摘要
由于轴对称的原因,理想半球壳的弯曲共振模态会形成具有相同频率的双简并模态。当有质量密度瑕疵时,此双简并模态的频率会出现分歧,损坏量测精度。通过哈密顿原理、Niordson薄壳理论和球壳中心轴面不可伸长的瑞利(Rayleigh)位移近似解公式可以推导出理想与瑕疵两状态下半球壳的动力学方程,并以该两模态的振幅为变数。理想半球壳的简并模态的轴可以为任意方向,并且可以证明瑕疵半球壳的共振模态轴不为任意方向,而是通过单一瑕疵点。理论的正解可得到特征频率(即两分歧的共振模态频率)与特征向量(两共振模态的方向)的显式公式。在消除频差时,通过在高频特征向量与半球壳开口端的相交点来以激光蚀刻一定深度的密度较大的铝金属共晶熔入修正质量,其不会改变特征向量的方向,并且理论分析得到蚀刻深度与瑕疵量的显式公式。
The ideal hemispherical shell have pairs of mode shapes,which are called degenerated eigenmodes,at the same frequency due to its axisymmetric structure.When the mass imperfections appear,the natural frequency of these two degenerated modes splits into two different values which destroys the measuring performance of the Gyroscope.The equations of motion for both the ideal and imperfect shells can be derived on the basis of Hamilton's principle,Niordson thin shell theory,and the approximate analytical Rayleigh displacement formula.based on the assumption that the middle surface of the shell is inextensible.The variables of these equations are the vibration amplitudes of the two degenerated modes.For ideal hemispherical shell,the pair of eigenmodes can present at any orientation,but at the presence of the mass imperfection,one of the pair of modes must pass through the point of mass imperfection.From the theoretical exact solution the explicit expressions for the eigenvalues(i.e.,the resonant frequencies)and the eigenvectors(i.e.,the orientations of the two eigenmodes)can be obtained.In order to eliminate the frequency splitting trimming the mass of the shell at the points where there are the intersection of the eigenvectors and the hemispherical shell at its open boundary will not alter the eigenvectors.Finally,the purpose of eliminating the frequency splitting the laser is used to etch out a hole for eutectic bonding of the added aluminum and the fused quartz shell.The equation relating the depth of laser etching and the amount of the mass imperfection is obtained through theoretical analysis.
作者
周闯
谭品恒
邓文高
张家欧
ZHOU Chuang;TAN Pin-heng;DENG Wen-gao;CHANG Chia-ou(College of Mechanical Engineering, Guangxi University, Nanning 530004, China)
出处
《广西大学学报(自然科学版)》
CAS
北大核心
2020年第5期1121-1134,共14页
Journal of Guangxi University(Natural Science Edition)
基金
国家自然科学基金资助项目(11672077)。
关键词
半球谐振子
哈密顿原理
特征向量
质量瑕疵
频率分歧修正
hemispherical harmonic resonator
Hamilton's principle
eigenvector
mass imperfection
frequency-splitting correction