View source for Modal analysis using FEM

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
modes.

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 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>
[M] [\ddot U] +
[C] [\dot U] +
[K] [U] =
[F]
</math>

where <math> [M] </math> is the mass matrix,
<math> [\ddot U] </math> is the 2nd time derivative of the displacement
<math> [U] </math> (i.e., the acceleration), <math> [\dot U] </math>
is the velocity, <math> [C] </math> is a damping matrix,
<math> [K] </math> is the stiffness matrix, and <math> [F] </math>
is the force vector.  The general problem, with nonzero damping, is a [[quadratic eigenvalue problem]]. However, for vibrational modal analysis, the damping is generally ignored, leaving only the 1st and 3rd terms on the left hand side:

:<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> This 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>).
Using 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>

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

In contrast, the equation for static problems is:

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

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
expressed as:

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

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

:<math>
\mu =  \frac{1}{\lambda}
</math>

but the eigenvectors are the same.

== See also ==

*[[Finite element method]]
*[[Finite element method in structural mechanics]]
*[[Modal analysis]]
*[[Seismic analysis]]
*[[Structural Dynamics]]
*[[Eigensystem]]
*[[Eigenmode]]
*[[Quadratic eigenvalue problem]]

== References ==

{{Reflist}}

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

{{DEFAULTSORT:Modal Analysis Using Fem}}
[[Category:Finite element method]]
[[Category:Numerical differential equations]]
[[Category:Numerical linear algebra]]