Propeller Inertia vs Torsional Amplitude
 

A few months back Ross asked a very good question; why did he observe a significant increase in torsional vibration when he swapped to a heavier propeller on a test subject? In retrospect, I didn't do a good job answering his question. At the time my mind was focused on change of natural frequency with a change of inertia or a change in a connecting stiffness. The question really dealt with amplitude, ie, an angular measurement of shaft twist between opposing inertias.

Not all bad however; we did explore some very simple math to demonstrate that a change in inertia did not shift frequency very much, a good rule of thumb to remember. We also looked at a change in connecting stiffness and established a similar rule; it has a larger effect on frequency than a change in inertia. You can find the the discussion here:
http://www.vansairforce.com/communit...t=19030&page=2

Anyway, I was left with a nagging question. Could I establish a simple rule-of-thumb for change in amplitude with a change of prop inertia? I am a student of vibration, not an expert. Such questions are how I learn, so it made my list of things to do in spare moments; bedtime reading if you will.

Well, I think I've got it. The nice thing is that understanding it requires practically no math at all, but it does assume you understand the concept of mode shape.

When inertias are connected to shafts (each shaft having a stiffness value) and set to oscillating torsionally, the direction of rotation for the inertias opposes each other; some rotate one way, some the opposite. The degree of rotation of each inertia (amplitude) is predictable (a function of the relative size of the inetias and the magnitude of the connecting stiffnesses); big ones rotate less, small ones rotate more. The number of available natural frequencies is equal to the number of inertias minus one; thus for a system with three inertias there are two natural frequencies. Terms vary depending on who you're reading, but I'll refer to them as F1 and F2. Another way to say F1 and F2 would be the "first mode of vibration" and the "second mode of vibration". Here's a drawing to illustrate:



It is important to establish the actual relative amplitude (the relative degree of rotation) for the inertias, and the classic way of doing so is called a "Holzer table". One inertia is assigned an angular displacement value of 1 and the others are compared to it. The results of a Holzer are often graphed for illustration, and that graph is called a mode shape. Here's an example from Den Hartog:



From left to right, you're looking at a seven-inertia model; a large-inertia generator, a flywheel, four diesel crankthrows and the accessory drive. The horizontal line is zero twist. If you wish, think of the curve below the line as being clockwise rotation and the curve above the line as being counterclockwise, although it really just means "in opposition". The little circle where the curve crosses the zero twist line is called the "node". The numbers are expressed in proportion to the amplitude of the accessory inertia. It tells you in the first mode (natural frequency 5300 VPM, or 88.33 hz) the generator oscillates with an amplitude 5 times less than the other end of the system. The second mode is kinda scary, because the curve is steep between the flywheel and the accessory end, meaning that if the system resonates (if its natural frequency of 10,950 VPM or 182.5 hz is matched by a driver of the same frequency), the crank will be very highly stressed in torsion.

So much for mode shape. Now here's the rule-of-thumb.

I ran some 3-inertia Holzer numbers (software, easy to do) for a purely theoretical system with inertia and stiffness values like we might find in the small engine-propeller systems we use. I picked 0.5 Kg-m^2 for the prop, 4500 Nm/radian for a rubber coupler or torsion spring forward of the flywheel, a flywheel inertia of 0.05 (1/10 of the prop inertia), 45000 Nm/rad (10x the rubber coupler) for the crank stub and again 0.05 for the crank inertia. The first mode result is +1 for the prop, -4.849 for the flywheel and -5.151 for the crank. Put another way, the flywheel-crank assembly has roughly five times as much amplitude as the prop.

Re-ran the Holzer, this time with double the prop inertia, 1.0 Kg-m^2. The new values were +1 for the prop, -9.725 for the flywheel, and -10.28 for the crank. Notice the amplitude almost exactly doubled.

Again, this time with 3x prop inertia; 1.5 Kg-m^2. Values were +1, -14.6, and -15.4. Three times the prop inertia resulted in three times the shaft twist.

The rule-of-thumb is obvious; shaft stress increases in almost direct proportion to propeller inertia.

Imagine a mode shape plot of the above. The curve would cross the zero line between the prop and the flywheel, then run almost flat between the flywheel and the crank inertia. This is good; it says almost all the twist is taking place in the rubber coupler, with very little torsional stress in the crank stub.

I also ran Holzer values for the same three prop inertias, but this time with 45,000 Nm/radian forward of the flywheel to simulate a "hard" system, ie, no soft coupler, just a steel shaft equal in stiffness to the crank stub. New values were:

0.5 prop: +1, -3.534, -6.466
1.0 prop: +1, -7.358, -12.64
1.5 prop: +1, -11.18, -18.82

Again, the rule-of-thumb holds true, but note the change in the mode shape. No surprise; with equal stiffness on both sides of the flywheel, the crank is now getting twisted much harder. Not only is the curve steep between the flywheel and the crank, but the maximum value is higher compared to the "soft", rubber-coupled system.

Comments welcome. As always, I appreciate "professional supervision" <g>

OH, Dan you make my head hurt------------

propeller inertia and tortional amplitude--
 

Dan, after reading your dissertation on the above subjects, I really decided I had to jump in here and try to see where you obviously were on the wrong track, and how I might enlighten you as to the real causes and effects of tortional amplitudes as well as propeller inertia and vibrations, and their effects on airframes and engine parts, such as crankshafts, camshafts, and all other moving parts. However, after deep deliberation and my best efforts, I am humbled to admit I have to agree with everything you said. :D:D

Interesting Article on Torsional Vibration
 

Dan, very interesting topic.

I am wondering if the importance on the relative amplitudes is interesting, but not telling much of the story. The amplitude that matters is the total amplitude between masses (stick to a two mass system for now....). The relative amplitudes do not tell that story. They really only tell you where the node is, which has no real physical significance (it might be the one location on the shaft which is rotating at a constant angular velocity). For example, the actual twist total between the masses could be the same for both cases, where the prop's moment doubles or triples, even though the node has moved a lot. The movement of the node alone is not something that will be externally noticed, at least in a two mass system.

The equation defining the resonant frequency for a two torsional mass system with no damping is:

W=(Kt/(I1*I2)*(I1 + I2)^.5,

where W= omega, circular frequency, radians per second (2*pi*f), f being frequency
I's are the mass moments of inertia (mmi)
and Kt the torsional spring constant between the masses.

If we put your example values in this equation to examine the natural frequency shift, we get:

W1 = 314 rad/s (.5, .05, 4500)
W2 = 307 rad/s (1.0, .05, 4500)
W3 = 304 rad/s (1.5, .05, 4500)

W4 = 994 rad/s (.5, .05, 45000)
W5 = 972 rad/s (1.0, .05, 45000)
W6 = 964 rad/s (1.5, .05, 45000)

These changes to resonant frequency aren't much, due to the 10:1 minimum ratio of the large mmi to the small mmi.

The vibration that could be greatly affected by the props mmi is the engine on the mounts. Each cylinder's firing torque pulse may impart more angular acceleration to the engine with a larger mmi prop. In other words, a lighter prop may have been the component undergoing relatively large swings in angular velocity with each firing, while the heavy one is varying less (which causes higher peak torques on the crank, which in turn pushes on the motor mounts). I think we would be in deep trouble if we could actually sense the torsional vibrations of the crankshaft itself. I suspect the problem is more "macro". I'm not sure about this concept yet, but will think further...

It would be interesting to compare the mmi of the props in your example to the mmi of the engine around the axis of the crank.

Man, vibrations are a difficult subject! :confused:

Hi Alex,
<<The amplitude that matters is the total amplitude between masses (stick to a two mass system for now....).>>

Correct, for any number of inertias. I'm not sure what you mean by a distinction between "relative amplitudes" and "total amplitude between masses".

Ahhh, wait......are you thinking about shaft twist during resonant operation, in degrees? Perhaps an example to illustrate (terminology in this game is often a problem). See if this fits what you had in mind:

Assume F1 (first mode) is at 35hz. The system will start resonating when excited at about 0.8 of 35, or 28hz, but it won't have significant amplitude. When excited at 32 hz it will have more amplitude, at 35 it will reach peak amplitude (which can be very high), at 38 it will again have less amplitude, and at about 42 hz (1.2 x 35) things will be almost back to normal. A mode shape won't tell us the actual amplitudes reached at any of these conditions, but in every one of them the relative amplitudes will be identical. Using a two mass system with a prop inertia 10 times the engine, the opposing rotation for the two masses may be 0.1 degree and 1 degree at 28 hz, 0.2 and 2 at 32 hz, 0.5 and 5 at 35, 0.2 and 2 at 38, and 0.1 and 1 at 42 hz.....but in all cases the relative amplitude is 10 to 1....the mode shape.

A Holzer table is an intermediate step in the process of determining resonant amplitude.

<<..node....(it might be the one location on the shaft which is rotating at a constant angular velocity).>>

It is.

<< For example, the actual twist total between the masses could be the same for both cases, where the prop's moment doubles or triples, even though the node has moved a lot.>>

No, the total twist will double or triple.

<< The equation defining the resonant frequency for a two torsional mass system....>>

Right. That was the discussion from the previous thread.

<<Each cylinder's firing torque pulse may impart more angular acceleration to the engine with a larger mmi prop.>>

Sure, but with a conceptual tweak. It is true for every point in the RPM range, just not the resonant periods. It doesn't have much to do with the mode shapes of F1, F2, F(etc). Remember, the F's are natural frequencies. If they're not matched by an exciting frequency, the corresponding mode shape angular deflections don't happen. When matched and resonating, the first mode shakes the block very hard via reaction to the cylinder walls. However, take a look at the example for the 2nd mode shape in the previous post. It will shake the block far less than the 1st mode (hmm, maybe not, needs some thinking) but crank stress is higher. Mode shape does tell interesting things.

<<It would be interesting to compare the mmi of the props in your example to the mmi of the engine around the axis of the crank.>>

Yes indeed. I'll bet total engine inertia is quite large. And I suspect the wet-dog shake of a Lyc when it fires up is at least partially a matter of firing impulses matching the natural frequency of the engine assembly on it's mounts.

<<Man, vibrations are a difficult subject!>>

Careful, you'll get hooked too. <g>

The relative amplitudes in eigenfrequency analysis don't mean much when looking for the real amplitudes. One exception (sort of) is when damping is included, and your results are in complex mumbers with non-zero imaginary values.

The actual amplitude at ressonance is directly related to energy dissipation due to friction in bearings, internal (material) damping and viscous effects (propeller rotating in air for instance). The damping due to the propeller is large, but also very complicated because it is dependent on many factors that are unrelated to the initial problem (actual pitch, rpm, velocity of the aircraft and so on).

Let's take a very simple example. Consider everything equal, except that one of two propellers have more mass than the other. They are mounted on torsional springs, and for simplicity let's assume they are not rotating, only oscillating (easier to visualize when they have zero mean rpm). The driving force is a constant power torsional shaker. The one with most mass will have a ressonant frequency below the other one, but what about the amplitude? The one with a small mass will oscillate faster, and this creates more dissipation than the heavier one, and so it will need smaller amplitudes for the same dissipation rate. If the ressonant frequencies were 2:1, then the amplitdes would typically be 1:4.

Hello Bjørnar,
<<The actual amplitude at resonance is directly related to energy dissipation due to friction in bearings, internal (material) damping and viscous effects (propeller rotating in air for instance).>>

True; the Holzer calculation does not consider damping.

<<The damping due to the propeller is large, but also very complicated because it is dependent on many factors that are unrelated to the initial problem (actual pitch, rpm, velocity of the aircraft and so on).>>

That goes back to a question David asked a few months ago (change in propeller pitch quieted a gear rattle). At the time I could find little about the damping value of an air propeller. I'd love to have more information; is there an available reference specific to the subject?

<<The one with a small mass will oscillate faster, and this creates more dissipation than the heavier one, and so it will need smaller amplitudes for the same dissipation rate. If the resonant frequencies were 2:1, then the amplitudes would typically be 1:4.>>

For clarity (remember, I'm a student) let's explore inertia first, then damping.

The goal here was to develop a practical rule-of-thumb specific to our practical application; change in amplitude due to change in propeller inertia. So, is it possible to get a 2:1 shift in frequency using only a change in propeller inertia?

Using a Holzer code and not considering damping, it looks like the propeller inertia change required to shift first mode frequency by a factor of 2 would be impossible in the practical world.

With the more-or-less realistic values from the three-inertia example in the first post:

0.5 Kg-m^2, 4500 Nm/radian, 0.05 Kg-m^2, 45000 Nm/rad, 0.05 Kg-m^2

......frequency would be 230 radians per second (36.6 hz).

Recalculating with higher and lower prop inertias, a first mode frequency half of 230 (115 rad/sec) is impossible even with an infinite prop inertia. Reducing prop inertia in order to double the first mode frequency (460 rad/sec) would require a prop inertia of 0.026 Kg-m^2. That would be a tiny little toothpick, about 5% of the 0.5 Kg-m^2 inertia for the example MT-7.

Clearly you can't change the frequency by a factor of 2 by changing prop inertia, at least not within the practical constraints of our application.

So, that leaves us with damping. I have no values for damping in our practical application, nor any software to explore the frequency effects of damping. At my student level I tend to do simple frequency calculations to find resonant intersections, then assume the actual natural frequencies found in operation will be a bit lower. So far, limited field experience has found them to be a few hundred RPM lower, in the range of 5 to 10 hz, perhaps a 25% shift at most, and I've attributed it to damping. I've never seen a 2:1 shift, and I have to wonder if there can be enough damping to drive frequency down by a factor that large. I'm happy to accept your 2:1/1:4 frequency-to-amplitude statement; I'm still too uneducated to really consider the point. However, what might it look like if the frequency reduction due to damping was more like 15%?

Later:

I was driving down the road thinking, and I may have been a bit brain dead about your comment:

<<If the resonant frequencies were 2:1, then the amplitudes would typically be 1:4.>>

In the above, does the "2:1" mean double or half the previous frequency? And what does the "1:4" represent?

I'm sorry, I was a bit brain dead myself when my cat woke me up much too early this morning :) I was just illustrating the general effect of dissipation of energy vs amplitude. If the frequency of the small inertia prop vs large inertia prop was a factor 2 (2 to 1 or 2:1), then the amplitude of the oscillations at ressonance for would be a (roughly) factor 1 to 4 (small vs large inertia). So if the small inertia prop oscillates twice as fast, it's amplitude would only be 1/4 of the large inertia prop. But, this was on a prop mounted on a torsional rod that is fixed in the other end, not a system of masses where the prop is the one with largest inertia, making this a very poor example in this thread.

Reading your post again, you are very correct about the relative angular displacements. The problem is that you have no real (actual) displacement to relate your system to. If you can measure the actual displacement or amplitude at one spot, then the other ones will be relative to that one for that system at that frequency. If you now change the inertia on the prop, you have created a new system, and you cannot relate the amplitudes of the old system to the new system, not without a new measurement on the new system. For instance, the example from Hartog shows the first two frequencies and the amplitudes are scaled using the right most node. If you measured the actual (real) amplitude at that node, I would believe that for the highest frequency, the amplitude would be only 1/2 or less compared with the amplitude of the lowest frequency. So even though the higher frequency oscillation looks worse in terms of stress looking only at the mode shapes, it most probably is much better than the low frequency when looking at the actual amplitudes.

What these results tells you:
0.5 prop: +1, -3.534, -6.466
1.0 prop: +1, -7.358, -12.64
1.5 prop: +1, -11.18, -18.82
is that the more inertia is put in the prop, the less the prop will oscillate relative to the other masses. But it is also natural that a large inertia prop is much more stable, you have to use a lot more of energy to accelerate it, so the amplitude of +1 for the 1.0 prop, is probably only 1/2 measured in actual displacement, and the +1 for the 1.5 prop is probablu closer to 1/3.

Added:
This can be seen more easely when looking at the energy is the system. Considering the dissipation is the same regardless of inertia on the prop, then the max energy must be the same regardless of inertia in the prop. For the 0.5 prop, the strain energy is 0.5*K1*(1 + 3.534)^2 between the prop and the first mass and 0.5*K2*(6.466 - 3.534)^2 between the last two masses. Obviously this is much less than for the 1.5 prop which is 0.5*K1*(1.0 + 11.18)^2 + 0.5*K2*(18.82 - 11.18)^2. So the comparison between the props cannot be correct energy-vise. For a rough comparison with similar frequencies you can scale the results using the strain energies, but this will not be 100% correct because it is the dissipation energy you have to compare, and this will be dependant on the frequencies, and you have to know the damping in the system.

Strain energy
 

Ok, this is not entirely correct, but nevertheless without numbers for the damping it gives a clue. When the total strain energies are equal for the vareous props the results will be:

0.5 prop: +1.000, -3.534, -6.466
1.0 prop: +0.507, -3.727, -6.403
1.5 prop: +0.339, -3.791, -6.382

All in all a larger prop doesn't really do that much. Probably because it is so much larger than the other masses to start with. What is does is to move some strain energy in the shaft between the prop and the first mass to the shaft between the two last masses.