Problem $1(30pts).$ Let $x_1(t)$ and $x_2(t)$ be the positions of two masses $m_1$ and $m_2,$

respectively. Non$-$linear springs are attached to these masses as shown below.

The elastic energy of the system is $E(x_1,x_2)=E_1+E_2,$ where

$E_1(x_1)=k{a}^{2}cosh(\frac{x_1}{a}),E_2(x_1,x_2)=k{a}^{2}cosh(\frac{x_1-x_2}{a})$

are the elastic energies of the two springs, respectively, and $k$ and a are constants.

$($a$)[5$pts$]$ Compute the force $f_1=-\frac{\text{d}\text{e}\text{l}\text{E}}{del{x}_1}$ exerted on mass $m_1,$ and the force $f_2=-\frac{\text{d}\text{e}\text{l}\text{E}}{x_2}$

exerted on mass $m_2.$

$($b$)[5$pts$]$ Write the system of non$-$linear second$-$order ODEs governing the motion of the

two masses $m_1$ and $m_2.$

$($c$)[7$pts$]$ Let $m_1=m_2=m,$ and consider small perturbations of $x_1$ and $x_2$ about their

equilibrium positions. Show that your answer in $($b$)$ reduces to the following system of

linear second$-$order ODEs

$x^{}=-Ax$

where

$x=\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]$

and $A$ is a constant matrix.

$($d$)[5$pts$]$ In order to determine the natural frequencies of vibrations of the system we let

$x=c{e}^{it}.$ Show that with this substitution $(2)$ turns into an eigenvalue problem.

$($e$)8pts$ Solve the characteristic equation of the eigenvalue problem to find the natural

frequencies of vibration. For each eigenvalue sketch the corresponding vibration mode

$($eigenvector$).$

$($f$)5pts$ Note that the matrix $A$ is symmetric positive definite $($SPD$).$ List appropriate

numerical methods to solve the following problems:

find the maximum eigenvalue$/$eigenvector of a SPD matrix;

find the minimum eigenvalue$/$eigenvector of a SPD matrix;

find all eigenvalues$/$eigenvectors of a SPD matrix.