諧振半球殼之質量瑕疵分析與修正

Analysis and Correction of Mass Imperfection of a Hemispherical Resonant Shell

林思嘉(Szu-Chia Lin)

指導教授 ： 張家歐

共同指導教授 ： 謝發華

國立臺灣大學/工學院/應用力學研究所/碩士(2015年)

https://doi.org/10.6342/NTU.2015.10578

摘要

半圓球殼陀螺儀之主要原件為半圓球殼，其毛胚在製造過程中多少都會伴隨著雜質或氣泡所產生密度的不均勻性，以及加工產生的半徑或厚度等參數的誤差，此時半圓球殼即呈瑕疵狀態。理想球殼相同自然頻率的兩個不同模態會因這些瑕疵產生自然頻率分歧(frequency bifurcation)。本文利用漢彌頓原理以及Niordson所採用的雷利近似解之方法推導出理想半圓球殼的運動方程式以及質量分布不均勻半圓球殼的運動方程式，觀察瑕疵存在對於半圓球殼頻率與特徵向量的影響，並且修正此頻率分歧之現象。
為了建立修正頻率分岐的理論模型，首先規劃有效微調質量修正方法，謝發華[9]曾利用驅動與感測模態做修正，本文則利用在半圓球殼之低頻特徵向量方向上做修正。選擇兩分岐自然頻率的低頻所對應的特徵向量與球殼的交線點做雷射質量燒蝕的位置。本文證明在特徵向量上的點做修正是不會產生耦合效應，使特徵向量在不斷的燒蝕過程中保持不變。由兩模態振幅來繪出Lissajous圖，藉調整激發與感測電極的角度及由Lissajous圖可判斷特徵向量之方向。由理論推導出的燒蝕質量與兩分岐自然頻率的函數關係公式可作為實務修正的參考依據。

關鍵字

半圓球殼 ； 漢彌頓原理 ； 自然頻率 ； 特徵向量 ； 橢圓軌跡圖

並列摘要

The major component of the hemispherical resonator gyroscope is the hemispherical shell (HS). More or less the voids, bubbles, and impurities are produced in the forming process of blanks which causes the non-uniformity in density，and it is inevitable to suffer the geometric errors in the manufacturing process of the HS. These situations cause the HS to be imperfection. In this thesis Hamilton principle and Lore Rayleigh’s approximate solution are employed to derive the equations of motion of both the ideal HS and the imperfection HS. The effect of the imperfection on the frequency bifurcation and eigenvectors is investigated.
In order to establish the theoretical model of correcting the frequency bifurcation Dr. Fa-Hua Hsieh [9] proposed the method of mass-trimming on the driving and sensing modes. Here a method of mass-trimming on the eigenvectors is invented. The positions for mass-trimming are located at the intersections of the eigenvector, corresponding to the lower one of the two bifurcated frequencies, and the great circle of the shell. It is proved that mass-trimming on the eigenvector eliminate the coupling effect, that is, eigenvector keeps unchanging during the successive corrections of mass trimming. The amplitudes of the modes are used to plot the Lissajous figure. By adjusting the angular positions of the driving and sensing electrodes and the judgment from the Lisajous figure the eigenvector can be determined. The derived equation which relates the amount of mass to be trimmed and the bifurcated frequencies can be used as reference for frequency correction.

並列關鍵字

Hemispherical shell ； Hamilton principle ； natural frequency ； bifurcate frequency ； eigenvectors

參考文獻

[1] Rayleigh, L, 1881, “On the Infinitesimal Bending of Surfaces of Revolution,” Proc. Math. Soc., London, Vol.13, pp. 4-16.

[2] Love, A. E. H., 1888, “On th Small Free Vibrations and Deformation on Thin Elastic Shells,” Phil. Transactions Roy. Soc., A179, pp. 491-546.

[3] Bryan, G. H., 1890, “On the Beats in the Vibrations of a Revolving Cylinder or Bell,” Proc. Cambridge Philos. Soc., Vol. VII, Nov. 24, pp. 101-111.

[4] Quick, W. H, 1964, “Theory of the Vibrating String as an Angular Motion Sensor,” Transactions ASME, J. Appl. Mech., pp. 523-534

[5] Friedland, Bernard and Maurice F. Hutton,1978, “Theory and Error Analysis of Vibrating-Member Gyroscope,” IEEE Transactions on Automatic Control, Vol. AC-2345, No. 4, pp. 545-556