effect of shaft torque on critical speed

[chetwynd]

I'm looking at a simple shaft problem, there are no distributed masses. The shaft is supported between crowned splined couplings. My questions,
1. is there are relationship between torque in the shaft and critical speed
2. are there standard formulae for the effect of initial curvature and eccentricity
3. my end constraints can rotate and slide axially, how do I factor this into the critical speed calculation
Many thanks for any pointers......

 

[desertfox]

Hi chetwynd

I am no expert in this area but from my mechanics book I would say that the torque would only play apart in the critical speed of the shaft if that torque was directly related to the shaft speed.The standard formula for critical speed of shaft's accounts for the static deflection of the shaft which would be the same if the shaft were a beam. The formula I have here are for various
cases ie:- uniform plain shaft with long and short bearings, uniform shaft with one end free etc.The formula for the first two cases these are as follows and are probably the
ones most useful for your situation:-



Nc (short bearings)= (1.57/L^2)*(E*I/m)

Nc (long bearings)= (3.57/L^2)*(E*I/m)


where L= length of shaft
E= modulus of elasticity of shaft material
I= second moment of area
m= mass per unit length of shaft


regards desertfox

 

[EnglishMuffin]

I would say that since the critical speed of the shaft is basically the same as its natural frequency, which is partly determined by the end conditions, then anything that affects those end conditions will affect the critical speed. The crowned couplings would tend to behave more like fixed joints than pinned joints as the torque increased, because of the greater friction on the coupling teeth, so I would say that the critical speed would go up with increasing torque, if anything.

 

[EnglishMuffin]

As far as the effect of eccenticity of the center of mass goes, elementary texts indicate that this only affects the vibration amplitude, not the critical speed. I would not have expected curvature itself to have much of an effect, unless the shaft was very flexible and there were some axial force effects present.
The effect of the end constraints is very significant. But the main thing you need to know is their rotational stiffness in the plane of the shaft.
See Blevins - Formulas for natural frequency and mode shape. See in particular "Natural frequencies of a pinned-pinned beam with unequal torsion springs at the pinned joints".

 

[chetwynd]

thanks for the above replies fellow engineers, I certainly try and track down Blevins formula suggested (any idea where I can track this case down online?). Regarding torsion, I had heard that, with some initial eccentricity, torque can tend to induce a "cork screw" shape and then the additional offset mass of the deformed shape intitiates critical speed at a lower frequency. Any comments?

 

[EnglishMuffin]

Actually, the Blevins "formula" includes a whole page of numbers (probably computer generated) - I hope it's relevent in some way - just a thought. As to the corkscrew theory, sounds interesting. But I'm a bit puzzled over how it works - since eccentricity doesn't normally have a big effect on frequency. Also, I would have thought it would have to be an awfully flimsy shaft for this to have much effect. Interesting to see if it's true in practice - it's exactly the opposite of what I would have thought - but we keep learning !

 

[electricpete]

I agree it's the opposite of what I would have thunk. I'd of thunk increasing torque increases shaft stiffness.

Guess that's why I'm not a roto-dynamic engineer ;-)

 

[GregLocock]

There's an analagous effect when analysing beams - as you increase the load in a structure the stiffness of the structure drops. This is because the structure softens as the compression members begin to approach buckling, this reduces their bending stiffness, and so increases rotations of joints etc.

Anyway, for 'normal' driveshafts the torque carried makes no difference to the actual resonant frequency.

I find it hard to believe that any single low friction bearing would behave as better than a pin joint, for these purposes.

Incidentally don't expect too much accuracy from hand calcs in this field, testing is the way to go.

Remember to account for foundation stiffnesses as well.

Cheers

Greg Locock

 

[EnglishMuffin]

Sorry - I don't see the analogy on a detail level - I fail to see how the buckling phenomenon is relevant here - although I'm ready to be convinced. It sounds somewhat difficult to verify experimentally also.

 

[EnglishMuffin]

Greg - by the way - I did not mean to imply by my first answer that the pinned joints would behave like fixed joints - only that they would tend to behave more like them - I'm thinking of a few percent increase in frequency at most. And it could quite possibly be immeasurable. I just don't believe the frequency goes down, that's all - but I could be wrong.