Page 28 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

CONCEPTS, DEFINITIONS AND METHODS 11

eigenvectors making up [A]. Hence, the coordinate transformation

(41 = [Al(Yl (2.12)
is introduced, in which yi, i = 1, . . . , N, are the normal or principal coordinates. Substi-
tuting (2.12) into (2.4), and pre-multiplying by [AIT leads to
[PIIYI + [SIIy) = [AITIQl = (FI, (2.13)

in which
[PI = [AITW1[Al1 [SI = [AITIKl[Al (2.14)

are diagonal, in view of the relations (2.10).
The system (2.13) has therefore been decoupled. Each row reads p;y; + s; y; = F, (t),
which is easily solvable, subject to the initial conditions (y(0)J = [A]-' {q(O)] and
(y(0)) = [A]-'{q(O)). The response in terms of the original coordinates may then be
obtained by application of (2.12).
In case of repeated eigenvalues, or if [MI or [K] are not symmetric but the eigenvalues
are still real, provided that linearly independent eigenvectors may be found,' one may
proceed as follows: (i) equation (2.4) is pre-multiplied by [MI-', (ii) transformation (2.12)
is introduced, and (iii) the equation is decoupled by pre-multiplication by [AI-'; this
leads to
IY} + [AIIYI = [Al-'[~l-'(Ql~ (2.15)
where [A]-'[W][A] = [A] has been utilized, and [A] is the diagonal matrix of the eigen-
values.
If damping is present, then the full form of equation (2.1) applies - provided, of
course, that the damping is viscous or that it may be approximated as such. In this case,
eigenvalues and eigenvectors are no longer real. The procedure that follows applies to
cases where [MI, [K] and [C] are symmetric - the latter being so if [C] is derived from
a dissipation function, for instance (Bishop & Johnson 1960). The following partitioned
matrices and vectors of order 2N are defined:

and equation (2.1) may now be reduced into the first-order form

[Bl(iJ + [El(z) = (@I. (2.17)
The procedure henceforth parallels that of the conservative system. Assuming solutions of
the form {z] = (A} exp(At) (A} exp(iQr), the reduced equation (2.17) eventually leads
to the eigenvalue problem
(2.18)
(Nil - [YI)(Al = to19
where [Y] = -[B]-'[E]. The eigenvalues, A;, and eigenvectors (A];, i = 1,2,. . . ,2N,
may now be determined. The A; occur in complex conjugate pairs,' and the eigenvectors

'Hence, in principle and if desired, a set of orthogonal eigenvectors may be determined via the Gram-Schmidt
procedure.
'Note that, even for a conservative mass-spring one-degree-of-freedom system, one obtains R = fm,
where the negative value is usually ignored (see Section 2.3); here f2i A, so A1.2 = Oi f (k/ni)''*.