Page 27 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

10 SLENDER STRUCTURES AND AXIAL FLOW

where (A] is a column of unknown amplitudes and R the circular frequency. Substi-
tuting (2.7) into (2.6) and defining A 02, leads to the standard eigenvalue problem,
(ALII - [Wl){Al = (2.8)

where [I] is the unit matrix. Nontrivial solution of (2.8) requires that
det ([W] - A[Z]) = 0, (2.9)

which is the characteristic equation, from which the eigenvalues, Ai, i = 1, 2, . . . , N,
and hence the corresponding eigenvectors, {A}i or Ai, may be found. The free-vibration
characteristics of the system are fully determined by the eigenvalues (and hence the eigen-
frequencies Qi = and the corresponding eigenvectors. The latter may be viewed as
shape functions. Thus, for the double pendulum of Figure 2.l(a), if MI = 2M, M2 = M
and 11 = 12 = I, one obtains hl = $(g/l) and 12 = 2(g/l). The first- and second-mode
eigenvectors are, respectively { 1, l)T and { 1, -2)T, which means that, for motions purely
in the first mode (at Rl), the second pendulum oscillates with the same angular ampli-
tude as the first, and in the same direction; while in the second mode (at Q2), the second
pendulum has twice the amplitude of the first, but in the opposite sense. Pure first-
mode motions could be generated via initial conditions {q(O)] = (1, ($0)) = [O),
and similarly for second-mode motions. Other initial conditions generate motions which
involve - can be synthesized from - both eigenvectors and both eigenfrequencies.
As a consequence of [MI and [K] being symmetric, the eigenvalues are real (as in
the foregoing example),+ and the following weighted orthogonality holds true for the
eigenvectors:

{A}; [K]{A]i = 0, {A); [M]{A]i = 0 for i # j; (2.10)
if [W] is symmetric too - recall that the product of two symmetric matrices is not neces-
sarily symmetric - then direct orthogonality also applies, i.e. {A}T(A)i = 0 for i # j.
Relations (2.10) hold true, provided that the eigenvalues are distinct; the case of repeated
eigenvalues will be treated later.
Since [MI is, or can be, derived from the kinetic energy, which is a positive definite
function, [MI is a positive definite matrix (Meirovitch 1967; Pipes 1963).$ If [K] is also
positive definite, then so is the system, and the eigenvalues are all positive. If [K] is only
positive, the system is said to be semidejnite, and it may have zero eigenvalues - e.g.
if the system as a whole is unrestrained.
For the forced response, equation (2.4) has to be solved. This may be done in many
ways, e.g. by the use of Laplace transforms or by modal analysis. This latter will be
reviewed briefly in what follows. First, the modal matrix is defined,

[AI = [{A)IIA)z. . . (AINI; (2.11)
then, the so-called expansion theorem is invoked, stating that any vector, including {q),
in the vector space spanned by [A] may be expressed (‘synthesized’) in terms of the

+This is physically reasonable - see equation (2.7).
*If the determinant of successive submatrices, each containing the left-hand corner element are all positive,
then the matrix is positive definite. That is, for a 3 x 3 matrix [MI : 1n11 > 0, nqlrn22 - n~1n112 > 0 and
det[M] > 0 and similarly for higher order matrices. If any of the determinants is zero, then [MI is said to be
only positive rather than positive definite.