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DISCUSSION and HINTS
Recall the following four-step plan outline in the lecture book and discussed in lecture:
Step 1: FBDs
Draw individual free body diagrams (FBDs) of the two disks and block A. Note that the tension forces in the different sections of the cable are NOT equal. It is important to temporarily define coordinates that describe the rotations of the two disks. Let’s call these θC and θO, and define these to be positive in the clockwise sense.
Step 2: Kinetics (Newton/Euler)
Using your FBDs from above, write down the Euler equations for the two disks, and the Newton equation for the block. Combine these three equations into a single equation through the elimination of the tension forces for the equations.
Step 3: Kinematics
You need to relate the angular acceleration of the disks and the acceleration of the block. How is this done? What are the results?
Step 4: EOM
From your equations in Steps 2 and 3, derive the equation of motion (EOM) of the system in terms of x.
Once you have determined the EOM for the system, identify the natural frequency from the EOM. Note that the static deformation of the system is found by setting x_ddot = 0 in the EOM.