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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
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>
:<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 onlygeneral termsproblem, keptwith arenonzero 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:
which give the following system:
:<math>
[M] [\ddot U] + [K] [U] = [0]
[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>
engineering using the [[FEM]]. Further, harmonic motion is typically assumed for the
structureis sotaken thatto equal <math>\lambda [\ddot U] </math>,
where <math>\lambda</math> is an eigenvalue (with units of reciprocal time squared, e.g., <math>\mathrm{s}^{-2}</math>).
is taken to equal <math> \lambda [U] </math>,
Using this, the equation reduces to:<ref>Clough, Ray W. and Joseph Penzien, ''Dynamics of Structures'', 2nd Ed.,
where <math> \lambda </math> is an eigenvalue,
McGraw-Hill Publishing Company, New York, 1993, page 201</ref>
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.
=== 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>
[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
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.,
Englewood Cliffs, 1987 page 582-584 </ref>
When this is done, the resulting eigenvalues, <math> \mu </math>, relate to that of the original by:
but the eigenvectors are the same.
== Methods of solution ==
For linear elastic problems that are properly set up (no rigid body rotation or translation),
the stiffness and mass matrices and the system in general are [[Positive-definite matrix|positive definite]].
These are the easiest matrices to deal with because the numerical methods commonly
applied are guaranteed to converge to a solution. When all the qualities of the system are
considered:
# Only the smallest eigenvalues and eigenvectors of the lowest modes are desired
# The mass and stiffness matrices are sparse and highly banded
# The system is positive definite
a typical prescription of solution is first to [[tridiagonal|tridiagonalize ]] the system using the
[[Lanczos algorithm]]. Next, use the [[QR algorithm ]] to find the eigenvectors and eigenvalues of
this tridiagonal system. If inverse iteration is used, the new eigenvalues will
relate to the old by <math> \mu = \frac{1}{\lambda} </math>, while the eigenvectors of the original can
be calculated from those of the tridiagonalized matrix by:
:<math>
[r^{n}] =
[Q] [v^{n}]
</math>
where <math> [r^{n}] </math> is a Ritz vector approximately equal to
the eigenvector of the original system, <math> [Q] </math> is the matrix
of Lanczos vectors, and <math> [v^{n}] </math> is the <math> n^{th} </math> eigenvector
of the tridiagonal matrix.
== Example ==
The mesh shown below is the frame of a building modeled as [[beam elements]], specifically
consisting of 930 elements and 385 nodal points. The building is constrained at
its base where displacements and rotations are zero. The next images are that
of the first 5 lowest modes of this building during free vibration. This problem can
be seen as a depiction of the likeliest deflections a building would take during an
earthquake. As expected, the first mode is a swaying of the building from
front to back. The next mode is swaying of the building side to side.
The third mode is a stretching and compression mode in the vertical <math > y </math>
direction. For the fourth mode, the building nearly assumes the shape
of half a sine wave. The fifth mode is a twisting mode.
[[Image:building_mode0.png|thumb|200px|left|original mesh]]
<br>
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<br>
<br>
<br>
<br>
<br>
<br>
<br>
<br>
<br>
<br>
<br>
<br>
[[Image:building_mode1.png|thumb|200px|left|mode 1 swaying front to back]]
[[Image:building_mode01.png|thumb|200px|center|mode 1 and original mesh]]
<br>
[[Image:building_mode2.png|thumb|200px|left|mode 2 swaying side to side]]
[[Image:building_mode02.png|thumb|200px|center|mode 2 and original mesh]]
<br>
[[Image:building_mode3.png|thumb|200px|left|mode 3 stretching and compression]]
[[Image:building_mode03.png|thumb|200px|center|mode 3 and original mesh]]
<br>
[[Image:building_mode4.png|thumb|200px|left|mode 4 sine shape]]
[[Image:building_mode04.png|thumb|200px|center|mode 4 and original mesh]]
<br>
[[Image:building_mode5.png|thumb|200px|left|mode 5 twisting]]
[[Image:building_mode05.png|thumb|200px|center|mode 5 and original mesh]]
== See also ==
*[[Eigensystem]]
*[[Eigenmode]]
*[[Quadratic eigenvalue problem]]
*[[Modal Analysis for deformation simulation]]
== Footnotes ==
<div class="references-small">
<references/>
</div>
== References ==
{{Reflist}}
*Clough, Ray W. and Joseph Penzien, '' Dynamics of Structures'', 2nd Ed., McGraw-Hill Publishing Company, New York, 1993.
== External links ==
*Golub, Gene H. and C.F. Van Loan, '' Matrix Computations'', 3rd Ed., The Johns Hopkins University Press, Baltimore, 1996.
*[https://frame3dd.sourceforge.net/ Frame3DD open source 3D structural modal analysis program]
*Hughes, Thomas J. R., '' The Finite Element Method '', Prentice-Hall Inc., Englewood Cliffs, 1987.
*Thomson, William T., '' Theory of Vibration with Applications'', 3rd Ed., Prentice-Hall Inc., Englewood Cliffs, 1988.
{{DEFAULTSORT:Modal Analysis Using Fem}}
[[Category:Finite element method]]
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[[Category:Numerical linear algebra]]
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