Use the finite elements method for the hanging cable with a lumped mass shown in the figure below to approximate its response $y(x,t)$ to the initial deflection given as $y(x,0)=y_0(x)=0\mathrm{.}25x(x-0\mathrm{.}6).$ Note that the tension in the cable is $T(x)=g(L-x)+mg.$ Assume that $y^{}(x,0)=0,L=1m,=1k\frac{g}{m},m=2kg,g=9\mathrm{.}81\frac{m}{s^2}.$

Choose one of the following two options:

Project Without Bonus

Compute $y(x,t)$ for $n=2$ and plot $y(\frac{L}{2},t)$ and $y(L,t)$ vs$.$ time for $10$ seconds.

Project With Bonus

Compute $y(x,t)$ for each of $n=2,3,4,5,$ and for each $n$ plot $y(L,t)$ vs$.$ time for $10$ seconds. Plot also ${}_1($lowest frequency$)$ vs$.n$ to show the convergence of the solution.

get solution by eigen values problems