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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}} ==
For the most basic problem involving a linear elastic material which obeys [[Hooke's
the [[Matrix (mathematics)|matrix]] equations take the form of a dynamic three
The generalized equation of motion is given as:<ref>
McGraw-Hill Publishing Company, New York, 1993, page 173
:<math>
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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
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>)
McGraw-Hill Publishing Company, New York, 1993, page 201
:<math>[M][U] \lambda + [K][U] = [0]</math>
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which is expected when all terms having a time derivative are set to zero.
===
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>
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>
Englewood Cliffs, 1987 page 582-584
When this is done, the resulting eigenvalues, <math> \mu </math>, relate to that of the original by:
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but the eigenvectors are the same.
== See also ==
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*[[Eigenmode]]
*[[Quadratic eigenvalue problem]]
== References ==
{{Reflist}}
== External links ==▼
▲==External links==
▲*[http://frame3dd.sourceforge.net/ Frame3DD open source 3D structural modal analysis program]
{{DEFAULTSORT:Modal Analysis Using Fem}}
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[[Category:Numerical differential equations]]
[[Category:Numerical linear algebra]]
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