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 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.

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Greg Locock

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[suviuuno]

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

 

[GregLocock]

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

Well, I've never come across that before.





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Greg Locock

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[Drej]

> 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 [ω]_n:

[ω]_n=1/2[π][×](K/M)^0.5

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

Cheers.


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[GregLocock]

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


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[suviuuno]

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

 

[Drej]

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 k_t (units of Force*Length/Radian):

[ω]_ntor=(K_t/I_M)^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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[GregLocock]

- 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.









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Greg Locock

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[ivymike]

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?

 

[ivymike]

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.

 

[ivymike]

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.

 

[Drej]

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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[ivymike]

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?

 

[GregLocock]

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




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Greg Locock

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[GregLocock]

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.





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Greg Locock

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[EnglishMuffin]

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.

 

[suviuuno]

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.

 

[ivymike]

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.

 

[EnglishMuffin]

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.