Modal analysis using FEM

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The goal of modal analysis in structural mechanics is to determine the natural shapes and frequencies of an object or structure during free vibration. It is common to use the finite element method(FEM) to perform this analysis because, like other calculations using the FEM, the object being analyzed can have arbitrary shape and the results of the calculations are acceptable. The types of equations which arise from modal analysis are those seen in eigensystems. The physical interpretation of the eigenvalues and eigenvectors which come from solving the system are that they represent the frequencies and corresponding mode shapes. Usually, the only desired modes are the smallest because they are the most prominent modes at which the object will vibrate dominating all the higher modes.

 FEA eigensystems

For the most basic problem involving a linear elastic material which obeys Hooke's Law, the matrix equations take the form of a dynamic three dimensional spring mass system. The generalized equation of motion is given as <ref> Clough, Ray W. and Joseph Penzien, Dynamics of Structures, 2nd Ed., McGraw-Hill Publishing Company, New York, 1993, page 173 </ref> :

$\begin{bmatrix} M \end{bmatrix} \ddot{ \begin{bmatrix} U \end{bmatrix} } + \begin{bmatrix} C \end{bmatrix} \dot{ \begin{bmatrix} U \end{bmatrix} } + \begin{bmatrix} K \end{bmatrix} \begin{bmatrix} U \end{bmatrix} = \begin{bmatrix} F \end{bmatrix}$

where $\begin{bmatrix} M \end{bmatrix}$ is the mass matrix, $\ddot{ \begin{bmatrix} U \end{bmatrix} }$ is the 2nd time derivative of the displacement $\begin{bmatrix} U \end{bmatrix}$ (i.e. the acceleration), $\dot { \begin{bmatrix} U \end{bmatrix} }$ is the velocity, $\begin{bmatrix} C \end{bmatrix}$ is a damping matrix, $\begin{bmatrix} K \end{bmatrix}$ is the stiffness matrix, and $\begin{bmatrix} F \end{bmatrix}$ is the force vector. The only terms kept are the 1st and 3rd terms on the left hand side which give the following system:

$\begin{bmatrix} M \end{bmatrix} \ddot{ \begin{bmatrix} U \end{bmatrix} } + \begin{bmatrix} K \end{bmatrix} \begin{bmatrix} U \end{bmatrix} = \begin{bmatrix} 0 \end{bmatrix}$

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 $\ddot{ \begin{bmatrix} U \end{bmatrix} }$ is taken to equal $\lambda \begin{bmatrix} U \end{bmatrix}$, where $\lambda$ is an eigenvalue, and the equation reduces to <ref> Clough, Ray W. and Joseph Penzien, Dynamics of Structures, 2nd Ed., McGraw-Hill Publishing Company, New York, 1993, page 201 </ref> :

$\begin{bmatrix} M \end{bmatrix} \begin{bmatrix} U \end{bmatrix} \lambda + \begin{bmatrix} K \end{bmatrix} \begin{bmatrix} U \end{bmatrix} = \begin{bmatrix} 0 \end{bmatrix}$

In contrast, the equation for static problems is:

$\begin{bmatrix} K \end{bmatrix} \begin{bmatrix} U \end{bmatrix} = \begin{bmatrix} F \end{bmatrix}$

which is expected when all terms having a time derivative are set to zero.

 Comparison to linear algebra

In linear algebra, it is more common to see the standard form of an eigensystem which is expressed as:

$\begin{bmatrix} A \end{bmatrix} \begin{bmatrix} x \end{bmatrix} = \begin{bmatrix} x \end{bmatrix} { \lambda }$

Both equations can be seen as the same because if the general equation is multiplied through by the inverse of the mass, $\begin{bmatrix} M \end{bmatrix}^{-1}$, it will take the form of the latter <ref> Thomson, William T., Theory of Vibration with Applications, 3rd Ed., Prentice-Hall Inc., Englewood Cliffs, 1988, page 165 </ref>. It should be noted that because only the lower modes are desired, solving the system more likely involves the equivalent of multiplying through by the inverse of the stiffness, $\begin{bmatrix} K \end{bmatrix}^{-1}$, a process called inverse iteration <ref> Hughes, Thomas J. R., The Finite Element Method, Prentice-Hall Inc., Englewood Cliffs, 1987 page 582-584 </ref>. When this is done, the resulting eigenvalues, $\mu$, relate to that of the original by:

$\mu = \frac{1}{\lambda}$

but the eigenvectors are the same.

 Methods of solution

For linear elastic problems that are properly set up(no rigid body rotation or translation), the stiffness and mass matrices and the system in general are positive definite . These are the easiest matrices to deal with because the numerical methods commonly applied are guaranteed to converge to a solution. When all the qualities of the system are considered:

1) Only the smallest eigenvalues and eigenvectors of the lowest modes are desired
2) The mass and stiffness matrices are sparse and highly banded
3) The system is positive definite

a typical prescription of solution is first to tridiagonalize the system using the Lanczos algorithm. Next, use the QR algorithm to find the eigenvectors and eigenvalues of this tridiagonal system. If inverse iteration is used, the new eigenvalues will relate to the old by $\mu = \frac{1}{\lambda}$, while the eigenvectors of the original can be calculated from those of the tridiagonalized matrix by:

$\begin{bmatrix} r^{n} \end{bmatrix} = \begin{bmatrix} Q \end{bmatrix} \begin{bmatrix} v^{n} \end{bmatrix}$

where $\begin{bmatrix} r^{n} \end{bmatrix}$ is a Ritz vector approximately equal to the eigenvector of the original system, $\begin{bmatrix} Q \end{bmatrix}$ is the matrix of Lanczos vectors, and $\begin{bmatrix} v^{n} \end{bmatrix}$ is the $n^{th}$ eigenvector of the tridiagonal matrix.

 Example

The mesh shown below is the frame of a building modeled as beam elements, specifically consisting of of 930 elements and 385 nodal points. The building is constrained at its base where displacements and rotations are zero. The next images are that of the first 5 lowest modes of this building during free vibration. This problem can be seen as a depiction of the likeliest deflections a building would take during an earthquake. As expected, the first mode is a swaying of the building from front to back. The next mode is swaying of the building side to side. The third mode is a stretching and compression mode in the vertical <math > y [/itex] direction. For the fourth mode, the building nearly assumes the shape of half a sine wave. The fifth mode is a twisting mode.

Image:Building mode1.png
mode 1 swaying front to back
Image:Building mode01.png
mode 1 and original mesh

Image:Building mode2.png
mode 2 swaying side to side
Image:Building mode02.png
mode 2 and original mesh

Image:Building mode3.png
mode 3 stretching and compression
Image:Building mode03.png
mode 3 and original mesh

Image:Building mode4.png
mode 4 sine shape
Image:Building mode04.png
mode 4 and original mesh

Image:Building mode5.png
mode 5 twisting
Image:Building mode05.png
mode 5 and original mesh

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 References

• Clough, Ray W. and Joseph Penzien, Dynamics of Structures, 2nd Ed., McGraw-Hill Publishing Company, New York, 1993.
• Golub, Gene H. and C.F. Van Loan, Matrix Computations, 3rd Ed., The John Hopkins University Press, Baltimore, 1996.
• Hughes, Thomas J. R., The Finite Element Method , Prentice-Hall Inc., Englewood Cliffs, 1987.
• Thomson, William T., Theory of Vibration with Applications, 3rd Ed., Prentice-Hall Inc., Englewood Cliffs, 1988.