本文提出一種兩桿、單自由度之開放式連桿機構，稱為Spirograph機構。此機構的特色在於，連桿透過鏈輪－鏈條或帶輪－皮帶等運動對加以約束，使得兩桿的運動維持一定的轉速比關係，其運動軌跡為圓內、外擺線。透過與圓內外擺\線運動方程式的分析與比較，說明機構參數與運動路徑的關係，在特定參數設定下，運動路徑會出現尖點(Cusps)的情況，表示機構運動產生暫停(Dwell)的現象，並建立不同參數下之軌跡圖形，作為路徑合成之型態參考。此外，提出以不同連桿與轉速比的搭配，可合成相同封閉路徑的概念。
考慮部分路徑合成問題時，藉由精確點合成方法，比較在不同已知參數之合成結果。另外，對於封閉路徑，利用可連續變速之非圓形鏈輪，搭配一圓形鏈輪，在不改變桿長或桿長已知的情況下，提出合成方法與步驟。文中說明可合成路徑範圍與設計限制，藉由運動瞬心法分析非圓形鏈輪之轉速比，以及相對運動關係。對於可合成之封閉路徑，透過牛頓法計算連桿在路徑合成之轉速比變化，判斷可能合成之運動型態(內擺線或外擺\線)，並藉由包絡線方法合成非圓形鏈輪輪廓，最後舉例說明。其轉速比合成與連桿角度分析，均以電腦程式撰寫並模擬路徑合成結果。

A three-link, single-degree-of-freedom mechanism, named the Spirograph linkage, is investigated in this thesis. What is special about this open-chain mechanism is that, by the use of a chain and two sprockets and therefore maintaining a constant angular velocity ratio between the two rotating links, the path traced by this mechanism can be shown to be either an epitrochoid or a hypotrochoid. Through the understanding of the formulations of the trochoids, the relations between the design parameters of the Spirograph linkage and the corresponding path patterns are derived.
With certain design parameter combinations, the Spirograph linkage is able to feature paths with cusps, which means dwells and are useful in applications. A design chart for such cusp-generating Spirograph linkages is then included in the thesis for the sake of convenience.
In addition, by matching the link-length ratio with the angular velocity ratio, ways for finding closed single-loop paths of the Spirograph linkage are also studied. As for the path generation problems concerning only a fraction of the entire path, we compare results given by the classical precision point scheme with assorted combinations of initial conditions.
In the latter part of the thesis, non-circular sprockets are also introduced to provide the Spirograph linkage additional freedom to cope with the demand of more flexible, i.e., non-symmetrical shapes of the path. However, many constraints are to be imposed on the design process of such linkages, and these design limitations are elaborated in the thesis. At last, the use of Newton’s method, the selection of the proper type of trochoids, the application of the envelop method for obtaining the profile of the non-circular sprocket, the analyses of angular velocity ratios, etc. are all exemplified in the several numerical examples of the thesis.

謝 誌 I
目 錄 III
圖目錄 V
表目錄 VIII
摘 要 IX
ABSTRACT X
第一章 緒論 - 1 -
1.1 知識背景 - 1 -
1.2 文獻回顧 - 4 -
1.3 研究動機 - 5 -
1.4 論文架構 - 6 -
第二章 圓內外擺線與SPIROGRAPH機構介紹 - 8 -
2.1 圓內、外擺線軌跡方程式 - 8 -
2.2 SPIROGRAPH機構軌跡方程式與擺線方程式的比較 - 10 -
2.3 運動軌跡型態分析 - 13 -
第三章 圓形鏈輪之SPIROGRAPH機構設計 - 20 -
3.1 圓形鏈輪的基本設計概念 - 20 -
3.2 SPIROGRAPH機構實際運動探討 - 24 -
3.2.1 路徑同構(Cognate)討論 - 25 -
3.2.2 軌跡節點(Cusps) - 27 -
3.3 機構設計之部份路徑合成 - 29 -
3.3.1 精確點合成~路徑跟隨(Path Following) - 32 -
3.3.2 方法比較與優缺點評估 - 34 -
第四章 非圓形鏈輪之SPIROGRAPH機構之運動分析 - 39 -
4.1 背景知識與設計需求 - 39 -
4.2 非圓形鏈輪的基本分析與設計概念 - 41 -
4.2.1 運動瞬心法分析 - 41 -
4.2.2 非圓形鏈輪運動分析 - 44 -
4.2.3 非圓形鏈輪之節曲線合成 - 47 -
4.3 非圓形鏈輪設計限制 - 50 -
4.4 轉速比圖集 - 51 -
第五章 非圓形鏈輪在SPIROGRAPH 機構上之路徑合成設計 - 58 -
5.1 文獻回顧與研究目標 - 58 -
5.2 非圓形鏈輪在SPIROGRAPH機構的應用與路徑合成 - 59 -
5.2.1 合成路徑判定與路徑正規化 - 59 -
5.2.2 等角度對應分析與包絡線合成 - 65 -
5.2.3 合成角度與轉速比關係 - 72 -
5.2.4 設計限制 - 75 -
5.3 合成例子 - 76 -
第六章 結論 - 94 -
參考文獻 - 96 -

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