Stochastic Analysis of Linear Systems

Synonyms

Evolutionary frequency response function; Evolutionary power spectral density function; Gaussian zero-mean random models of seismic accelerations; Non-geometric spectral moments; Stochastic analysis

Introduction

The stochastic analysis of structural vibrations deals with the description and characterization of structural loads and responses that are modeled as stochastic processes. The probabilistic characterization of the input process could be extremely complex in time domain where the probability density functions depend on the autocorrelation functions which experimentally have to be specified over given set points. Since this approach is difficult to be used in applications, stochastic vibration analysis of structural linear systems subjected to Gaussian input processes is quite often performed in the frequency domain by means of the spectral analysis. This analysis is a very powerful tool for the analytical and experimental treatment of a large class of physical as well as structural problems subjected to random excitations. The main reasons are (a) the spectrum has an immediate physical interpretation as a power-frequency distribution; (b) the spectrum provides information on the stochastic structure of the process; and (c) the spectrum may be estimated by fairly simple numerical techniques which do not require any specific assumption of the structure of the process.

In the framework of earthquake engineering, the stationary non-white input models were suggested first. These models, which account for site properties and for the dominant frequency in ground motion, fail to reproduce the time-varying intensity typical of real earthquakes ground-motion accelerograms. In order to overcome this drawback, the so-called quasi-stationary (or uniformly modulated) random processes have been introduced (see, e.g., Shinozuka and Sato 1967; Jennings et al. 1969; Hsu and Bernard 1978). These processes are constructed as the product of a stationary zero-mean Gaussian random process by a deterministic function of time; for this reason they are also called separable nonstationary stochastic processes.

Furthermore, a time-varying frequency content is observed in actual accelerogram records. This nonstationary frequency is prevalently due to different arrival times of the primary, secondary, and surface waves that propagate at different velocities through the Earth’s crust. To take into account both the simultaneous amplitude and frequency non-stationarity, Spanos and Solomos (1983) proposed a non-separable model introducing a particular evolutionary power spectral density (EPSD) function; Fan and Ahmadi (1990) proposed a generalization of the Kanai-Tajimi filter model with time-dependent coefficients; Conte and Peng (1997) defined the ground-motion accelerations as the sum of a finite number of pairwise independent uniformly modulated zero-mean Gaussian stochastic process, the so-called sigma-oscillatory process.

Once the problem is formulated from a mathematical point of view, the further step deals with the evaluation of the structural response to perform the prediction of the safety of structural systems. In this framework, the maximum absolute peak of stationary or nonstationary stochastic responses may be useful in design information of several engineering situations (see, e.g., Lin 1976; Lutes and Sarkani 2004; Muscolino and Palmeri 2005; Li and Chen 2009). Approximate procedures to calculate the statistics of the maximum absolute peak of the response have been proposed. These procedures lead to the probabilistic assessment of structural failure as a function of barrier crossing rates, distribution of peaks, and extreme values. The latter quantities can be evaluated, for stationary input process, as a function of the well-known geometric spectral moments (GSMs) introduced by Vanmarcke (1972). For stationary stochastic response processes, the GSMs are defined as the geometric moments of the one-sided power spectral density (PSD) of the response process. Application of spectral methods to nonstationary random processes is more difficult than for the stationary ones; indeed for nonstationary processes the geometric approach fails (Di Paola 1985; Muscolino 1991). To perform the structural reliability in the latter cases, the so-called nongeometric spectral moments (NGSMs) have been introduced (Michaelov et al. 1999a, b).

In this study the before outlined topics will be addressed in order to evaluate the spectral characteristics of the structural response that are useful to perform the reliability assessment of linear systems subjected to stationary or nonstationary mono-/multi-correlated excitations.

Spectral Representation of Stochastic Processes

To carry out the spectral analysis, it is necessary to determine the spectral properties of the involved functions. These properties may be determined through the Fourier-Stieltjes transform (Priestley 1999).

Let us consider now a zero-mean stationary stochastic process, F(t). This process is characterized by the feature that its statistical moments do not change over time and generally arise from any “stable” system which has achieved a “steady state.” Moreover, the probabilistic structure of a stationary process is invariant under a shift of the time origin. This is a consequence of the fact that a sample of the process will almost certainly not “decay” to zero at infinity. Then the stationary processes possess infinite energy. Since a stationary process possesses infinite energy, its kth sample, F^(k)(t), cannot be represented by the Fourier transform. In fact in this case, the Dirichlet condition is not satisfied. It follows that the spectral representation of a sample of a stationary stochastic process can be performed only by the Fourier-Stieltjes integral (Priestley 1999):

$$ {F}^{(k)}(t)={\displaystyle \underset{-\infty }{\overset{+\infty }{\int }} \exp \left(\mathrm{i} \omega t\right)}\;\mathrm{d} {N}^{(k)}\left(\omega \right) $$

(1)

where $\mathrm{i}=\sqrt{-1}$ is the imaginary unit and N^(k)(ω) is the kth sample of the complex stochastic process N(ω), satisfying the condition

$$ \mathrm{E}\left\langle \mathrm{d} N\left({\omega}_1\right)\;\mathrm{d} {N}^{*}\left({\omega}_2\right)\right\rangle =\delta \left({\omega}_1-{\omega}_2\right)\;{S}_{NN}\left({\omega}_1\right)\mathrm{d} {\omega}_1\mathrm{d} {\omega}_2 $$

(2)

where δ(•) is the Dirac delta, the symbol E〈•〉 means stochastic average, and the asterisk indicates the complex conjugate quantity. Notice that in Eq. 2 sometimes the Kronecker delta is introduced instead of the Dirac delta; this is to avoid the inconsistence of this relationship for ω₁ = ω₂ (Spanos and Solomos 1983). The relationship (2) shows that the stochastic process N(ω) is a process with orthogonal increments, in the sense that its increments d N(ω₁) and d N(ω₂) at any two distinct points ω₁ and ω₂ are uncorrelated random variables. Furthermore, in Eq. 2S _NN (ω), which is a real and symmetric function, S _NN (−ω) = S _NN (ω), is the power spectral density (PSD) function of the process N(ω). According to the theory of stationary stochastic process, the autocorrelation function, R _FF (τ), of the zero-mean stationary stochastic process F(t) is a real function given as (Lin 1976; Lutes and Sarkani 2004; Li and Chen 2009)

$$\begin{array}{ll} {\widehat{\mathbf{f}}}_{3D}(x)&=\left[\begin{array}{c}\hfill {\widehat{F}}_x(x)\hfill \\ {}\hfill {\widehat{V}}_y(x)\hfill \\ {}\hfill {\widehat{V}}_z(x)\hfill \\ {}\hfill {\widehat{M}}_x(x)\hfill \\ {}\hfill {\widehat{M}}_y(x)\hfill \\ {}\hfill {\widehat{M}}_z(x)\hfill \end{array}\right]\\&=\left[\begin{array}{cc}\hfill {\boldsymbol{\Phi}}_f^{+}\hfill & \hfill {\boldsymbol{\Phi}}_f^{-}\hfill \end{array}\right]\;\left[\begin{array}{cc}\hfill {\boldsymbol{\Lambda}}_x^{+}\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill {\boldsymbol{\Lambda}}_x^{-}\hfill \end{array}\right]\;\left[\begin{array}{c}\hfill {\mathbf{A}}^{+}\hfill \\ {}\hfill {\mathbf{B}}^{-}\hfill \end{array}\right], \end{array}$$

(3)

which in virtue of Eq. 2 leads to

$$ {R}_{FF}\left(\tau \right) = \mathrm{E}\left\langle F\left(t+\tau \right)F(t)\right\rangle ={\displaystyle \underset{-\infty }{\overset{\infty }{\int }} \exp \left(\mathrm{i}\;\omega\;\tau \right){S}_{FF}\left(\omega \right)\;\mathrm{d}\omega } $$

(4)

In this equation S _FF (ω) is the PSD function of the stationary process F(t).

In postulating the stationarity of the stochastic process, very strong assumptions regarding the structure of the process are made. Once these assumptions are dropped, the process can become nonstationary in many different ways. In the framework of the spectral analysis of nonstationary processes, Priestley (see, e.g., Priestley 1999) introduced the evolutionary power spectral density (EPDS) function. The EPSD function has essentially the same type of physical interpretation of the PSD function of stationary processes. The main difference is that whereas the PSD function describes the power-frequency distribution for the whole stationary process, the EPSD function is time dependent and describes the local power-frequency distribution at each instant time. The theory of EPSD function is the only one which preserves this physical interpretation for the nonstationary processes. Moreover, since the spectrum may be estimated by fairly simple numerical techniques, which do not require any specific assumption of the structure of the process, this model, based on the EPSD function, is nowadays the most adopted model for the analysis of structures subjected to nonstationary processes as the seismic motion due to earthquakes.

In the Priestley spectral representation of nonstationary processes, a sample of the nonstationary stochastic process is defined by the Fourier-Stieltjes integral as follows:

$$ {F}^{(k)}(t)={\displaystyle \underset{-\infty }{\overset{+\infty }{\int }} \exp \left(i \omega t\right)\;a\left(\omega,,,,t\right)}\;\mathrm{d} {N}^{(k)}\left(\omega \right) $$

(5)

where a(ω, t) is a slowly varying complex deterministic time-frequency modulating function which has to satisfy the condition a(ω, t) ≡ a^∗(−ω, t) and N(ω) is an orthogonal process satisfying the condition (2). In Eq. 2S _NN (ω) is the PSD function of the so-called “embedded” stationary counterpart process, N(ω) (Michaelov et al. 1999a). It follows that the autocorrelation function of the zero-mean Gaussian nonstationary random process F(t) can be obtained as

$$\begin{array}{lll} {R}_{FF}\left({t}_1,{t}_2\right)&=\mathrm{E}\left\langle F\left({t}_1\right)F\left({t}_2\right)\right\rangle \\&=\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }} \exp \left[\mathrm{i}\left({\omega}_1\;{t}_1-{\omega}_2\;{t}_2\right)\right]\;a\left({\omega}_1,t\right)} {a}^{*}\left({\omega}_2,{t}_2\right)\;\mathrm{E}\left\langle \mathrm{d} N\left({\omega}_1\right) \mathrm{d} {N}^{*}\left({\omega}_2\right)\right\rangle \mathrm{d}{\omega}_1\;\mathrm{d}{\omega}_2\end{array} $$

(6)

It is a real function which, in virtue of Eq. 2, leads to

$$\begin{array}{lll} {R}_{FF}\left({t}_1,{t}_2\right) = \displaystyle \underset{-\infty }{\overset{\infty }{\int }} \exp \left[\mathrm{i}\omega \left({t}_1-{t}_2\right)\right]a\left(\omega, {t}_1\right)\;{a}^{*}\left(\omega, {t}_2\right)\\{S}_{NN}\left(\omega \right)\;\mathrm{d}\omega ={\displaystyle \underset{-\infty }{\overset{\infty }{\int }} \exp \left[\mathrm{i}\omega \left({t}_1-{t}_2\right)\right]\;{S}_{FF}\left(\omega, {t}_1,{t}_2\right)\;\mathrm{d}\omega }\end{array} $$

(7)

where

$$ {S}_{FF}\left(\omega, {t}_1,{t}_2\right)=a\left(\omega, {t}_1\right)\;{a}^{*}\left(\omega, {t}_2\right)\;{S}_{NN}\left(\omega \right) $$

(8)

According to the Priestley evolutionary process model (Priestley 1999), the function

$$ {S}_{FF}\left(\omega,,,, t\right)={\left|a\left(\omega,,,, t\right)\right|}^2{S}_{NN}\left(\omega \right) $$

(9)

is the so-called EPSD function of the nonstationary process F(t). In the previous equations the symbol | • | denotes the modulus of the function in brackets. The processes characterized by the EPSD function S _FF (ω, t) are called fully nonstationary or non-separable random process, since both time and frequency content change, and they cannot be decoupled. If the modulating function is a time-dependent function, a(ω, t) ≡ a(t), the nonstationary process is called quasi-stationary or uniformly modulated or separable random process. In this case the time content change is independent by the frequency content change; indeed, the EPSD function assumes the following expression:

$$ {S}_{FF}\left(\omega,,,, t\right)={a}^2(t)\;{S}_{NN}\left(\omega \right) $$

(10)

In the stochastic analysis the one-sided PSD is generally used; the latter can be suitably defined in the Priestley representation by the following equation:

$$ {G}_{FF}\left(\omega, {t}_1,{t}_2\right)=\left\{\begin{array}{c}\hfill a\left(\omega, {t}_1\right)\;{a}^{*}\left(\omega, {t}_2\right)\;{G}_{NN}\left(\omega \right)\equiv 2{S}_{FF}\left(\omega, {t}_1,{t}_2\right),\omega \ge 0;\hfill \\ {}\hfill 0,\omega <0\hfill \end{array}\right. $$

(11)

where G _NN (ω) (G _NN (ω) = 2S _NN (ω), ω ≥ 0; G _NN (ω) = 0, ω < 0) is the one-sided PSD function of the stationary counterpart of the input process F(t). In this case the autocorrelation function of the process F(t) is given by the following relationship:

$$\begin{array}{ll} {\overline{R}}_{FF}\left({t}_1,{t}_2\right)&=\displaystyle \underset{-\infty }{\overset{\infty }{\int }} \exp \left[\mathrm{i}\omega \left({t}_1-{t}_2\right)\right]a\left(\omega, {t}_1\right) \\ &{a}^{*}\left(\omega, {t}_2\right)\;{G}_{NN}\left(\omega \right)\mathrm{d}\omega \end{array} $$

(12)

Note that since the one-sided PSD function G _NN (ω) is not symmetric, the corresponding autocorrelation function, ${\overline{R}}_{FF}\left({t}_1,{t}_2\right),$ is a complex function (Di Paola 1985), whose real part coincides with the function defined in Eq. 6: $\mathrm{R}\mathrm{e}\left\{{\overline{R}}_{FF}\left({t}_1,{t}_2\right)\right\}\equiv {R}_{FF}\left({t}_1,{t}_2\right).$It can be easily proved that the complex function (12) is the autocorrelation function of a complex process $\overline{F}(t)$ defined as (Di Paola and Petrucci 1990)

$$ \overline{F}(t)=\sqrt{2}{\displaystyle \underset{0}{\overset{+\infty }{\int }} \exp \left(\mathrm{i} \omega t\right)\;a\left(\omega, t\right)}\;\mathrm{d} N\left(\omega \right) $$

(13)

The real part of $\overline{F}(t)$ is proportional to the process F(t), while the imaginary part of $\overline{F}(t)$ is a nonstationary process having stationary counterpart proportional to Hilbert transform of the real part of the stationary counterpart of the process itself (Di Paola 1985; Di Paola and Petrucci 1990. Muscolino 1991). The complex process, $\overline{F}(t)$, which generates the complex autocorrelation function (12), has been called pre-envelope process by Di Paola (1985).

Reliability of Linear Structural Systems Subjected to Stochastic Excitations

The structural systems are conceived and designed to survive to natural actions. If the excitations are modeled as random processes, the dynamic responses are random processes too, and the structural safety needs to be evaluated in a probabilistic sense. Among the models of failure, the simplest one, which is also the most widely used in practical analyses, is based on the assumption that a structure fails as soon as the response at a critical location exits a prescribed safe domain for the first time. The probability of failure, in this case, coincides with the first passage probability, i.e., the probability that the absolute value of the random response process X(t) of a selected structural response (e.g., strain or stress at a critical point) will exceed a specified safety bound, b, within a specified time interval (Lin 1976). In random vibration theory, the problem of probabilistically predicting this event is termed first passage problem. Unfortunately, this is one of the most complicated problems in computational stochastic mechanics. The solution of this problem has not been derived in exact form, even in the simplest case of the stationary response of a single-degree-of-freedom (SDoF) linear oscillator under zero-mean Gaussian white noise (Lin 1976; Lutes and Sarkani 2004; Muscolino and Palmeri 2005). Hence, a large number of approximated techniques have been proposed in literature, which differ in generality, complexity, and accuracy. In the framework of approximate methods, the time-dependent reliability of the structure, based on the first passage failure criterion, can be expressed, for a symmetric barrier, as (Lutes and Sarkani 2004)

$$ {L}_{\left|X\right|}\left(b,t\right)={L}_{\left|X\right|}\left(b,0\right) \exp \left[-{\displaystyle \underset{0}{\overset{t}{\int }}{\eta}_X\left(b,\rho \right)\;\mathrm{d}\rho}\right] $$

(14)

where η _X (b, t) is the so-called hazard function and L_|X|(b, 0) is the reliability at time t = 0, which for the nonstationary case can be assumed as unity.

For narrow-band zero-mean stationary Gaussian process, the hazard function has been derived by Vanmarcke (1972) as

$$ {\eta}_X(b)=\frac{1}{\uppi}\sqrt{\frac{\lambda_{\;2,X}}{\lambda_{\;0,X}}}\left[\frac{1- \exp \left(-b\;{\delta}_X^{\;1.2}\sqrt{\frac{\uppi}{2\;{\lambda}_{\;0,X}}}\right)}{ \exp \left(\frac{b^2}{2\;{\lambda}_{\;0,X}}\right)-1}\right] $$

(15)

with

$$ {\delta}_X=\sqrt{1-\frac{\lambda_{\;1,\;X}^2}{\lambda_{\;0,X} {\lambda}_{\;2,X}}} $$

(16)

In this equation δ _X is the so-called bandwidth parameter of the process X(t) (Vanmarcke 1972, 1975). This parameter measures the variation of the narrowness of the stochastic process X(t). Usually, a stochastic process with 0 ≤ δ _X  ≤ 0.35 is called a narrow-band stochastic process. Finally, in the previous equations λ _i,X (i = 0, 1, 2) are the so-called geometric spectral moments (GSMs), introduced by Vanmarcke (1972) as the geometric moments of the one-sided PSD of the response process:

$$ {\lambda}_{\;i,X}\equiv {\lambda}_{\;i,X}^G={\displaystyle \underset{0}{\overset{\infty }{\int }}{\omega}^i {G}_{XX}\left(\omega \right) \mathrm{d}\omega } $$

(17)

In this equation the apex G emphasizes the geometric evaluation of the GSMs. Notice that λ_0,X coincides with the variance of the zero-mean process X(t), λ_0,X ≡ σ ² _X  = E〈X²(t)〉; λ_2,X coincides with the variance of the zero-mean process $\dot{X}(t)$, ${\lambda}_{\;2,X}\equiv {\sigma}_{\dot{\dot{X}}}^2=\mathrm{E}\left\langle {\dot{X}}^2(t)\right\rangle$; while λ_1,X does not coincide with the cross-covariance of the processes X(t) and $\dot{X}(t)$.

For narrow-band zero-mean nonstationary Gaussian process, the hazard function has been derived by Corotis et al. (1972) as

$$ {\eta}_X\left(b,,,,, t\right)=\frac{1}{\uppi}\sqrt{\frac{\lambda_{\;2,X}(t)}{\lambda_{\;0,X}(t)}}\left[\frac{1- \exp \left(-b\;{\delta}_X^{\;1+d}(t)\sqrt{\frac{\uppi}{2\;{\lambda}_{\;0,X}(t)}}\right)}{ \exp \left(\frac{b^2}{2\;{\lambda}_{\;0,X}(t)}\right)-1}\right] $$

(18)

where d (d = 0 or d = 0.2) is an empirical parameter and δ _X (t) is the bandwidth parameter which in the nonstationary case is time dependent. Because of the non-stationarity of random process, this parameter involves complex functions, and it is defined by Michaelov et al. (1999a, b) as

$$ {\delta}_X(t)=\sqrt{1-\frac{\mathrm{Re}{\left\{{\lambda}_{\;1,X}(t)\right\}}^2}{\lambda_{\;0,X}(t){\lambda}_{\;2,X}(t)}} $$

(19)

The evaluation of the time-dependent quantities λ_i,X(t) is conceptually more complicated than for stationary processes; indeed for these processes the geometric approach fails for i = 1 and i = 2 (Di Paola 1985; Di Paola and Petrucci 1990; Muscolino 1991; Michaelov et al. 1999a, b), that is

$$\begin{array}{lll} {\lambda}_{\;i,X}(t)\ne &{\lambda}_{\;i,X}^G(t)\\&={\displaystyle\underset{0}{\overset{\infty }{\int }}{\omega}^i{G}_{XX}\left(\omega,,,, t\right) \mathrm{d}\omega }, i=1,2\end{array}$$

(20)

The physical inconsistency on the evaluation of the geometric GSMs, in the nonstationary case, as the moments of the one-sided EPSD function was pointed out by Corotis et al. (1972). In fact they discovered that for the case of the transient response of an oscillator subjected to stationary Gaussian white noise processes, the second GSM does not exist because it is unbounded. At same time in the stationary case, this GSM, which is the limit of the transient as the time approaches infinity, is finite. The first that considered the problem of spectral characteristics from a nongeometric point of view was Di Paola (1985). The basic idea of this approach is to establish a time-domain interpretation of the SM. In order to do this, the pre-envelope covariances as the covariances of structural systems subjected to a complex-valued random process (the so-called pre-envelope process) have been introduced (Di Paola and Petrucci 1990). The real part of this process is proportional to the original nonstationary process, while the imaginary part is an auxiliary random process related to the real part in such a way that the complex process exhibits power in the positive frequency range only. Since the use of complex pre-envelope process is not very intuitive, Michaelov et al. (1999a, b) evaluated the pre-envelope covariances as a function of the EPSD of the response and recalled them as nongeometric spectral moments (NGSMs). It has to be emphasized that the NGSMs contain more information than the “conventional” covariances. Indeed, the NGSMs have been proved to be more appropriate for describing nonstationary process and can be effectively employed in structural reliability applications (Di Paola 1985; Di Paola and Petrucci 1990; Muscolino 1991; Michaelov et al. 1999a, b).

It has to be emphasized that in the framework of nonstationary analysis of structures, other time-dependent parameters, very useful in describing the time-variant spectral properties of the stochastic process, are (i) the mean frequency, ν ⁺ _X (t), which evaluate the variation in time of the mean up-crossing rate of the time axis, and (ii) the central frequency, ω_C,X(t), which scrutinizes the variation of the frequency content of the stochastic process with respect to time. The two functions introduced before can be evaluated as a function of NGSMs and have been defined, respectively, as (Michaelov et al. 1999a, b)

$$ {\nu}_X^{+}(t)=\frac{1}{2\;\uppi}\sqrt{\frac{\lambda_{\;2,X}(t)}{\lambda_{\;0,X}(t)}}; {\omega}_{\mathrm{C},X}(t)=\frac{\mathrm{Re}\left\{{\lambda}_{\;1,X}(t)\right\}}{\lambda_{\;0,X}(t)} $$

(21)

In stationary case the mean frequency, ν ⁺ _X , and the central frequency, ω_C,X, are not time dependent. Moreover, in the latter case, because of the Im{λ_1,X} = 0, it follows Re{λ_1,X} ≡ λ_1,X.

Response of Single-Degree-of-Freedom (SDoF) Oscillators

Fundamental of Deterministic Analysis

The theory of deterministic linear systems plays a fundamental role in the dynamic analysis of structures subjected to stochastic excitations. For this reason in this section the fundamental of deterministic analysis of SDoF subjected to deterministic excitation is synthetically reviewed. Particular care has been devoted to the state-space approach. This approach is the best suited for the development of formulations in the framework of random vibrations. In fact, its adaptability to numerical method of solution of differential equations and its extension to multi-degree-of-freedom (MDoF) systems are very straightforward.

The equation of motion of a linear oscillator with mass, m; viscous damping, c; stiffness, k; and subjected to the excitation f (t) and at rest at initial time of motion, t = t₀, can be written as

$$ m \ddot{u}(t)+c \dot{u}(t)+k u(t)=f(t); u\left({t}_0\right)=0, \dot{u}\left({t}_0\right)=0 $$

(22)

or in canonical form as follows:

$$ \ddot{u}(t)+2\;{\xi}_0\;{\omega}_0 \dot{u}(t)+{\omega}_0^2 u(t)=F(t); u\left({t}_0\right)=0, \dot{u}\left({t}_0\right)=0 $$

(23)

where u(t) is the displacement response of the mass, ${\omega}_0=\sqrt{k/m}$ is the natural circular frequency, ${\xi}_0=c/2\sqrt{m\;k}$ is the damping ratio, and F(t) = f(t)/m; a dot over a variable denotes differentiation with respect to time t. In state variables the equation of motion of the oscillator, in canonical form, can be written as a set of two first-order differential equations:

$$ \dot{\mathbf{y}}(t)={\mathbf{D}}_0 \mathbf{y}(t)+{\mathbf{v}}_0 F(t); \mathbf{y}\left({t}_0\right)=\mathbf{0} $$

(24)

where

$$ \mathbf{y}(t)=\left[\begin{array}{c}\hfill u(t)\hfill \\ {}\hfill \dot{u}(t)\hfill \end{array}\right], {\mathbf{D}}_0=\left[\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ {}\hfill -{\omega}_0^2\hfill & \hfill -2\;{\xi}_0\;{\omega}_0\hfill \end{array}\right], {\mathbf{v}}_0=\left[\begin{array}{c}\hfill 0\hfill \\ {}\hfill 1\hfill \end{array}\right] $$

(25)

Denoting by y_p(t) the particular solution vector, the solution of Eq. 24 can be evaluated as (Borino and Muscolino 1986)

$$ \mathbf{y}(t)={\mathbf{y}}_{\mathrm{p}}(t)+{\Theta}_0\left(t-{t}_0\right)\left[\mathbf{y}\left({t}_0\right)-{\mathbf{y}}_{\mathrm{p}}\left({t}_0\right)\right] $$

(26)

where Θ₀(t) is the so-called transition matrix:

$$ {\Theta}_0(t)= \exp \left({\mathbf{D}}_0\;t\right)=\left[\begin{array}{cc}\hfill -{\omega}_0^2\;{g}_0(t) \hfill & \hfill {h}_0(t)\hfill \\ {}\hfill -{\omega}_0^2\;{h}_0(t) \hfill & \hfill {\dot{h}}_0(t)\hfill \end{array}\right] $$

(27)

with

$$ \begin{array}{l}{g}_0(t)=-\frac{1}{\omega_0^2} \exp \left(-{\xi}_0 {\omega}_0 t\right)\left[ \cos \left({\overline{\omega}}_0 t\right)+\frac{\xi_0 {\omega}_0}{{\overline{\omega}}_0} \sin \left({\overline{\omega}}_0 t\right)\right];\\ {}{h}_0(t)={\dot{g}}_0(t)=\frac{1}{{\overline{\omega}}_0} \exp \left(-{\xi}_0 {\omega}_0 t\right) \sin \left({\overline{\omega}}_0 t\right);\\ {}{\dot{h}}_0(t)= \exp \left(-{\xi}_0 {\omega}_0 t\right)\left[ \cos \left({\overline{\omega}}_0 t\right)-\frac{\xi_0 {\omega}_0}{{\overline{\omega}}_0} \sin \left({\overline{\omega}}_0 t\right)\right]\end{array} $$

(28)

and ${\overline{\omega}}_0={\omega}_0\sqrt{1-{\xi}_0}$ is the damped natural circular frequency. Notice that the contribution of the last term in the right member of Eq. 26 decreases in the time because the transition matrix satisfies the following condition:

$$ \underset{t\to \infty }{ \lim }{\boldsymbol{\Theta}}_0(t)=\mathbf{0} $$

(29)

Alternatively, by applying the so-called parameter variation method, the solution of Eq. 24 can be written in integral form as follows:

$$ \mathbf{y}(t)={\boldsymbol{\Theta}}_0\left(t-{t}_0\right)\mathbf{y}\left({t}_0\right)+{\displaystyle \underset{t_0}{\overset{t}{\int }}{\boldsymbol{\Theta}}_0\left(t-\tau \right)} \mathbf{v} F\left(\tau \right)\mathrm{d}\tau $$

(30)

For quiescent systems, because the relationship y(t₀) = 0 is satisfied, Eqs. 26 and 30 become

$$\begin{array}{ll}\mathbf{y}(t)&={\mathbf{y}}_{\mathrm{p}}(t)-{\boldsymbol{\Theta}}_0\left(t-{t}_0\right){\mathbf{y}}_{\mathrm{p}}\left({t}_0\right)\\&={\displaystyle\underset{t_0}{\overset{t}{\int}}{\boldsymbol{\Theta}}_0\left(t-\tau \right)} \mathbf{v}F\left(\tau \right)\mathrm{d}\tau \end{array}$$

(31)

Notice that in this case the first element of vector y(t), written in integral form, coincides with the well-known Duhamel integral. Starting by the integral form solution in state variables, it is possible to derive a very powerful unconditionally stable numerical procedure for the evaluation of the structural response (see, e.g., Borino and Muscolino 1986).

Stochastic Response

Mathematically strictly speaking, as a consequence of the introduction of the one-sided PSD, the input process is a complex one. It follows that the response processes, u(t), is a complex function too. After some algebra, for the quiescent oscillator (Eq. 23), the NGSMs can be evaluated, in time domain, as (Di Paola 1985; Di Paola and Petrucci 1990, Muscolino1991)

$$ \begin{array}{l}{\lambda}_{0,uu}(t)\equiv \mathrm{E}\left\langle u(t)\;{u}^{*}(t)\right\rangle ={\displaystyle \underset{0}{\overset{t}{\int }}{\displaystyle \underset{0}{\overset{t}{\int }}h\left(t-{\tau}_1\right)h\left(t-{\tau}_2\right){\overline{R}}_{FF}\left({\tau}_1,{\tau}_2\right)\;\mathrm{d}{\tau}_1\;\mathrm{d}{\tau}_2}};\\ {}{\lambda}_{1,uu}(t)=-\mathrm{i}\;{\displaystyle \underset{0}{\overset{t}{\int }}{\displaystyle \underset{0}{\overset{t}{\int }}h\left(t-{\tau}_1\right)\dot{h}\left(t-{\tau}_2\right){\overline{R}}_{FF}\left({\tau}_1,{\tau}_2\right)\;\mathrm{d}{\tau}_1\;\mathrm{d}{\tau}_2}};\\ {}{\lambda}_{2,uu}(t)\equiv \mathrm{E}\left\langle \dot{u}(t)\;{\dot{u}}^{*}(t)\right\rangle ={\displaystyle \underset{0}{\overset{t}{\int }}{\displaystyle \underset{0}{\overset{t}{\int }}\dot{h}\left(t-{\tau}_1\right)\dot{h}\left(t-{\tau}_2\right){\overline{R}}_{FF}\left({\tau}_1,{\tau}_2\right)\;\mathrm{d}{\tau}_1\;\mathrm{d}{\tau}_2}}\end{array} $$

(32)

where ${\overline{R}}_{FF}\left({t}_1,{t}_2\right)$ is the complex autocorrelation function defined in Eq. 12. Moreover, the presence of the imaginary unit in the second of Eq. 32 inverts the roles of the real and imaginary parts of λ_1,uu(t) with respect to the variances λ_0,uu(t) and λ_2,uuℓ(t); furthermore, while λ_0,uu(t) and λ_2,uu(t) are real functions, λ_1,uu(t) is a complex one. Substituting Eq. 12 into Eq. 32, the following relationships are obtained:

$$ \begin{array}{l}{\lambda}_{0,uu}(t)={\displaystyle \underset{0}{\overset{\infty }{\int }}{Z}_0^{\ast}\left(\omega,,,, t\right){Z}_0\left(\omega,,,,t\right){G}_{NN}\left(\omega \right)\mathrm{d}\omega };\\ {}{\lambda}_{1,uu}(t)=-\mathrm{i}{\displaystyle \underset{0}{\overset{\infty }{\int }}{Z}_0^{\ast}\left(\omega,,,, t\right){\dot{Z}}_0\left(\omega,,,, t\right){G}_{NN}\left(\omega \right)\mathrm{d}\omega };\\ {}{\lambda}_{2,uu}(t)={\displaystyle \underset{0}{\overset{\infty }{\int }}{\dot{Z}}_0^{\ast}\left(\omega,,,, t\right){\dot{Z}}_0\left(\omega,,,, t\right){G}_{NN}\left(\omega \right)\mathrm{d}\omega}\end{array} $$

(33)

where

$$ \begin{array}{l}{Z}_0\left(\omega, t\right)={\displaystyle \underset{0}{\overset{t}{\int }}h\left(t-\tau \right) \exp \left(\mathrm{i}\omega \tau \right)a\left(\omega, \tau \right)\;\mathrm{d}\tau };\\ {}{\dot{Z}}_0\left(\omega, t\right)={\displaystyle \underset{0}{\overset{t}{\int }}\dot{h}\left(t-\tau \right) \exp \left(\mathrm{i}\omega \tau \right)a\left(\omega, \tau \right)\;\mathrm{d}\tau}\end{array} $$

(34)

Notice that the function Z₀(ω, t) is the so-called evolutionary frequency response function of the oscillator (Li and Chen 2009). Remarkably, since the integrals (Eq. 34) are convolution integrals of Duhamel’s type, they can be interpreted as the response, in terms of state variables, of the quiescent oscillator, at time t = 0, subjected to the deterministic complex function f(ω, t) = exp(iωt) a(ω, t). By introducing the state variables, the evolutionary frequency response vector function can be defined as

$$ {\mathbf{Y}}_0\left(\omega,,,, t\right)=\left[\begin{array}{c}\hfill {Z}_0\left(\omega,,,, t\right)\hfill \\ {}\hfill {\dot{Z}}_0\left(\omega,,,, t\right)\hfill \end{array}\right] $$

(35)

It follows that relationships (33) can be rewritten in compact form as follows:

$$ {\Sigma}_{uu}(t) =\left[\begin{array}{cc}\hfill {\lambda}_{ 0,uu}(t)\hfill & \hfill \mathrm{i}{\lambda}_{1,uu}(t)\hfill \\ {}\hfill -\mathrm{i}{\lambda}_{1,uu}^{*}(t)\hfill & \hfill {\lambda}_{2,uu}(t)\hfill \end{array}\right] \equiv {\displaystyle \underset{0}{\overset{\infty }{\int }}{G}_{NN}\left(\omega \right) {\mathbf{Y}}_0^{\ast}\left(\omega,,,, t\right){\mathbf{Y}}_0^T\left(\omega,,,,t\right)\mathrm{d}\omega } $$

(36)

This matrix coincides with the so-called pre-envelope covariance matrix introduced by Di Paola and Petrucci (1990) which is a complex matrix. As a conclusion, for input processes characterized by one-sided EPSD function, the cross-covariance matrix (Eq. 36) of an oscillator is a complex matrix whose elements are the NGSMs.

Generalizing Eq. 32, it can be easily proved that the complex cross-correlation function matrix of the zero-mean response process, which collects the cross-correlation of the pre-envelope response processes, can be evaluated as

$$ {\mathbf{R}}_{uu}\left({t}_1,{t}_2\right)={\displaystyle \underset{0}{\overset{\infty }{\int }}{G}_{NN}\left(\omega \right) {\mathbf{Y}}_0^{\ast}\left(\omega, {t}_1\right){\mathbf{Y}}_0^T\left(\omega, {t}_2\right)\mathrm{d}\omega } $$

(37)

Finally, according to the Priestley evolutionary process model (Priestley 1999), this complex function matrix R _uu (t₁, t₂) can be also rewritten as

$$ {\mathbf{R}}_{uu}\left({t}_1,{t}_2\right)={\displaystyle \underset{0}{\overset{\infty }{\int }} \exp \left[\mathrm{i}\omega \left({t}_1-{t}_2\right)\right]{\mathbf{G}}_{uu}\left(\omega, {t}_1,{t}_2\right)\mathrm{d}\omega } $$

(38)

where

$$ {\mathbf{G}}_{uu}\left(\omega, {t}_1,{t}_2\right)={G}_{NN}\left(\omega \right) {\mathbf{Y}}_0^{\ast}\left(\omega, {t}_1\right){\mathbf{Y}}_0^T\left(\omega, {t}_2\right) $$

(39)

Consequently the NGSMs of the oscillator response are the elements of the pre-envelope covariance matrix, given as

$$ {\Sigma}_{uu}(t)\equiv {\mathbf{R}}_{uu}\left(t,t\right)\equiv {\displaystyle \underset{0}{\overset{\infty }{\int }}{\mathbf{G}}_{uu}\left(\omega, t\right) \mathrm{d}\omega } $$

(40)

Let us assume now the modulating function $a\left(\omega, t\right)=\mathit{\mathsf{U}}(t)$. Starting from the nonstationary formulation, it is possible to deduce the formulation in the case of stationary input. This result is obtained by performing the limit as t → ∞ into Eq. 34:

$$ \begin{array}{l}\underset{t\to \infty }{ \lim}\;{Z}_0\left(\omega, t\right)={\displaystyle \underset{0}{\overset{\infty }{\int }}{h}_0\left(t-\tau \right) \exp \left(\mathrm{i}\omega \tau \right)\mathrm{d}\tau }= \exp \left(\mathrm{i}\omega t\right) {H}_0\left(\omega \right);\\ {}\underset{t\to \infty }{ \lim }{\dot{Z}}_0\left(\omega, t\right)={\displaystyle \underset{0}{\overset{\infty }{\int }}{\dot{h}}_0\left(t-\tau \right) \exp \left(\mathrm{i}\omega \tau \right)\mathrm{d}\tau =\mathrm{i}\omega \exp \left(\mathrm{i}\omega t\right) {H}_0\left(\omega \right)}\end{array} $$

(41)

where h₀(t) and ${\dot{h}}_0(t)$ are the functions defined in Eq. 28 and H₀(ω) is the frequency response function of the oscillator defined as

$$ {H}_0\left(\omega \right)={\displaystyle \underset{0}{\overset{\infty }{\int }}{h}_0\left(\tau \right) \exp \left(-\mathrm{i}\omega \tau \right)\mathrm{d}\tau }=\frac{1}{\omega_0^2-{\omega}^2+\mathrm{i}\;2{\xi}_0{\omega}_0\omega } $$

(42)

By substituting Eq. 41 into Eq. 35 and the results into Eq. 36, the following relationship is obtained:

$$ {\Sigma}_{uu}={\displaystyle \underset{0}{\overset{\infty }{\int }}{G}_{NN}\left(\omega \right) {\mathbf{H}}_0^{*}\left(\omega \right){\mathbf{H}}_0^T\left(\omega \right)\;\mathrm{d}\omega} \equiv \left[\begin{array}{cc}\hfill {\lambda}_{0,uu}\hfill & \hfill \mathrm{i}{\lambda}_{1,uu}\hfill \\ {}\hfill -\mathrm{i}{\lambda}_{1,uu}^{*}\hfill & \hfill {\lambda}_{2,uu}\hfill \end{array}\right] $$

(43)

where

$$ {\mathbf{H}}_0\left(\omega \right)=\left[\begin{array}{c}\hfill {H}_0\left(\omega \right)\hfill \\ {}\hfill \mathrm{i}\omega {H}_0\left(\omega \right)\hfill \end{array}\right] $$

(44)

The elements of this matrix Σ _uu are the GSMs, introduced by Vanmarcke (1972), that is

$$ {\lambda}_{i,uu}\equiv {\lambda}_{ i,uu}^G={\displaystyle \underset{0}{\overset{\infty }{\int }}{\omega}^i\;{\left|{H}_0\left(\omega \right)\right|}^2\;{G}_{NN}\left(\omega \right)\mathrm{d}\omega }; i=0,1,2 $$

(45)

Response of Multi-degree-of-Freedom (MDoF) Systems

Fundamental of Deterministic Analysis

Let us consider the equation of motion of a linear quiescent n-degree-of-freedom (n-DoF) classically damped structural system whose dynamic behavior is governed by the equation of motion:

$$ \mathbf{M}\;\ddot{\mathbf{u}}(t)+\mathbf{C}\dot{\mathbf{u}}(t)+\mathbf{K}\mathbf{u}(t)=\mathbf{f}(t) $$

(46)

where M, C, and K are the (n × n) mass, damping, and stiffness matrices of the structure; u(t) is the (n × 1) vector of displacements, having for ith element u _i (t); and f(t) denotes the external load vector. Under the assumption of classically damped system, the equation of motion can be decoupled by applying the modal analysis. To this aim let us introduce the modal coordinate transformation:

$$ \mathbf{u}(t)=\boldsymbol{\Phi}\;\mathbf{q}(t)={\displaystyle \sum_{j=1}^m{\boldsymbol{\upphi}}_j {q}_j(t)}\ \Rightarrow\ {u}_i(t)={\displaystyle \sum_{j=1}^m{\phi}_{i j} {q}_j(t)} $$

(47)

In this equation, $\boldsymbol{\Phi} =\left[\begin{array}{cccc}\hfill {\boldsymbol{\upphi}}_{\;1}\;\hfill & \hfill {\boldsymbol{\upphi}}_{\;2}\;\hfill & \hfill \dots \hfill & \hfill {\boldsymbol{\upphi}}_{\;m}\;\hfill \end{array}\right]$ is the modal matrix, of order n × m, collecting the m eigenvectors ϕ _j , normalized with respect to the mass matrix M, solutions of the following eigenproblem:

$$ {\mathbf{K}}^{-1}\mathbf{M}\boldsymbol{\Phi } =\boldsymbol{\Phi} {\boldsymbol{\Omega}}^{-2}; {\boldsymbol{\Phi}}^T\mathbf{M}\boldsymbol{\Phi } ={\mathbf{I}}_m $$

(48)

In this equation Ω is a diagonal matrix listing the undamped natural circular frequency ω _j , I _m is the identity matrix of order m, and the apex T means transpose operator. Once the modal matrix Φ is evaluated, by applying the coordinate transformations (47) to Eq. 46, the following set of decoupled second-order differential equations is obtained:

$$ \ddot{\mathbf{q}}(t)+\boldsymbol{\Xi}\;\dot{\mathbf{q}}(t)+{\boldsymbol{\Omega}}^2\mathbf{q}(t)={\boldsymbol{\Phi}}^T\mathbf{f}\;(t) $$

(49)

in which Ξ is a generalized damping matrix given by

$$ \boldsymbol{\Xi} ={\boldsymbol{\Phi}}^T\mathbf{C}\boldsymbol{\Phi } $$

(50)

For classically damped structures the modal damping matrix Ξ is a diagonal matrix listing the quantities 2ξ _j ω _j , ξ _j being the modal damping ratio. It follows that the jth differential Eq. 49 can be written as

$$ {\ddot{q}}_j(t)+2{\xi}_j{\omega}_j {\dot{q}}_j(t)+{\omega}_j^2{q}_j(t)={\boldsymbol{\upphi}}_j^T\mathbf{f}\;(t) $$

(51)

In the state space, Eq. 51 can be written in the first-order form as

$$ {\dot{\mathbf{y}}}_j(t)={\mathbf{D}}_j\;{\mathbf{y}}_j(t)+{\mathbf{V}}_j\;\mathbf{f}(t) $$

(52)

where

$$ {\mathbf{y}}_j(t)=\left[\begin{array}{c}\hfill {q}_j(t)\hfill \\ {}\hfill {\dot{q}}_j(t)\hfill \end{array}\right]; {\mathbf{D}}_j=\left[\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ {}\hfill -{\omega}_j^2\hfill & \hfill -2{\xi}_j{\omega}_j\hfill \end{array}\right]; {\mathbf{V}}_j=\left[\begin{array}{c}\hfill \mathbf{0}\hfill \\ {}\hfill {\boldsymbol{\upphi}}_j^T\hfill \end{array}\right] $$

(53)

Stochastic Response for Mono-correlated Stochastic Input Processes

Let us assume now that the forcing term is a mono-correlated zero-mean Gaussian random process vector given by the relationship:

$$ \mathbf{f}(t)=\mathbf{b} \overline{F}(t) $$

(54)

where b is the (n × 1) vector of spatial distribution of loads and $\overline{F}(t)$ is a zero-mean Gaussian nonstationary random process. It follows that the jth differential Eq. 51 can be written as

$$ {\ddot{q}}_j(t)+2{\xi}_j{\omega}_j {\dot{q}}_j(t)+{\omega}_j^2{q}_j(t)={p}_j \overline{F}(t), j = 1,2,\dots, m $$

(55)

where

$$ {p}_j={\boldsymbol{\upphi}}_j^T\;\mathbf{b} $$

(56)

After very simple algebra it can be shown that the jth NGSMs, ${\lambda}_{\;j,{u}_i{u}_i}(t)$ (i = 0, 1, 2), of the ith nodal displacement response, u _i (t), are given as a function of modal NGSMs, p _j  λ_j,kℓ(t) (ℓ, k = 1 …, m), by the following relationships:

$$ \begin{array}{l}{\lambda}_{\;0,{u}_i {u}_i}(t)={\displaystyle \sum_{k=1}^m{\displaystyle \sum_{\ell =1}^m\;{p}_k\;{p}_{\ell}\;{\phi}_{ik}\;{\phi}_{i\ell}\;{\lambda}_{0,k\ell }(t)}};\;\\ {}{\lambda}_{1,{u}_{\;i} {u}_{ i}}(t)={\displaystyle \sum_{k=1}^m{\displaystyle \sum_{\ell =1}^m\;{p}_k\;{p}_{\ell}\;{\phi}_{ik}\;{\phi}_{i\ell}\;{\lambda}_{1,k\ell }(t)}};\\ {}{\lambda}_{\;2,{u}_{\;i} {u}_{ i}}(t)={\displaystyle \sum_{k=1}^m{\displaystyle \sum_{\ell =1}^m\;{p}_k\;{p}_{\ell}\;{\phi}_{ik}\;{\phi}_{i\ell}\;{\lambda}_{2,k\ell }(t)}}\end{array} $$

(57)

It has to be emphasized that the zeroth NGSM, ${\lambda}_{\;0,{u}_i{u}_i}(t)$, and the second-order NGSM, ${\lambda}_{\;2,{u}_i{u}_i}(t)$, are real functions that coincide with the variances of the response in terms of displacement and velocity, respectively, while the first-order NGSM, ${\lambda}_{1,{u}_i{u}_i}(t)$, is a complex quantity whose real part can be evaluated as the cross-covariance between the response process and the response velocity process of the same linear system subjected to a nonstationary input whose stationary counterpart is proportional to its Hilbert transform (Di Paola 1985; Langley 1986; Di Paola and Petrucci 1990; Muscolino 1991). As shown in Eq. 57 the nodal NGSMs can be evaluated as a function of λ_r,kℓ(t), r = 0, 1, 2, which are the so-called time-dependent modal NGSMs “purged” by p _j factors. After some algebra, these quantities, which are complex ones, can be evaluated, in time domain, for quiescent structural systems as (Di Paola 1985; Di Paola and Petrucci 1990; Muscolino 1991)

$$ \begin{array}{l}{\lambda}_{0,k\ell }(t)={\displaystyle \underset{0}{\overset{t}{\int }}{\displaystyle \underset{0}{\overset{t}{\int }}{h}_k\left(t-{\tau}_1\right){h}_{\ell}\left(t-{\tau}_2\right){\overline{R}}_{FF}\left({\tau}_1,{\tau}_2\right)\mathrm{d}{\tau}_1\mathrm{d}{\tau}_2}};\\ {}{\lambda}_{1,k\ell }(t)=-\mathrm{i}\;{\displaystyle \underset{0}{\overset{t}{\int }}{\displaystyle \underset{0}{\overset{t}{\int }}{h}_k\left(t-{\tau}_1\right){\dot{h}}_{\ell}\left(t-{\tau}_2\right){\overline{R}}_{FF}\left({\tau}_1,{\tau}_2\right)\mathrm{d}{\tau}_1\mathrm{d}{\tau}_2}}; k=1,2,\dots, m; \ell =1,2,\dots, m;\\ {}{\lambda}_{2,k\ell }(t)={\displaystyle \underset{0}{\overset{t}{\int }}{\displaystyle \underset{0}{\overset{t}{\int }}{\dot{h}}_k\left(t-{\tau}_1\right){\dot{h}}_{\ell}\left(t-{\tau}_2\right){\overline{R}}_{FF}\left({\tau}_1,{\tau}_2\right)\mathrm{d}{\tau}_1\mathrm{d}{\tau}_2}}\end{array} $$

(58)

where ${\overline{R}}_{FF}\left({t}_1,{t}_2\right)$ is the complex autocorrelation function defined in Eq. 12. Notice that the presence of the imaginary unit in the second term of the second of Eq. 58 inverts the roles of the real and imaginary parts of λ_1,kℓ(t) with respect to the cross-covariance λ_0,kℓ(t) and λ_2,kℓ(t); furthermore for k = ℓ while λ_0,kℓ(t) and λ_2,kℓ(t) become real quantities, λ_1,kℓ(t) remains a complex one.

It can be easily proved that the “purged” NGSMs, given in Eq. 58, can be evaluated as a function of the statistics of the response of a dummy oscillator whose motion is governed by

$$ {\ddot{z}}_j(t)+2{\xi}_j{\omega}_j {\dot{z}}_j(t)+{\omega}_j^2{z}_j(t)=\overline{F}(t) $$

(59)

where $\overline{F}(t)$ is a complex process whose imaginary part is a process having stationary counterpart proportional to Hilbert transform of the real part of the stationary counterpart of the complex process itself; it follows that z _j (t) = q _j (t)/p _j is a complex process too. Notice that, with the position p _j  = 1, Eq. 59 coincides with Eq. 55. Then by substituting the complex function ${\overline{R}}_{FF}\left({t}_1,{t}_2\right)$, defined in Eq. 12, into Eq. 58, after very simple algebra, it is possible to evaluate the “purged” NGSMs as

$$ \begin{array}{l}{\lambda}_{0,k\ell }(t)={\displaystyle \underset{0}{\overset{\infty }{\int }}{Z}_k^{*}\left(\omega, t\right){Z}_{\ell}\left(\omega, t\right){G}_{NN}\left(\omega \right)\mathrm{d}\omega };\\ {}{\lambda}_{1,k\ell }(t)=-\mathrm{i}{\displaystyle \underset{0}{\overset{\infty }{\int }}{Z}_k^{*}\left(\omega, t\right){\dot{Z}}_{\ell}\left(\omega, t\right){G}_{NN}\left(\omega \right)\mathrm{d}\omega };\\ {}{\lambda}_{2,k\ell }(t)={\displaystyle \underset{0}{\overset{\infty }{\int }}{\dot{Z}}_k^{*}\left(\omega, t\right){\dot{Z}}_{\ell}\left(\omega, t\right){G}_{NN}\left(\omega \right)\mathrm{d}\omega}\end{array} $$

(60)

where, for j = k, ℓ, the following positions have been made:

$$ \begin{array}{l}{Z}_j\left(\omega,,, t\right)={\displaystyle \underset{0}{\overset{t}{\int }}{h}_j\left(t-\tau \right) \exp \left(\mathrm{i}\omega \tau \right)a\left(\omega, \tau \right)\mathrm{d}\tau };\\ {}{\dot{Z}}_j\left(\omega,,, t\right)={\displaystyle \underset{0}{\overset{t}{\int }}{\dot{h}}_j\left(t-\tau \right) \exp \left(\mathrm{i}\omega \tau \right)a\left(\omega, \tau \right)\mathrm{d}\tau}\end{array} $$

(61)

with

$$ {h}_j(t)=\frac{1}{{\overline{\omega}}_j} \exp \left(-{\xi}_j{\omega}_jt\right) \sin \left({\overline{\omega}}_j\;t\right) $$

(62)

and ${\overline{\omega}}_j={\omega}_j\sqrt{1-{\xi}_j^2}$ (j = k, ℓ) is the damped circular frequency of the jth dummy oscillator. Moreover, introducing the state variable, the modal evolutionary frequency response function vector is defined as

$$ {\mathbf{Y}}_j\left(\omega,,, t\right)=\left[\begin{array}{c}\hfill {Z}_j\left(\omega, t\right)\hfill \\ {}\hfill {\dot{Z}}_j\left(\omega, t\right)\hfill \end{array}\right]; j=k,\ell $$

(63)

Then, relationships (33) can be rewritten in compact form as follows:

$$ {\Sigma}_{k\ell }(t) =\left[\begin{array}{cc}\hfill {\lambda}_{ 0,k\ell }(t)\hfill & \hfill \mathrm{i}{\lambda}_{1,k\ell }(t)\hfill \\ {}\hfill -\mathrm{i}{\lambda}_{1,k\ell}^{*}(t)\hfill & \hfill {\lambda}_{2,k\ell }(t)\hfill \end{array}\right] \equiv {\displaystyle \underset{0}{\overset{\infty }{\int }}{G}_{NN}\left(\omega \right) {\mathbf{Y}}_k^{*}\left(\omega, t\right){\mathbf{Y}}_{\ell}^T\left(\omega, t\right)\mathrm{d}\omega } $$

(64)

which coincides with the cross-modal pre-envelope covariance matrix which is a complex matrix (Di Paola and Petrucci 1990). As a conclusion for input processes characterized by one-sided PSD function, the cross-covariance matrix (Eq. 64) between the k, ℓth dummy oscillators is a complex matrix whose elements are the “purged” NGSMs. It can be easily proved that the complex cross-correlation function matrix of the zero-mean modal “purged” response process, which collects the cross-correlation of the pre-envelope response processes, according to the Priestley evolutionary process model (Priestley 1999), can be evaluated as

$$ {\mathbf{R}}_{k\ell}\left({t}_1,{t}_2\right)={\displaystyle \underset{0}{\overset{\infty }{\int }}{G}_{NN}\left(\omega \right) {\mathbf{Y}}_k^{*}\left(\omega, {t}_1\right){\mathbf{Y}}_{\ell}^T\left(\omega, {t}_2\right)\mathrm{d}\omega}\equiv {\displaystyle \underset{0}{\overset{\infty }{\int }} \exp \left[\mathrm{i}\omega \left({t}_1-{t}_2\right)\right]{\mathbf{G}}_{k\ell}\left(\omega, {t}_1,{t}_2\right)\mathrm{d}\omega } $$

(65)

where

$$ {\mathbf{G}}_{k\ell}\left(\omega, {t}_1,{t}_2\right)={G}_{NN}\left(\omega \right) {\mathbf{Y}}_k^{*}\left(\omega, {t}_1\right){\mathbf{Y}}_{\ell}^T\left(\omega, {t}_2\right) $$

(66)

is the cross-modal EPSD function matrix between the k, ℓth dummy oscillators. By means of the modal transformation (Eq. 47), the nodal autocorrelation and the EPSD function matrices of the displacement response, u _i (t), can be evaluated, after very simple algebra, respectively, as follows:

$$ \begin{array}{l}{\mathbf{R}}_{u_{\;i} {u}_{ i}}\left({t}_1,{t}_2\right)={\displaystyle \sum_{k=1}^m{\displaystyle \sum_{\ell =1}^m\;{p}_k\;{p}_{\ell}\;{\phi}_{ik}\;{\phi}_{i\ell}\;{\mathbf{R}}_{k\ell}\left({t}_1,{t}_2\right) }},\\ {}{\mathbf{G}}_{u_{\;i} {u}_{ i}}\left(\omega, {t}_1,{t}_2\right)={\displaystyle \sum_{k=1}^m{\displaystyle \sum_{\ell =1}^m\;{p}_k\;{p}_{\ell}\;{\phi}_{ik}\;{\phi}_{i\ell}\;{\mathbf{G}}_{k\ell}\left(\omega, {t}_1,{t}_2\right) }}\end{array} $$

(67)

Consequently the nodal displacement NGSMs are the elements of the nodal pre-envelope covariance matrix, given as

$$ {\Sigma}_{u_{\;i} {u}_{ i}}(t)\equiv {\mathbf{R}}_{u_{\;i} {u}_{ i}}\left(t,t\right)\equiv {\displaystyle \underset{0}{\overset{\infty }{\int }}{\mathbf{G}}_{u_{\;i} {u}_{ i}}\left(\omega, t\right) \mathrm{d}\omega }={\displaystyle \sum_{k=1}^m{\displaystyle \sum_{\ell =1}^m{p}_k\;{p}_{\ell}\;{\phi}_{ik}\;{\phi}_{i\ell}\;{\Sigma}_{k\ell }(t)}} $$

(68)

Let us assume now the modulating function $a\left(\omega, t\right)=\mathit{\mathsf{U}}(t)$. Starting from the nonstationary formulation, it is possible to deduce the formulation in the case of stationary input. This result is obtained by performing the limit as t → ∞ into Eq. 61:

$$ \begin{array}{l}\underset{t\to \infty }{ \lim}\;{Z}_j\left(\omega, t\right)={\displaystyle \underset{0}{\overset{\infty }{\int }}{h}_j\left(t-\tau \right) \exp \left(\mathrm{i}\omega \tau \right)\mathrm{d}\tau} = \exp \left(\mathrm{i}\;\omega\;t\right) {H}_j\left(\omega \right);\\ {}\underset{t\to \infty }{ \lim }{\dot{Z}}_j\left(\omega, t\right)={\displaystyle \underset{0}{\overset{\infty }{\int }}{\dot{h}}_j\left(t-\tau \right) \exp \left(\mathrm{i}\omega \tau \right)\mathrm{d}\tau =\mathrm{i}\;\omega \exp \left(\mathrm{i}\;\omega\;t\right) {H}_j\left(\omega \right)}\end{array} $$

(69)

where h _j (t) is the functions defined in Eq. 62 and H _j (ω) is the frequency response function of the jth modal oscillator:

$$ {H}_j\left(\omega \right)={\displaystyle \underset{0}{\overset{\infty }{\int }}{h}_j\left(\tau \right) \exp \left(-\mathrm{i}\omega \tau \right)\mathrm{d}\tau }=\frac{1}{\omega_j^2-{\omega}^2+\mathrm{i}\;2{\xi}_j{\omega}_j\omega } $$

(70)

By substituting Eq. 69 into Eq. 63 and the results into Eq. 64, the following relationship is obtained:

$$ {\Sigma}_{k\ell }={\displaystyle \underset{0}{\overset{\infty }{\int }}{G}_{NN}\left(\omega \right) {\mathbf{H}}_k^{*}\left(\omega \right){\mathbf{H}}_{\ell}^T\left(\omega \right)\;\mathrm{d}\omega}\equiv \left[\begin{array}{cc}\hfill {\lambda}_{ 0,k\ell}\hfill & \hfill \mathrm{i}{\lambda}_{1,k\ell}\hfill \\ {}\hfill -\mathrm{i}{\lambda}_{1,k\ell}^{*}\hfill & \hfill {\lambda}_{2,k\ell}\hfill \end{array}\right] $$

(71)

where

$$ {\mathbf{H}}_j\left(\omega \right)=\left[\begin{array}{c}\hfill {H}_j\left(\omega \right)\hfill \\ {}\hfill \mathrm{i}\omega {H}_j\left(\omega \right)\hfill \end{array}\right] j=k,\ell $$

(72)

The elements of the matrix Σ _kℓ are the cross GSMs, introduced by Vanmarcke (1972), that is,

$$ {\lambda}_{ i,k\ell}\equiv {\lambda}_{ i,k\ell}^G={\displaystyle \underset{0}{\overset{\infty }{\int }}{\omega}^i {H}_k^{\ast}\left(\omega \right)\;{H}_{\ell}\left(\omega \right)\;{G}_{NN}\left(\omega \right)\mathrm{d}\omega }; i=0,1,2 $$

(73)

Stochastic Response for Multi-correlated Stochastic Input Processes

In earthquake engineering it is common to assume that the entire base of a structure is subjected to a uniform ground motion. This hypothesis inherently implies that the ground motion is a result of spatially uniform motion. Thus it is certainly true when the base dimensions of the structure are small relative to the seismic vibration wavelengths. From an analytical point of view, this hypothesis leads to mono-correlated zero-mean Gaussian models of the ground-motion acceleration. It follows that this model of earthquake excitations is undoubtedly advantageous, because it substantially facilitates the stochastic analysis of dynamic problem. Indeed, in this case the stochastic analysis requires the knowledge of only one PSD of the input process.

The assumption of uniform ground motion is inappropriate for structures, such as long span bridges, which require accounting for spatial variability of the support motions. The main sources of ground-motion spatial variability are the loss of coherency of seismic waves with distance, the difference in the arrival times of waves at separate supports, and the difference in soil conditions underneath the supports. In these cases the multi-correlated model of the seismic excitation is more appropriate.

In order to take into account of the spatial variability of earthquake-induced ground motions, let us consider an n-DoF structural system subjected to an N support motion. It follows that the stochastic forcing vector process has to be modeled as a multi-correlated zero-mean Gaussian random vector process:

$$ \mathbf{f}(t)=\mathbf{B}\;\overline{\mathbf{F}}(t) $$

(74)

where B is the (n × N) matrix of spatial distribution of loads and $\overline{\mathbf{F}}(t)$ is a zero-mean Gaussian nonstationary multi-correlated random vector process of order N. Following the Priestley spectral representation of nonstationary processes (Priestley 1999), this vector can be defined by the Fourier-Stieltjes integral as follows (Di Paola and Petrucci 1990):

$$ \overline{\mathbf{F}}(t)=\sqrt{2}{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }} \exp \left(i \omega t\right)\;\mathbf{A}\left(\omega, t\right)}\;\mathrm{d} \mathbf{N}\left(\omega \right) $$

(75)

where A(ω, t) is a slowly varying complex deterministic time-frequency modulating diagonal function matrix, of order N × N, which has to satisfy the condition A(ω, t) ≡ A^∗(−ω, t), while N(ω) is an orthogonal vector process satisfying the following conditions:

$$ \mathrm{E}\left\langle \mathrm{d} \mathbf{N}\left({\omega}_1\right)\;\mathrm{d} {\mathbf{N}}^{*T}\left({\omega}_2\right)\right\rangle =\delta \left({\omega}_1-{\omega}_2\right)\;{\mathbf{G}}_{\mathbf{N}\mathbf{N}}\left({\omega}_1\right)\mathrm{d} {\omega}_1\mathrm{d} {\omega}_2 $$

(76)

This relationship shows that the stationary counterpart of the multi-correlated stochastic process is a vector process, N(ω) (of order N), with orthogonal increments. Furthermore, G _NN (ω) is an (N × N) Hermitian matrix function which describes the one-sided PSD function matrix of the so-called “embedded” stationary counterpart vector process, N(ω). After some algebra it can be proved that the autocorrelation function matrix of the zero-mean Gaussian nonstationary random vector process $\overline{\mathbf{F}}(t)$ can be obtained as

$$ \begin{array}{l}{\overline{\mathbf{R}}}_{\mathbf{F}\mathbf{F}}\left({t}_1,{t}_2\right)=\mathrm{E}\left\langle \overline{\mathbf{F}}\left({t}_1\right){\overline{\mathbf{F}}}^{*T}\left({t}_2\right)\right\rangle ={\displaystyle \underset{0}{\overset{\infty }{\int }} \exp \left[\mathrm{i}\omega \left({t}_1-{t}_2\right)\right]\mathbf{A}\left(\omega, {t}_1\right)\;{\mathbf{G}}_{\mathbf{NN}}\left(\omega \right)\;{\mathbf{A}}^{*}\left(\omega, {t}_2\right)\mathrm{d}\omega}\\ {} ={\displaystyle \underset{-\infty }{\overset{\infty }{\int }} \exp \left[\mathrm{i}\omega \left({t}_1-{t}_2\right)\right]{\mathbf{G}}_{\mathbf{F}\mathbf{F}}\left(\omega, {t}_1,{t}_2\right)\mathrm{d}\omega }\ \end{array} $$

(77)

where

$$ {\mathbf{G}}_{\mathbf{FF}}\left(\omega, {t}_1,{t}_2\right)=\mathbf{A}\left(\omega, {t}_1\right)\;{\mathbf{G}}_{\mathbf{NN}}\left(\omega \right)\;{\mathbf{A}}^{*}\left(\omega, {t}_2\right) $$

(78)

According to the Priestley evolutionary process model (Priestley 1999), the EPSD function matrix of the nonstationary multi-correlated process $\overline{\mathbf{F}}(t)$ is given as

$$ {\mathbf{G}}_{\mathbf{FF}}\left(\omega,,,, t\right)=\mathbf{A}\left(\omega,,,, t\right)\;{\mathbf{G}}_{\mathbf{NN}}\left(\omega \right)\;{\mathbf{A}}^{*}\left(\omega,,,,, t\right) $$

(79)

In the framework of stochastic seismic analysis, the main distinct phenomena that give rise to the spatial variability of earthquake-induced ground motions are (i) the incoherence effect associated to loss of coherency of seismic waves due to the differential superpositioning of waves arriving from an extended source, (ii) the wave-passage effect due to difference in the arrival times of waves at separate supports, (iii) the attenuation effect due to gradual decay of wave amplitudes with distance due to energy dissipation in the ground medium, and (iv) the site-response effect associated to spatially varying of the local soil profiles. These effects are taken into account by the so-called coherency function, γ_NN,s r(ω), defined as

$$ {\upgamma}_{\mathbf{NN},s\;r}\left(\omega \right)=\frac{G_{\mathbf{NN},s\;r}\left(\omega \right)}{\sqrt{G_{\mathbf{NN},r\;r}\left(\omega \right) {G}_{\mathbf{NN},s\;s}\left(\omega \right)}} $$

(80)

where G_NN,s r(ω) is the sth and rth element of the matrix G _NN (ω) that is the one-sided PSD at the sth and the rth supports. The coherency function is usually written as

$$ {\upgamma}_{\mathbf{NN},s\;r}\left(\omega \right)={\rho}_{s\;r}\left(\omega \right) \exp \left(-\mathrm{i}\;\omega {d}_{s\;r}/c\right) $$

(81)

in which exp(−i ω d _s r /c) is a measure of the wave-passage delay due to the apparent velocity of waves, c represents the velocity of the seismic waves through the ground (c decreases as the distance between the support points increases or the soil is softer), d _sr is the distance between the rth and sth support points, and ρ _sr (ω) is the real frequency-dependent spatial correlation function between the two support points. Several models of the coherency function have been proposed in literature (see, e.g., Harichandran and Vanmarcke 1986; Zerva 1991). Moreover, the one-sided cross-PSD of ground accelerations in a particular direction between surface points r and s, G_NN,sr(ω), may be written as the product of the coherence function by the target one-sided PSD of ground acceleration G₀(ω) as follows:

$$ {G}_{\mathbf{NN},s\;r}\left(\omega \right)=\left\{\begin{array}{c}\hfill {\upgamma}_{\mathbf{NN},s\;r}\left(\omega \right)\;{G}_0\left(\omega \right), \mathrm{f}\mathrm{o}\mathrm{r} r\ne s\hfill \\ {}\hfill {G}_0\left(\omega \right), \mathrm{f}\mathrm{o}\mathrm{r} r=s\hfill \end{array}\right. $$

(82)

Let us evaluate now the NGSMs of the structural response. In the case of multi-correlated excitations, the forcing vector f(t) assumes the expression (Eq. 74); it follows that the differential equation (49), governing the motion in modal subspace, can be written as

$$ \ddot{\mathbf{q}}(t)+\boldsymbol{\Xi}\;\dot{\mathbf{q}}(t)+{\boldsymbol{\Omega}}^2\mathbf{q}(t)=\mathbf{P}\;\overline{\mathbf{F}}\ (t) $$

(83)

where

$$ \mathbf{P}={\boldsymbol{\Phi}}^T\;\mathbf{B} $$

(84)

After very tedious algebra it can be shown that the jth NGSMs, ${\lambda}_{\;j,{u}_i{u}_i}(t)$ (j = 0, 1, 2), of the jth nodal displacement response, u _i (t), are given as a function of modal NGSMs, p _j v  λ_j,kℓrs(t) (ℓ, k = 1, …, m ; r, s = 1, ⋯, N), by the following relationships:

$$ \begin{array}{l}{\lambda}_{\;0,{u}_i{u}_i}(t)={\displaystyle \sum_{k=1}^m{\displaystyle \sum_{\ell =1}^m{\displaystyle \sum_{r=1}^N{\displaystyle \sum_{s=1}^N\;{p}_{k\;r}\;{p}_{\ell\;s}\;{\phi}_{\;i\;k}\;{\phi}_{\;i\;\ell}\;{\lambda}_{\;0,k\;\ell\;r\;s}(t)}}}};\;\\ {}{\lambda}_{\;1,{u}_i{u}_i}(t)={\displaystyle \sum_{k=1}^m{\displaystyle \sum_{\ell =1}^m{\displaystyle \sum_{r=1}^N{\displaystyle \sum_{s=1}^N\;{p}_{k\;r}\;{p}_{\ell\;s}\;{\phi}_{\;i\;k}\;{\phi}_{\;i\;\ell}\;{\lambda}_{\;1,k\;\ell\;r\;s}(t)}}}};\\ {}{\lambda}_{\;2,{u}_i{u}_i}(t)={\displaystyle \sum_{k=1}^m{\displaystyle \sum_{\ell =1}^m{\displaystyle \sum_{r=1}^N{\displaystyle \sum_{s=1}^N\;{p}_{k\;r}\;{p}_{\ell\;s}\;{\phi}_{\;i\;k}\;{\phi}_{\;i\;\ell}\;{\lambda}_{\;2,k\;\ell\;r\;s}(t)}}}}\end{array} $$

(85)

where p _jv is the jth and vth element of the matrix P(j = k, ℓ; v = r, s). Once again it has to be emphasized that the zeroth NGSM, ${\lambda}_{\;0,{u}_i{u}_i}(t),$ and the second-order NGSM, ${\lambda}_{\;2,{u}_i{u}_i}(t)$, are real functions that coincide with the variances of the response in terms of displacement and velocity, respectively, while the first-order NGSM, ${\lambda}_{1,{u}_i{u}_i}(t)$, is a complex quantity (Di Paola 1985; Langley 1986; Di Paola and Petrucci 1990; Muscolino 1991).

As shown in Eq. 85 the nodal NGSMs can be evaluated as a function of λ_j,kℓrs(t), j = 0, 1, 2, which are the so-called time-dependent modal NGSMs “purged” by p _jv factors. After some algebra, these quantities, which are complex ones, can be evaluated, in frequency domain, for quiescent structural systems, at time t = 0, as (Di Paola 1985; Di Paola and Petrucci 1990, Muscolino1991):

$$ \begin{array}{l}{\lambda}_{\;0,k\;\ell\;r\;s}(t)={\displaystyle \underset{0}{\overset{\infty }{\int }}{Z}_{k\;r}^{*}\left(\omega, t\right){Z}_{\ell\;s}\left(\omega, t\right){G}_{\mathbf{NN},s\;r}\left(\omega \right)\;\mathrm{d}\omega };\\ {}{\lambda}_{\;1,k\;\ell\;r\;s}(t)=-\mathrm{i}{\displaystyle \underset{0}{\overset{\infty }{\int }}{Z}_{k\;r}^{*}\left(\omega, t\right){\dot{Z}}_{\ell\;s}\left(\omega, t\right){G}_{\mathbf{NN},s\;r}\left(\omega \right)\;\mathrm{d}\omega };\\ {}{\lambda}_{\;2,k\;\ell\;r\;s}(t)={\displaystyle \underset{0}{\overset{\infty }{\int }}{\dot{Z}}_{k\;r}^{*}\left(\omega, t\right){\dot{Z}}_{\ell\;s}\left(\omega, t\right){G}_{\mathbf{NN},s\;r}\left(\omega \right)\;\mathrm{d}\omega}\end{array} $$

(86)

where for j = k, ℓ, the following positions have been made:

$$ \begin{array}{l}{Z}_{j\;v}\left(\omega, t\right)={\displaystyle \underset{0}{\overset{t}{\int }}{h}_j\left(t-\tau \right) \exp \left(\mathrm{i}\omega \tau \right){a}_{vv}\left(\omega, \tau \right) \mathrm{d}\tau };\\ {}{\dot{Z}}_{j\;v}\left(\omega, t\right)={\displaystyle \underset{0}{\overset{t}{\int }}{\dot{h}}_j\left(t-\tau \right) \exp \left(\mathrm{i}\omega \tau \right){a}_{vv}\left(\omega, \tau \right) \mathrm{d}\tau }; j=k,\ell; v=r,s\end{array} $$

(87)

with h _j (t) the function defined in Eq. 62 and a _vv (ω, t) the vth element of the diagonal matrix A(ω, t). Moreover, introducing the state variable, the modal evolutionary frequency response function vector is defined as

$$ {\mathbf{Y}}_{j\;v}\left(\omega,,, t\right)=\left[\begin{array}{c}\hfill {Z}_{j\;v}\left(\omega,,, t\right)\hfill \\ {}\hfill {\dot{Z}}_{j\;v}\left(\omega,,, t\right)\hfill \end{array}\right]; j=k,\ell; v=r,s $$

(88)

Then relationships (33) can be rewritten in compact form as follows:

$$\begin{array}{lll} {\Sigma}_{\;k\;\ell\;r\;s}(t) &=\left[\begin{array}{cc}\hfill {\lambda}_{\;0,k\;\ell\;r\;s}(t)\hfill & \hfill \mathrm{i}{\lambda}_{\;1,k\;\ell\;r\;s}(t)\hfill \\ {}\hfill -\mathrm{i}{\lambda}_{1,k\;\ell\;r\;s}^{*}(t)\hfill & \hfill {\lambda}_{2,k\;\ell\;r\;s}(t)\hfill \end{array}\right] \\&\equiv {\displaystyle \underset{0}{\overset{\infty }{\int }}{\mathbf{Y}}_{k\;r}^{*}\left(\omega, t\right)\;{G}_{\mathbf{NN},s\;r}\left(\omega \right)\;{\mathbf{Y}}_{\ell\;s}^T\left(\omega, t\right)\mathrm{d}\omega } \end{array}$$

(89)

where Σ _{k ℓ r s} (t) coincides with the cross-modal pre-envelope covariance matrix (Di Paola and Petrucci 1990). The one-sided PSD G_NN, s r(ω) can be evaluated as a function of the coherence function introduced in Eq. 81. Notice that according to the Priestley evolutionary process model (Priestley 1999), the complex cross-correlation function matrix of the zero-mean modal “purged” response process can be evaluated as

$$ {\mathbf{R}}_{k\;\ell\;r\;s}\left({t}_1,{t}_2\right)={\displaystyle \underset{0}{\overset{\infty }{\int }}{\mathbf{Y}}_{k\;r}^{*}\left(\omega, {t}_1\right)\;{G}_{\mathbf{NN}, sr}\left(\omega \right)\;{\mathbf{Y}}_{\ell\;s}^T\left(\omega, {t}_2\right)\mathrm{d}\omega}\equiv {\displaystyle \underset{0}{\overset{\infty }{\int }} \exp \left[\mathrm{i}\omega \left({t}_1-{t}_2\right)\right]{\mathbf{G}}_{k\;\ell\;r\;s}\left(\omega, {t}_1,{t}_2\right)\mathrm{d}\omega } $$

(90)

where

$$ {\mathbf{G}}_{k\;\ell\;r\;s}\left(\omega, {t}_1,{t}_2\right)={\mathbf{Y}}_{k\;r}^{*}\left(\omega, {t}_1\right)\;{G}_{\mathbf{NN}, sr}\left(\omega \right)\;{\mathbf{Y}}_{\ell\;s}^T\left(\omega, {t}_2\right) $$

(91)

is the cross-modal EPSD function matrix between the k, ℓth dummy oscillators at the rth and sth support points. By means of the modal transformation (47), the nodal autocorrelation and the EPSD function matrices of the displacement response, u _i (t), can be evaluated, after very simple algebra, respectively, as follows:

$$ \begin{array}{l}{\mathbf{R}}_{u_{\;i} {u}_{ i}}\left({t}_1,{t}_2\right)={\displaystyle \sum_{k=1}^m{\displaystyle \sum_{\ell =1}^m{\displaystyle \sum_{r=1}^N{\displaystyle \sum_{s=1}^N{p}_{k\;r}\;{p}_{\ell\;s}\;{\phi}_{\;i\;k}\;{\phi}_{\;i\;\ell}\;{\mathbf{R}}_{k\;\ell\;r\;s}\left({t}_1,{t}_2\right)}}} },\\ {}{\mathbf{G}}_{u_{\;i} {u}_{ i}}\left(\omega, {t}_1,{t}_2\right)={\displaystyle \sum_{k=1}^m{\displaystyle \sum_{\ell =1}^m{\displaystyle \sum_{r=1}^N{\displaystyle \sum_{s=1}^N{p}_{k\;r}\;{p}_{\ell\;s}\;{\phi}_{\;i\;k}\;{\phi}_{\;i\;\ell}\;{\mathbf{G}}_{k\;\ell\;r\;s}\left(\omega, {t}_1,{t}_2\right)}} }}\end{array} $$

(92)

Consequently the nodal displacement NGSMs are the elements of the nodal pre-envelope covariance matrix, given as

$$ {\Sigma}_{u_{\;i} {u}_{ i}}(t)\equiv {\mathbf{R}}_{u_{\;i} {u}_{ i}}\left(t,t\right)\equiv {\displaystyle \underset{0}{\overset{\infty }{\int }}{\mathbf{G}}_{u_{\;i} {u}_{ i}}\left(\omega, t\right) \mathrm{d}\omega }={\displaystyle \sum_{k=1}^m{\displaystyle \sum_{\ell =1}^m{\displaystyle \sum_{r=1}^N{\displaystyle \sum_{s=1}^N{p}_{k\;r}\;{p}_{\ell\;s}\;{\phi}_{\;i\;k}\;{\phi}_{\;i\;\ell}\;{\Sigma}_{k\;\ell\;r\;s}(t)}}}} $$

(93)

Calculating Nonstationary Stochastic Responses

In this section, in order to evidence the main differences between the stochastic responses of structural systems subjected to both uniformly and fully nonstationary models of seismic excitations, the spectral characteristics of the response of two SDoF oscillators are evaluated.

Comparison Between Steady-State and Nonstationary Responses of SDoF Systems

Let us consider an oscillator, whose differential equation governing the motion is written in canonical form in Eq. 23, forced by the uniformly modulate Gaussian zero-mean nonstationary process F(t), defined as

$$ F(t)=a(t) N(t) $$

(94)

where N(t) is a Gaussian white noise process, with one-sided PSD G _NN (ω) = 1 cm²/sec³, and a(t) is the normalized to one modulating function proposed by Hsu and Bernard (1978):

$$ a(t)=\alpha \exp (1)\;t \exp \left(-\alpha\;t\right) \mathit{\mathsf{U}}(t) $$

(95)

which takes its maximum value at time, t_max = 1/α, and $\mathit{\mathsf{U}}(t)$ is the unit step function defined as

$$ \mathit{\mathsf{U}}\left(t-{t}_0\right)=\left\{\begin{array}{c}\hfill 0, t\le {t}_0; \hfill \\ {}\hfill 1, t>{t}_0\;\hfill \end{array}\right. $$

(96)

In Fig. 1 the modulating function (95) is depicted for different value of α = 1/t_max. In the same figure the EPSD function G _FF (ω, t) = |a(t)|² G _NN (ω) of the quasi-stationary (separable) Hsu and Bernard (1978) model is also shown assuming α = 1/5 and G _NN (ω) = 1 cm²/sec³.

Stochastic Analysis of Linear Systems, Fig. 1

Hsu and Bernard (1978) model of the uniformly modulated nonstationary excitation: (a) modulating function for different value of the time instant, t_max, in which it takes the maximum value; (b) EPSD function G _FF (ω, t) = |a(t)|² G _NN (ω) with G _NN (ω) = 1 cm²/sec³ and α = 1/5

In Fig. 2 the NGSM, λ_j,uu(t) (j = 0, 1, 2), and bandwidth parameter, δ _uu (t), functions of the response of an oscillator with natural circular frequency ω₀ = 2π rad/sec and damping ratio ξ₀ = 0.05 for the Hsu and Bernard (1978) model of the uniformly modulated nonstationary excitation are depicted. These NGSMs are compared to the transient NGSMs of the stationary case ($a(t)=\mathit{\mathsf{U}}(t)$), for different value of the time instant, t_max. In Fig. 3 the comparisons are performed for an oscillator with natural circular frequency ω₀ = 8π rad/sec and damping ratio ξ₀ = 0.05. These figures shows that when the response approaches to its steady-state condition, defined by the steady-state time t_SS ≥ 3/(ξ₀ ω₀), the maximum values of the NGSM functions coincide with the corresponding values of the NGSM for stationary input processes. If this condition is not satisfied, the stationary approximation of input process leads to overestimated results. In fact, for the first oscillator for which the steady-state time is t_SS ≈ 9.55 sec, the NGSMs with t_max < t_SS possess maximum values lesser than the corresponding stationary case. Similar results are obtained for the second oscillator for which t_SS ≈ 2.39 sec. Moreover, the analysis of the bandwidth parameters shows that if t_max < t_SS the parameter δ _uu (t) is narrower than the value obtained in the relative stationary case.

Stochastic Analysis of Linear Systems, Fig. 2

NGSM functions, λ_j,uu(t) (j = 0, 1, 2) and bandwidth parameter, δ _uu (t), of the transient response of an oscillator (ω₀ = 2π rad/sec, ξ₀ = 0.05) in the stationary case (solid line) and adopting the Hsu and Bernard (1978) model, for different value of the time instant, t_max, in which the modulating a(t) function take the maximum value (dashed lines)

Stochastic Analysis of Linear Systems, Fig. 3

NGSMs, λ_j,uu(t) (j = 0, 1, 2) and bandwidth parameter, δ _uu (t), of the transient response of an oscillator (ω₀ = 8π rad/sec, ξ₀ = 0.05) in the stationary case (solid line) and adopting the Hsu and Bernard (1978) model, for different value of the time instant, t_max, in which the modulating a(t) function take the maximum value (dashed lines)

Comparison Between Uniformly and Fully Nonstationary Responses of SDoF Systems

Let us consider now the Spanos and Solomos (1983) model of the fully nonstationary Gaussian zero-mean process. For this model the normalized to one evolutionary modulating function can be written as

$$ a\left(\omega, t\right)=\varepsilon \left(\omega \right)\;t\; \exp \left(-\alpha \left(\omega \right)\;t\right)\mathit{\mathsf{U}}(t) $$

(97)

Selecting the following parameters $\varepsilon \left(\omega \right)=\omega\;\sqrt{2}/5\;\uppi\;{a}_{\max }$ and α(ω) = 0.15/2 + ε²(ω)/4, the normalizing to one coefficient is a_max = 1.34. The unitary maximum is reached at ω = 1.937 π rad/sec and at t = 6.667 sec. In Fig. 4, the section of the modulating function at different values of abscissa ω is depicted together with the one-sided EPSD function G _FF (ω, t) = ∣a(ω, t)∣² G _NN (ω) with G _NN (ω) = 1 cm²/sec³. This figure evidences the frequency dependence of this model of nonstationary input process.

Stochastic Analysis of Linear Systems, Fig. 4

Spanos and Solomos (1983) model of the fully nonstationary excitation model: (a) sections of the normalized to one modulating function a(ω, t) at different values of abscissa ω; (b) EPSD function G _FF (ω, t) = |a(ω, t)|² G _NN (ω) with G _NN (ω) = 1 cm²/sec³

In Figs. 5 and 6 the NGSMs, λ_j,uu(t) (j = 0, 1, 2), of the two oscillators with natural circular frequency ω₀ = 2π rad/sec and ω₀ = 8π rad/sec and damping ratio ξ₀ = 0.05, obtained by means of the two modulating functions before described, are depicted and compared. Analyzing these figures it can be evidenced that the temporal variation of the frequency content of the EPSD function, often neglected for mathematic convenience, has substantial effects on the structural response. In fact the maximum values of the NGSM functions depend on the form of modulating function as well as the dynamic characteristics of the structural systems. Moreover, in some cases, the quasi-stationary modellation of the modulating function can lead to very unconservative results (see Fig. 5) of the stochastic structural response.

Stochastic Analysis of Linear Systems, Fig. 5

NGSMs of the response of an oscillator with ω₀ = 2π rad/sec, ξ₀ = 0.05 for the Hsu and Bernard (1978) model (solid line) and for the Spanos and Solomos (1983) model (dashed line)

Stochastic Analysis of Linear Systems, Fig. 6

NGSMs of the response of the oscillator with ω₀ = 8π rad/sec, ξ₀ = 0.05 for the Hsu and Bernard (1978) model (solid line) and for the Spanos and Solomos (1983) model (dashed line)

In Figs. 7 and 8 the mean frequencies, ν ⁺ _uu (t), and the normalized time-varying central frequencies, ω_C,uu(t)/ω₀, defined in Eq. 21, of the response of the two oscillators before analyzed for the Hsu and Bernard (1978) and Spanos and Solomos (1983) modulating functions of the nonstationary zero-mean Gaussian input process, are depicted.

Stochastic Analysis of Linear Systems, Fig. 7

Mean frequencies of the response of the oscillator for the Hsu and Bernard (1978) model (solid line) and for the Spanos and Solomos (1983) model (dashed line); (a) oscillator with ω₀ = 2π rad/sec and ξ₀ = 0.05; (b) oscillator with ω₀ = 8π rad/sec and ξ₀ = 0.05

Stochastic Analysis of Linear Systems, Fig. 8

Time-varying central frequency, normalized by the natural circular frequency, of the oscillator for the Hsu and Bernard (1978) model (solid line) and for the Spanos and Solomos (1983) model (dashed line); (a) oscillator with ω₀ = 2π rad/sec and ξ₀ = 0.05; (b) oscillator with ω₀ = 8π rad/sec and ξ₀ = 0.05

Figure 7 shows that for the Hsu and Bernard (1978) modulating function, the mean frequencies, ν ⁺ _uu (t), get to an asymptotic value very close to the natural frequency of the oscillators. This behavior is not verified for the oscillator with higher natural circular frequency subjected to the Spanos and Solomos (1983) model of the nonstationary input process, where the mean frequency decreases evidencing the frequency dependence of the structural response. Similar results are obtained for the normalized time-varying central frequencies, ω_C,uu(t)/ω₀, depicted in Fig. 8.

Summary

The dynamic behavior of structural systems subjected to uncertain dynamic excitations can be performed through the stochastic analysis, which requires the probabilistic characterization of both input and output processes. The characterization of output processes can be extremely complex, when nonstationary and/or non-Gaussian input processes are involved. However, in several cases the approximate description of the dynamic structural response based on its spectral characteristics may be sufficient.

In this study a unitary approach to evaluate the spectral characteristics of the structural response, to perform the reliability assessment, of classically damped linear systems subjected to stationary or nonstationary mono-/multi-correlated zero-mean Gaussian excitations, is described.

The main steps of the described approach are (i) the use of modal analysis to decouple the equation of motion; (ii) the determination, in state variable, of the evolutionary frequency response vector functions and of the evolutionary power spectral density function matrix of the structural response; and (iii) the evaluation of the nongeometric spectral moments as well as the spectral characteristics of the stochastic response of linear systems subjected to stationary or nonstationary mono-/multi-correlated zero-mean Gaussian seismic excitations.

Cross-References

Probability Density Evolution Method in Stochastic Dynamics

Stochastic Analysis of Nonlinear Systems

Stochastic Ground Motion Simulation