Intuition of Eigenvalues and their relation to vibration and buckling

[Bharathsf]

Hi,
Can anybody please give me the intuition of Eigenvalues and their relations to vibrations and buckling analysis. I am just not able to get my head around how they are related. I know what eigenvalues are in a matrix but I am not able to understand it in a physical way and their relation to vibrations and buckling.

Bharat

 

[electricpete]

There was discussion of buckling and relationship to the variouis forms of potential energy here:
thread384-312306

The total PE is PEbending - PEcompression
For small axial compression the associated PE is much less than the PE from bending.
As we increase axial compression, total PE decreases, resonant frequency decreases.
The critical point comes where PEcompression = PEbending, at which point total PE = 0 and resonant frequency is zero and this is the theoretically-predicted onset of buckling.

From a purely simplistic point of view, an undamped resonant frequency means we can have an ongoing finite natural response at that frequency even without any ongoing excitation. A system oscilating at some finite natural frequency is not necessarily a big concern. A system exhibiting response at 0 natural frequency implies the system will continue to distort and never return to original position.


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(2B)+(2B)' ?

 

[electricpete]

Further contrast between finite undamped natural frequency and 0-frequency natural frequency:

The finite undamped natural frequency response will oscillate, returning to the same maximum amplitude each cycle.

The 0-frequency natural frequency response has an amplitude which continuously increases. Continous increase in vibration amplitude (which is a measure of deformation) is not sustainable since material limits will eventually be reached

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(2B)+(2B)' ?