Modal analysis using FEM: Difference between revisions

Revision as of 14:10, 20 July 2009

Revision as of 18:23, 20 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 for the

structure 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>

:<math>

[M] [U] \lambda +

[K] [U] =

[0]

</math>

In contrast, the equation for static problems is:

:<math> [K][U] = [F] </math>

[K] [U] =

[F]

</math>

which is expected when all terms having a time derivative are set to zero.

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expressed as:

:<math>[A][x] = [x]\lambda</math>

[A] [x] =

[x] { \lambda }

</math>

Both equations can be seen as the same because if the general equation is

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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>

It should be noted that because only 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.,