critical speed for uniform shaft including gyroscopic effects

[electricpete]

Let's say we are looking at a simply-supported uniform shaft.

If it is very thin, the Euler Bernoulli model tells us the resonant frequency.

But let's say the beam is not very thing - maybe diameter is 20% of the length. There are additional effects in the Timoshenko beam model:
shear deformation and rotary inertia.

I am not interested in shear deformation.

I believe I understand how to calculate the resonant frequency including effects of "rotary inertia" if the beam is simply vibrating in one plane. It is given in Rao's Mechanical Vibrations example 8.10 equation E.6. It will be a lower frequency than the one we calculate using the Euler beam model.

But what if the beam is rotating? Now we also have a gyroscopic effect which should (imo) cause the natural frequency to be higher than the Euler beam model. Does anyone have a formula to calculate this?


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[GregLocock]

Just for refernece that is probably example 8.9 in my 1986 copy.

No. Is it in Blevins? I'd try an energy method I think.

One way of looking at it is that you are stiffening the beam against bending by adding a series of precessing flywheels that will need an additional transverse moment to move their rotational axis to the deflected shape of the beam. Can you (conceptually) mount a series of massless gyroscopes along the length of the beam and work out their contribution to the KE side of the equation?

So does it boil down to, how much work is done to precess a gyro by x radians in 1/4 of a cycle?

At that, I'm afraid my brain has exploded.









Cheers

Greg Locock

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[desertfox]

Hi electricpete

I hope this might help file might help.
The way I see it you can work out the static deflection
first then calculate the deflection due to rotation.

Regards

desertfox

 

[electricpete]

Thanks for the responses.

I am trying to validate a critical speed program that I have programmed in excel vba using the transfer matrix method. It matches the output of a commercial program including gyroscopic effects both for overhung rotors and between-bearings rotors. I have also validated it against analytical solutions for overhung rotors including gyro effects in Den Hartog. But I don't have any solutions for between-bearings rotors including gyro effects.

The geometry posed by desertfox would be a good one to benchmark, since if the disk is not centered between the bearings, gyroscopic forces will come into play for the first critical speed. But the solution posted in the link above includes no gyroscopic effects. It would give you the same solution whether you had a point mass W or a great big disk W.

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[electricpete]

For desertfox' geometry, I think I did find an analytical solution contained in Timoshenko's Mechanical Vibrations (link posted on the forum previously) page 294. I'll try that out.

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[suviuuno]

Kinematics and Dynamics of Machines by Martin.

This is derived from energy method what Greg was referring.

w=sqrt((g*sum(My))/sum(My^2))

M - mass of disk i
y - distance from horizontal to deflected shape location of disk i

 

[desertfox]

Hi electricpete

I think I am slightly confused by your terminology regarding gyroscopic effect.
To get the above, you need to get your spinning axes to
change direction, for example a rotor of a turbine which
is pitching in a ship.
I am not sure from your posts whats causing gyroscopic effects, if its just rotation between to bearings then its a
case of static deflection and centrifugal effects.
Please explain as I might well be missing something.

regards

desertfox

 

[GregLocock]

As the shaft bends the shafts own rotational inertia needs to be precessed (most strongly at the ends).

I must admit I have a hard time believing it would be significant, but then my problems are usually calculable on an old envelope, when I've got the right equation.



Cheers

Greg Locock

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[desertfox]

Hi Greg

Thanks for the response, I wasn't thinking on such a small
scale, what you said now makes sense. I have never come across gyroscopic effects being taken into account on a spinning axes due to shaft bending.
I worked on several occasions for a pump manufacturer and
they used to look at the static deflection of the shaft based on simply supported beam and checked the critical speed and that was basicly it.
That said in some applications it may well be very important.

regards

desertfox

 

[WCFoiles]

Rao's (and others) book on rotordynamics talks about this, as I recall.

Gyroscopics, tends to raise the forward 'criticals' and lower the backwards ones. The equations end up with (It-Ip) terms for the cross terms in the equations of motion. The It is the transverse inertia term) - The Ip or polar moment term modulates the effect of the It. These will be muliplied by rotation frequency and whirl frequency.

I believe you have some material on the transfer matrix method; this would show these terms.

Regards,

Bill

 

[rob768]

in Den Hartog, there is a description and i believe a formula for whirling of disks, mounted on a shaft.
Perhaps that is useful. In theory, the basic frequency splits in a higher and a lower frequency (sorry, can't recall the correct ebhlish terms at the moment)