Gordon Arnaut wrote:
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These various excitation forces work to induce vibration in the drivetrain
system and its various pieces.

Restraining forces counteract the tendency toward vibration. They consist of
three factors: stiffness, mass and dampening.
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As mentioned, every object in the engine -- or every other object for that
matter -- has its own natural "resonant" frequency. That is the frequency at
which a particular object will vibrate if it is acted upon by a single
excitation of sufficient force to displace it from its resting state.

I'm going to have to disagree.

Stiffness, and mass are almost irrelevant to torsional vibration (and
any other sort).
Stiffness and mass change the resonant frequency.

I'd approach this from the 'Q' perspective. Q is 'quality factor', which
basically means how good something is at resonating.
A Q of 100 means that it will oscillate 100 times at its resonant
frequency before the oscilation decays to 27% of its original level (not
completely sure of the 27%).

Consider a mass on the end of a stiff wire hanging down from a fixed
point.
Twist and hold it.
There is now energy stored in the wire, in the form of deformation. If
you try to twist it too far, well, any structural element can only take so
much energy stored in it, and if you exceed that, it bends or breaks.

Let the weight go, and it'll spin clockwise, come to rest, then spin
back anticlockwise.
This is energy being traded from potential (stored in the wire) to
kinetic (movement) and back.
Let's call this amount of energy stored 'E'.

If the wire was a springy steel one, then the Q is probably quite high,
say 20.

In order to keep the mass spinning clockwise and anticlockwise, to the
same amount, you need to supply some energy.
Now, the higher the Q is, the less energy. E/Q, or E/20.
If you supply (at just the right times) 1 unit of torsional energy, the
system will build up so that it's storing 20 units in the vibration.

It's just like a swing - if it's got a heavy weight on it, but you push
it a little every time, it'll build up to a decent swing.

The other side to this is that as it's only storing 20 units of energy,
after a time, somewhere else that 1 unit is coming out, probably as heat,
maybe in another part of the system as vibration.

There are several ways to address this.

You can make sure that the resonant frequencies are never hit - which is
horribly hard in many cases, or make sure that the Q is very low, which
means low stored energy.

Adding mass, or stiffness does not alone change Q.

Taking the earlier posted example of the overheating clutch.

Let's say a 50Kw engine was pumping in 10KW of vibrational power, to a long
springy shaft, connected to a clutch.
Say the Q is 10, at the frequency that the engine is running at.

After a few moments, the vibrational power in the shaft is not 10Kw, but
100Kw, the vibrational torque stored in the shaft is 10 times higher than the
input torque, and the shaft is twisting back and forth much more than
expected.
But, the clutch is only rated for 70Kw maximum of torque, and starts to
slip back and forwards.
This damps the vibration in the shaft somewhat, but in the end, it ends
up with most of the 10Kw vibration output heating the clutch.

(I'm mixing power and energy, but I hope it's clear)

Now, you can fix this several ways.

Double the clutch size.
This will fix the immediate problem, but as the energy in the shaft now
is not being damped in the clutch, the vibration there will increase.
This may cause other things to break in the system, maybe the prop,
maybe the PSRU, maybe the pilot

Add a big sticker to the panel saying "Do not run for more than 10
seconds at between 2300 and 2400 RPM"
Maybe OK in some applications.

Make the shaft more or less stiff, or more or less massive.
This will change the resonant frequency, and may avoid the problem at
2340 RPM, but may add a new one at 3200.

Or, you add some sort of damping to the system, reducing the Q.

I need to go to bed, will add more tomorrow.