Three-dimensional dynamics of a cantilevered pipe conveying pulsating fluid

Abstract

This paper investigates the stability and three-dimensional (3-D) nonlinear dynamics of a cantilevered pipe with an internal fluid having a harmonic component of flow velocity superposed on a constant mean value. The nonlinear equations of motion for an inextensible cantilevered pipe with the consideration of internal pulsating flow are presented. The nonlinear inertial terms in the governing equations are replaced by equivalent displacement and velocity terms by using a perturbation method. The partial differential equations are then transformed into a set of ordinary differential equations (ODEs) by using the Galerkin method. The instability regions of the subcritical and supercritical resonances of a linear system are determined via the Floquet theory. The effects of mean flow velocity and mass ratio are investigated. The resulting coupled nonlinear differential equations are numerically solved using a fourth-order Runge-Kutta integration scheme for the subcritical and supercritical flow velocities. The nonlinear dynamical responses are presented in the form of bifurcation diagrams, time histories, phase portraits, power spectral densities (PSDs) and Poincaré maps. Some interesting and sometimes unexpected results have been observed with different flow velocities. The analytical model is found to exhibit rich and variegated dynamical behaviors which include 2-D or 3-D periodic, quasiperiodic and chaotic motions. The convergence analysis of the number of truncating modes in the Galerkin approach is also conducted.