Problem $4$: Elastodynamics $(30$ Points$)$

The Newmark predictor$-$corrector algorithm for dynamics was presented in class:

Consider a $1-$D bar made of $2$ linear elements shown above. No loads or boundary conditions are applied to the bar. The initial conditions are $d_0={\left[\begin{array}{ccc}0& 0\mathrm{.}05& 0\mathrm{.}1\end{array}\right]}^T$ for the displacements and $v_0={\left[\begin{array}{ccc}0& 0& 0\end{array}\right]}^T$ for the velocities. The assembled stiffness and consistent mass matrix are: $K=\left[\begin{array}{ccc}1& -1& 0\\ -1& 2& -1\\ 0& -1& 1\end{array}\right]$

$M_C=\frac{1}{6}\left[\begin{array}{ccc}2& 1& 0\\ 1& 4& 1\\ 0& 1& 2\end{array}\right]$

First, compute the lumped mass matrix $M_L.$ Then, for each of the following cases, compute the initial accelerations $a_0$ and perform the required number of time steps using the Newmark algorithm. Report the displacements, velocities, and accelerations in tabular form $($each node and each time step$).$

$($a$)$ Compute the lumped mass matrix $M_L$

$($b$)$ Use $t=0\mathrm{.}5,=\frac{1}{4},=\frac{1}{2},$ and the consistent mass matrix, perform $6$ time steps

$($c$)$ Use $t=0\mathrm{.}5,=0,=\frac{1}{2},$ and the lumped mass matrix, perform $6$ time steps

$($d$)$ Use $t=0\mathrm{.}25,=0,=\frac{1}{2},$ and the lumped mass matrix, perform $12$ time steps Briefly discuss your results. Since no loads or BCs are applied, the result should look like oscillatory free vibration. Note that the solutions are NOT expected to be identical. Also, why is the lumped mass matrix more appropriate for the analysis with $=0,=\frac{1}{2}?$