% 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: info@chikrii.com% License: CSL#001781\documentclass{article}\usepackage{latexsym}\usepackage{graphicx}\begin{document}example2：具有三個自由度的``彈簧－質量''系統，如圖4{\-}8所示，取$k=1$、$m=1$，系統的初始條件為：\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}求系統的響應。\begin{figure}[htbp]\centerline{\includegraphics[width=2.89in,height=1.14in]{example2的說明1.eps}}\label{fig1}\end{figure}\begin{center}圖4{\-}8 三自由度``彈簧－質量''系統\end{center}該系統的剛度陣和質量陣為：\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}精確解為：\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}