Nonlinear interactions between unstable oscillatory modes in a cantilevered pipe conveying fluid

Abstract

Intense interest has been expressed in high-codimensional bifurcations and the nonlinear interactions between unstable modes. Nonlinear interactions between oscillatory modes can produce numerous complex motions. Such motions are caused by the double Hopf bifurcation. A cantilevered pipe conveying fluid is a typical non-conservative continuous system. When the flow velocity exceeds a critical value, a certain mode becomes unstable due to Hopf bifurcation which can be caused by the non-orthogonality of the eigenfunctions. Linear stability analyses have also revealed that another mode can experience an oscillatory instability as the flow velocity is increased further. Therefore, nonlinear interactions between two unstable modes become a problem. We focus on the double Hopf bifurcation of a pipe conveying fluid and investigate the nonlinear interactions between unstable second and third modes. We derive the amplitude equations governing the time evolution of the amplitudes of two unstable modes from a nonlinear nonself-adjoint partial differential equation and its boundary conditions. The theoretical results show that the self-excited planar pipe vibration can be produced either in the second or the third mode in a certain range of flow velocity, whereas the mixed-modal self-excited vibration is inhibited. Experiments were also conducted to verify the theoretical results. The theoretical results give a qualitatively good account of the typical features of double Hopf interactions in experiments.