Another common direct step$-$by$-$step integration method is called Central Difference

Method wherein the acceleration and velocity terms are approximated by finite difference

quotients:

$x_t^{}=\frac{1}{t^2}(x_{t-t}-2x_t+x_{t+t})$

$x_t^{}=\frac{1}{2t}(-x_{t-t}+x_{t+t})$

By examining the equation of motion $M{x}^{}+x{x}^{}+Kx=f(t)$ at the time t with the

foregoing finite difference method.

$($a$)$ Develop a formula by which one can find the displacement $x_{t+t}$ in terms of

displacement response$/$loading at the two past time steps.

$($b$)$ Derive $x_{t-t}=x_0-t{x}_0^{}+\frac{t^2}{2}{x}_0^{}$ as an equation for the starting procedure at $t=0$

$($because of the $2-$step nature of the algorithm$)$ from $(1)$ and $(2)$ and the equation of

motion at $t=0$ under any prescribed initial conditions $x_0(0)$ and $x_0^{}(0).$

A soil layer is $80$ feet thick, has a unit weight of $100$ pcf $,$ and an average shear modulus

G of $0\mathrm{.}25{10}^{6}psf$ sitting on top of bedrock.

$($a$)$ Determine the parameters of a $1$ DOF lumped mass model for the soil stratum

$($distribute $50\%$ of the mass to the top and take the damping ratio equal $0\mathrm{.}2).$

$($b$)$ Estimate basic site period $($and its natural frequency$).$

$($c$)$ Determine the governing equation of motion for the displacement $x(t)$ relative to

bedrock which has a horizontal motion with acceleration $x_g^{}(t).$

$($d$)$ Determine and plot the real and imaginary parts as well as amplitude of $($i$)$ the

transfer function $C()$ which relates the Fourier transform tilde$(x)$ to that of the bedrock

acceleration tilde$(tilde(x){)}_g$ and $($ii$)$ the transfer function $H_{\text{a}\text{c}\text{c}}()$ which relates the absolute

acceleration $((tilde(tilde(x)))+$tilde$(x^{}{)}_g)$ at the soil surface to tilde$(x^{}{)}_g$ in the frequency $($or Fourier

transform$)$ domain.

$($e$)$ Suppose there is an earthquake which induces a ground acceleration at the base

level with a Fourier amplitude spectrum as shown, determine the corresponding

Fourier amplitude spectrum of the absolute ground surface acceleration of the soil

stratum. Plot the two spectra on the same graph and comment on the engineering

implication. Solve it step by step