How does shaft deflection affect rotating machinery?

[Spimon]

Hey everyone,

After a little help with some deflection caused by a rotating offset shaft.

This is the scenario.
We have a vertical shaft (165kg ~3.3m, supported at one end only) rotating at 14.67rad/s and getting some undesirable deflection and odd dynamic behaviour.

I'm trying to establish if this is caused by the shaft not being vertical before it even starts rotating. There is some static deflection (relatively small, ~2mm) and I'd like to see if this has a significant effect once it starts running.

My calculations with the actual data give a smaller than expected deflection. I'm wondering if this is my units or my process...or just not a big deflection.

My thought process.
1. Assuming the shaft itself is straight (it's been checked to within 0.05mm/m)
2. Force = mrw^2 ( mass x radius x square of angular velocity )
3. Use this force and standard beam deflection formulae to establish deflection caused by rotation.

My 'Worries'
1. There should be a force due to the modulus of elasticity in the shaft to prevent this deflection (to a point). Otherwise any deflection would accelerate and behave similar to when the rotation reaches the critical frequency. I don't know how to calculate this force.

2. Is there something else I'm missing?

3. Am I making any unfair/false assumptions?

Would you guys please take a look at my calculations in the attached pdf and see if:
a. My process is valid
b. My assumptions are fair and reasonable
c. My units are correct

Thank you all very much for any assistace!

 

[edgepflow]

That is a creative solution. If the shaft were perfectly straight, I think the forces would all be radial and uniform over the entire length of the shaft and hence no net deflection. The 2mm of static deflection may be causing your dynamic deflection. I will think about this some more...

 

[Spimon]

The shaft could well have been slightly bent (and probably was at some stage). The run out would have been less than 1mm over the 3m length. I was just using a straight example to try to grasp the concept.
I initially figured if it was bent I could simply sub in (an approximation of) the equation of the bend to describe the radius. Maybe it's more complicated than this.

I also think it may have been bent at some stage becasue a video of this shaft seems to trace an oval shape rather than a perfect circle. I've been told this is a sign of a bent shaft but I'm unsure how to handle this mathematically.

At the end of the day I'm trying to establish the cause of the run out I'm seeing. Nothing we've tested/measured (alignment, operating speed, etc) screams out any obvious answers so now I'm testing the 'what if' scenarios (what if it was bent by 1mm/1m?).

 

[SteamKing]

The phenomenon you are encountering with this shaft layout is known as whirling. If you Google 'shaft whirling' or consult a text on shaft design, this phenomenon is covered in detail since the vibrations caused by whirling can lead to shaft failure, if not from gross overstress of the shaft, then ultimately from fatigue. Whirling can be a problem for any type of rotating machinery, regardless of how the shaft is supported.

 

[Spimon]

Does this phenonema occur below the natural frequency? This shaft is rotating well below its natural frequency.

From what I've read so far this seems to be the same as resonance at critical speed.

 

[SteamKing]

Whirling occurs, like the name suggests, anytime the shaft is rotating.

 

[Spimon]

Thanks! That's given me something to do a little research into. I'd initially discounted it as the speed is onlyl about 50% of critical speed.
I'll look into it further. Thanks again.