Part $1[40\%]$: Accuracy and stability analysis of Newmark's method

For a given undamped free vibration dynamic system $u^{}+{}^{2}u=0$ with a zero initial

displacement and a nonzero initial velocity $u^{}(0)=u_0^{}($choose any values for $$ and $u_0^{}),$

assess the accuracy of the algorithm by plotting the error of the computed period for a

wide range of different time steps.

The stability of the algorithm is accessed by following the procedure: $1)$ choose a time

step; $2)$ record the maximum kinetic energy of the system over a time integration interval

of $0,100T,$ where $T$ is the natural period of the system; and $3)$ if a significant increase of

the energy is observed, it would indicate that the algorithm is unstable for the

corresponding and larger time steps. The procedure should be repeated for a large range

of different time steps. MATLAB program for this question