半球谐振子动力学方程的建立及固有频率的计算

Establishment of dynamics equation of HRG resonator and calculation of natural frequency

为了建立半球谐振陀螺仪谐振子的动力学方程,由基希霍夫-李雅夫假设分析了半球谐振子上任一点的应变及应力,然后在谐振子上取一薄壳单元,对其进行受力分析,并考虑了三轴角速率和角加速度的情况下引起的惯性力,建立了薄壳单元的平衡方程及薄壳的边缘运动方程,利用解微分方程的布勃诺夫-加廖尔金法建立了谐振子的动力学方程,最后根据动力学方程求出了谐振子的固有频率.通过与各种计算固有频率的方法的计算值以及实测值的比较,证明了本文推导的半球谐振子动力学方程和固有频率的正确性。

In order to establish dynamics equation of resonator in HRG,the strains and stresses of an arbitrary point in the hemispherical thin shell were deduced under Kirchhoff-Lyav hypothesis.The force and torque equilibrium equations of the thin shell element selected from the resonator were gained on the basis of the calculated strains and stresses,then the border motion equations of the HRG were obtained in consideration of the subjected 3-D angular rate and the 3-D angular acceleration.Finally the dynamics equations of resonator were established by Bubonov-Galerkin method which is common used for solving differential equations,thus the natural frequency is computed by the established dynamics equations.By contrast with other methods of calculating natural frequency and the measured frequency values of practical HRG,the correctness of the established dynamics equations and the method of frequency calculation are proved.