Frequency

The (generalized) eigenvalue problem is handled in the Frequency step. To define a valid step, please use the following command.

step frequency (1) (2)
# (1) int, unique step tag
# (2) int, number of eigen modes to be solved

To successfully run an eigen analysis, the system shall be symmetric, otherwise complex eigen values are computed. This typically requires the stiffness matrix to be symmetric. Besides, it must have

1. a positive definite stiffness matrix,

2. a semi-positive definite mass matrix.

Remarks

1. The symmetric banded storage uses _pbsv solver which only accepts symmetric positive definite banded matrix. If the Frequency fails to compute the required eigen modes, please use other storage schemes.

2. The computed eigenvalue is the eigenvalue of the system. In the field of structural dynamics, it is $\omega^2$. The (angular) frequency and period can be computed accordingly.

3. The constrained (generalized) eigenvalue problems cannot be handled when the constraints are implemented via Lagrange multiplier method. If the system contains constraints, users shall make sure they are applied via the penalty function method.

FEAST Solver

By default, the ARPACK solver is used to solve the generalized eigen problem.

The FEAST Eigenvalue Solver can also be used. To switch, one can use

solver FEAST (1) (2) (3) [4]
# (1) int, unique solver tag
# (2) int, number of eigen modes
# (3) double, centre
# [4] double, radius, default: centre

Currently, the FEAST solver can be applied to full, banded and sparse storage. For banded storage, it is necessary to use the SPIKE solver.

set system_solver SPIKE

For the given centre $c$ and radius $r$, the FEAST solver seeks eigenvalues within the bracket $[c-r,c+r]$. If the radius is not assigned, it defaults to $c$, thus the bracket becomes $[0,2c]$.

Example

Consider a massless elastic cantilever beam with lumped end mass. Assume the length is $L=2.84$, the elastic modulus is $E=94.13$, the moment of inertia is $I=1.34$ and the lumped mass is $M=5.76$ so that

$$\omega^2=\dfrac{3EI}{ML^3}=\dfrac{3\times94.13\times1.34}{5.76\times2.84^3}=2.8680.$$

node 1 0 0
node 2 0 2.84
material Elastic1D 1 94.13
element EB21 1 1 2 1 1.34 1
mass 2 2 5.76 1
fix 1 E 1
recorder 1 hdf5 Eigen
step frequency 1 1
analyze
peek eigenvalue
exit

The output is

+--------------------------------------------------+
|   __        __        suanPan is an open source  |
|  /  \      |  \          FEM framework (64-bit)  |
|  \__       |__/  __   __          Acrux (0.1.0)  |
|     \ |  | |    |  \ |  |                        |
|  \__/ |__| |    |__X |  |     maintained by tlc  |
|                             all rights reserved  |
+--------------------------------------------------+

Eigenvalues:
   2.8680

Finished in 0.005 seconds.