At the suggestion of Paul J, I started doing research on vibration analysis today. There are several online tutorials regarding vibration analysis. I'm currently reading this one:
This paragraph taken from the "Natural Frequencies" section is somewhat noteworthy:
'The multitude of spring-mass-damper systems that make up a mechanical system are called "degrees of freedom", and the vibration energy put into a machine will distribute itself among the degrees of freedom in amounts depending on their natural frequencies and damping, and on the frequency of the energy source. For this reason, the vibration will not be uniformly distributed in the machine. For instance, in a machine driven by an electric motor, a major source of vibration energy is residual imbalance in the motor rotor. This will result in a measurable vibration at the motor bearings. But if the machine has a degree of freedom with a natural frequency close to the RPM of the rotor, its vibration level can be very high, even though it may be a long distance from the motor. It is important to be aware of this fact when evaluating the vibration of a machine -- the location of the maximum vibration level may not be close to the source of the vibration energy. Vibration energy frequently travels great distances along pipes, and can wreak havoc when it encounters a remote structure with a natural frequency near that of its source.'
Based on the paragraph above, I suspect that the AC motor is providing a small excitation around 5500 RPM and that is causing major vibrations out in the extremities of the transmission. The transmission mounts are original and cracked, so replacing those might help a bit.
One of my coworkers said he would let me borrow an accelerometer to attach to my oscilloscope. I'll probably take him up on the offer and see what I can find.