External and internal resonances of the pipe conveying fluid in the supercritical regime

Nonlinear forced vibration of a viscoelastic pipe conveying fluid around the curved equilibrium configuration resulting from the supercritical flow speed is investigated with the emphasis on dynamics of external and internal resonance. The governing equation for the pipe system, a nonlinear integro-partial-differential model with variable coefficients, is truncated into a discrete perturbed gyroscopic system via the Galerkin method. A condition for the two-to-one internal resonance is established, and the condition implies that internal resonance is possible in the supercritical regime. The method of multiple scales is developed to present the solvability condition of approximate solutions. The modulation equations of the amplitude and the phase are derived from the condition. The first two primary resonances in the presence of the two-to-one internal resonance are examined. Steady-state solutions and their stabilities are determined. Some typical steady-state responses are demonstrated via the amplitude–frequency curves. In addition to jumping, hysteresis and saturation, other dynamical behaviors are observed to highlight the effects of the modal interaction in the internal resonance. The analytical results are confirmed by the numerical integrations.