ABSTRACT
Consider a specific system for which K 1 = 40N/cm, K2 = K 3 = K4 = 20N/cm, and Ks = 90 N/ em, and mI= m2 = m3 = 2 kg. Let Kref = ION/em and mrer = 2 kg. For these values, Eq. (2.8) becomes:
(8 - A)X1 - 2X2 - 2X3 = 0 -2X1 + (4 - },)X2 - 2X3 = 0 -2X1 - 2X2 + (13 - },)X3 = 0
(2.9a) (2.9b) (2.9c)
Equation (2.9) is a system of three homogeneous linear algebraic equations. There are four unknowns: XI' X2, X3, and A (i.e., w). Clearly unique values of the four unknowns cannot be determined by three equations. In fact, the only solution, other than the trivial solution X = 0, depends on the special values of }" called eigenvalues. Equation (2.9) is a classical eigenproblem. The values of }, that satisfy Eq. (2.9) are called eigenvalues. Unique values of XT = [XI X2 X3] cannot be determined. However, for every value of }" relative values of XI' X2, and X3 can be determined. The corresponding values of X are called eigenvectors. The eigenvectors determine the mode of oscillation (i.e., the relative values of XI' X2, X3). Equation (2.9) can be written as
I (A - AI)X = 0 I (2.10)
