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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
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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
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Mode % s#aying front to bac!
Mode % and original mesh
Mode s#aying side to side
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Mode and original mesh
Mode stretching and compression
Mode and original mesh
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Mode ; sine shape
Mode ; and original mesh
Mode : t#isting
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Mode : and original mesh