ac

cele

ratio

n (m

s2 ) A B E F GDC

(c)

0 2 4 6 8 10 12 14 16 18Time (s)


0102030

Z-ac

cele

ratio

n (m

s2 ) A B E F GDC

(d)

0 2 4 6 8 10 12 14 16 18Time (s)

400

5005060

403020100

300

Freq

uenc

y (H

z)

200

100

0

(e)


(a) (b)

Tension sensor

(c)













0 5 10 15 20 25 30 35 40 45 500365

0370375

0380385

0390395

040405


Fric

tion

coef

ficie

nt


k

yx

m

kc

yc





y 2F

ω22 minusω2

1( 1113857m(2Δ + 1)+

3F

4ω2m(2Δ + 1)21113890 1113891sinωt

minusπF


F

πω2m(2Δ + 1)

middot 1113944infin

n135


nminus

F(ωtminus π)

πω2m(2Δ + 1)

middot 1113944infin

n135


n

(2)

inwhichω1 km

radicω2

(k + kc)m

1113968ω0 2ω1ω2






1( 1113857

m n2 minus 1( )2ω4

sinωt

+2F ω2

2 minusω21( 1113857



minus 1113944

infin

n234

2F ω22 minusω2

1( 1113857



(3)



y 4lF


middot 1113944infin

n135


l 2 4 6

(4)













c2 c1cy c2cy


F 1000N

Ff1 Ff2 μsF

(5)








(6)



0 δxi gtLi (7)



σ δxi∣∣∣∣∣∣∣∣minus Li( )( )2

(8)




Knuckle

Caliper

Piston

Disc

Brake pad

Brake fluid

(a)

mc

c2cx c1cxc1dx

c2cy c1cy

ccy

ccx

c2dx

cdx

k2cx k1cxk2dx

kdx

k1dx

k2cy k1cy

kcy

kcx

m2 m1md

v0

F F

Y

X

(b)





k5 15times 106 c5 50k6 15times 106 c6 50




σ = 1000σ = 100σ = 10

S (δx i)

|δxi| ndash Li040302010 05 06 07 08 09 1

0

02

04

06

08

1

(a)

σ = 10e6σ = 10e5

σ = 10e4σ = 1000

|δxi| ndash Li

S (δx i)

0

02

04

06

08

1


(b)


ndash2 0 2 4 6 8 10 12yc (m)

ndash3

ndash2

ndash1

0

1

2

3

4times10ndash4

times10ndash4

y c (m

middotsndash1)

(a)


10

ndash4ndash5

ndash3ndash2ndash1

01234

times10ndash4

times10ndash4

y c (m

middotsndash1)

(b)


dycd

t (m

middotsndash1)

times10ndash5yc (m)

ndash003

ndash002

ndash001

0

001

002

003

(c)






5 Conclusions






Data Availability




Acknowledgments


References












































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia
















0 2 4 6 8 10 12 14 16 18Time (s)

0

4

8

12

16

Brak

e pre

ssur

e (ba

r)

(a)

Time (s)0 2 4 6 8 10 12 14 16 18


0102030

X-ac

cele

ratio

n (m

s2 )

A B DC FE G

(b)

0 2 4 6 8 10 12 14 16 18Time (s)


0102030

Y-ac

cele

ratio

n (m

s2 ) A B E F GDC

(c)

0 2 4 6 8 10 12 14 16 18Time (s)


0102030

Z-ac

cele

ratio

n (m

s2 ) A B E F GDC

(d)

0 2 4 6 8 10 12 14 16 18Time (s)

400

5005060

403020100

300

Freq

uenc

y (H

z)

200

100

0

(e)


(a) (b)

Tension sensor

(c)













0 5 10 15 20 25 30 35 40 45 500365

0370375

0380385

0390395

040405


Fric

tion

coef

ficie

nt


k

yx

m

kc

yc





y 2F

ω22 minusω2

1( 1113857m(2Δ + 1)+

3F

4ω2m(2Δ + 1)21113890 1113891sinωt

minusπF


F

πω2m(2Δ + 1)

middot 1113944infin

n135


nminus

F(ωtminus π)

πω2m(2Δ + 1)

middot 1113944infin

n135


n

(2)

inwhichω1 km

radicω2

(k + kc)m

1113968ω0 2ω1ω2






1( 1113857

m n2 minus 1( )2ω4

sinωt

+2F ω2

2 minusω21( 1113857



minus 1113944

infin

n234

2F ω22 minusω2

1( 1113857



(3)



y 4lF


middot 1113944infin

n135


l 2 4 6

(4)













c2 c1cy c2cy


F 1000N

Ff1 Ff2 μsF

(5)








(6)



0 δxi gtLi (7)



σ δxi∣∣∣∣∣∣∣∣minus Li( )( )2

(8)




Knuckle

Caliper

Piston

Disc

Brake pad

Brake fluid

(a)

mc

c2cx c1cxc1dx

c2cy c1cy

ccy

ccx

c2dx

cdx

k2cx k1cxk2dx

kdx

k1dx

k2cy k1cy

kcy

kcx

m2 m1md

v0

F F

Y

X

(b)





k5 15times 106 c5 50k6 15times 106 c6 50




σ = 1000σ = 100σ = 10

S (δx i)

|δxi| ndash Li040302010 05 06 07 08 09 1

0

02

04

06

08

1

(a)

σ = 10e6σ = 10e5

σ = 10e4σ = 1000

|δxi| ndash Li

S (δx i)

0

02

04

06

08

1


(b)


ndash2 0 2 4 6 8 10 12yc (m)

ndash3

ndash2

ndash1

0

1

2

3

4times10ndash4

times10ndash4

y c (m

middotsndash1)

(a)


10

ndash4ndash5

ndash3ndash2ndash1

01234

times10ndash4

times10ndash4

y c (m

middotsndash1)

(b)


dycd

t (m

middotsndash1)

times10ndash5yc (m)

ndash003

ndash002

ndash001

0

001

002

003

(c)






5 Conclusions






Data Availability




Acknowledgments


References












































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia












0 5 10 15 20 25 30 35 40 45 500365

0370375

0380385

0390395

040405


Fric

tion

coef

ficie

nt


k

yx

m

kc

yc





y 2F

ω22 minusω2

1( 1113857m(2Δ + 1)+

3F

4ω2m(2Δ + 1)21113890 1113891sinωt

minusπF


F

πω2m(2Δ + 1)

middot 1113944infin

n135


nminus

F(ωtminus π)

πω2m(2Δ + 1)

middot 1113944infin

n135


n

(2)

inwhichω1 km

radicω2

(k + kc)m

1113968ω0 2ω1ω2






1( 1113857

m n2 minus 1( )2ω4

sinωt

+2F ω2

2 minusω21( 1113857



minus 1113944

infin

n234

2F ω22 minusω2

1( 1113857



(3)



y 4lF


middot 1113944infin

n135


l 2 4 6

(4)













c2 c1cy c2cy


F 1000N

Ff1 Ff2 μsF

(5)








(6)



0 δxi gtLi (7)



σ δxi∣∣∣∣∣∣∣∣minus Li( )( )2

(8)




Knuckle

Caliper

Piston

Disc

Brake pad

Brake fluid

(a)

mc

c2cx c1cxc1dx

c2cy c1cy

ccy

ccx

c2dx

cdx

k2cx k1cxk2dx

kdx

k1dx

k2cy k1cy

kcy

kcx

m2 m1md

v0

F F

Y

X

(b)





k5 15times 106 c5 50k6 15times 106 c6 50




σ = 1000σ = 100σ = 10

S (δx i)

|δxi| ndash Li040302010 05 06 07 08 09 1

0

02

04

06

08

1

(a)

σ = 10e6σ = 10e5

σ = 10e4σ = 1000

|δxi| ndash Li

S (δx i)

0

02

04

06

08

1


(b)


ndash2 0 2 4 6 8 10 12yc (m)

ndash3

ndash2

ndash1

0

1

2

3

4times10ndash4

times10ndash4

y c (m

middotsndash1)

(a)


10

ndash4ndash5

ndash3ndash2ndash1

01234

times10ndash4

times10ndash4

y c (m

middotsndash1)

(b)


dycd

t (m

middotsndash1)

times10ndash5yc (m)

ndash003

ndash002

ndash001

0

001

002

003

(c)






5 Conclusions






Data Availability




Acknowledgments


References












































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




y 2F

ω22 minusω2

1( 1113857m(2Δ + 1)+

3F

4ω2m(2Δ + 1)21113890 1113891sinωt

minusπF


F

πω2m(2Δ + 1)

middot 1113944infin

n135


nminus

F(ωtminus π)

πω2m(2Δ + 1)

middot 1113944infin

n135


n

(2)

inwhichω1 km

radicω2

(k + kc)m

1113968ω0 2ω1ω2






1( 1113857

m n2 minus 1( )2ω4

sinωt

+2F ω2

2 minusω21( 1113857



minus 1113944

infin

n234

2F ω22 minusω2

1( 1113857



(3)



y 4lF


middot 1113944infin

n135


l 2 4 6

(4)













c2 c1cy c2cy


F 1000N

Ff1 Ff2 μsF

(5)








(6)



0 δxi gtLi (7)



σ δxi∣∣∣∣∣∣∣∣minus Li( )( )2

(8)




Knuckle

Caliper

Piston

Disc

Brake pad

Brake fluid

(a)

mc

c2cx c1cxc1dx

c2cy c1cy

ccy

ccx

c2dx

cdx

k2cx k1cxk2dx

kdx

k1dx

k2cy k1cy

kcy

kcx

m2 m1md

v0

F F

Y

X

(b)





k5 15times 106 c5 50k6 15times 106 c6 50




σ = 1000σ = 100σ = 10

S (δx i)

|δxi| ndash Li040302010 05 06 07 08 09 1

0

02

04

06

08

1

(a)

σ = 10e6σ = 10e5

σ = 10e4σ = 1000

|δxi| ndash Li

S (δx i)

0

02

04

06

08

1


(b)


ndash2 0 2 4 6 8 10 12yc (m)

ndash3

ndash2

ndash1

0

1

2

3

4times10ndash4

times10ndash4

y c (m

middotsndash1)

(a)


10

ndash4ndash5

ndash3ndash2ndash1

01234

times10ndash4

times10ndash4

y c (m

middotsndash1)

(b)


dycd

t (m

middotsndash1)

times10ndash5yc (m)

ndash003

ndash002

ndash001

0

001

002

003

(c)






5 Conclusions






Data Availability




Acknowledgments


References












































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



c2 c1cy c2cy


F 1000N

Ff1 Ff2 μsF

(5)








(6)



0 δxi gtLi (7)



σ δxi∣∣∣∣∣∣∣∣minus Li( )( )2

(8)




Knuckle

Caliper

Piston

Disc

Brake pad

Brake fluid

(a)

mc

c2cx c1cxc1dx

c2cy c1cy

ccy

ccx

c2dx

cdx

k2cx k1cxk2dx

kdx

k1dx

k2cy k1cy

kcy

kcx

m2 m1md

v0

F F

Y

X

(b)





k5 15times 106 c5 50k6 15times 106 c6 50




σ = 1000σ = 100σ = 10

S (δx i)

|δxi| ndash Li040302010 05 06 07 08 09 1

0

02

04

06

08

1

(a)

σ = 10e6σ = 10e5

σ = 10e4σ = 1000

|δxi| ndash Li

S (δx i)

0

02

04

06

08

1


(b)


ndash2 0 2 4 6 8 10 12yc (m)

ndash3

ndash2

ndash1

0

1

2

3

4times10ndash4

times10ndash4

y c (m

middotsndash1)

(a)


10

ndash4ndash5

ndash3ndash2ndash1

01234

times10ndash4

times10ndash4

y c (m

middotsndash1)

(b)


dycd

t (m

middotsndash1)

times10ndash5yc (m)

ndash003

ndash002

ndash001

0

001

002

003

(c)






5 Conclusions






Data Availability




Acknowledgments


References












































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




σ = 1000σ = 100σ = 10

S (δx i)

|δxi| ndash Li040302010 05 06 07 08 09 1

0

02

04

06

08

1

(a)

σ = 10e6σ = 10e5

σ = 10e4σ = 1000

|δxi| ndash Li

S (δx i)

0

02

04

06

08

1


(b)


ndash2 0 2 4 6 8 10 12yc (m)

ndash3

ndash2

ndash1

0

1

2

3

4times10ndash4

times10ndash4

y c (m

middotsndash1)

(a)


10

ndash4ndash5

ndash3ndash2ndash1

01234

times10ndash4

times10ndash4

y c (m

middotsndash1)

(b)


dycd

t (m

middotsndash1)

times10ndash5yc (m)

ndash003

ndash002

ndash001

0

001

002

003

(c)






5 Conclusions






Data Availability




Acknowledgments


References












































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





5 Conclusions






Data Availability




Acknowledgments


References












































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


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