Problem $1(25$ pts$).$ Consider the system of three identical masses and four identical

springs below. The coordinates of these masses are $x_1,x_2,$ and $x_3,$ respectively. All springs

have stiffness constant $k.$

$($a$)[5$pts$]$ The equation of motion of the three masses can be written in matrix form as

$m{x}^{}=-Kx$

where

$x=\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]$

Find the stiffness matrix $K.$ What property of $K$ can you observe?

$($b$)[3$pts$]$ The elastic energy of the system is $E=\frac{1}{2}{x}^{T}Kx.$ Energy is a positive number

if $||x||>0,$ and $E=0$ if $||x||=0.$ This implies what other property of $K?$ As a

consequence, what properties of the eigenvalues and eigenvectors of $K$ are expected?

$($c$)7pts$ Assume harmonic motion and derive the eigenvalue problem from which the

natural frequencies and modes of vibration can be determined.

$($d$)[5$pts$]$ The eigenvalue$/$eigenvector pairs are

${}_1=2-\sqrt[\phantom{2}]{2},q_1=[1,\sqrt[\phantom{2}]{2},1]$

${}_2=2,q_2=[-1,0,1]$

${}_3=2+\sqrt[\phantom{2}]{2},q_3=[-1,\sqrt[\phantom{2}]{2},-1]$

Find the corresponding vibration frequencies and sketch the modes of vibration.

$($e$)[5$pts$]$ Sketch how the matrix $K$ looks like if instead of three masses and four springs

we have a system of $N$ masses and $N+1$ springs. How many eigenvalue$/$eigenvector

pairs can be found in this case? For large $N,$ what numerical method would you use

to find all eigenvalue$/$eigenvector pairs?