半球陀螺谐振子的理论建模方法综述

A Review of Theoretical Methods for Resonator Modeling of Hemispherical Resonator Gyro

半球谐振子的动力学行为是半球谐振陀螺(HRG)功能的物理基础。针对目前国内半球谐振子动力学建模研究中普遍存在的缺乏具体参考来源,以及推导不严密甚至谬误等问题,详细推导了谐振子的动力学建模过程,总结了建模通用的学科背景知识,并梳理出了两大类常用的建模思路,以期为后续的进一步的研究和工程应用提供理论支撑。首先,对于半球谐振子动力学建模所通用的弹性薄壳理论进行了概括,包括描述应变和位移关系的弹性几何方程和将应变和应力联系起来的胡克定律;然后介绍了在早期理论研究中常用的基于达朗贝尔原理的建模方法,通过建立平衡方程以及载荷分析求解得到谐振子的二阶振动方程;最后介绍了由国外引入的基于拉格朗日力学的建模方法,通过计算谐振子整体的动能以及应变势能,并代入拉格朗日方程推导出谐振子的二阶运动方程,该方法在最近几年内也开始被国内的研究者所重视。

The dynamic behavior of hemispherical resonator comprises the physical basis of hemispherical resonator gyros(HRG).This paper concerns the common problems that exists in the domestic researches of dynamic modeling of HRG to date,such as the lack of specific sources of reference,plenty of ambiguous deductions(some are even incorrect),hence the derivation process of dynamic modeling of resonator is carefully presented in detail.The paper summarizes the general background knowledge of relevant disciplines,and teases out two main types of modeling thoughts,in the sake of providing a theoretical support for the subsequent researches and engineering applications.Firstly,the elastic shell theory used as a general basis in dynamic modeling of hemispherical resonator is introduced,including the elastic geometric equations that describe the relationship between strain and displacement and the Hooke's law that relates strain to stress;then the modeling method based on the d'Alembert principle commonly used in early theoretical researches is presented,and the second-order motion equations of resonator is obtained by establishing equilibrium equations and load analysis;finally,the modeling method based on Lagrangian mechanics which is inspired from aboard research is introduced,and the second-order motion equations are derived by calculating the kinetic and strain energy of the overall resonator with substitution of the Lagrangian equations.The latter method is increasingly popular by domestic researchers in recent years.