$5\mathrm{.}1$ In Section $5\mathrm{.}2$ we developed recurrence formulas for numerical solution of the equation of

motion of a linear SDF system based on linear interpolation of the forcing function $p(t)$

over each time step. Develop a similar procedure using a piecewise$-$constant representation

of the forcing function wherein the value of the force in the interval $t_i$ to $t_{i+1}$ is a constant

equal to tilde$(p{)}_i($Fig$.$ P$5\mathrm{.}1).$ Show that the recurrence formulas for the response of an undamped

system are:

$u_{i+1}=u_{i}cos({}_n{t}_i)+u_i^{}\frac{sin({}_{n}i)}{{}_n}+\frac{\text{t}\text{i}\text{l}\text{d}\text{e}(p{)}_i}{k}[1-cos({}_{n}i)]$

$u_{i+1}^{}=u_i[-{}_{n}sin({}_{n}i)]+u_i^{}cos({}_{n}i)+\frac{\text{t}\text{i}\text{l}\text{d}\text{e}(p{)}_i}{k}{}_{n}sin({}_n{t}_i)$

Specialize the recurrence formulas for the following definition of the piecewise$-$constant

force: tilde$(p{)}_i=\frac{p_i+p_{i+1}}{2}.$ Write the recurrence formulas in the following form:

$u_{i+1}=A{u}_i+B{u}_i^{}+C{p}_i+D{p}_{i+1}$

$u_{i+1}^{}=A^{\text{'}}{u}_i+B^{\text{'}}{u}_i^{}+C^{\text{'}}{p}_i+D^{\text{'}}{p}_{i+1}$

with equations for the constants $A,B,C,$dots,$D^{\text{'}}.$

Figure P$5\mathrm{.}1$