TABLE OF CONTENTS It was shown in Chapter 6 that provided the eigenvalues and eigenvectors of a system can be found, it is possible to transform the coordinates of the system from local or global coordinates to coordinates consisting of normal or 'principal' modes.
Depending on the damping, the eigenvalues and eigenvectors of a system can be real or complex, as discussed in Chapter 6. However, real eigenvalues and eigenvectors, derived from the undamped equations of motion, can be used in most practical cases, and will be assumed here, unless stated otherwise.
In Example 6.3, we used a very basic 'hand' method to demonstrate the derivation of the eigenvalues and eigenvectors of a simple 2-DOF system; solving the characteristic equation for its roots, and substituting these back into the equations to obtain the eigenvectors. In this chapter, we look at methods that can be used with larger systems.
In Chapter 6, it was shown that the undamped, homogeneous equations of a multi-DOF system can be written as:
| (7.1) |  |
or concisely,
| (7.2) |  |
where
[ M] is the ( n n), symmetric, mass matrix, [ K] the ( n n), symmetric, stiffness matrix, ? any one of the n natural frequencies and { z} the corresponding ( n 1) eigenvector.
Putting ? = ? 2, Eq. (7.2) can be written as:
| (7.3) |  |
where ? is any one of the n eigenvalues.
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