俯仰运动圆柱贮箱中液体的非线性晃动

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尹立中，邹经湘，王本利
（哈尔滨工业大学航天工程与力学系，哈尔滨 150001）

摘要：首次对俯仰运动圆柱贮箱中液体的有限幅值晃动问题进行解析研
究。首先建立了描述俯仰和/或偏航运动贮箱中液体晃动的非线性偏微
分方程组，而后提出了相应的变分原理，建立了压力体积分形式的Lag- range函数，通过变分方程，最终得到描述俯仰和/或偏航运动圆柱
贮箱中液体晃动的非线性动力学微分方程组，该动力学方程组自然满足
液体自由表面的运动学和动力学边界条件。而后运用多尺度法求解了所
得的动力学方程组，对非线性液体晃动的动力学特性进行了详细的理论
分析。

关键词：非线性晃动，俯仰/偏航，变分原理，圆柱贮箱

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NONLINEAR SLOSHING OF LIQUID IN A CIRCLE CYLINDRICAL CONTAINER UNDER PITCHING EXCITATION

Yin Lizhong, Zou Jingxiang, Wang Benli
(Harbin Institute of Technology, Harbin 150001, China)

Abstract: In this paper, the nonlinear sloshing of liquid in a circle cylindrical tank under pitching excitation is studied analytically for the first time. Owing to the complexity of problem, it is very difficult to solve the nonlinear sloshing of liquid in a container subjected to forced pitching and (or) yawing oscillation by existing methods. Therefore, a method used to analyze this problem is presented. Firstly, the nonlinear initial-boundary problem of PDE system for liquid sloshing in a container under pitching and (or) yawing excitation is established. With regard to the previous problem, variational principle and Lagrange function in the form of the volume integration of liquid pressure are obtained. Based on the variation equation and new Lagrange function proposed, nonlinear dynamic system for sloshing of liquid in a circle cylindrical container under pitching and (or) yawing excitation is derived. Subsequently, the nonlinear dynamic system gives the free surface kinematics and dynamic boundary condition. At the same time, the present method greatly reduces the work of formula derivation. Finally, the nonlinear dynamic system is solved by the multiple scale method. The dynamic characteristic of nonlinear liquid sloshing is analyzed in detail. Some kinds of motion which may appear in the liquid sloshing are discussed. Response curves and stable-unstable regions of liquid motion are determined. As to two-dimensional sloshing in a rigid, rectangular, open tank, the comparison between theoretical result by the present method and experiments shows good agreement. So, the present method is proved feasible. By means of the present method, the coupled dynamics of liquid in a tank and structure may also be investigated analytically.

Key words: nonlinear sloshing, pitching/yawing, variational principle, a circle cylindrical container