Page 184 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Properties of

Vibrating Systems

The elastic behavior of a system can be expressed either in terms of the stiffness or
the flexibility. So far, we have written the equations of motion for the normal mode
vibration in terms of the stiffness K\
(-co^[M] + [/:]){A-} = {0} (a)

In the stiffness formulation, the force is expressed in terms of the displacement:
{F}=[/C]{A'} (b)
The flexibility is the inverse of the stiffness. The displacement is here written
in terms of the force:
{X) = [ K] ' { F)
= [« K n (c)
The equation of motion in terms of the flexibility is easily determined by premulti­
plying Eq. (a) by [K]~^ = [a]:
i-co^[a][M]+I)[X} ={0} (d)
where K~ = I = unit matrix.
The choice as to which approach to adopt depends on the problem. Some
problems are more easily pursued on the basis of stiffness, and for others, the
flexibility approach is desirable. The inverse property of one or the other is an
important concept that is used throughout the theory of vibration.
The orthogonal property of normal modes is one of the most important
concepts in vibration analysis. The orthogonality of normal modes forms the basis
of many of the more efficient methods for the calculation of the natural frequen­
cies and mode shapes. Associated with these methods is the concept of the modal
matrix, which is essential in the matrix development of equations.

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