Problem #1 (40%) Consider the simplified schematic shown below that represents the movement of a car.

The input to the system is the engine torque r that is transmitted to the wheel m, . The uniform wheel is connected to the car body mass m2 through a spring with a stiffness of K, which represents the flexibility of the support structure of the car body. The damping force B12 represents the wind dragging force acting on the car body.

As shown, the wheel has a mass of inp a moment of inertia I with respect to the wheel center, and a radius of R. During the normal driving condition, the wheel rolls without slipping on the ground. The reference positions for displacements are defined when the spring does not have any deflection.

To be able to design a cruise controller for the car, it is necessary to obtain the input/output model for the system with the system output being the velocity of the car body (i.e., y =1.2 ). To solve the problem, follow the following procedure:

a. Draw FBDs for each basic element and find the equations of motion for the system.

b. Obtain the transfer function of the system by using Laplace transforms. The input to the system is the engine torque and the output is the car velocity.

c. Write down input/output model based on the transfer function obtained in part c.