Page 260 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 8.7 Transformation of Coordinates (Standard Computer Form) 247

Although A and A are different, they are both called the dynamic matrix
because the dynamic properties of the system are defined by A ox A. Again, matrix
A is generally not symmetric.
If a given system is solved by either Eq. (8.6-2) or (8.6-3), the eigenvalues will
be reciprocally related, but will result in the same natural frequencies. The
eigenvectors for the two equations will also be identical.

8.7 TRANSFORMATION OF COORDINATES (STANDARD
COMPUTER FORM)

In Eqs. (8.6-2) or (8.6-3), dynamic matrices A and A are usually unsymmetric. To
obtain the standard form of the equation of motion for the computer, the following
transformation of coordinates
= f / " 'y (8.7-1)
is introduced into the equation
[-AM + K ]X = 0
which results in the transformed equation

[ -AMf/ -‘ + = 0
Premultiplying this equation by the transpose of U \ which is designated as

= U~
we obtain the equation
+ U -’KV-^]Y= 0 (8.7-2)
It is evident here that if we decompose either M ox K into U ^ U in the preceding
equation, we would obtain the standard form of the equation of motion.
With M = U^U, Eq. (8.7-2) becomes

[-A7 + U-^KU-^]Y= 0 A (8.7-3)
whereas if K = the equation is
- A I ] Y = 0 X = \/(o^ (8.7-4)
Both equations are in the standard form

[ -A/ + A ]Y = 0
where the dynamic matrix A is symmetric. We now define the transformation
matrix U.