諧振半球殼缺陷分析與控制

半圓球殼振動陀螺儀生產過程中容易造成結構上或材料上的不均，造成使用上的誤差與不便。本研究使用薄殼理論有最簡潔表示式的Niordson的半圓球薄殼理論為基礎，採用Kirchhoff-Love的薄殼假設，將二維曲面座標延伸至三維簡正座標，利用三維簡正座標與球座標系統的關係進行轉換，配合漢彌頓原理與半圓球殼振動模態的雷利近似解求得半圓球殼含旋轉運動的完整運動方程式，其中運動方程式包括理想半圓球殼振動陀螺儀的運動方程式與有誤差之半圓球殼振動陀螺儀的運動方程式，分析各種誤差產生時對半圓球殼自然頻率的影響。由於半圓球殼作自由振動時，會因為阻尼消耗能量而衰減，為了維持半圓球殼的振動，必須另外設立電場供應能量，除此之外也希望可以藉由設立電場的方式改善由於誤差造成的影響並利用電場控制半圓球殼的運動行為。利用漢彌頓原理求得分離靜電場驅動半球殼之運動方程式；求得其對半球殼的影響。設計分離靜電場的設置，用以調整因為生產上造成的微小誤差對陀螺儀之影響，使兩廣義座標的自然頻率調整至相近。最後欲達到控制半圓球殼陀螺儀的目的，將包含分離靜電場驅動的半圓球殼運動方程式利用微擾法(perturbation theory)中的多重尺度法(multiple-scale method)作分析，透過分析結果設計分離靜電場的擺設、施加電壓的大小與波形對陀螺儀兩廣義座標上振幅的穩定性，藉此穩定性之結果達到對陀螺儀控制的目的。

關鍵字

半圓球殼；諧振器； HRG ；漢彌頓原理；自然頻率；微擾法；電場驅動

並列摘要

The purpose of the study was to construct the dynamics equations to describe the hemispherical resonator gyroscope which may in perfect or imperfect structure. It is easy to have defects within manufacturing process. For instance, some impurities in fused quartz like sand or air bladder, engineer tolerance (permissible limit(s) of variation in an object). This research used shell theory written by Niordson, Kirchhoff-Love, Rayleigh’s approximate solution and Hamilton principle to derive the equation of motion. When there are defects in the hemispherical shell, the natural frequency will be influenced. Because of the air damping, the vibrations decayed. In order to solve these problems, we use electrode sheets to control the hemispherical shell which was plated thin gold on the outer surface. Similarly, using Hamilton principle to derive the equation of the motion.The electrode changes the stiffness of the hemispherical shell. According to this feature, the natural frequency could be modified. As the electrode sheets were set on the different place, the effect to hemispherical shell were different. Using perturbation method (multiple-scale method) can find the stability boundary. Based on the stability boundary, it could control the vibration of the hemispherical resonator gyroscope.

並列關鍵字

Hemispherical shell ； Resonator ； Hamilton principle ； Perturbation ； natural frequency

參考文獻

[1] David M. Rozell, 2009, "The Hemispherical Resonator Gyro: From Wineglass to the Planets," Proc. 19th AAS/AIAA Space Flight Mechanics Meeting, pp.1157-1178.

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[2] A.D. Meyer and D.M. Rozelle, 2012, “Milli-HRG inertial navigation system,” Position Location and Navigation Symposium (PLANS), IEEE/ION, 23-26 April 2012, pp.24-29.