基于三阶剪切变形理论的悬臂输流管道自由振动

采用三阶剪切变形理论，结合有限元法研究了悬臂输流管道的自由振动问题．利用虚功原理建立了输流管系统的有限元方程，同时将悬臂端弹性支承以势能的形式引入到系统方程中，求解了系统前三阶的复频率．分别探讨了流体速度和弹簧刚度对系统复频率实部和虚部的影响，重点分析了弹簧刚度与前三阶固有频率间的关系．在弹性支承刚度为零的特例下，对比了本文结果与Timoshenko梁理论的结果，证明了本文方法的可靠性．研究发现系统固有频率的实部恒为负值，表明一端带有弹性支承的约束形式有利于提高悬臂输流管道自由振动的稳定性；流体的流动对管道振动起到了阻尼作用，在流动速度足够大的情况下，各阶振动固有频率均趋于零；当弹簧刚度为无穷大，且流体速度足够大时，输流管道将发生失稳．

Free Vibration Analyses of Cantilevered Fluid Transmission Pipelines Based on the Third-Order Shear Deformation Theory

The free vibration of cantilever pipeline is studied by using the third-order shear deformation theory and finite element method. The finite element equation of the fluid transmission pipelines is established by using the virtual work principle. At the same time, the elastic supports at the cantilever end is introduced into the system equation in the form of potential energy, and the first three complex frequencies of the system are solved. The effects of fluid velocity and spring stiffness on the real and imaginary parts of the complex frequencies of the system are discussed respectively, and the relationship between the spring stiffness and the first three natural frequencies is emphatically analyzed. In the special case of zero elastic support stiffness, the results of the proposed method are compared with those of Timoshenko beam theory, to verify the reliability of the present method. It is found that the real part of the natural frequency of the system is always negative, indicating that the constraint form with elastic supports at one end is conducive to improve the stability of free vibration of cantilever fluid transmission pipeline. The fluid flow has a damping effect on pipeline vibration, and the natural frequencies of each order tend to zero when the flow velocity is large enough. When the spring stiffness is infinitely large and the fluid velocity is large enough, the fluid transmission pipelines will be unstable.