Consider that the velocity of the base has a jump from zero

Question

For a spring-mass system with base excitation, derive the following expression for the relative motion z(t) = x(t) - y(t), for an undamped system:

x(t) = 1/ω_n∫_o^t ysinω_n(t-τ)dτ

Now, assume that the base motion is specified by a velocity pulse of the base given by:

v(t) = (vo - vo(t/to)) = vo(U(t) - t/to)

where U(t) is the Unit Step Function

Consider that the velocity of the base has a jump from zero to v0 instantly at t=0.

Derive an expression for the acceleration for the base motion from the velocity equation above

Derive an expression for z(t) using equation and the acceleration term you have calculated for the base

If the peak amplitude of z occurs at a t < t1, derive an expression for the maximum amplitude z.