Question $1)$

In the given figure, the system is in the static equilibrium position. The mass $m$ is connected to

a fixed ground with the spring $k,$ and connected to a table that moves $y(t)=$Ysin$t$ with the

damping element $c.$

a$)$ Obtain the differential equation of the system using the Lagrange method.

b$)$ Obtain the differential equation of the system using Newton's $2^{nd}$ Law.

b$)$ Check whether the system is at resonance or not?

c$)$ Obtain the transient response of the system?

d$)$ Obtain the steady state response of the system?

e$)$ Write the general equation of motion for system and $x^{}(0)=0\frac{m}{sec}$

d$)$ Plot the system behavior in Matlab and interpret the graph you get.