NONLINEAR ANALYSIS OF THE PARAMETRIC RESONANCES OF A PLANAR FLUID-CONVEYING CANTILEVERED PIPE

Abstract

This paper deals with the nonlinear dynamics and the stability of cantilevered pipes conveying fluid, where the fluid has a harmonic component of flow velocity, assumed to be small, superposed on a constant mean value. The mean flow velocity is near the critical value for which the pipe becomes unstable by flutter through a Hopf bifurcation. The partial differential equation is transformed into a set of ordinary differential equations (ODEs) using the Galerkin method. The equations of motion contain nonlinear inertial terms, and hence cannot be put into standard form for numerical integration. Various approaches are adopted to tackle the problem: (a) the centre manifold theory applied on the set of non-autonomousequations, followed by the normal form method, yielding both the principal and the fundamental resonances; (b) a perturbation method via which the nonlinear inertial terms are removed by finding an equivalent term using the linear equation; the system is then put into first-order form and integrated using a Runge-Kutta scheme; (c) a finite difference method based on Houbolt's scheme, which leads to a set of nonlinear algebraic equations that is solved with a Newton-Raphson approach; (d) periodic solutions and stability boundaries are obtained using an incremental harmonic balance method as proposed by S. L. Lau. Using the four methods, the dynamics of the pipe conveying fluid are investigated in detail. For example, the effects of (i) the forcing frequency, (ii) the perturbation amplitude, and (iii) the flow velocity are considered. Particular attention is paid to the effect of the nonlinear terms. These results are compared with experiments undertaken in our laboratory, utilizing elastomer pipes conveying water. The pulsating component of the flow is generated by a plunger pump, and the motions are monitored by a noncontacting optical follower system. It is shown analytically, numerically and experimentally, that periodic and quasiperiodic oscillations can exist, depending on the parameters.