Three-dimensional dynamics of a cantilevered pipe conveying fluid, additionally supported by an intermediate spring array

In this paper, the three-dimensional (3-D) non-linear dynamics of a cantilevered pipe conveying fluid, constrained by arrays of four springs attached at a point along its length is investigated. In the theoretical analysis, the 3-D equations are discretized via Galerkin's technique. The resulting coupled non-linear differential equations are solved numerically using a finite difference method. The dynamic behaviour of the system is presented in the form of bifurcation diagrams, along with phase-plane plots, time-histories, PSD plots, and Poincaré maps for five different spring configurations. Interesting dynamical phenomena, such as 2-D or 3-D flutter, divergence, quasiperiodic and chaotic motions, have been observed with increasing flow velocity. Experiments were performed for the cases studied theoretically, and good qualitative and quantitative agreement was observed. The experimental behaviour is illustrated by video clips (electronic annexes). The effect of the number of beam modes in the Galerkin discretization on accuracy of the results and on convergence of the numerical solutions is discussed.