Question $1[26$ marks$]$

The drilling device used by a large oil exploration company essentially consists of a DC motor driving a $1$ km long drilling shaft rotating in a sleeve. A drill bit is fitted to the far end of the shaft. The purpose of the non$-$rotating sleeve is to protect the shaft from damage that might be caused by the sediment through which the drill has to penetrate in order to extract crude oil. A graphical depiction of the drilling system is shown in Figure $1.$

For simplicity the inertia of the shaft and its damping coefficient relative to the sleeve are to be discarded. The remaining parameters of interest are:

$e_a(t)=$ applied armature voltage

$i_a(t)=$ armature current

$e_b(t)=$ back emf

$R_a,=$ armature resistance

$L_a,=$ armature inductance

$K_b,=$ motor back emf constant

$K_i,=$ motor torque constant

$T_m(t)=$ motor torque generated

$I_m,=$ moment of inertia of the rotor

${}_m(t)=$ angular displacement of the rotorFigure $1$: Graphical depiction of the drilling system.a$.$ Write all the equations describing the dynamics of the drilling system.b$.$ Using these equations, draw a block diagram of the drilling system.c$.$ Derive the transfer function G$($s$)=($Omega$\_($b$)($s$))/($E$\_($a$)($s$)).$