Perhaps it would also be helpful for us non-engineers to put example frequency numbers to the principles.

For instance, with a 'soft' system, if we idle a 4 cyl 4 stroke engine at 1500 rpm, would lowest fundamental be 3000/minute / 60 = 50 Hz? Does that mean resonance should be somewhere below ~20 Hz (>1 octave below)?

Correct - here's the detail. A single cylinder four stroke engine makes one firing pulse every other revolution of the crank shaft. An even fire four cylinder makes two power pulses per rev. ff= n*c/2/60. At 1500rpm * 4/2 /60 = 50 Hz. That is also the firing frequency of a Wankel two rotor at the same rpm.

To be safely below that you generally need to be at least half an octave below. A full octave above is double the frequency, and full octave below is half the frequency. So you will need to be about a quarter of 50 hz lower, or about 37.5 Hz or lower. Ideal is about as many octaves below idle as it is octaves above cranking rpm. Cranking rpm is 150 for an engine of this size, so 5 Hz. Maybe you want you soft system frequency at more like 15-20 Hz.

For a 'stiff' system on the same engine, if it runs at 5000+ to 8000+ with the Yamaha sled engines, would the fundamental be somewhere above:

16000/minute / 60 = 266 Hz? Do we also need to consider 3rd order, 4th order, etc frequencies? If only the fundamental is of concern does the math show that resonance should be 266 * 2 * 2 = somewhere above 1100 Hz(>2 octaves above)?

I do not know the cylinder count of the engine you have in mind... 267 Hz is firing frequency at 8000 rpm of a four cylinder four stroke engine. The reason we usually talk in terms of having the first fundamental frequency 2-1/2 octaves higher is that four stroke engines have the pistons going up and down in the bores, giving the second largest vibe at twice firing frequency. This engine will have big vibratory input at firing frequency of 267 Hz and another that is 1/4 or so as big as firing at 533 Hz. All internal engine resonance in durable engines must be safely above both of these, or 667 Hz.

Making a stiff system downstream? It too must have its first frequency above 667 Hz.

Difference in spring rates for this soft vs this stiff system? f is proportional to square root of k/m. IF we can keep inertia the same with the stiff system (usually inertia will go up some too). So we want frequency to go up 667/20 = 33.3. So this system will have to be 1100 times stiffer torsionally to go from safely soft to safely stiff if it does not gain any weight in the process. Conversely, to go from a safely stiff system to a safely soft system means we would need to be 1/1100 as stiff... Hmmm. Fortunately, we know how to design springs made from everything from titanium alloys to rubber.

One other thing to talk about - The Wankel rotary engine is a bit unusual. I know that it has one firing pulse per rotation of the eccentric per rotor, so a two rotor has the same firing frequency vs rpm relationship as a four cylinder four stroke. But I do not know about its other significant orders. I have looked and not found it documented anywhere, and never had occasion to put a Rotec on one in my whole career... So maybe the stiff 7000 rpm Wankel made by Mazda needed 350 Hz (1.5 octaves) or maybe it needed 583 Hz. But I do not know...

Now, which method is harder/heavier/more expensive to achieve? My money is on a soft spring between the engine flywheel and the rest of the gears, shafts, bearings, etc.

If one or more cylinders' output fails while in flight, does either method offer any better protection, or would it even matter?

Well, that depends on how we lose the cylinder. If either ignition or fuel is interrupted, we still get a compression stroke before firing. Our 4 cylinder engine still has a 2nd order or rotation vibe on all four cylinders. But one of them is smaller than the other three, so now we also get a negative pulse at 1/2 order:

- One cylinder drops out at 8000 rpm has content at 67Hz. The soft system with resonance at 25 Hz will not resonate. And our stiff system at 667 Hz is cool too;
- If instead, we drop a valve on that cylinder, we lose the compression stroke as well as the power stroke, but we still have the two per rev and 1/2 per rev safely away;
- Now let's have a steady misfire at 3200 rpm. Bang, the 1/2 order input is right at 25 Hz, and we have a potentially destructive resonance;
- The soft system can still have resonance. The stiff system hardly knows a cylinder has dropped out.

Now let's look at the Mazda built Wankel. If we lose spark or fuel to one of two rotors, we too still have two per rev plus a one per rev. If we have a soft system, have a one octave margin below idle, and we go to one rotor, we may be only at idle, but we are also bang on for resonance. There is a story out there about a guy with a 13B that had a steady misfire on one and tore things up RIGHT NOW!

Maybe the stiff system seems like the way to go, but let's remember that several things happen to get to a stiff system:

- Making all of the soft elements stiffer adds weight to them;
- Making a stiff system means that the entire system sees all of the firing pulses, and that adds weight to everything;
- Making it a stiff system means that lash anywhere in the system makes impact through the system, so we need as close to a zero-lash system as we can get.

Have fun guys. I sure ain't building one...

Billski