Beam Natural Frequencies 2

[mechj]

I have a question regarding natural frequencies of cantilever beams. Under what conditions does loading affect the natural frequencies of the beam? Does this happen when the loading shows up in the boundary condition (ie an axial load applied to the free end)? What about when the beam is loaded in the transverse direction. I believe that transverse loading does not effect natural frequency, but I am not sure. Any clarification would be greatly appreciated.

Regards

jm

"Education is what remains after one has forgotten everything he learned in school." Albert Einstein

 

[GregLocock]

Pure force type loads don't affect modal frequencies of simple systems where linear elasticity is the restoring stiffness.

That probably seems counterintuitive, but it is a bit of a chicken and egg thing. In order to get those simple systems you make assumptions that eliminate the effect of those loads.




Cheers

Greg Locock

Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.

 

[MikeyP]

I would add that for and axial load, there may be an influence on frequency (even in a linear model) if that load is great enough that the restoring force due to tension is significant compared to the restoring force due to the bending stiffness of the beam (shear stiffness too if you are using a Timoshenko model).

I seem to remember that "Formulas for natural frequency and mode shape" by R D Blevins gives a method to estimate the change in natural frequency due to axial load.

M

--
Dr Michael F Platten

 

[GregLocock]

In which case it is no longer a simple system. Chickens, eggs.

Cheers

Greg Locock

Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.

 

[sgwhip]

Ok, I'll weigh in because I find the question interesting, and I wish to understand the concept. But I am a plant engineer, not a design engineer, so my thought process below is qualitative, not quantitative.

I think the natural frequency might be affected in 2 ways. First, if the transverse load is not a pure force load, then it will affect the natural frequencies in the other orthogonal directions. For example, if the load was a mass (obvious influence) or if the load impeded movement in the other directions due to friction, etc. I work in plants with large scale production machinery, so it is easier for me to imagine loads having some influence in the "secondary" orthoganal directions than not.

Secondly, perhaps this was said previously, if the load caused a significant increase in tension on one side of the beam and compression on the other, I suspect there would be a change in natural frequency in all orthogonal directions, much like a guitar string changes its natural frequency with changes in tension.

Have I got it?

 

[GregLocock]

Yes, those are real effects, generally constraining the system will increase the frequency (not always, there's at least one case where the frequency drops).

Adding secondary forces will tend to increase frequencies, I can't think of a case where adding a static force would decrease the frequency.

One useful way I think about modes is Rayleigh Ritz. KE=PE and all that. If you increase the PE then the frequency will increase. If the secondary force does not appear in that equation then I don't think it will affect the frequency.





Cheers

Greg Locock

Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.

 

[Dinosaur]

This is not as straightforward as some may think.

Transverse Loading: If it is a loading that has no mass then there would be no effect on the natural frequency; however, except for a steady wind or some kind of magnetic field, how do you apply a force without applying mass? An increase in mass changes the natural frequency as in the SDOF lumped mass equation w = sqrt(k/m)

Longitudinal (axial) Loading: If the force is constant and again without mass, you may recompute the beams stiffness to account for the effect of the axial force using stability functions. In this situation, tension increases the stability of the beam and compression reduces it. I don't recall exactly where in my notes stability functions appear, but a web search may turn up an answer.

Good Luck. Dinosaur

 

[Jolilechat]

How can a load change the modal properties of a structure? Two quick answers:

Answer 1: if the load is in the boundary conditions.
Answer 2: if the structure is non-linear.

I hope this helps.

 

[mechj]

Thank you all for your insight. I have a much better understanding. Here is how I take it:

Ralieghs Quotient R is the ratio of the potential to the kinetic energy. So if the forcing does not show up in the boundary conditions it will not affect the Natural Frequency.

A force in the boundary will either effect the boundary terms, or linear operator of PE. This therefore has an affect on natural frequency. But distributed loads, or other nonconservative type loads have no affect on natural frequency because they are not included in the PE term. It helps to think of it in terms of Energy.

Regards,
jm

"Education is what remains after one has forgotten everything he learned in school." Albert Einstein

 

[tomirvine]

Here is a paper on The Natural Frequencies of Beams Subjected to a Uniform Load:

http://www.vibrationdata.com/beam_axial_load.pdf

Tom Irvine

http://www.vibrationdata.com