# Natural Frequency of Stepped Shaft

• minger
In summary, a set of pressure rakes has natural frequencies that may cross some of their excitation lines. It might be a good idea to plink them before using them, but it is also possible to do a numerical analysis first.

#### 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,

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 $$l_1 \,\mbox{and}\, l_2$$ 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?

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

## What is the Natural Frequency of a Stepped Shaft?

The natural frequency of a stepped shaft is the frequency at which the shaft naturally vibrates when no external forces are applied. It is determined by the material properties, geometry, and boundary conditions of the shaft.

## How is the Natural Frequency of a Stepped Shaft calculated?

The natural frequency of a stepped shaft can be calculated using the equation f = (1/2π) x √(k/m), where f is the natural frequency, k is the stiffness of the shaft, and m is the mass of the shaft.

## What factors affect the Natural Frequency of a Stepped Shaft?

The natural frequency of a stepped shaft is influenced by several factors, including the material properties of the shaft, such as density and stiffness, the geometry of the shaft, and the boundary conditions, such as the supports and applied loads.

## Why is it important to know the Natural Frequency of a Stepped Shaft?

Knowing the natural frequency of a stepped shaft is important for ensuring the structural integrity and stability of the shaft. It can also help identify potential vibration issues that could lead to failure or malfunction of the shaft.

## How can the Natural Frequency of a Stepped Shaft be controlled?

The natural frequency of a stepped shaft can be controlled by altering the material properties, geometry, or boundary conditions of the shaft. This can be achieved through design modifications or the use of damping materials to reduce vibrations.