半球谐振陀螺的拉格朗日力学建模方法

摘要

陀螺仪是惯性技术的核心元件,半球谐振陀螺已成为空间应用的首选之一.对半球谐振子进行详细的动力学模型建立是研究其原理的基础.将半球谐振子视为半球弹性薄壳,求取其弹性伸长量及角变从而求得弹性薄壳的弹性势能,进而采用基于虚功原理和变分法的拉格朗日力学方法,求取弹性薄壳的动能,代入拉格朗日算子建立拉格朗日方程,得到系统整体的动力学模型.建模过程中通过广义坐标的选取,使最终模型等效为质量块在二维空间的简谐振动的等效模型,从而简化对半球谐振陀螺的分析.%As an essential element of inertial technology,hemispherical resonator gyro (HRG) has become a prior choice of space applications.Dynamic modeling for HRG is the foundation of explicit analysis for HRG's theories.Based on LOVE's basic hypothesis,harmonic oscillator can be treated as hemispherical elastic thin shell.Elongation and angle changes can be calculated by using elastic thin shell theory.Then the whole kinetic energy and potential energy can be calculated by using Lagrangian Mechanics,thus establishing the Lagrangian equations,as which consequence,the dynamic model of hemispherical harmonic oscillator is obtained.The result shows that the obtained model is equivalent to a 2-D simple harmonic vibration model of a mass block,as which result,the dynamic model of HRG is simplified.