Natural Frequency of Stepped Shaft

[minger]

Hi guys, I need to find the natural frequencies of a series of pressure rakes. We have a numerical model, but I'd like to confirm with a hand calculation. I would like to model the pressure rakes as a cantilevered beam with varying cross section (i.e. as each tube "stops" the beam decreases in area).

I looked on Ohiolink for journal articles and through Timoshenko's Vibration Problems in Engineering but can't come up with anything. If anyone has an article, or a link somewhere I would appreciate it. Thansk,

 

[FredGarvin]

Your best bet is to plink them. That's what I do. You can get in the area or do a numerical analysis, but it is very difficult to match the boundary conditions they actually see. This way you will get the exact natural frequencies from a quick test that is easy to do.

 

[minger]

Well we're designing them. Ideally we wouldn't want to buy a set, then find out that the parts we just bought have natural frequencies that cross certain excitation lines in our range of operation.

Either way it might be a good idea to plink them after we receive them, but I'd like to have an idea of their response before.

 

[timmay]

If I remember correctly you can treat a stepped shaft as a set of torsional springs arranged in series. Have a look at the first chapter of Vibration Problems in Engineering, by Weaver, Timoshenko and Young. I'm pretty sure it covers it in there.

 

[minger]

I 'think' I seen what you are thinking of, but IIRC it didn't quite apply to me. Let me see if I can find it.

Yea,

If the shaft consists of two parts having lengths l_1 \,\mbox{and}\, l_2 with diameters d_1\, \mbox{and}\, d_2, the separate torsional constants may be calculated from eq. (c).
k_r = \frac{GJ}{l} = \frac{\pi d^4 G}{32l}
The equivalent spring constant can then be obtained from eq. (k)
k_{eq} = \frac{k_1 k_2}{k_1 + k_2}

OK, so this directly applies to torsional vibration, but I can assume it applies to lateral as well. If so, then how can I calculate equivalent spring constants for the bar?

Would the individual spring constants simply be a long bar, with several hollow cylinders?

 

[timmay]

For lateral vibration can't you treat it as a series of masses rotating on a shaft, but with mass and stiffness distributed over the same area? Try a later chapter in that book (or any in the series)...I think it's chapter 5.

Edit: try finding the following paper if your subscription covers it - the approach and references should give you some help.

Yu, S.D. and Cleghorn, W.L.
Free Vibration of a Spinning Stepped Timoshenko Beam
J. Appl. Mech. -- December 2000 -- Volume 67, Issue 4, 839 (3 pages)
DOI:10.1115/1.1331282