Determine equations of motion using newtonian and lagrange

Question

1. In the figure below a vehicle is modeled as a twoDdegree of freedom system to allow for bounce and pitch motion.  This model can be determined from the schematic shown below the figure. Determine:

a) equations of motion using Newtonian and Lagrange method 

b) equations in state space form,

c) The transfer function θ(s)/F(s), X(s)/F(s) d) and determine, using Simulink, the response to the engine shut off, which is modeled as an impulse moment with a magnitude of M(t) of 10³ Nm radius of gyration r²=0.64m²  m=4000kg  c1=c2=2000 Ns/m k1=k2=20000N/m  l₁=0.9m  l₂=1.4m.

2)  The DCDmotor driven rack and pinion gear system is shown in the Figure below. m is the mass of the rack, R is the radius of the pinion and T is the torque delivered to the motor by the electrical current shown below.  The parameters are the following:

La=0.001H, Ra=2.60, Ke=0.008 Vs/rad and Kt=0.008Nm/A  R=.007m, m=0.5 kg

a) Derive the equation of motion of the system

b) Choose the armature current i_a, the rack displacement x, motor rotation θ_m, pinion rotation θ_p and the rack velocity x as state variables and find the state space form of the system

c) Assuming zero initial conditions, find the transfer function X(s)/Va(s)

d) build a simulink block diagram using the differential equations obtained in a)

e) build a simscape model of the rack and pinion and find displacement x(t) when the voltage applied to the DC motor is a pulse function, v_a(t)=2V for 1