Modal analysis using FEM: Difference between revisions

Revision as of 23:36, 20 July 2009

Revision as of 11:38, 23 July 2009

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This is the general form of the eigensystem encountered in structural

engineering using the [[FEM]]. Further, harmonic motion is typically assumed{{byBy whom|date=July 2009}}<!-- I assume it is assumed because that's actually what you get for a linear (small deflection) system if you don't make the assumption. That is, you wind up with a second-order ODE and get e^{-j \omega t} or whatnot.--> for the

structure so that <math>[\ddot U]</math>

is taken to equal <math>\lambda [U]</math>,

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# The system is positive definite

a typical prescription of solution is first to [[tridiagonal|tridiagonalize ]]ize the system using the

[[Lanczos algorithm]]. Next, use the [[QR algorithm ]] to find the eigenvectors and eigenvalues of

this tridiagonal system. If inverse iteration is used, the new eigenvalues will

relate to the old by <math> \mu = \frac{1}{\lambda} </math>, while the eigenvectors of the original can

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of half a sine wave. The fifth mode is a twisting mode.

[[Image:building_mode0building mode0.png|thumb|200px|left|original mesh]]

<br>

<br>

<br>

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<br>

[[Image:building_mode1building mode1.png|thumb|200px|left|mode 1 swaying front to back]]

[[Image:building_mode01building mode01.png|thumb|200px|center|mode 1 and original mesh]]

<br>

[[Image:building_mode2building mode2.png|thumb|200px|left|mode 2 swaying side to side]]

[[Image:building_mode02building mode02.png|thumb|200px|center|mode 2 and original mesh]]

<br>

[[Image:building_mode3building mode3.png|thumb|200px|left|mode 3 stretching and compression]]

[[Image:building_mode03building mode03.png|thumb|200px|center|mode 3 and original mesh]]

<br>

[[Image:building_mode4building mode4.png|thumb|200px|left|mode 4 sine shape]]

[[Image:building_mode04building mode04.png|thumb|200px|center|mode 4 and original mesh]]

<br>

[[Image:building_mode5building mode5.png|thumb|200px|left|mode 5 twisting]]

[[Image:building_mode05building mode05.png|thumb|200px|center|mode 5 and original mesh]]

== See also ==

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*Thomson, William T., '' Theory of Vibration with Applications'', 3rd Ed., Prentice-Hall Inc., Englewood Cliffs, 1988.

{{DEFAULTSORT:Modal Analysis Using Fem}}

[[Category:Finite element method]]

[[Category:Numerical differential equations]]