3.2.9. eigen Command

This command is used to perform the eigenvalue analysis.

eigen <$solver> $numEigenvalues

Argument	Type	Description
$numEigenvalues	integer	number of eigenvalues required.
$solver	string	optional string detailing type of solver: -genBandArpack, -symmBandLapack, -fullGenLapack (default: -genBandArpack).

Returns

A list containing the eigenvalues

Note

1. The eigenvectors are stored at the nodes and can be printed out using a Node Recorder, the nodeEigenvector command, or the Print command.

2. The default eigensolver is able to solve only for N-1 eigenvalues, where N is the number of inertial DOFs. When running into this limitation the -fullGenLapack solver can be used instead of the default Arpack solver.

3.2.9.1. Theory

A generalized eigenvalue problem for two symmetric matrices \(K\) and \(M\) of size \(n \times n\) is given by:

\[\left (K - \lambda M \right ) \Phi = 0\]

where:

• \(K\) is the stiffness matrix

• \(M\) is the mass matrix

• \(\lambda\) is the eigenvalue

• and \(\Phi\) is the associated eigenvector

Example

The following example shows how to use the eigen command to obtain a list of eigenvalues.

1. Tcl Code

# obtain 10 eigenvalues using the default solver (-genBandArpack)
set eigenvalues [eigen 10]

# or, obtain 10 eigenvalues explicitly specifying the solver
set eigenvalues [eigen -fullGenLapack 10]

2. Python Code

# obtain 10 eigenvalues using the default solver (-genBandArpack)
eigenvalues = eigen(10)

# or, obtain 10 eigenvalues explicitly specifying the solver
eigenvalues = eigen('-fullGenLapack', 10)

Code Developed by: fmk