example2的说明.tex

动力学计算程序

% Converted from Microsoft Word to LaTeX
% by Chikrii Softlab Word2TeX converter (version 3.0)
% Copyright (C) 1999-2003 Chikrii Softlab. All rights reserved.
% http://www.chikrii.com
% mailto: [e-mail]
% License: CSL#001781

\documentclass{article}
\usepackage{latexsym}
\usepackage{graphicx}

\begin{document}
example2：具有三个自由度的``弹簧－质量''系统，如图4{\-}8所示，取$k=1$、$m=1$，系统的初始条件为：
\begin{equation}
\label{eq1}
\boldsymbol{u}_0 =\left\{ {{\begin{array}{*{20}c}
 2 \hfill \\
 2 \hfill \\
 0 \hfill \\
\end{array} }} \right\},\quad \boldsymbol{v}_0 =\left\{ 
{{\begin{array}{*{20}c}
 0 \hfill \\
 0 \hfill \\
 0 \hfill \\
\end{array} }} \right\}
\end{equation}
求系统的响应。

\begin{figure}[htbp]
\centerline{\includegraphics[width=2.89in,height=1.14in]{example2的说明1.eps}}
\label{fig1}
\end{figure}

\begin{center}
图4{\-}8 三自由度``弹簧－质量''系统
\end{center}

该系统的刚度阵和质量阵为：
\begin{equation}
\label{eq2}
\boldsymbol{K}=\left[ {{\begin{array}{*{20}c}
 {3k} \hfill & {-k} \hfill & 0 \hfill \\
 {-k} \hfill & {2k} \hfill & {-k} \hfill \\
 0 \hfill & {-k} \hfill & {3k} \hfill \\
\end{array} }} \right]
\end{equation}
\begin{equation}
\label{eq3}
\boldsymbol{M}=\left[ {{\begin{array}{*{20}c}
 m \hfill & 0 \hfill & 0 \hfill \\
 0 \hfill & m \hfill & 0 \hfill \\
 0 \hfill & 0 \hfill & m \hfill \\
\end{array} }} \right]
\end{equation}
精确解为：
\begin{equation}
\label{eq4}
\boldsymbol{u}(t)=\left\{ {{\begin{array}{*{20}c}
 {u_1 (t)} \hfill \\
 {u_2 (t)} \hfill \\
 {u_3 (t)} \hfill \\
\end{array} }} \right\}=\left\{ {{\begin{array}{*{20}c}
 {\cos (\omega _1 t)+\cos (\omega _2 t)} \hfill \\
 {2\cos (\omega _1 t)} \hfill \\
 {\cos (\omega _1 t)-\cos (\omega _2 t)} \hfill \\
\end{array} }} \right\}
\end{equation}
其中
\begin{equation}
\label{eq5}
\omega _1 =\sqrt {k \mathord{\left/ {\vphantom {k m}} \right. 
\kern-\nulldelimiterspace} m} ,\quad \omega _2 =\sqrt {{3k} \mathord{\left/ 
{\vphantom {{3k} m}} \right. \kern-\nulldelimiterspace} m} 
\end{equation}
\end{document}