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
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?
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 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?
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?
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?
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 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?
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?
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?
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?
RE: Effect of prestress on torsional vibration?
RE: Effect of prestress on torsional vibration?
> - 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?
RE: Effect of prestress on torsional vibration?
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 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?
RE: Effect of prestress on torsional vibration?
RE: Effect of prestress on torsional vibration?
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?
RE: Effect of prestress on torsional vibration?
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?
RE: Effect of prestress on torsional vibration?
BK
RE: Effect of prestress on torsional vibration?
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?
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?
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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Online tools for structural design
RE: Effect of prestress on torsional vibration?
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?
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?
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?
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