Discussion: The above is an animation of the particular solution for this problem using a frequency of excitation that is different from that given in the problem statement. For this frequency, is this excitation frequency greater or less than the natural frequency?

Deriving the EOM: the four-step plan

1. FBD: Since the motion of the cart is prescribed in time, we need to focus on the analysis of the disk. Draw a free body diagram of the disk. Be reminded that there will be a friction force on the disk at the contact point C that needs to be included in the FBD.
2. Newton-Euler equations: Note that the no-slip point C on the disk has both vertical and horizontal components of acceleration. Because of this, you CANNOT use sum(M_C) = I_C*theta_ddot, where theta is the angle of rotation for the disk. You are recommended to use both the summation of forces in the x-direction, and the moment equation about point O. Using these two equations together, you can eliminate the friction force from the dynamical equations.
3. Kinematics: Put some careful thought into this. Note that C is NOT the IC for the disk. As a result the angular speed of the disk is not proportional to x_dot, rather it is proportional to the difference between x_dot and x_B_dot.
4. EOM: Combine your kinematics equations with the equations from Step 2 to arrive at the single differential EOM for the system.

Any questions?

Any questions?

Discussion and hints:

Shown above are animations of the response of this system from simulations performed with the exciting frequency omega being 0.5*omega_n and 1.5*omega_n. Note that for 0.5*omega_n the response is in-phase with the base motion x_B(t), whereas for 1.5*omega_n  the response is 180° out-of-phase with the base motion. This phase difference should be apparent from both the visualization of the motion, as well as from the plots provided for x(t) and x_B(t). Can you observe this difference?

Derivation of the EOM: the four-step plan

1. FBD: Define a coordinate x that represents the displacement of the block as measured from its position when the springs are unstretched.  Draw a free body diagram (FBD) of the block. In doing so, take care to get the directions of the spring forces correct, and that the force in the spring on the left depends on the relative motion between B and the block. Also, draw the FBDs of the disks. Define some rotation coordinates, one for each disk.
2. Newton/Euler: It is recommended that you use Euler's equations for the disks, choosing their centers as the reference points. Use Newton's second law for the block.
3. Kinematics: You will need to relate x_ddot to the angular accelerations of the disks. Be sure to get the signs correct on these as you look back over your definitions of rotation coordinates in Step 1.
4. EOM: Combine the kinetics equation from Step 2 with the kinematics from Step 3 to arrive at the single differential equation of motion for the system in terms of the x coordinate.

Any questions?

IMPORTANT ADDITION/CORRECTION TO THE PROBLEM STATEMENT:

• The applied force is  F(t) = F_0*sin(omega*t).
• This is a mismatch of the masses specified for the blocks between the problem statement and the figure. Please use the mass values given in the figure (mass of A = 2m and the mass of B = m).

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