modal analysis using fem

8/12/2019 Modal Analysis Using FEM

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Modal analysis using FEM

The goal of modal analysis in structural mechanics is to determine the natural shapes andfrequencies of an object or structure during free vibration. It is common to use the finite

element method (FEM) to perform this analysis because li!e other calculations using theFEM the object being analy"ed can have arbitrary shape and the results of thecalculations are acceptable. The types of equations #hich arise from modal analysis are

those seen in eigensystems. The physical interpretation of the eigenvalues and

eigenvectors #hich come from solving the system are that they represent the frequenciesand corresponding mode shapes. $sually the only desired modes are the smallest

because they are the most prominent modes at #hich the object #ill vibrate dominating

all the higher modes.

Contents

% FE& eigensystems

%.% 'omparison to linear algebra Methods of solution

E*amples

FEA eigensystems

For the most basic problem involving a linear elastic material #hich obeys +oo!e,s -a#

the matri* equations ta!e the form of a dynamic three dimensional spring mass system.

The generali"ed equation of motion is given as

#here is the mass matri* is the nd time derivative of the displacement

(i.e. the acceleration) is the velocity is a damping matri* is the stiffness

matri* and is the force vector. The only terms !ept are the %st and rd terms on theleft hand side #hich give the follo#ing system

This is the general form of the eigensystem encountered in structural engineering using

the FEM. Further harmonic motion is typically assumed for the structure so that is

ta!en to equal #here / is an eigenvalue and the equation reduces to



by #hile the eigenvectors of the original can be calculated from those of thetridiagonali"ed matri* by

#here is a 6it" vector appro*imately equal to the eigenvector of the original

system is the matri* of -anc"os vectors and is the nth eigenvector of the

tridiagonal matri*.

Example

The mesh sho#n belo# is the frame of a building modeled as beam elements specifically

consisting of 78 elements and 9: nodal points. The building is constrained at its base

#here displacements and rotations are "ero. The ne*t images are that of the first : lo#estmodes of this building during free vibration. This problem can be seen as a depiction of

the li!eliest deflections a building #ould ta!e during an earthqua!e. &s e*pected the first

mode is a s#aying of the building from front to bac!. The ne*t mode is s#aying of the

building side to side. The third mode is a stretching and compression mode in the vertical y direction. For the fourth mode the building nearly assumes the shape of half a sine

#ave. The fifth mode is a t#isting mode.

3riginal mesh



Mode % s#aying front to bac!

Mode % and original mesh

Mode s#aying side to side



Mode and original mesh

Mode stretching and compression

Mode and original mesh



Mode ; sine shape

Mode ; and original mesh

Mode : t#isting



Mode : and original mesh

modal analysis using fem

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