NONLINEAR PHENOMENAIN A STOCHASTICALLY EXCITED DYNAMIC SYSTEM

N. van de Wouw, A. de Kraker, D.H. van Campen

Eindhoven University of TechnologyFaculty of Mechanical EngineeringP.O. Box 513, 5600 MB Eindhoven

The NetherlandsEmail: nathan@wfw.wtb.tue.nl

ABSTRACT

The response of strongly nonlinear dynamic systems tostochastic excitation exhibits many interesting characteristics.In this paper, excitation forms like white noise and Gaussianband limited noise are applied to a bilinear system. The non-linear characteristics cannot be represented sufficiently accurateusing statistical linearization techniques. Integration techniqueshave to be used. The emphasis lies on frequency domain phe-nomena. Phenomena like multiple resonance frequencies andmultiple solutions are investigated along with their origin. Thestochastic response characteristics are clarified by comparingthem to periodic response features of the system. Furthermore,stochastic response phenomena can provide information on theperiodic response characteristics of the system.

INTRODUCTION

Nonlinear dynamic systems forced by random processesare often encountered in practice. The source of the ran-domness can vary from surface randomness in vehicle mo-tion, environmental changes, such as earthquakes and windexciting high rise buildings or wave motions at sea excit-ing offshore structures or ships, to electric or acoustic noiseexciting mechanical structures.

When choosing a method for the analysis of the res-ponse of a nonlinear system one should consider three as-pects: (1) the class of systems one would like to be able tocope with; (2) the forms of random excitation that can beapplied; and (3) the response characteristics that should be-come available. The class of systems and excitations shouldbe as broad as possible. It should include multi-degree-of-freedom, strongly nonlinear systems, with nonlinearities of

arbitrary kind, and systems excited by parametric or exter-nal noise. For the purpose of this paper this noise might bewhite or band limited. The response information pursuedis not only information regarding the probability densityfunction or the statistical moments, but also informationregarding the power spectral density.

Well-known response analysis methods like stochasticaveraging (Roberts and Spanos, 1986) and closure methods(Wu and Lin, 1984) do not provide power spectral densityinformation. Moreover, these methods are not applicable tothe broad class of systems described above. A widely ap-plied method that can provide frequency domain informa-tion is statistical linearization (Roberts and Spanos, 1990).This method, however, cannot provide accurate responseinformation for strongly nonlinear systems. Furthermore,it lacks some essential nonlinear response characteristics, aswill be shown in this paper.

Simulation methods using (stochastic) integration tech-niques (Kloeden and Platen, 1992) are suitable to tacklethe wide class of problems described above. The non-trivialsubject of integration will be covered by the next section. Insection 3 the nonlinear dynamic system that is investigatedwill be described. The results of the application of whiteand band limited noise excitations to this system are pre-sented in section 4. Finally, in section 5 some conclusionswill be presented.

SIMULATION TECHNIQUES

When discussing the numerical simulation of nonlineardynamical systems excited by random signals, two classes

of excitations should be distinguished: white noise excita-tion and Gaussian random excitation whose power spectraldensity eventually goes to zero for some frequency. The lat-ter form of excitation will be termed Gaussian band limitednoise.

White Noise Excitation

In this case, the solution of a differential equation ofthe following form is sought:

dX

dt= a(t, X) + b(t, X) ξ(t) (1)

in which a(t, X) is the deterministic (and in our case non-linear) part and ξ(t) represents a white noise excitation.It is quite common in literature to rewrite Eq. (1) in thefollowing form:

dX = a(t, X) dt + b(t, X) ξ(t) dt (2)

This symbolic differential can be interpreted as an integralequation:

X(t) = X(t0) +

t∫t0

a(s, X(s)) ds

+

t∫t0

b(s, X(s)) ξ(s) ds (3)

The white noise process cannot be a stochastic process inthe usual sense. Namely, the white noise process has aconstant power spectral density for all frequencies up toinfinity. The consequence of this is that this process is ofinfinite variance and that the probability density functionis not well-defined.

The white noise process can be seen as the time-derivative of the Wiener process W (t). However, the Wienerprocess is differentiable nowhere, so once again strictlyspeaking ξ(t) does not exist as a conventional function oftime. Due to the fact that the white noise process is theformal derivative of the Wiener process W (t), Eq. (3) canbe written as:

X(t) = X(t0) +

t∫t0

a(s, X(s)) ds

+

t∫t0

b(s, X(s)) dW (s) (4)

The first integral of Eq. (3) & (4) can be interpreted as aRiemann integral. The second integral in Eq. (3) cannotbe a Riemann integral because ξ(t) is not a conventionalfunction of time. The second integral of Eq. (4) cannoteven be a Riemann-Stieltjens integral due to the fact thatthe Wiener process is almost sure not of bounded variation.The second integral of Eq. (4) is termed an Ito integral.

For the numerical solution of this Ito integral spe-cific integration schemes have been developed (Kloeden andPlaten, 1992), because numerical integration schemes de-veloped to solve Riemann and Riemann-Stieltjens integralsare not suitable to handle Ito integrals. For the white noiseproblems in this paper an explicit second order weak schemewas used (Kloeden and Platen, 1992).

Gaussian Band Limited Excitation

Let η(t) be a Gaussian band limited process. This formof random excitation does not contain (theoretically) in-finitely high frequencies. In such a case, ξ(t) is replaced byη(t) in the second integral of Eq. (3), which is then Riemannintegrable. This integral can now be solved by using classi-cal (deterministic) integration techniques. For the problemshandled in this paper Runge-Kutta schemes were used.

A band limited process has its energy concentratedbetween two frequencies fmin and fmax. For a specificpower spectral density function Pηη(f) of the excitation pro-cess one can simulate realizations of that Gaussian randomprocess by using the following method (Shinozuka, 1972;Yang, 1972). The idea behind the method is that a one-dimensional Gaussian random process η(t) with zero meanand one-sided power spectral density Pηη(f) (frequency f)can be represented by a sum of cosine functions with a uni-formly distributed random phase Φ. A realization η(t) ofη(t) can be simulated by:

η(t) =√

∆f Re{F (t)} (5)

in which Re{F (t)} is the real part of F (t) and

F (t) =N∑

k=1

{√2 Pηη(fk) eiφk

}ei2πfkt (6)

is the finite complex Fourier transform of

√2 Pηη(f) eiφ (7)

in which φ are the realized values of Φ.

THE NONLINEAR DYNAMIC SYSTEM

The nonlinear dynamic system, investigated in this pa-per, is a bilinear SDOF model of a beam with a nonlinearsupport (Fey, 1992). The model is governed by the followingdimensionless nonlinear differential equation:

x(t) + 2 ζ x(t) + x(t) + α ρ(x(t)) x(t) = e(t) (8)

in which

ρ(x(t)) ={

0 if x ≥ 01 if x < 0 (9)

Equation (9) shows that the system is bilinear. e(t) can be awhite noise excitation process ξ(t) or a Gaussian band lim-ited excitation process η(t). ζ and α are the dimensionlessdamping parameter and the nonlinearity parameter, respec-tively.

In the case that e(t) = ξ(t), the resulting Ito stochasticdifferential equation is:

dX = F (X, t) dt + G(X, t) dW (t) (10)

in which

X = [x x]T (11)

F (X, t) =[

x−2ζ x − x − α ρ(x) x

]

G(X, t) = [0 1]T

and W (t) is a Wiener process.

RESULTS

In this section the nonlinear response phenomena ofthe dynamic system are investigated. To study differentresponse characteristics, the system is excited by white noiseand Gaussian band limited excitation processes.

White Noise Excitation

The nonlinear dynamic system is simulated using anexplicit second order weak scheme (Kloeden and Platen,1992). This scheme also exhibits a first order convergencein the strong sense. The computed realizations of the res-ponse can be used to estimate the invariant measures of thestationary solutions, such as statistical moments, probabil-ity density function and power spectral density.

1 2 3 4 5 61

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

Nonlinearity parameter α

Mea

n µ

x

[−]

Simulation

Linearization

Figure 1. Estimation of the mean µx by simulation and linearization (ζ =0.01)

1 2 3 4 5 63.2

3.4

3.6

3.8

4

4.2

4.4

Nonlinearity parameter α

Sta

ndar

d de

viat

ion

σx

[−

]

Simulation

Linearization

Figure 2. Estimation of the standard deviation σx by simulation and lin-

earization (ζ = 0.01)

For non-stationary responses many computationally ex-pensive simulations would have to be executed in order toensure an accurate estimate of the invariant measures ateach point of time. However, the necessity of a large numberof records can be eliminated when the response is station-ary, as is the case here. In this case ergodicity with respectto a particular statistical moment can be assumed. This as-sumption allows the determination of this specific ensemblestatistical moment by using its temporal counterpart.

Statistical Moments. The estimates for the mean µx

and the standard deviation σx of the dimensionless displace-ment variable x (for ζ = 0.01 and varying values of α) areshown in Fig. 1 and 2. In these figures the simulation resultsare compared to the results of statistical linearization. Themaximum errors (with 95 % probability) of the simulationestimates µx and σx are 0.3 % and 0.4 %, respectively. Itis clear that both µx and σx are estimated too low by thelinearization technique and that this error becomes larger

1 2 3 4 5 60.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

Nonlinearity parameter α

Ske

wne

ss γ

x [−

]

Estimated values

95 % confidence intervals

Figure 3. Estimation of the skewness κx by simulation (ζ = 0.01)

1 2 3 4 5 62.95

3

3.05

3.1

3.15

3.2

3.25

Nonlinearity parameter α

Kur

tosi

s κ

x [

−]

Estimated values

95 % confidence intervals

Figure 4. Estimation of the kurtosis κ by simulation (ζ = 0.01)

for a stronger nonlinearity.Besides these Gaussian response characteristics, also

higher order moments, such as skewness γx and kurtosisκx, are investigated. The response of a nonlinear dynamicsystem to white noise excitation is generally known to benon-Gaussian. Therefore, the estimates of the skewness andkurtosis, which are defined by

γx =∑

(xi − x)3

(n − 1) σ3x

(12)

κx =∑

(xi − x)4

(n − 1) σ4x

(13)

will deviate from the Gaussian values γx = 0 and κx = 3.In Fig. 3 and 4 the estimates for the skewness and kur-tosis are plotted. These figures make clear that γx andκx deviate more and more from the Gaussian values for a

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.610

−4

10−3

10−2

10−1

100

101

102

103

104

frequency f [−]

Pow

er s

pect

ral d

ensi

ty P

xx (

f)

[−] Simulation

Linearization

Figure 5. Power spectral density for α = 1 and ζ = 0.01 estimated by

linearization and simulation

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.810

−4

10−3

10−2

10−1

100

101

102

103

104

frequency f [−]

Pow

er s

pect

ral d

ensi

ty P

xx (

f)

[−] Simulation

Linearization

Figure 6. Power spectral density for α = 6 and ζ = 0.01 estimated by

linearization and simulation

stronger nonlinearity. Hence, the response process becomesless Gaussian when the nonlinearity in the system increases.A deviation from zero for the skewness points at an asym-metry of the probability density function. This asymmetryof the response is a nonlinear characteristic of the system.

Power Spectral Density. In literature little attentionhas been paid to the frequency domain characteristics ofnonlinear dynamic systems excited by stochastic processes.In Fig. 5 and 6 the one-sided power spectral density of theresponse Pxx(f) is plotted for two different levels of non-linearity. For weakly nonlinear systems the resonance fre-quency of the linearized system appears near the main res-onance peak of the nonlinear system. However, for strongernonlinearities the linearization technique overestimates thisfrequency. Moreover, the power spectral density of the res-ponse of the nonlinear system contains ’extra’ frequenciesat which a significant amount of energy is located. Thisbecomes more evident for stronger nonlinearities and lower

10 20 30 40 50

10-3

10-2

10-1

fe [Hz]

|ymax| [m]harmonic

1/2 subharmonic

1/3 subharmonic

solid line: ξ = 0.01

dotted line: ξ = 0.10

s

s

s

s

s

s

s

uu

6 7 8 9

s

ss

u

fe

1/4, 1/8,1/16 subh

2nd superh

1/2 subh

Figure 7. Response amplitude versus excitation frequency for periodic (har-

monic) excitation and α = 6, ζ = 0.01 (Fey, 1992)

damping. For higher damping (ζ = 0.1) and a weak non-linearity (α = 1) the ’extra’ frequency peaks can almosttotally vanish. It should be noted that the ’extra’ frequen-cies are multiples of the main resonance frequency. More-over, these frequencies coincide with the excitation frequen-cies at which, in case of harmonic excitation, the periodicresponse features subharmonic solutions, see Fig. 7 (Fey,1992). Furthermore, the response contains a significant lowfrequency component. Both phenomena cannot be repro-duced by the linearization technique.

The energy located at these ’extra’ frequencies can beseen as an explanation of the fact that the linearizationtechnique structurally underestimates the variance of theresponse. In fact, the area beneath the power spectral den-sity is equal to the variance of the response. Estimatingthe variance too low can be dangerous when this estimateis used in system failure criteria.

Gaussian Band Limited Excitation

In order to investigate which frequencies in the excita-tion are mainly responsible for specific response character-istics, simulations with Gaussian band limited excitationare performed for α = 6 and ζ = 0.01. In particular,an explanation for the appearance of the ’extra’ frequencypeaks in the power spectral density of the response to whitenoise is sought. For each of the simulations, the frequen-cies fmin and fmax are chosen to cover the major part ofa specific resonance peak without overlapping other reso-nance peaks. Figure 8 shows the power spectral density ofa stochastic equivalent of a harmonic solution for an exci-tation frequency band of 0.18 ≤ fe ≤ 0.28. Note that the

0 0.2 0.4 0.6 0.8 1 1.210

−6

10−4

10−2

100

102

104

frequency f [−]

Pow

er s

pect

ral d

ensi

ty P

xx (

f)

[−] (a) fmin=0.18, fmax=0.28

(b) fmin=0.28, fmax=0.41

Figure 8. Power spectral density of the response (α = 6 and ζ = 0.01) to

Gaussian band limited excitation with (a) fmin = 0.18 and fmax = 0.28and (b) fmin = 0.28 and fmax = 0.41

’harmonic’ solution of a nonlinear system to harmonic ex-citation with frequency fe has a specific period time 1

febut

comprises multiple frequencies (fe, 2fe, 3fe, . . .). Clearly,this is also the case for stochastically excited systems. Thepower spectral density of the response for an excitation fre-quency band of 0.28 ≤ fe ≤ 0.41 does not completely fill theconvex ’valley’ of the power spectral density of the responseto an excitation in the frequency band 0.18 ≤ fe ≤ 0.28,see Fig. 8. This explains the fact that the response to whitenoise excitation (for these system parameters) exhibits ’ex-tra’ frequency peaks. For weaker nonlinearities and higherdamping, this convex ’valley’ is filled in more and more un-til the response to white noise consists of a single resonancepeak (α = 1, ζ = 0.1).

Figure 9 shows that a stochastic equivalent of a 12 sub-

harmonic solution exists. A stochastic equivalent of a 13

subharmonic solution also exists. These stochastic equiva-lents of subharmonic solutions also contribute to the ’extra’frequency peaks in Fig. 6.

So, there are three reasons for the extra frequency peaksin the power spectral density of the response to white noiseexcitation. Firstly, each frequency band in the excitation,within resonance peaks of the system, results in more fre-quency bands in the response, see Fig. 8. Secondly, there arealso subharmonic effects present. For example, a 1

2 subhar-monic effect is responsible for the fact that the excitationfrequency band fmin ≤ fe ≤ fmax also results in an im-portant response in the frequency range fmin

2 ≤ f ≤ fmax

2 ,see Fig. 9. Finally, due to the nonlinearity of the system athird effect should be acknowledged: the frequencies in theresponse ’interact’. It is well-known that when the excita-tion, and therefore the response, contains two frequencies f1

0 0.2 0.4 0.6 0.8 1 1.2

10−8

10−6

10−4

10−2

100

102

frequency f [−]

Pow

er s

pect

ral d

ensi

ty P

xx (

f)

[−]

Figure 9. Power spectral density of the response (α = 6 and ζ = 0.01) to

Gaussian band limited excitation with fmin = 0.41 and fmax = 0.51

and f2, the response can also contain the frequency f2 − f1

when the system is nonlinear. This ’difference’-frequencyappears in the response as a consequence of the asymmetryof the nonlinear solution. Note that white noise excitationcontains (theoretically) an infinite number of frequencies.Hence, a lot of interaction can be expected in that case.When those excitation frequencies lie in a resonance peakof the system, the ’difference’-frequencies can be expectedto contain a significant amount of energy.

The third phenomenon, mentioned above, is also ex-pected to be responsible for the presence of considerablelow-frequency energy in the stochastic response, see for ex-ample Fig. 6, because white noise excitation contains a greatnumber nearby frequencies. Once again, if those excitationfrequencies lie close to each other in the same resonancepeak, the resulting low frequency response component canbecome important.

The response of the bilinear system to periodic (har-monic) excitation exhibits another important nonlinear fea-ture, which is the possibility of multiple solutions. Forα = 6.41 and ζ = 0.142 a stable harmonic and a sta-ble 1

3 subharmonic solution coexist in the frequency band0.65 ≤ fe ≤ 0.87 (Van de Vorst, 1996). In which attractoran orbit of a system will settle only depends on the initialstate of the system for harmonic excitation. The set of allinitial states of orbits which approach a specific attractor istermed the basin of attraction of the attractor. The basinsof attraction in the state space of a dissipative system arebordered and separated by, in this case 1-dimensional, basinboundaries, see Van de Vorst (1996). The stable manifoldsare responsible for the structure of the phase space portrait.They form the basin boundaries, since they do not approachan attractor, but a saddle solution.

−0.1 −0.05 0 0.05 0.1 0.15

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

x [−]

x

[−

]

Stable manifold

Unstable manifoldHarmonic 1/3 subharmonic

attractorattractor

Figure 10. Manifolds of the stable harmonic and stable and unstable 13

sub-

harmonic attractor of the bilinear system (α = 6.41, ζ = 0.0142), harmon-

ically excited with fe = 0.76

At this point, an important question is whether multiplesolutions and basins of attraction still exist (in the practi-cal sense) when the excitation is not periodic but narrowbanded Gaussian noise. The stable and unstable manifoldsof this system, when harmonically excited with frequencyfe = 0.76, are shown in Fig. 10.

In the following, the stochastic response of this system(with equal system parameters) to ’white’ narrow bandedGaussian noise with fmin = 0.759 and fmax = 0.761 is in-vestigated. It should be noted that the positions of the at-tractors and the manifolds in the phase space, for excitationfrequencies fe = 0.759 or fe = 0.761, do not differ signifi-cantly from those of fe = 0.76. In this case, ’white’ meansthat each frequency within the limited frequency band hasan (in practice approximately) equal energy contribution tothe excitation signal. The initial condition X0 = [0.0 0.0]T

is used. For harmonic excitation with frequency fe = 0.76,X0 would lie in the basin of attraction of the stable har-monic attractor.

Due to the bilinearity of the system, the level of the ex-citation does not influence the character of the response.It is very remarkable that the computations resulted in’stochastic harmonic’ solutions as well as ’stochastic 1

3 sub-harmonic’ solutions. These solutions have nearly the sameform in the phase space as their periodic equivalents. Thefact that a solution with its initial condition in the basinof attraction (as defined in the periodically excited case) ofthe harmonic solution ends up as a subharmonic solutioncan be explained by the following mechanism. The bandedGaussian excitation consists of the central frequency (in thiscase f = 0.76) and other frequencies. The other frequen-cies in the excitation can seen as disturbances on a periodic

excitation signal. These disturbances can cause a jump ofthe solution from the basin of attraction of the harmonicsolution to the basin of attraction of the 1

3 subharmonicsolution.

In case of harmonic excitation, only the initial condi-tion determines to which solution the system will converge.However, in case of stochastic excitation, one can only de-termine a probability with which each of the solutions canoccur. In the simulations only 17% (of a total of 500 so-lutions) converged to a stochastic harmonic solution. Eachof the following characteristics appears to be important forthe proportion (probability) of ’stochastic harmonic’ and’stochastic 1

3 subharmonic’ solutions:

1. The global stability of the harmonic and 13 subharmonic

solutions.The global stability of an attractor is determined by thestructure of its basin boundaries, the stable manifolds.If those basin boundaries lie close to the attractor, arelatively small perturbation can cause a jump to an-other attractor. In this case, the perturbation is a devi-ation from periodicity of the excitation, which is due tothe randomness of the excitation. So, the percentage ofharmonic solutions will drop when the magnitude of thebasin of attraction of the harmonic attractor decreases.In our example, the global stability of the harmonic at-tractor is rather low, despite the fact that the attractoris locally stable.

2. A non-uniform power spectral density for the excitation.Until now we only used a constant (’white’) power spec-tral density. When the contribution of the central fre-quency is much higher than those of the other frequen-cies the percentage of 1

3 subharmonic solutions is ex-pected to drop. In order to investigate this influence, a’colored’ narrow banded Gaussian noise excitation wasapplied to the system. ’Colored’ means that the fre-quencies within the limited frequency band have differ-ent energy contributions to the excitation signal. Thepower spectral density of the excitation at the centralfrequency (f = 0.76) is taken approximately 10 times ashigh as at the frequencies f = 0.759 and f = 0.761. Thepercentage of solution (of a total of 500) that convergedto a stochastic harmonic solution is 22%. So, the per-centage of harmonic solutions increased (17% → 22%).This is a consequence of the fact that the disturbances(the extra frequencies) in the excitation are weaker thanthe central frequency in case of the applied ’colored’narrow banded Gaussian excitation. For both ’white’and ’colored’ narrow banded Gaussian excitation, themajority of the trajectories immediately ’locks’ to a 1

3subharmonic or harmonic solution type after a shorttransient. Those trajectories did not switch from one

solution type to another during the simulated time in-terval. However, a small percentage of the trajectoriesthat ended up as a harmonic solution first ’locks’ to a 1

3subharmonic solution after the transient and switchedlater to the harmonic solution type.

3. The initial condition of the solution.When the initial condition lies in the basin of attrac-tion of the 1

3 subharmonic solution, the percentage ofstochastic 1

3 subharmonic solutions is expected to benear 100%. This is a consequence of the fact that theglobal stability of this attractor is so much higher thanthe global stability of the harmonic attractor. This isconfirmed by the outcome of a numerical experimentin which 500 trajectories were computed using ’white’narrow banded noise excitation with initial conditionX0 = [−0.1 0.0]T . The percentage (number) of 1

3 sub-harmonic solutions appeared to be 95 %. It should benoted that despite the chosen initial condition and thefact that the basin of attraction of the harmonic attrac-tor is much smaller than the basin of attraction of the13 subharmonic attractor (global stability), see Fig. 10,the disturbances of the extra frequencies in the excita-tion can still lead to stochastic harmonic solutions (5%).

4. The bandwidth of the excitation frequency band.It is obvious that the near periodicity assumption,which means that the stochastic excitation can be seenas a periodic basic excitation (the central frequency)and disturbances due to the other frequencies in fre-quency band of the excitation, is only useful (for thepurpose of interpretation of the results) when the exci-tation is really very narrow banded, as assumed untilnow. The influence of the enlargement of the excita-tion bandwidth on the percentage of 1

3 subharmonicsolutions is difficult to predict due to the following ob-servations. Firstly, at a certain point (as one gradu-ally broadens the excitation bandwidth, for examplefmin = 0.75 and fmax = 0.77) the bandwidth con-tains frequencies at which the manifold configurationin phase space differs significantly from the manifoldconfiguration of the central frequency. So, the globalstability of the different harmonic solutions (at each sin-gle frequency) changes. Secondly, when the bandwidthof the excitation is enlarged even further (for examplefmin = 0.71 fmax = 0.81) the form of the stochasticsolution in the phase space does not resemble the formof the periodic solutions anymore. Moreover, the man-ifold configurations of the frequencies in the excitationbandwidth can change dramatically when other typeof solutions (for example a 1

5 subharmonic, see Van deVorst (1996)) coexist with the 1

3 subharmonic and theharmonic solution.

CONCLUSIONS

Many interesting specifically nonlinear stochastic res-ponse phenomena have been investigated and discussed.These response phenomena occurred while applying whitenoise and Gaussian band limited noise excitations to thebilinear system. Nonlinear phenomena like multiple reso-nance peaks in the power spectral density of the responseand a significant low frequency response component cannotbe generated by a statistical linearization technique. Animportant consequence is that the linearization techniqueestimates the variance of the response structurally too low.

The response to Gaussian band limited noise excita-tions showed stochastic equivalents of harmonic and sub-harmonic solutions for certain system parameters. Thesesubharmonic solutions appear and disappear for the samesystem parameters as their periodic response equivalents. Asingle sample simulation with Gaussian band limited noiseexcitation can thus provide information on whether periodicsubharmonic solutions exist for certain excitation frequen-cies. Additionally, it can give an indication on the resonancefrequencies of the harmonic and subharmonic solutions.

Furthermore, the response of the bilinear system to nar-row banded Gaussian noise was investigated. This band ofnoise was defined in a frequency range in which the systemexhibits multiple solutions when excited by a harmonic sig-nal. The response to the stochastic excitation also showedmultiple solutions with the same form in phase space astheir periodic equivalents. However, which solution is foundis not only dependent on the initial condition (as is the casefor harmonic excitations). For one specific initial conditionand excitation bandwidth all solutions can appear with adefinite probability. This probability is expected to dependon both the global stability of the stable attractors and therelative magnitude of the contributions of different frequen-cies to the excitation.

References

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Kloeden, P.E., and Platen, E. 1992. Numerical Solution ofStochastic Differential Equations. Springer-Verlag.

Roberts, J.B., and Spanos, P.D. 1986. Stochastic Averaging:An Approximate Method of Solving Random VibrationProblems. Int. J. Non-Linear Mech., 21(2), 111–134.

Roberts, J.B., and Spanos, P.D. 1990. Random Vibrationand Statistical Linearization. New York: John Wileyand Sons.

Shinozuka, M. 1972. Monte Carlo Solution of structuralDynamics. Computers & Structures, 2, 855–874.

Van de Vorst, E.L.B. 1996. Long term dynamics and stabi-

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Wu, W.F., and Lin, Y.K. 1984. Cumulant-neglect Closurefor Non-linear Oscillators under Random Parametricand External Excitations. Int. J. Non-Linear Mech.,19(4), 349–362.

Yang, J.-N. 1972. Simulation of Random Envelope Pro-cesses. Journal of Sound and Vibration, 21(1), 73–85.