This question has FIVE parts: $($a$),($b$),($c$),($d$),$ and $($e$).$

$(5)$ Question is missing tests or variants.

The system shown in the figure below is composed of three discs, each with mass $m=8\mathrm{.}5$

kg $,$ radius $r=2\mathrm{.}0m$ and moment of inertia $I=14\mathrm{.}39kg*m^2$ about their respective centres. The discs rotate

about their centres, with their angular position defined by the coordinates ${}_1,{}_2$ and ${}_3,$ respectively. All

rotations are considered positive when CCW$.$

The position of the centre of mass of each disc differs:

The two discs on the top have their centres of mass located in their centres $C_1$ and $C_2,$

respectively

The disc on the bottom$-$right has a centre of mass $G_3$ located at a distance $d=0\mathrm{.}6m$ from its centre $C_3$

The disc on the left is connected to the ground by a flexible cable, which behaves as a spring of stiffness $k=$

$111\frac{N}{m}.$ A second flexible cable of stiffness $k=111\frac{N}{m}$ connects the disc on the left with the top$-$right

disc. The two discs on the right are connected to each other by an inextensible cable.

An external harmonic moment $M(t)=M_0*cos(t)$ is applied to the top$-$right disc. The moment has an

amplitude $M_0=0\mathrm{.}84N*m$ and a frequency $=0\mathrm{.}22ra\frac{d}{s}.$

When ${}_1={}_2={}_3=0,$ all springs are unstretched and the centre of mass $G_3$ of the bottom$-$right disc is

horizontally aligned to the disc centre $C_3($as in figure$).$

a$)$ In your PDF solutions, show that the nonlinear equations of motion of the system in ${}_1$ and ${}_3$

are the following:

$14\mathrm{.}39*{}_1^{}+888*{}_1-444*{}_3=0$

$28\mathrm{.}78*{}_3^{}-444*{}_1+444*{}_3+50\mathrm{.}03*cos({}_3)=M(t)$

b$)$ In your PDF solutions and starting from the equations of motion given in part $($a$),$ show that the position

${}_1=-0\mathrm{.}11$ rad and ${}_3=-0\mathrm{.}22$ rad is a stable equilibrium position $($you need to show that the given

position is an equilibrium position and also that it is stable$)$

c$)$ In your PDF solutions, linearise the equations of motion given in part $($a$)$ around the equilibrium position

given in part $($b$).$ Then find and report here the numerical values of the Mass and Stiffness matrices

d$)$ Find the steady state amplitude of vibration for ${}_1$ and ${}_3$ in degrees.

e$)$ In your PDF solutions, draw the particular solution $($steady state solution$){}_{1,p}(t)$ and ${}_{3,p}(t)$ as

a function of time and annotate your diagrams with the key quantities.