Discussion and hints
Shown below is the particular (steady-state) solution of the EOM for this system with the frequency of excitation less than the natural frequency: ω < ωn . Note that the response in blue is completely in phase with the base motion in red. This is expected based on what we have see in the lecture book and in lecture.
Now we look at the response with the frequency of excitation being greater than the natural frequency: ω > ωn . Note that the response in blue is completely 180° out of phase with the base motion in red. This is also expected based on what we have see in the lecture book and in lecture.
Recall the following four-step plan outline in the lecture book and discussed in lecture for deriving the EOM for the system:
Step 1: FBDs
Draw an FBD of the block. Take care in expressing the forces on the block due to the springs. For two of the springs, the forces depend in magnitude on the relative displacements of the block and the cart, with the direction of the spring force found by the thought process covered in lecture.
Step 2: Kinetics (Newton/Euler)
Write down the Newton equation for the block in the x-direction.
Step 3: Kinematics
Do you need any additional kinematics here?
Step 4: EOM
Your single differential equation of motion for the system was found in Step 2.
For this problem, you need to determine the particular solution for the EOM corresponding to the excitation.
Any questions?