disc brake vibration model based on stribeck effect and its ...mathematicalproblemsinengineering 5...

Research ArticleDisc Brake Vibration Model Based on Stribeck Effect and ItsCharacteristics under Different Braking Conditions

Shugen Hu and Yucheng Liu

College of Mechanical Engineering Zhejiang University Hangzhou 310027 China

Correspondence should be addressed to Shugen Hu hsgdjzjueducn

Received 9 March 2017 Revised 17 April 2017 Accepted 23 April 2017 Published 28 May 2017

Academic Editor Oleg V Gendelman

Copyright copy 2017 Shugen Hu and Yucheng LiuThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

A 7-degree-of-freedom (DOF) vibration model of a fixed-caliper disc brake system was developed herein based on the Stribeckeffect Furthermore a dynamometer brake test was conducted to determine the characteristic system parameters of the 7-DOFvibration model This model was developed to study the effects of braking conditions such as disc rotational speed and brakepressure on brake noise The complex eigenvalues of the system were also calculated to analyze the brake model stability underdifferent braking conditions The acceleration time history diagrams and phase plots were obtained by solving the equations of thesystemThe numerical calculation results showed that the brake noise increased with an increasing braking force and a decreasingbreaking speed These numerical findings were verified by the results of the dynamometer tests

1 Introduction

Fixed-caliper disc brakes are widely used in passenger andlight commercial vehicles One ormore pairs of opposing pis-tons are used in such brake systems to clamp both sides of thedisc with the caliper carrying the pistons fixed on a knucklesuch that its position does not change relative to the discHowever with the increasing demand for driving comfort agreater need to address the noise vibration and harshness(NVH) issues of such brake systems is required Brake noiseespecially ldquobrake squealrdquo which occurs with frequencies of1ndash16 kHz and reaches the upper human auditory threshold isof particular importance in current vehicle NVH engineering[1ndash3]

Studies on brake noise began in the 1930s [4] whenresearchers conducted diverse investigations from differentperspectives including frictional properties of utilized mate-rials and structures of the brake system parts [5 6] Thesestudies revealed a close relationship between brake noise andvibrational excitation caused by friction between the disc andthe lining [7 8]The theory of this relationship is based on thetwo following basic characteristics of friction coefficients (1)the static friction coefficient is higher than the sliding friction

coefficient and (2) the friction coefficients decrease withthe increasing relative velocity within a certain range Theformer characteristic produces the system stick-slip effectwhich causes vibration [9] Meanwhile the latter inducesnegative damping which results in system instability [10ndash12]However previous studies on brake noise that investigatedthe friction characteristics mainly considered the self-excitedvibration at a specific friction coefficient or unstable vibrationcaused by the negative slope in the relative velocity versus thefriction coefficient curve [13ndash15]

Accordingly 2-DOF vibration models were employed inmost of the previous studies in which brake squeal wasexamined based on frictionThis made it difficult to simulatethe working conditions during braking thereby hinderingthe investigation of the interaction between the different partsof the brake assembly [16ndash18] Vibration models with morethan two DOFs have consequently been developed for thebrake squeal investigation For example Kinkaid designeda 4-DOF brake model that captured some of the dynamicsof a set of brake pads used to stop a rotor [19] Ahmedbuilt a 10-DOFmathematical prediction model to investigatethe effects of different brake component parameters on aventilated disc brake squeal [20] Wang et al developed a

HindawiMathematical Problems in EngineeringVolume 2017 Article ID 6023809 13 pageshttpsdoiorg10115520176023809

2 Mathematical Problems in Engineering

Brake pad

Disc

Brake uid

Piston

Caliper

Knuckle

Figure 1 Structure of a fixed-caliper disc brake

4-DOF model of a disc brake with friction and contact lossnonlinearities to investigate the mechanism and the dynamiccharacteristics of a brake squeal [21] The present studydeveloped a 7-DOF vibration model of a fixed-caliper brakesystem based on the Stribeck effect A dynamometer test wasconducted to determine the characteristic systemparametersFurthermore the effects of the braking conditions on thesystem stability were studied

2 Establishment of the BrakeVibration System

21 Establishment of the State Equations Figure 1 shows atypical structure of a fixed-caliper disc brake for a vehi-cle During the braking process the brake fluid pressureincreases and the rotating disc is clamped by both sides ofthe brake linings thereby resulting in the deceleration of themoving vehicle The vibration model of the brake systemwhich basically comprises the caliper disc and both sides ofthe brake linings can be simplified as shown in Figure 2

In Figure 2 119898119889 is the mass of the disc 119898119888 is the massof the caliper 1198981 and 1198982 are the masses of the inner andouter brake linings respectively V0 is the absolute linearvelocity of the brake disc assumed to be in the 119910+ directionand 119865 is the brake pressure on either lining of both sidesconsidering that the inner and outer pistons are of the samesize The movement along the 119909-axis was considered for thedisc while simultaneous movements along the 119909- and 119910-axes were considered for the caliper and both brake liningsSuppose that the disc moves in the x+ direction during thebraking process the inner lining would move in an xminus119910+direction the outer liningwouldmove in an x+119910+ directionand the caliper would move in an x+119910+ direction Thishypothesis was used to define the initial state of the systemto facilitate analysis such that the directions of the reactionforces on the springs and dampers can be obtained The finalresults would not be affected by these assumptions becausethe obtained values of the reaction forces would be negativeif their directions are opposite the assumed direction

mc

md

F F

c2cx

k2cx

k2cy

c2dx

k2dx

m2 m1

cdx

kdx

c1dx

k1dx

k1cy

c1cx

k1cx

c1cy

ccx

kcx

kcy ccyY

X

0

c2cy

Figure 2 Vibration model of a fixed-caliper disc brake

Based on the abovementioned assumptions and the rela-tionships among the different parts in the brake system theequations of the vibration system shown in Figure 2 can beobtained as follows by using Newtonrsquos law

119898119888119888 = minus119888119888119909119888 + 1198882119888119909 (2 minus 119888) minus 1198881119888119909 (119888 minus 1) minus 119896119888119909119909119888+ 1198962119888119909 (1199092 minus 119909119888) minus 1198961119888119909 (119909119888 minus 1199091)

119898119888 119910119888 = minus119888119888119910 119910119888 + 1198882119888119910 ( 1199102 minus 119910119888) + 1198881119888119910 ( 1199101 minus 119910119888) minus 119896119888119910119910119888+ 1198962119888119910 (1199102 minus 119910119888) + 1198961119888119910 (1199101 minus 119910119888)

119898119889119889 = minus119888119889119889 + 1198882119889119909 (2 minus 119889) minus 1198881119889119909 (1 minus 119889)minus 119896119889119909119889 + 1198962119889119909 (1199092 minus 119909119889) minus 1198961119889119909 (119909119889 minus 1199091)

11989811 = minus119865 + 1198881119889119909 (119889 minus 1) + 1198881119888119909 (119888 minus 1)+ 1198961119889119909 (119909119889 minus 1199091) + 1198961119888119909 (119909119888 minus 1199091)

1198981 1199101 = 1198651198911 minus 1198881119888119910 ( 1199101 minus 119910119888) minus 1198961119888119910 (1199101 minus 119910119888) 11989822 = 119865 minus 1198882119889119909 (2 minus 119889) minus 1198882119888119909 (2 minus 119888)

minus 1198962119889119909 (1199092 minus 119909119889) minus 1198962119888119909 (1199092 minus 119909119888) 1198982 1199102 = 1198651198912 minus 1198882119888119910 ( 1199102 minus 119910119888) minus 1198962119888119910 (1199102 minus 119910119888)

(1)

where 1198651198911 and 1198651198912 are the frictional forces between thedisc and the two brake linings respectively The expressionsof which were obtained using a Stribeck model and thedynamometer test results (Section 22 for a detailed descrip-tion) Considering the structural symmetry of a fixed-caliperdisc brake system it was assumed that 1198881 = 1198881119889119909 = 1198882119889119909

Mathematical Problems in Engineering 3

1198961 = 1198961119889119909 = 1198962119889119909 1198882 = 1198881119888119910 = 1198882119888119910 1198962 = 1198961119888119910 = 11989621198881199101198883 = 1198881119888119909 = 1198882119888119909 1198963 = 1198961119888119909 = 1198962119888119909 1198884 = 119888119889119909 1198964 = 1198961198891199091198885 = 119888119888119909 1198965 = 119896119888119909 1198886 = 119888119888119910 and 1198966 = 119896119888119910 Furthermorethe generalized displacement 119902 and the generalized velocity119902 were introduced by defining 1199101 = 1199021 1199102 = 1199022 119910119888 = 11990231199091 = 1199024 1199092 = 1199025 119909119888 = 1199026 and 119909119889 = 1199027 Thus the vibrationequations could be derived based on generalized coordinates

Finally the state variable 119911 was used to rewrite the equationsas follows

119911119894 = 119902119894 119894 = 1 2 7119911119894+7 = 119902119894 119894 = 1 2 7 (2)

The system state equations were also obtained as follows

1 = 119911119894+7 (119894 = 1 2 7) 8 = [1198651198911 minus 1198882 (1199118 minus 11991110) minus 1198962 (1199111 minus 1199113)]1198981

9 = [1198651198912 minus 1198882 (1199119 minus 11991110) minus 1198962 (1199112 minus 1199113)]1198982

10 = [minus119888611991110 + 1198882 (1199119 minus 11991110) minus 11989661199113 + 1198962 (1199111 + 1199112 minus 21199113)]119898119888

11 = [minus119865 + 1198881 (11991111 minus 11991114) minus 1198883 (11991113 minus 11991111) + 1198961 (1199117 minus 1199114) + 1198963 (1199116 minus 1199114)]1198981

12 = [119865 minus 1198881 (11991112 minus 11991114) minus 1198883 (11991112 minus 11991113) minus 1198961 (1199115 minus 1199117) minus 1198963 (1199115 minus 1199116)]1198982

13 = [minus119888511991113 + 1198883 (11991111 + 11991112 minus 211991113) minus 11989651199116 + 1198963 (1199114 + 1199115 minus 21199116)]119898119888


(3)

22 Representation of the Braking Force Based on the StribeckEffect In 1903 Stribeck found that the Coulomb frictionmodel could not be used to properly describe the actual fric-tion behavior especially the relationship between the frictioncoefficient and the normal pressure [22]The relative velocityin the classic Coulomb friction model was not considered inthe sliding friction The transition between the static frictionand the sliding friction was discrete Furthermore the valueof the sliding friction coefficient was always smaller thanthe maximum static friction coefficient However Stribeckproposed that the sliding friction coefficient decreased withthe increasing relative velocity and presented the former asa continuous function of the latter for low velocities Thefriction behavior in the low-velocity region was referred to asthe negative-slope friction phenomenon because of thenegative slope of the velocity-friction curve The followingexponential model of the phenomenon was proposed by Boand Pavelescu [23]

119891 (V) = 119891119888 + (119891119904 minus 119891119888) 119890minus(VV119904)120575 (4)

where 119891119888 is the Coulomb friction force V is the relativevelocity between the two contacting surfaces V119904 is theStribeck velocity and V119904 and 120575 are the empirical constantsBo and Pavelescu proposed a range of 05ndash1 for 120575 [23] while

some other scholars considered 120575 = 1 or 120575 = 2 as reasonable[24 25]The frictional force was directly related to the normalpressure Therefore (4) can be rewritten as follows

119891 (V) = 119865 sdot 120583 (V) (5)

120583 (V) = 120583119888 + Δ120583 sdot 119890minus(VV119904)120575 (6)

The friction conditions of the brake linings on both sideswere similar because of the structural symmetry of the fixed-caliper brake Hence 1198651198911 and 1198651198912 can be expressed as followsby substituting the variables of the state equations into (5) and(6) and setting 120575 = 1

1198651198911 = sgn(V0 minus 6011991182120587119877119864) sdot 119865

sdot [120583119888 + Δ120583 sdot exp(minus1003816100381610038161003816V0 minus (6011991182120587119877119864)1003816100381610038161003816

V119904)]

1198651198912 = sgn(V0 minus 6011991192120587119877119864) sdot 119865


V119904)]

(7)


Table 1 Parameters determined by fitting of the Stribeck frictionmodel

Parameter Value Standard error120583119888 027090 000278Δ120583 014479 000230V119904 25775876 1003653

045

040

035

030

025

Fric

tion

coe

cien

t

0 50 100 150 200 250 300 350 400Rotational speed (RPM)

Fitting dataRaw data

Figure 3 n-120583 curve fitted using the dynamometer test results

The friction forces between the disc and the innerouterbrake linings in the vibration system presented as (7) werebased on the Stribeck friction model where sgn is the signfunction used to determine the direction of the friction onthe contacting surfaces 1199118 and 1199119 are the linear velocitiesof the contacting surfaces in ms (a unit conversion factoris required because V0 is the rotational speed of the disc inRPM) and 119877119864 is the effective radius of the brake systemThevalues of120583119888Δ120583 V119904 and119877119864were required for calculation usingthe abovementioned equations119877119864 is a design parameter witha value of 0018m and provided by the vehicle manufacturerThe values of the other parameters can be determined by thedynamometer test results

The tests in the present study were conducted using amodel 3900 brake NVH dynamometer (manufactured byLINK Engineering Company) and based on the SAE J2521(Ver 2013Apr) test procedure Hence the dynamic character-istics of the break friction coefficient during the entire brak-ing process from 50 kmh to rest can be determined Figure 3presents the obtained raw dataThe curve obviously matchedwell the characteristics of the Stribeck effect although somefluctuations in the friction which were caused by creepingexisted All 1412 sets of friction coefficient and rotationalspeed data were substituted into (4) and (6) by fitting the rawdata shown in Figure 3 The values of 120583119888 Δ120583 and V119904 can thenbe determined Table 1 and Figure 3 show the obtainedparameter values and the fitted curve respectively

3 Numerical Calculations

31 Determination of the System Stability Using the Complex-Eigenvalue Method A point at which = 0 is referred to

as an ordinary point in the state space while one that satisfies(8) is referred to as a balance point

= 1 2 14119879 = 119885119894 (1199111 1199112 11991114) = 0 (8)

The coordinates of a balance point should be calculated todetermine the stability The change rates 119894 (119894 = 1 2 14)of all the state variables should be 0 at a balance point Hencethe coordinates can be calculated as follows by substituting(8) into the system state equations

1199111119890 = 1199112119890 = 1198651198910 ( 21198966 +11198962)

1199113119890 = 211986511989101198966

1199114119890 = minus119865(1198961 + 1198963)

1199115119890 = 119865(1198961 + 1198963)

119911119898119890 = 0 119898 = 6 7 14

(9)

1198651198910 = 119865 sdot (120583119888 + Δ120583 sdot 119890minusV0V119904) (10)

Equation (9) obviously showed that a balance pointcannot occur at the origin of the coordinate in this systemTherefore coordinate transformationwas necessary to satisfythe requirements for the system linearization in the process ofascertaining the system stability McLaughlinrsquos expansion wasapplied with items of the second or higher orders ignoredbecause the functions of 1198651198911 and 1198651198912 were the only nonlinearterms in the system state equations The approximate expres-sions of 1198651198911 and 1198651198912 near the origin are presented as follows

1198651198911 = 119865 sdot (120583119888 + Δ120583 sdot 119890minusV0V119904) + 119865 sdot Δ120583 sdot 602120587119877119864V119904 sdot 119890

minusV0V119904

sdot 11991181198651198911 = 119865 sdot (120583119888 + Δ120583 sdot 119890minusV0V119904) + 119865 sdot Δ120583 sdot 60

2120587119877119864V119904 sdot 119890minusV0V119904

sdot 1199119

(11)

The linearized state equations near the origin can beobtained by substituting (11) into the state equations aftercoordinate transformation This can also be used to deter-mine the Jacobian matrix and the equations of the systemcharacteristics shown as (12) The existence of the positivereal parts of the eigenvalues is the necessary and sufficientcondition for establishing a balance point to be unstable


Table 2 Vibration model parameters of the fixed-caliper disc brake

Mass Value (kg) Stiffness Value (Nsdotmminus1) Damping Value (Nsdotssdotmminus1)1198981 035 1198961 15 times 105 1198881 51198982 035 1198962 75 times 105 1198882 30m119888 556 1198963 75 times 105 1198883 15m119889 704 1198964 10 times 106 1198884 40

1198965 15 times 106 1198885 501198966 15 times 106 1198886 50

Table 3 Number of eigenvalues with positive real parts

Disc speed (RPM) Brake pressure (N)300 400 500 600 700 800 1000 1200 1500

15 0 0 2 2 4 4 6 6 625 0 0 2 2 2 4 6 6 642 0 0 0 2 4 4 6 6 683 0 0 0 0 2 2 4 4 6100 0 0 0 0 0 2 4 4 6150 0 0 0 0 0 0 2 4 4200 0 0 0 0 0 0 2 2 4250 0 0 0 0 0 0 0 2 2

Hence the stability of the system balance point can bedetermined by calculating the eigenvalues of the system

119886119894119895 = 120597119885119894 (1199111 1199112 11991114)12059711991111989510038161003816100381610038161003816100381610038161003816100381610038161199111=1199112=sdotsdotsdot=11991114=0

119894 119895 = 1 2 14

det ([119886] minus 120582 [1]) =10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

11988611 minus 120582 11988612 sdot sdot sdot 11988611411988621 11988622 minus 120582 sdot sdot sdot 119886214 d

119886141 119886142 sdot sdot sdot 1198861414 minus 120582

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816= 0

(12)

In the present study themajor working condition param-eters of the brake system included the initial disc speed V0and the brake pressure119865 Different values of these parameterswere applied to the system vibration model to numericallyascertain the stability of the balance point under variousworking conditions A total of 72 working conditions wereconsidered in the present study based on the actual brake con-ditions comprising various combinations of brake pressure 119865values of 300 400 500 600 700 800 1000 1200 and 1500Nand initial disc speed V0 values of 15 25 42 83 100 150 200and 250 RPM (25 42 83 and 250 RPM correspond to thelinear speeds of 3 5 10 and 30 kmh in SAE J2521 resp)Table 2 shows the employed values of the other parametersof the vibration system based on the structure symmetry andthe connection relationships among the different parts of thebrake system

MATLAB was used to calculate the eigenvalues of thesystem to qualitatively investigate the stability of the systemunder different working conditions Table 3 presents thenumber of eigenvalues with positive real parts under eachworking condition

Several balance points with differing stabilities couldbe present for a nonlinear vibration system Accordinglythe stability of a nonlinear system cannot be determinedbased only on the stability of one balance point ThereforeMcLaughlinrsquos expansion was used to linearize the nonlinearterms such that the system equations could be linearizedand approximately maintained at the balance point Thestability of the vibration system after linearization could bedetermined based on the stability of the balance point becausethere can only be one balance point for a liner vibrationsystem

The stability of the brake vibration system at any rota-tional speed was closely related to the pressure applied on thebrake linings according to the data in Table 3The systemmaybecome unstable when the brake pressure 119865 reaches a certainthresholdThe number of eigenvalues with positive real partsincreased with the increase of 119865 In addition the value of119865 at which the system became unstable varied with therotational speed This result indicated that the probability ofinstability of a fixed-caliper disc brake system increased withthe increasing brake pressure at a given rotational speed

Furthermore the system eigenvalues with positive realparts only occurred when the brake pressure reached thethreshold and the threshold increased with the increasingrotational speed In other words the brake systemmore easilybecame unstable at a lower vehicle speed Moreover a higherbrake pressure was required to cause an instability at a higherspeed


y1

(mmiddotＭminus2)

times101

1 2 30t (s)

0

20

40

60

x1

(mmiddotＭminus2)

times102

1 2 30t (s)

0

04

08

12

y2

(mmiddotＭminus2)

times101

1 2 30t (s)

0

20

40

60

x2

(mmiddotＭminus2)

times102

1 2 30t (s)

0

04

08

12

yc

(mmiddotＭminus2)

times101

1 2 30t (s)

0

05

10

15

Figure 4 System acceleration time history diagrams for V0 = 150RPM and 119865 = 500N

32 System Time History Diagrams and Phase Plots Thedetermined equations of the brake system were used toobtain the corresponding acceleration time history diagramsand the phase plots for further analysis of the relationshipbetween the working conditions and the system stability All72 working conditions were considered Three representativeworking conditions were selected to reflect the influence of

the disc rotation speed and the brake pressure on the systemstability with the obtained diagrams in Figures 4ndash9 Sevenacceleration time history diagrams and seven phase plotswere obtained for each working condition considering theseven DOFs and 14-dimensional phase spaces in the systemThe caliper and the disc remained static along the 119909-axisof the vibration system because of the structural symmetry


y1

(mmiddotＭminus1)

times10minus3

minus04

minus02

0

02

04

05 100

y1 (m)

times10minus3

y2

(mmiddotＭminus1)

minus04

minus02

0

02

04

05 100

y2 (m)

yc (m) times10minus4minus50 0 50

yc

(mmiddotＭminus1)

minus02

minus01

0

01

02

times10minus3

x1

(mmiddotＭminus1)

minus10

minus05

0

05

10

minus10 minus05 0minus15

x1 (m)

times10minus3

x2

(mmiddotＭminus1)

minus10

minus05

0

05

10

05 10 150

x2 (m)

Figure 5 System phase plots for V0 = 150RPM and 119865 = 500N


y1

(mmiddotＭminus2)

times102

1 2 30t (s)

0

10

20

30

x1

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

y2

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

x2

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

yc

(mmiddotＭminus2)

times102

1 2 30t (s)

0

04

08

12


of the fixed-caliper brake system thereby causing the forcesapplied on the caliper and the disc along the 119909-axis to alwaysoccur in pairs of equal magnitude but opposing directionsConsequently the displacement velocity and acceleration of119909119888 and 119909119889 remained 0 during the braking process whichcomprised a horizontal line at 0 in the acceleration timehistory diagrams and a single dot at the origin in the phaseplots Therefore 119909119888 and 119909119889 figures were omitted

All the unstable DOFs of the vibration system wereconcentrated on the 119910-axis in each acceleration time historydiagram However though the vibration on the 119909-axis ofthe inner and outer linings began after the excitation by thebrake pressure119865 but the amplitude of the acceleration rapidlyattenuated and finally reached the steady state

Figures 4ndash7 present the effects of the brake pressureon the system vibration characteristics at a given rotational


times10minus3

x1

(mmiddotＭminus1)


x1 (m)

minus20

minus10

0

10

20

times10minus3

y2

(mmiddotＭminus1)

minus20

minus10

0

10

20

0 50minus50

y2 (m) times10minus3

x2

(mmiddotＭminus1)

minus20

minus10

0

10

20

10 20 300

x2 (m)


yc

(mmiddotＭminus1)

minus20

minus10

0

10

20

y1

(mmiddotＭminus1)

times10minus3

minus20

minus10

0

10

20

0 50

y1 (m)minus50



y1

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

x1

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

y2

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

x2

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

yc

(mmiddotＭminus2)

times101

1 2 30t (s)

0

05

10

15


speed The vibrations for all the DOFs of the system can beattenuated and stabilized with the trajectory rotating inwardand gradually converging on the focus of the phase plots inFigure 5 when the brake pressure was not sufficiently highHowever the stabilities of both linings and the caliper alongthe 119910-axis were lost when the brake pressure increased tothe stability threshold for a given speed Figure 6 shows thattheir acceleration increased to a certain value Meanwhile

the trajectory rotates outward and limit cycles were formedin Figure 7 Thus the phase plots were consistent with thesystem stability characteristics determined by calculating thesystem eigenvalues

The system stability transition at a given speed wasdirectly related to the negative slope of the Stribeck frictioneffect which caused the friction coefficient to decrease withthe increasing relative speed Consequently in contrary to the


y1

(mmiddotＭminus1)

times10minus3

minus02

minus01

0

01

02

050 10


x1

(mmiddotＭminus1)


x1 (m)

minus10

minus05

0

05

10

times10minus3

y2

(mmiddotＭminus1)

minus02

minus01

0

01

02

050 10


x2

(mmiddotＭminus1)

minus10

minus05

0

05

10

05 10 150

x2 (m)

yc (m) times10minus3minus05 0 05 10

yc

(mmiddotＭminus1)

minus02

minus01

0

01

02



806040200SP

L (d

B)

0 2 4 6 8 10 12 14 16 18 20Frequency (Hz)

minus20

3 kmh amp 15－０

3 kmh amp 10－０

10 kmh amp 15－０

Figure 10 Brake noise under different dragging conditions

case of the viscous damping the equivalent damping causedby the velocity feedback was negative with the negativeequivalent damping effect becoming more significant withthe increasing brake pressure because the friction force wasdirectly proportional to the normal force Finally the stabilityof the system was lost and a self-excited vibration wasinduced when the total damping of the system was negative

Figures 4 5 8 and 9 show the effect of the disc rotationalspeed on the system vibration for a given brake pressureThe stability of the vibration system was lost at a certainbrake pressure with the decreasing disc rotational speedThisobservation was related to the trend of the friction coefficientversus the speed curve According to (4) the Stribeck frictionmodel curve was actually a transformed exponential curve(Figure 3)The slope of the curve decreased with the decreas-ing disc rotational speed Hence the negative equivalentdamping caused by the velocity feedback decreased as thespeed decreased thereby consequently decreasing the totaldamping with the system becoming unstable when the totaldamping is negative

4 Dynamometer Tests

The abovementioned assertion that the stability of a fixed-caliper brake system decreases with an increasing brakepressure and a decreasing disc rotational speed was based ona numerical investigation using a vibration model Anappropriate fixed-caliper disc brake was used to conducta dynamometer test using the standard SAE J2521 testprocedure and validate the numerical findings

The test data presented in Figure 10 were obtained fromthree stops along the corresponding drag cycles using thesame initial brake temperature condition as prescribed inSection 3 Module 4 of SAE J2521 A brake squeal with a fre-quency of 11100Hz and a sound pressure level (SPL) of 83 dBwas observed at a drag speed of 3 kmh and a brake pressureof 15MPa However no squeal was observed and themaximum sound pressure level was 57 dB when the brakepressure was decreased to 10MPa or the drag speed wasincreased to 10 kmh This finding indicated that the proba-bility of brake noise generation decreased with the decreasingbrake pressure and the increasing brake speed Figure 11shows the SPL data for all the stops prescribed in Section 3Module 4 of SAE J2521 which substantiates the abovemen-tioned observations on the brake noise occurrence

00 05 10 15 20 25 30

Brake pressure (MPa)

3 ＥＧＢ

10 ＥＧＢ

0

20

40

60

80

100

0

20

40

60

80

100

Max

imum

SPL

(dB)

Figure 11 Maximum SPL data obtained as prescribed in Section 3Module 4 of SAE J2521

80604020

0SPL

(dB)

0 2 4 6 8 10 12 14 16 18 20Frequency (kHz)

12

34

56

Tim

e (s)

minus20

Figure 12 Brake noise during the deceleration braking process

The effects of the disc rotational speed on the brake noisewere not only investigated in the drag section of thedynamometer test but also in the deceleration sectionFigure 12 shows the complete brake noise spectrum for theninth stop prescribed in Section 7 Module 7 of SAE J2521The disc was forced-braked from 50 kmh to rest under abrake pressure of 2MPa at this stop The figure showed nonoise at the beginning of the braking process However thebrake noise with an 11000Hz frequency was generated in thefinal stage of the process as the rotation speed of the discdecreased with time

5 Conclusion

This study developed a 7-DOF vibration model of a fixed-caliper disc brake based on the Stribeck effect The eigen-values of the corresponding Jacobian matrix were calculatedto analyze the system stability Furthermore the accelerationtime history diagrams and the phase plots of each DOF in thevibration system were obtained by solving the systemequations revealing regular relationships among the brakepressure disc rotational speed and generated noise Thedynamometer tests were used to determine the systemparameters and verify the numerical analysis results Themain conclusions drawn from the findings of the study areas follows


(1) The probability of brake noise generation increaseswith the increasing brake pressure because the veloc-ity feedback causes a negative equivalent dampingwhich increases with the increasing brake pressure

(2) The probability of brake noise generation increaseswith the decreasing disc rotational speed becausethe slope of the curve decreases with the decreasingdisc rotational speed and the negative equivalentdamping caused by the velocity feedback decreasesas the speed decreases which also decreases the totalsystem damping

(3) The 7-DOF brake vibration model based on theStribeck effect can be used to design and investigatethe brake system parameters because its brake noisepredictions agree well with the dynamometer testresults

Conflicts of Interest

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

References

[1] P Liu H Zheng C Cai et al ldquoAnalysis of disc brake squealusing the complex eigenvalue methodrdquo Applied Acoustics vol68 no 6 pp 603ndash615 2007

[2] J G P da Silva E R Fulco P E D Varante V Do NascimentoF N Diesel and D L Boniatti ldquoNumerical and experimentalevaluation of brake squealrdquo SAE Technical Papers 2013-36-0030 2013

[3] A Belhocine and N M Ghazaly ldquoEffects of material propertieson generation of brake squeal noise using finite elementmethodrdquo Latin American Journal of Solids and Structures vol12 no 8 pp 1432ndash1447 2015

[4] A Papinniemi J C S Lai J Zhao and L Loader ldquoBrake squeala literature reviewrdquoApplied Acoustics vol 63 no 4 pp 391ndash4002002

[5] L T Matozo A Tamagna H A Al-Qureshi A R Menetrierand P E Varante ldquoInvestigation of the relation between frictionmaterial abrasive properties and the squeal noise tendencyin automotive disc brake systemsrdquo International Journal ofAcoustics and Vibrations vol 17 no 4 pp 200ndash203 2012

[6] S-K Yang Z Sun Y-J Liu B-W Lu T Liu and H-S HouldquoAutomotive brake squeal simulation and optimizationrdquo SAETechnical Paper 2016-01-1298 2016

[7] F Chen ldquoDisc brake squeal an overviewrdquo SAETechnical Papers2007-01-0587 2007

[8] J Kang ldquoFinite element modeling for stick-slip pattern ofsqueal modes in disc brakerdquo Journal of Mechanical Science andTechnology vol 28 no 10 pp 4021ndash4026 2014

[9] S Hegde and B-S Suresh ldquoStudy of friction induced stick-slip phenomenon in a minimal disc brake modelrdquo Journal ofMechanical Engineering and Automation vol 5 pp 100ndash1062015

[10] C-Y Teoh and Z-M Ripin ldquoTransient analysis of drum brakesqueal with binary flutter and negative friction-velocity insta-bility mechanismsrdquo Journal of Vibroengineering vol 13 no 2pp 275ndash287 2011

[11] T Nakae T Ryu A Sueoka Y Nakano and T Inoue ldquoSquealand chatter phenomena generated in a mountain bike discbrakerdquo Journal of Sound andVibration vol 330 no 10 pp 2138ndash2149 2011

[12] A R AbuBakar and H Ouyang ldquoA prediction methodology ofdisk brake squeal using complex eigenvalue analysisrdquo Interna-tional Journal of Vehicle Design vol 46 no 4 pp 416ndash435 2008

[13] N-P Hoffmann and L Gaul ldquoFriction induced vibrations ofbrakes research fields and activitiesrdquo SAE Technical Papers2008-01-2579 2008

[14] N M Kinkaid O M OrsquoReilly and P Papadopoulos ldquoAutomo-tive disc brake squealrdquo Journal of Sound and Vibration vol 267no 1 pp 105ndash166 2003

[15] F Chen M-K Abdelhamid P Blaschke and J Swayze ldquoOnautomotive disc brake squeal part III test and evaluationrdquo SAETechnical Papers 2003-01-1622 2003

[16] K Shin M J Brennan J-E Oh and C J Harris ldquoAnalysis ofdisc brake noise using a two-degree-of-freedommodelrdquo Journalof Sound and Vibration vol 254 no 5 pp 837ndash848 2002

[17] F-H Yang W Zhang and J Wang ldquoSliding bifurcations andchaos induced by dry friction in a braking systemrdquo ChaosSolitons amp Fractals vol 40 no 3 pp 1060ndash1075 2009

[18] M Paliwal A Mahajan J Don T Chu and P Filip ldquoNoiseand vibration analysis of a disc-brake system using a stick-slipfriction model involving coupling stiffnessrdquo Journal of Soundand Vibration vol 282 no 3-5 pp 1273ndash1284 2005

[19] N-M Kinkaid O-M OReilly and P Papadopoulos ldquoOnthe transient dynamics of a multi-degree-of-freedom frictionoscillator a new mechanism for disc brake noiserdquo Journal ofsound and vibration vol 287 no 4 pp 901ndash917 2005

[20] I Ahmed ldquoAnalysis of ventilated disc brake squeal using a 10dof modelrdquo SAE Technical Papers 2012-01-1827 2012

[21] H Wang X Liu Y Shan and T He ldquoNonlinear behaviorevolution and squeal analysis of disc brake based on differentfriction modelsrdquo Journal of Vibroengineering vol 16 no 5 pp2593ndash2609 2014

[22] R Stribeck ldquoThe key qualities of sliding and roller bearingsrdquoZeitschrift des Vereines Seutscher Ingenieure vol 46 no 38 pp1342ndash1348 1902

[23] L-C Bo and D Pavelescu ldquoThe friction-speed relation and itsinfluence on the critical velocity of stick-slipmotionrdquoWear vol82 no 3 pp 277ndash289 1982

[24] A Tustin ldquoThe effects of backlash and of speed-dependentfriction on the stability of closed-cycle control systemsrdquo Journalof the Institution of Electrical EngineersmdashPart IIA AutomaticRegulators and Servo Mechanisms vol 94 no 1 pp 143ndash1511947

[25] B Armstrong-Helouvry ldquoStick-slip arising from Stribeck fric-tionrdquo in Proceedings of the IEEE International Conference onRobotics and Automation pp 1377ndash1382 IEEE CincinnatiOhio USA May 1990

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Journal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Brake pad

Disc

Brake uid

Piston

Caliper

Knuckle

Figure 1 Structure of a fixed-caliper disc brake

4-DOF model of a disc brake with friction and contact lossnonlinearities to investigate the mechanism and the dynamiccharacteristics of a brake squeal [21] The present studydeveloped a 7-DOF vibration model of a fixed-caliper brakesystem based on the Stribeck effect A dynamometer test wasconducted to determine the characteristic systemparametersFurthermore the effects of the braking conditions on thesystem stability were studied

2 Establishment of the BrakeVibration System

21 Establishment of the State Equations Figure 1 shows atypical structure of a fixed-caliper disc brake for a vehi-cle During the braking process the brake fluid pressureincreases and the rotating disc is clamped by both sides ofthe brake linings thereby resulting in the deceleration of themoving vehicle The vibration model of the brake systemwhich basically comprises the caliper disc and both sides ofthe brake linings can be simplified as shown in Figure 2

In Figure 2 119898119889 is the mass of the disc 119898119888 is the massof the caliper 1198981 and 1198982 are the masses of the inner andouter brake linings respectively V0 is the absolute linearvelocity of the brake disc assumed to be in the 119910+ directionand 119865 is the brake pressure on either lining of both sidesconsidering that the inner and outer pistons are of the samesize The movement along the 119909-axis was considered for thedisc while simultaneous movements along the 119909- and 119910-axes were considered for the caliper and both brake liningsSuppose that the disc moves in the x+ direction during thebraking process the inner lining would move in an xminus119910+direction the outer liningwouldmove in an x+119910+ directionand the caliper would move in an x+119910+ direction Thishypothesis was used to define the initial state of the systemto facilitate analysis such that the directions of the reactionforces on the springs and dampers can be obtained The finalresults would not be affected by these assumptions becausethe obtained values of the reaction forces would be negativeif their directions are opposite the assumed direction

mc

md

F F

c2cx

k2cx

k2cy

c2dx

k2dx

m2 m1

cdx

kdx

c1dx

k1dx

k1cy

c1cx

k1cx

c1cy

ccx

kcx

kcy ccyY

X

0

c2cy

Figure 2 Vibration model of a fixed-caliper disc brake

Based on the abovementioned assumptions and the rela-tionships among the different parts in the brake system theequations of the vibration system shown in Figure 2 can beobtained as follows by using Newtonrsquos law

119898119888119888 = minus119888119888119909119888 + 1198882119888119909 (2 minus 119888) minus 1198881119888119909 (119888 minus 1) minus 119896119888119909119909119888+ 1198962119888119909 (1199092 minus 119909119888) minus 1198961119888119909 (119909119888 minus 1199091)

119898119888 119910119888 = minus119888119888119910 119910119888 + 1198882119888119910 ( 1199102 minus 119910119888) + 1198881119888119910 ( 1199101 minus 119910119888) minus 119896119888119910119910119888+ 1198962119888119910 (1199102 minus 119910119888) + 1198961119888119910 (1199101 minus 119910119888)

119898119889119889 = minus119888119889119889 + 1198882119889119909 (2 minus 119889) minus 1198881119889119909 (1 minus 119889)minus 119896119889119909119889 + 1198962119889119909 (1199092 minus 119909119889) minus 1198961119889119909 (119909119889 minus 1199091)

11989811 = minus119865 + 1198881119889119909 (119889 minus 1) + 1198881119888119909 (119888 minus 1)+ 1198961119889119909 (119909119889 minus 1199091) + 1198961119888119909 (119909119888 minus 1199091)

1198981 1199101 = 1198651198911 minus 1198881119888119910 ( 1199101 minus 119910119888) minus 1198961119888119910 (1199101 minus 119910119888) 11989822 = 119865 minus 1198882119889119909 (2 minus 119889) minus 1198882119888119909 (2 minus 119888)

minus 1198962119889119909 (1199092 minus 119909119889) minus 1198962119888119909 (1199092 minus 119909119888) 1198982 1199102 = 1198651198912 minus 1198882119888119910 ( 1199102 minus 119910119888) minus 1198962119888119910 (1199102 minus 119910119888)

(1)

where 1198651198911 and 1198651198912 are the frictional forces between thedisc and the two brake linings respectively The expressionsof which were obtained using a Stribeck model and thedynamometer test results (Section 22 for a detailed descrip-tion) Considering the structural symmetry of a fixed-caliperdisc brake system it was assumed that 1198881 = 1198881119889119909 = 1198882119889119909




119911119894 = 119902119894 119894 = 1 2 7119911119894+7 = 119902119894 119894 = 1 2 7 (2)









(3)


119891 (V) = 119891119888 + (119891119904 minus 119891119888) 119890minus(VV119904)120575 (4)



119891 (V) = 119865 sdot 120583 (V) (5)

120583 (V) = 120583119888 + Δ120583 sdot 119890minus(VV119904)120575 (6)


1198651198911 = sgn(V0 minus 6011991182120587119877119864) sdot 119865


V119904)]

1198651198912 = sgn(V0 minus 6011991192120587119877119864) sdot 119865


V119904)]

(7)




045

040

035

030

025

Fric

tion

coe

cien

t









= 1 2 14119879 = 119885119894 (1199111 1199112 11991114) = 0 (8)


1199111119890 = 1199112119890 = 1198651198910 ( 21198966 +11198962)

1199113119890 = 211986511989101198966

1199114119890 = minus119865(1198961 + 1198963)

1199115119890 = 119865(1198961 + 1198963)

119911119898119890 = 0 119898 = 6 7 14

(9)




minusV0V119904


2120587119877119864V119904 sdot 119890minusV0V119904

sdot 1199119

(11)





1198965 15 times 106 1198885 501198966 15 times 106 1198886 50



15 0 0 2 2 4 4 6 6 625 0 0 2 2 2 4 6 6 642 0 0 0 2 4 4 6 6 683 0 0 0 0 2 2 4 4 6100 0 0 0 0 0 2 4 4 6150 0 0 0 0 0 0 2 4 4200 0 0 0 0 0 0 2 2 4250 0 0 0 0 0 0 0 2 2


119886119894119895 = 120597119885119894 (1199111 1199112 11991114)12059711991111989510038161003816100381610038161003816100381610038161003816100381610038161199111=1199112=sdotsdotsdot=11991114=0

119894 119895 = 1 2 14

det ([119886] minus 120582 [1]) =10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816



10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816= 0

(12)







y1

(mmiddotＭminus2)

times101

1 2 30t (s)

0

20

40

60

x1

(mmiddotＭminus2)

times102

1 2 30t (s)

0

04

08

12

y2

(mmiddotＭminus2)

times101

1 2 30t (s)

0

20

40

60

x2

(mmiddotＭminus2)

times102

1 2 30t (s)

0

04

08

12

yc

(mmiddotＭminus2)

times101

1 2 30t (s)

0

05

10

15





y1

(mmiddotＭminus1)

times10minus3

minus04

minus02

0

02

04

05 100

y1 (m)

times10minus3

y2

(mmiddotＭminus1)

minus04

minus02

0

02

04

05 100

y2 (m)


yc

(mmiddotＭminus1)

minus02

minus01

0

01

02

times10minus3

x1

(mmiddotＭminus1)

minus10

minus05

0

05

10


x1 (m)

times10minus3

x2

(mmiddotＭminus1)

minus10

minus05

0

05

10

05 10 150

x2 (m)



y1

(mmiddotＭminus2)

times102

1 2 30t (s)

0

10

20

30

x1

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

y2

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

x2

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

yc

(mmiddotＭminus2)

times102

1 2 30t (s)

0

04

08

12






times10minus3

x1

(mmiddotＭminus1)


x1 (m)

minus20

minus10

0

10

20

times10minus3

y2

(mmiddotＭminus1)

minus20

minus10

0

10

20

0 50minus50


x2

(mmiddotＭminus1)

minus20

minus10

0

10

20

10 20 300

x2 (m)


yc

(mmiddotＭminus1)

minus20

minus10

0

10

20

y1

(mmiddotＭminus1)

times10minus3

minus20

minus10

0

10

20

0 50

y1 (m)minus50



y1

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

x1

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

y2

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

x2

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

yc

(mmiddotＭminus2)

times101

1 2 30t (s)

0

05

10

15






y1

(mmiddotＭminus1)

times10minus3

minus02

minus01

0

01

02

050 10


x1

(mmiddotＭminus1)


x1 (m)

minus10

minus05

0

05

10

times10minus3

y2

(mmiddotＭminus1)

minus02

minus01

0

01

02

050 10


x2

(mmiddotＭminus1)

minus10

minus05

0

05

10

05 10 150

x2 (m)


yc

(mmiddotＭminus1)

minus02

minus01

0

01

02



806040200SP

L (d

B)

0 2 4 6 8 10 12 14 16 18 20Frequency (Hz)

minus20

3 kmh amp 15－０

3 kmh amp 10－０

10 kmh amp 15－０




4 Dynamometer Tests



00 05 10 15 20 25 30


3 ＥＧＢ

10 ＥＧＢ

0

20

40

60

80

100

0

20

40

60

80

100

Max

imum

SPL

(dB)


80604020

0SPL

(dB)

0 2 4 6 8 10 12 14 16 18 20Frequency (kHz)

12

34

56

Tim

e (s)

minus20



5 Conclusion








References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







119911119894 = 119902119894 119894 = 1 2 7119911119894+7 = 119902119894 119894 = 1 2 7 (2)









(3)


119891 (V) = 119891119888 + (119891119904 minus 119891119888) 119890minus(VV119904)120575 (4)



119891 (V) = 119865 sdot 120583 (V) (5)

120583 (V) = 120583119888 + Δ120583 sdot 119890minus(VV119904)120575 (6)


1198651198911 = sgn(V0 minus 6011991182120587119877119864) sdot 119865


V119904)]

1198651198912 = sgn(V0 minus 6011991192120587119877119864) sdot 119865


V119904)]

(7)




045

040

035

030

025

Fric

tion

coe

cien

t









= 1 2 14119879 = 119885119894 (1199111 1199112 11991114) = 0 (8)


1199111119890 = 1199112119890 = 1198651198910 ( 21198966 +11198962)

1199113119890 = 211986511989101198966

1199114119890 = minus119865(1198961 + 1198963)

1199115119890 = 119865(1198961 + 1198963)

119911119898119890 = 0 119898 = 6 7 14

(9)




minusV0V119904


2120587119877119864V119904 sdot 119890minusV0V119904

sdot 1199119

(11)





1198965 15 times 106 1198885 501198966 15 times 106 1198886 50



15 0 0 2 2 4 4 6 6 625 0 0 2 2 2 4 6 6 642 0 0 0 2 4 4 6 6 683 0 0 0 0 2 2 4 4 6100 0 0 0 0 0 2 4 4 6150 0 0 0 0 0 0 2 4 4200 0 0 0 0 0 0 2 2 4250 0 0 0 0 0 0 0 2 2


119886119894119895 = 120597119885119894 (1199111 1199112 11991114)12059711991111989510038161003816100381610038161003816100381610038161003816100381610038161199111=1199112=sdotsdotsdot=11991114=0

119894 119895 = 1 2 14

det ([119886] minus 120582 [1]) =10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816



10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816= 0

(12)







y1

(mmiddotＭminus2)

times101

1 2 30t (s)

0

20

40

60

x1

(mmiddotＭminus2)

times102

1 2 30t (s)

0

04

08

12

y2

(mmiddotＭminus2)

times101

1 2 30t (s)

0

20

40

60

x2

(mmiddotＭminus2)

times102

1 2 30t (s)

0

04

08

12

yc

(mmiddotＭminus2)

times101

1 2 30t (s)

0

05

10

15





y1

(mmiddotＭminus1)

times10minus3

minus04

minus02

0

02

04

05 100

y1 (m)

times10minus3

y2

(mmiddotＭminus1)

minus04

minus02

0

02

04

05 100

y2 (m)


yc

(mmiddotＭminus1)

minus02

minus01

0

01

02

times10minus3

x1

(mmiddotＭminus1)

minus10

minus05

0

05

10


x1 (m)

times10minus3

x2

(mmiddotＭminus1)

minus10

minus05

0

05

10

05 10 150

x2 (m)



y1

(mmiddotＭminus2)

times102

1 2 30t (s)

0

10

20

30

x1

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

y2

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

x2

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

yc

(mmiddotＭminus2)

times102

1 2 30t (s)

0

04

08

12






times10minus3

x1

(mmiddotＭminus1)


x1 (m)

minus20

minus10

0

10

20

times10minus3

y2

(mmiddotＭminus1)

minus20

minus10

0

10

20

0 50minus50


x2

(mmiddotＭminus1)

minus20

minus10

0

10

20

10 20 300

x2 (m)


yc

(mmiddotＭminus1)

minus20

minus10

0

10

20

y1

(mmiddotＭminus1)

times10minus3

minus20

minus10

0

10

20

0 50

y1 (m)minus50



y1

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

x1

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

y2

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

x2

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

yc

(mmiddotＭminus2)

times101

1 2 30t (s)

0

05

10

15






y1

(mmiddotＭminus1)

times10minus3

minus02

minus01

0

01

02

050 10


x1

(mmiddotＭminus1)


x1 (m)

minus10

minus05

0

05

10

times10minus3

y2

(mmiddotＭminus1)

minus02

minus01

0

01

02

050 10


x2

(mmiddotＭminus1)

minus10

minus05

0

05

10

05 10 150

x2 (m)


yc

(mmiddotＭminus1)

minus02

minus01

0

01

02



806040200SP

L (d

B)

0 2 4 6 8 10 12 14 16 18 20Frequency (Hz)

minus20

3 kmh amp 15－０

3 kmh amp 10－０

10 kmh amp 15－０




4 Dynamometer Tests



00 05 10 15 20 25 30


3 ＥＧＢ

10 ＥＧＢ

0

20

40

60

80

100

0

20

40

60

80

100

Max

imum

SPL

(dB)


80604020

0SPL

(dB)

0 2 4 6 8 10 12 14 16 18 20Frequency (kHz)

12

34

56

Tim

e (s)

minus20



5 Conclusion








References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







045

040

035

030

025

Fric

tion

coe

cien

t









= 1 2 14119879 = 119885119894 (1199111 1199112 11991114) = 0 (8)


1199111119890 = 1199112119890 = 1198651198910 ( 21198966 +11198962)

1199113119890 = 211986511989101198966

1199114119890 = minus119865(1198961 + 1198963)

1199115119890 = 119865(1198961 + 1198963)

119911119898119890 = 0 119898 = 6 7 14

(9)




minusV0V119904


2120587119877119864V119904 sdot 119890minusV0V119904

sdot 1199119

(11)





1198965 15 times 106 1198885 501198966 15 times 106 1198886 50



15 0 0 2 2 4 4 6 6 625 0 0 2 2 2 4 6 6 642 0 0 0 2 4 4 6 6 683 0 0 0 0 2 2 4 4 6100 0 0 0 0 0 2 4 4 6150 0 0 0 0 0 0 2 4 4200 0 0 0 0 0 0 2 2 4250 0 0 0 0 0 0 0 2 2


119886119894119895 = 120597119885119894 (1199111 1199112 11991114)12059711991111989510038161003816100381610038161003816100381610038161003816100381610038161199111=1199112=sdotsdotsdot=11991114=0

119894 119895 = 1 2 14

det ([119886] minus 120582 [1]) =10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816



10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816= 0

(12)







y1

(mmiddotＭminus2)

times101

1 2 30t (s)

0

20

40

60

x1

(mmiddotＭminus2)

times102

1 2 30t (s)

0

04

08

12

y2

(mmiddotＭminus2)

times101

1 2 30t (s)

0

20

40

60

x2

(mmiddotＭminus2)

times102

1 2 30t (s)

0

04

08

12

yc

(mmiddotＭminus2)

times101

1 2 30t (s)

0

05

10

15





y1

(mmiddotＭminus1)

times10minus3

minus04

minus02

0

02

04

05 100

y1 (m)

times10minus3

y2

(mmiddotＭminus1)

minus04

minus02

0

02

04

05 100

y2 (m)


yc

(mmiddotＭminus1)

minus02

minus01

0

01

02

times10minus3

x1

(mmiddotＭminus1)

minus10

minus05

0

05

10


x1 (m)

times10minus3

x2

(mmiddotＭminus1)

minus10

minus05

0

05

10

05 10 150

x2 (m)



y1

(mmiddotＭminus2)

times102

1 2 30t (s)

0

10

20

30

x1

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

y2

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

x2

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

yc

(mmiddotＭminus2)

times102

1 2 30t (s)

0

04

08

12






times10minus3

x1

(mmiddotＭminus1)


x1 (m)

minus20

minus10

0

10

20

times10minus3

y2

(mmiddotＭminus1)

minus20

minus10

0

10

20

0 50minus50


x2

(mmiddotＭminus1)

minus20

minus10

0

10

20

10 20 300

x2 (m)


yc

(mmiddotＭminus1)

minus20

minus10

0

10

20

y1

(mmiddotＭminus1)

times10minus3

minus20

minus10

0

10

20

0 50

y1 (m)minus50



y1

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

x1

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

y2

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

x2

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

yc

(mmiddotＭminus2)

times101

1 2 30t (s)

0

05

10

15






y1

(mmiddotＭminus1)

times10minus3

minus02

minus01

0

01

02

050 10


x1

(mmiddotＭminus1)


x1 (m)

minus10

minus05

0

05

10

times10minus3

y2

(mmiddotＭminus1)

minus02

minus01

0

01

02

050 10


x2

(mmiddotＭminus1)

minus10

minus05

0

05

10

05 10 150

x2 (m)


yc

(mmiddotＭminus1)

minus02

minus01

0

01

02



806040200SP

L (d

B)

0 2 4 6 8 10 12 14 16 18 20Frequency (Hz)

minus20

3 kmh amp 15－０

3 kmh amp 10－０

10 kmh amp 15－０




4 Dynamometer Tests



00 05 10 15 20 25 30


3 ＥＧＢ

10 ＥＧＢ

0

20

40

60

80

100

0

20

40

60

80

100

Max

imum

SPL

(dB)


80604020

0SPL

(dB)

0 2 4 6 8 10 12 14 16 18 20Frequency (kHz)

12

34

56

Tim

e (s)

minus20



5 Conclusion








References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







1198965 15 times 106 1198885 501198966 15 times 106 1198886 50



15 0 0 2 2 4 4 6 6 625 0 0 2 2 2 4 6 6 642 0 0 0 2 4 4 6 6 683 0 0 0 0 2 2 4 4 6100 0 0 0 0 0 2 4 4 6150 0 0 0 0 0 0 2 4 4200 0 0 0 0 0 0 2 2 4250 0 0 0 0 0 0 0 2 2


119886119894119895 = 120597119885119894 (1199111 1199112 11991114)12059711991111989510038161003816100381610038161003816100381610038161003816100381610038161199111=1199112=sdotsdotsdot=11991114=0

119894 119895 = 1 2 14

det ([119886] minus 120582 [1]) =10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816



10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816= 0

(12)







y1

(mmiddotＭminus2)

times101

1 2 30t (s)

0

20

40

60

x1

(mmiddotＭminus2)

times102

1 2 30t (s)

0

04

08

12

y2

(mmiddotＭminus2)

times101

1 2 30t (s)

0

20

40

60

x2

(mmiddotＭminus2)

times102

1 2 30t (s)

0

04

08

12

yc

(mmiddotＭminus2)

times101

1 2 30t (s)

0

05

10

15





y1

(mmiddotＭminus1)

times10minus3

minus04

minus02

0

02

04

05 100

y1 (m)

times10minus3

y2

(mmiddotＭminus1)

minus04

minus02

0

02

04

05 100

y2 (m)


yc

(mmiddotＭminus1)

minus02

minus01

0

01

02

times10minus3

x1

(mmiddotＭminus1)

minus10

minus05

0

05

10


x1 (m)

times10minus3

x2

(mmiddotＭminus1)

minus10

minus05

0

05

10

05 10 150

x2 (m)



y1

(mmiddotＭminus2)

times102

1 2 30t (s)

0

10

20

30

x1

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

y2

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

x2

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

yc

(mmiddotＭminus2)

times102

1 2 30t (s)

0

04

08

12






times10minus3

x1

(mmiddotＭminus1)


x1 (m)

minus20

minus10

0

10

20

times10minus3

y2

(mmiddotＭminus1)

minus20

minus10

0

10

20

0 50minus50


x2

(mmiddotＭminus1)

minus20

minus10

0

10

20

10 20 300

x2 (m)


yc

(mmiddotＭminus1)

minus20

minus10

0

10

20

y1

(mmiddotＭminus1)

times10minus3

minus20

minus10

0

10

20

0 50

y1 (m)minus50



y1

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

x1

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

y2

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

x2

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

yc

(mmiddotＭminus2)

times101

1 2 30t (s)

0

05

10

15






y1

(mmiddotＭminus1)

times10minus3

minus02

minus01

0

01

02

050 10


x1

(mmiddotＭminus1)


x1 (m)

minus10

minus05

0

05

10

times10minus3

y2

(mmiddotＭminus1)

minus02

minus01

0

01

02

050 10


x2

(mmiddotＭminus1)

minus10

minus05

0

05

10

05 10 150

x2 (m)


yc

(mmiddotＭminus1)

minus02

minus01

0

01

02



806040200SP

L (d

B)

0 2 4 6 8 10 12 14 16 18 20Frequency (Hz)

minus20

3 kmh amp 15－０

3 kmh amp 10－０

10 kmh amp 15－０




4 Dynamometer Tests



00 05 10 15 20 25 30


3 ＥＧＢ

10 ＥＧＢ

0

20

40

60

80

100

0

20

40

60

80

100

Max

imum

SPL

(dB)


80604020

0SPL

(dB)

0 2 4 6 8 10 12 14 16 18 20Frequency (kHz)

12

34

56

Tim

e (s)

minus20



5 Conclusion








References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





y1

(mmiddotＭminus2)

times101

1 2 30t (s)

0

20

40

60

x1

(mmiddotＭminus2)

times102

1 2 30t (s)

0

04

08

12

y2

(mmiddotＭminus2)

times101

1 2 30t (s)

0

20

40

60

x2

(mmiddotＭminus2)

times102

1 2 30t (s)

0

04

08

12

yc

(mmiddotＭminus2)

times101

1 2 30t (s)

0

05

10

15





y1

(mmiddotＭminus1)

times10minus3

minus04

minus02

0

02

04

05 100

y1 (m)

times10minus3

y2

(mmiddotＭminus1)

minus04

minus02

0

02

04

05 100

y2 (m)


yc

(mmiddotＭminus1)

minus02

minus01

0

01

02

times10minus3

x1

(mmiddotＭminus1)

minus10

minus05

0

05

10


x1 (m)

times10minus3

x2

(mmiddotＭminus1)

minus10

minus05

0

05

10

05 10 150

x2 (m)



y1

(mmiddotＭminus2)

times102

1 2 30t (s)

0

10

20

30

x1

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

y2

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

x2

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

yc

(mmiddotＭminus2)

times102

1 2 30t (s)

0

04

08

12






times10minus3

x1

(mmiddotＭminus1)


x1 (m)

minus20

minus10

0

10

20

times10minus3

y2

(mmiddotＭminus1)

minus20

minus10

0

10

20

0 50minus50


x2

(mmiddotＭminus1)

minus20

minus10

0

10

20

10 20 300

x2 (m)


yc

(mmiddotＭminus1)

minus20

minus10

0

10

20

y1

(mmiddotＭminus1)

times10minus3

minus20

minus10

0

10

20

0 50

y1 (m)minus50



y1

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

x1

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

y2

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

x2

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

yc

(mmiddotＭminus2)

times101

1 2 30t (s)

0

05

10

15






y1

(mmiddotＭminus1)

times10minus3

minus02

minus01

0

01

02

050 10


x1

(mmiddotＭminus1)


x1 (m)

minus10

minus05

0

05

10

times10minus3

y2

(mmiddotＭminus1)

minus02

minus01

0

01

02

050 10


x2

(mmiddotＭminus1)

minus10

minus05

0

05

10

05 10 150

x2 (m)


yc

(mmiddotＭminus1)

minus02

minus01

0

01

02



806040200SP

L (d

B)

0 2 4 6 8 10 12 14 16 18 20Frequency (Hz)

minus20

3 kmh amp 15－０

3 kmh amp 10－０

10 kmh amp 15－０




4 Dynamometer Tests



00 05 10 15 20 25 30


3 ＥＧＢ

10 ＥＧＢ

0

20

40

60

80

100

0

20

40

60

80

100

Max

imum

SPL

(dB)


80604020

0SPL

(dB)

0 2 4 6 8 10 12 14 16 18 20Frequency (kHz)

12

34

56

Tim

e (s)

minus20



5 Conclusion








References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





y1

(mmiddotＭminus1)

times10minus3

minus04

minus02

0

02

04

05 100

y1 (m)

times10minus3

y2

(mmiddotＭminus1)

minus04

minus02

0

02

04

05 100

y2 (m)


yc

(mmiddotＭminus1)

minus02

minus01

0

01

02

times10minus3

x1

(mmiddotＭminus1)

minus10

minus05

0

05

10


x1 (m)

times10minus3

x2

(mmiddotＭminus1)

minus10

minus05

0

05

10

05 10 150

x2 (m)



y1

(mmiddotＭminus2)

times102

1 2 30t (s)

0

10

20

30

x1

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

y2

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

x2

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

yc

(mmiddotＭminus2)

times102

1 2 30t (s)

0

04

08

12






times10minus3

x1

(mmiddotＭminus1)


x1 (m)

minus20

minus10

0

10

20

times10minus3

y2

(mmiddotＭminus1)

minus20

minus10

0

10

20

0 50minus50


x2

(mmiddotＭminus1)

minus20

minus10

0

10

20

10 20 300

x2 (m)


yc

(mmiddotＭminus1)

minus20

minus10

0

10

20

y1

(mmiddotＭminus1)

times10minus3

minus20

minus10

0

10

20

0 50

y1 (m)minus50



y1

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

x1

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

y2

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

x2

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

yc

(mmiddotＭminus2)

times101

1 2 30t (s)

0

05

10

15






y1

(mmiddotＭminus1)

times10minus3

minus02

minus01

0

01

02

050 10


x1

(mmiddotＭminus1)


x1 (m)

minus10

minus05

0

05

10

times10minus3

y2

(mmiddotＭminus1)

minus02

minus01

0

01

02

050 10


x2

(mmiddotＭminus1)

minus10

minus05

0

05

10

05 10 150

x2 (m)


yc

(mmiddotＭminus1)

minus02

minus01

0

01

02



806040200SP

L (d

B)

0 2 4 6 8 10 12 14 16 18 20Frequency (Hz)

minus20

3 kmh amp 15－０

3 kmh amp 10－０

10 kmh amp 15－０




4 Dynamometer Tests



00 05 10 15 20 25 30


3 ＥＧＢ

10 ＥＧＢ

0

20

40

60

80

100

0

20

40

60

80

100

Max

imum

SPL

(dB)


80604020

0SPL

(dB)

0 2 4 6 8 10 12 14 16 18 20Frequency (kHz)

12

34

56

Tim

e (s)

minus20



5 Conclusion








References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





y1

(mmiddotＭminus2)

times102

1 2 30t (s)

0

10

20

30

x1

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

y2

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

x2

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

yc

(mmiddotＭminus2)

times102

1 2 30t (s)

0

04

08

12






times10minus3

x1

(mmiddotＭminus1)


x1 (m)

minus20

minus10

0

10

20

times10minus3

y2

(mmiddotＭminus1)

minus20

minus10

0

10

20

0 50minus50


x2

(mmiddotＭminus1)

minus20

minus10

0

10

20

10 20 300

x2 (m)


yc

(mmiddotＭminus1)

minus20

minus10

0

10

20

y1

(mmiddotＭminus1)

times10minus3

minus20

minus10

0

10

20

0 50

y1 (m)minus50



y1

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

x1

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

y2

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

x2

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

yc

(mmiddotＭminus2)

times101

1 2 30t (s)

0

05

10

15






y1

(mmiddotＭminus1)

times10minus3

minus02

minus01

0

01

02

050 10


x1

(mmiddotＭminus1)


x1 (m)

minus10

minus05

0

05

10

times10minus3

y2

(mmiddotＭminus1)

minus02

minus01

0

01

02

050 10


x2

(mmiddotＭminus1)

minus10

minus05

0

05

10

05 10 150

x2 (m)


yc

(mmiddotＭminus1)

minus02

minus01

0

01

02



806040200SP

L (d

B)

0 2 4 6 8 10 12 14 16 18 20Frequency (Hz)

minus20

3 kmh amp 15－０

3 kmh amp 10－０

10 kmh amp 15－０




4 Dynamometer Tests



00 05 10 15 20 25 30


3 ＥＧＢ

10 ＥＧＢ

0

20

40

60

80

100

0

20

40

60

80

100

Max

imum

SPL

(dB)


80604020

0SPL

(dB)

0 2 4 6 8 10 12 14 16 18 20Frequency (kHz)

12

34

56

Tim

e (s)

minus20



5 Conclusion








References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





times10minus3

x1

(mmiddotＭminus1)


x1 (m)

minus20

minus10

0

10

20

times10minus3

y2

(mmiddotＭminus1)

minus20

minus10

0

10

20

0 50minus50


x2

(mmiddotＭminus1)

minus20

minus10

0

10

20

10 20 300

x2 (m)


yc

(mmiddotＭminus1)

minus20

minus10

0

10

20

y1

(mmiddotＭminus1)

times10minus3

minus20

minus10

0

10

20

0 50

y1 (m)minus50



y1

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

x1

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

y2

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

x2

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

yc

(mmiddotＭminus2)

times101

1 2 30t (s)

0

05

10

15






y1

(mmiddotＭminus1)

times10minus3

minus02

minus01

0

01

02

050 10


x1

(mmiddotＭminus1)


x1 (m)

minus10

minus05

0

05

10

times10minus3

y2

(mmiddotＭminus1)

minus02

minus01

0

01

02

050 10


x2

(mmiddotＭminus1)

minus10

minus05

0

05

10

05 10 150

x2 (m)


yc

(mmiddotＭminus1)

minus02

minus01

0

01

02



806040200SP

L (d

B)

0 2 4 6 8 10 12 14 16 18 20Frequency (Hz)

minus20

3 kmh amp 15－０

3 kmh amp 10－０

10 kmh amp 15－０




4 Dynamometer Tests



00 05 10 15 20 25 30


3 ＥＧＢ

10 ＥＧＢ

0

20

40

60

80

100

0

20

40

60

80

100

Max

imum

SPL

(dB)


80604020

0SPL

(dB)

0 2 4 6 8 10 12 14 16 18 20Frequency (kHz)

12

34

56

Tim

e (s)

minus20



5 Conclusion








References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





y1

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

x1

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

y2

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

x2

(mmiddotＭminus2)

times102

1 2 30t (s)

0

06

12

18

yc

(mmiddotＭminus2)

times101

1 2 30t (s)

0

05

10

15






y1

(mmiddotＭminus1)

times10minus3

minus02

minus01

0

01

02

050 10


x1

(mmiddotＭminus1)


x1 (m)

minus10

minus05

0

05

10

times10minus3

y2

(mmiddotＭminus1)

minus02

minus01

0

01

02

050 10


x2

(mmiddotＭminus1)

minus10

minus05

0

05

10

05 10 150

x2 (m)


yc

(mmiddotＭminus1)

minus02

minus01

0

01

02



806040200SP

L (d

B)

0 2 4 6 8 10 12 14 16 18 20Frequency (Hz)

minus20

3 kmh amp 15－０

3 kmh amp 10－０

10 kmh amp 15－０




4 Dynamometer Tests



00 05 10 15 20 25 30


3 ＥＧＢ

10 ＥＧＢ

0

20

40

60

80

100

0

20

40

60

80

100

Max

imum

SPL

(dB)


80604020

0SPL

(dB)

0 2 4 6 8 10 12 14 16 18 20Frequency (kHz)

12

34

56

Tim

e (s)

minus20



5 Conclusion








References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





y1

(mmiddotＭminus1)

times10minus3

minus02

minus01

0

01

02

050 10


x1

(mmiddotＭminus1)


x1 (m)

minus10

minus05

0

05

10

times10minus3

y2

(mmiddotＭminus1)

minus02

minus01

0

01

02

050 10


x2

(mmiddotＭminus1)

minus10

minus05

0

05

10

05 10 150

x2 (m)


yc

(mmiddotＭminus1)

minus02

minus01

0

01

02



806040200SP

L (d

B)

0 2 4 6 8 10 12 14 16 18 20Frequency (Hz)

minus20

3 kmh amp 15－０

3 kmh amp 10－０

10 kmh amp 15－０




4 Dynamometer Tests



00 05 10 15 20 25 30


3 ＥＧＢ

10 ＥＧＢ

0

20

40

60

80

100

0

20

40

60

80

100

Max

imum

SPL

(dB)


80604020

0SPL

(dB)

0 2 4 6 8 10 12 14 16 18 20Frequency (kHz)

12

34

56

Tim

e (s)

minus20



5 Conclusion








References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





806040200SP

L (d

B)

0 2 4 6 8 10 12 14 16 18 20Frequency (Hz)

minus20

3 kmh amp 15－０

3 kmh amp 10－０

10 kmh amp 15－０




4 Dynamometer Tests



00 05 10 15 20 25 30


3 ＥＧＢ

10 ＥＧＢ

0

20

40

60

80

100

0

20

40

60

80

100

Max

imum

SPL

(dB)


80604020

0SPL

(dB)

0 2 4 6 8 10 12 14 16 18 20Frequency (kHz)

12

34

56

Tim

e (s)

minus20



5 Conclusion








References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of










References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of











Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




disc brake vibration model based on stribeck effect and its ...mathematicalproblemsinengineering 5...

Documents