Modal analysis using FEM: Difference between revisions

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The goal of [[modal analysis]] in structural mechanics is to determine the natural mode shapes and frequencies of an object or structure during free [[vibration]]. It is common to use the [[finite element method]] (FEM) to perform this analysis because, like other calculations using the FEM, the object being analyzed can have arbitrary shape and the results of the

calculations are acceptable. The types of equations which arise from modal analysis are those seen in [[eigensystem]]s. The physical interpretation of the [[eigenvalues]] and [[eigenvectors]] which come from solving the system are that

they represent the frequencies and corresponding mode shapes. Sometimes, the only desired modes are the lowest frequencies because they can be the most prominent modes at which the object will vibrate, dominating all the higher frequency

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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.,

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

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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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but the eigenvectors are the same.

== After FEA eigensystems ==

{{Expand section| The article not complete|date=May 2012}}

== See also ==

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{{Reflist}}

== External links ==

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

{{DEFAULTSORT:Modal Analysis Using Fem}}