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/community/showthread.php?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>
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/community/showthread.php?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>
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