Modal analysis using FEM: Difference between revisions

Revision as of 18:05, 2 June 2009

Revision as of 13:49, 20 July 2009

Line 14:

:<math>

[M] [\ddot U] +

\begin{bmatrix} M \end{bmatrix} \ddot{ \begin{bmatrix} U \end{bmatrix} } +

[C] [\dot U] +

\begin{bmatrix} C \end{bmatrix} \dot{ \begin{bmatrix} U \end{bmatrix} } +

[K] [U] =

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

[F]

\begin{bmatrix} F \end{bmatrix}

</math>

where <math> \begin{bmatrix} [M \end{bmatrix}] </math> is the mass matrix,

<math> [\ddot{ \begin{bmatrix} U \end{bmatrix} }] </math> is the 2nd time derivative of the displacement

<math> \begin{bmatrix} [U \end{bmatrix}] </math> (i.e. the acceleration), <math> [\dot { \begin{bmatrix} U \end{bmatrix} }] </math>

is the velocity, <math> \begin{bmatrix} [C \end{bmatrix}] </math> is a damping matrix,

<math> \begin{bmatrix} [K \end{bmatrix}] </math> is the stiffness matrix, and <math> \begin{bmatrix} [F \end{bmatrix}] </math>

is the force vector. The only terms kept are the 1st and 3rd terms on the left hand side

which give the following system:

:<math>

[M] [\ddot U] +

\begin{bmatrix} M \end{bmatrix} \ddot{ \begin{bmatrix} U \end{bmatrix} } +

[K] [U] =

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

[0]

\begin{bmatrix} 0 \end{bmatrix}

</math>

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 <math> [\ddot{ \begin{bmatrix} U \end{bmatrix} }] </math>

is taken to equal <math> \lambda \begin{bmatrix} [U \end{bmatrix}] </math>,

where <math> \lambda </math> is an eigenvalue,

and the equation reduces to:<ref> Clough, Ray W. and Joseph Penzien, ''Dynamics of Structures'', 2nd Ed.,

Line 43:

:<math>

[M] [U] \lambda +

\begin{bmatrix} M \end{bmatrix} \begin{bmatrix} U \end{bmatrix} \lambda +

[K] [U] =

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

[0]

\begin{bmatrix} 0 \end{bmatrix}

</math>

Line 51:

:<math>

[K] [U] =

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

[F]

\begin{bmatrix} F \end{bmatrix}

</math>

Line 63:

:<math>

[A] [x] =

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

\begin{bmatrix} [x \end{bmatrix}] { \lambda }

</math>

Both equations can be seen as the same because if the general equation is

multiplied through by the inverse of the mass,

<math> \begin{bmatrix} [M \end{bmatrix}]^{-1} </math>,

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,

<math> \begin{bmatrix} [K \end{bmatrix}]^{-1} </math>, 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, <math> \mu </math>, relate to that of the original by:

Line 92:

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 [[tridiagonal|tridiagonalize ]] the system using the

Line 103:

:<math>

[r^{n}] =

\begin{bmatrix} r^{n} \end{bmatrix} =

[Q] [v^{n}]

\begin{bmatrix} Q \end{bmatrix} \begin{bmatrix} v^{n} \end{bmatrix}

</math>

where <math> \begin{bmatrix} [r^{n} \end{bmatrix}] </math> is a Ritz vector approximately equal to

the eigenvector of the original system, <math> \begin{bmatrix} [Q \end{bmatrix}] </math> is the matrix

of Lanczos vectors, and <math> \begin{bmatrix} [v^{n} \end{bmatrix}] </math> is the <math> n^{th} </math> eigenvector

of the tridiagonal matrix.