Q$4(8$ pts$)$

$\backslash (\backslash$Omega $\backslash )$ Protocol is a manufacturing company that owns a drilling machine $($shown in figure $1-$a$)$ with a mass of $\backslash ($ m$\_\{1\}=500\backslash$mathrm$\{$~kg$\}\backslash ).$ The drilling machine is mounted on a floor that vibrates due to the operation of another machine in the vicinity. This causes the drilling machine to exhibit an unwanted amplitude of vibration, which needs to be mitigated.

As an engineer in the company, you did some initial investigations and found out the following:

$-$ The motion of the floor has a sinusoidal behavior with an amplitude of $0\mathrm{.}1$ mm and a frequency of $\backslash (1\mathrm{.}414\backslash$mathrm$\{$rad$\}/\backslash$mathrm$\{$s$\}.\backslash$mathrm$\{$L$\}\backslash )$

$-$ The drilling machine supports can be assumed as a spring$-$damper system with stiffness $\backslash ($ k$\_\{1\}=\backslash )\backslash (1000\backslash$mathrm$\{$~N$\}/\backslash$mathrm$\{$m$\}\backslash )$ and damping coefficient $\backslash ($ c$\_\{1\}=1\backslash$mathrm$\{$~N$\}.\backslash$mathrm$\{$s$\}/\backslash$mathrm$\{$m$\}\backslash ).$

An idea of attaching a small device with a mass of $\backslash ($ m$\_\{2\}=10\backslash$mathrm$\{$~kg$\}\backslash ),$ stiffness of $\backslash ($ k$\_\{2\}=10\backslash$mathrm$\{$~N$\}/\backslash$mathrm$\{$m$\}\backslash )$ and damping coefficient of $\backslash ($ c$\_\{2\}=0\mathrm{.}02\backslash$mathrm$\{$~N$\}.\backslash$mathrm$\{$s$\}/\backslash$mathrm$\{$m$\}\backslash )$ as shown in figure $1-$b was proposed to reduce the response amplitude of the drilling machine. Therefore, you are required to perform the following:

a$)$ Determine the amplitude of the machine before attaching the device and explain why the floor vibration is causing this high amplitude to the drilling machine.

b$)$ Evaluate the effect of attaching the proposed device on the amplitude of the drilling machine.

c$)$ Based on your results, was this a good idea or not? Explain.

Figure $1$: The drilling machine $($a$)$ before adding the proposed device $($b$)$ after adding the proposed, this is the full question no more information, please solve with the given information