Modal analysis using FEM: Difference between revisions

Revision as of 18:05, 23 May 2012

Revision as of 13:21, 11 December 2012

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For the most basic problem involving a linear elastic material which obeys [[Hooke's Law]],

the [[Matrix (mathematics)|matrix]] equations take the form of a dynamic three dimensional spring mass system.

The generalized equation of motion is given as:<ref> Clough, Ray W. and Joseph Penzien, ''Dynamics of Structures'', 2nd Ed.,

McGraw-Hill Publishing Company, New York, 1993, page 173 </ref>

:<math>

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This is the general form of the eigensystem encountered in structural

engineering using the [[FEM]]. To represent the free-vibration solutions of the structure harmonic motion is assumed ,<ref> Bathe, Klau Jürgen, '' Finite Element Procedures'', 2nd Ed., Prentice-Hall Inc., New Jersey, 1996, page 786 </ref>, so that <math>[\ddot U]</math>

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

where <math>\lambda</math> is an eigenvalue (with units of reciprocal time squared, e.g., <math>\mathrm{s}^{-2}</math>),

and the equation reduces to:<ref> Clough, Ray W. and Joseph Penzien, ''Dynamics of Structures'', 2nd Ed.,

McGraw-Hill Publishing Company, New York, 1993, page 201 </ref>

:<math>[M][U] \lambda + [K][U] = [0]</math>

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multiplied through by the inverse of the mass,

<math> [M]^{-1} </math>,

it will take the form of the latter.<ref> Thomson, William T., '' Theory of Vibration with

Applications'', 3rd Ed., Prentice-Hall Inc., Englewood Cliffs, 1988, page 165 </ref>

Because the lower modes are desired, solving the system

more likely involves the equivalent of multiplying through by the inverse of the stiffness,

<math> [K]^{-1} </math>, a process called [[inverse iteration]].<ref> Hughes, Thomas J. R., ''The Finite Element Method'', Prentice-Hall Inc.,

Englewood Cliffs, 1987 page 582-584 </ref>

When this is done, the resulting eigenvalues, <math> \mu </math>, relate to that of the original by:

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== Footnotes ==

{{Reflist}}

<div class="references-small">

<references/>

</div>

== References ==

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==External links==

*[http://frame3dd.sourceforge.net/ Frame3DD open source 3D structural modal analysis program]