Modal analysis using FEM: Difference between revisions

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It is also possible to test a physical object to determine its natural frequencies and mode shapes. This is called an [[modal analysis|Experimental Modal Analysis]]. The results of the physical test can be used to calibrate a finite element model to determine if the underlying assumptions made were correct (for example, correct material properties and boundary conditions were used).

== FEA eigensystems{{Technical inline|date=September 2024|reason='FEA' has not been defined or mentioned elsewhere in this article}} ==

== FEA eigensystems ==

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

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.,

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:<math>

[M] [\ddot U] + [K] [U] = [0]

</math>

This is the general form of the eigensystem encountered in structural

engineering using the [[Finite element method|FEM]]. To represent the free-vibration solutions of the structure, harmonic motion is assumed,.<ref>Bathe, Klaus Jürgen, '' Finite Element Procedures'', 2nd Ed., Prentice-Hall Inc., New Jersey, 1996, page 786</ref> soThis assumption means 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>),.

andUsing this, 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>

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

=== Comparison to linear algebra ===

In [[linear algebra]], it is more common to see the standard form of an eigensystem which is

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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>

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,

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{{Reflist}}

== External links ==

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

{{DEFAULTSORT:Modal Analysis Using Fem}}