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Effect of prestress on torsional vibration?
2

Effect of prestress on torsional vibration?

Effect of prestress on torsional vibration?

(OP)
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

RE: Effect of prestress on torsional vibration?

I don't quite understand your description of the system, but in a linear system preloads and static loads  have no   effect on the frequencies, or mode shapes.

Cheers

Greg Locock

Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.

RE: Effect of prestress on torsional vibration?

(OP)
Greg,
I mean if you take into account centripetal force from the disk and resulting prestress. This does have effect on bending and axial frequencies. Yes, mode shapes are the same. Apply inertial load to the rotor, then run normal mode analysis along with static case. Compare this with plain normal mode analysis for same thing.

suviuuno

RE: Effect of prestress on torsional vibration?

Due to gyroscopic effects? Which would actually look like   extra inertia, thinking about it.

Well, I've never come across that before.



Cheers

Greg Locock

Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.

RE: Effect of prestress on torsional vibration?

> Why are torsional vibration frequencies higher if considering prestress in disk due to shaft rotation?

It is the same for a linear (translational) system. To satisfy yourself, model a simple cantilever beam both with/without an axial end load. Then extract the mode shapes and eigenvalues of the system for both cases. You will find that the mode shape of each system will be the same, but the eigenvalues will be slightly higher (proportional to the load) in the axially loaded beam. The loaded beam's extra stiffness is added to its initial stiffness (matrix), hence when extracting the eigenvalue ωn:

ωn=1/2π×(K/M)0.5

because of the higher stiffness (K) the eigenvalue is higher.

Cheers.


------------
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RE: Effect of prestress on torsional vibration?

I don't get it. What extra stiffness? E doesn't change. I doesn't change.

Cheers

Greg Locock

Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.

RE: Effect of prestress on torsional vibration?

(OP)
Thanks Drej,
I am still not satisfied. I agree with your example on bending and axial frequencies. Regardless of how "stiff" the disk is, torsional frequencies should be not be affected?

Thanks,
suviuuno

RE: Effect of prestress on torsional vibration?

The stiffness (rotational or otherwise) has a direct relationship with the torsional/linear frequencies (see my equation above for the translational system). For torsional frequencies there is an analagous equation including rotational stiffness kt (units of Force*Length/Radian):

ωntor=(Kt/IM)0.5

The rotational stiffness for a simple shaft is just GJ/L. For your rotational system you just replace the mass M with the mass moment of inertia, I.

Two of the best examples I can think of of how pre-stressing a structure affects its modal response:

- A guitar string or a drum head will vibrate at higher frequencies as its tension is increased.

- When a turbine blade spins, its natural frequencies tend to be higher because of the prestress caused by centrifugal forces.

The same must be true of any simple rotational system. The best way to proove this, and to ultimately satisfy yourself (the important thing), is to do some simple modelling work.  Bear in mind also that dynamically the system stiffness will change as well of course


------------
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RE: Effect of prestress on torsional vibration?

- A guitar string or a drum head will vibrate at higher frequencies as its tension is increased.

That is a different mechanism entirely.

- When a turbine blade spins, its natural frequencies tend to be higher because of the prestress caused by centrifugal forces.

Not really.The restoring force is greater, so in a non linear analysis the frequency will increase. In a linear analysis, which is the only place where your equation is accurate, it will have no effect.

That is, arbitrarily fudging k to include a centrifugal straightening stiffness destroys the validity of the linear analysis - it will only be accurate at one level of excitation.







Cheers

Greg Locock

Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.

RE: Effect of prestress on torsional vibration?

It is the same for a linear (translational) system. To satisfy yourself, model a simple cantilever beam both with/without an axial end load. Then extract the mode shapes and eigenvalues of the system for both cases. You will find that the mode shape of each system will be the same, but the eigenvalues will be slightly higher (proportional to the load) in the axially loaded beam.

Would you also expect, then, that a point mass suspended from a simple spring would have a different natural frequency if a constant force (gravity perhaps) is applied to it than if the constant force is not applied to it?  

RE: Effect of prestress on torsional vibration?

nevermind the above - I see that you said "axial," which means that you have an additional restoring moment that increases with beam deflection, which I agree would increase the nf of the system.

RE: Effect of prestress on torsional vibration?

in the case of the disk-on-a-stick example, the only thing that comes to mind is that if the constant rotational component is enough to cause "stretching" of the disk, then you will have a situation where trying to increase the velocity increases disk inertia, and conservation of momentum results in an extra "slow down" torque, while slowing of the disk reduces inertia and results in an extra "speed up" torque.  If the stretch is slight, then I would guess that it would change the nf of the system a bit.  

RE: Effect of prestress on torsional vibration?

Greg/others - With all due respect, I think you need to re-read my post a little more closely. In the context of pre-stressing, and how this affects system eigenvalues, the two examples I gave are entirely correct.

> - When a turbine blade spins, its natural frequencies tend to be higher because of the prestress caused by centrifugal forces.
> Not really.The restoring force is greater, so in a non linear analysis the frequency will increase. In a linear analysis, which is the only place where your equation is accurate, it will have no effect.

No, not true. Again, in linear analyses stress stiffening -- pre-stressing -- will affect frequencies. Refer to my example above.

Don't just say that something isn't true, prove it. Go and model these systems for yourself if necessary.


------------
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RE: Effect of prestress on torsional vibration?

in that case, back to my example of the simple spring/mass system.  If you "pre-stress" the spring by adding a constant force to the mass, do you expect the nf to change?

RE: Effect of prestress on torsional vibration?

The guitar string example is irrelevant to this discussion. It is also misleading.


Cheers

Greg Locock

Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.

RE: Effect of prestress on torsional vibration?

I'll try again. The system as described is a simple flywheel on a shaft, suspended in two bearings.

I can see three likely modes of vibration.

1) bending of the shaft

2) torsion of the shaft

3) modes of the flywheel.

OP says "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?"

From this I assume that 1 and 2 were the effect that the original poster was concerned with, not 3.

The torsional modes of the shaft are not, for any reasonable geometry, affected by the state of stress in the flywheel.

I'm still struggling with why the OP thinks even the bending modes are affected, in a linear analysis, but I'm just about prepared to accept that the additional restoring force you'd get from tilting the flywheel might increase the frequency of certain higher order modes (I'm still tempted to say that it is more of a 'mass like' effect, without doing the maths).

But torsion? no way.

 

Cheers

Greg Locock

Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.

RE: Effect of prestress on torsional vibration?

So called "Gyroscopic stiffening" occurs with large diameter heavy rotors on flexible shafts - the higher the speed the greater will be the resistance to local beam rotation at the rotor connection points, and hence the bending natural frequency will increase with speed. It is not caused by stress, but since centrifugal stresses also increase with speed there would be a correlation, although this is irrelevent to the discussion. Like Greg, I can't see what any of this has to do with torsion.

RE: Effect of prestress on torsional vibration?

(OP)
Thank you all for inputs. GREG, I am concerned with torsion of the shaft and torsion of the flywheel. Flywheel is rigidly attached to the shaft. If torsional nor bending frequencies are not directly affected by the state of stress in the flywheel, then why does torsional frequencies increase in linear analysis with increasing rotational speed? EnglishMuffin's response does provide answer for torsional vibration question in hand, although physical understanding for me is in handwaving sense.

RE: Effect of prestress on torsional vibration?

EnglishMuffin's response does provide answer for torsional vibration question in hand, although physical understanding for me is in handwaving sense.

More handwaving perhaps, but the way I picture this is as though you have a bunch of "fibers" connected to the shaft, and as the shaft twists locally, the fibers wrap around it slightly. The more outward force is pulling against the outer ends of the fibers, the more they will resist this twisting.

RE: Effect of prestress on torsional vibration?

suviuuno : My response does NOT explain why torsional natural frequency is related to stress. It does not even intimate that bending natural frequency is related to stress - only that it might erroneously appear to be related because of an incidental correlation. For the sake of argument, it is possible to envisage an imaginary system consisting of a shaft with rotors mounted on it in which each rotor was stiffly coupled about it's out of plane axes to an auxiliary rotor geared so that it turned in the opposite direction. Such a sytem would then exhibit no gyroscopic stiffening, but the centrifugal stresses in the rotors would be just as high. But in any case, none of this has anything to do with the original question about torsion. The only way I can see torsional stiffness being measurably affected by stress would be if the system were significantly non linear - for example you might have couplings or joints with clearances etc.

RE: Effect of prestress on torsional vibration?

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

Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.

RE: Effect of prestress on torsional vibration?

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).

RE: Effect of prestress on torsional vibration?

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

BK

RE: Effect of prestress on torsional vibration?

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

Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.

RE: Effect of prestress on torsional vibration?

(OP)
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.

RE: Effect of prestress on torsional vibration?

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

http://www.xcalcs.com
Online tools for structural design

RE: Effect of prestress on torsional vibration?

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".
 

RE: Effect of prestress on torsional vibration?

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

http://www.xcalcs.com
Online tools for structural design

RE: Effect of prestress on torsional vibration?

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

Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.

RE: Effect of prestress on torsional vibration?

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

http://www.xcalcs.com
Online tools for structural design

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