Chapter 6.4, Problem 13P

Consider the two-mass, three-spring system whose equations of motion are Eqs.(11) in the text. Let $m_1=m_2=1$ and $k_1=k_2=k_3=1$.

(a) Using the values of parameters given in the previous statement, find the eigenvalues and eigenvectors of the matrix $\text{A}$.

(b) Write down the general solution of the system.

(c) For each fundamental mode, draw graphs of $x_1$ versus $t$ and $x_2$ versus $t$ and describe the fundamental modes of viberation.

(d) Consider the initial conditions $\text{x}\left(0\right)={\left(\begin{array}{cccc}-1& 3& 0& 0\end{array}\right)}^T$. Evaluate the arbitrary constants in the general solution in part (b) and plot graphs of $x_1$ versus $t$ and $x_2$ versus $t$.

(e) Consider other initial conditions of your own choice, and plot graphs of $x_1$ versus $t$ and $x_2$ versus $t$ in each case.