Question $1[35$ points$]$

A machine component is modeled as a pendulum connected to a spring and driven by a motor that applies a sinusoidal moment, $.$ Pi$.$t$)$ Nm $,$ about its axis of rotation $($as shown in Figure $1).$ The pendulum has a mass of $2$ kg and a length of $0\mathrm{.}5$ m $,$ with a spring stiffness of $20\frac{N}{m}$ and a damping coefficient of $20N*\frac{s}{m}.$ The rod of the pendulum is assumed to be massless. The system starts with an initial angular displacement of $0\mathrm{.}175$ rad $($approximately $10)$ from the vertical, with no initial angular velocity.

$1\mathrm{.}1$ Derive the equation of motion and find the natural frequency of the system

$1\mathrm{.}2$ Plot the total response and find the maximum angular displacement of the pendulum $(05)$

$1\mathrm{.}3$ Find the angular frequency of the applied moment that would lead to steady$-$state vibrations of the maximum amplitude.

$(10)$

Figure $1$: Spring Pendulum.

Please assist urgent.