Page 323 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

304 SLENDER STRUCTURES AND AXIAL FLOW

0, is
afi”” + q”” + u - f - y(1 - 6) - id (v’)~ dt} q”

{2
(5.80)

with
d = AL2/I, (5.81)

all other quantities being as in (3.71), (5.38) and (5.44). Most of the detailed work is
done with a simplified form of (5.80) by taking r = y = 0 and neglecting the nonlinear
dissipative term, namely

+ $”’ + { *2 - I’ (q’)’ d:} q” + 2p1I2ulj’ + (TG + ij = 0, (5.82)

for a simply-supported (pinned-pinned) pipe - thus satisfying q = q” = 0 at 6 = 0, 1.
Holmes considers the dynamics of the system in two ways, via (a)$nite dimensional
analysis and (b) in$nite dimensional analysis, to be outlined in what follows. Then, some
interesting work by Lunn (1982) is discussed in (c), leading to (d) the final conclusion.
(a) Finite dimensional analysis

For equation (5.82), a two-mode Galerkin discretization of the simply-supported pipe
system is obtained via ~(6, t) = E[& sin(rn()]q,(t), r = I, 2. Converting this to first-
order form, leads to the following simple four-dimensional system:

41 = Ply 42 = P2>
2
2
2
= -X (n - U )SI - (an4 + a)pl + ~p up2 - ;dn2(q: + 4q;)q1, (5.83)
16 112
It is seen that the nonlinearities are of the stiffening cubic type (of the same sign as the
linear stiffness), which helps explain the global stability of the system.
It is useful here to refresh the reader’s mind as to the linear dynamics of the system.
Since damping is present (a, o # 0), the eigenvalues for u = 0 are complex conjugate
pairs with negative real parts (Figure 2.10). Here the notation introduced by Holmes is
utilized, in which such eigenvalues are denoted by the quartet h = I-. -, -, -}, the
signs being those of the real parts of the eigenvalues; a + means that one of the eigen-
values has positive real part, while 0 denotes a zero real part. As u is increased, the
first bifurcation occurs at u = n (Section 3.4.1); it is a pitchfork bifurcation, leading
to divergence, as shown in Figure 5.4 (cf. Figures 3.9 and 3.14). Hence, for u > n we
have h = [+, -, -, -). The solution of equations (5.83) shows that two new fixed points
are generated for u > n, located increasingly farther away from the origin. The numer-
ical solutions for a = (T = 0.01, = 0.2, are shown in Figure 5.5, together with centre
manifold predictions, which will be discussed next.
At the critical point where the bifurcation occurs, the four-dimensional system (5.83)
is projected onto the centre manifold (Appendices F.2 and H.l), which in this case is