$(30$ points$)$

The damping ratio and the natural frequency of an accelerometer is $0\mathrm{.}01$ and $50,000Hz,$ respectively.

$($a$)$ Determine the maximum error over the frequency range from $0$ to $7,000Hz.$ Do not assume zero damping.

$($b$)$ The accelerometer adopted a new MEMS $($micro$-$electromechanical system$)$ mechanism such that the damping ratio is now $0\mathrm{.}7.$ What is the maximum error for this new system assuming the same natural frequency and the frequency range?

$($c$)$ By sketching an amplitude response plot, explain the result in part $($b$).$