非线性支撑下悬臂输流管道的 Hopf 分叉特性

摘要

研究了外激励作用下非线性支撑悬臂输流管道系统的 Hopf 分叉特性，建立了外激励作用下非线性支撑悬臂输流管道系统的动力学方程，并采用 Galerkin 方法离散动力学方程，由增量谐波平衡（IHB）法推导了方程的近似解析解，由 Floquet 理论判定了解的稳定性，同时给出了系统的 Hopf 分叉点。利用数值算法和IHB 法研究了支撑位置、支撑结构刚度和阻尼对系统 Hopf 分叉特性的影响规律。研究表明：系统的幅频特性在 Hopf 分叉前后发生了改变，响应频率由外激励频率变为系统的自激振动频率，且系统 Hopf 分叉后，幅值显著增大。该研究结果可为悬臂管道的振动控制提供理论基础。%This paper investigates the Hopf bifurcations of the nonlinear-supporting cantilever fluid-conveying pipeline under the action of external excitation and establishes its dynamical equation.The paper makes use of the Galerkin method to discretize the equation,the incremental harmonic balance (IHB)method to derive the approximate analytical solution and the Floquet theory to determine the stability of the solution and meanwhile offers Hopf bifurcation points of the periodic solution.Then the numerical algorithm and the IHB method are employed to analyze the influence of the supporting point and the stiffness and damping of the support structure on the Hopf bifurcation of the system. The results show that changes take place in the amplitude-frequency characteristic of the system be-fore and after the Hopf bifurcation,that the response frequency is changed from the excitation fre-quency to the self-excited frequency in the system and that the amplitude is increased obviously after the Hopf bifurcation.The research which has been made can provide a theoretical basis for controlling the vibration of the cantilever fluid-conveying pipeline.