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Consider a system made of a cylindrical column (fixed on the ground) with a mass attached on top of it (B). Both the mass of the column and the mass are not negligible. Depending on the value of the mass I could have self-buckling of the column. What I would like to compare is the natural frequency at the top of the column as a function of mass and presence of self-buckling.

1) If there isn't self buckling, then I can compute the natural frequency of the column+mass system as a cylindrical cantilever (still don't know how, but it seems standard).

2) If I have self-buckling, the column bends and I expect the system to have a natural frequency around the z-axis due to system symmetry (in red). Is that true? How can I compute that frequency?

https://en.wikipedia.org/wiki/Self-buckling

I need help especially for "2)". Any reference/help for "1)" is appreciated.

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  • $\begingroup$ I edited "proper frequency" to "natural frequency" - I assume "proper" was a poor translation of the German "eigen". Actually, in English we sometimes use the words "eigenfrequency", "eigenvalues" and "eigenvectors." $\endgroup$ – alephzero Oct 21 '18 at 13:50
  • $\begingroup$ Ups I wrote that too fast! You're right... I actually took it from french ;)! $\endgroup$ – Worldsheep Oct 21 '18 at 16:06
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(1) What you say is correct, but you need to include the effect of the compressive stress in the column on its lateral stiffness. Names for this are the "stress stiffness" term (as compared with the "elastic stiffness" which depends on the material properties), or "geometric nonlinearity". Explaining the theory would be far too long to do in an answer here.

If the column's mass is significant, the compressive stress will increase from the top of the column to the bottom, and you would need to make a computer (finite element) model to get a numerical answer - I doubt whether anyone has bothered to work out and publish the details of an "exact" solution of the equations of motion for that situation.

(2) The frequency for "orbiting around" the initial configuration (which is what you seem have drawn) is the same as the frequency of side-to-side vibration. You can consider it as two side-to-side vibrations oriented 90 degrees apart, with a phase shift between them. Look up Lissajous figures for more explanation. As the mass at the top is increased until the column buckles, this frequency will decrease until it is zero.

The higher modes of vibration of the column and mass will still exist in theory if the mass is increased beyond that point, but in practice (and after all, engineering is a practical subject), knowing "what the vibration frequency would have been if some magic had stopped the column from bucking" isn't very interesting.

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  • $\begingroup$ (1) Yes, I didn't find any solution on the web for the problem as described. Do you know of any analytical standard solution giving a good approximation of the order of magnitude of the resonant frequency? I don't actually need an exact solution. $\endgroup$ – Worldsheep Oct 21 '18 at 13:30
  • $\begingroup$ (2) I understand that an orbiting frequency can be decomposed in two oscillations x-y. Do you think the column cannot rotate after buckling? I would say yes, but the bending involved in the column is different. Which side "the column" choses to buckle to since the system is symmetric? Any direction is good right? The set of points in 3D space where the top of the column "B" can buckle to is a circle (with same energy I would say). What happens if I push the mass at that point? I think it would rotate around the z-axis but without going through the "straight column" configuration. $\endgroup$ – Worldsheep Oct 21 '18 at 13:30
  • $\begingroup$ Not sure it is clear. But if the energy doesn't change I would say it can rotate at any frequency it pleases to. P.S. I always consider 1st modes :). $\endgroup$ – Worldsheep Oct 21 '18 at 13:30

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