Question $1(25$ points$)$:

A common example of base motion is the single$-$degree$-$of$-$freedom model of an automobile driving over a road, indicated in the following figure. Assume that the suspension system provides an equivalent stiffness of $\backslash (2\mathrm{.}8\backslash$times $10^\{4\}\backslash$mathrm$\{$~N$\}/\backslash$mathrm$\{$m$\}\backslash ),$ damping of $\backslash (14\backslash$times $10^\{2\}\backslash$mathrm$\{$~N$\}\backslash$mathrm$\{$~s$\}/\backslash$mathrm$\{$m$\}\backslash ),$ and mass of $1100$ kg $.$

$(1)$ Determine the natural frequency and damping ratio

$(2)$ Find the relation between displacement transmission ratio $\backslash (\backslash$mathrm$\{$X$\}/\backslash$mathrm$\{$Y$\}\backslash )$ and the driving speed v $.$

$(3)$ Find the relation between force transmission ratio and the driving speed $\backslash ($ v $\backslash ).$

$(4)$ Plot the relations in $(2)$ and $(3)$