Wear 301 (2013) 47–56

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Dynamic transient analysis of squealing vibration of a reciprocatingsliding system

W.J. Qian, G.X. Chen n, Z.R. Zhou

Tribology Research Institute, Southwest Jiaotong University, Chengdu, Sichuan 610031, China

a r t i c l e i n f o

Article history:

Received 31 August 2012

Received in revised form

26 December 2012

Accepted 28 December 2012Available online 11 January 2013

Keywords:

Friction

Squeal

Vibration

Finite element

Transient dynamics

48/$ - see front matter & 2013 Elsevier B.V. A

x.doi.org/10.1016/j.wear.2012.12.057

esponding author. Tel./fax: þ86 28 87634122

ail address: chen_guangx@163.com (G.X. Che

a b s t r a c t

A dynamic transient model of a reciprocating sliding system is established and a numerical study of

friction-induced vibrations of the model is performed. Firstly, the propensity of friction-induced

vibrations and noise during a reciprocating sliding is predicted using the finite element complex

eigenvalue method. Secondly, a dynamic transient analysis of the model is carried out using the

ABAQUS software. The result predicted by the complex eigenvalue analysis is compared with the

experimental test result. It is found that these two results have a good agreement in frequency.

A comparison between the dynamic transient analysis results and the experimental results is carried

out, and it is found that the agreement between the simulation results and the experimental results is

generally good (in time and frequency domains). The dynamic transient analysis results demonstrate

that when squeal occurred, the normal accelerations and the tangential accelerations have the same

vibration frequency. This phenomenon indicates that the coupling of normal and tangential vibration is

a main cause resulting in squeal occurrence. Moreover, parameter sensitivity analysis shows that the

normal load, the frequency and the displacement of reciprocating sliding have important influences on

the friction-induced vibrations of the metal reciprocating sliding system.

& 2013 Elsevier B.V. All rights reserved.

1. Introduction

Friction-induced vibrations can be excited when a metalcounterface slides against another metal counterface under cer-tain conditions. These vibrations are responsible for differentnoises and stress concentrations. Friction induced noise is gen-erally classified into two major categories: (a) low frequencynoise (about 10–500 Hz), termed chatter, moan or groan; and(b) medium and high frequency noise (around 500–18,000 Hz),called squeal or squeak. The friction-induced noise has beenstudied over years. Several excellent reviews have been pub-lished. Ibrahim [1,2], Papinniemi et al. [3] and Kinkaid et al. [4]presented a wide panorama of vibration dynamics caused byfriction. Akay [5] has given an overview of friction acoustics. Fourpossible mechanisms of friction-induced noise in disc brakesystems were reported in the literature. Mills [6] showed thatfriction-induced instabilities can occur when the friction coeffi-cient increases as relative velocity goes to zero. Spurr [7] high-lighted the importance of contact kinematics; he obtained aninstability condition on the contact’s angle of incidence with aconstant friction coefficient by using the Sprag–Slip model. North[8] found that friction-induced vibrations are due to a coalescence

ll rights reserved.

.

n).

of two eigenfrequencies of the system in an analytical andexperimental study of a commercial braking system. However,until now there is no single theory that can be generalized toanalyze various noise phenomena [2].

Finite element analysis has been widely used in numericalstudies of the friction-induced noise. There are typically twodifferent analysis methodologies available to predict friction-induced noise using the finite element method, which are complexeigenvalue analysis and dynamic transient analysis [9]. The linearcomplex eigenvalue analysis is based on a linear hypothesis in thecontact zone. It permits detection of the stability limit of the system,by analysing its eigenvalues and eigenvectors around the steadysliding state [10–13]. The positive real parts of the complexeigenvalues indicate the degree of instability of the linear modelof a friction system and are thought to show the likelihood offriction-induced noise occurrence or the noise intensity. However,the non-linear effects of the contact cannot be neglected wheninstability occurs. Thus, the dynamic transient analysis is used tostudy the evolution of the vibration of the system during instability.In the process of dynamic transient analysis, the non-linear aspect ofcontact with friction can be fully considered. It permits obtainingthe values of displacements, velocities and accelerations, as well asthe values of the forces and area of the contact during systemvibrations [14–16]. Recently, several researchers have performedboth types of analysis in their numerical study of the friction-induced vibrations and found that they are complementary [17–19].

Fig. 2. Model of the reciprocating sliding system (a) finite element model and

(b) load and boundary conditions of the finite element model.

W.J. Qian et al. / Wear 301 (2013) 47–5648

The correlation between the numerical and experimentalresults is an interesting issue on the research of friction-inducedvibrations. Many publications presented the comparison betweenthe numerical and experimental results of friction-induced vibra-tions. Most publications showed that the numerical and experi-mental results have a good agreement in unstable frequencies[20]. Besides, some publications showed that the numerical andexperimental results have a good agreement in the level ofvibrations [21]. As far as the authors know, however, the pub-lications on the comparison in both time and frequency domainsbetween the numerical and experimental results of friction-induced vibrations are very few.

This paper presents a numerical study of the friction-inducedvibration under the reciprocating sliding condition. An elasticvibration model of a reciprocating sliding system is established.Complex eigenvalue and dynamic transient analyses were per-formed by using the ABAQUS software. These simulation resultswere compared with the experimental test results. It is found thatthe simulation results are consistent with the experimentalresults both in the time and frequency domains. The dynamictransient analysis methodology can also be used in the research ofbreak squeal or other friction-induced vibration problems. Thispaper’s results demonstrate that if the finite element model of thefriction system is established correctly, we can get the accurateprediction results of friction-induced vibration in both the timeand the frequency domains by using the dynamic transientanalysis methodology.

2. Finite element modeling of a metal reciprocating slidingsystem

2.1. Experimental apparatus and measuring instruments

The experiment was carried out with a reciprocating slidingtest system. The test apparatus is shown in Fig. 1. This test systemdoes not contain any rotational part. It does not yet produce hightemperature at the sliding surfaces during testing. But it is veryeasy to produce squealing noise [22]. The apparatus consists of aflat specimen (passive specimen) attached to a static frame of thematerials test system through a load cell and another flat speci-men (active specimen) mounted on a frame that is connected toan actuator ram of the materials test system. The actuator ramwas driven sinusoidally by a hydraulic power supply. Duringtesting, the normal (x-direction) and tangential (y-direction)vibration accelerations of the passive specimen were measuredseparately with two piezoelectric accelerometers. All flat speci-mens used in the present test were made from AISI 1045 steel.The contact surfaces of the flat specimen were sanded andpolished carefully before the test. First, we use different types ofsandpapers (P400, P600, P800, P1000, and P1500) sanded the

Y

X

Z

Bearing

Accelerometer (y-direction)

Accelerometer (x-direction)

Passive specimen Active specimen

Actuator

Up

Down

Pulley

Weight

Passive specimen frameActive specimen frame

Load cell

Fig. 1. Schematic of the test apparatus.

contact surfaces carefully. During the sanding process, the sand-papers were laid on a flat glass surface. It ensured that the contactsurface is flat. Then, the flat specimen was put into the polishingmachine for polishing the contact surface. The polishing processensured that the initial surface roughness values of the contactsurfaces were about Ra¼0.25 mm. All specimens were carefullydegreased with alcohol and acetone before the tests. The fre-quency of reciprocating sliding (f) was 1 and 2 Hz. The number ofcycles ranged (N) from 1 to 5000. The normal load (Fn) was set to100 N. The amplitude of oscillation of the active specimen (D) was1 mm (corresponding to a stroke length of 2 mm) and withoutlubrication.

2.2. Finite element model of the reciprocating sliding system

The finite element model of the reciprocationg sliding systemshown in Fig. 1 was established as shown in Fig. 2. This modelconsists of five parts: a load cell, a passive specimen frame, an activespecimen frame, a passive specimen (10�10�20 mm3 in size) andan active specimen (40�40�40 mm3 in size). The passive speci-men frame is tied on the load cell, the passive specimen is tied onthe passive specimen frame and the active specimen is tied on theactive specimen frame. The contact between the two specimens(passive and active) is frictional contact. The contact surfaces areconsidered as perfectly flat. The thermal effects are neglected.The material parameters of active and passive specimens are thesame. Density (r) is 7800 kg/m3, Young’s modulus (E) is210,000 MPa, and Poisson’s ratio (n) is 0.3. The load and boundaryconditions of the finite element model are shown in Fig. 2b. Anencastre boundary condition is applied on the center hole in the topof the load cell. The bottom of the active specimen frame is fixed inthe x- and z-directions, and the velocity boundary condition isapplied on it in the y-direction. The normal load is applied on theback of the passive specimen in the x-direction. There are a total of154618 3D continuum elements in the finite element model.

3. Complex eigenvalue analysis

The complex eigenvalues analysis is frequently adopted toidentify the unstable propensity of the friction systems. This is

velocity=-12.566mm/s

velocity=12.566mm/s

down strokeup strokeZ

X

Y

Fig. 3. Difference between the up stroke and down stroke in the complex

eigenvalue analysis.

0.2 0.3 0.4 0.5 0.6 0.7 0.8

2000

3000

4000

5000

6000

7000

8000

9000

10000

34

35

2319.6Hz

3407.6Hz

8909.3Hz 3534

1514

8

Freq

uenc

y f

R (Hz)

Friction coefficient µ

9

8974.7Hz

µ = 0.39

µ = 0.3

µ = 0.37

µ = 0.69

Fig. 4. Mode coalescences of the metal reciprocating sliding system in the up

stroke: Fn¼100N, vs¼12.566 mm/s.

W.J. Qian et al. / Wear 301 (2013) 47–56 49

possible because squeal instability, at its onset, occurs in thelinear field. The finite element analysis software ABAQUS versions6.4 and above provide a complex eigenvalue solution to thestability analysis of friction induced vibration problems. In thepresent paper, ABAQUS’ complex eigenvalue analysis capability isapplied to study stability of the reciprocating sliding system.

3.1. Finite element equations of complex eigenvalue analysis

In the linear stability analysis, the complex eigenvalue pro-blem is solved using the subspace projection method, thus anatural frequency extraction must be performed first in order todetermine the projection subspace. The governing equation of thesystem is as follows:

½M� €xþ½C� _xþ½K�x¼ 0 ð1Þ

where ½M� is the mass matrix, which is symmetric and positivedefinite. ½C� is the damping matrix, which can include friction-induced damping effects as well as material damping contribu-tion. ½K� is the stiffness matrix, which is asymmetric due tofriction. The eigenavlue equation of Eq. (1) can be written asfollows:

l2½M�þl½C�þ½K�

� �F¼ 0 ð2Þ

where l is the eigenvalue, F is the corresponding eigenvector.Because the eigenvalue extraction is performed at a deformedconfiguration, the stiffness matrix ½K� can include initial stress andload stiffness. Both eigenvalues and eigenvectors may be com-plex. This system becomes symmetric when ignoring asymmetriccontributions to the stiffness matrix ½K�. In addition, when damp-ing is ignored, l becomes a pure imaginary eigenvalue, l¼ io, andthe eigenvalue problem now becomes:

�o2½M�þ½KS�� �

f¼ 0 ð3Þ

where ½KS� is the symmetric part of the stiffness matrix, o is aneigenfrequency of the system. This symmetric eigenvalue pro-blem is solved using the subspace iteration eigensolver. The nextstep is that the original matrices are projected in the subspace ofreal eigenvectors f and given as follows:

½Mn� ¼ ½f1,. . .,fN �

T ½M�½f1,. . .,fN �, ð4aÞ

½Cn� ¼ ½f1,. . .,fN�

T ½C�½f1,. . .,fN�, ð4bÞ

½Kn� ¼ ½f1,. . .,fN�

T ½K�½f1,. . .,fN�: ð4cÞ

Now the reduced eigenvalue problem is expressed in thefollowing form:

l2½Mn�þl½Cn

�þ½Kn�

� �Fn¼ 0 ð5Þ

This problem is solved using the QZ method for generalizednonsymmetric eigen-problems. Finally, the eigenvectors of theoriginal system are recovered in the following equation:

Fi ¼ ½f1,. . .,fN�Fn

i ð6Þ

where Fi is the approximation of the ith eigenvector of theoriginal system.

The general solution of Eq. (5) is

uðtÞ ¼X

Fiexp litð Þ ¼X

Fiexp aiþ joið ÞtÞð ð7Þ

where aiþ joi is the ith eigenavlue of Eq. (5) and j is the imaginaryunit.

From Eq. (7), it is seen that when the real part of an eigenvalueis larger than zero, the nodal displacement uðtÞ will increase withtime, which means the vibration of the system is growing and thesystem will become unstable.

The effective damping ratio (z) is a parameter to measure thepropensity of self-excited vibration occurrence. It is defined as

z¼�2ReðlÞ=9ImðlÞ9 ð8Þ

If the effective damping ratio is negative, the system becomesunstable and has a tendency to radiate squealing. Generallyspeaking, the smaller the effective damping ratio, the easier thecorresponding unstable vibration.

3.2. Complex eigenvalue analysis results

The complex eigenvalue analysis can only predict the instabilitypropensity of the friction system under the steady sliding state.Therefore, the sliding direction of the active specimen cannot bechanged during the complex eigenvalue analysis process. As aconsequence, the sliding process of the active specimen was dividedinto two parts, the up stroke and the down stroke. These two slidingprocesses were analyzed separately. The main difference betweenthe up stroke and down stroke analysis is the sliding direction of theactive specimen. In the up stroke analysis, the sliding direction ofthe active specimen is in the y-axis positive direction. In the downstroke analysis, the sliding direction of the active specimen is in they-axis negative direction. Fig. 3 shows the difference between the upstroke and down stroke analysis. In the complex eigenvalue analysis,

0.2 0.3 0.4 0.5 0.6 0.7 0.8

2200

2400

2600

2800

3000

3200

13

12

11

89

10

2447.4Hz µ = 0.68

Freq

uenc

y

f R (Hz)

Friction coefficient µ

Fig. 6. Mode coalescence of the metal reciprocating sliding system in the down

stroke.

W.J. Qian et al. / Wear 301 (2013) 47–5650

the element type was C3D8I (8-node linear brick, incompatiblemodes), the normal force (Fn) was set to 100 N. The contactformulation was finite sliding with penalty method. The frictionformulation was penalty method. The slip velocities (vs) of activespecimen were 12.566 mm/s (up stroke) and �12.566 mm/s (downstroke). The value of slip velocity is equal to the maximum speed ofthe active specimen under the conditions of oscillation amplitudeD¼1 mm and reciprocating sliding frequency f¼2 Hz.

3.2.1. Analysis results in the up stroke

According to Chen et al. [22], the experimental result showsthat the friction-induced squeal may occur when friction coeffi-cient (m) is greater than 0.28. Fig. 4 shows the complex eigenvalueanalysis results of the system as a function of the frictioncoefficient (m). In the frequency domain considered (o10 kHz)and for friction coefficient (m) between 0.2 and 0.8, three modecoalescences are observed. Their unstable vibration frequencies(fR) and critical friction coefficients (mc) are: fR¼2319.6 Hz,mc¼0.69; fR¼3407.6 Hz, mc¼0.37; fR¼8974.7 Hz, mc¼0.3. FromFig. 4, it is found that mode 34 and mode 35 separate whenm¼0.39. This suggests the corresponding unstable vibration(8974.7 Hz) is unlikely to occur during the up stroke process.Mode 8 and mode 9 coalesce at 2319.6 Hz when m¼0.69.The experimental study indicated that when a metal counterfaceslides against another metal counterface, it is difficult for thefriction coefficient (m) to reach 0.69. The squeal at 2319.6 Hz isalso difficult to occur. This suggests that the unstable vibration at3407.6 Hz most easily occurs in the up stroke process. Underthe same conditions (Fn¼100 N, D¼1 mm, f¼2 Hz), the experi-mental result is 3415.53 Hz. The percentage error between theeigenvalue analysis and experimental results is only 0.23%. Fig. 5shows the mode shape of the unstable vibration at 3407.6 Hz.Fig. 5a–c show the oscillation amplitudes in x-, y- and z-directionsrespectively. It can be found that the main vibrations occurredboth in the x- and y-directions. This phenomenon demonstratesthat when unstable vibrations occurred, the normal vibration(in x-direction) was coupled with the tangential vibration(in y-direction).

3.2.2. Analysis results in the down stroke

In the down stroke process, the complex eigenvalue analysisshows that no mode coalescence occurs when friction coefficient(m) is less than 0.68. When m¼0.68, mode 10 and mode 11coalesce at 2442.7 Hz as shown in Fig. 6. However, experimental

Fig. 5. Model shape of the unstable vibration in the up stroke process, Fn¼100 N, vs¼12

the x-direction; (b) oscillation amplitudes in the y-direction and (c) oscillation amplitu

result shows that the friction coefficient of a metal counterfaceagainst another metal counterface is always less than 0.68.This suggests that squeal is difficult to occur in the down strokeprocess. This is consistent with the experimental result. Fig. 7shows the unstable mode shape whose frequency is 2442.7 Hz.It is seen that self-excited vibration most probably takes place onthe mounting bolt.

4. Dynamic transient analysis and discussion

In this study, the explicit dynamic finite element code ABA-QUS/Explicit is used to simulate the vibration behavior of themetal reciprocating sliding system in time domain. This dynamicanalysis procedure is based on the implementation of an explicitcentral-difference time integration rule together with the use ofdiagonal lumped element mass matrices. This method is able toevaluate the normal and tangential contact stresses along thecontact region and to determine whether the contact surfacesstick, slide or separate locally. The friction effect between theactive specimen and passive specimen can be fully considered inthe analysis process.

.566 mm/s, m¼0.38, fR¼3407.6 Hz and z¼�0.01034: (a) Oscillation amplitudes in

des in the z-direction.

Fig. 7. Mode shape of unstable vibration in the down stroke: Fn¼100 N,

vs¼�12.566 mm/s, m¼0.68, fR¼2442.7 Hz,and z¼�0.00541.

Down stroke

Time (s)

Up stroke

Velo

city

(m

m/s

)Nor

mal

load

(N)

0

0

t0

f0

Fig. 8. Time history of the normal load and the velocity of the active specimen in

the dynamic transient analysis.

W.J. Qian et al. / Wear 301 (2013) 47–56 51

4.1. Finite element equations of dynamic transient analysis

In the process of dynamic transient analysis, the followingfinite element equation of motion is solved:

½M� €xðtÞ ¼ PðtÞ�IðtÞ ð9Þ

At the beginning of the increment, accelerations are computedas follows:

€xðtÞ ¼ ½M��1 PðtÞ�IðtÞ� �

ð10Þ

where ½M� is the diagonal lumped element mass matrix, PðtÞ is theapplied load vector, IðtÞ is the internal force vector, €xðtÞ is theacceleration vector, and t is the time.

Eq. (10) is integrated using the explicit central-differenceintegration rule. The velocity and the displacement vector of thebody are given in the following equations:

_x tþDt=2ð Þ ¼_x t�Dt=2ð Þþ

Dt tþDtð Þ þDtðtÞ� �

2€xðtÞ ð11Þ

x tþDtð Þ ¼ xðtÞ þDt tþDtð Þ_x tþDt=2ð Þ ð12Þ

where x tþDtð Þ is the displacement vector, _xðtþDt=2Þ is the velocityvector, Dt is the increment of time, the subscript tþDt=2

� �and

t�Dt=2� �

refer to mid-increment values. Since the central differ-ence operator is not self-starting because of the mid-increment ofvelocity, the initial values at time t¼0 for velocity and accelera-tion need to be defined. In this case, both values are set to zero asthe active specimen is stationary at time t¼0.

According to the velocity vector _xðtþDt=2Þ, the explicit solver cancalculate the element strain rate _e. The element strain increment defrom element strain rate _e can be computed. The element internal

stress sðtþDtÞ can be obtained by solving the constitutive equation ofthe element.

s tþDtð Þ ¼ f sðtÞ,de� �

ð13Þ

Then explicit solver can calculate the internal force vector I tþDtð Þ

from element internal stress sðtþDtÞ. Finally, we can set the time t astþDt, and calculate the next increment step.

As opposed to the implicit dynamic integration, the explicitdynamic integration does not need a convergent solution beforeattempting the next increment step. The central differenceoperator used in the explicit time integration is conditionallystable, and the stability limit for the operator is given in terms ofthe highest eigenvalue (omax) in the system. Abaqus/Explicit usesan adaptive algorithm to determine conservative bounds for thehighest element frequency.

Dtr2

omaxð14Þ

From formula (14), it is clearly found that the Dt is very small.That is why the explicit dynamic transient analysis needs a longcomputing time.

4.2. Time domain simulation results

Fig. 8 shows the time history of the normal load and thevelocity of the active specimen. At the first stage, a normal load isapplied gradually until it reaches the prescribed value f0 and thenis kept constant. At time t0, a sinusoidally varying velocity in they-direction is applied on the bottom of the active specimen frameto set a reciprocating motion of the active specimen. In thedynamic transient analysis, the element type was C3D8R (8-nodelinear brick, reduced integration), the contact formulation waskinematic method, the sliding formulation was finite sliding andthe friction formulation was the penalty method. The measure-ment point of the acceleration was laid on the back of the passivespecimen frame, which is the same as in the experimental study.

Fig. 9 shows several experimental and simulation results ofvibration acceleration in time and frequency domains. The normalload and the motion parameters of the active specimen are given asfollowed. Normal load Fn¼100 N, oscillation amplitude of the activespecimen D¼1 mm, frequency of reciprocating sliding f¼2 Hz, fric-tion coefficient m¼0.38. Fig. 9a shows the displacement of the activespecimen in the y-direction. Fig. 9b and d show the experimental anddynamic transient analysis results of the acceleration of the measure-ment point in the x- and y-directions. The experimental results showthat in most cases, the unstable vibrations only occur in the up strokeprocess. The dynamic transient analysis demonstrates similar results.Furthermore, comparing the dynamic transient analysis result withthe experimental result, it can be found that the exponential growth

0.0 0.2 0.4 0.6 0.8 1.0-1.0

-0.5

0.0

0.5

1.0Down stroke

Dis

plac

emen

t (m

m)

Time (s)

Up stroke

0.0 0.2 0.4 0.6 0.8 1.0-240

-120

0

120

2400.0 0.2 0.4 0.6 0.8 1.0

-240

-120

0

120

240

Acc

eler

atio

ns in

the

x-d

irect

ion

(m/s

2 )

Time (s)

Dynamic transient analysis result

Experimental result

0 2000 4000 6000 8000 100000

6

12

18

240 2000 4000 6000 8000 10000

0

2

4

6

Dynamic transient analysis result

Experimental result

PSD

of t

he a

ccel

erat

ions

in th

e x-

dire

ctio

n (m

2 /s3 )

Frequency (Hz)

3415.53Hz

3281.25Hz

0.0 0.2 0.4 0.6 0.8 1.0-100

-50

0

50

1000.0 0.2 0.4 0.6 0.8 1.0

-100

-50

0

50

100

Dynamic transient analysis result

Experimental result

Acc

eler

atio

ns in

the

y-di

rect

ion

(m/s

2)

Time (s)

0 2000 4000 6000 8000 100000.00.51.01.52.02.5

0 2000 4000 6000 8000 100000.00.51.01.52.02.5

Dynamic transient analysis result

Experimental result

PSD

of t

he a

ccel

erat

ions

in th

e y-

dire

ctio

n (m

2 /s3 )

Frequency (Hz)

3281.25Hz

3386.23Hz

Fig. 9. Experimental and simulation results, Fn¼100 N, D¼1 mm, f¼2 Hz, m¼0.38: (a) displacement of the active specimen in the y-direction; (b) accelerations in the

x-direction; (c) PSD of the accelerations in the x-direction; (d) accelerations in the y-direction and (e) PSD of the accelerations in the y-direction.

W.J. Qian et al. / Wear 301 (2013) 47–5652

of the simulation acceleration in its initiation formation stage isconsistent with the squealing vibration evolution observed in theexperiments. And the oscillation amplitudes of the simulation accel-eration of the measurement point in both x- and y-directions have thesame order as measured during the experiment.

Fig. 9c shows the power spectral density (PSD) of the simula-tion and measured accelerations in the x-direction. The experi-mental result is characterized by one main frequency of3415.53 Hz. The dynamic transient simulation result is alsocharacterized by one main frequency of 3281.25 Hz, which isslightly different from the experimental result. The percentageerror between them is 3.9%. Fig. 9e shows the PSD of thesimulation and measured accelerations in the y-direction.The main vibration frequency of the experimental result in they-direction is 3386.23 Hz, and the main vibration frequency of thesimulation result in the y-direction is 3281.25 Hz. The percentageerror between them is 3.1%.

For the dynamic transient analysis results, the PSD analyses(Fig. 9c and e) show that the frequency of unstable vibration inthe x-direction is the same as that in the y-direction. Thisphenomenon demonstrates that the normal vibration is coupledwith tangential vibration at the frequency of 3281.25 Hz. In the

complex eigenvalue analysis, the mode shape of the unstablevibration also shows that the normal vibration is coupled with thetangential vibration as shown in Fig. 5. In the experimentalresults, the main frequency of unstable vibration in the x-direc-tions is very close to that in the y-direction (Fig. 9c and e).This slight difference in the experimental results is due to thenormal and tangential piezoelectric accelerometers were installedon different locations. The location difference may cause somemeasurement errors in the experimental process. This suggeststhat the normal vibration is also coupled with the tangentialvibration in the experimental process. Therefore, we consider thatthe coupling of normal and tangential vibrations is a main causeresulting in squeal occurrence.

Fig. 10 shows another experimental and dynamic transientsimulation results under the conditions of Fn¼100 N, D¼1 mm,f¼1 Hz and m¼0.38. It is seen that the experimental and thedynamic transient simulation results also have a good agreementin this case. From Fig. 10a, b and d, it is found that the unstablevibrations only occur in the up stroke process. The simulation andmeasured accelerations are in the same order in magnitude.Fig. 10c and e shows the PSD of the simulation and measuredaccelerations in the x- and y-directions. The percentage errors of

0.0 0.2 0.4 0.6 0.8 1.0-1.0

-0.5

0.0

0.5

1.0

Down stroke

Dis

plac

emen

t (m

m)

Time (s)

Up stroke

0.0 0.2 0.4 0.6 0.8 1.0-180

-90

0

90

180

0.0 0.2 0.4 0.6 0.8 1.0-180

-90

0

90

180

Dynamic transient analysis result

Experimental result

Acc

eler

atio

ns in

the

x-d

irect

ion

(m/s

2 )

Time (s)

0 2000 4000 6000 8000 100000

5

10

15

20

0 2000 4000 6000 8000 100000

1

2

3

4

5

PSD

of t

he a

ccel

erat

ions

in th

e x-

dire

ctio

n (m

2 /s3 )

PSD

of t

he a

ccel

erat

ions

in th

e y-

dire

ctio

n (m

2 /s3 )

Dynamic transient analysis result

Experimental result

Frequency (Hz)

3395.96Hz

3349.61Hz

0.0 0.2 0.4 0.6 0.8 1.0-80

-40

0

40

80

0.0 0.2 0.4 0.6 0.8 1.0-80

-40

0

40

80

Acc

eler

atio

ns in

the

y-d

irect

ion

(m/s

2 )

Time (s)

Dynamic transient analysis result

Experimental result

0 2000 4000 6000 8000 100000.0

0.4

0.8

1.2

1.60 2000 4000 6000 8000 10000

0.0

0.1

0.2

0.3

0.4

Dynamic transient analysis result

Experimental result

Frequency (Hz)

3349.61Hz

3425.29Hz

Fig. 10. Experimental and simulation results, Fn¼100 N, D¼1 mm, f¼1 Hz, m¼0.38: (a) displacement of the active specimen in the y-direction, (b) accelerations in the

x-direction; (c) PSD of the accelerations in the x-direction; (d) accelerations in the y-direction and (e) PSD of the accelerations in the y-direction.

W.J. Qian et al. / Wear 301 (2013) 47–56 53

the vibration frequency between the experimental and dynamictransient simulation results are 1.4% (in x-direction) and 2.2%(in y-direction), respectively. Based on these results, it can beconcluded that the dynamic transient simulation can accuratelypredict the frequencies and amplitudes of the unstable vibrationsof the friction system.

4.3. Parameter sensitivity analysis

4.3.1. Influence of the normal load

The dynamic transient simulations were performed for differ-ent normal loads (Fn). Fig. 11 shows simulation accelerations inthe x- and y-directions for normal loads (Fn) of 100 N, 200 N and300 N. From Fig. 11a and b, it is found that with the increase ofthe normal load, the oscillation amplitude of the accelerationincreases. But the duration of unstable vibration decreases withthe increasing of the normal load. Fig. 12 shows the variation ofthe vibration frequency with normal load. It can be observed thatwith increasing normal load, the frequency corresponding to theunstable vibration increases slightly. This suggests that withincreasing normal load, the duration of the squealing vibrationreduces and the unstable vibration of the system becomes moresevere.

4.3.2. Influence of the oscillation amplitude of the active specimen

Fig. 13 shows the influence of the oscillation amplitude of theactive specimen on the accelerations in the x- and y-directions.It can be observed that with an increase in the oscillationamplitude of active specimen (D), the oscillation amplitude ofthe acceleration corresponding to the unstable vibration isincreased. And the duration of vibration increases with theincreasing of the oscillation amplitude of active specimen.Fig. 14 shows the variation of the vibration frequency with theoscillation amplitude of the active specimen. It is seen that withthe increase of the oscillation amplitude of the active specimen,the frequency of the unstable vibration varies slightly. But thisvariation is not regular. These analysis results show that theduration of the squealing vibration increases as the oscillationamplitude of the active specimen is increased.

4.3.3. Influence of the reciprocating sliding frequency

Fig. 15 shows the variation of the vibration acceleration withtime for several different reciprocating sliding frequencies (f). It isseen that the amplitude of the vibration acceleration increaseswith the increasing of the reciprocating sliding frequency. But theincreasing rate is relatively small. Fig. 16 shows the variation of

0.0 0.2 0.4 0.6 0.8 1.0-600

-300

0

300

600

Acc

eler

atio

ns in

the

x-di

rect

ion (m

/s2 )

Time (s)

Fn = 100NFn = 200N Fn = 300N

0.0 0.2 0.4 0.6 0.8 1.0-200

-100

0

100

200

Acc

eler

atio

ns in

the

y-di

rect

ion (m

/s2 )

Time (s)

Fn = 100NFn = 200N Fn = 300N

Fig. 11. Acceleration variation with time for Fn of 100 N, 200 N and 300 N, D¼1 mm,

f¼2 Hz, m¼0.38: (a) accelerations in the x-direction; and (b) accelerations in the

y-direction.

100 200 3003250

3300

3350

3400

Vib

ratio

n fr

eque

ncy

(Hz)

Normal force (N)

3281.25Hz

3320.31Hz

3369.14Hz

Fig. 12. Variation of the vibration frequency with normal load.

0.0 0.2 0.4 0.6 0.8 1.0-400

-200

0

200

400

Acc

eler

atio

ns in

the

x-di

rect

ion (m

/s2 )

Time (s)

D = 2mmD = 1mmD = 0.5mm

0.0 0.2 0.4 0.6 0.8 1.0-150

-100

-50

0

50

100

150

Acc

eler

atio

ns in

the

y-di

rect

ion (m

/s2 )

Time (s)

D = 2mmD = 1mmD = 0.5mm

Fig. 13. Variation of acceleration with time under the conditions of D¼0.5 mm,

1 mm and 2 mm, Fn¼100 N, f¼2 Hz, m¼0.38: (a) accelerations in the x-direction;

(b) accelerations in the y-direction.

0.0 0.5 1.0 1.5 2.0 2.53240

3280

3320

3360

Vib

ratio

n fr

eque

ncy

(Hz)

Oscillation amplitude (mm)

3344.73Hz

3281.25Hz

3310.55Hz

Fig. 14. Variation of the vibration frequency with oscillation amplitude of active

specimen.

0.0 0.2 0.4 0.6 0.8 1.0

-300-150

0150300

-300-150

0150300

-300-150

0150300

f =4Hz

Time (s)

f =2HzA

ccel

erat

ions

in th

e x-

dire

ctio

n (m

/s2 )

f =1Hz

Fig. 15. Time history of accelerations in the x-direction under the conditions of

f¼1, 2 and 4 Hz, Fn¼100 N, D¼1 mm, m¼0.38.

0 1 2 3 4 53240

3280

3320

3360

Vib

ratio

n fr

eque

ncy

(Hz)

Reciprocating sliding frequency (Hz)

3349.61Hz

3281.25Hz

3267.43.Hz

Fig. 16. Variation of the vibration frequency with reciprocating sliding frequency.

W.J. Qian et al. / Wear 301 (2013) 47–5654

the vibration frequency with the reciprocating sliding frequency.It is seen that the frequency of the unstable vibration decreasesslightly with the increasing of the reciprocating sliding frequency.This suggests that when the reciprocating sliding frequency issignificantly below the lowest natural frequency of the system, ithas a little influence on the amplitude and frequency of squealingvibration.

W.J. Qian et al. / Wear 301 (2013) 47–56 55

4.4. Relation between the wear of contact surfaces and the contact

force

As we know, friction-induced vibration has a significant effecton sliding surface topography. Some observations of the correla-tion between the friction-induced vibration and wear of slidingsurfaces were reported [23,24]. Massi et al. [25] studied the effectof squealing vibration on the contact surface topography of abreak system. They found that the large oscillations of the contactstresses due to the high frequency vibrations of the system causesthe rise of fatigue cracks and the fragmentation of the contactmaterial.

Chen et al. [26] studied the correlation between the surfacetopography and the formation of squeal by using a reciprocatingsliding system. His experimental results show that the roughness ofthe sliding surfaces with squeal is larger than that without squeal.He also reported that for flat–flat contact the area with squeal ischaracterized by uneven pit-like material removal. Fig. 17 shows thesimulation result of the contact force between the two flat speci-mens. It can be found that the oscillation amplitudes of contact forcein the up stroke process are larger than that in the down strokeprocess. This suggests that the contact force varies more severewhen squealing vibration occurs. This phenomenon may be appliedto explain the roughness of the sliding surfaces with squeal largerthan that without squeal. From Fig. 17, it can also be found that inthe up stroke process the contact forces grow sharply until thepassive specimen starts to separate completely from the activespecimen, in this case the contact force becomes zero. The authorsconsider that this phenomenon may also exist in the experimentalprocess and the duration time is very short. The separation andimpact between the two contact surfaces may be one causeresulting in the uneven pit-like material removal on the contactsurface. Of course, more endeavor needs to be made to verify thisassumption.

0.0 0.2 0.4 0.6 0.8 1.00

100

200

300Down strokeDown stroke Up stroke

Con

tact

forc

e (N

)

Time (s)

Up stroke

Fig. 17. Normal contact force between two flat specimens, Fn¼100 N, D¼1 mm,

f¼2 Hz, m¼0.38.

Fig. 18. Loading methods of the reciprocating sliding syst

4.5. Effect of the loading method

The effect of load application method on the squealing beha-vior is an interesting issue in the research of friction-inducedvibration [27]. We modified the finite element model by using thespring element to apply the load on the friction system as shownin Fig. 18. Three springs were used and the total stiffness was150 kN/m. Fig. 19 shows simulation results of two differentloading methods. Fig. 19a shows a simulation result of loadingthrough the spring. Fig. 19b shows a PSD analysis of the vibrationshown in Fig. 19a. Fig. 19c shows a simulation result of staticloading (dead weight). Fig. 19d shows a PSD analysis of thevibration shown in Fig. 19c. Comparing Fig. 19a with Fig. 19c,it is found that the amplitudes of unstable vibrations weresignificantly reduced when the load was applied through thespring. From Fig. 19c and d, it is found that using different loadingmethods, the frequencies of unstable vibrations changed slightly.This illustrates that the spring loading method has a small effecton the frequencies of unstable vibrations. Of course, thesesimulation results need to be verified in experimental tests. Thesimulation results demonstrate that the loading methods have asignificant effect on the squealing vibration. Improving the load-ing method may suppression the unstable vibration.

5. Conclusions

This paper explores a prediction way of unstable friction-induced vibration of a reciprocating sliding system by using thedynamic transient analysis. The simulation results were firstcorrelated with experimental results in both the time and thefrequency domains. It is found that the frequencies and theamplitudes of unstable vibrations obtained by the transientdynamic simulation are consistent with those of the squealingvibrations measured in the experimental tests. It suggests that thedynamic transient simulation can accurately predict unstablefriction-induced vibration both in the time and frequencydomains. The following conclusions can be drawn.

(1)

em:

Both the transient dynamic analysis and the complex eigenvalueanalysis results show that when a unstable vibration occurs, itsnormal vibration component is coupled with the tangentialvibration component. This phenomenon indicates that thecoupling between the normal and tangential vibration compo-nents is a main cause resulting in squeal occurrence.

(2)

The parameter sensitivity analysis shows that the normal loadand the oscillation amplitude of the active specimen havesignificant effect on the amplitude and duration time of thesquealing vibration. But they have a little influence on the

(a) loading through the spring (b) static loading.

0.0 0.1 0.2 0.3 0.4 0.5-240

-120

0

120

240

ehtni

noitareleccA x-

dire

ctio

n (m

/s2 )

Time (s)

Loading through the spring

0 2000 4000 6000 8000 100000.0

0.2

0.4

0.6

0.8

PSD

of t

he a

ccel

erat

ion

in th

e x-

dire

ctio

n (m

2 /s3 )

Frequency (Hz)

3515.63HzLoading through the spring

0.0 0.1 0.2 0.3 0.4 0.5-240

-120

0

120

240

ehtni

noitareleccA x-

dire

ctio

n (m

/s2 )

Time (s)

Static loading

0 2000 4000 6000 8000 100000

5

10

15

20

253281.25Hz

PSD

of t

he a

ccel

erat

ion

in th

e X

dire

ctio

n (

m2 /s

3 )

Frequency (Hz)

Static loading

Fig. 19. Simulation results of two different loading methods, Fn¼100 N, D¼1 mm, f¼2 Hz, m¼0.38: (a) vibration acceleration of loading through the spring; (b) PSD

analysis of vibration acceleration of loading through the spring; (c)vibration acceleration of static loading and (d) PSD analysis of vibration acceleration of static loading.

W.J. Qian et al. / Wear 301 (2013) 47–5656

frequency of unstable vibration. When the reciprocatingsliding frequency is significantly below the lowest naturalfrequency of the system, it has a little influence on theamplitude and frequency of squealing vibration.

(3)

The comparison between the experimental results [26] ofsurface topography and the behavior of the contact loadcalculated by numerical simulations shows that the largeoscillations of the contact force due to the high frequencyvibrations of the system may causes the rise of the roughnessand wear of the sliding surfaces.

(4)

The simulation results show that the loading methods havesignificantly effect on the squealing vibration. Improving theloading method may suppress the unstable vibration.

Acknowledgments

The authors thank financial supports from the National NaturalScience Foundation of China (Nos. U1134103 and 51275429).

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