CAO Qianying, HU Sau-Lon James, and LI Hewenxuan

A Novel Approach for Evaluating Nonstationary Response of Dynamic Systems to Stochastic Excitation

CAO Qianying1), HU Sau-Lon James2), *, and LI Hewenxuan3)

1) College of Engineering, Ocean University of China, Qingdao 266100, China 2) Department of Ocean Engineering, University of Rhode Island, Narragansett, RI 02882, USA 3) College of Engineering, University of Rhode Island, Kingston, RI 02881, USA

The transient state of a dynamic system, such as offshore structures, to random excitation is always nonstationary. Many studies have contributed to evaluating response covariances at the transient state of a linear multi-degree-of-freedom (MDOF) system to random excitations, but a closed-form solution was not available unless the excitation was assumed to be a physically unrealizable white noise process. This study derives explicit, closed-form solutions for the response covariances at the transient state by using a pole-residue (PR) approach operated in the Laplace domain when the excitations are assumed to be stationary random processes described by physically realizable spectral density functions. By using the PR method, we can analytically solve the triple integral in evaluating the nonstationary response covariance. As this approach uses the poles and residues of system transfer functions, rather than the conventional mode superposition technique, the method is applicable to MDOF systems with non-classical damping models. Particular application of the proposed method is demonstrated for multi-story shear buildings to stochastic ground acceleration characterized by the Kanai–Tajimi spectral density function model, and a numerical example is provided to illustrate the detailed steps. No numerical integrations are required for computing the response covariances as the exact closed-form solution has been derived. The correctness of the proposed method is numerically verified by Monte Carlo simulations.

linear systems; transient state; nonstationary response; pole; residue

1 Introduction

For offshore structures located in seismically active regions, earthquake motions and sea waves are two main design loads that are random in nature (Yamada, 1989). Owing to the influence of the initial conditions, the transient state of a dynamic system to random excitation is always nonstationary regardless of whether the excitation is stationary or not. This study derives explicit closed-form solutions for the response covariances at the transient state of a multi-degree-of-freedom (MDOF) system to stationary random excitations described by physically realizable power spectral density (PSD) functions. A Laplace-frequency domain approach based on pole-residue (PR) formulations is employed.

Although evaluating response covariances at the transient state of linear systems to random excitations has been extensively studied for several decades, under various assumptions, idealizations, and formulations, by using time/frequency domain approaches (Hammond, 1968; Shinozuka, 1968; Roberts, 1971; Fujimori and Lin, 1973; Gasparini, 1979; Sakata and Kimura, 1979; Gas- parini and DebChaudhury, 1980; To, 1982, 1984, 1986; Di Paola, 1984; Sun and Kareem, 1989), only a few articles focused on obtaining a closed-form solution (Cau- ghey and Stumpf, 1961; Curtis and Boykin Jr., 1961; Lin, 1967; Iwan and Hou, 1989). Addressing MDOF systems, Grigoriu (1992) developed a two-phase method for calculating the probabilistic characteristics of the transient response by using a state space formulation for time-invariant stable linear systems to stationary Gaussian processes. Applying the modal superposition approach for classically damped systems, Conte and Peng (1996) obtained explicit closed-form solutions for the correlation matrix (or covariances) of the response of a MDOF system excited by a modulated white noise. Overall, traditional solution methods for evaluating the nonstationary response have been conducted in the time and/or frequency domains, and no explicit, closed-form solution is available unless the excitation was assumed to be a physically unrealizable white/colored noise process (Lin, 1967; Madsen, 2006).

Mathematically, even for a single-degree-of-freedom (SDOF) system subjected to a stationary stochastic excitation, evaluating the nonstationary response covariance requires carrying out a triple integral (Caughey and Stumpf, 1961; Lin, 1967). This triple integral involves two sequential steps: first, operating a double integral to compute the evolutionary response PSD from a given excitation PSD; second, taking the integral of the evolutionary response PSD with respect to the frequency to evaluate the time-dependent response covariance. For step 1, most traditional methods operated in the time or frequency domain (Caughey and Stumpf, 1961; Barnoski and Maurer, 1969; Iwan and Hou, 1989). Spanos(2016) developed a time-frequency approach based on generalized harmonic wavelets to determine the response evolutionary PSD of SDOF systems subjected to nonstationary excitations. To obtain a response covariance from the response evolutionary PSD at step 2, Spanos(2016) must rely on a numerical approach. Limited to SDOF systems subjected to stochastic loading governed by a simple PSD function, a recent study (Hu, 2018) derived explicit closed-form solutions for the mean-square response at the transient state by using a pole-residue (PR) approach operated in the Laplace domain. The study demonstrated that the PR method was more efficient than other existing approaches. Through the PR method, the double integral in step 1 was converted to simple algebraic pole-residue operations, which avoided conducting convolution integrals. In addition, the outcome of step 1 was in a PR form, which facilitated carrying out step 2 by the method of residues (Lin, 1967). The novelty of the PR method was mainly at step 1, while the method of residues at step 2 has been a well-established technique in performing the contour integral. In both deterministic and stochastic dynamic analyses, the key concepts of a PR method include: 1) the poles and residues of the response can be obtained from those of the input and system transfer functions by simple algebraic operations, and 2) once the output poles and residues are available, the response time history can be obtained immediately (Hu, 2016). Extending to MDOF systems subjected to stationary random excitations described by physically realizable PSD functions, the present study derives explicit closed-form solutions for the response covariances at the transient state by using a PR approach. As this approach uses the poles and residues of system transfer functions rather than the conventional mode superposition technique, the method is applicable to MDOF systems with non-classical damping models.

To demonstrate the effectiveness of the proposed method, both detailed analytical derivation and numerical investigation aim to focus on multi-story shear building models to stochastic ground acceleration characterized by the well-known Kanai-Tajimi PSD model.

2 Preliminaries

2.1 MDOF Systems

The mathematical model of an MDOF system is often written in the following second-order matrix differential equation form (Craig and Kurdila, 2006):

or

where

where

2.2 Stationary Random Excitation

This study assumes that random excitationsF() andF(), acting respectively on coordinatesand, are both zero-mean, stationary, Gaussian random processes. The cross-PSD function ofF() andF() is denoted byS(), and the cross-correlation function byR(). These two functions are related to each other through the specific Fourier transform pair

and

These relationships also hold for auto-PSD and auto-correlation functions, namely, for=.

2.3 Shear Building to Earthquake PSD

Although the proposed method is applicable to any linear MDOF system for a broad stochastic loading, special attention is given to a multi-story shear building to a strong earthquake (see Fig.1).

This study assumes that no rotation of the base occurs and that the base moves as a rigid body with displacement

Fig.1 Discretized MDOF system with rigid-base translation.

x(). The equations of motion of the multi-story shear building can be written as (Clough and Penzien, 1995)

where the total displacement() is the sum of the relative motions() plus the displacements resulting directly from the support motionsx(). For the system shown in Fig.1, this relationship may be written as

where {1} represents a column of ones. This vector expresses the fact that a unit static translation of the base of this structure directly produces a unit displacement of all degrees of freedom.

In earthquake engineering, the Kanai-Tajimi model has been widely adopted to characterize the spectral density,S(), of the stochastic ground acceleration of earthquakes, as follows (Kanai, 1957):

whereis spectral density at bedrock, and0and0are parameters depending on local geology. The PR form ofS() can be derived as

where

and

3 Nonstationary Random Responses

When an MDOF system subjected to random excitationF() is initially at rest, the displacement at coordinateis

Thus, the joint second moment|X(1)X(2)| of random responsesX(1) and X(2) is

where[·] denotes the expected value. AsF() and F() are both assumed to be zero mean, the random responsesX(1) and X(2) are also zero mean. As a result,|X(1)X(2)| is equal to the covariance ofX(1) and X(2).

The total responseX() at coordinatecontributed by multiple loadingsF(), (=1, 2, ···,) is expressed as

Then, the joint second moment ofX(1) and X(2) responses at two coordinates and two instants of time is obtained as

A closed-form solution of|X(1)X(2)| can be obtained as long as that of|X(1)X(2)| has been derived. If excitationsF() and F() at two distinct coordinates are uncorrelated, namely,|X(1)X(2)|=0 for1, then Eq. (22) is simplified as

For simplicity in presentation, the rest of this article focuses on computing the second joint moment at a single coordinate, that is,=, knowing that the derivation for the general|X(1)X(2)| at two coordinates can follow the same procedure.

3.1 Single Output Coordinate j: Joint Second Moment of Xjk(t1) and Xjl(t2)

Letting=in Eq. (20), and replacing the cross-correlation function ofF() and F() with its cross-spectral density functionS() by using Eq. (11), we can write

This formulation allows the time variables1and2to be separated. Regrouping the above equation leads to

The interchange of the order of integration in Eq. (24) is permissible as explained by Lin (1967). We denote the first bracket term in Eq. (25) by

We can show the following-dependent residues:

and

Substituting Eqs. (7) and (27) into Eq. (29), we obtain

Similarly, substituting Eqs. (7) and (27) into Eq. (30), we obtain

or the complex frequency response function

Now,1(1,) in Eq. (34) not only has a simple explicit form but also contains a clear physical meaning; the summation term is the natural (transient) response governed by system poles, and the last term is the forced (steady-state) response governed by the excitation pole, ?(Hu and Gao, 2017).

Let the second bracket term in Eq. (25) be denoted by

Taking the Laplace transform of Eq. (35) yields

We show the following-dependent residues:

and

Denoting(1,2,)=1(1,)2(2,), Eq. (25) becomes

A simple multiplication of Eqs. (34) and (40) provides

Upon substituting Eq. (42) into Eq. (41), we write

where

Note that4(1,2) is a stationary term, which is a function of time difference2?1.

3.2 Single Instant of Time: Joint Second Moment of Xjk(t) and Xjl(t)

When1=2=, Eq. (43) becomes

whereE() has been denoted forG(,). Thus,

An explicit closed-form solution of[X()X()] can be obtained if the integrals shown in Eqs. (49)–(52) have been evaluated analytically. AsA() andB() have been in a PR form (see Eqs. (31) and (38), respectively), the integrals in Eqs. (49)–(52) can be evaluated by the method of residues (Lin, 1967) as long as the cross-PSD functionS() can also be expressed in a PR form.

Given the specific cross-PSD function of the effective earthquake forcesS() in Eq. (18), by using the method of residues, we obtain the closed-form solutions of the integrals in Eqs. (49)–(52) as follows:

In summary, by using Eqs. (48) and Eqs. (53)–(56), we obtained the exact closed-form joint second moment[X()X()] for an MDOF shear building model to the Kanai-Tajimi earthquake PSD.

4 Numerical Studies

For the three-story shear building model, the mass at the-th story is denoted bymand the mass matrix is a diagonal matrix, diag{}=[1,2,3]. Referring to Fig.2 for the definition ofk, one has the stiffness matrix

The damping matrixshares the same form as, but replacing kwith the correspondingc.

Fig.2 Three-DOF lumped mass shear building model.

Table 1 Analytical poles, modal frequencies, damping ratios, and corresponding complex modes

Table 2 Residues βjk, n, n=1,2,3

Fig.3 Magnitude plots of complex frequency response functions for a three-DOF lumped mass model.

Fig.4 Kanai-Tajimi PSD, Sg(ω).

Fig.5 along with its individual terms for Kanai-Tajimi PSD.

The computation for each of the six constituents is similar. For illustration purposes,[31()32()] is selected as an example. From Eq. (48),[31()32()] is the sum of four terms: nonstationary terms1()?3() and a stationary term4. These four terms are computed by Eqs. (53)–(56) whenS() has been specified by Eq. (18) based on the Kanai-Tajimi PSD model. These four individual terms, along with their sums, are shown in Fig.6. Note that2() and3() are almost the same for the entire time history. Whenbecomes large, nonstationary terms1()?3() diminish due to the presence of system damping, and[31()32()] eventually becomes stationary, which is completely governed by4. From Fig.6, we also note the features at time=0, including1(0)=?2(0)=?3(0)=4and[31(0)32(0)]=0. These features are attributed to the imposed at-rest initial conditions.

Fig.6 E[X31(t)X32(t)] along with its individual terms for Kanai-Tajimi PSD.

Fig.7 Comparison of based on Monte Carlo simulations and proposed method for Kanai-Tajimi PSD.

5 Concluding Remarks

Considering MDOF systems to stationary random pro- cesses described by physically realizable (non-white) spectral density functions, this study developed a novel Lap- lace-frequency method that could effectively obtain the exact closed-form solutions for the response covariances at the transient state. Unlike many traditional time/frequency domain methods, the proposed approach operated in the Laplace-frequency domain did not require the viscous damping matrix of an MDOF system to follow a proportional damping model. Another significant advantage of the proposed approach was that meaningful physical insights were obtained in the solution procedure. A specific analytical and numerical demonstration of the proposed method was given to multi-story shear buildings subjected to stochastic ground acceleration characterized by the Kanai-Tajimi spectral density function model. As the exact closed-form solution was derived for computing the response covariances, no numerical integrations were required. A numerical example was presented to illustrate the detailed steps of the proposed method, and its correctness was verified by Monte Carlo simulations.

Acknowledgements

This study was financially supported by the National Natural Science Foundation of China (No. 51879250). Thefirst author was supported by the China Scholarship Council while conducting her research in the United States.

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. E-mail: jameshu@uri.edu

September 6, 2019;

December 4, 2019;

December 6, 2019

(Edited by Xie Jun)