臺灣博碩士論文加值系統

本文是探討壓電驅動半圓球殼振動陀螺儀的設計與操作原理，以作為設計、製造及控制的依據，並可進一步作為微小化基礎。
首先以Niordson的彈性薄殼理論為基礎，採用Kirchhoff-Love的薄殼假設，先延伸Niordson的彈性薄殼理論至均質、厚度方向極化、上下表面鍍上電極的壓電陶瓷薄殼，由三維簡正坐標中的組成律開始，推導至二維曲面坐標中的組成律。
對於彈性半球形薄殼外面貼著壓電薄殼的機電耦合系統，亦以Niordson的彈性薄殼理論為基礎，配合漢彌頓原理及Lord Rayleigh的半圓球殼假設，據以推導包含旋轉的壓電驅動半圓球殼振動陀螺儀的運動方程式，以及探討影響運動方程式中各項係數的因素，以作為設計與製造的依據。
在考量實際存在材質與製造上的容許誤差，因此建立一般情況的振動陀螺儀誤差模型，並以多重時間尺度法(multiple time scale)，求解在無角速度輸入及自由振動時的振動行為，並分別在時域與橢圓軌跡域予以解析，以作為控制上的依據。
對有誤差存在的非理想振動陀螺儀，先做頻率域分析，在依據頻率域分析結果，探討以線性方式補償各項誤差的程序。
對於振動陀螺儀的兩大感測模式，全圓周角度 (whole-angle-tracking)與力平衡 (force-to-balance)量測模式，分別探討其感測原理、振動控制的方式及誤差模式。

The paper considers the theory of a new type of vibrating hemispherical gyroscope, which comprises a hemispherical shell with discrete piezoceramic actuation and sensing elements bonded on the outer side and close rim of the shell.
Constitutive equations for a thin piezoceramic shell with thickness polarization and electrode-covered faces are derived using Niordson’s shell theory.
Coupled electromechanical equations of motion, describing the dynamics of the vibrating hemispherical gyroscope, are derived using Hamilton’s principle, Lord Rayleigh’s solution of the shell and Rayleigh-Ritz method.
The linear error model due to imperfections of materials or manufactural of tolorance of vibrating gyroscopes is established. The operation of the gyroscope without and with imperfections are described under free vibration.
The equation of motion of two operation modes of vibrating gyroscopes, whole-angle-tracking and forced-to-balance modes, are derived.

誌謝i
中文摘要ii
Abstractiii
目錄iv
圖目錄viii
第1章 緒論1
1.1前言1
1.2文獻回顧1
1.2.1振動陀螺儀1
1.2.2薄殼動力與壓電材料3
1.2.3薄殼振動陀螺儀動力分析5
1.3本文目的與內容介紹6
第2章 壓電殼組成律8
2.1Niordson的線性薄殼理論8
2.2簡正坐標中的壓電殼組成律14
2.3壓電殼應變能密度函數與電位能密度函數20
2.4曲面坐標系統中的壓電殼組成律24
第3章壓電驅動半圓球殼振動陀螺儀運動方程式30
3.1機電耦合系統30
3.1.1簡正坐標系統中的旋轉殼動能31
3.1.2曲面坐標系統中的旋轉殼動能32
3.1.3薄球殼應變能與電位能34
3.1.4外部負載38
3.2漢彌頓原理40
3.2.1半圓球殼振動陀螺儀與球坐標40
3.2.2變分原理的應用46
3.3運動方程式48
3.3.1近似解法的雷利解48
3.3.2致動方程式與感測方程式53
3.3.3感測係數61
3.3.4角頻率63
3.3.5機電耦合係數67
3.3.6壓電殼電容與電荷係數69
第4章 振動陀螺儀之誤差分析72
4.1理想振動陀螺儀感測原理72
4.1.1橢圓軌跡73
4.1.2感測原理74
4.2非理想振動陀螺儀自由振動81
4.2.1橢圓軌跡變化率81
4.2.2橢圓軌跡變化率中的常數項84
4.2.3橢圓軌跡變化率中的線性與非線性項87
4.3角頻率變化對軌跡的影響89
4.3.1僅有角頻率變化時的影響89
4.3.2彈性不均效應102
4.4衰減率變化對軌跡的影響111
4.4.1僅有衰減率變化時的影響111
4.4.2阻尼效應116
4.5一般性變化對軌跡的影響122
第5章 振動陀螺之振動控制128
5.1非理想振動陀螺儀頻率域分析128
5.2非理想振動陀螺儀線性補償132
5.3振動陀螺儀全圓周角度量測模式139
5.3.1全圓周角度量測模式的振動控制139
5.3.2全圓周角度量測模式時的角度量測144
5.4振動陀螺儀力平衡量測模式146
5.4.1力平衡量測模式的振動控制146
5.4.2力平衡量測模式時的角速度量測149
第6章 總結151
參考文獻153
附錄3-1 動能項廣義坐標計算162
附錄3-2 感測係數168
附錄3-3 角頻率175
附錄3-4 機電耦合係數188
附錄3-5 壓電殼電容193
附錄4-1 誤差cs項推導198
附錄4-2 誤差ks項推導201
附錄4-3 誤差ha項推導204
附錄4-4 誤差ca項推導207
附錄4-5 誤差gs項推導210
附錄4-6 誤差ka項推導213
附錄4-7 誤差hs項推導216
附錄4-8 一般項誤差推導219
附錄4-9 僅有角頻率變化時橢圓軌跡長軸傾角224
附錄4-10 僅有衰減率變化時橢圓軌跡長軸傾角228
附錄4-11 一般情形時橢圓軌跡長軸傾角232
附錄5-1 軟體模擬hyx與qyx補償程序233
附錄5-2 全圓周角度量測模式的結構配置251
附錄5-3 全圓周角度量測模式的振動控制推導254
附錄5-4 力平衡量測模式的結構配置257
符號說明259

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