Three-dimensional parametric resonance of fluid-conveying pipes in the pre-buckling and post-buckling states

The paper presents a model of describing three-dimension parametric vibrations of a simply supported pipe conveying pulsating fluid. The three-dimensional motion equation of the system is a set of two nonlinear partial differential equations developed on the basis of Euler-Bernoulli beam theory, geometric nonlinearity and Kelvin-Voigt damping model. The three-dimensional motion equation is discretized by the Galerkin method and the nonlinear responses are solved by a fourth order Runge-Kutta integration algorithm. The three-dimensional model is validated for the natural frequencies from the pre-buckling state to the post-buckling state. Results of the nonlinear parametric resonance responses are shown in the form of time histories, spectrograms, phase-plane portraits, Poincaré map sections and motion trajectories. The vibration mode of sub-harmonic resonance is similar to a certain order free vibration mode, and the vibration mode of combination resonance is a superposition of two adjacent modes. The motion trajectories indicate the vibration of sub-harmonic resonance is planar but the vibration of combination resonance is non-planar. The lock-in phenomenon occurs in the pre-buckling and post-buckling states when the natural frequencies are locked to the pulsation frequency. The amplitudes of parametric resonance in the post-buckling state is much larger in comparison to parametric resonance in the pre-buckling state. Jump phenomenon is highlighted for the parametric resonance in the post-buckling state in comparison to the pre-buckling state.