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.
![408_vehicle.png](https://secure.tutorsglobe.com/CMSImages/408_vehicle.png)
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 ia, the rack displacement x, motor rotation θm, pinion rotation θp 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, va(t)=2V for 1
![1028_motor.png](https://secure.tutorsglobe.com/CMSImages/1028_motor.png)