### 统计代写|随机控制代写Stochastic Control代考| Modeling of Random Dynamic Excitations

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## 统计代写|随机控制代写Stochastic Control代考|Random Seismic Ground Motion

Performance-based design and control of structures not only relies upon the structural model and the computational method but also relies upon the rationality of the modeling of random dynamis excitations of structures. Classical random process theory usually employs the power spectral density to describe the random excitations, such as the Kanai-Tajimi spectrum (Kanai 1957; Tajimi 1960) used in the earthquake engineering community, the Davenport spectrum (Davenport 1961) used in the wind engineering community, and the Pierson-Moskowitz spectrum (Pierson and Moskowiz 1964) used in the marine engineering community. One might recognize that the power spectral density denotes the second-order statistics of stationary processes in essence, which hardly reveals, however, the complete probabilistic information of original random processes. Moreover, the measure on the power spectral density of random excitations cannot be accurately delivered to the stochastic response through nonlinear structural systems, not mentioned to carry out the logical control of structural performance. However, a family of physically motivated random excitation models has been developed in recent years by exploring the physical mechanism of engineering excitations (Li 2006; 2008). For illustrative purposes, the modeling of random seismic ground motion and of spatial fluctuating wind-velocity field are investigated herein, and the pertinent theory and methods are introduced.

It is well understood that the behaviors of seismic ground motions rely upon a series of critical factors such as the fault mechanism, propagation medium, and properties of the local site (Boore 2003). Due to the uncontrollability of these factors, the observed seismic ground motion arises to have a significant randomness. An efficient means for exploring the seismic wave and its propagation is to establish a wave equation with boundary conditions in conjunction with the seismic source motion (Aki and Richards 1980).

## 统计代写|随机控制代写Stochastic Control代考|Spectral Transfer Function

Assuming that the propagation medium is homogenous, elastic, and timeindependent, the one-dimensional seismic ground motion field is governed by a wave cquation as follows (Wang and Li 2011):
$$\sum_{j=0}^{n} \sum_{k=0}^{m} a_{j k} \frac{\partial^{j+k}}{\partial x^{j} \partial t^{k}} u(x, t)=0$$

where $a_{j k}$ is a medium-relevant parameter; $u(x, t)$ denotes the wave displacement of seismic ground motion. The initial and boundary conditions are given by
$$u(0, t)=u_{0}(t),\left.\quad \frac{\partial^{i} u(x, t)}{\partial t^{i}}\right|{t \rightarrow 0}=0,\left.\quad \frac{\partial^{i} u(x, t)}{\partial t^{i}}\right|{t \rightarrow+\infty}=0, i=0,1, \ldots, n$$
By virtue of the Fourier transform, the partial differential equation shown in Eq. (2.5.1) can be transformed into an ordinary differential equation, of which the solution has a formulation as follows:
$$U(x, \omega)=\sum_{j=0}^{n} b_{j}(\omega) \exp \left(-\mathrm{i} k_{j}(\omega) x\right)$$
where $k_{j}(\omega)$ is the eigenvalue of wave displacement, which relies upon the propagation medium; $b_{j}(\omega)$ denotes the synthetic effect of seismic source and propagation path.
Inverse Fourier transform on the wave displacement $U(x, \omega)$, yields
$$u(x, t)=\frac{1}{2 \pi} \sum_{j=0}^{n} \int_{-\infty}^{\infty} B_{j}(\omega, x) \exp \left[\mathrm{i} \omega\left(t-\frac{x}{c_{j}(\omega)}\right)\right] \mathrm{d} \omega$$
where $c_{j}(\omega)=\omega / \operatorname{Re}\left[k_{j}(\omega)\right] ; \operatorname{Re}[\cdot]$ denotes real component.
Equation (2.5.4) can be further expanded as
\begin{aligned} u(x, t)=& \frac{1}{2 \pi} \int_{-\infty}^{\infty} A\left(b_{0}(\omega), \ldots, b_{n}(\omega) ; k_{0}(\omega), \ldots, k_{n}(\omega) ; \omega, x\right) \ & \cdot \cos \left[\omega t+\Phi\left(b_{0}(\omega), \ldots, b_{n}(\omega) ; k_{0}(\omega), \ldots, k_{n}(\omega) ; \omega, x\right)\right] \mathrm{d} \omega \end{aligned}
It is indicated that the seismic ground motion field can be represented as a formulation of superposition harmonics, of which the amplitude and phase both are influenced by the boundary condition and the characteristics of propagation medium.
Assuming that the specific engineering site is far from the seismic source and the fault develops extensively fast, the dislocation process of seismic source can be viewed as irrelevance with the behaviors of the propagation path of seismic wave. Meanwhile, the scale of the local engineering site is far less than that of the propagation path of seismic wave, and the frequeney seatter effect of local site on the seismic ground motion can be ignored safely. The amplitude spectrum $A(\omega, x)$ and the phase angle $\Phi(\omega, x)$ in Eq. (2.5.5) can be thus written in a separation formulation (Wang and Li 2011).

## 统计代写|随机控制代写Stochastic Control代考|Seismic Source Model

Seismic source models in seismology are mainly classified into the kinematic models and dynamic models (Aki and Richards 1980). The former describes the kinematic characteristics of seismic source and focuses on the modeling of motion amplitude of seismic source. The latter describes the dynamic characteristics of seismic source and focuscs on the modeling of dislocation and dynamic devclopment of scismic source. The kinematic model of seismic source is widely used in the earthquake engineering community. The most celebrated spectral models pertaining to the kinematics of seismic source are the $\omega^{-3}$ model based on the Haskell rectangular dislocation mechanism of seismic source (Haskell 1964,1966 ), the $\omega^{-2}$ model based on the

Haskell rectangular dislocation mechanism of seismic source (Aki 1967), and the Brune source model based on the Brune circle dislocation mechanism of seismic source (Brune 1970). Among these models, the Brune source model has the benefits of less parameters and solid physical background, in which the fault surface is assumed to be circular and the dislocation distributes uniformly on the fault surface, and the shear stress wave caused by the shear stress drop propagates perpendicular to the dislocation surface. The Fourier amplitude spectrum and the Fourier phase spectrum on the Brune source model are thus denoted as follows (Brune 1970):
$$A_{s}\left(\alpha_{E}, \omega\right)=\frac{A_{0}}{\omega \sqrt{\omega^{2}+\left(\frac{1}{\tau}\right)^{2}}}, \Phi_{s}\left(\alpha_{E}, \omega\right)=\arctan \left(\frac{1}{\omega \tau}\right)$$
where $\alpha_{E}=\left(A_{0}, \tau\right)$ denotes the random vector of physical parameters relevant to the seismic source; $A_{0}$ denotes the amplitude parameter which is a random variable pertaining to intensity of seismic source; $\tau$ denotes the source parameter which is a random variable pertaining to the characteristics of seismic source.

## 统计代写|随机控制代写Stochastic Control代考|Spectral Transfer Function

∑j=0n∑ķ=0米一种jķ∂j+ķ∂Xj∂吨ķ在(X,吨)=0

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