Show me the steps to solve, Note: This is a practice question university gave can't be wrong doesnt need more information, send to other experts. Question

The system in Figure Q$3$ is composed of two discs connected by a flexible cable with linear stiffness coefficient

$k=1,000\frac{N}{m}.$ The two discs rotate about their centres $O_1$ and $O_2,$ which are pinned to the ground. Their

rotations are defined by the angles ${}_1$ and ${}_2,$ respectively. These two angles are considered positive counter$-$

clockwise and are equal to zero in the instant shown in Figure Q$3.$

Figure Q$3$

The first disc has mass $m_1=10kg,$ radius $r_1=0\mathrm{.}5m,$ moment of inertia $I_{O1}=1\mathrm{.}65kg*m^2$ with respect to its

centre $O_1.$ The disc's center of mass $G_1$ is located at a distance $d_1=0\mathrm{.}2m$ from $O_1$ and it is vertically aligned

below $O_1$ when ${}_1=0($as in Figure Q$3).$

The second disc has mass $m_2=1kg,$ radius $r_2=0\mathrm{.}2m,$ moment of inertia $I_{O2}=0\mathrm{.}03kg*m^2$ with respect to

its centre $O_2.$ The disc's center of mass $G_2$ is located at a distance $d_2=0\mathrm{.}1m$ from $O_2$ and it is horizontally

aligned to the left of $O_2$ when ${}_2=0($as in Figure Q$3).$

The spring is unstretched when ${}_1=0$ and ${}_2=0($as in Figure Q$3).$ An external moment $M=M_{0}cos(t)$ is

applied to the second disc, with $M_0=2N*m$ and $=6ra\frac{d}{s}.$

Answer all the following subquestions:

$($a$)$ Write the non$-$linear equations of motion of the system.

$($b$)$ Demonstrate that the position ${}_{1,eq}=-0\mathrm{.}1189$rad and ${}_{2,eq}=0\mathrm{.}3206$rad is a stable equilibrium

position. You have to show that this is both an equilibrium position and that it is stable.

$($c$)$ Derive the linearised equation of motion about the stable equilibrium position indicated in $($b$)$ and show

the numerical values of the mass and stiffness matrices.

$($d$)$ Find the steady state angular vibration amplitudes for ${}_1$ and ${}_2.$

$($e$)$ Qualitatively draw how the system vibrates at steady state $($show this by drawing the discs in a vibrating

configuration$).$