Effect of prestress on torsional vibration? 2

[suviuuno]

Greetings,

Consider simple shaft supported at both ends with disk in the middle. One end free in axial direction, transverse rotations are free. Other end is constrained in axial rotation, all translations constrained.
Why are torsional vibration frequencies higher if considering prestress in disk due to shaft rotation?
Higher axial and bending modes in prestressing I understand, but as far as I know torsional frequencies should be about same. Beam elements for shaft, shell elements for disk. What am I missing?

Thanks,
suviuuno

 

[GregLocock]

I suppose one option is that a torsional mode may not be pure, and so has some bending, and so is affected by the gyroscopic straightening action. The frequency of torsional and bending modes in automotive crankshafts, for example, are often quite similar in frequency, so it is conceivable that one could get a mixed mode.

Cheers

Greg Locock

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[EnglishMuffin]

This discussion brings to mind an experiment I once did, although it's largely irrelevant. I was having a discussion with a fellow engineer about axially pre-tensioning ball screws to increase their natural frequency (which from the standpoint of "whirling" is an important consideration sometimes for high speed and long travels). Wouldn't it be neat, I said, if you could sell hollow ball screws with a rod through the middle and put the rod in compression, thus putting the screw in tension and increasing it's natural frequency (the guitar string effect mentioned above)- thereby making large amounts of money. (Actually, this doesn't work very well in practice for various reasons). But anyway, said this friend of mine, why make it so complicated ? Why not take a hollow ball screw, and fill it with pressurized oil ?. The pressure on the ends will put the screw in tension and you will get the same effect. So one lunchtime, I got a piece of hollow mechanical steel tubing about 2 inches diameter and about 12 feet long with a 1 inch hole in the center, and placed the ends on simple supports. I checked the transverse bending natural frequency, then pressurized the inside to 10000 psi, and checked the natural frequency again. Would anyone care to say what they think the change in frequency was ? (Unfortunately, I didn't check torsion).

 

[bklauba]

I would think the natural frequency would be higher, but the resonance peak would be lower.

BK

 

[GregLocock]

Nice experiment, but there's about a zillion variables in there.

1) more oil due to compressibility hence higher mass per unit length

2) more oil due to expansion of the cylinder hence higher mass per unit length

3) higher moment of inertia due to said expansion. hence higher I

Well that's three biggies, if not a zillion.

Finally we have what we were after

4) increased static axial stress in the tube

My argument is that 4 won't matter, 3) is more powerful than 2) and I haven't got the faintest idea about the relative effect of 1), which would need the tube thickness and the bulk modulus of the oil.






Cheers

Greg Locock

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[suviuuno]

There is small change in mass, and like Greg indirectly pointed out longitudinal stress is half of hoop stress.
My guess is that longitudinal stress does matter and well overrides increased mass due to oil. As a result nf increases, I am going to say it almost doubles.

 

[prex]

My guess is that there was a very slight decrease in frequency (if measurable) in the EnglishMuffin's experiment, due to the increase of mass of pressurized oil.
The argument, that was brought by various posters, that an axial force causes restoring moments, is true (for example for a guitar string), but only if that force is external, so that it keeps its axial direction when the beam bends. In the experiment the force on the end caps stays axial with the deformed shape of the tubing, so there is no restoring moment and no effect on frequency.
Coming back to suviuuno's question I understand that he finds a change in torsional frequency in a FEM model: so I would double check the assumptions made in the model (and how the code treats prestress and dynamic effects), as I too agree that there shouldn't be any effect due to rotational speed.

prex

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[EnglishMuffin]

Hmmm - 4 replies to my question. At last I have time to reply - I have been so busy lately.
Greg Locock mentions four effects, all of which I think I considered when I did this experiment almost 20 years ago. I think I gave all the relevant data, except for the bulk modulus of the oil - I have always used the rule of thumb of .5%/1000 psi, or 5*10^-6 (psi)^-1 for rough calculations. Assuming this value, for effect #1, the frequency change due to the oil compressibility alone would amount to a reduction of less than one part per thousand. Effect #3 produces an increase in frequency of only 3 parts per hundred thousand. I did not bother to calculate the result for effect #2, but it also is extremely small. All these effects are insignificant. We now come to effect #4. Blevins (formulas for natural frequency & mode shape) gives the following formula for the frequency ratio increase under a tension P, for this particular case : sqrt(1 + P*L^2/(E*I*pi^2)). This effect is NOT negligible. If this were the only significant effect, it would produce a frequency increase of about 32%. I remember checking the axial extension of the tube with a dial indicator when I did this experiment to confirm that I really had the correct axial load, which in this case is 7854 pounds.
And now to the result of the experiment: The pressurization produced no measurable change in frequency whatsoever !
Greg Locock is therefore correct when he says "my argument is that 4 won't matter", although apparently he does not say exactly what his argument is. Prex's conclusion is also correct, although his explanation is somewhat misleading in my view.
I believe the correct explanation is as follows :

If one analyzes the problem from first principles, along the lines of the classic derivation of Blevins' formula, one procedes by considering the integration of a number of infinitesimal segments of the tube, each of which, with the tube in its deformed state, would look like a very short banana shape, or toroidal segment, having a curvature coincident with the local beam curvature. The two end faces of the segment will have forces applied to them, normal to the faces, each equal to 7854 pounds (ignoring any minute variation in tension along the length of the beam, which is valid to a first order of approximation). If the element is of finite size, the two forces are at a slight angle to each other, and it is this slight inclination that produces a radial force component which tends to pull the element back towards the undeflected position. The situation as considered so far does not differ at all from the Blevins case with externally applied tension. However, there are two other forces due to the internal oil pressure which act opposite to each other in the radial direction. These forces are not equal to each other if the beam is deflected, since because of the curvature, the projected areas on the two halves of the inner pressurized wall are not the same. It turns out that this effect produces a radial force (transverse to the beam) that exactly cancels out the axial tension effect. Actually, this is immediately obvious if one assumes the oil volume is in equilibrium, but it helps to understand the whole picture.
Now I ask this : Assuming it were practicable, if the inside wall of the tube were isolated from the radial oil pressure, by inserting an additional thin wall tube inside and pressurizing that, what would then be the resulting change in natural frequency ? (I have not done this experiment).
Although none of this is directly relevant to the original question, it does underscore the need to think about each situation in detail - rather than just making general statements to the effect that "stress increases natural frequency".

 

[prex]

EnglishMuffin, I'm afraid that you are over complicating things: I must admit I didn't try to fully understand your reasoning, but simply because I can't see how it could explain why, if the axial force was applied externally (by means of a spring, to keep it constant when the tube deforms), you would indeed find the expected change in frequency.
I will propose an energy argument, besides my argument above that I fully confirm, to explain the discrepancy.
When you pressurize the oil, the energy you spend in doing this goes almost entirely into strain of the tubing wall, and, when you bend the tube to cause vibrations to occur, the internal axial force won't be able of doing any more work. Or, in other words, there is no restoring effect due to the internal pressure, the deformed tube will come back to its initial straight shape solely because of the elastic spring back due to bending, there is no additional spring back due to inside pressure.
If on the contrary the axial force is applied externally as a force (not as a deformation, otherwise we come back to the preceding case), then this force will be able to do work, and this will correspond to the reduction in distance of the beam ends when the beam is forced to bend. I think it is intuitive that under such condition the beam will resist more to a transverse deformation and will spring back more actively when released: this is exactly the reason why frequency will change under such a condition.

prex

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[GregLocock]

That's a useful way of thinking, but I think it does not explain a guitar string very well. The bridge and the neck do not need to move relative to each other, so the external tensioning force is not doing any work.



Cheers

Greg Locock

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[prex]

True, the way a guitar string behaves is a bit different, but only a bit.
In this case the ends of the beam (of course a string is not a beam, because it has no bending stiffness, but this is unrelevant to the discussion) are fixed in their axial position, so that, when the string or beam is moved away of its rest position, the energy won't be stored into the work of an external force, but will go into the internal axial strain of the string (additional to bending strain, if it's a true beam).
From this point on the reasoning remains the same.

prex

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