瑕疵谐振石英半球壳固有刚性轴的定位

Location of normal mode axes of imperfect quartz hemispherical resonant shell

石英谐振半球壳是半球谐振陀螺仪(HRG)的核心结构,瑕疵半球壳的固有刚性轴测量定位是对其进行频差修正的前置任务,因而这也影响到半球谐振陀螺仪的工作性能和精度。为准确定位含瑕疵石英半球壳的固有刚性轴,通过理论推导和数值仿真两种方式对含密度和阻尼不均匀瑕疵谐振半球壳在强迫振动条件下的运动状态进行分析。含瑕疵石英半球壳运动方程等价于二维弹簧—阻尼—质点系统的物理模型,以此模型证明了稳态时壳体上两点的位移李萨如图(Lissajous-Figure)是一个椭圆。同时推导出了此椭圆在瑕疵状态下的长、短半轴及其倾角的显式表达式;另外在数值仿真上,采用Mathematica软件绘制了位移的李萨如图与椭圆倾角随激励电极角度的变化图。结果表明:含密度和阻尼不均匀瑕疵石英半球壳的固有刚性轴采用此种方案可方便快速完成其定位。因此评判石英瑕疵半球壳固有刚性轴的准则是:调整不同的激发电极角度,若稳态李萨如图的椭圆倾角为0°时,则此时的激发电极角度就为固有刚性轴所在方位角。理论结果和数值仿真已验证此结论的正确性。

The quartz hemispherical shell is the core structure of Hemispherical Resonator Gyroscope (HRG),and location of its normal mode axes is the predecessor task for eliminating the frequency splitting,which,of course,affects the performance and precision of HRG.In order to precisely locate the normal mode axes of the quartz hemispherical shell,both theoretical derivation and numerical simulation under the condition of forced vibration are used to analyze the motion behavior of the hemispherical resonant shell with imperfection of nonuniformity in density and damping.The equations of motion of the imperfect shell are equivalent to a two-dimensional spring-mass-dashpot system.The steady-state trajectory of these two displacements is proved to be an ellipse Lissajous Figure.The expressions relating the semi-major/minor axes and inclination angle of the ellipse to the imperfect parameters of the shell are derived in the explicit form.Besides these,numerical simulations based on the software Mathematica are used to plot the various elliptical Lissajous figures corresponding to the different angles of exciting electrode.It turn out that the normal mode axes of the hemispherical resonant shell with imperfection of nonuniformity in density and damping can be determined quickly and easily by this scheme.The criterion for locating the normal mode axis is to adjust the excitation angles,if the inclination angle of the ellipse is 0°,then,the angle of exciting electrode is the angle of normal mode axis.This result is verified by the theoretical solution and the numerical simulation.