Users:General FEM Analysis/Analyses Reference/Eigenfrequency Analysis

From Carat++ Public Wiki
(Difference between revisions)
Jump to: navigation, search
(Example of a Complete Input Block with consideration of External Loads)
Line 88: Line 88:
 
MASS_LUMPED = 0
 
MASS_LUMPED = 0
 
INITIAL_STATIC_CALCULATION=TRUE
 
INITIAL_STATIC_CALCULATION=TRUE
 +
INITIAL_LOAD_STEPS = 1
 +
MAX_ITER_EQUILIBRIUM = 100
 +
EQUILIBRIUM_ACCURACY = 1e-12
 
</pre>
 
</pre>
  

Revision as of 09:29, 8 November 2017


Contents

General Description

This analysis is intended for the solution of dynamic eigenvalue problems. It is suitable for analyzing the vibration characteristics of structures and to determine their eigenfrequencies.


Structure of governing equation

A dynamic eigenvalue 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 eigenvalue problem defined by the sparse matrices K and M, which can be solved by an eigenvalue solver.

The equation shows that the problem does not depend on external loads. This underlines that the eigenfrequency analysis refers to the homogeneous solution of the underlying PDE only.


Inclusion of External Loads

Alternatively, an dynamic eigenvalue 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 from an a priori static analysis. A solver for this static step necessitates the 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 eigenvalue of the dynamic eigenvalue problem defined by the system matrices. What the user usually is interested in is the eigenfrequeny of the system. The relation between these values is the following:

Relation between eigenvalue and eigenfrequency

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 of a Complete Input Block with consideration of External Loads

PC-ANALYSIS 1: EIGENFREQUENCY
EIGEN_SOLVER = PC-SOLVER 1
LINEAR_SOLVER = PC-SOLVER 2
OUTPUT = PC-OUT 1
COMPCASE = LD-COM 1
DOMAIN = EL-DOMAIN 1
NUM_ROOT = 4
MASS_LUMPED = 0
INITIAL_STATIC_CALCULATION=TRUE
INITIAL_LOAD_STEPS = 1
MAX_ITER_EQUILIBRIUM = 100
EQUILIBRIUM_ACCURACY = 1e-12

Examples

Example of shell-discretized beam

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 corresponding 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 eigenfrequency with an error of only 0.3%. The fourth eigenfrequency is still approximated within a tolerance of 5%.

The full imput file belonging to this example can be found in the current Carat revision:

  • ..\examples\benchmark_examples\analyses\eigenfrequency_shell8_beam\shell8_beam_eigenfreq.dat

The following animation illustrates the system and the corresponding eigen vectors.

EigenDynamics Animation.GIF

Benchmark examples

  • eigenfrequency of a single-span beam discretized by (SHELL8)-elements: ..\examples\benchmark_examples\analyses\eigenfrequency_shell8_beam\shell8_beam_eigenfreq.dat
  • eigenfrequency of a plate discretized by (SHELL8)-elements: ..\examples\benchmark_examples\analyses\eigenvalue_shell8_plate\cbm_eigenvalue_shell8_plate.dat
  • eigenfrequency of cable discretized by (TRUSS1)-elements: ..\examples\benchmark_examples\analyses\eigenvalue_truss1_I\cbm_vib_row_50_elm.txt
  • eigenfrequency of a flat plate discretized by (MEMBRANE1)-elements with prestress: ..\examples\benchmark_examples\analyses\eigenvalue_membrane1_I\cbm_Dynamic_Membrane_200_elm.txt




Whos here now:   Members 0   Guests 0   Bots & Crawlers 1
 
Personal tools
Content for Developers