Natural Frequency of Stepped Shaft

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Discussion Overview

The discussion revolves around determining the natural frequencies of pressure rakes modeled as cantilevered beams with varying cross sections. Participants explore both numerical and hand calculation methods, as well as the implications of design choices on frequency response.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant seeks to confirm numerical model results with hand calculations for natural frequencies of pressure rakes.
  • Another suggests using a plucking method to obtain exact natural frequencies, emphasizing the difficulty of matching boundary conditions in numerical analysis.
  • A different participant expresses concern about potential resonance issues if natural frequencies align with operational excitation lines, advocating for preliminary calculations before testing.
  • One participant proposes modeling a stepped shaft as a series of torsional springs, referencing a specific engineering text for guidance.
  • Another participant discusses calculating torsional constants for different shaft segments and questions the applicability of these calculations to lateral vibrations.
  • A suggestion is made to treat lateral vibrations as a series of masses on a shaft, with a reference to a later chapter in the same engineering text for further insights.
  • A specific academic paper is recommended for additional approaches and references related to the topic.

Areas of Agreement / Disagreement

Participants express various methods and considerations for calculating natural frequencies, but no consensus is reached on a single approach or solution. Multiple competing views on modeling techniques and testing methods remain present.

Contextual Notes

Participants reference specific equations and concepts related to torsional and lateral vibrations, but the discussion includes assumptions and conditions that are not fully resolved. The applicability of certain methods to different vibration types is also uncertain.

minger
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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,
 
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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.
 
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.
 
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.
 
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 [tex]l_1 \,\mbox{and}\, l_2[/tex] with diameters [tex]d_1\, \mbox{and}\, d_2[/tex], the separate torsional constants may be calculated from eq. (c).
[tex]k_r = \frac{GJ}{l} = \frac{\pi d^4 G}{32l}[/tex]
The equivalent spring constant can then be obtained from eq. (k)
[tex]k_{eq} = \frac{k_1 k_2}{k_1 + k_2}[/tex]
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?
 
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
 

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