Co-dimension 2 bifurcations and chaos in cantilevered pipe conveying time varying fluid with three-to-one internal resonances

The nonlinear behavior of a cantilevered fluid conveying pipe subjected to principal parametric and internal resonances is investigated in this paper. The flow velocity is divided into constant and sinusoidal parts. The velocity value of the constant part is so adjusted such that the system exhibits 3:1 internal resonances for the first two modes. The method of multiple scales is employed to obtain the response of the system and a set of four first-order nonlinear ordinary-differential equations for governing the amplitude of the response. The eigenvalues of the Jacobian matrix are used to assess the stability of the equilibrium solutions with varying parameters. The co-dimension 2 derived from the double-zero eigenvalues is analyzed in detail. The results show that the response amplitude may undergo saddle-node, pitchfork, Hopf, homoclinic loop and period-doubling bifurcations depending on the frequency and amplitude of the sinusoidal flow. When the frequency of the sinusoidal flow equals exactly half of the first-mode frequency, the system has a route to chaos by period-doubling bifurcation and then returns to a periodic motion as the amplitude of the sinusoidal flow increases.