Page 425 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: NONLINEAR AND CHAOTIC DYNAMICS 397

an analysis of more limited scope, essentially redoing some of the same work as by
Namachchivaya and co-workers, but via more standard and easily accessible terms, in fact
following Namachchivaya & Ariaratnam (1987). What is presented below is a melange of
all of this, but differentiating according to authors as appropriate; the results are discussed
separately.
In all these studies the nonlinear equation used is similar to Holmes’ (Sections 5.2.9(b)
and 5.5.2), the only nonlinearity taken into account being due to the deformation-induced
tension between laterally and axially fixed supports; the rest of the equation, including the
terms associated with the flow pulsation (Le. the iC terms), is linear and as in Section 4.5
and Pai‘doussis & Issid (1974). Thus, the nonlinear dimensionless equation of motion is
given by variants of equation (5.80):

in which the parameters are defined in (3.71) and (5.81), subject to

u = UO(1 + p cos or). (5.144)

Assuming y = ad = a = 0 and taking u as in (5.144), Namachchivaya (1989) discretizes
equation (5.143) into one of two degrees of freedom,

= [p,91/2u~w(C D) sin wt - p(2u;C + 2,9’/’uoB) cos wtlq - aAq, (5.145)
-
in which @ = $dcijcklqiqjqkqi, the ckl being terms of the type making up C, A is a
diagonal matrix with elements h;, where hj are the dimensionless beam eigenvalues - cf.
equation (4.70); p and a are assumed to be small (<< 1).+ To accentuate its structure, this
equation may be written in simplified notation as

4 + Gq + Kq = p(EI cos ot + E2 sin os)q - i- f(q). (5.146)

This equation is transformed into standard form by an elegant Hamiltonian symplectic
transformation (Namachchivaya 1989), a more standard technique via the solutions of the
unperturbed system (Namachchivaya & Tien 1986b), and a standard method by Chang &
Chen (1 994), all leading to variants of the following equation:

u = (Bo + aBl)u + ,u(Kl cos (or + sin on)u + f(u), (5.147)

+The notation here differs from that in many of this group of papers where, to put in evidence the smallness
of and a, they are scaled by E, E << 1; thus, IL = EW* and a = <a*.