The equation of motion can then be expressed as

(40.8)

Where,

(40.8)

 is a 2n dimensional positive definite matrix whereas is neither positive definite nor symmetric. Considering the Cholesky decomposition of as  and introducing the transformation , the above state space equation can be written as

(40.9)

Where,  and . The left and right eigenvalue problems associated with the transformed state equation are:

In order to decouple the equations of motion, assuming the solution of Equation (40.9) of the form , where,   is the modal matrix, a model transformation can be carried out to get decoupled states as

	(40.10)
	(40.11)

Noting the above equation could also be written as

(40.12)

Retaining only a finite numbers of frequencies, approximating the response of distributed parameter system the above equation can be expressed as:

(40.9)

Where,   and are the numbers of frequencies to be retained. The finite element routine and the described solutions scheme could be implemented in MATLAB environment taking advantage of the built in application routines for linear algebraic operations.

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