Sparky,
In this day and age, any such evaluation would probably be done using finite element analysis. There are any number of these general analysis programs available, but they would require a degree of expertise and experience to use productively. Most likely, a 3-d CAD model of the crank and attachments (flywheels/dampers etc.) would be loaded into the program, automatically meshed, and then analyzed. Alternatively, with the sophisticated instrumentation available today, one could probably detect the oscillations in a running engine, although it might be a trick to sort that out of the various torsional loads being imparted by the rods and pistons.
Rather than a full 3-d model, a simplified version using discrete approximated masses, inertias, and stiffnesses could also be done with FEA, or for that matter, by hand. However, estimating the torsional stiffnesses of the various sections of something as convoluted as a crankshaft would be somewhat challenging, and in the end the results would be only as good as those estimates. (However, this is how it must have been done prior to the advent of FEA.)
Another possibility is to do it experimentally, if all you want to know is the fundamental frequency of vibration--higher modes would be more difficult to determine. This would probably require some electronic monitoring (accelerometer(?)) to discern the frequency adequately. In effect, one would be ringing a bell and monitoring the tone.
Unless the engine is being bodily rotated, as opposed to remaining in essentially the same orientation, gyroscopic forces would not enter into it. I suspect you meant the inertias of the flywheels etc. In that case, the sizes and locations of the added inertias would most certainly affect the behavior. A big flywheel at one end of the crank would effectively “ground” that end, and the crank would oscillate at a certain frequency with the free end showing the greatest deflection. Sort of a rotary version of a diving board. If inertia is then added to the front end of the crank, the zero oscillation “node”, which was at the flywheel, will move forward to some location in the crank, and the front and back portions of the assembly will oscillate in opposite directions about that nodal location. And this frequency would be lower than the original. Like holding either end of a dumbbell in each hand and alternately torqueing it about its axis.
So, by apportioning the inertias at either end, one can control where the node is and what the fundamental frequency is. One doesn’t want the natural frequency to be near the operating frequency, although that may be difficult to achieve. Enter.....dampers... to keep the resonance from getting out of hand.
