Page 106 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

88 SLENDER STRUCTURES AND AXIAL FLOW

integration over the domain [0,1] yields

(3.86)

in which the orthonormality of the eigenfunctions was utilized (Le. the fact that
JJ @,@, dt = S,,, S,, being Kronecker’s delta), as well as the fact that @r h:@,, A,
=
being the rth dimensionless eigenvalue of the beam. The definite integrals may be
evaluated in closed form, defining the following set of constants:

Their values for some sets of boundary conditions are given in Table 3.1, in which the
a, are the constants associated with the 4, [Bishop & Johnson 1960; cf. equation (2.28)].
The method for evaluating h,,, c,, and d,, analytically is illustrated in Appendix B.
Equation (3.86) may be written in matrix form as follows:

q + [F + 2/3”2uB]Q + {A + yB + [u2 - r + n(l - 214 - y]C + yD}q = 0, (3.88)

where q = (41, q2, . . . , q~)~, F and A are diagonal matrices with elements (ah: + a)
and (A: + k), respectively, and B, C and D are matrices with elements b,,, c,, and d,,,
respectively. This equation may be written in standard form,

[Mlq + [CIQ + [Klq = 0 (3.89)

cf. equation (2.1), Section 2.1. Its eigenvalues may be found in various ways; e.g. by
transforming it into first-order form by the procedure leading from equation (2.15) to
(2.17), and then to the standard eigenvalue problem of equation (2.18). The eigenvalues
may be obtained numerically, e.g. by the IMSL library subroutines or those given by
Press et aE. (1992).

3.4 PIPES WITH SUPPORTED ENDS
3.4.1 Main theoretical results

We first consider the simplest possible system: a simply-supported (or ‘pinned-pinned’)
horizontal pipe (y = 0) with zero dissipation, and with /? = 0.1, r = I7 = k = 0 in equa-
tion (3.70). The dynamical behaviour of this system with increasing dimensionless flow
velocity, u, is illustrated by the Argand diagram of Figure 3.9. It is recalled that %e(w) is
the dimensionless oscillation frequency, while 9am(w) is related to damping, the damping
ratio being ( = 9am(w)/%e(w). The general dynamical features already remarked upon in
Sections 3.2.1 and 3.2.3 are clearly seen: (i) since dissipation is absent in this example, the