An Analysis of Transient Response Moments of a Linear System Subjected to Non-Gaussian Random Excitation Using Higher-Order Autocorrelation Functions of Excitation

Fukushima, Hideto; Tsuchida, Takahiro

doi:10.1115/1.4055079

Abstract

We present the analytical solutions of the second-, third-, and fourth-order response moments of a single-degree-of-freedom linear system subjected to a class of non-Gaussian random excitation. The non-Gaussian excitation is a zero-mean stationary stochastic process prescribed by an arbitrary probability density and a power spectrum whose peak is located at zero frequency. The excitation is described by an Itô stochastic differential equation in which the drift and diffusion coefficients are determined from the probability density and spectral density of the excitation. In order to obtain the analytical solutions of the response moments, first, we derive the third- and fourth-order autocorrelation functions of the non-Gaussian excitation using its Markov and detailed balance properties. The third-order correlation function is given by the same expression regardless of the difference in the probability density function of the excitation. On the other hand, the fourth-order correlation function is derived under the assumption that the excitation probability density belongs to the Pearson distribution family, which is one of the widest classes of probability distributions. Then, combining the autocorrelation functions of the excitation and the convolution representation of the response, we obtain the exact solutions of the response moments, and it is shown what kind of components the response moments are composed of. Finally, we investigate the dominant time-varying components of the response moments for several different excitation bandwidths.

1 Introduction

It is important to determine and examine the stochastic response of dynamic systems subjected to random excitations from the viewpoint of enhancement of reliability, safety, and comfortability of machines and structures. When analyzing stochastic dynamic systems, in many cases, a random excitation has been modeled by a Gaussian process. This is due to the fact that real random excitations often have the probability densities similar to a Gaussian distribution, and the analytical advantage that the probabilistic properties of Gaussian processes can be completely described by the statistics up to the second order, namely, the mean function and the autocorrelation function (or equivalently the power spectral density for stationary processes). However, some engineering systems are subjected to highly non-Gaussian random excitations, for example, vehicles running on rough road surfaces [1,2], trains running on tracks with irregularities [1,3], marine structures excited by wave forces [4,5], and low-rise buildings under wind pressure [6,7]. The response of a system under non-Gaussian excitation is generally also non-Gaussian even though the system is linear. Thus, if we assume a Gaussianity for a random excitation that is actually non-Gaussian, the assumption may lead to large errors in response analysis results. Therefore, the stochastic response analysis taking into account the non-Gaussianity of random excitations properly and the elucidation of the effects of the excitation non-Gaussianity on system responses have become one of the key issues in the community of random vibration.

A mathematically rigorous description of a non-Gaussian process requires its finite-dimensional distribution of arbitrary order, which is unrealistic in modeling real non-Gaussian phenomena. For this reason, in many practical situations, two essential probabilistic properties, the first-order probability density and the power spectral density, are used. In the last few decades, considerable effort has been devoted to developing models for a non-Gaussian process prescribed by the probability density and the power spectrum. And now, there are various models such as translation process [8,9], spectral representation method with translation process theory [10–15], Karhunen–Loève expansion model [16–18], polynomial chaos expansion model [19–21], stochastic differential equation model [22–28], etc.

Among such non-Gaussian models, this paper focuses on a stochastic differential equation model presented by Cai and Lin [25]. In this model, a non-Gaussian process is assumed to be zero-mean and stationary, and described by a one-dimensional Itô stochastic differential equation. The drift and diffusion coefficients in the Itô equation are adjusted to match the spectral density and the probability density, respectively. The advantages of the Cai and Lin’s model are as follows: (i) this model is very versatile in terms that it can treat a non-Gaussian process with an arbitrary probability density, which is useful to find the general characteristics of system response that commonly appear for various non-Gaussian excitations. (ii) No approximations are involved in modeling, and a model that exactly fits the given non-Gaussian distribution and power spectrum can be obtained. (iii) Since the random process is represented as a Markov process, it is possible to perform an analysis using the well-known Markov property.

So far, some studies have been conducted for linear and nonlinear systems under non-Gaussian random excitation described by the Cai and Lin’s model. Tsuchida and Kimura carried out Monte Carlo simulation to examine the stationary response probability density of single-degree-of-freedom (SDOF) linear and nonlinear-stiffness systems under the non-Gaussian excitation [29]. It was found that the shape of the response distribution is highly dependent on the bandwidth of the excitation power spectrum. Especially for a linear system, the displacement response distribution has a shape close to the excitation probability density when the excitation bandwidth is small compared to the bandwidth of the frequency response function of the system. Conversely, if the excitation bandwidth is larger, then the response distribution becomes almost Gaussian. Such a relationship between the excitation bandwidth and the stationary response of the system was also investigated in terms of the moments [30–32].

In many engineering problems, it is important to understand not only the stationary response but also the transient response. A few studies have addressed the transient response of systems subjected to non-Gaussian excitation modeled by the Cai and Lin’s procedure. Wu and Cai investigated the effects of the excitation probability density on the transient response properties for various linear and nonlinear dynamical systems in the case of wide-band excitation [33]. It was shown that the excitation probability distribution has a great effect on the transient behavior of the response for a dynamical system, either linear or nonlinear, and the effect reduces as the system response approaches the stationary state. Recently, Fukushima and Tsuchida also studied the transient response of a SDOF linear system subjected to the non-Gaussian excitation with a variety of bandwidths using Monte Carlo simulation [34]. The simulation results demonstrated that as in the case of the stationary response [29], the bandwidth of the excitation spectral density influences on the transient response characteristics significantly. Immediately after the excitation starts, the effect of non-Gaussianity of the excitation appears in the response distribution clearly, but when the excitation bandwidth is large, the response distribution approaches a Gaussian distribution almost monotonically and rapidly. On the other hand, for the small excitation bandwidth, the effect of the excitation non-Gaussianity remains in the response until the response reaches the stationary state, and the width and degree of non-Gaussianity of the response distribution change periodically in the transient state. In the above studies, however, it was not sufficiently investigated what kind of time-varying components the probability distribution and moments of the transient response are composed of, and which of them are dominant. A better understanding of such time-varying characteristics of the transient response is expected to contribute to improving the reliability and safety of non-Gaussian randomly excited systems. For non-Gaussian response, its probability density function generally has a shape with asymmetry and/or heavy tails. In terms of statistical moments, the asymmetry and tail shape of the probability density are characterized by the third- and fourth-order moments, which are related to the skewness and kurtosis of the response, respectively. Thus, the non-Gaussian characteristics of the response can be found by analyzing these higher-order response moments in detail.

In the present paper, we derive the analytical solutions of the second-, third-, and fourth-order time-dependent response moments of a SDOF linear system subjected to non-Gaussian random excitation expressed by the Cai and Lin’s model. In this regard, the higher-order autocorrelation functions of the non-Gaussian excitation, which are necessary for obtaining the response statistics, are also derived using its Markov and detailed balance properties. The response moments are then decomposed into their components to see what time-varying components are included. Furthermore, we also observe the relationship between the dominant components of the response moments and the excitation bandwidth.

2 Models of System and Non-Gaussian Random Excitation

2.1 Equation of Motion.

Consider a single-degree-of-freedom linear system described by the following non-dimensional equation of motion:

\ddot{X} + 2 ζ \dot{X} + X = U (t)

(1)

where X denotes the displacement, ζ is the damping ratio, and U(t) is a non-Gaussian random excitation. t is the time variable non-dimensionalized by the undamped natural angular frequency of the system. The dot symbol is the differentiation with respect to t. We assume that the system is initially at rest and the excitation U(t) is added to the system at t = 0.

2.2 Non-Gaussian Random Excitation.

The non-Gaussian random excitation U(t) considered in this paper is a zero-mean stationary stochastic process described by a model developed by Cai and Lin [25]. In this model, U(t) is prescribed by the probability density function p_U(u) and the following power spectrum S_U(ω):

S_{U} (ω) = \frac{α E [U^{2}]}{π (ω^{2} + α^{2})}

(2)

where α > 0 is the bandwidth parameter and E[U²] is the mean square value of U(t). S_U(ω) for five types of α and E[U²] = 1 is shown in Fig. 1. This spectral density has one peak at ω = 0. In the case of small α, the zero frequency component is dominant; on the other hand, S_U(ω) is wide-band for large α. Although α is not often referred to as bandwidth when the peak of the spectrum is located at ω = 0, in this paper, we refer to α as bandwidth for the sake of convenience.

Fig. 1

View large Download slide

Power spectrum of random excitation

In Cai and Lin’s procedure, U(t) with an arbitrary probability density p_U(u) and the power spectrum S_U(ω) given by Eq. is expressed by the following one-dimensional Itô stochastic differential equation:

d U (t) = - α U (t) d t + D (U (t)) d B (t)

(3)

where α is the same parameter in Eq. , B(t) is a standard Brownian motion, and −αu and D²(u) are the drift and diffusion coefficients, respectively. Multiplying Eq. by U(t − τ) and taking the ensemble average, we obtain

\frac{d R_{U} (τ)}{d τ} = - α R_{U} (τ)

(4)

where R_U(τ) = E[U(t)U(t − τ)] is the autocorrelation function of U(t), and in deriving this equation, the independent increments property of the Brownian motion B(t) was used. Since R_U(0) corresponds to E[U²], the solution of Eq. is given by

R_{U} (τ) = E [U^{2}] \exp (- α | τ |)

(5)

Thus, by the Wiener–Khinchin theorem, it can be seen that the spectral density of U(t) is expressed by Eq. .

The stationary probability density p_U(u) of U(t) is governed by the reduced Fokker–Planck equation corresponding to Eq.

\frac{d}{d u} G (u) = \frac{d}{d u} {α u p_{U} (u) + \frac{1}{2} \frac{d}{d u} [D^{2} (u) p_{U} (u)]} = 0

(6)

where G(u) is known as the probability flow. In the present one-dimensional case, G(u) must vanish everywhere [35], and Eq. reduces to

α u p_{U} (u) + \frac{1}{2} \frac{d}{d u} [D^{2} (u) p_{U} (u)] = 0

(7)

Finally, Eq. leads to

D^{2} (u) = - \frac{2 α}{p_{U} (u)} \int_{- \infty}^{u} r p_{U} (r) d r

(8)

From Eq. , the diffusion coefficient D²(u) in Eq. is determined according to the probability density function p_U(u). It is also noted that U(t) belongs to the class of detailed balance because Eq. holds [35]. This fact will be used to derive the higher-order autocorrelation functions of U(t) in the next section.

Thus, the non-Gaussian random excitation U(t) described by Eq. (3) with D(u) in Eq. (8) possesses a given probability density p_U(u) and the spectral density S_U(ω) given by Eq. (2). For this non-Gaussian excitation, we analyze the second-, third-, and fourth-order response moments of the SDOF linear system (1).

3 Derivation of Higher-Order Autocorrelation Functions of Non-Gaussian Excitation

In this section, the third- and fourth-order autocorrelation functions of the non-Gaussian random excitation U(t) are derived.

3.1 Third-Order Autocorrelation Function.

For three time instants t₁, t₂, and t₃, let t₁ ≤ t₂ ≤ t₃. To simplify the notation, the excitation at each time is denoted as U(t₁) = U₁, U(t₂) = U₂, and U(t₃) = U₃. Since the random excitation U(t) is described by the stochastic differential equation , U(t) is a diffusive Markov process. Therefore, the third-order joint probability density

p_{U_{1} U_{2} U_{3}} (u_{1}, u_{2}, u_{3})

of U(t) can be written as

p_{U_{1} U_{2} U_{3}} (u_{1}, u_{2}, u_{3}) = p_{U_{3} | U_{2}} (u_{3} | u_{2}) p_{U_{1} | U_{2}} (u_{1} | u_{2}) p_{U_{2}} (u_{2})

(9)

where

p_{U_{i} | U_{j}} (u_{i} | u_{j})

is the conditional probability density function of U_i under the condition that U_j = u_j. By using Eq. , the third-order autocorrelation function E[U₁U₂U₃] of U(t) can be calculated as follows:

\begin{aligned} E [U_{1} U_{2} U_{3}] & = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} u_{1} u_{2} u_{3} p_{U_{1} U_{2} U_{3}} (u_{1}, u_{2}, u_{3}) d u_{1} d u_{2} d u_{3} \\ = \int_{- \infty}^{\infty} {\int_{- \infty}^{\infty} u_{3} p_{U_{3} | U_{2}} (u_{3} | u_{2}) d u_{3}} \\ \times {\int_{- \infty}^{\infty} u_{1} p_{U_{1} | U_{2}} (u_{1} | u_{2}) d u_{1}} u_{2} p_{U_{2}} (u_{2}) d u_{2} \\ = \int_{- \infty}^{\infty} E [U_{3} | U_{2} = u_{2}] E [U_{1} | U_{2} = u_{2}] u_{2} p_{U_{2}} (u_{2}) d u_{2} \end{aligned}

(10)

where E[·|U₂ = u₂] is the conditional expectation conditioned on U₂ = u₂ at time t₂.

We consider the two conditional expectations E[U₃|U₂ = u₂] and E[U₁|U₂ = u₂] in Eq. . On both sides of Eq. , taking the conditional average E[·|U₂ = u₂] under t ≥ t₂ and dividing by dt lead to

\frac{d E [U (t) | U_{2} = u_{2}]}{d t} = - α E [U (t) | U_{2} = u_{2}]

(11)

Equation has the solution

E [U (t) | U_{2} = u_{2}] = u_{2} \exp (- α (t - t_{2}))

(12)

Hence, since t₃ ≥ t₂, we obtain

E [U_{3} | U_{2} = u_{2}] = u_{2} \exp {- α (t_{3} - t_{2})}

(13)

For E[U₁|U₂ = u₂] (t₁ ≤ t₂), Eq. is unable to use directly. Accordingly, we utilize the fact that because the excitation U(t) satisfies the detailed balance as mentioned in Sec. 2.2, the following equation for its joint probability density function

p_{U_{1} U_{2}} (u_{1}, u_{2})

holds [36].

p_{U_{1} U_{2}} (u_{1}, u_{2}) = p_{U_{1} U_{2}} (u_{2}, u_{1})

(14)

Equation leads to

E [U_{1} | U_{2} = u_{2}] = E [U_{2} | U_{1} = u_{2}]

(15)

Since t₂ ≥ t₁, the conditional expectation E[U₂|U₁ = u₂] in the right-hand side of Eq. can be obtained by the same procedure as in Eqs. –

E [U_{2} | U_{1} = u_{2}] = u_{2} \exp {- α (t_{2} - t_{1})}

(16)

From Eqs. and ,

E [U_{1} | U_{2} = u_{2}] = u_{2} \exp {- α (t_{2} - t_{1})}

(17)

Finally, using Eqs. , , and , the third-order autocorrelation function E[U₁U₂U₃] of the excitation U(t) is derived as

\begin{aligned} E [U_{1} U_{2} U_{3}] & = \int_{- \infty}^{\infty} u_{2} \exp {- α (t_{3} - t_{2})} \\ \times u_{2} \exp {- α (t_{2} - t_{1})} u_{2} p_{U_{2}} (u_{2}) d u_{2} \\ = E [U^{3}] \exp {- α (t_{3} - t_{1})} \end{aligned}

(18)

where

E [U^{3}] (= E [U_{2}^{3}])

is the third-order moment of U(t). Although Eq. is valid only for t₁ ≤ t₂ ≤ t₃, the third-order correlation function with different order of t₁, t₂, and t₃ can be obtained by replacing the three indices of time t in Eq. . For example, E[U₁U₂U₃] with t₃ ≤ t₁ ≤ t₂ is given by

E [U_{1} U_{2} U_{3}] = E [U^{3}] \exp {- α (t_{2} - t_{3})}

(19)

We also note that the third-order correlation function derived here is established for any probability density p_U(u) of the excitation U(t).

3.2 Fourth-Order Autocorrelation Function.

For four time instants t₁, t₂, t₃, and t₄, let t₁ ≤ t₂ ≤ t₃ ≤ t₄. The excitation at each time is denoted as U(t₁) = U₁, U(t₂) = U₂, U(t₃) = U₃, and U(t₄) = U₄. As in Sec. 3.1, using the Markov property of the excitation U(t), the fourth-order joint probability density

p_{U_{1} U_{2} U_{3} U_{4}} (u_{1}, u_{2}, u_{3}, u_{4})

of U(t) can be expressed as

p_{U_{1} U_{2} U_{3} U_{4}} (u_{1}, u_{2}, u_{3}, u_{4}) = p_{U_{4} | U_{3}} (u_{4} | u_{3}) p_{U_{3} | U_{2}} (u_{3} | u_{2}) p_{U_{1} | U_{2}} (u_{1} | u_{2}) p_{U_{2}} (u_{2})

(20)

Thus, the fourth-order autocorrelation function E[U₁U₂U₃U₄] of U(t) is calculated as follows:

\begin{aligned} E [U_{1} U_{2} U_{3} U_{4}] = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \\ u_{1} u_{2} u_{3} u_{4} p_{U_{1} U_{2} U_{3} U_{4}} (u_{1}, u_{2}, u_{3}, u_{4}) d u_{1} d u_{2} d u_{3} d u_{4} \\ = \int_{- \infty}^{\infty} {\int_{- \infty}^{\infty} {\int_{- \infty}^{\infty} u_{4} p_{U_{4} | U_{3}} (u_{4} | u_{3}) d u_{4}} u_{3} p_{U_{3} | U_{2}} (u_{3} | u_{2}) d u_{3}} \\ \times {\int_{- \infty}^{\infty} u_{1} p_{U_{1} | U_{2}} (u_{1} | u_{2}) d u_{1}} u_{2} p_{U_{2}} (u_{2}) d u_{2} \\ = \int_{- \infty}^{\infty} {\int_{- \infty}^{\infty} E [U_{4} | U_{3} = u_{3}] u_{3} p_{U_{3} | U_{2}} (u_{3} | u_{2}) d u_{3}} E \\ [U_{1} | U_{2} = u_{2}] u_{2} p_{U_{2}} (u_{2}) d u_{2} \end{aligned}

(21)

where the conditional expectation E[U₁|U₂ = u₂] is given by Eq. , and E[U₄|U₃ = u₃] can also be obtained through the same procedure as in Eqs. –

E [U_{4} | U_{3} = u_{3}] = u_{3} \exp {- α (t_{4} - t_{3})}

(22)

Substituting Eqs. and into Eq. leads to

\begin{aligned} E [U_{1} U_{2} U_{3} U_{4}] & = \exp {- α (t_{4} - t_{3} + t_{2} - t_{1})} \\ \times \int_{- \infty}^{\infty} {\int_{- \infty}^{\infty} u_{3}^{2} p_{U_{3} | U_{2}} (u_{3} | u_{2}) d u_{3}} u_{2}^{2} p_{U_{2}} (u_{2}) d u_{2} \\ = \exp {- α (t_{4} - t_{3} + t_{2} - t_{1})} \\ \times \int_{- \infty}^{\infty} E [U_{3}^{2} | U_{2} = u_{2}] u_{2}^{2} p_{U_{2}} (u_{2}) d u_{2} \end{aligned}

(23)

We now calculate the conditional expectation

E [U_{3}^{2} | U_{2} = u_{2}]

in Eq. . By using Eq. and applying Itô’s formula [36], we obtain an Itô stochastic differential equation for U²(t)

d (U^{2} (t)) = {- 2 α U^{2} (t) + D^{2} (U (t))} d t + 2 U (t) D (U (t)) d B (t)

(24)

On both sides of Eq. , taking the conditional expectation E[·|U₂ = u₂] under t ≥ t₂ and dividing by dt yield

\frac{d}{d t} E [U {(t)}^{2} | U_{2} = u_{2}] = - 2 α E [U {(t)}^{2} | U_{2} = u_{2}] + E [D^{2} (U (t)) | U_{2} = u_{2}]

(25)

Equation cannot be solved when the diffusion coefficient D²(u) is of the general form. However, if D²(u) is expressed by a quadratic polynomial in u, we can find the solution of Eq. . Therefore, hereafter, consider the case where D²(u) is written in quadratic form as D²(u) = Au² + Bu + C. This type of diffusion coefficient corresponds to the case in which the excitation probability distribution p_U(u) belongs to the Pearson distribution family [25]. Substituting D²(u) = Au² + Bu + C into Eq. , we have

\begin{aligned} \frac{d}{d t} E [U {(t)}^{2} | U_{2} & = u_{2}] = (- 2 α + A) E [U {(t)}^{2} | U_{2} = u_{2}] \\ + B E [U (t) | U_{2} = u_{2}] + C \end{aligned}

(26)

which is a linear ordinary differential equation with inhomogeneous terms in the second and third terms on the right-hand side. E[U(t)|U₂ = u₂] is given by Eq. . The solution of Eq. when A ≠ α, 2α is

\begin{aligned} E [U {(t)}^{2} | U_{2} = u_{2}] & = u_{2}^{2} e^{(A - 2 α) (t - t_{2})} \\ + \frac{B u_{2}}{A - α} {e^{(A - 2 α) (t - t_{2})} - e^{- α (t - t_{2})}} \\ + \frac{C}{A - 2 α} {e^{(A - 2 α) (t - t_{2})} - 1} \end{aligned}

(27)

Thus, the conditional expectation

E [U_{3}^{2} | U_{2} = u_{2}]

in Eq. is expressed as

\begin{aligned} E [U_{3}^{2} | U_{2} = u_{2}] & = u_{2}^{2} e^{(A - 2 α) (t_{3} - t_{2})} + \frac{B u_{2}}{A - α} {e^{(A - 2 α) (t_{3} - t_{2})} - e^{- α (t_{3} - t_{2})}} \\ + \frac{C}{A - 2 α} {e^{(A - 2 α) (t_{3} - t_{2})} - 1} \end{aligned}

(28)

Combining Eqs. and yields the fourth-order autocorrelation function E[U₁U₂U₃U₄] of U(t) as follows:

\begin{aligned} E [U_{1} U_{2} U_{3} U_{4}] = e^{- α (t_{4} - t_{3} + t_{2} - t_{1})} \\ \times [E [U^{4}] e^{(A - 2 α) (t_{3} - t_{2})} + \frac{B E [U^{3}]}{A - α} {e^{(A - 2 α) (t_{3} - t_{2})} - e^{- α (t_{3} - t_{2})}} \\ + \frac{C E [U^{2}]}{A - 2 α} {e^{(A - 2 α) (t_{3} - t_{2})} - 1}] \end{aligned}

(29)

where

E [U^{2}], E [U^{3}]

⁠, and E[U⁴] are the second-, third-, and fourth-order moments of U(t), respectively. Similar to the third-order case in Sec. 3.1, E[U₁U₂U₃U₄] with different orders of t₁, t₂, t₃, and t₄ can be obtained by replacing the four indices of time t in Eq. . Incidentally, it is possible to get the autocorrelation function in the case of A = α and A = 2α by taking the limit of A → α and A → 2α in Eq. , respectively. The solutions are provided in Appendix A.

4 Analysis of Transient Response Moments

The displacement response X(t) of the system is written as a convolution integral of the excitation U(t) and the impulse response function h(t) of the system.

X (t) = \int_{0}^{t} U (τ) h (t - τ) d τ

(30)

The impulse response function h(t) is given by

h (t) = \frac{1}{ω_{d}} e^{- ζ t} \sin (ω_{d} t)

(31)

where ω_d is the damped natural angular frequency

ω_{d} = \sqrt{1 - ζ^{2}}

⁠. The velocity response

\dot{X} (t)

is also described as

\dot{X} (t) = \int_{0}^{t} U (τ) \dot{h} (t - τ) d τ

(32)

From these relationships between the responses and the excitation, the nth-order response moments E[Xⁿ(t)] and

E [{\dot{X}}^{n} (t)]

are

\begin{aligned} E [X^{n} (t)] & = \int_{0}^{t} \dots \int_{0}^{t} E [U (t_{1}) U (t_{2}) \dots U (t_{n})] h (t - t_{1}) h (t - t_{2}) \\ \dots h (t - t_{n}) d t_{1} d t_{2} \dots d t_{n} \end{aligned}

(33)

\begin{aligned} E [{\dot{X}}^{n} (t)] & = \int_{0}^{t} \dots \int_{0}^{t} E [U (t_{1}) U (t_{2}) \dots U (t_{n})] \dot{h} (t - t_{1}) \dot{h} (t - t_{2}) \\ \dots \dot{h} (t - t_{n}) d t_{1} d t_{2} \dots d t_{n} \end{aligned}

(34)

where E[U(t₁)U(t₂) · · · U(t_n)] is the nth-order autocorrelation function of the excitation U(t).

In the following sections, we find the analytical solutions of the second-, third-, and fourth-order response moments by using the autocorrelation functions of U(t) derived in Sec. 3. Furthermore, based on the analytical solutions, we investigate what kind of time-varying components each moment consists of.

4.1 Second-Order Moments.

From Eqs. and , the second-order displacement moment E[X²(t)] and the second-order velocity moment

E [{\dot{X}}^{2} (t)]

are expressed, respectively, as follows:

E [X^{2} (t)] = \int_{0}^{t} \int_{0}^{t} E [U (t_{1}) U (t_{2})] h (t - t_{1}) h (t - t_{2}) d t_{1} d t_{2}

(35)

E [{\dot{X}}^{2} (t)] = \int_{0}^{t} \int_{0}^{t} E [U (t_{1}) U (t_{2})] \dot{h} (t - t_{1}) \dot{h} (t - t_{2}) d t_{1} d t_{2}

(36)

where E[U(t₁)U(t₂)] is the autocorrelation function of U(t), and by Eq. , E[U(t₁)U(t₂)] = E[U²]exp ( − α|t₂ − t₁|). Substituting it into Eqs. and , and performing integration, we can obtain the analytical solutions of E[X²(t)] and

E [{\dot{X}}^{2} (t)]

⁠. The integral calculation was done using Matlab (R2021a) in this study. Since the analytical solutions are given by very long expressions, they are included in Appendix B. These solutions are not dependent on the shape of the probability density p_U(u) of the excitation U(t) although they are proportional to E[U²]. From the analytical solutions, it is found that both E[X²(t)] and

E [{\dot{X}}^{2} (t)]

consist of the following terms (we assume ω_d ≃ 1, i.e., the system is lightly damped):

Exponentially decaying oscillatory terms with period 2π
$e^{- (ζ + α) t} \sin (ω_{d} t), e^{- (ζ + α) t} \cos (ω_{d} t)$
Exponentially decaying oscillatory terms with period π
$e^{- 2 ζ t} \sin (2 ω_{d} t), e^{- 2 ζ t} \cos (2 ω_{d} t)$
Monotonically decaying term
$e^{- 2 ζ t}$
Stationary term (constant term)

where the stationary term is constant regardless of time and is the term that remains when t → ∞ in the analytical solution, which corresponds to the stationary moment.

4.2 Third-Order Moments.

The third-order displacement moment E[X³(t)] and the third-order velocity moment

E [{\dot{X}}^{3} (t)]

are given, respectively, as follows:

\begin{aligned} E [X^{3} (t)] & = \int_{0}^{t} \int_{0}^{t} \int_{0}^{t} E [U (t_{1}) U (t_{2}) U (t_{3})] h (t - t_{1}) h (t - t_{2}) h (t - t_{3}) \\ d t_{1} d t_{2} d t_{3} \end{aligned}

(37)

\begin{aligned} E [{\dot{X}}^{3} (t)] & = \int_{0}^{t} \int_{0}^{t} \int_{0}^{t} E [U (t_{1}) U (t_{2}) U (t_{3})] \dot{h} (t - t_{1}) \dot{h} (t - t_{2}) \dot{h} (t - t_{3}) \\ d t_{1} d t_{2} d t_{3} \end{aligned}

(38)

These moments are calculated by substituting E[U(t₁)U(t₂)U(t₃)] derived in Sec. 3.1 into Eqs. and and then performing integration. When integrating, we need to take into account that Eq. is valid only for t₁ ≤ t₂ ≤ t₃, and that there are a total of six combinations of the order of t₁, t₂, and t₃. As mentioned at the end of Sec. 3.1, E[U(t₁)U(t₂)U(t₃)] with different order of t₁, t₂, and t₃ can be obtained by replacing the three indices of time t in Eq. . With the above in mind, we divide the integral into six parts

\begin{aligned} E [X^{3} (t)] & = \int_{0}^{t} \int_{0}^{t_{3}} \int_{0}^{t_{2}} E [U^{3}] e^{- α (t_{3} - t_{1})} h (t - t_{1}) h (t - t_{2}) h (t - t_{3}) d t_{1} d t_{2} d t_{3} \\ + \int_{0}^{t} \int_{0}^{t_{2}} \int_{0}^{t_{3}} E [U^{3}] e^{- α (t_{2} - t_{1})} h (t - t_{1}) h (t - t_{2}) h (t - t_{3}) d t_{1} d t_{3} d t_{2} \\ + \int_{0}^{t} \int_{0}^{t_{3}} \int_{0}^{t_{1}} E [U^{3}] e^{- α (t_{3} - t_{2})} h (t - t_{1}) h (t - t_{2}) h (t - t_{3}) d t_{2} d t_{1} d t_{3} \\ + \int_{0}^{t} \int_{0}^{t_{1}} \int_{0}^{t_{3}} E [U^{3}] e^{- α (t_{1} - t_{2})} h (t - t_{1}) h (t - t_{2}) h (t - t_{3}) d t_{2} d t_{3} d t_{1} \\ + \int_{0}^{t} \int_{0}^{t_{2}} \int_{0}^{t_{1}} E [U^{3}] e^{- α (t_{2} - t_{3})} h (t - t_{1}) h (t - t_{2}) h (t - t_{3}) d t_{3} d t_{1} d t_{2} \\ + \int_{0}^{t} \int_{0}^{t_{1}} \int_{0}^{t_{2}} E [U^{3}] e^{- α (t_{1} - t_{3})} h (t - t_{1}) h (t - t_{2}) h (t - t_{3}) d t_{3} d t_{2} d t_{1} \end{aligned}

(39)

Comparing the intervals of integration and the integrands among the above six terms, it can be seen that all the terms yield the same integral result. Consequently, Eq. reduces to

E [X^{3} (t)] = 6 E [U^{3}] \int_{0}^{t} \int_{0}^{t_{3}} \int_{0}^{t_{2}} e^{- α (t_{3} - t_{1})} h (t - t_{1}) h (t - t_{2}) h (t - t_{3}) d t_{1} d t_{2} d t_{3}

(40)

In the same way, the velocity moment

E [{\dot{X}}^{3} (t)]

can be calculated from the following equation:

E [{\dot{X}}^{3} (t)] = 6 E [U^{3}] \int_{0}^{t} \int_{0}^{t_{3}} \int_{0}^{t_{2}} e^{- α (t_{3} - t_{1})} \dot{h} (t - t_{1}) \dot{h} (t - t_{2}) \dot{h} (t - t_{3}) d t_{1} d t_{2} d t_{3} |

(41)

From Eqs. and , E[X³(t)] and

E [{\dot{X}}^{3} (t)]

are proportional to the third-order excitation moment E[U³]. The integrals in Eqs. and were calculated using Matlab to obtain the analytical solutions of the third-order response moments. These solutions are provided in Appendix B. We observe from the analytical solutions that both E[X³(t)] and

E [{\dot{X}}^{3} (t)]

consist of the following terms:

Exponentially decaying oscillatory terms with period 2π
$\begin{aligned} e^{- (ζ + α) t} \sin (ω_{d} t), e^{- (ζ + α) t} \cos (ω_{d} t) \\ e^{- 3 ζ t} \sin (ω_{d} t), e^{- 3 ζ t} \cos (ω_{d} t) \end{aligned}$
Exponentially decaying oscillatory terms with period π
$e^{- (2 ζ + α) t} \sin (2 ω_{d} t), e^{- (2 ζ + α) t} \cos (2 ω_{d} t)$
Exponentially decaying oscillatory terms with period 2π/3
$e^{- 3 ζ t} \sin (3 ω_{d} t), e^{- 3 ζ t} \cos (3 ω_{d} t)$
Monotonically decaying term
$e^{- (2 ζ + α) t}$
Stationary term (constant term)

where the stationary term is the constant term that remains when t → ∞ in the analytical solution (i.e., the stationary third-order moment).

4.3 Fourth-Order Moments.

The fourth-order displacement moment E[X⁴(t)] and the fourth-order velocity moment

E [{\dot{X}}^{4} (t)]

are written, respectively, as

\begin{aligned} E [X^{4} (t)] & = \int_{0}^{t} \int_{0}^{t} \int_{0}^{t} \int_{0}^{t} E [U (t_{1}) U (t_{2}) U (t_{3}) U (t_{4})] h (t - t_{1}) h (t - t_{2}) \\ h (t - t_{3}) h (t - t_{4}) d t_{1} d t_{2} d t_{3} d t_{4} \end{aligned}

(42)

\begin{aligned} E [{\dot{X}}^{4} (t)] & = \int_{0}^{t} \int_{0}^{t} \int_{0}^{t} \int_{0}^{t} E [U (t_{1}) U (t_{2}) U (t_{3}) U (t_{4})] \dot{h} (t - t_{1}) \dot{h} (t - t_{2}) \\ \dot{h} (t - t_{3}) \dot{h} (t - t_{4}) d t_{1} d t_{2} d t_{3} d t_{4} \end{aligned}

(43)

which are calculated through the similar manner as in Sec. 4.2. First, the integration is performed under t₁ ≤ t₂ ≤ t₃ ≤ t₄ in which Eq. is applicable. Then, multiplying the integral result by 24, which is the total number of permutations of t₁, t₂, t₃, and t₄, leads to E[X⁴(t)] and

E [{\dot{X}}^{4} (t)]

⁠, namely,

\begin{aligned} E [X^{4} (t)] & = 24 \int_{0}^{t} \int_{0}^{t_{4}} \int_{0}^{t_{3}} \int_{0}^{t_{2}} E [U (t_{1}) U (t_{2}) U (t_{3}) U (t_{4})] h (t - t_{1}) h (t - t_{2}) \\ h (t - t_{3}) h (t - t_{4}) d t_{1} d t_{2} d t_{3} d t_{4} \end{aligned}

(44)

\begin{aligned} E [{\dot{X}}^{4} (t)] & = 24 \int_{0}^{t} \int_{0}^{t_{4}} \int_{0}^{t_{3}} \int_{0}^{t_{2}} E [U (t_{1}) U (t_{2}) U (t_{3}) U (t_{4})] \dot{h} (t - t_{1}) \dot{h} (t - t_{2}) \\ \dot{h} (t - t_{3}) \dot{h} (t - t_{4}) d t_{1} d t_{2} d t_{3} d t_{4} \end{aligned}

(45)

By substituting Eq. into Eqs. and and calculating them using Matlab, we obtained the analytical solutions of E[X⁴(t)] and

E [{\dot{X}}^{4} (t)]

⁠. These solutions are much longer than those of the second- and third-order moments, and thus, the solutions of the fourth-order moments in their complete forms are not shown in this paper. However, the terms contained in E[X⁴(t)] and

E [{\dot{X}}^{4} (t)]

are common and indicated below.

Exponentially decaying oscillatory terms with period 2π
$\begin{aligned} e^{- (ζ + α) t} \sin (ω_{d} t), e^{- (ζ + α) t} \cos (ω_{d} t), \\ e^{- (3 ζ + α) t} \sin (ω_{d} t), e^{- (3 ζ + α) t} \cos (ω_{d} t) \end{aligned}$
Exponentially decaying oscillatory terms with period π
$\begin{aligned} e^{- 2 ζ t} \sin (2 ω_{d} t), e^{- 2 ζ t} \cos (2 ω_{d} t), \\ e^{- (2 ζ + α) t} \sin (2 ω_{d} t), e^{- (2 ζ + α) t} \cos (2 ω_{d} t), \\ e^{- 4 ζ t} \sin (2 ω_{d} t), e^{- 4 ζ t} \cos (2 ω_{d} t), \\ e^{- (2 ζ + 2 α - A) t} \sin (2 ω_{d} t), e^{- (2 ζ + 2 α - A) t} \cos (2 ω_{d} t) \end{aligned}$
Exponentially decaying oscillatory terms with period 2π/3
$e^{- (3 ζ + α) t} \sin (3 ω_{d} t), e^{- (3 ζ + α) t} \cos (3 ω_{d} t)$
Exponentially decaying oscillatory terms with period π/2
$e^{- 4 ζ t} \sin (4 ω_{d} t), e^{- 4 ζ t} \cos (4 ω_{d} t)$
Monotonically decaying terms
$e^{- 2 ζ t}, e^{- (2 ζ + α) t}, e^{- 4 ζ t}, e^{- (2 ζ + 2 α - A) t}$
Stationary term (constant term)

where the stationary term corresponds to the stationary fourth-order moment. It is also worth mentioning that unlike the second- and third-order moments, E[X⁴(t)] and

E [{\dot{X}}^{4} (t)]

depend upon not only the excitation moment of the same order E[U⁴] but also the second-order moment E[U²] and the third-order moment E[U³]. This feature is due to the fact that the fourth-order autocorrelation function of the excitation (Eq. ) includes E[U²] and E[U³] as well as E[U⁴].

5 Analysis Results

In this section, the magnitude of each component in the analytical solution of the response moment is compared quantitatively to examine its dominant time-varying component. The values of the damping ratio ζ of the system and the bandwidth α of the excitation power spectrum S_U(ω) are given as follows:

Damping ratio: ζ = 0.05
Bandwidth of excitation power spectrum: α = 0.01, 0.05, 1

For the non-dimensional linear system , ζ corresponds to the bandwidth of its frequency response function. α is also the non-dimensional bandwidth parameter of S_U(ω). Hence, ζ = α indicates that the bandwidth of the frequency response function of the system and that of S_U(ω) are equal. In the previous papers [29,34], we investigated the stationary and transient responses of the same linear system as in this study, and it was revealed that the response characteristics change significantly according to whether α is larger or smaller than ζ. Based on this knowledge, we consider three types of α, which are smaller, equal, and larger values compared to ζ, respectively.

In order to illustrate the accuracy of the analytical solutions, they are compared with the results obtained by Monte Carlo simulation. As a numerical example, we carried out the simulation in the case of gamma-distributed excitation. Since we assume the excitation is zero-mean, the gamma distribution is shifted so that its mean value is zero and is written as

p_{U} (u) = \frac{(u + a b)^{a - 1} \exp (- (u / b) - a)}{b^{a} Γ (a)} (u \geq - a b)

(46)

where a > 0 and b > 0 are the shape and scale parameters, respectively. The diffusion coefficient D²(u) corresponding to the shifted gamma distribution is

D^{2} (u) = 2 α b (u + a b) (u \geq - a b)

(47)

which is derived from Eq. . In this simulation, the values of a and b were chosen as follows:

a = 1, b = 1

(48)

The shifted gamma distribution with a = b = 1 is shown in Fig. 2. In the Monte Carlo simulation procedure, first, the sample functions of the excitation U(t) are generated by solving Eq. with numerically using the Euler–Maruyama scheme [37]. Then, by the use of the generated sample functions of U(t), the linear system is numerically integrated by the fourth-order Runge–Kutta algorithm to obtain the sample functions of the displacement X(t) and the velocity

\dot{X} (t)

⁠, which are employed to evaluate the second-, third-, and fourth-order response moments. The simulation conditions are as follows:

Time-step: Δt = 0.1 (α = 0.01 and 0.05), Δt = 0.01 (α = 1)
Number of sample functions: 2 × 10⁷

It should be noted that although the Monte Carlo simulation was conducted in the case of a non-Gaussian excitation with a specific probability density (i.e., gamma distribution), the analytical solutions derived in this paper are applicable to non-Gaussian excitation with an arbitrary probability density for E[X²(t)],

E [{\dot{X}}^{2} (t)]

⁠, E[X³(t)], and

E [{\dot{X}}^{3} (t)]

⁠, and to excitation with the Pearson distribution family for E[X⁴(t)] and

E [{\dot{X}}^{4} (t)]

⁠.

Fig. 2

View large Download slide

Shifted gamma distribution

5.1 Second-Order Moments.

Figure 3 shows the temporal change of each component of the second-order response moments E[X²(t)] and $E [{\dot{X}}^{2} (t)]$ derived in Sec. 4.1. Since E[X²(t)] and $E [{\dot{X}}^{2} (t)]$ are proportional to the second-order moment E[U²] of the excitation U(t), in Fig. 3, the results of E[X²(t)] and $E [{\dot{X}}^{2} (t)]$ are normalized by E[U²]. If there are multiple terms in each component, the sum of them is plotted. The black solid line (total) denotes the sum of all components, which is namely E[X²(t)]/E[U²] itself or $E [{\dot{X}}^{2} (t)] / E [U^{2}]$ itself. The circles indicate the corresponding Monte Carlo simulation results for comparison. The analytical results (total) are in good agreement with the simulation results, which prove that the analytical solutions derived in this study are accurate.

Fig. 3

Second-order response moments E[X2(t)] (left) and E[X˙2(t)] (right) normalized by second-order excitation moment E[U2] (upper: α = 0.01, middle: α = 0.05, and lower: α = 1)

View large Download slide

Second-order response moments E[X²(t)] (left) and $E [{\dot{X}}^{2} (t)]$ (right) normalized by second-order excitation moment E[U²] (upper: α = 0.01, middle: α = 0.05, and lower: α = 1)

The characteristics of the time variation of E[X²(t)] and $E [{\dot{X}}^{2} (t)]$ ⁠, which can be seen from Fig. 3, are described below. First, we observe the results in the case that the excitation bandwidth parameter α is smaller than the damping ratio ζ which corresponds to the bandwidth of the frequency response function. Comparing the components of E[X²(t)] for α = 0.01, the exponentially decaying oscillation with period 2π is dominant. Correspondingly, the period of E[X²(t)] (total) is also 2π. On the other hand, for $E [{\dot{X}}^{2} (t)]$ ⁠, its π-period component is superior to other components, so that the period of $E [{\dot{X}}^{2} (t)]$ is also π. For α = 0.05 (α = ζ), as in the case of α = 0.01, the 2π- and π-period components dominate in E[X²(t)] and $E [{\dot{X}}^{2} (t)]$ ⁠, respectively. It can also be seen that there is no monotonically decaying component in either E[X²(t)] or $E [{\dot{X}}^{2} (t)]$ (“decay” line is always 0). For α = 1 (α > ζ), in both E[X²(t)] and $E [{\dot{X}}^{2} (t)]$ ⁠, the monotonic decay component is dominant. Comparing between the oscillatory components, the π-period component is larger, although its magnitude itself is small. Therefore, the second-order response moments for α = 1 increase with a small oscillation with period π.

5.2 Third-Order Moments.

Figure 4 shows each component of the third-order response moments E[X³(t)] and $E [{\dot{X}}^{3} (t)]$ derived in Sec. 4.2. Similar to the second-order moments, E[X³(t)] and $E [{\dot{X}}^{3} (t)]$ are proportional to the third-order excitation moment E[U³]. Hence, in Fig. 4, E[X³(t)]/E[U³] and $E [{\dot{X}}^{3} (t)] / E [U^{3}]$ are plotted assuming E[U³] ≠ 0. (When E[U³] = 0, E[X³(t)] and $E [{\dot{X}}^{3} (t)]$ are equal to 0). The analytical results (total) agree very well with the simulation results.

Fig. 4

Third-order response moments E[X3(t)] (left) and E[X˙3(t)] (right) normalized by third-order excitation moment E[U3] (upper: α = 0.01, middle: α = 0.05, and lower: α = 1)

View large Download slide

Third-order response moments E[X³(t)] (left) and $E [{\dot{X}}^{3} (t)]$ (right) normalized by third-order excitation moment E[U³] (upper: α = 0.01, middle: α = 0.05, and lower: α = 1)

From Fig. 4, the oscillation component with period 2π is dominant for both E[X³(t)] and $E [{\dot{X}}^{3} (t)]$ irrespective of the excitation bandwidth α. Since the other oscillatory and monotonic decay components are all smaller than the 2π-period component, these terms do not contribute much to the time variation of E[X³(t)] and $E [{\dot{X}}^{3} (t)]$ ⁠.

5.3 Fourth-Order Moments.

The fourth-order response moments E[X⁴(t)] and

E [{\dot{X}}^{4} (t)]

are not proportional to the fourth-order excitation moment E[U⁴]. In addition, E[X⁴(t)] and

E [{\dot{X}}^{4} (t)]

are dependent on not only E[U⁴] but also the second-order moment E[U²] and the third-order moment E[U³]. Therefore, for the fourth-order response moments, it is not possible to show general results that apply to any excitation distribution as those in Figs. 3 and 4. For this reason, we show the results for two types of the excitation probability densities, which are a gamma distribution and a uniform distribution. The gamma distribution considered here is given by Eqs. and , and the corresponding diffusion coefficient is expressed by Eq. . The uniform distribution, on the other hand, is written as

p_{U} (u) = \frac{1}{2 Δ} (- Δ \leq u \leq Δ)

(49)

where Δ > 0 is the parameter, and let

Δ = \sqrt{3}

in this example. The diffusion coefficient for the uniform distribution is

D^{2} (u) = α (Δ^{2} - u^{2}) (- Δ \leq u \leq Δ)

(50)

The second-, third-, and fourth-order moments of the gamma and uniform distributions with a = b = 1 and

Δ = \sqrt{3}

are shown below.

Gamma distribution E[U²] = 1 E[U³] = 2 E[U⁴] = 9
Uniform distribution E[U²] = 1 E[U³] = 0 E[U⁴] = 1.8

Since the diffusion coefficients D²(u) corresponding to the gamma and uniform distributions are linear and quadratic, respectively (see Eqs. and ), the fourth-order autocorrelation function of random excitations with these non-Gaussian distributions is expressed by Eq. (A = 0, B = 2αb, C = 2αab² for gamma distribution and A = −α, B = 0, C = αΔ² for uniform distribution). Thus, the analytical solutions of E[X⁴(t)] and

E [{\dot{X}}^{4} (t)]

can be obtained by the manner described inSec. 4.3.

Figures 5 and 6 show each component of the fourth-order response moments E[X⁴(t)] and $E [{\dot{X}}^{4} (t)]$ for the gamma and uniform distributions, respectively. Note that Figs. 5 and 6 show the fourth-order response moments themselves rather than the normalized ones. The simulation results in Fig. 6 were obtained with the pertinent Monte Carlo simulation using 2 × 10⁷ realizations of the uniformly distributed excitation. These two figures demonstrate that the analytical solutions derived in this study are precise.

Fig. 5

Fourth-order response moments E[X4(t)] (left) and E[X˙4(t)] (right) for gamma-distributed excitation (upper: α = 0.01, middle: α = 0.05, and lower: α = 1)

View large Download slide

Fourth-order response moments E[X⁴(t)] (left) and $E [{\dot{X}}^{4} (t)]$ (right) for gamma-distributed excitation (upper: α = 0.01, middle: α = 0.05, and lower: α = 1)

Fig. 6

Fourth-order response moments E[X4(t)] (left) and E[X˙4(t)] (right) for uniformly distributed excitation (upper: α = 0.01, middle: α = 0.05, and lower: α = 1)

View large Download slide

Fourth-order response moments E[X⁴(t)] (left) and $E [{\dot{X}}^{4} (t)]$ (right) for uniformly distributed excitation (upper: α = 0.01, middle: α = 0.05, and lower: α = 1)

From Figs. 5 and 6, it is found that the temporal change of the fourth-order response moments has some common features regardless of the excitation probability distribution. For α = 0.01 and 0.05, the 2π- and π-period components are dominant in E[X⁴(t)] and $E [{\dot{X}}^{4} (t)]$ ⁠, respectively. In the case of α = 1, the monotonic decay component is dominant. Among the oscillatory components, the π-period component is slightly larger, and the remaining three oscillatory components (period 2π, 2π/3, and π/2) are almost zero. Therefore, the fourth-order response moments for α = 1 increase with a small oscillation with period π. Finally, the above features are similar to those observed in the second-order response moments. Hence, the characteristics of the temporal change in the response moments are common for the second- and fourth-order moments and different for the third-order moment.

6 Conclusions

We have derived the analytical solutions of the response statistics of a single-degree-of-freedom linear system subjected to a class of non-Gaussian random excitation. The non-Gaussian excitation considered in this paper is a zero-mean stationary stochastic process prescribed by an arbitrary probability density and a power spectrum with bandwidth parameter. The excitation is modeled by a one-dimensional Itô-type stochastic differential equation whose drift and diffusion coefficients are determined according to the probability density and the spectral density.

In order to obtain the analytical solutions of the response statistics, first, the third- and fourth-order autocorrelation functions of the non-Gaussian excitation have been derived using its Markov and detailed balance properties. The third-order correlation function is valid for any probability density function of the excitation. The fourth-order correlation function, on the other hand, has been derived under the assumption that the diffusion coefficient of the stochastic differential equation for the excitation is expressed as a quadratic polynomial. However, the quadratic-type diffusion coefficient corresponds to the fact that the probability density belongs to the Pearson distribution family, which is one of the widest classes of probability distributions. Hence, the fourth-order correlation function derived in this study is applicable to excitation with a wide variety of probability density functions. Then, using the autocorrelation functions of the excitation, the second-, third-, and fourth-order response moments have been found based on the convolution integral of the excitation and the impulse response function of the system. It has been shown that the response moments consist of the several exponentially decaying oscillatory components with different periods, the monotonically decaying components and the stationary component.

Finally, we have investigated the dominant time-varying components of the response moments in the case of three different excitation bandwidths. The findings are as follows:

Second- and fourth-order moments

When the excitation bandwidth is small or equal to the system damping ratio corresponding to the bandwidth of the system, the displacement moment is dominated by the decaying oscillation component with period 2π. On the other hand, the decaying oscillation with period π predominates in the velocity moment.
When the excitation bandwidth is large compared to the damping ratio, the monotonically decaying component becomes the most dominant in both the displacement and velocity moments. Among the oscillation components, the π-period component is slightly larger, and other components are nearly zero. Therefore, the moments increase with a small oscillation with period π.

Third-order moment

Irrespective of the excitation bandwidth, the decaying oscillation component with period 2π dominates for both the displacement and velocity moments.

In this paper, a stationary non-Gaussian process has been considered as a random excitation acting on a system. In some engineering problems, random excitations may have not only non-Gaussianity but also non-stationarity. Therefore, it is of importance to analyze dynamic systems under non-stationary non-Gaussian excitations. The analytical solutions derived in this paper can be applied to the case of non-stationary non-Gaussian excitation if the excitation is given in the form of the stationary non-Gaussian process described by Cai and Lin’s model multiplied by a modulating function introducing amplitude non-stationarity. This topic will be examined concretely in our future work and reported in a later paper.

2

https://github.com/iqiukp/ZoomPlot

Acknowledgment

This work was supported by JSPS KAKENHI Grant Number JP18K13712. The figures were drawn using ZoomPlot (Matlab extension) provided by Qiu, K.^²

Conflict of Interest

The authors have no competing interests to declare that are relevant to the content of this paper.

Data Availability Statement

The authors attest that all data for this study are included in the paper.

Appendix A: The Fourth-Order Autocorrelation Function of Excitation When A = α and 2α

Case of A = α

E [U_{1} U_{2} U_{3} U_{4}] = e^{- α (t_{4} - t_{3} + t_{2} - t_{1})} [E [U^{4}] e^{- α (t_{3} - t_{2})} + B E [U^{3}] e^{- α (t_{3} - t_{2})} (t_{3} - t_{2}) + \frac{C E [U^{2}]}{α} {1 - e^{- α (t_{3} - t_{2})}}]

(A1)

Case of A = 2α

E [U_{1} U_{2} U_{3} U_{4}] = e^{- α (t_{4} - t_{3} + t_{2} - t_{1})} [E [U^{4}] + \frac{B E [U^{3}]}{α} {1 - e^{- α (t_{3} - t_{2})}} + C E [U^{2}] (t_{3} - t_{2})]

(A2)

These equations are valid for t₁ ≤ t₂ ≤ t₃ ≤ t₄.

Appendix B: Analytical Solutions of the Second- and Third-Order Response Moments

The second-order displacement moment

\begin{aligned} \frac{E [X^{2} (t)]}{E [U^{2}]} & = \sum_{i = 1}^{7} A_{i} \\ A_{1} & = e^{- 2 ζ t} \frac{{ω_{d}^{2} - ζ (α + ζ)} \cos (2 ω_{d} t) + ω_{d} (α + 2 ζ) \sin (2 ω_{d} t)}{2 ω_{d}^{2} (α^{2} + 2 α ζ + 1)} \\ A_{2} & = e^{- 2 ζ t} \frac{α + ζ}{2 ω_{d}^{2} ζ (α^{2} + 2 α ζ + 1)} A_{3} = - 2 e^{- (ζ + α) t} \frac{(α^{2} + ω_{d}^{2} - ζ^{2}) \cos (ω_{d} t) + 2 ω_{d} ζ \sin (ω_{d} t)}{2 ω_{d}^{2} (α^{2} - 2 α ζ + 1) (α^{2} + 2 α ζ + 1)} \\ A_{4} & = 2 α e^{- 2 ζ t} \frac{ζ (α^{2} + 3 ω_{d}^{2} - ζ^{2}) \cos (2 ω_{d} t) - ω_{d} (α^{2} + ω_{d}^{2} - 3 ζ^{2}) \sin (2 ω_{d} t)}{2 ω_{d}^{2} (α^{2} - 2 α ζ + 1) (α^{2} + 2 α ζ + 1)} \\ A_{5} & = 2 e^{- (ζ + α) t} \frac{(- α^{2} + 1) \cos (ω_{d} t) + 2 α ω_{d} \sin (ω_{d} t)}{2 ω_{d}^{2} (α^{2} - 2 α ζ + 1) (α^{2} + 2 α ζ + 1)} \\ A_{6} & = 2 α e^{- 2 ζ t} \frac{α^{2} + ω_{d}^{2} - ζ^{2}}{2 ω_{d}^{2} ζ (α^{2} - 2 α ζ + 1) (α^{2} + 2 α ζ + 1)} \\ A_{7} & = \frac{α (α^{2} + ω_{d}^{2} - 3 ζ^{2}) + 2 ζ}{2 ζ (α^{2} - 2 α ζ + 1) (α^{2} + 2 α ζ + 1)} \end{aligned}

The second-order velocity moment

\begin{aligned} \frac{E [{\dot{X}}^{2} (t)]}{E [U^{2}]} & = \sum_{i = 1}^{7} B_{i} \\ B_{1} & = 2 α ζ C_{1} (\frac{ω_{d}}{{(α + ζ)}^{2} - ζ^{2} + 1} - \frac{e^{- (α + ζ) t} (\cos (ω_{d} t) ω_{d} + \sin (ω_{d} t) (α + ζ))}{{(α + ζ)}^{2} - ζ^{2} + 1}) \\ B_{2} & = - \frac{1}{2} α C_{2} (ζ (\cos (2 ω_{d} t) e^{- 2 ζ t} - 1) + \frac{e^{- 2 ζ t} - 1}{ζ} - \sin (2 ω_{d} t) e^{- 2 ζ t} ω_{d}) \\ B_{3} & = - \frac{1}{2} C_{1} (e^{- 2 ζ t} (ζ \sin (2 ω_{d} t) + \cos (2 ω_{d} t) ω_{d}) - ω_{d}) \\ B_{4} & = - ζ C_{3} (\frac{e^{- 2 ζ t} - 1}{2 ζ} - ζ (\frac{\cos (2 ω_{d} t) e^{- 2 ζ t}}{2} - \frac{1}{2}) + \frac{\sin (2 ω_{d} t) e^{- 2 ζ t} ω_{d}}{2}) \\ B_{5} & = - 2 α C_{2} (\frac{α + ζ}{{(α + ζ)}^{2} - ζ^{2} + 1} + \frac{e^{- (α + ζ) t} (\sin (ω_{d} t) ω_{d} - \cos (ω_{d} t) (α + ζ))}{{(α + ζ)}^{2} - ζ^{2} + 1}) \\ B_{6} & = α ζ C_{1} (e^{- 2 ζ t} (ζ \sin (2 ω_{d} t) + \cos (2 ω_{d} t) ω_{d}) - ω_{d}) \\ B_{7} & = α ζ^{2} C_{3} (\frac{e^{- 2 ζ t} - 1}{2 ζ} - ζ (\frac{\cos (2 ω_{d} t) e^{- 2 ζ t}}{2} - \frac{1}{2}) + \frac{\sin (2 ω_{d} t) e^{- 2 ζ t} ω_{d}}{2}) \\ C_{1} & = \frac{1}{ω_{d} (α^{2} - 2 α ζ + 1)}, C_{2} = \frac{1}{α^{2} - 2 α ζ + 1}, C_{3} = \frac{1}{(ζ^{2} - 1) (α^{2} - 2 α ζ + 1)} \end{aligned}

The third-order displacement moment

\begin{aligned} \frac{E [X^{3} (t)]}{E [U^{3}]} & = \sum_{i = 1}^{23} D_{i} \\ D_{1} & = E_{1} (\frac{6 ω_{d}}{α^{2} + 2 ζ α + 1} - \frac{6 e^{- t (α + ζ)} (\cos (ω_{d} t) ω_{d} + \sin (ω_{d} t) (α + ζ))}{α^{2} + 2 ζ α + 1}) \\ D_{2} & = - E_{1} (\frac{6 ω_{d}}{α^{2} + 4 ζ α + 4} - \frac{6 e^{- t (α + 2 ζ)} (\sin (2 ω_{d} t) (α + 2 ζ) + 2 \cos (2 ω_{d} t) ω_{d})}{2 α^{2} + 8 ζ α + 8}) \\ D_{3} & = - E_{2} (\frac{6 ω_{d}}{16 ζ^{2} + 2} + 3 e^{- 3 ζ t} (\frac{ζ \sin (3 ω_{d} t)}{3} - \frac{3 ζ \sin (ω_{d} t) + \cos (ω_{d} t) ω_{d}}{8 ζ^{2} + 1} + \frac{\cos (3 ω_{d} t) ω_{d}}{3}) - ω_{d}) \\ D_{4} & = - 6 E_{3} ζ (\frac{α + 2 ζ}{2 α^{2} + 8 ζ α + 8} - \frac{1}{2 α + 4 ζ} + \frac{e^{- t (α + 2 ζ)}}{2 α + 4 ζ} - \frac{e^{- t (α + 2 ζ)} (\cos (2 ω_{d} t) (α + 2 ζ) - 2 \sin (2 ω_{d} t) ω_{d})}{2 α^{2} + 8 ζ α + 8}) \\ D_{5} & = - E_{2} (\frac{6 ω_{d}}{α^{2} + 2 ζ α + 1} - \frac{6 e^{- t (α + ζ)} (\cos (ω_{d} t) ω_{d} + \sin (ω_{d} t) (α + ζ))}{α^{2} + 2 ζ α + 1}) \\ D_{6} & = 12 E_{4} ζ (\frac{ω_{d}}{α^{2} + 2 ζ α + 1} - \frac{e^{- t (α + ζ)} (\cos (ω_{d} t) ω_{d} + \sin (ω_{d} t) (α + ζ))}{α^{2} + 2 ζ α + 1}) \\ D_{7} & = - 12 E_{4} ζ^{3} (\frac{ω_{d}}{16 ζ^{2} + 2} + \frac{e^{- 3 ζ t}}{2} (\frac{ζ \sin (3 ω_{d} t)}{3} - \frac{3 ζ \sin (ω_{d} t) + \cos (ω_{d} t) ω_{d}}{8 ζ^{2} + 1} + \frac{\cos (3 ω_{d} t) ω_{d}}{3}) - \frac{ω_{d}}{6}) \\ D_{8} & = - 12 E_{4} ζ^{3} (\frac{ω_{d}}{α^{2} + 2 ζ α + 1} - \frac{e^{- t (α + ζ)} (\cos (ω_{d} t) ω_{d} + \sin (ω_{d} t) (α + ζ))}{α^{2} + 2 ζ α + 1}) \\ D_{9} & = - 12 E_{4} α (\frac{ω_{d}}{α^{2} + 2 ζ α + 1} - \frac{e^{- t (α + ζ)} (\cos (ω_{d} t) ω_{d} + \sin (ω_{d} t) (α + ζ))}{α^{2} + 2 ζ α + 1}) \\ D_{10} & = 3 E_{4} α^{3} (\frac{ω_{d}}{16 ζ^{2} + 2} + \frac{e^{- 3 ζ t}}{2} (\frac{ζ \sin (3 ω_{d} t)}{3} - \frac{3 ζ \sin (ω_{d} t) + \cos (ω_{d} t) ω_{d}}{8 ζ^{2} + 1} + \frac{\cos (3 ω_{d} t) ω_{d}}{3}) - \frac{ω_{d}}{6}) \end{aligned}

\begin{aligned} D_{11} & = 24 E_{4} α ζ^{2} (\frac{ω_{d}}{16 ζ^{2} + 2} + \frac{e^{- 3 ζ t}}{2} (\frac{ζ \sin (3 ω_{d} t)}{3} - \frac{3 ζ \sin (ω_{d} t) + \cos (ω_{d} t) ω_{d}}{8 ζ^{2} + 1} + \frac{\cos (3 ω_{d} t) ω_{d}}{3}) - \frac{ω_{d}}{6}) \\ D_{12} & = - 15 E_{4} α^{2} ζ (\frac{ω_{d}}{16 ζ^{2} + 2} + \frac{e^{- 3 ζ t}}{2} (\frac{ζ \sin (3 ω_{d} t)}{3} - \frac{3 ζ \sin (ω_{d} t) + \cos (ω_{d} t) ω_{d}}{8 ζ^{2} + 1} + \frac{\cos (3 ω_{d} t) ω_{d}}{3}) - \frac{ω_{d}}{6}) \\ D_{13} & = ζ E_{5} e^{- 3 ζ t} (9 ζ \cos (ω_{d} t) - ζ \cos (3 ω_{d} t) - 8 ζ^{3} \cos (3 ω_{d} t) - 8 ζ e^{3 ζ t} - 3 \sin (ω_{d} t) ω_{d} + \sin (3 ω_{d} t) ω_{d} + 8 ζ^{3} e^{3 ζ t} + 8 ζ^{2} \sin (3 ω_{d} t) ω_{d}) \\ D_{14} & = 12 α ζ^{2} E_{4} (\frac{ω_{d}}{α^{2} + 2 ζ α + 1} - \frac{e^{- t (α + ζ)} (\cos (ω_{d} t) ω_{d} + \sin (ω_{d} t) (α + ζ))}{α^{2} + 2 ζ α + 1}) \\ D_{15} & = - \frac{1}{2} α E_{5} e^{- 3 ζ t} (9 ζ \cos (ω_{d} t) - ζ \cos (3 ω_{d} t) - 8 ζ^{3} \cos (3 ω_{d} t) - 8 ζ e^{3 ζ t} - 3 \sin (ω_{d} t) ω_{d} + \sin (3 ω_{d} t) ω_{d} + 8 ζ^{3} e^{3 ζ t} + 8 ζ^{2} \sin (3 ω_{d} t) ω_{d}) \\ D_{16} & = - 3 E_{7} α^{3} e^{- 3 ζ t} (3 ζ \sin (ω_{d} t) - e^{3 ζ t} ω_{d} + \cos (ω_{d} t) ω_{d}) \\ D_{17} & = - α^{2} E_{6} e^{- 3 ζ t} (9 ζ \cos (ω_{d} t) - ζ \cos (3 ω_{d} t) - 8 ζ^{3} \cos (3 ω_{d} t) - 8 ζ e^{3 ζ t} - 3 \sin (ω_{d} t) ω_{d} + \sin (3 ω_{d} t) ω_{d} + 8 ζ^{3} e^{3 ζ t} + 8 ζ^{2} \sin (3 ω_{d} t) ω_{d}) \\ D_{18} & = - 2 ζ^{2} E_{6} e^{- 3 ζ t} (9 ζ \cos (ω_{d} t) - ζ \cos (3 ω_{d} t) - 8 ζ^{3} \cos (3 ω_{d} t) - 8 ζ e^{3 ζ t} - 3 \sin (ω_{d} t) ω_{d} + \sin (3 ω_{d} t) ω_{d} + 8 ζ^{3} e^{3 ζ t} + 8 ζ^{2} \sin (3 ω_{d} t) ω_{d}) \\ D_{19} & = - 12 α E_{7} e^{- 3 ζ t} (3 ζ \sin (ω_{d} t) - e^{3 ζ t} ω_{d} + \cos (ω_{d} t) ω_{d}) \\ D_{20} & = 12 ζ E_{7} e^{- 3 ζ t} (3 ζ \sin (ω_{d} t) - e^{3 ζ t} ω_{d} + \cos (ω_{d} t) ω_{d}) \\ D_{21} & = - 12 α ζ^{2} E_{7} e^{- 3 ζ t} (3 ζ \sin (ω_{d} t) - e^{3 ζ t} ω_{d} + \cos (ω_{d} t) ω_{d}) \\ D_{22} & = 15 α^{2} ζ E_{7} e^{- 3 ζ t} (3 ζ \sin (ω_{d} t) - e^{3 ζ t} ω_{d} + \cos (ω_{d} t) ω_{d}) \\ D_{23} & = 3 α ζ E_{6} e^{- 3 ζ t} (9 ζ \cos (ω_{d} t) - ζ \cos (3 ω_{d} t) - 8 ζ^{3} \cos (3 ω_{d} t) - 8 ζ e^{3 ζ t} - 3 \sin (ω_{d} t) ω_{d} + \sin (3 ω_{d} t) ω_{d} + 8 ζ^{3} e^{3 ζ t} + 8 ζ^{2} \sin (3 ω_{d} t) ω_{d}) \end{aligned}

\begin{aligned} E_{1} & = \frac{1}{ω_{d} (α^{2} - 2 α ζ + 1)}, E_{2} = \frac{1}{ω_{d} (α^{2} - 2 α ζ + 1) (α^{2} - 4 α ζ + 4)}, E_{3} = \frac{1}{(ζ^{2} - 1) (α^{2} - 2 α ζ + 1)} \\ E_{4} = \frac{1}{ω_{d}^{3} (α - 2 ζ) (α^{2} - 2 α ζ + 1) (α^{2} - 4 α ζ + 4)}, E_{5} = \frac{1}{(ζ^{2} - 1) (8 ζ^{2} + 1) (α^{2} - 2 α ζ + 1) (α^{2} - 4 α ζ + 4)} \\ E_{6} = \frac{1}{(ζ^{2} - 1) (8 ζ^{2} + 1) (α - 2 ζ) (α^{2} - 2 α ζ + 1) (α^{2} - 4 α ζ + 4)} E_{7} = \frac{1}{ω_{d}^{3} (8 ζ^{2} + 1) (α - 2 ζ) (α^{2} - 2 α ζ + 1) (α^{2} - 4 α ζ + 4)} \end{aligned}

The third-order velocity moment

\begin{aligned} \frac{E [{\dot{X}}^{3} (t)]}{E [U^{3}]} & = \sum_{i = 1}^{51} F_{i} \\ F_{1} & = G_{1} (\frac{6 (α + ζ)}{α^{2} + 2 ζ α + 1} + \frac{6 e^{- (α + ζ) t} (\sin (ω_{d} t) ω_{d} - \cos (ω_{d} t) (α + ζ))}{α^{2} + 2 ζ α + 1}) \\ F_{2} & = - G_{1} (\frac{6 ζ (4 ζ^{2} + 5)}{24 ζ^{2} + 3} + \frac{24 \cos (ω_{d} t) e^{- 3 ζ t} (ζ - ζ^{3}) (\cos (2 ω_{d} t) - 1)}{24 ζ^{2} + 3} - \frac{2 \cos (2 ω_{d} t) \sin (ω_{d} t) e^{- 3 ζ t} (4 ζ^{4} - 5 ζ^{2} + 1)}{ω_{d} (8 ζ^{2} + 1)}) \\ F_{3} = - G_{1} (- \frac{6 ζ \cos (ω_{d} t) \cos (2 ω_{d} t) e^{- 3 ζ t} (4 ζ^{2} + 5)}{24 ζ^{2} + 3} + \frac{12 \sin (ω_{d} t) e^{- 3 ζ t} ω_{d} (2 ζ^{2} + 1) (\cos (2 ω_{d} t) + 1)}{24 ζ^{2} + 3}) \\ F_{4} = - 6 α ζ G_{2} (\frac{α + 2 ζ}{2 α^{2} + 8 ζ α + 8} - \frac{1}{2 α + 4 ζ} + \frac{e^{- t (α + 2 ζ)}}{2 α + 4 ζ} - \frac{e^{- t (α + 2 ζ)} (\cos (2 ω_{d} t) (α + 2 ζ) - 2 \sin (2 ω_{d} t) ω_{d})}{2 α^{2} + 8 ζ α + 8}) \\ F_{5} = 3 α^{2} G_{1} (\frac{ζ (4 ζ^{2} + 5)}{24 ζ^{2} + 3} + \frac{4 \cos (ω_{d} t) e^{- 3 ζ t} (ζ - ζ^{3}) (\cos (2 ω_{d} t) - 1)}{24 ζ^{2} + 3} - \frac{\cos (2 ω_{d} t) \sin (ω_{d} t) e^{- 3 ζ t} (4 ζ^{4} - 5 ζ^{2} + 1)}{3 ω_{d} (8 ζ^{2} + 1)}) \\ F_{6} = 3 α^{2} G_{1} (- \frac{ζ \cos (ω_{d} t) \cos (2 ω_{d} t) e^{- 3 ζ t} (4 ζ^{2} + 5)}{24 ζ^{2} + 3} + \frac{2 \sin (ω_{d} t) e^{- 3 ζ t} ω_{d} (2 ζ^{2} + 1) (\cos (2 ω_{d} t) + 1)}{24 ζ^{2} + 3}) \\ F_{7} = - 6 ζ^{2} G_{1} (\frac{ζ (4 ζ^{2} + 5)}{24 ζ^{2} + 3} + \frac{4 \cos (ω_{d} t) e^{- 3 ζ t} (ζ - ζ^{3}) (\cos (2 ω_{d} t) - 1)}{24 ζ^{2} + 3} - \frac{\cos (2 ω_{d} t) \sin (ω_{d} t) e^{- 3 ζ t} (4 ζ^{4} - 5 ζ^{2} + 1)}{3 ω_{d} (8 ζ^{2} + 1)}) \\ F_{8} = - 6 ζ^{2} G_{1} (- \frac{ζ \cos (ω_{d} t) \cos (2 ω_{d} t) e^{- 3 ζ t} (4 ζ^{2} + 5)}{24 ζ^{2} + 3} + \frac{2 \sin (ω_{d} t) e^{- 3 ζ t} ω_{d} (2 ζ^{2} + 1) (\cos (2 ω_{d} t) + 1)}{24 ζ^{2} + 3}) \\ F_{9} = - 6 α ω_{d} G_{2} (\frac{ω_{d}}{α^{2} + 4 ζ α + 4} - \frac{e^{- t (α + 2 ζ)} (\sin (2 ω_{d} t) (α + 2 ζ) + 2 \cos (2 ω_{d} t) ω_{d})}{2 α^{2} + 8 ζ α + 8}) \\ F_{10} = - 3 α^{2} G_{1} (\frac{α + ζ}{α^{2} + 2 ζ α + 1} + \frac{e^{- (α + ζ) t} (\sin (ω_{d} t) ω_{d} - \cos (ω_{d} t) (α + ζ))}{α^{2} + 2 ζ α + 1}) \end{aligned}

\begin{aligned} F_{11} = 6 ζ^{2} G_{1} (\frac{α + ζ}{α^{2} + 2 ζ α + 1} + \frac{e^{- (α + ζ) t} (\sin (ω_{d} t) ω_{d} - \cos (ω_{d} t) (α + ζ))}{α^{2} + 2 ζ α + 1}) \\ F_{12} = 6 α G_{3} (\cos (ω_{d} t) (\frac{e^{- (α + ζ) t} (α + ζ)}{2 α^{2} + 4 ζ α + 2} - \frac{3 ζ e^{- 3 ζ t}}{16 ζ^{2} + 2}) - \frac{α + ζ}{2 α^{2} + 4 ζ α + 2} + \frac{3 ζ}{16 ζ^{2} + 2} + \sin (ω_{d} t) (\frac{e^{- 3 ζ t} ω_{d}}{16 ζ^{2} + 2} - \frac{e^{- (α + ζ) t} ω_{d}}{2 α^{2} + 4 ζ α + 2})) \\ F_{13} = - 6 ζ G_{3} (\cos (ω_{d} t) (\frac{e^{- (α + ζ) t} (α + ζ)}{2 α^{2} + 4 ζ α + 2} - \frac{3 ζ e^{- 3 ζ t}}{16 ζ^{2} + 2}) - \frac{α + ζ}{2 α^{2} + 4 ζ α + 2} + \frac{3 ζ}{16 ζ^{2} + 2} + \sin (ω_{d} t) (\frac{e^{- 3 ζ t} ω_{d}}{16 ζ^{2} + 2} - \frac{e^{- (α + ζ) t} ω_{d}}{2 α^{2} + 4 ζ α + 2})) \\ F_{14} = - 6 ζ ω_{d} G_{1} (\frac{ω_{d}}{α^{2} + 2 ζ α + 1} - \frac{e^{- (α + ζ) t} (\cos (ω_{d} t) ω_{d} + \sin (ω_{d} t) (α + ζ))}{α^{2} + 2 ζ α + 1}) \\ F_{15} = 6 α ζ^{3} G_{4} (\frac{α + 2 ζ}{2 α^{2} + 8 ζ α + 8} - \frac{1}{2 α + 4 ζ} + \frac{e^{- t (α + 2 ζ)}}{2 α + 4 ζ} - \frac{e^{- t (α + 2 ζ)} (\cos (2 ω_{d} t) (α + 2 ζ) - 2 \sin (2 ω_{d} t) ω_{d})}{2 α^{2} + 8 ζ α + 8}) \\ F_{16} = - 12 ζ^{3} G_{5} (\frac{ω_{d}}{16 ζ^{2} + 2} + \frac{e^{- 3 ζ t}}{2} (\frac{ζ \sin (3 ω_{d} t)}{3} - \frac{3 ζ \sin (ω_{d} t) + \cos (ω_{d} t) ω_{d}}{8 ζ^{2} + 1} + \frac{\cos (3 ω_{d} t) ω_{d}}{3}) - \frac{ω_{d}}{6}) \\ F_{17} = - 6 ζ^{5} G_{6} (\frac{ω_{d}}{16 ζ^{2} + 2} + \frac{e^{- 3 ζ t}}{2} (\frac{ζ \sin (3 ω_{d} t)}{3} - \frac{3 ζ \sin (ω_{d} t) + \cos (ω_{d} t) ω_{d}}{8 ζ^{2} + 1} + \frac{\cos (3 ω_{d} t) ω_{d}}{3}) - \frac{ω_{d}}{6}) \\ F_{18} = - 6 α ζ^{2} G_{7} (\frac{ω_{d}}{α^{2} + 4 ζ α + 4} - \frac{e^{- t (α + 2 ζ)} (\sin (2 ω_{d} t) (α + 2 ζ) + 2 \cos (2 ω_{d} t) ω_{d})}{2 α^{2} + 8 ζ α + 8}) \\ F_{19} = 9 α ζ G_{1} (\frac{ζ (4 ζ^{2} + 5)}{24 ζ^{2} + 3} + \frac{4 \cos (ω_{d} t) e^{- 3 ζ t} (ζ - ζ^{3}) (\cos (2 ω_{d} t) - 1)}{24 ζ^{2} + 3} - \frac{\cos (2 ω_{d} t) \sin (ω_{d} t) e^{- 3 ζ t} (4 ζ^{4} - 5 ζ^{2} + 1)}{3 ω_{d} (8 ζ^{2} + 1)}) \\ F_{20} = 9 α ζ G_{1} (- \frac{ζ \cos (ω_{d} t) \cos (2 ω_{d} t) e^{- 3 ζ t} (4 ζ^{2} + 5)}{24 ζ^{2} + 3} + \frac{2 \sin (ω_{d} t) e^{- 3 ζ t} ω_{d} (2 ζ^{2} + 1) (\cos (2 ω_{d} t) + 1)}{24 ζ^{2} + 3}) \\ F_{21} = - 6 ζ^{4} G_{8} (\frac{α + ζ}{α^{2} + 2 ζ α + 1} + \frac{e^{- (α + ζ) t} (\sin (ω_{d} t) ω_{d} - \cos (ω_{d} t) (α + ζ))}{α^{2} + 2 ζ α + 1}) \end{aligned}

\begin{aligned} F_{22} & = - 12 ζ^{3} G_{5} (\frac{ω_{d}}{α^{2} + 2 ζ α + 1} - \frac{e^{- (α + ζ) t} (\cos (ω_{d} t) ω_{d} + \sin (ω_{d} t) (α + ζ))}{α^{2} + 2 ζ α + 1}) \\ F_{23} & = - 6 ζ^{5} G_{6} (\frac{ω_{d}}{α^{2} + 2 ζ α + 1} - \frac{e^{- (α + ζ) t} (\cos (ω_{d} t) ω_{d} + \sin (ω_{d} t) (α + ζ))}{α^{2} + 2 ζ α + 1}) \\ F_{24} & = 6 ζ^{2} G_{9} (\frac{ω_{d}}{16 ζ^{2} + 2} - \frac{ω_{d}}{2 α^{2} + 4 ζ α + 2} + \sin (ω_{d} t) (\frac{e^{- (α + ζ) t} (α + ζ)}{2 α^{2} + 4 ζ α + 2} - \frac{3 ζ e^{- 3 ζ t}}{16 ζ^{2} + 2}) - \cos (ω_{d} t) (\frac{e^{- 3 ζ t} ω_{d}}{16 ζ^{2} + 2} - \frac{e^{- (α + ζ) t} ω_{d}}{2 α^{2} + 4 ζ α + 2})) \\ F_{25} & = 6 ζ^{4} G_{10} (\frac{ω_{d}}{16 ζ^{2} + 2} - \frac{ω_{d}}{2 α^{2} + 4 ζ α + 2} + \sin (ω_{d} t) (\frac{e^{- (α + ζ) t} (α + ζ)}{2 α^{2} + 4 ζ α + 2} - \frac{3 ζ e^{- 3 ζ t}}{16 ζ^{2} + 2}) - \cos (ω_{d} t) (\frac{e^{- 3 ζ t} ω_{d}}{16 ζ^{2} + 2} - \frac{e^{- (α + ζ) t} ω_{d}}{2 α^{2} + 4 ζ α + 2})) \\ F_{26} & = 9 α ω_{d} G_{1} (\frac{ω_{d}}{16 ζ^{2} + 2} - \frac{e^{- 3 ζ t}}{2} (\frac{ζ \sin (3 ω_{d} t)}{3} + \frac{3 ζ \sin (ω_{d} t) + \cos (ω_{d} t) ω_{d}}{8 ζ^{2} + 1} + \frac{\cos (3 ω_{d} t) ω_{d}}{3}) + \frac{ω_{d}}{6}) \\ F_{27} & = - 9 α ζ G_{1} (\frac{α + ζ}{α^{2} + 2 ζ α + 1} + \frac{e^{- (α + ζ) t} (\sin (ω_{d} t) ω_{d} - \cos (ω_{d} t) (α + ζ))}{α^{2} + 2 ζ α + 1}) \\ F_{28} & = - 6 ζ ω_{d} G_{1} (\frac{ω_{d}}{16 ζ^{2} + 2} + \frac{e^{- 3 ζ t}}{2} (\frac{ζ \sin (3 ω_{d} t)}{3} - \frac{3 ζ \sin (ω_{d} t) + \cos (ω_{d} t) ω_{d}}{8 ζ^{2} + 1} + \frac{\cos (3 ω_{d} t) ω_{d}}{3}) - \frac{ω_{d}}{6}) \\ F_{29} & = 6 ζ^{4} G_{8} (\frac{ζ (4 ζ^{2} + 5)}{24 ζ^{2} + 3} + \frac{4 \cos (ω_{d} t) e^{- 3 ζ t} (ζ - ζ^{3}) (\cos (2 ω_{d} t) - 1)}{24 ζ^{2} + 3} - \frac{\cos (2 ω_{d} t) \sin (ω_{d} t) e^{- 3 ζ t} (4 ζ^{4} - 5 ζ^{2} + 1)}{3 ω_{d} (8 ζ^{2} + 1)}) \\ F_{30} & = 6 ζ^{4} G_{8} (- \frac{ζ \cos (ω_{d} t) \cos (2 ω_{d} t) e^{- 3 ζ t} (4 ζ^{2} + 5)}{24 ζ^{2} + 3} + \frac{2 \sin (ω_{d} t) e^{- 3 ζ t} ω_{d} (2 ζ^{2} + 1) (\cos (2 ω_{d} t) + 1)}{24 ζ^{2} + 3}) \\ F_{31} & = 6 ζ^{3} G_{11} (\cos (ω_{d} t) (\frac{e^{- (α + ζ) t} (α + ζ)}{2 α^{2} + 4 ζ α + 2} - \frac{3 ζ e^{- 3 ζ t}}{16 ζ^{2} + 2}) - \frac{α + ζ}{2 α^{2} + 4 ζ α + 2} + \frac{3 ζ}{16 ζ^{2} + 2} + \sin (ω_{d} t) (\frac{e^{- 3 ζ t} ω_{d}}{16 ζ^{2} + 2} - \frac{e^{- (α + ζ) t} ω_{d}}{2 α^{2} + 4 ζ α + 2})) \end{aligned}

\begin{aligned} F_{32} & = - 3 α^{2} ζ^{2} G_{8} (\frac{α + ζ}{α^{2} + 2 ζ α + 1} + \frac{e^{- (α + ζ) t} (\sin (ω_{d} t) ω_{d} - \cos (ω_{d} t) (α + ζ))}{α^{2} + 2 ζ α + 1}) \\ F_{33} & = - 3 α^{2} ζ^{3} G_{6} (\frac{ω_{d}}{α^{2} + 2 ζ α + 1} - \frac{e^{- (α + ζ) t} (\cos (ω_{d} t) ω_{d} + \sin (ω_{d} t) (α + ζ))}{α^{2} + 2 ζ α + 1}) \\ F_{34} & = - 9 α ζ^{3} G_{8} (\frac{ζ (4 ζ^{2} + 5)}{24 ζ^{2} + 3} + \frac{4 \cos (ω_{d} t) e^{- 3 ζ t} (ζ - ζ^{3}) (\cos (2 ω_{d} t) - 1)}{24 ζ^{2} + 3} - \frac{\cos (2 ω_{d} t) \sin (ω_{d} t) e^{- 3 ζ t} (4 ζ^{4} - 5 ζ^{2} + 1)}{3 ω_{d} (8 ζ^{2} + 1)}) \end{aligned}

\begin{aligned} F_{35} & = - 9 α ζ^{3} G_{8} (- \frac{ζ \cos (ω_{d} t) \cos (2 ω_{d} t) e^{- 3 ζ t} (4 ζ^{2} + 5)}{24 ζ^{2} + 3} + \frac{2 \sin (ω_{d} t) e^{- 3 ζ t} ω_{d} (2 ζ^{2} + 1) (\cos (2 ω_{d} t) + 1)}{24 ζ^{2} + 3}) \\ F_{36} & = - 6 α ζ^{2} G_{11} (\cos (ω_{d} t) (\frac{e^{- (α + ζ) t} (α + ζ)}{2 α^{2} + 4 ζ α + 2} - \frac{3 ζ e^{- 3 ζ t}}{16 ζ^{2} + 2}) - \frac{α + ζ}{2 α^{2} + 4 ζ α + 2} + \frac{3 ζ}{16 ζ^{2} + 2} + \sin (ω_{d} t) (\frac{e^{- 3 ζ t} ω_{d}}{16 ζ^{2} + 2} - \frac{e^{- (α + ζ) t} ω_{d}}{2 α^{2} + 4 ζ α + 2})) \\ F_{37} & = 9 α ζ^{2} G_{5} (\frac{ω_{d}}{16 ζ^{2} + 2} - \frac{e^{- 3 ζ t}}{2} (\frac{ζ \sin (3 ω_{d} t)}{3} + \frac{3 ζ \sin (ω_{d} t) + \cos (ω_{d} t) ω_{d}}{8 ζ^{2} + 1} + \frac{\cos (3 ω_{d} t) ω_{d}}{3}) + \frac{ω_{d}}{6}) \\ F_{38} & = - 6 α^{2} ζ G_{5} (\frac{ω_{d}}{16 ζ^{2} + 2} - \frac{e^{- 3 ζ t}}{2} (\frac{ζ \sin (3 ω_{d} t)}{3} + \frac{3 ζ \sin (ω_{d} t) + \cos (ω_{d} t) ω_{d}}{8 ζ^{2} + 1} + \frac{\cos (3 ω_{d} t) ω_{d}}{3}) + \frac{ω_{d}}{6}) \\ F_{39} & = 9 α ζ^{2} G_{5} (\frac{ω_{d}}{16 ζ^{2} + 2} + \frac{e^{- 3 ζ t}}{2} (\frac{ζ \sin (3 ω_{d} t)}{3} - \frac{3 ζ \sin (ω_{d} t) + \cos (ω_{d} t) ω_{d}}{8 ζ^{2} + 1} + \frac{\cos (3 ω_{d} t) ω_{d}}{3}) - \frac{ω_{d}}{6}) \\ F_{40} & = 3 α^{2} ζ G_{5} (\frac{ω_{d}}{16 ζ^{2} + 2} + \frac{e^{- 3 ζ t}}{2} (\frac{ζ \sin (3 ω_{d} t)}{3} - \frac{3 ζ \sin (ω_{d} t) + \cos (ω_{d} t) ω_{d}}{8 ζ^{2} + 1} + \frac{\cos (3 ω_{d} t) ω_{d}}{3}) - \frac{ω_{d}}{6}) \\ F_{41} & = 9 α ζ^{4} G_{6} (\frac{ω_{d}}{16 ζ^{2} + 2} + \frac{e^{- 3 ζ t}}{2} (\frac{ζ \sin (3 ω_{d} t)}{3} - \frac{3 ζ \sin (ω_{d} t) + \cos (ω_{d} t) ω_{d}}{8 ζ^{2} + 1} + \frac{\cos (3 ω_{d} t) ω_{d}}{3}) - \frac{ω_{d}}{6}) \\ F_{42} & = - 6 α ζ G_{9} (\frac{ω_{d}}{16 ζ^{2} + 2} - \frac{ω_{d}}{2 α^{2} + 4 ζ α + 2} + \sin (ω_{d} t) (\frac{e^{- (α + ζ) t} (α + ζ)}{2 α^{2} + 4 ζ α + 2} - \frac{3 ζ e^{- 3 ζ t}}{16 ζ^{2} + 2}) - \cos (ω_{d} t) (\frac{e^{- 3 ζ t} ω_{d}}{16 ζ^{2} + 2} - \frac{e^{- (α + ζ) t} ω_{d}}{2 α^{2} + 4 ζ α + 2})) \\ F_{43} & = 3 α^{2} ζ^{2} G_{8} (\frac{ζ (4 ζ^{2} + 5)}{24 ζ^{2} + 3} + \frac{4 \cos (ω_{d} t) e^{- 3 ζ t} (ζ - ζ^{3}) (\cos (2 ω_{d} t) - 1)}{24 ζ^{2} + 3} - \frac{\cos (2 ω_{d} t) \sin (ω_{d} t) e^{- 3 ζ t} (4 ζ^{4} - 5 ζ^{2} + 1)}{3 ω_{d} (8 ζ^{2} + 1)}) \\ F_{44} & = 3 α^{2} ζ^{2} G_{8} (- \frac{ζ \cos (ω_{d} t) \cos (2 ω_{d} t) e^{- 3 ζ t} (4 ζ^{2} + 5)}{24 ζ^{2} + 3} + \frac{2 \sin (ω_{d} t) e^{- 3 ζ t} ω_{d} (2 ζ^{2} + 1) (\cos (2 ω_{d} t) + 1)}{24 ζ^{2} + 3}) \end{aligned}

\begin{aligned} F_{45} = 9 α ζ^{3} G_{8} (\frac{α + ζ}{α^{2} + 2 ζ α + 1} + \frac{e^{- (α + ζ) t} (\sin (ω_{d} t) ω_{d} - \cos (ω_{d} t) (α + ζ))}{α^{2} + 2 ζ α + 1}) \\ F_{46} = 9 α ζ^{2} G_{5} (\frac{ω_{d}}{α^{2} + 2 ζ α + 1} - \frac{e^{- (α + ζ) t} (\cos (ω_{d} t) ω_{d} + \sin (ω_{d} t) (α + ζ))}{α^{2} + 2 ζ α + 1}) \\ F_{47} = 3 α^{2} ζ G_{5} (\frac{ω_{d}}{α^{2} + 2 ζ α + 1} - \frac{e^{- (α + ζ) t} (\cos (ω_{d} t) ω_{d} + \sin (ω_{d} t) (α + ζ))}{α^{2} + 2 ζ α + 1}) \\ F_{48} = 9 α ζ^{4} G_{6} (\frac{ω_{d}}{α^{2} + 2 ζ α + 1} - \frac{e^{- (α + ζ) t} (\cos (ω_{d} t) ω_{d} + \sin (ω_{d} t) (α + ζ))}{α^{2} + 2 ζ α + 1}) \\ F_{49} = - 3 α^{2} ζ^{3} G_{6} (\frac{ω_{d}}{16 ζ^{2} + 2} + \frac{e^{- 3 ζ t}}{2} (\frac{ζ \sin (3 ω_{d} t)}{3} - \frac{3 ζ \sin (ω_{d} t) + \cos (ω_{d} t) ω_{d}}{8 ζ^{2} + 1} + \frac{\cos (3 ω_{d} t) ω_{d}}{3}) - \frac{ω_{d}}{6}) \\ F_{50} = - 6 α ζ^{3} G_{10} (\frac{ω_{d}}{16 ζ^{2} + 2} - \frac{ω_{d}}{2 α^{2} + 4 ζ α + 2} + \sin (ω_{d} t) (\frac{e^{- (α + ζ) t} (α + ζ)}{2 α^{2} + 4 ζ α + 2} - \frac{3 ζ e^{- 3 ζ t}}{16 ζ^{2} + 2}) - \cos (ω_{d} t) (\frac{e^{- 3 ζ t} ω_{d}}{16 ζ^{2} + 2} - \frac{e^{- (α + ζ) t} ω_{d}}{2 α^{2} + 4 ζ α + 2})) \\ F_{51} = (3 α ζ G_{12} - \frac{3}{2} α ζ^{3} G_{13} + α^{2} ζ^{2} G_{13}) e^{- 3 ζ t} \times (9 ζ \cos (ω_{d} t) - ζ \cos (3 ω_{d} t) - 8 ζ^{3} \cos (3 ω_{d} t) - 8 ζ e^{3 ζ t} - 3 \sin (ω_{d} t) ω_{d} + \sin (3 ω_{d} t) ω_{d} \\ + 8 ζ^{3} e^{3 ζ t} + 8 ζ^{2} \sin (3 ω_{d} t) ω_{d}) \\ G_{1} = \frac{1}{(α^{2} - 2 α ζ + 1) (α^{2} - 4 α ζ + 4)}, G_{2} = \frac{1}{α^{2} - 2 α ζ + 1}, \\ G_{3} = \frac{1}{(α - 2 ζ) (α^{2} - 2 α ζ + 1)} G_{4} = \frac{1}{(ζ^{2} - 1) (α^{2} - 2 α ζ + 1)}, G_{5} = \frac{1}{ω_{d} (α^{2} - 2 α ζ + 1) (α^{2} - 4 α ζ + 4)} \\ G_{6} = \frac{1}{ω_{d}^{3} (α^{2} - 2 α ζ + 1) (α^{2} - 4 α ζ + 4)}, G_{7} = \frac{1}{ω_{d} (α^{2} - 2 α ζ + 1)} G_{8} = \frac{1}{(ζ^{2} - 1) (α^{2} - 2 α ζ + 1) (α^{2} - 4 α ζ + 4)}, \\ G_{9} = \frac{1}{ω_{d} (α - 2 ζ) (α^{2} - 2 α ζ + 1)} G_{10} = \frac{1}{ω_{d}^{3} (α - 2 ζ) (α^{2} - 2 α ζ + 1)}, G_{11} = \frac{1}{(ζ^{2} - 1) (α - 2 ζ) (α^{2} - 2 α ζ + 1)} \\ G_{12} = \frac{1}{(16 ζ^{2} + 2) (α^{2} - 2 α ζ + 1) (α^{2} - 4 α ζ + 4)}, G_{13} = \frac{1}{(ζ^{2} - 1) (8 ζ^{2} + 1) (α^{2} - 2 α ζ + 1) (α^{2} - 4 α ζ + 4)} \end{aligned}

References

1.

Grigoriu

,

M.

,

1995

,

Applied Non-Gaussian Processes

,

Prentice Hall

,

Englewood Cliffs, NJ

.

2.

Kotulski

,

Z.

, and

Sobczyk

,

K.

,

1981

, “

Linear Systems and Normality

,”

J. Stat. Phys.

,

24

(

2

), pp.

359

–

373

.

Google Scholar

Crossref

3.

Iyengar

,

R. N.

, and

Jaiswal

,

O. R.

,

1993

, “

A New Model for Non-Gaussian Random Excitations

,”

Probab. Eng. Mech.

,

8

(

3

), pp.

281

–

287

.

Google Scholar

Crossref

4.

Moshchuk

,

N.

, and

Ibrahim

,

R. A.

,

1996

, “

Response Statistics of Ocean Structures to Nonlinear Hydrodynamic Loading. Part II: Non-Gaussian Ocean Waves

,”

J. Sound Vib.

,

191

(

1

), pp.

107

–

128

.

Google Scholar

Crossref

5.

Zheng

,

X. Y.

, and

Liaw

,

C. Y.

,

2004

, “

Response Cumulant Analysis of a Linear Oscillator Driven by Morison Force

,”

Appl. Ocean Res.

,

26

(

3

), pp.

154

–

161

.

Google Scholar

Crossref

6.

Sadek

,

F.

, and

Simiu

,

E.

,

2002

, “

Peak Non-Gaussian Wind Effects for Database-Assisted Low-Rise Building Design

,”

J. Eng. Mech.

,

128

(

5

), pp.

530

–

539

.

Google Scholar

Crossref

7.

Zhao

,

H.

,

Grigoriu

,

M.

, and

Gurley

,

K. R.

,

2019

, “

Translation Processes for Wind Pressures on Low-Rise Buildings

,”

J. Wind Eng. Ind. Aerodyn.

,

184

(

1

), pp.

405

–

416

.

Google Scholar

Crossref

8.

Grigoriu

,

M.

,

1984

, “

Crossings of Non-Gaussian Translation Processes

,”

J. Eng. Mech.

,

110

(

4

), pp.

610

–

620

.

Google Scholar

Crossref

9.

Grigoriu

,

M.

,

1998

, “

Simulation of Stationary Non-Gaussian Translation Processes

,”

J. Eng. Mech.

,

124

(

2

), pp.

121

–

126

.

Google Scholar

Crossref

10.

Yamazaki

,

F.

, and

Shinozuka

,

M.

,

1988

, “

Digital Generation of Non-Gaussian Stochastic Fields

,”

J. Eng. Mech.

,

114

(

7

), pp.

1183

–

1197

.

Google Scholar

Crossref

11.

Deodatis

,

G.

, and

Micaletti

,

R. C.

,

2001

, “

Simulation of Highly Skewed Non-Gaussian Stochastic Processes

,”

J. Eng. Mech.

,

127

(

12

), pp.

1284

–

1295

.

Google Scholar

Crossref

12.

Shields

,

M. D.

,

Deodatis

,

G.

, and

Bocchini

,

P.

,

2011

, “

A Simple and Efficient Methodology to Approximate a General Non-Gaussian Stationary Stochastic Process by a Translation Process

,”

Probab. Eng. Mech.

,

26

(

4

), pp.

511

–

519

.

Google Scholar

Crossref

13.

Shields

,

M. D.

, and

Deodatis

,

G.

,

2013

, “

Estimation of Evolutionary Spectra for Simulation of Non-stationary and Non-Gaussian Stochastic Processes

,”

Comput. Struct.

,

126

(

1

), pp.

149

–

163

.

Google Scholar

Crossref

14.

Shields

,

M.

, and

Deodatis

,

G.

,

2013

, “

A Simple and Efficient Methodology to Approximate a General Non-Gaussian Stationary Stochastic Vector Process by a Translation Process With Applications in Wind Velocity Simulation

,”

Probab. Eng. Mech.

,

31

(

1

), pp.

19

–

29

.

Google Scholar

Crossref

15.

Kim

,

H.

, and

Shield

,

M. D.

,

2015

, “

Modeling Strongly Non-Gaussian Non-stationary Stochastic Processes Using the Iterative Translation Approximation Method and Karhunen–Loève Expansion

,”

Comput. Struct.

,

161

(

1

), pp.

31

–

42

.

Google Scholar

Crossref

16.

Phoon

,

K. K.

,

Huang

,

S. P.

, and

Quek

,

S. T.

,

2002

, “

Simulation of Second-Order Processes Using Karhunen–Loeve Expansion

,”

Comput. Struct.

,

80

(

12

), pp.

1049

–

1060

.

Google Scholar

Crossref

17.

Phoon

,

K. K.

,

Huang

,

H. W.

, and

Quek

,

S. T.

,

2005

, “

Simulation of Strongly Non-Gaussian Processes Using Karhunen–Loeve Expansion

,”

Probab. Eng. Mech.

,

20

(

2

), pp.

188

–

198

.

Google Scholar

Crossref

18.

Zhao

,

T.

, and

Wang

,

Y.

,

2018

, “

Simulation of Cross-Correlated Random Field Samples From Sparse Measurements Using Bayesian Compressive Sensing

,”

Mech. Syst. Signal Process.

,

112

(

1

), pp.

384

–

400

.

Google Scholar

Crossref

19.

Sakamoto

,

S.

, and

Ghanem

,

R.

,

2002

, “

Simulation of Multi-dimensional Non-Gaussian Nonstationary Random Fields

,”

Probab. Eng. Mech.

,

17

(

2

), pp.

167

–

176

.

Google Scholar

Crossref

20.

Sakamoto

,

S.

, and

Ghanem

,

R.

,

2002

, “

Polynomial Chaos Decomposition for the Simulation of Non-Gaussian Nonstationary Stochastic Processes

,”

J. Eng. Mech.

,

128

(

2

), pp.

190

–

201

.

Google Scholar

Crossref

21.

Field Jr.

,

R. V.

, and

Grigoriu

,

M.

,

2004

, “

On the Accuracy of the Polynomial Chaos Approximation

,”

Probab. Eng. Mech.

,

19

(

1

), pp.

65

–

80

.

Google Scholar

Crossref

22.

Kontorovich

,

V. Y.

, and

Lyandres

,

V. Z.

,

1995

, “

Stochastic Differential Equations: An Approach to the Generation of Continuous Non-Gaussian Processes

,”

IEEE Trans. Signal Process.

,

43

(

10

), pp.

2372

–

2385

.

Google Scholar

Crossref

23.

Kontorovich

,

V.

,

Lyandres

,

V.

, and

Primak

,

S.

,

1996

, “

The Generation of Diffusion Markovian Processes With Probability Density Function Defined on Part of the Real Axis

,”

IEEE Signal Process. Lett.

,

3

(

1

), pp.

19

–

21

.

Google Scholar

Crossref

24.

Primak

,

S.

,

Lyandres

,

V.

,

Kaufman

,

O.

, and

Kliger

,

M.

,

1999

, “

On the Generation of Correlated Time Series With a Given Probability Density Function

,”

Signal Process.

,

72

(

2

), pp.

61

–

68

.

Google Scholar

Crossref

25.

Cai

,

G. Q.

, and

Lin

,

Y. K.

,

1996

, “

Generation of Non-Gaussian Stationary Stochastic Processes

,”

Phys. Rev. E

,

54

(

1

), pp.

299

–

303

.

Google Scholar

Crossref

26.

Cai

,

G. Q.

, and

Wu

,

C.

,

2004

, “

Modeling of Bounded Stochastic Processes

,”

Probab. Eng. Mech.

,

19

(

3

), pp.

197

–

203

.

Google Scholar

Crossref

27.

Zhu

,

W. Q.

, and

Cai

,

G. Q.

,

2014

, “

Generation of Non-Gaussian Stochastic Processes Using Nonlinear Filters

,”

Probab. Eng. Mech.

,

36

(

1

), pp.

56

–

62

.

Google Scholar

Crossref

28.

Tsuchida

,

T.

, and

Kimura

,

K.

,

2015

, “

Simulation of Narrowband Non-Gaussian Processes Using Envelope Distribution

,”

Proceedings of 12th International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP12)

,

Vancouver, Canada

,

July 12–15

, p.

331

.

29.

Tsuchida

,

T.

, and

Kimura

,

K.

,

2013

, “

Response Distribution of Nonlinear Systems Subjected to Random Excitation With Non-Gaussian Probability Densities and a Wide Range of Bandwidth

,”

Proceedings of 15th Asia Pacific Vibration Conference (APVC2013)

,

Jeju, South Korea

,

June 2–6

, pp.

1199

–

1204

.

30.

Cai

,

G. Q.

, and

Suzuki

,

Y.

,

2006

, “

Response of Systems Under Non-Gaussian Random Excitations

,”

Nonlinear Dyn.

,

45

(

1

), pp.

95

–

108

.

Google Scholar

Crossref

31.

Tsuchida

,

T.

, and

Kimura

,

K.

,

2015

, “

Equivalent Non-Gaussian Excitation Method for a Linear System Subjected to a Non-Gaussian Random Excitation

,”

Theor. Appl. Mech. Japan

,

63

(

1

), pp.

81

–

90

.

32.

Kanno

,

K.

,

Tsuchida

,

T.

, and

Kimura

,

K.

,

2018

, “

Moment Analysis of a Duffing Oscillator Subjected to Non-Gaussian Random Excitation by Using the Equivalent Non-Gaussian Excitation Method and the Equivalent Linearization

,”

Theor. Appl. Mech. Japan

,

64

(

1

), pp.

115

–

130

.

33.

Wu

,

C. Q.

, and

Cai

,

G. Q.

,

2004

, “

Effects of Excitation Probability Distribution on System Responses

,”

Int. J. Non-Linear Mech.

,

39

(

9

), pp.

1463

–

1472

.

Google Scholar

Crossref

34.

Fukushima

,

H.

, and

Tsuchida

,

T.

,

2021

, “

Transient Response Characteristics of Linear Systems Subjected to Non-Gaussian Random Excitation With a Wide Range of Bandwidth (in Japanese)

,”

Trans. JSME

,

87

(

899

), pp.

1

–

16

.

Google Scholar

Crossref

35.

Cai

,

G. Q.

, and

Zhu

,

W. Q.

,

2016

,

Elements of Stochastic Dynamics

,

World Scientific Publishing

,

Hackensack, NJ

.

Google Scholar

Crossref

36.

Gardiner

,

C.

,

2009

,

Stochastic Methods: A Handbook for the Natural and Social Sciences

, 4th ed.,

Springer-Verlag

,

Berlin

.

37.

Kloeden

,

P.

, and

Platen

,

E.

,

1992

,

Numerical Solution of Stochastic Differential Equations

,

Springer

,

Berlin

.

Google Scholar

Crossref

An Analysis of Transient Response Moments of a Linear System Subjected to Non-Gaussian Random Excitation Using Higher-Order Autocorrelation Functions of Excitation

Abstract

1 Introduction

2 Models of System and Non-Gaussian Random Excitation

2.1 Equation of Motion.

2.2 Non-Gaussian Random Excitation.

3 Derivation of Higher-Order Autocorrelation Functions of Non-Gaussian Excitation

3.1 Third-Order Autocorrelation Function.

3.2 Fourth-Order Autocorrelation Function.

4 Analysis of Transient Response Moments

4.1 Second-Order Moments.

4.2 Third-Order Moments.

4.3 Fourth-Order Moments.

5 Analysis Results

5.1 Second-Order Moments.

5.2 Third-Order Moments.

5.3 Fourth-Order Moments.

6 Conclusions

Acknowledgment

Conflict of Interest

Data Availability Statement

Appendix A: The Fourth-Order Autocorrelation Function of Excitation When A = α and 2α

Appendix B: Analytical Solutions of the Second- and Third-Order Response Moments

References

Get Email Alerts

Cited By

ASME Journals

ASME Conference Proceedings

ASME eBooks

Resources

Opportunities

An Analysis of Transient Response Moments of a Linear System Subjected to Non-Gaussian Random Excitation Using Higher-Order Autocorrelation Functions of Excitation

Abstract

1 Introduction

2 Models of System and Non-Gaussian Random Excitation

2.1 Equation of Motion.

2.2 Non-Gaussian Random Excitation.

3 Derivation of Higher-Order Autocorrelation Functions of Non-Gaussian Excitation

3.1 Third-Order Autocorrelation Function.

3.2 Fourth-Order Autocorrelation Function.

4 Analysis of Transient Response Moments

4.1 Second-Order Moments.

4.2 Third-Order Moments.

4.3 Fourth-Order Moments.

5 Analysis Results

5.1 Second-Order Moments.

5.2 Third-Order Moments.

5.3 Fourth-Order Moments.

6 Conclusions

Acknowledgment

Conflict of Interest

Data Availability Statement

Appendix A: The Fourth-Order Autocorrelation Function of Excitation When A = α and 2α

Appendix B: Analytical Solutions of the Second- and Third-Order Response Moments

References

Get Email Alerts

Cited By

Related Articles

Related Proceedings Papers

Related Chapters

ASME Journals

ASME Conference Proceedings

ASME eBooks

Resources

Opportunities

This Feature Is Available To Subscribers Only