轴对称椭球形贮箱液体晃动频率解析研究

Analytical Study on Liquid Sloshing Frequency of Axisymmetric Ellipsoidal Tank

摘要: 现代航天事业的发展对航天器的运载能力和在轨时间提出了更高的要求，导致液体燃料所占比重不断加大，液体燃料的运动对航天器的姿态控制和稳定性将产生重大影响。本文对微重力环境下轴对称椭球腔的静液面构型和液体晃动固有频率进行解析研究。在微重力环境中，充液航天器中的静液面呈弯月形，求解静液面形状是研究液体晃动的前提。对于表示静液面的二阶非线性微分方程，利用Runge-Kutta法获得轴对称椭球腔的充液汽–液分界面构型。基于本文建立的球坐标系，用变分原理和最小势能原理推导了液体晃动的控制方程，由高斯超几何级数表达速度势和波高的模态函数，用Galerkin方法把控制方程转变为标准特征值问题形式的频率方程，求出不同情况下轴对称椭球贮箱液体晃动频率，并对Bond数和贮箱几何尺寸的变化对液体晃动频率的影响进行研究。研究结果表明：在参数条件相同的情况下，随着Bond数的增加，液体晃动频率和无量纲频率会呈现不同的变化规律，而储腔的几何形状也会对液体晃动无量纲频率的变化规律造成显著影响。该研究能够为微重力环境下液体晃动和航天器总体设计提供参考意义。

Abstract: The development of modern aerospace has put forward higher requirements for the carrying ca-pacity and on-orbit time of spacecraft, resulting in an increasing proportion of liquid fuel, and the sloshing of liquid fuel will have a significant impact on the attitude control and stability of the spacecraft. In this paper, the hydrostatic surface configuration and the natural frequency of liquid sloshing of axisymmetric ellipsoidal tanks in microgravity environment are analyzed analytically. In the microgravity environment, the hydrostatic surface is meniscus, and obtaining the configuration of the hydrostatic surface of liquid-filled fuel container is the premise of studying liquid sloshing. The liquid-filled vapor-liquid interface configuration of the axisymmetric ellipsoidal tanks is ob-tained by solving the nonlinear ordinary differential equation using the Runge-Kutta method. Based on the spherical coordinate system established in this paper, the governing equations of liq-uid sloshing are deduced by the variational principle and the principle of minimum potential ener-gy. The mode functions of velocity potential and liquid surface displacement are expressed by Gauss hypergeometric series, and the governing equations are transformed into standard eigenvalue problem by Galerkin method. The frequency equation in the form of a numerical problem is used to obtain the liquid sloshing frequency of the axisymmetric ellipsoidal tanks under different working conditions, and the influence of the Bond number and the geometric size of the tank on the liquid sloshing frequency are studied. This research conclusion can provide reference for liquid sloshing and the overall design of spacecraft in microgravity environment.