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’s 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, Poincaré 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.

中文翻译：

---

两端柔性约束的输液管的非线性参数振动

本文研究了两端受挠性约束并受到脉动流激励的输油管非线性参数振动。使用汉密尔顿原理，通过考虑凯文-沃格特粘弹性阻尼，几何非线性以及两端支撑的平移和旋转弹簧，得出非线性运动方程。根据弹性边界条件，通过假设的模态方法确定模态函数和本征频率。实施Galerkin方法是为了获得具有不同流速的输送流体的管道的固有频率和振型。分析了柔性约束条件对管道稳定性的影响。用直接数值方法求解了脉动激励下管道的非线性响应。详细讨论了振动行为，例如时间历史，频谱，相平面肖像，庞加莱图和运动轨迹。结果表明，亚谐谐振和组合谐振的响应也可以反映在刚性支撑的管道中。1 / 5、1 / 8和1/13次谐波谐振可以在非线性参数振动系统的某些激励频率下发生。稳态响应幅度大幅度增加，并显着影响管道的稳定性。提出了不同的弹簧刚度系数对参数共振响应的影响。对于较大的平移弹簧和旋转刚度系数，共振频率会移至更高的区域，并且共振幅度可能会根据刚体运动而有所降低。这项研究可以为受振动的管道系统的分析和设计提供有用的指导。