Page 243 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

224 SLENDER STRUCTURES AND AXIAL FLOW

model, wherefore Young's modulus E and the shear modulus G become complex: E* =
E( 1 + ip) and G" = G( 1 + ip), with p being the hysteretic damping constant.
The boundary conditions for a free end are Q = A= 0; for a clamped end, w = @ = 0.
Thus, in dimensionless terms, we have
(i) for a clamped-clamped pipe:

(4.39a)

(ii) for a cantilevered pipe:

= 0. (4.39b
E= 1

4.4.2 Method of analysis
The modal analysis method is utilized for the solution of the equations of motion. The
motion being free, let

where w is a dimensionless frequency, related to the dimensional radian frequency of
motion, Q, by
M+m 'I2
w= L2Q. (4.41)

Furthermore, the fluid-dynamic force fA is assumed to vary temporally in the same manner,
i.e.
f~ = fAeior. (4.42)

As in previous analyses, w is generally complex, and the system is stable or unstable
accordingly as the imaginary part of w is positive or negative.
The modal analysis method proceeds by expressing V(t) and $(e) as the superposition
of an infinite set of comparison functions (Galerkin's technique), i.e.

(4.43)

where a, and h, are dimensionless generalized coefficients, and Yn(t) and Pn(t) are
the eigenfunctions of a Timoshenko beam, with the appropriate boundary conditions,
expressed in dimensionless form; this solution then inherently satisfies the boundary condi-
tions. Substitution of equations (4.40), (4.42) and (4.43) into (4.38) and application of the