Discussion and hints:

The derivation of the dynamical equation of motion (EOM) for a system is a straight-forward application of what we have learned from Chapter 5 in using the Newton-Euler equations. The goal in deriving the EOM is to end up with a single differential equation in terms of a single dependent variable that describes the motion of the system. Here in this problem, we want our EOM to be in terms of x(t).

Recall the following four-step plan outline in the lecture book and discussed in lecture:

Step 1: FBDs
Draw an FBD of the disk. Define a rotational coordinate θ for the disk.

Step 2: Kinetics (Newton/Euler)
Take care in expressing the force on the disk due to the springs. The forces in those springs depend in magnitude on the relative displacements of the disk and the cart, with the direction of the spring force found by the thought process covered in lecture.

Step 3: Kinematics
Use the no-slip condition at C to relate x and θ.

Step 4: EOM
Combine your Newton/Euler equations and your kinematics equations to arrive at the single differential equation of motion for the system.

For this problem, you need to determine the particular solution for the EOM corresponding to the excitation.

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Any questions?