response distribution of a single-degree-of-freedom linear ...response distribution of a...

13th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP13Seoul, South Korea, May 26-30, 2019

Response distribution of a single-degree-of-freedom linear systemsubjected to non-Gaussian random excitation by usingcross-correlation function

Takahiro TsuchidaAssistant Professor, Dept. of Systems and Control Engineering, Tokyo Institute ofTechnology, Tokyo, Japan

Koji KimuraProfessor, Dept. of Systems and Control Engineering, Tokyo Institute of Technology,Tokyo, Japan

ABSTRACT: Response probability density function of a single-degree-of-freedom linear system sub-jected to non-Gaussian random excitation is investigated. The excitation is assumed to be a zero-meanstationary stochastic process prescribed by both the non-Gaussian probability density and the powerspectral density with bandwidth and dominant frequency parameters. In this study, bimodal and Laplacedistributions are used as the non-Gaussian distribution of the excitation. Monte Carlo simulations are per-formed to obtain the stationary probability distributions of the displacement and velocity of the system.Then, we calculate the maximum absolute value of the real part of cross-correlation function between theexcitation and the response, which is considered as an evaluation index of waveform similarity betweenthe excitation and the response. It is shown that the response probability distributions vary with the maxi-mum value. With the aid of this maximum value, the shapes of the probability distributions of the systemresponse can be predicted roughly without Monte Carlo simulations.

The response of machines and structures subjectedto random excitation have been widely analyzed us-ing probabilistic methods for many decades. In alarge number of studies so far, the random excita-tion has been assumed to be a Gaussian process.This assumption is due to the fact that many ran-dom processes in engineering problems have theprobability density functions similar to the Gaus-sian distribution. However, it is known that somerandom excitations such as wind pressure (Ko etal., 2005), shallow water waves (Ochi, 1986), un-evenness on the road surface (Grigoriu, 1995) andvertical acceleration of traveling automobile (Stein-wolf, 2012) exhibit highly non-Gaussianity.　 Most non-Gaussian random excitations have acommon feature that their probability densities haswider tails than those of a Gaussian distribution.This is a serious problem from the viewpoint of en-gineering because the probability of occurrence of

excitation with large-amplitude is higher than thatof the Gaussian random excitation. When such anon-Gaussian random excitation acts on machines,structures or equipment inside them, the responsewith large magnitude and/or unique characteristicsnot found in the case of Gaussian excitation mayoccur, which has a great influence on the system.Therefore, in order to achieve a good understandingof the behavior of such systems and enhancementof safety, reliability and comfortability of mechani-cal and structural systems, acquiring knowledge ofresponse characteristics of non-Gaussian randomlyexcited systems is crucial.　 The authors examined the response probabilitydensity of single-degree-of-freedom linear and non-linear systems subjected to non-Gaussian randomexcitation by Monte Carlo simulation (Tsuchidaand Kimura, 2013). The simulation results showedthat when the bandwidth of the excitation power

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spectrum is narrow, the shape of the response distri-bution is close to the shape of the excitation proba-bility density, while in the case of wide-band excita-tion the response distribution resembles that of thesystem under Gaussian white noise. Subsequently,the influence of the dominant frequency of non-Gaussian excitation on response probability distri-bution was investigated (Uehara et al., 2015). It wasrevealed that the response distribution becomes ashape close to the non-Gaussian distribution of theexcitation when the excitation dominant frequencyis low and becomes a shape close to a Gaussian dis-tribution when the dominant frequency is high. Fur-thermore, it was also observed that for some com-binations of the bandwidth and dominant frequencyof the excitation, the response distribution exhibitsan intermediate shape between the excitation distri-bution and a Gaussian distribution.　 In this study, the change in shape of the responseprobability distribution of a SDOF linear systemunder non-Gaussian excitation is examined usingcross-correlation functions. First, Monte Carlosimulations are carried out to obtain the probabilitydistributions of the stationary displacement and ve-locity responses. Then, in order to study the prop-erties of the response distribution from the view-point of the waveform similarity between the ex-citation and the response, the maximum absolutevalue of the real part of the cross-correlation func-tion is employed. The correspondence of this maxi-mum value to the shape of the response distributionis illustrated. The maximum value can be evaluatedeasily and enables us to roughly estimate the shapesof the probability distributions of the displacementand velocity responses without Monte Carlo simu-lation.

1. ANALYTICAL MODEL1.1. Equation of motionConsider a single-degree-of-freedom linear systemgoverned by

Ẍ +2ζ Ẋ +X =U(t) (1)

where ζ is the damping ratio, t is a non-dimensionaltime parameter and U(t) is non-Gaussian randomexcitation described in detail below.

1.2. Non-Gaussian random excitationThe non-Gaussian excitation U(t) is assumed tobe a zero-mean stationary stochastic process pre-scribed by both the non-Gaussian probability den-sity pU(u) and the power spectrum SU(ω).

1.2.1. Non-Gaussian probability densityBimodal and Laplace distributions are used as thenon-Gaussian probability density pU(u) to clearlyobserve the influence of the non-Gaussian nature ofU(t) on system response. These two distributionsare expressed as follows:

•Bimodal distribution

pU(u) =1

24√

πΩ

(512Ω3

u6 +192Ω2

u4 +72Ω

u2 +15)

× exp(− 4

Ωu2)

(2)

•Laplace distribution

pU(u) =β2

exp(−β |u|) (3)

where Ω and β are the parameters which govern thevariance of these distributions. In this study, thesetwo parameters are given as Ω = 2 and β =

√2 so

that both the probability distributions have the samevariance: σ2U = 1. The bimodal and Laplace distri-butions are shown in Figs. 1 and 2. They are quitedifferent from each other and a Gaussian distribu-tion.

1.2.2. Power spectrumThe power spectral density SU(ω) of U(t) consid-ered in this study is given by

SU(ω) =E[U2]

πα(α2 +ρ2 +ω2)

(α2 +ρ2 −ω2)2 +4α2ω2(4)

where α and ρ are the bandwidth and dominant fre-quency parameters, respectively. In Fig.3, SU(ω) isshown. When α is small, U(t) is narrow-band, onthe other hand, U(t) is wide-band for large α . Thedominant frequency shifts with the value of ρ .

2


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-6 -4 -2 0 2 4 6

pU

(u)

u

Figure 1: Bimodal distribution (Ω = 2)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-6 -4 -2 0 2 4 6

pU

(u)

u

Figure 2: Laplace distribution (β =√

2)

0.0001

0.001

0.01

0.1

1

10

100

0 0.5 1 1.5 2 2.5 3

SU

(ω)

ω

α=0.01, ρ=0.2 α=0.05, ρ=1 α=0.25, ρ=2

Figure 3: Power spectrum of excitation (E[U2] = 1)

1.2.3. Generation method of excitationIn order to carry out Monte Carlo simulations, thesample functions of U(t) with pU(u) given in sec-tion 1.2.1 and SU(ω) given in section 1.2.2 are gen-erated through the method presented by Tsuchidaand Kimura (2015). This method is summarizedbriefly below.

A zero-mean stationary stochastic process U(t)with the power spectrum in Eq.(4) can be expressedby the following two-dimensional Itô stochastic dif-ferential equation:

dU = (−αU +ρV )dt +D1(U,V )dB1(t)dV = (−ρU −αV )dt +D2(U,V )dB2(t)

(5)

where α and ρ are the same as the bandwidth anddominant frequency parameters of SU(ω). B1(t)and B2(t) are mutually independent unit Wienerprocesses. V (t) is a zero-mean stationary processwhich is orthogonal to U(t) (i.e., E[U(t)V (t)] = 0).

D21(u,v) and D22(u,v) are the diffusion coeffi-

cients given by

D21(u,v) =−2α

pUV (u,v)

∫ u−∞

spUV (s,v)ds

D22(u,v) =−2α

pUV (u,v)

∫ v−∞

spUV (u,s)ds

(6)

where pUV (u,v) is the stationary joint probabilitydistribution of U(t) and V (t), which is given by

pUV (u,v) =1

2πpA(

√u2 + v2)√

u2 + v2)(7)

where pA(a) is the probability density of the enve-lope of U(t). pA(a) can be derived from pU(u) asfollows: (Rytov et al., 1988)

pA(a)a

=∫ ∞

0xJ0(xa)

∫ ∞−∞

pU(u)eiuxdudx (8)

where J0(·) is the Bessel function of the first kindof order 0.

By using Eqs. (5) - (8), the sample functions ofU(t) can be generated. The procedure is shown be-low.

1. pU(u), α and ρ are given.2. pA(a) is derived from pU(u) through Eq. (8).3. Substituting pA(a) into Eq. (7), pUV (u,v) is

obtained.4. Using α and pUV (u,v), D21(u,v) and D

22(u,v)

are calculated from Eq. (6).5. After substituting α , ρ , D1(u,v) and D2(u,v)

into Eq. (5), Eq. (5) is solved numericallyby the Euler-Maruyama scheme (Kloeden andPlaten, 1992) to generate the sample functions.

The diffusion coefficients D21(u,v) and D22(u,v)

for the bimodal distribution and the Laplace distri-bution are given as follows:

3


• Case of bimodal distribution

D21(u,v) = D22(u,v)

= α{

Ω4+

3Ω2

161

u2 + v2+

3Ω3

321

(u2 + v2)2

+3Ω4

1281

(u2 + v2)3

} (9)

• Case of Laplace distribution

D21(u,v) = D22(u,v) = αβ

√u2 + v2

K1(β√

u2 + v2)

K0(β√

u2 + v2)(10)

where K0(·) and K1(·) are the modified Bessel func-tions of the second kind of order 0 and 1, respec-tively.

2. MONTE CARLO SIMULATIONMonte Carlo simulations are performed to obtainthe probability distributions of the stationary dis-placement and velocity of the system describedin section 1. First, the sample functions of thenon-Gaussian excitation are generated by using themethod in section 1.2.3. Then, the response samplefunctions are calculated from Eq. (1) using the 4th-order Runge-Kutta method. Finally, the responseprobability distributions are obtained from the re-sponse sample functions. The computational con-ditions are as follows: Number of sample functions: 200, Number of sample points per 1 sample func-tion : 217, Time step size : 0.002

Hereafter, let us introduce new parameters A andB associated with the parameters of the excitationpower spectrum SU(ω).

A =αζ, B =

ρ√1−ζ 2

(11)

A represents the bandwidth ratio of the excitationpower spectrum to the frequency response functionof the linear system. When A > 1 (or A < 1), theexcitation bandwidth is broader (or narrower) thanthat of the frequency response function. B repre-sents the ratio of the dominant frequency of theexcitation to the damped natural frequency of thesystem. When B = 1, the resonance occurs. Inthis study, the response distributions are examinedwhile fixing the damping ratio ζ to 0.05 and chang-ing A and B in the following wide range:

• 0.1 ≤ A ≤ 20,∆A = 0.1 for 0.1 ≤ A ≤ 2,∆A = 1 for 2 < A ≤ 4,∆A = 2 for 4 < A ≤ 20

• 0 ≤ B ≤ 2, ∆B = 0.1

3. RESULTS3.1. Relationship between response distribution

and waveform similarityFigure 4 shows the sample functions of the excita-tion u(t) and the stationary displacement responsex(t) and the displacement probability density pX(x)for the bandwidth ratio parameter A = 0.1 and thedominant frequency ratio parameter B = 1.3. Theresults for A = 10, B = 0.2 are also shown in Fig-ure 5. In these figures, the black-solid line indicatesthe sample function (upper figure) or probabilitydistribution (lower figure) of the response, and thered-dashed line indicates those of the excitation. Inaddition, the response variance is normalized so asto be equal to the excitation variance to facilitatethe comparison between the shapes of the responsewaveform and distribution and those of the excita-tion.

Fig. 4 indicates that when the response proba-bility distribution is similar to the excitation prob-ability distribution, both the waveforms of the ex-citation and the response fluctuate sinusoidally andresemble each other, although there is a phase dif-ference. On the other hand, from Fig.5, when theresponse distribution has a shape close to a Gaus-sian distribution, the response fluctuates almost pe-riodically while the excitation fluctuates like noise,and the waveforms of the response and the excita-tion are greatly different. These observations on theshape of the response distribution and the similar-ity of the waveform also apply to the case of thevelocity response (not shown in the figures).

From the above results, it is inferred that thewaveform similarity between the excitation and theresponse and the shape of the response distributionare strongly related. Therefore, in the next section,an evaluation index of the waveform similarity willbe introduced in order to discuss the shape of the re-sponse distribution from the viewpoint of the simi-larity of the waveform.

4


-6

-4

-2

0

2

4

6

150 155 160 165 170 175 180

u(t

) o

r x

(t)

t

excitationresponse

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

-6 -4 -2 0 2 4 6p

X(x

)x

(a) Bimodal-distributed excitation

-6

-4

-2

0

2

4

6

150 155 160 165 170 175 180

u(t

) o

r x

(t)

t

excitationresponse

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

-6 -4 -2 0 2 4 6

pX(x

)

x

(b) Laplace-distributed excitation

Figure 4: Upper : Sample functions of stationary dis-placement (black solid line) and excitation (red dashedline). Lower : Probability densities of stationary dis-placement (black solid line) and excitation (red dashedline) for A = 0.1 and B = 1.3

-6

-4

-2

0

2

4

6

150 155 160 165 170 175 180

u(t

) o

r x

(t)

t

excitationresponse

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

-6 -4 -2 0 2 4 6

pX(x

)

x

(a) Bimodal-distributed excitation

-6

-4

-2

0

2

4

6

150 155 160 165 170 175 180

u(t

) o

r x

(t)

t

excitationresponse

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

-6 -4 -2 0 2 4 6

pX(x

)

x

(b) Laplace-distributed excitation

Figure 5: Upper : Sample functions of stationary dis-placement (black solid line) and excitation (red dashedline). Lower : Probability densities of stationary dis-placement (black solid line) and excitation (red dashedline) for A = 10 and B = 0.2

3.2. Evaluation index of the waveform similarityusing cross-correlation function

Normalized cross-correlation function ρUX(τ) be-tween the excitation and the stationary displace-ment and normalized cross-correlation functionρUẊ(τ) between the excitation and the stationaryvelocity are expressed by

ρUX (τ) =RUX (τ)√

E[U2]E[X2], ρUẊ (τ) =

RUẊ (τ)√E[U2]E[Ẋ2]

(12)

where RUX(τ) and RUẊ(τ) are the correspondingnon-normalized cross-correlation functions, whichcan be written in terms of the cross-power spectraldensities SUX(ω) and SUẊ(ω) as follows:

RUX(τ) =∫ ∞−∞

SUX(ω)eiωτdω (13)

RUẊ(τ) =∫ ∞−∞

SUẊ(ω)eiωτdω (14)

SUX(ω) = H(ω)SU(ω)SUẊ(ω) = iωH(ω)SU(ω)

H(ω) =1−ω2 −2iζω

(1−ω2)2 +4ζ 2ω2

where H(ω) and iωH(ω) are the frequency re-sponse functions of displacement and velocity, re-spectively. E[X2] and E[Ẋ2] in Eq. (12) are themean square values of the stationary displacementand velocity, respectively given by

E[X2] =∫ ∞−∞

|H(ω)|2SU(ω)dω

E[Ẋ2] =∫ ∞−∞

|iωH(ω)|2SU(ω)dω

When Re[ρUX(τ)] (or Re[ρUẊ(τ)]) is close to 1 or−1, there is a strong positive or negative correlationbetween the two waveforms of the excitation andthe displacement (or the velocity), respectively. Onthe other hand, when Re[ρUX(τ)] (or Re[ρUẊ(τ)])is close to 0, these waveforms are uncorrelated.

From the results in section 3.1, it is presumedthat as long as the excitation waveform and the re-sponse waveform resemble each other, even if thereis a phase difference between them, the responseprobability distribution has a shape similar to theexcitation probability distribution. Therefore, in

5


this study, in order to evaluate the similarity de-gree of the waveforms between the excitation andthe response quantitatively, the maximum absolutevalues of the real parts of the normalized cross-correlation functions, i.e., max(|Re[ρUX(τ)]|) andmax(|Re[ρUẊ(τ)]|), are employed. Taking the ab-solute value corresponds to ignoring the phase dif-ference between the excitation and the response.Since it is difficult to obtain max(|Re[ρUX(τ)]|) andmax(|Re[ρUẊ(τ)]|) analytically, they are calculatednumerically using Eqs. (12)-(14).

3.3. Relationship between response distributionand maximum absolute value of real part ofcross-correlation function

The evaluation indexes of the waveform similar-ity max(|Re[ρUX(τ)]|) and max(|Re[ρUẊ(τ)]|) areused to examine the change in shape of the responseprobability distributions. The displacement prob-ability densities pX(x) and the velocity probabil-ity densities pẊ(ẋ) for the bimidal-distributed ex-citation obtained by the Monte Carlo simulationsare shown in Figs. 6, 8, and 10. The responseprobability distributions for the Laplace-distributedexcitation are also shown in Figs. 7, 9 and 11.In these figures, the black-solid line and the red-dashed line indicate the response distribution andthe excitation distribution, respectively, and in or-der to facilitate the comparison of the shapes ofthe distributions, the response variance is normal-ized so as to be equal to the excitation variance. Inthe figure caption, the value of max(|Re[ρUX(τ)]|)is written in the figures of pX(x), and the valueof max(|Re[ρUẊ(τ)]|) is written in the figures ofpẊ(ẋ), respectively.

It can be found from Fig. 6 that when thevalues of max(|Re[ρUX(τ)]|) are about the same,the displacement response distributions pX(x) be-come similar shapes, regardless of the values ofthe bandwidth ratio parameter A and the domi-nant frequency ratio parameter B. This can also befound from Fig. 7. Similarly, the velocity responsedistributions pẊ(ẋ) with almost the same valuesof max(|Re[ρUẊ(τ)]|) are close to each other, asshown in Figs. 8 and 9. Furthermore, Figs. 10and 11 are the figures comparing pX(x) and pẊ(ẋ),and these figures indicate that provided that the val-

ues of max(|Re[ρUX(τ)]|) and max(|Re[ρUẊ(τ)]|)are about the same, pX(x) and pẊ(ẋ) also have al-most the same shapes as each other.　Figs. 6 and 7 show that when max(|Re[ρUX(τ)]|)and max(|Re[ρUẊ(τ)]|) are close to 1, the shapesof pX(x) and pẊ(ẋ) look like that of the excitationprobability distribution pU(u). From Figs. 8 and 9,when max(|Re[ρUX(τ)]|) and max(|Re[ρUẊ(τ)]|)decrease to about 0.85, the response distribution be-comes an intermediate shape between pU(u) anda Gaussian distribution. Then, when these maxi-mum values further decrease and reach a value ofabout 0.6 or less, as shown in Figs. 10 and 11,the response distribution becomes almost Gaussianregardless of the difference in non-Gaussianity ofthe excitation. These facts hold true for both thedisplacement response and the velocity responsewith combinations (A, B) of a wide range of thebandwidth ratio A and the dominant frequency ra-tio B shown in Section 2. max(|Re[ρUX(τ)]|)and max(|Re[ρUẊ(τ)]|) can be evaluated from thefrequency response function of the system andthe excitation power spectral density in a simplemanner, as described in section 3.2. In Fig.12,max(|Re[ρUX(τ)]|) and max(|Re[ρUẊ(τ)]|) for awide range of the parameters A and B are shown.By using this figure, it is possible to readily es-timate the rough shapes of the probability distri-butions of the displacement and velocity responsesfor given A and B without performing Monte Carlosimulations.

4. CONCLUSIONSThe response probability distributions of a SDOFlinear system subjected to non-Gaussian randomexcitation have been investigated. The excitationis a zero-mean stationary stochastic process pre-scribed by both the non-Gaussian probability den-sity and the power spectrum with bandwidth anddominant frequency parameters.

First, Monte Carlo simulations have been carriedout to obtain the stationary probability distributionsof the displacement and velocity of the system. Thesimulation results have revealed that the shape ofthe response distribution is relevant to the wave-form similarity of the excitation and the response.Then, we have calculated the maximum absolute

6


Figure 6: Probability density of stationary dis-placement response (black solid line) and bimodaldistribution of excitation (red dashed line). Left :(A, B, max(|Re[ρUX(τ)]|) = (0.1, 0.5, 0.971), Right: (A, B, max(|Re[ρUX(τ)]|)) = (0.2, 0.3, 0.951)

Figure 7: Probability density of stationary dis-placement response (black solid line) and Laplacedistribution of excitation (red dashed line). Left :(A, B, max(|Re[ρUX(τ)]|) = (0.1, 0.5, 0.971), Right: (A, B, max(|Re[ρUX(τ)]|)) = (0.2, 0.3, 0.951)

Figure 8: Probability density of stationary veloc-ity response (black solid line) and bimodal dis-tribution of excitation (red dashed line). Left :(A, B, max(|Re[ρUẊ(τ)]|)) = (0.6, 1.3, 0.856), Right :(A, B, max(|Re[ρUẊ(τ)]|)) = (0.2, 0.5, 0.84)

Figure 9: Probability density of stationary veloc-ity response (black solid line) and Laplace dis-tribution of excitation (red dashed line). Left :(A, B, max(|Re[ρUẊ(τ)]|)) = (0.6, 1.3, 0.856), Right :(A, B, max(|Re[ρUẊ(τ)]|)) = (0.2, 0.5, 0.84)

Figure 10: Probability density of stationary response(black solid line) and bimodal distribution of excita-tion (red dashed line). Left : displacement response(A, B, max(|Re[ρUX(τ)]|)) = (4.5, 0.4, 0.597), Right: velocity response (A, B, max(|Re[ρUẊ(τ)]|)) =(5, 0.6, 0.511)

Figure 11: Probability density of stationary response(black solid line) and Laplace distribution of excita-tion (red dashed line). Left : displacement response(A, B, max(|Re[ρUX(τ)]|)) = (4.5, 0.4, 0.597), Right: velocity response (A, B, max(|Re[ρUẊ(τ)]|)) =(5, 0.6, 0.511)

5 10 15 20A

0.0

0.5

1.0

1.5

2.0

B

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

max

(|Re[ρUX(τ

)]|)

0.1

5 10 15 20A

0.0

0.5

1.0

1.5

2.0

B

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

max

(|Re[ρUẊ(τ

)]|)

0.1

Figure 12: max(|Re[ρUX(τ)]|) ( Upper ) andmax(|Re[ρUẊ(τ)]|) ( Lower )

7


value of the real part of the cross-correlation func-tion between the excitation and the response, whichhas been considered as an evaluation index of sim-ilarity of the waveforms between them. It has beenshown that the stationary response distributions be-come similar to each other when the maximum val-ues are nearly equal to each other. When the max-imum value is close to 1, the shape of the proba-bility distribution of the system response looks likethe shape of the distribution of the non-Gaussianexcitation. For the maximum value around 0.85,the response distribution becomes the middle shapebetween the excitation probability density and aGaussian distribution. In the case of the maximumvalue less than 0.6, the response distribution is al-most Gaussian, irrespective the excitation probabil-ity density.

The maximum absolute value of the real part ofthe cross-correlation function between the excita-tion and the response can be calculated readily us-ing the frequency response function of a linear sys-tem and the excitation power spectrum, regardlessof the excitation probability distribution. With theaid of this maximum value, the shapes of the prob-ability distributions of the displacement and veloc-ity of the system can be predicted roughly withoutMonte Carlo simulation, which generally requiresexcessive computational resources and effort.

5. REFERENCESGrigoriu, M. (1995). "Applied Non-Gaussian Processes:

Examples, Theory, Simulation, Linear Random Vi-bration, and MATLAB Solutions"，Prentice Hall,Englewood Cliffs, New Jersey.

Kloeden, P.E. and Platen, E. (1992). "Numerical Solu-tion of Stochastic Differential Equations", Springer,Berlin.

Ko, N.H., You, K.P. and Kim, Y.M. (2005). "The effectof non-Gaussian local wind pressures on a side faceof a square building", Journal of Wind Engineeringand Industrial Aerodynamics, 93 (5), 383–397.

Ochi, M.K. (1986). "Non-Gaussian random processes inocean engineering", Probabilistic Engineering Me-chanics, 1 (1), 28–39.

Rytov, S. M., Kravtsov, Y. A. and Tatarskii, V. I. (1988)."Principles of Statistical Radiophysics, vol. 2, Cor-

relation Theory of Random Processes", Springer-Verlag, New York.

Steinwolf, A. (2012). "Random vibration testing withkurtosis control by IFFT phase manipulation", Me-chanical Systems and Signal Processing, 28, 561–573.

Tsuchida, T., and Kimura, K. (2013). "Response Distri-bution of Nonlinear Systems Subjected to RandomExcitation with Non-Gaussian Probability Densitiesand a Wide Range of Bandwidth", In: Proceedingsof the 15th Asia Pacific Vibration Conference, Jeju,Korea, 1199–1204.

Tsuchida, T., and Kimura, K. (2015). "Simulation ofNarrowband Non-Gaussian Processes Using Enve-lope Distribution", In: Proceedings of 12th Inter-national Conference on Applications of Statisticsand Probability in Civil Engineering, Vancouver,Canada, 331.

Uehara, D., Tsuchida, T. and Kimura, K. (2015). "Re-sponse distributions of a linear system subjected tonon-Gaussian random excitations with bandwidthand dominant frequency" In: Proceedings of JSMEDynamic and Design Conference 2015, 147 (inJapanese).

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ANALYTICAL MODELEquation of motionNon-Gaussian random excitationNon-Gaussian probability densityPower spectrumGeneration method of excitation

MONTE CARLO SIMULATIONRESULTSRelationship between response distribution and waveform similarityEvaluation index of the waveform similarity using cross-correlation functionRelationship between response distribution and maximum absolute value of real part of cross-correlation function

CONCLUSIONSREFERENCES

response distribution of a single-degree-of-freedom linear ...response distribution of a...

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