Page 339 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: NONLINEAR AND CHAOTIC DYNAMICS 319

(a) 2-D motions
The first nonlinear study is due to Rousselet & Henmann (1977), dealing with a system of
two rigid pipes (N = 2) hanging vertically, with ideal articulations of zero stiffness and
damping. The equations of motion are (5.64) and (5.65), but with kl = k2 = 0. Hence,
in the nondimensionalized equations, different parameters from (5.68) are utilized, e.g.
and
u = lJ/m so on. Fluctuations in the flow velocity due to varying acceleration and
gravity head are taken into account, as per equations (5.66) and (5.67); they are taken to
be small, so that if uo is the mean flow velocity,

u = uo + Au, AU << 1. (5.102)

The main objective is to obtain information on the dynamics in the vicinity of the Hopf
bifurcation which leads to flutter; in particular, whether the predicted limit cycle is stable
or unstable (supercritical or subcritical Hopf bifurcation), and what is its amplitude. With
(5.102), (5.64) and (5.65), the equations of motion may be written in the form
+
[MI141 + [m) [me1 = IF), (5.103)
where {e) = {el, O2IT; {F} = IF1, F2JT contains all the nonlinear terms, Le.

~01,&, &,el, 42, AU, AL; system parameters),
~
(F~, 2 = f(el, (5.104)
)
where the ‘system parameters’ are a, B, y, no, h and h* - see equations (5.68). Similarly,
(5.102), (5.66) and (5.67) lead to a ‘flow equation’,
Ai = g(Au, el,€$, e:, 6;; system parameters). (5.105)

The ingenious procedure adopted to solve equations (5.103)-(5.105) (Rousselet 1975;
Rousselet & Henmann 1977) is outlined in what follows.
(i) The linear part of (5.103) is solved first, yielding the eigenvalues and eigenvectors,
and hence also the critical value for flutter uo = uof, if it exists; it is noted that for
-
fi > 0.51 it does not, and only divergence is then possible.
(ii) The equation of motion is then transformed into first-order form,

[Blk} + [EIIz) = {@I> (5.106)
T
(z} = {{e}, (e)} - cf. equations (2.16) and (2.17). Then, the homogeneous form of
(5.106) and its adjoint [see equation (2.20)] are solved simultaneously for uo = ucj,
yielding the same eigenvalues but different eigenvectors from those of the original system.
The use of the biorthogonality property [equation (2.21)] then allows the decoupling of
the system. Attention is thenceforth devoted exclusively to the mode associated with the
Hopf bifurcation, ignoring the other (stable one), thus reducing the fourth-order system
in (5.106) to one of order 2. Nowadays, the same would have been accomplished via the
centre manifold method (Appendices F and H). The resulting equation has the form
ji + (a2 + w2)x = 2ai + f(F1, Fz, UO, a, w; system parameters, modal form), (5.107)

where h = a + iw, la1 << 1, it being understood that uo is close to u,~; ‘modal form’
signifies the eigenvector information for the mode undergoing the Hopf bifurcation.