在廣義橢圓座標下分析石英半球殼振子的瑕疵與控制器

Abstract

半圓球殼諧振陀螺儀會因為石英毛胚的密度非均勻特性以及製造過程產生幾何的誤差而使陀螺儀呈現瑕疵狀態，而瑕疵會造成兩振動模態自然頻率分歧的問題，因此，本研究提出一控制方法來消除瑕疵對半圓球殼諧振陀螺儀的影響。 本研究將廣義座標下之含質量-勁度瑕疵和阻尼瑕疵的運動方程式轉換到橢圓座標中，並透過微擾法中的平均法來近似方程式，然後透過數值解來觀察瑕疵對橢圓變數的影響。了解瑕疵對橢圓變數的影響後，本研究隨即透過強迫振盪的技術找出質量-勁度主軸角度位置，並可使用雷射消融技術以及藉由對電極施加電壓進行補償來消除質量-勁度瑕疵的頻率分歧。 欲達到同時控制阻尼瑕疵和殘餘的質量-勁度瑕疵對角速率量測的效應之目的，本研究將透過設計與分析控制器來控制橢圓變數至期望值，最後將控制力從橢圓座標轉回至廣義座標中以控制兩廣義座標主軸上的振幅，而因為本研究是利用球殼上鍍的金屬電極和驅動座內側電極間因電位差所產生的靜電場來驅動，所以須將廣義座標中的控制力轉成電壓值，然後透過適當地配置電極對的方式，分配電壓來控制瑕疵對半圓球殼諧振陀螺儀造成的效應。

The non-uniformity in density of the quartz blanks and the geometrical error due to manufacturing cause the hemispherical resonator gyroscope (HRG) to be imperfect. This imperfection will result in the frequency-splitting problem. Therefore, the purpose of the study is to propose a control method to eliminate the influence of the imperfection on the performance of the HRG. In this study, we transform the equations of motion concluding mass-stiffness and damping imperfections from general coordinates into elliptical coordinates, and then use the method of averaged perturbation theory to approximate these equations. After that, we observe the influence of imperfection on the elliptical variables through the numerical solution. Afterwards, we find the angles of mass-stiffness principal axes by the technique of forced vibration, so that we can use the technique of Laser-ablation as well as the compensation by applied voltage to the electrodes to eliminate the frequency-splitting of the mass-stiffness imperfection. For the purpose of simultaneously controlling the effect of both the damping and the residual mass-stiffness imperfections on the angular-rate measurement of HRG, this study will design and analyze the controller to control elliptical variables to those desired values. Finally, we transform the control force from elliptical coordinates backward to the generalized coordinates to control the amplitudes of the two principal axes in generalized coordinates. However, the shell is driven by the electrostatic field generated by the potential difference between the metal electrode on the hemispherical shell and the inner electrode of the driving seat, so the control force in the generalized coordinates must be converted into voltage. Therefore, we adjust the voltage to control the imperfection by proper layout of the electrode pairs.