Hopf-Hopf interactions in a spring-supported pipe conveying fluid

Abstract

A cantilevered pipe conveying fluid with a spring support has long been an object of discussion. The linear and nonlinear dynamics of this system have been investigated by numerous researchers, and this problem has been studied extensively. This is because the additional spring support makes the cantilevered pipe dynamically rich, and the modified system shows a sequence of bifurcations, such as double Hopf bifurcation and doubly degenerate pitchfork and Hopf bifurcations, resulting in two- or three-dimensional complex motions. Moreover, an additional mass on the system can induce complex bifurcating motions. In the present paper, we investigate the dynamics of a pipe with a spring support and an end mass. In particular, we focus on Hopf-Hopf interactions and conduct nonlinear analyses to clarify the evolutions of the amplitudes of two unstable modes. In a certain range of the parameters, there are mixed-mode self-excited vibrations, which are produced by the Hopf-Hopf interactions. Experiments were also conducted in order to verify the theoretical predictions. Moreover, preliminary experiments were extensively conducted to capture the three-dimensional pipe motions for a future study.