Users:General FEM Analysis/Analyses Reference/Eigenfrequency Analysis

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Contents

General Description

This analysis is intended for the solution of eigen dynamic problems. It is suitable for analysing the vibration characteristics of structures and to determine their eigen frequencies.


Structure of governing equation

An eigen dynamics problem is defined by the equation

Governing equation of an eigen dynamic problem

with the unknown eigen mode φ and the corresponding eigen value λ, which is unknown, too. The equation shown above is a classical eigen problem defined by the sparse matrices K and M, which can be solved by an eigen solver.

The equation shows that the problem is not depending on external loads. This underlines that eigen frequency analysis refers to the homogeneous solution of the underlying PDE only.

Inclusion of External Loads

Alternatively, an eigen dynamics analysis of a structure under external loads, e.g. self weight, could be performed by definition of external loads and triggering of the "INITIAL_STATIC_CALCULATION" flag in the PC-ANALYSIS section. Such an analysis considers the loading by making use of the geometric stiffness contribution by performing an a priori static analysis. A solver for this static step necessitates a definition of an additional solver. An example input block is presented in the "Input Parameters" section

Relation between Eigenvalue and Eigenfrequency

The equation above refers to the eigen value of the eigen problem defined by the system matrices. What the user usually is interested in is the eigen frequeny of the system. The relation between this values is the following:

Relation between eigen alue and eigen frequency

Input Parameters

Parameter Description

Compulsory Parameters
Parameter Values, Default(*) Description
PC-ANALYSIS int : EIGENFREQUENCY Analysis ID
EIGEN_SOLVER PC-SOLVER int Linking to an eigen solver
OUTPUT PC-OUT int Linking to output objects (specifies the type of output format, e.g. GiD)
COMPCASE LD-COM int Linking to computation case objects which specify the boundary conditions (supports only!)
DOMAIN EL-DOMAIN int Linking to the domain the analysis should work on
NUM_ROOT int Number of eigen values to be computed
MASS_LUMPED 0, 1 indicates if mass matrix is lumped (1) or not (0)

Example of a Complete Input Block

PC-ANALYSIS 1: EIGENFREQUENCY
EIGEN_SOLVER = PC-SOLVER 1
OUTPUT = PC-OUT 1
COMPCASE = LD-COM 1
DOMAIN = EL-DOMAIN 1
NUM_ROOT = 4
MASS_LUMPED = 0


Example

The standard benchmark for eigen dynamics is a fixed supported beam modeled by SHELL8-elements. Due to its mass, stiffness and geometry the first four eigen frequencies can be computed analytically to

1st: 5.878 Hz
2nd: 23.513 Hz
3rd: 52.904 Hz
4th: 94.052 Hz

The related Carat output reads

     ##########################################
     ### Results of eigenfrequency analysis ###
     ##########################################
   Number of eigenfrequencies to compute: 4

   Number |   Eigenfrequency    |      Eigenvalue
 -------------------------------------------------------
     1         5.896354e+00           1.372546e+03
     2         2.380388e+01           2.236945e+04
     3         5.439391e+01           1.168047e+05
     4         9.884036e+01           3.856811e+05
 -------------------------------------------------------

 Eigenfrequency Analysis finished!

The finite element computation approximates the first eigen frequancy with an error of only 0.3%. The fourth eigen frequency is still approximated within a tolerance of 5%.

The full imput file belonging to this example can be found in the current Carat revision under http://www.develop.caratplusplus.de/upload_from_server/benchmark_examples/analyses/eigenfrequency_shell8_beam/shell8_beam_eigenfreq.dat. The following animation illustrates the system and the corresponding eigen vectors.

EigenDynamics Animation.GIF




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