## Multiple Degree of Freedom Systems: Eigenvalue Problem

The problem of finding the natural frequencies and mode shapes for a multiple degree of freedom system is essentially an eigenvalue problem, although we have so far not presented it as such. To see this recall that the equations of motion for an MDOF system can be written as

(8.26)

where and are the mass and stiffness matrices respectively and is a column vector containing the coordinates. We look for solutions in which all of the coordinates are undergoing simple simultaneous harmonic motion of the form

so that

Substituting these results into the equations of motion gives

or

(8.27)

Pre-multiplying both sides of (8.27) by the inverse of the mass matrix gives

or

(8.28)

which is a standard eigenvalue problem

- is known as the
*dynamic matrix* - the natural frequencies (squared) are the
*eigenvalues*of - the mode shapes are the associated eigenvectors of

The dynamic matrix can be used with standard software packages such as Matlab.

Alternatively, starting with equation (8.27) and premultiplying both side by gives

or

which is another formulation of the standard eigenvalue problem.