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DISCUSSION and HINTS
The particular solution for the equation of motion for this system is shown below for two values of the frequency of excitation, ω. As we have seen in lecture, and as shown below, when the excitation frequency is less than the natural frequency,ωn, the response is in phase with the excitation, and when the frequency is greater than ωn, the response is 180° out of phase with the excitation. Can you see this in the animations below?
Recall the following four-step plan outline in the lecture book and discussed in lecture:
Step 1: FBDs
Draw a free body diagram (FBD) of the disk. It is important to temporarily define a translation coordinate that describes the position of the center of the disk, O. Let's call that variable x, and have it defined as being positive to the right.
Step 2: Kinetics (Newton/Euler)
Use Euler's equation about the no-slip contact point C.
Step 3: Kinematics
Relate x_ddot to θ_ddot through kinematics.
Step 4: EOM
Combine your equations from Steps 2 and 3 to arrive at your EOM in terms of θ.
Once you have determined the EOM for the system, identify the natural frequency from the EOM. From the EOM we know that the particular solution of the EOM is given by: xP(t) = A*sin(ωt)+ B*sin(ωt). How do you find the forced response coefficients A and B?
Does your result here agree with the expected relationship between excitation frequency and response with regard to the "phase" of the solution?