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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]]ize 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]]
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[[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]]
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[[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]]
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[[Image:building mode3.png|thumb|200px|left|mode 3 stretching and compression]]
[[Image:building mode03.png|thumb|200px|center|mode 3 and original mesh]]
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[[Image:building mode4.png|thumb|200px|left|mode 4 sine shape]]
[[Image:building mode04.png|thumb|200px|center|mode 4 and original mesh]]
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[[Image:building mode5.png|thumb|200px|left|mode 5 twisting]]
[[Image:building mode05.png|thumb|200px|center|mode 5 and original mesh]]
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== See also ==
| 