Nonlinear Dynamics of Slender Cylinders Supported at Both Ends and Subjected to Axial Flow

Abstract

In this paper the weakly nonlinear equations of motion, correct to third order of magnitude, are presented for a slender cylinder subjected to axial flow. The cylinder is considered to be extensible and two coupled nonlinear equations describe its motions, involving both longitudinal and transverse displacements. The inviscid component of the fluid force is modeled by an extension of Lighthill’s slender-body work, and viscous, hydrostatic, gravity and pressure-loss forces are added in a similar manner as for cantilevered inextensible cylinders. However, both the derivation and the final equations have many different and distinctive features. The equations are discretized via Galerkin’s method and solved by Houbolt’s finite difference method. Bifurcation diagrams with flow velocity as the independent variable, supported by phase- plane plots, show that the system loses stability via a supercritical pitchfork bifurcation leading to divergence. At higher flow velocities, a secondary Hopf bifurcation leads to flutter and, at higher flow velocities, the limit cycle evolves into chaotic oscillation. In some cases, an oscillatory large-amplitude limit cycle is found with no clear physical origination.