ARTICLE IN PRESS

International Journal of Non-Linear Mechanics 45 (2010) 474–481

Contents lists available at ScienceDirect

International Journal of Non-Linear Mechanics

0020-74

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/nlm

Resonant response of a non-linear vibro-impact system to combineddeterministic harmonic and random excitations

Haiwu Rong a,�, Xiangdong Wang a, Wei Xu b, Tong Fang b

a Department of Mathematics, Foshan University, Guangdong 528000, PR Chinab Department of Mathematics, Northwestern Polytechnical University, Xi’an 710072, PR China

a r t i c l e i n f o

Article history:

Received 26 July 2008

Received in revised form

2 January 2010

Accepted 26 January 2010

Keywords:

Non-linear vibro-impact system

Random responses

Multiple scales method

62/$ - see front matter & 2010 Elsevier Ltd. A

016/j.ijnonlinmec.2010.01.005

esponding author.

ail address: ronghw@foshan.net (H. Rong).

a b s t r a c t

The resonant resonance response of a single-degree-of-freedom non-linear vibro-impact oscillator, with

cubic non-linearity items, to combined deterministic harmonic and random excitations is investigated.

The method of multiple scales is used to derive the equations of modulation of amplitude and phase.

The effects of damping, detuning, and intensity of random excitations are analyzed by means of

perturbation and stochastic averaging method. The theoretical analyses verified by numerical

simulations show that when the intensity of the random excitation increases, the non-trivial steady-

state solution may change from a limit cycle to a diffused limit cycle. Under certain conditions, impact

system may have two steady-state responses. One is a non-impact response, and the other is either an

impact one or a non-impact one.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

An impact oscillator, often named vibro-impact system, is theterm used to represent a system, which is driven in some way andwhich also undergoes intermittent or a continuous sequence ofcontacts with motion limiting constraints [1]. Analyses of impactsystems may be important in various engineering applications.Certain useful applications of vibration may be involved withimpacts, such as vibratory pile drivers, tie placers, etc. Analyses ofimpact motions are important for properly designing the impact-related machines and devices [2]. However, it is very difficult toinvestigate these systems, because they are usually of non-smooth types. In practice, engineering structures are oftensubjected to time dependent loadings of both deterministic andstochastic nature, such as natural phenomena due to wind gusts,earthquakes, ocean waves, and random disturbances or noiseswhich always exist in a physical system. The influence of randomdisturbance on the dynamical behavior of an impact dynamicalsystem has caught the attention of many researchers. Someanalysis methods, e.g. linearization method [3], quasistaticapproach method [4], Markov processes method [5,6], stochasticaveraging method [7–9], variable transformation method [10,11],energy balance method [12], and mean impact Poincare mapmethod [13], have been developed. In Ref. [2], the authors tried toreview and summarize the existing methods, results and

ll rights reserved.

literatures available for solving problem of stochastic vibro-impactsystems. For the new development of vibro-impact system, werefer readers to [14–16]. However, most researches focused on theresponses of linear impact oscillator (here, ‘‘linear’’ means that thedifferential equations of motion among impacts are linear) underbroad-band random excitations; only few of them focused on theresponses of non-linear impact oscillator under narrow-bandrandom excitations.

In this paper, the resonant response of a single-degree-of-freedom non-linear oscillator impact oscillator to combineddeterministic harmonic and random excitations, which can betaken as narrow-band random excitations, is investigated. Theimpact considered here is an instantaneous one with restitutionfactor e. The method of multiple scales is used to obtain twoaveraged first-order differential equations for modulation of theamplitude and phase of the response with slow time scale.Perturbation method and stochastic averaging method are usedto determine the mean value or the mean-square value of theamplitude and phase by these modulation equations. It is foundthat under certain conditions, the system may have two steady-state responses. One is a non-impact response, and the other iseither an impact or a non-impact one, thus forming an interestingphenomenon of random impact system.

2. System description

Consider a SDOF vibro-impact system with cubic non-linearityand one-sided constraint under harmonic and random excitations,

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whose differential equation can be expressed as

€xþeb _xþxþea1x3þea2 _x2x¼ ehðcosOtþxðtÞÞ; xoD;

_xþ ¼�e _x�; x¼D;

(ð1Þ

where, x stands for response and overhead dot indicatesdifferentiation with respect to time t; e a small parameter,0oeoo1; b the damping coefficient; a1 and a2 represent theintensities of non-linear terms; D stands for the constraintdistance between the system’s static equilibrium position andrigid barrier; h40 and O40, h and O stand for the amplitudeand frequency of the harmonic excitation, respectively; 0oer1,e the restitution factor for an ideal instantaneous impact; thesubscripts ‘‘minus’’ and ‘‘plus’’ for _x refer to response velocity justbefore and just after the instantaneous impact. In Eq. (1), x(t) is awhite noise with zero mean and correlation function

RxðtÞ ¼ 2pS0dðtÞ; ð2Þ

with S0 being the spectral density of x(t), and d(t) is the Diracfunction. When S0 is small enough, the combined harmonic andwhite noise excitation can be approximately taken as a narrow-band random excitation.

In case D=N, there will be no impact happening in the system.Non-impact systems have been discussed by Rajan and Davies[17], Nayfeh and Serhan [18], and the present authors of thispaper [19,20], by using the method of multiple scales [21,22].However, if there are impacts in the steady state in the case whereD is small, the response of system (1) has not been studied. Theaim of this paper is to develop a method applicable to this type ofproblems.

3. Response of non-impact system

Consider the case when the constraint distance D is largeenough so that there is no impact in the steady state of system (1).Hence, Eq. (1) can be reduced into

€xþeb _xþxþea1x3þea2 _x2x¼ ehðcosOtþxðtÞÞ: ð3Þ

In order to use the method of multiple scales [21,22] foranalyzing the case of primary resonance, i.e., OE1.0, we describethe nearness of the resonance by introducing the detuningparameter s defined by to O=1+es. Then, we seek a uniformapproximate solution of Eq. (3) in the form

xðT; eÞ ¼ x0ðT0; T1Þþex1ðT0; T1Þþ � � � ; ð4Þ

where T0=t, T1=et are fast and slow time scales, respectively.Denoting

D0 ¼ ð@=@T0Þ; D1 ¼ ð@=@T1Þ

the ordinary-time derivatives can be transformed into partialderivatives as

d

dt¼D0þeD1þ � � � ;

d2

dt2¼D2

0þ2eD0D1þ � � � : ð5Þ

By substituting Eqs. (4) and (5) into Eq. (3) and equatingcoefficients of e and e2 to zeros, the following equations can bederived:

D20x0þx0 ¼ 0; ð6Þ

D20x1þx1 ¼�2D0D1x0�bD0v0�a1x3

0�a2ðD0x0Þ2x0þhcosOtþxðtÞ:

ð7Þ

The general solution of Eq. (6) can be written as

x0ðT0; T1Þ ¼ acosðT0þjÞ; ð8Þ

where a=a(T1) and j=j(T1) are functions of the slow time scale.Substituting Eq. (8) into Eq. (7) and eliminating the secular

producing terms, we get that a and j vary in the slow time scaleaccording to

a0 ¼ �ðb=2Þaþðh=2Þsin gþx1ðT1Þ;

ag0 ¼ sa�ð3=8Þa3�ðh=2Þcos gþx2ðT1Þ;

(ð9Þ

where prime is derivative with respect to T1, g=sT0�j,a¼ a1þð1=3Þa2, and

x1ðT1Þ ¼ �1

2p

Z 2p

0xðT1Þcoscdc; x2ðT1Þ ¼

1

2p

Z 2p

0xðT1Þsincdc;

ð10Þ

where c¼Ot�g. After solving a and g, the first-order uniformexpansion for the solution of Eq. (3) is given by

xðtÞ ¼ aðeT0Þ cosðOt�gðeT0ÞÞþOðeÞ: ð11Þ

In general, Eq. (9) is difficult to solve exactly due to its non-linear nature, hence perturbation method is used to obtain anapproximate solution of Eq. (9). Assume that x1 and x2 are smallcompared with h, we are to determine steady-state solutions ofthe amplitude and phase when S0=0 ðx1 ¼ 0; x2 ¼ 0Þ. In this caseEq. (9) can be written as

a0 ¼ �ðb=2Þaþðh=2Þsin g;ag0 ¼ sa�ð3=8Þa3�ðh=2Þcos g:

(ð12Þ

The steady-state solutions of Eq. (12) can be found by puttinga=a0, g=g0, a0=0 and g0=0. These substitutions lead to thefollowing result

ðb=2Þa0 ¼ ðh=2Þ sin g0;

sa0�ð3=8Þa30 ¼ ðh=2Þ cos g0

:

(ð13Þ

Squaring and adding Eq. (13) yields the frequency-responseequation

½ðb2=4Þþðs�ð3=8Þaa20Þ

2�a2

0 ¼ ðh2=4Þ: ð14Þ

If ba0 and h=0, from Eq. (14) one obtains a0=0, which showsthat when there are no external excitations, dissipation systemdose not have any stationary periodic solution except trivial one.Applying the Floquet theory [23], we obtain the necessary andsufficient condition for the stable periodic solution of Eq. (12) asfollows:

ðð3=8Þaa20�sÞðð9=8Þaa2

0�sÞþðb2=4Þ40: ð15Þ

Condition (15) shows that not all the branches given byEq. (14) are stable. If there are three branches, among them onlythe largest one a0=aM and smallest one a0=am are stable andrealizable, meanwhile, switching back and forth between thesetwo stable branches forms the jump phenomenon.

Then, we determine the effect of the noise, i.e., x1a0; x2a0,on the deterministic steady-state motion. To this end, we let

a¼ a0þa1; g¼ g0þg1; ð16Þ

where a0 and g0 are determined by Eq. (13), and a1 and g1 areperturbation terms. Substituting Eq. (16) into Eq. (9) andneglecting the non-linear terms, we obtain the linearization ofthe modulation Eq. (9) at a0, g0,

x0ðT1Þ ¼ CxðT1Þþ f ðT1Þ; ð17Þ

where

xðT1Þ ¼x1ðT1Þ

x2ðT1Þ

( )¼

a1ðT1Þ

g1ðT1Þ

( ); f ðT1Þ ¼

f1ðT1Þ

f2ðT1Þ

( ); C ¼

c1 c2

c3 c4

" #;

ð18Þ

f1ðT1Þ ¼ x1ðT1Þ; f2ðT1Þ ¼ x2ðT1Þ; ð19Þ

c1 ¼ c4 ¼�ðb=2Þ; c2 ¼ ð3=8Þaa30�sa0; c3 ¼

sa0�ð9=8Þaa0: ð20Þ

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The general solutions of Eq. (18) can be obtained by modalanalysis method [24]. The eigenvalues and eigenvectors of C aregiven by Cx=lx while the eigenvectors of the adjoin problems aregiven by CT y¼ ly. Then, the modal matrices are

X ¼

d1 d2

d1ðl1�c1Þ

c2

d2ðl2�c1Þ

c2

264

375; Y ¼

1 1l1�c1

c3

l2�c1

c3

24

35; ð21Þ

where

dn ¼c2c3

c2c3þðln�c1Þ2; n¼ 1;2:

It is convenient to introduce a transformation that decouples Eq.(17). To this end, we let xðT1Þ ¼ XzðT1Þ. Substituting this equationinto Eq. (17) and premultiplying by YT yields

z01ðT1Þ ¼ l1z1ðT1Þþg1ðT1Þ;

z02ðT1Þ ¼ l1z2ðT1Þþg2ðT1Þ;

(ð22Þ

where

gnðT1Þ ¼ x1ðT1Þþln�c1

c3x2ðT1Þ; n¼ 1;2: ð23Þ

Then, the statistics of the uncoupled coordinates z1 and z2 are

Ezm ¼ 0;

E½zmðt1Þznðt2Þ� ¼ eðlmt1 þlnt2ÞR t1

0

R t2

0 e�ðlmt1 þlnt2ÞE½gmðt1Þgnðt2Þ�dt1 dt2; m;n¼ 1;2:

(

ð24Þ

Using Eqs. (2), (10), (23) and (24), we can express the expectedvalues and the mean-square values of the uncoupled coordinateszn as

Ezm ¼ 0; E½zmzn� ¼ �pS0

lmþln1þðlm�c1Þðln�c1Þ

a20c2

3

" #; m;n¼ 1;2:

ð25Þ

4. Response of impact system

Now we consider the case when the constraint distance D isnot large so that there are impacts in the steady state of system(1), and determine steady-state solutions of the amplitude andphase when S0=0 ðx1 ¼ 0; x2 ¼ 0Þ. In this case Eq. (1) can bewritten as

€xþeb _xþxþea1x3þea2 _x2x¼ h cosOt; xoD;

_xþ ¼�e _x�; x¼D;

(ð26Þ

where h ¼ eh. Since e is a small parameter, by neglecting non-linear terms and damping term, Eq. (26) can be reduced to

€xþx¼ h cosOt; xoD;_xþ ¼�e _x�; x¼D:

(ð27Þ

It is easy to obtain the response of system (27) [1], whichsatisfies the following conditions:

xð0Þ ¼D; _xð0Þ ¼ _xþ ; x2pl

O

� �¼D; _x

2pl

O

� �¼ _x�: ð28Þ

where l is a positive integer. This kind of steady-state response ofsystem (27) can be obtained as

xðtÞ ¼ x0ðtÞ ¼ ao cos t�pl

O

� �þaO cosðOtþjÞ; ð29Þ

where

ao ¼D�aO cosjcosðpl=OÞ

; aO ¼h

j1�O2j:

The impact velocity is

_x� ¼½17

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�ð1�ða2

O=D2ÞÞ

2ð1þB2Þ

q�D

ð1þB2ÞD; ð30Þ

where

B¼ ð1=OÞ1�e

1þetan

pl

O; D¼�ð1=2Þð1þeÞ cot

pl

O: ð31Þ

Now we use perturbation method to obtain the responses ofsystem (26). To this end, we let

xðtÞ ¼ x0ðtÞþex1ðtÞ; ð32Þ

where x0 is determined by Eq. (29), and ex1(t) is a perturbationterm. By substituting Eq. (32) into Eq. (26) and equatingcoefficient of e to zero, we get the following equation:

€x1þx1 ¼�b _x0�a1x30�a2 _x

20x0: ð33Þ

Substituting Eq. (29) into Eq. (33) and holding on the termsonly of the frequencies O=2p and ð1=2pÞ, we obtain

€x1þx1 ¼ k1 cos t�pl

O

� �þk2 sin t�

pl

O

� �þk3 cosðOtþjÞþk4 sinðOtþjÞ;

ð34Þ

where

k1 ¼� a1

�ð3=4Þa3

oþð3=2Þaoa2O

�þð3=4Þa2a3

o

h i; k2 ¼ bao;

k3 ¼� a1

�ð3=4Þa3

oþð3=2Þa2oaO

�þð3=4Þa2O

2a3O

h i; k4 ¼ bOaO:

The solution of Eq. (34) is

x1 ¼ C1 cos tþC2 sin tþk1

2t sin t�

pl

O

� ��

k2

2t cos t�

pl

O

� �

þk3

1�O2cosðOtþjÞþ k4

1�O2sinðOtþjÞ; ð35Þ

where arbitrary constants C1 and C2 are determined by theconditions x1(0)=0 and x1ð2pl=OÞ ¼ 0.

When x1a0, x2a0, we try to determine the effect of the noiseon the deterministic steady-state response, which is determinedby Eq. (32). To this end, we let

xðtÞ ¼ x0ðtÞþex1ðtÞþffiffiffiep

U; ð36Þ

be a solution of Eq. (1), where x0(t) and x1(t) are determined byEqs. (29) and (35) respectively, and

ffiffiffiep

U is a perturbation term.Substituting Eq. (36) into Eq. (1) and neglecting the non-linearterms, we obtain the following linearization equation

€Uþef½bþ2a2 _x0ðtÞx0ðtÞ� _Uþ3a1x20ðtÞUgþU ¼

ffiffiffiep

xðtÞ: ð37Þ

Since e is a small parameter, the steady-state response ofEq. (37) can be obtained by the method of stochastic averaging[24]. By introducing the following transformation:

UðtÞ ¼ AðtÞ cosF; _U ðtÞ ¼ �AðtÞ sinF; F¼ tþY; ð38Þ

and using the method of stochastic averaging, we obtain thefollowing Ito equations:

dA¼ �bA

2eþ pS0

2Ae

� �dtþ

ffiffiffiffiffiffiffiffipS0

p ffiffiffiep

dW1ðtÞ;

dY¼3a1ða

2oþa2

OÞ

4edtþ

ffiffiffiffiffiffiffiffipS0

pA

ffiffiffiep

dW2ðtÞ;

8>><>>: ð39Þ

where W1(t) and W2(t) are independent standard Wienerprocesses.

It is clear that A(t) is a Markov process, whose steady-stateprobability density function p(a) is governed by the following FPKequation:

d

da

ba

2e�pS0

2ae

� �p

� �þpS0e

2

d2p

da2¼ 0: ð40Þ

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The solution of Eq. (40) is

pðaÞ ¼ ða=r2Þ exp �a2

2s2

� �; r2 ¼

pS0

b; ð41Þ

The first and second steady-state moments of A(t) are

EAðtÞ ¼

Z þ1=0

apðaÞda¼

ffiffiffiffip2

rr; EA2ðtÞ ¼

Z þ1=0

a2pðaÞda¼ 2r2 ð42Þ

From Eq. (42), we have

P 0rAðtÞr2ffiffiffiffiffiffiffiffiffiffiffiffiffiEA2ðtÞ

q ¼ Pf0rAðtÞr2

ffiffiffi2p

rg

¼

Z 2ffiffi2p

r

0pðaÞda¼ 1�e�2 ¼ 98:17%:

Then from Eq. (38), we have

PfjUðtÞjr2ffiffiffi2p

rgZ98:17%: ð43Þ

In fact, Eq. (36) and inequality (43) gives the confidenceinterval of the response of system (1).

Fig. 1. Frequency response of system (1) (D=6.0, S0=0.0): —— stable solution;

– – – unstable solution and JJJ numerical solution.

Fig. 2. Numerical results of Eq. (1) (D=6.0, S0=0.0025, x(0)=�

5. Numerical simulation

For the method of numerical simulation, we refer the readersto Zhu [24] and Shinozuka [25,26]. In this paper, the powerspectrum of x(t) is taken as

SðoÞ ¼S0; 0oor2O0; o42O:

(

For numerical simulation it is more convenient to use thepseudorandom signal given by [24]

xðtÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffi4OS0

N

r XN

k ¼ 1

cosONð2k�1Þtþjk

� �; ð44Þ

where jk‘s are independent and uniformly distributed in (0, 2p],and N is a large integer. In numerical simulation, parameters insystem (1) and (44) are chosen as follows:

a1 ¼ 0:09; a2 ¼ 0:03; b¼ 0:3; h¼ 2; e¼ 0:8; e¼ 0:1; N¼ 1000:

The governing Eq. (1) is numerically integrated by the fourthorder Runge–Kutta algorithm between impacts, which is validuntil the first encounter with the barriers, that is until the equalityx=D is satisfied. The impact condition _xþ ¼�e _x� is then imposed,using the numerical solution _x�. This gives the rebound velocity_xþ , thereby providing the initial value for the next step’snumerical calculation. The numerical results are shown fromFigs. 1–9.

First we consider the case when the constraint distance D islarge enough (D=6.0), such that there is no impact in the steady-state of system (1). The variations of the steady-state responsea0 with O¼ 1þes are shown in Fig. 1 when xðtÞ ¼ 0, and thetheoretical results given by Eq. (14) are also shown in Fig. 1 forcomparison. Fig. 1 shows that the deterministic responsepredicted by the method of multiple scales is in good agreementwith that obtained by numerical results.

Next, we determine the effect of the noise term x(t) on theprimary responses. Eq. (14) has two stable solutions a0=am=1.03and a0=aM=5.44 when s=1.0 and S0=0.0025, hence the stationaryresponses of the deterministic system (12) may be different fordifferent initial values. The numerical results of Eq. (1) are shownin Figs. 2 and 3, where yðtÞ ¼ _xðtÞ is the velocity of the mass.

The initial values are xð0Þ ¼�5:0; _xð0Þ ¼ 4:5 in Fig. 2, andxð0Þ ¼ 1:0; _xð0Þ ¼ 1:5 in Fig. 3. Figs. 2 and 3 show that when S0 is

5.0, _xð0Þ ¼ 4:5): (a) Time history of x(t) and (b) phase plot.

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Fig. 3. Numerical results of Eq. (1) (D=6.0, S0=0.0025, x(0)=1.0, _xð0Þ ¼ 1:5): (a) Time history of x(t) and (b) phase plot.

Fig. 4. Frequency response of system (1) (D=6.0, S0=0.0025): —— stable

solution;– – – unstable solution and JJJ numerical solution.Fig. 5. Numerical results of Eq. (26) (D=6.0, s=1.0): —— theoretical solution andJJJ numerical solution.

H. Rong et al. / International Journal of Non-Linear Mechanics 45 (2010) 474–481478

small enough, the stationary responses of system (1) may bedifferent for different initial values in some parameter area of O.The random noise x(t) will change the steady-state response ofsystem (1) from a limit cycle to a diffused limit cycle. Obviously,the responses shown in Figs. 2 and 3 are non-impact ones.Variations of the steady-state response with s are shown in Fig. 4when S0=0.0025, and theoretical results are also shown forcomparison. Calculation shows that the stationary responses ofsystem (1) may be different for different initial values in the area0:61oso1:57.

Now we consider the case when the constraint distance D issmall (D=0.6), so that there are impacts in the steady stateof system (1). Eq. (14) has two stable solutions a0=am=1.03 anda0=aM=5.44 when x(t)=0 and s=1.0, hence the stationaryresponses of the deterministic system (14) may be different fordifferent initial values. Since Doam, the steady-state response ofsystem (1) should be an impact one which is determined byEq. (32), but cannot be a non-impact one which is determined byEq. (11). Numerical results are shown in Figs. 5 and 6, and

theoretical results given by Eq. (32) are also shown in Fig. 5 forcomparison. Fig. 5 shows that the deterministic responsespredicted by the perturbation method are in good agreementwith that obtained by numerical results.

The numerical results of system (1) are shown in Fig. 7 whenS0=0.0025 and s=1.0, for comparison with the results shown inFigs. 5 and 6. Numerical simulations in Figs. 7–9 show that theresponses of system are in the confidence interval given byEq. (36) and inequality (43).

When D takes the value in the area amoDoaM, system (1)may have two steady-state responses for different initial values,the non-impact one a0=am which is determined by Eq. (11) andthe impact one which is determined by Eq. (36). The numericalresults of Eq. (1) are shown in Figs. 8 and 9 when D=2.5,S0=0.0025 and s=1.0. Note again that system (12) may have twostable steady-state responses a0=am=1.03 and a0=aM=5.44 fordifferent initial values.

Figs. 8 and 9 show that when S0 is small enough, the stationaryresponses of system (1) may be different for different initial

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Fig. 6. Numerical results of Eq. (26) (D=6.0, s=1.0): (a) Time history of x(t) and (b) time history of y(t).

Fig. 7. Numerical results of Eq. (1) (D=6.0, s=1.0, S0=0.0025): (a) Phase plot and (b) time history of x(t).

H. Rong et al. / International Journal of Non-Linear Mechanics 45 (2010) 474–481 479

values in some parameter area of O. The random noise x(t) willchange the steady-state response of system (1) from a limit cycleto a diffused limit cycle. Obviously, the response shown in Fig. 8 isa non-impact one which is determined by Eq. (11), while theresponse shown in Fig. 9 is an impact one which is determined byEq. (36). It is an interesting phenomenon found in random impactsystem, that such system may have both non-impact and impactsteady-state responses under certain conditions.

6. Conclusions and discussion

In this paper, the method of multiple scales is used to analyzethe response of a non-linear impact system under deterministicharmonic and random excitations. The perturbation method andstochastic averaging method are used to analyze the effect of therandom noise on the response of the impact oscillator. So far,exact solutions of non-linear impact system under randomexcitation are only available for a very limited number ofproblems. Thus, approximate methods have been developed.

These methods include the method of equivalent or stochasticlinearization, perturbation methods, stochastic averaging andseries expansions, etc. In fact, it is difficult or impossible to solveexactly even for SDOF non-linear deterministic system. Henceapproximate methods have been widely used in the analysis ofdeterministic non-linear system. These methods include smallparameter method, method of coordinate transformation, multi-ple scales method, method of slowly varying parameter, KBMmethod, method of equivalent linearization, method of harmonicbalance, etc. For a comprehensive survey of the approximatemethod in deterministic system, we refer readers to Nayfeh[21,22], Nayfeh and Mook [27], and Hagedorn [28]. We also knowthat approximate methods in deterministic system can beextended to random system. For example, in recent year, Rajanand Davies [17], Nayfeh and Serhan [18] have extended themethod of multiple scales to the analysis of non-linear systemsunder random external excitations, and the authors of this paper[19,20] extended this method to the analysis of non-linearsystems under random parameter excitation. In this paper, themethod of multiple scales is extended to the analysis of the

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Fig. 8. Numerical results of Eq. (1) (D=2.5, s=1.0, S0=0.0025, x(0)=0.5, _xð0Þ ¼ 0:5): (a) Time history of x(t) and (b) phase plot.

Fig. 9. Numerical results of Eq. (1) (D=2.5, s=1.0, S0=0.0025, x(0)=0.5, _xð0Þ ¼ 4:5): (a) Time history of x(t) and (b) phase plot.

H. Rong et al. / International Journal of Non-Linear Mechanics 45 (2010) 474–481480

response of non-linear impact oscillator under combined deter-ministic harmonic and random excitations. Numerical simulationshows that this method is valid.

Theoretical analyses and numerical simulations show thatwhen S0 is small enough, the stationary response of system (1)may be different for different initial values in some parameterareas of O. The random noise x(t) will change the steady-stateresponse of system (1) from a limit cycle to a diffused limit cycle.When the intensity of the random excitation increases, the widthof the diffused limit cycle will increase too. It is found that impactsystem may have two steady-state responses under someconditions, one is a non-impact response, and the other is animpact or non-impact one, which is an interesting phenomenonfor random impact system.

It is well known from the theory of non-linear oscillation, thatif an oscillator with hardening non-linear stiffness is subjected tosinusoidal excitation, the response may exhibit sharp jumps inamplitude. This jump behavior is associated with the fact that,over a range of the values of the ratio of excitation frequency to

the natural frequency of the degenerated linear oscillator, theresponse amplitude is triple-valued. Therefore, the system shouldhave two stationary responses which depend on the initialcondition. However, it is a disputable problem if there are morethan one stationary responses if an oscillator with hardening non-linear stiffness is subjected to narrow-band random excitations[29,30]. Zhu [29] and his cooperators concluded that there are twomore probable motions in the stationary response of a Duffingoscillator to narrow-band Gaussian excitation, and jumps mayoccur in a certain domain of the space of the parameters of theoscillator and excitation. The jump of the Duffing oscillator undernarrow-band random excitation is essentially a transition of theresponse from one more probable motion to another or vice versa.In the case when jumps occur, all the statistics, including thevariance of the displacement, of the stationary response areunique and independent of initial conditions. While some otherauthors [17,30] pointed that the system should have twostationary responses which depend on the initial condition.Similar dissensions also exist in the oscillator under combined

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H. Rong et al. / International Journal of Non-Linear Mechanics 45 (2010) 474–481 481

deterministic harmonic and random excitations [18,19]. In thispaper, system (1) is subjected to combined deterministicharmonic and random excitations, and will have two stationaryresponses, which depend on the initial condition in the extremecase when the random excitation equal to zero (x(t)=0), hencetwo stationary responses, which depend on the initial conditionmay be expected when x(t) is small enough. However, when x(t)increases, transition of the response from one more probable motionto another or vice versa is found in the numerical simulation.

Acknowledgements

The work reported in this paper was supported by the NationalNatural Science Foundation of China under Grant no. 10772046,and the Natural Science Foundation of Guangdong Province underGrant no. 7010407.

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