### 统计代写|随机控制代写Stochastic Control代考|Random Vibration of Structures

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## 统计代写|随机控制代写Stochastic Control代考|Modal Superposition Method

In practical applications, the complexity of attaining the analytical solution of the unit impulse response function $\mathbf{h}(t)$ is far more than that of the frequency response transfer function $\mathbf{H}(\omega)$ for a multiple-degree-of-freedom system. Moreover, the solution procedure of mean-square responses often involves high-dimensional integrals on the unit impulse response function and on the frequency response transfer function. The computational cost is unacceptable in most cases. In fact, for the linear stochastic dynamical system, a workload-reduced way refers to the so-called modal superposition method. The basic idea is that the original multiple-degree-of-freedom

stochastic system is decoupled into a series of single-degree-of-freedom stochastic systems, so as to significantly reduce the computational cost.

According to the principle of the modal superposition method, the equation of motion of the stochastic dynamical system shown in Eq. (2.3.1) can be rewritten as
$$\overline{\mathbf{M}} \ddot{\mathbf{U}}(t)+\overline{\mathbf{C}} \dot{\mathbf{U}}(t)+\overline{\mathbf{K}} \mathbf{U}(t)=\boldsymbol{\Phi}^{\mathrm{T}} \mathbf{F}(\boldsymbol{\Theta}, t)$$
where $\overline{\mathbf{M}}=\boldsymbol{\Phi}^{\mathrm{T}} \mathbf{M} \boldsymbol{\Phi}, \overline{\mathbf{C}}=\boldsymbol{\Phi}^{\mathrm{T}} \mathbf{C} \boldsymbol{\Phi}, \overline{\mathbf{K}}=\boldsymbol{\Phi}^{\mathrm{T}} \mathbf{K} \boldsymbol{\Phi}$ are the $n \times n$ modal mass, modal damping, and modal stiffness matrices, respectively; $\mathbf{U}=\boldsymbol{\Phi}^{\mathrm{T}} \mathbf{X}$ is the $n$-dimensional column vector denoting modal displacement; $\boldsymbol{\Phi}=\left[\phi_{1}, \phi_{2}, \ldots, \phi_{q}\right]=\left[\phi_{i j}\right]_{n \times q}$ $(q \leq n)$ is the modal matrix.

Assuming that the damping matrix $\mathbf{C}$ is a proportional damping matrix, Eq. (2.3.14) can be then decomposed into $q$ mutually independent single-degreeof-freedom systems, of which the equation of motion of the $j$ th-order mode is shown as follows:
$$\ddot{u}{i}(t)+2 \zeta{i} \omega_{i} \dot{u}{i}(t)+\omega{j}^{2} u_{i}(t)=\frac{1}{\bar{m}{j}} \phi{j}^{\mathrm{T}} \mathbf{F}(\boldsymbol{\Theta}, t)=\frac{1}{\bar{m}{j}} \sum{k=1}^{n} \phi_{i k} F_{k}(\boldsymbol{\Theta}, t), j=1,2, \ldots, q$$
where $\bar{m}{j}$ is the $j$ th-order modal mass; $\omega{j}$ is the $j$ th-order modal frequency; $\zeta_{j}$ is the $j$ th-order modal damping ratio.

By means of the Duhamel integral, the componental formulation of the displacement of the linear system in the modal space can be derived as
$$u_{j}(t)=\frac{1}{\bar{m}{j}} \int{0}^{t} h_{j}(t-\tau) \phi_{j}^{\mathrm{T}} \mathbf{F}(\boldsymbol{\Theta}, \tau) \mathrm{d} \tau$$
where $u_{j}(t)$ is referred to as the $j$ th-order modal displacement.
The displacement solution of the linear system in the original state space is then given by
$$\mathbf{X}(t)=\sum_{j=1}^{q} \frac{1}{\bar{m}{j}} \int{0}^{t} h_{j}(t-\tau) \phi_{j} \phi_{j}^{\mathrm{T}} \mathbf{F}(\boldsymbol{\Theta}, \tau) \mathrm{d} \tau$$
Further, the mean and correlation function of the displacement can be deduced as follows:
$$\mu_{\mathbf{X}}(t)=E[\mathbf{X}(t)]=\sum_{j=1}^{q} \frac{1}{\bar{m}{j}} \int{0}^{t} h_{j}(t-\tau) \phi_{j} \phi_{j}^{\mathrm{T}} \mu_{\mathbf{F}}(\tau) \mathrm{d} \tau$$

## 统计代写|随机控制代写Stochastic Control代考|Pseudo-Excitation Method

When the linear system exhibits a high dimension, solving the power spectral density (PSD) of the system response shown in Eq. (2.3.27) involves a complicated procedure. The pseudo-excitation method (PEM) could be employed to obtain the PSD solution in an elegant manner (Lin et al. 2001). This method decomposes the solving procedure into a series of deterministic harmonic analysis through constructing a

pseudo-harmonic excitation. This treatment can enhance the efficiency of numerical schemes significantly.

Denoting the PSD of the random excitation $\mathbf{F}(\boldsymbol{\Theta}, t)$ as $\mathbf{S}{\mathbf{F}}(\omega)$, a pseudo-harmonic excitation $\mathbf{F}=\widetilde{\mathbf{F}}{\sqrt{\mathbf{s}}} \mathrm{e}^{\mathrm{i} \omega t}$ can be readily constructed, where $\widetilde{\mathbf{F}}{\sqrt{\mathbf{s}}}$ satisfies $\widetilde{\mathbf{F}}{\sqrt{\mathbf{s}}} \cdot \widetilde{\mathbf{F}}{\sqrt{\mathbf{s}}}^{*}=$ $\mathbf{S}{\mathbf{F}}(\omega)$, i denotes the imaginary unit. Replacing the excitation in Eq. (2.3.14) by the pseudo-excitation yields
$$\overline{\mathbf{M}} \tilde{\mathbf{U}}(t)+\overline{\mathbf{C}} \tilde{\mathbf{U}}(t)+\overline{\mathbf{K}} \tilde{\mathbf{U}}(t)=\mathbf{F}^{\mathrm{T}} \tilde{\mathbf{F}}_{\sqrt{\mathbf{s}}} \mathrm{e}^{\mathrm{i} \omega t}$$
where $\tilde{\tilde{U}}(t), \tilde{\mathbf{U}}(t), \tilde{\mathbf{U}}(t)$ are the $n$-dimensional column vectors denoting the corresponding acceleration, velocity, and displacement to the system subjected to the pseudo-excitation, respectively.

According to the classical random vibration theory, the stationary solution of Eq. (2.3.28) can be deduced as
$$\tilde{U}{j}(\omega, t)=\frac{1}{\bar{m}{j}} H_{j}(\omega) \phi_{j}^{\mathrm{T}} \tilde{\mathbf{F}}{\sqrt{\mathbf{s}}} \mathrm{e}^{\mathrm{i} \omega t}$$ The auto-power spectral density of system response is then derived as follows: \begin{aligned} S{U_{j}} \widetilde{U}{k}(\omega) &=\widetilde{U}{j}(\omega, t) \widetilde{U}{k}^{*}(\omega, t)=\frac{1}{\bar{m}{j} \bar{m}{k}} H{j}(\omega) H_{k}(\omega) \phi_{j}^{\mathrm{T}} \widetilde{\mathbf{F}}{\sqrt{\mathbf{s}}} \widetilde{\mathbf{F}}{\sqrt{\mathbf{s}}} \phi_{k} \ &=\frac{1}{\bar{m}{j} \bar{m}{k}} H_{j}(\omega) H_{k}(\omega) \phi_{j}^{\mathrm{T}} \mathbf{S}{\mathbf{F}}(\omega) \phi{k}=S_{U_{j} U_{k}}(\omega) \end{aligned}
It is shown that in the calculation of spectral density function, the factor of pseudoharmonic excitation $\mathrm{e}^{\mathrm{i} \omega t}$ is always paired with its complex conjugate $\mathrm{e}^{-\mathrm{i} \omega t}$ which are eventually counteracted by multiplication, revealing the time-independent behaviors of auto- and cross-power spectral densities of stationary processes.
Further, one can attain the mean-square solution of system responses:
$$E\left[\mathbf{U}(t) \mathbf{U}^{\mathrm{T}}(t)\right]=\frac{1}{2 \pi} \int_{-\infty}^{\infty} \mathbf{S}{\mathbf{U}}(\omega) \mathrm{d} \omega$$ Projecting the generalized coordinate space onto the original coordinate space, then $$E\left[\mathbf{X}(t) \mathbf{X}^{\mathrm{T}}(t)\right]=\boldsymbol{\Phi} E\left[\mathbf{U}(t) \mathbf{U}^{\mathrm{T}}(t)\right] \boldsymbol{\Phi}^{\mathrm{T}}=\frac{1}{2 \pi} \int{-\infty}^{\infty} \boldsymbol{\Phi} \mathbf{S}_{\mathbf{U}}(\omega) \boldsymbol{\Phi}^{\mathrm{T}} \mathrm{d} \omega$$

## 统计代写|随机控制代写Stochastic Control代考|Nonlinear Random Vibration

Without loss of generality, a nonlinear stochastic dynamical system is investigated, of which the equation of motion is given by
$$\mathbf{M} \ddot{\mathbf{X}}(t)+\mathbf{f}(\mathbf{X}(t), \dot{\mathbf{X}}(t))=\mathbf{F}(\boldsymbol{\Theta}, t)$$
where $\mathbf{f}(\cdot)$ is the $n$-dimensional column vector denoting nonlinear internal force.
The nonlinear internal force is assumed to be denoted by a polynomial function of velocity and displacement. In fact, this is a weak hypothesis, and a large family of dynamical systems can be represented by the formulation, such as the Duffing oscillator with nonlinear stiffness force and the van der Pol oscillator with coupling nonlinearities between stiffness and damping forces. The componental form of the equation is then written as
$$\sum_{i=1}^{n} m_{j i} \ddot{x}{i}(t)+\sum{i=1}^{n} \sum_{k=0}^{q} \alpha_{j i, k} \dot{x}{i}^{q-k}(t) x{i}^{k}(t)=F_{j}(\boldsymbol{\Theta}, t)$$
where $j=1,2, \ldots, n, m_{j i}$ denotes the element of mass matrix; $\ddot{x}{i}(t), \dot{x}{i}(t), x_{i}(t)$ denote the acceleration, velocity, and displacement pertaining to the $i$ th component, respectively; $q$ denotes the highest order of the polynomial function of the internal

force; $\alpha_{j i, k}$ denotes the coefficient of the polynomial function. As the highest order $q$ is set as 1, Eq. (2.3.35) is reduced to a linear formulation:
$$\sum_{i=1}^{n} m_{j i} \ddot{x}{i}(t)+\sum{i=1}^{n} \alpha_{j i, 0} \dot{x}{i}(t)+\sum{i=1}^{n} \alpha_{j i, 1} x_{i}(t)=F_{j}(\Theta, t)$$
where $\alpha_{j i, 0}, \alpha_{j i, 1}$ denote the coefficients relevant to damping force and the restoring force, respectively.

## 统计代写|随机控制代写Stochastic Control代考|Modal Superposition Method

X(吨)=∑j=1q1米¯j∫0吨Hj(吨−τ)φjφj吨F(θ,τ)dτ

μX(吨)=和[X(吨)]=∑j=1q1米¯j∫0吨Hj(吨−τ)φjφj吨μF(τ)dτ

## 统计代写|随机控制代写Stochastic Control代考|Nonlinear Random Vibration

∑一世=1n米j一世X¨一世(吨)+∑一世=1n∑ķ=0q一种j一世,ķX˙一世q−ķ(吨)X一世ķ(吨)=Fj(θ,吨)

∑一世=1n米j一世X¨一世(吨)+∑一世=1n一种j一世,0X˙一世(吨)+∑一世=1n一种j一世,1X一世(吨)=Fj(θ,吨)

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## MATLAB代写

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