Nonlinear Parametric Vibration of a Fluid-Conveying Pipe Flexibly Restrained at the Ends

In this paper,the nonlinear parametric vibration of fluid-conveying pipes flexibly restrained at both ends and subjected to the pulsation flow excitation is investigated.The nonlinear equation of motion is derived using Hamilton^principle by considering the Kevin-Voigt viscoelastic damping,the geometric nonlinearity and the translational and rotational springs supported at the ends.The mode functions and eigen-frequencies are determined by the assumed mode method according to the elastic boundary conditions.The Galerkin method is implemented to obtain the natural frequencies and mode shapes of the pipe conveying fluid with different flow velocities.The effects of flexibly restrained conditions on stability of the pipe are analyzed.The nonlinear responses of the pipe under pulsating flow excitation are solved by the direct numerical method.The vibration behaviors are discussed in details,such as time history,frequency spectrum,phase-plane portrait,Poincare map and motion trajectory.The results show that the responses of sub-harmonic resonance and combination resonance can also be reflected in the rigidly supported pipes.The 1/5,1/8 and 1/13 sub-harmonic resonances can occur at certain excitation frequencies of the nonlinear parametric vibration system.The steady-state response amplitudes increase by a large margin and significantly affect the stability of the pipe.The effects of different spring stiffness coefficients on the parametric resonance responses are presented.For larger translational springs and rotational stiffness coefficients,the resonance frequencies shift to higher regions and the resonance amplitudes may reduce by a certain extent in accordance with the rigid-body motion.This study can provide helpful guidance on the analysis and design of piping systems subject to vibrations.