Problem $31$

A base isolation system is installed to minimize vibration of a floor slab in a laboratory. The isolation system consists of a spring, $k,$ and a damper, $c.$ The mass of the slab is $m.$

c$.$ Formulate the equation of motion in terms of the relative displacement, $w(t),$ for the mass, spring, and damper system subjected to harmonic vertical acceleration as shown in the figure.

d$.$ Consider the following structural parameters and ground excitation parameters: $m=5000kg,k=500k\frac{N}{m},=50ra\frac{d}{sec},$ and $u_{go}^{}=0\mathrm{.}3g.$ Find the range of damping which will result in a steady$-$state maximum absolute acceleration $u^{}{?}^a(t)$ of no more than $0\mathrm{.}1g.$

e$.$ If the ratio $\frac{}{{}_n}$ varies from $0\mathrm{.}1$ to $10,$ what are the advantages and$/$or disadvantages using a large damping in terms of vibration control? Use Figure $2$ to answer this question.