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ARTICLE IN PRESS
International Journal of Non-Linear Mechanics 45 (2010) 474–481
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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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H. Rong et al. / International Journal of Non-Linear Mechanics 45 (2010) 474–481476
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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