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DISCUSSION and HINTS

The animation below shows the vibrational motion of this system. Take note of the relationship between the angular motion of the two gears.  Note that neither the magnitudes nor the directions of these angular accelerations are the same. We will focus on this in Step 3 of deriving the equation of motion.

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. Be sure to include the equal-and-opposite contact force between the gears on each gear.

Step 2: Kinetics (Newton/Euler)
Write down the Euler equations for the two gears. Combine these two equations through the elimination of the gear-to-gear contact force.

Step 3: Kinematics
You need to relate the angular accelerations of the two gears. How is this done? What are the results? Also, how do you relate the stretch/compression in the two springs in terms of θ?

Step 4: EOM
From your equations in Steps 2 and 3, derive the equation of motion (EOM) of the system in terms of θ.