Yooper Tales: Transfer matrices and torsional vibes continued...

One of the engineering things I really enjoy is engine and drive train integration and synthesis whether that's considering vehicle mobility, tractive effort, efficiency, hybrid drives or just good old gear box design! Some of the more interesting challenges power train and drive train engineers face these days is what to do with a big rotor (called an electric motor)... But I digress. Here's sort of a intro into transfer matrices and how they work as applied to torsional vibration modeling.

So where did this idea come from? The oldest reference I have to solving this class of problem with matrix methods is from a 1942 Applied Physics paper written by one L.A. Pipes. A paper titled: "The Matrix Theory of Torsional Oscillations." Targoff looked at it again in 1947 and included bending with torsion which I thought was pretty cool. In practice however it's hard to get good uncoupling of torsion from bending because a lot of the structure your amidst has its own modal characteristics. I have used methods to de-couple these... it can be as simple as doing an fft on a shaft mounted stain gage to capture run-up, or critical mode frequencies. If you design something that has the ability to run through bending and torsional modes concomitantly, be mindful that barring certain operational speeds may be prudent... gosh I like this stuff!

So it's origins as far as I know are in WW2 era engineering efforts. I would also add here that W. Ker Wilson, one of my heroes... was very active in this area around the same time. His motivation was learning how to keep from breaking crankshafts and or busting gearboxes in radial engines in the aircraft of the day and in marine propulsion systems using higher power density diesels. The forcing functions of the new engines made excitation at various speed ranges something that could cause amplitude magnification if you dwelled on a certain speed. Dwelling on a resonance is usually a very bad thing unless your playing a viola... in which case it's a pretty amazing thing.

So where to begin. As with any dynamical system you will need to descritize the assembly into discreet parts. There's some value to experience in this. Deciding how equivalent spring rates are calculated from classical methods or by high fidelity FEA really fit into the real world system often involves some judgement. I'll leave this discussion for another time, but I do want to mention the mathematical solution you come up with is dependent on reasonable approximations for inertia (perhaps entrained inertia) and stiffness which could include part of a spline... and it's after all an approximation. But enough of that. Here's the deal!

Let us take the system in my earlier post: This is a 5 inertia, 4 spring system. Think of it as a 4 cylinder engine with generator bolted to the flywheel. It could be lots of things, but this I think is easy to visualize.

The inertia components are in units of [lb-in-s**2] and the torsional springs are in units of [lb-in/radian] They can be in N-m-s**2 or a dozen other inertial units as long as the associated spring rate units are correct.

In this little engine we have the following:

I1 = 10     K1 = 1.5E6
I2 = 10     K2 = 1.5E6
I3 = 10     K3 = 1.5E6
I4 = 10     K4 = 2.0E6
I5 = 20

These physical values will eventually be incorporated into a column vector called the state vector [z] and transfer matrix [f]. The state vector has two components: the angle of twist and torque that exist at a discreet point. Call this initial point, 'i.' So [z]i = [f]*[z]i-1

A little math...