在重力的牽引下，任何的機構或機械皆需要一個支撐力，才能使其在任意可到達的位置上保持停駐的平衡狀態，平衡也可以使該機構具有省力或省能源的效果，因此，「平衡」的概念在機構設計中是不容忽略的。平衡的方式簡單可區分為主動式與被動式，主動式平衡是指藉由額外能源的輸入來達到平衡；被動式平衡則是純粹藉由機構本身搭配特殊元件的設計，使整體機構在帶動後仍然可於所有的新位置上保持停駐的平衡狀態，並且只需要施加微小的外力，即能夠慢速地移動具有一定重量的機構，也稱其為「重力平衡」或「靜(力)平衡」。因此，被動式平衡由於設計簡單、低成本、高安全性等優勢，已被應用於各領域中。
本論文首先回顧過去被研究過的靜平衡機構類型，並且以具有最多優勢且被廣泛應用的「彈簧型」靜平衡做為研究方向。首先，參考於文獻「空間彈簧平衡理論」，利用總位能守恆推導空間二自由度(旋轉-旋轉關節)靜平衡模型於單、雙彈簧連接的平衡方程式。接著，使用SolidWorks建模與模擬，針對雙彈簧於空間中不同位置的靜平衡效果與其他影響進行比較和分析。於此研究過程中，發現三點有關於該文獻上缺失的考量，將於本研究中提出並作為主要貢獻，包含 : 1. 修正單彈簧連接中，平衡方程式的不完全推導。 2. 根據平衡方程式的推算，對彈簧位置的限制條件進行分析與統整，例如彈簧連接角度及其範圍。 3. 推導上事先假設連桿質心位置落於中心線上，可能會導致部分雙彈簧機構產生不平衡的結果，可以於彈簧位置的佈置上來解決。最後，將上述空間二自由度靜平衡模型的彈簧位置分類、限制約束與模擬結果的分析等，應用於MATLAB (APP Designer)，開發一電腦輔助設計程式。期望提供工程師一個使用者友善的環境介面，以應對各種類型的彈簧位置設計。

Because of the gravity, mechanism needs a force support to keep its stationary status in any reachable position, also known as "balance". Balance can also allow the mecha-nism owns the features of labor-saving and reduction of energy consumption. Conse-quently, the concept of balance in the mechanism design cannot be ignored. Basically, there are two approaches, active and passive, to accomplish balance. The passive bal-ance method, i.e., static balancing or gravity compensation, has the advantages of easy installation, low cost, and high security so that it has been applied to various fields.
In this thesis, many different types of statically balanced mechanism in the past are reviewed at the beginning and the type with “springs” in static balancing, which is the most advantageous and widely used, is selected as the study’s direction. Referring to the literature “Spatial Spring Equilibrator Theory”, the balancing equations in spatial two-DOF (rotary-rotary joint) static balancing models with single and double springs are carefully derived. Balancing performance is then investigated and analyzed by simulations using SolidWorks with different spring positions. Along this research pro-cess, three facts regarding lack of consideration in the literature were found and will be presented as the major contribution of this thesis. 1. Incomplete derivation in bal-ancing equations for single-spring framework is amended. 2. The restrictions on the spring position such as spring connection angles and their ranges are analyzed and or-ganized. 3. Some assumptions are not feasible in real world and may lead to unbal-anced results in some situations for the double-spring mechanism and can be fixed by arrangement of springs. Finally, according to mechanism classification, mathematical constraints and analysis of simulation results, a computer-aided design program is also developed by MATLAB with APP Designer to provide engineers a user-friendly inter-face environment to cope with arrangement of springs in the spring-supported static balancing mechanism.

論文審定書 ..………………………………......………………………...……... i
誌謝 ..………………………..………………………………………………….. ii
摘要 ..…………………………..………………………………………………. iii
Abstract ...………………………....…………………………………………… iv
目錄 ..……………………………...…………………………….………...……. v
圖目錄 ..…………………………...………………………...………..……….. vii
表目錄 …………………………………………………………….……...……. xi
第一章 緒論 ……………………………………………………………...…….. 1
1.1 研究背景 ………………………………………….……………....…........ 1
1.2 文獻回顧 ………......……………………………………………………... 4
1.3 研究動機與目的 ………………………..………………………....…….. 7
1.4 論文架構 ……………………………………………..………………...… 9
第二章 研究理論 ………………………………………………………..…… 10
2.1 零自由長度彈簧 ……………………………………………………….. 10
2.2 靜平衡原理 …………………………………………….…………….… 12
2.3 平衡公式推導 ………………………………………………………...... 18
2.3.1 單彈簧 ………………………………………………………......….. 19
2.3.2 雙彈簧 ………………………………..……………………..……… 21
2.4 彈簧連接之分類與限制 ……………………………………………..... 26
第三章 模擬驗證與結果分析 …………………………………………….... 30
3.1 參考模型與假設 ………………………………………....……………. 30
3.2 模擬方法 ………………………………………………….……….…… 31
3.3 模型展示與平衡驗證 ……………………………………....…………. 37
3.3.1 單彈簧模型模擬 ……………………………………….………….. 38
3.3.2 雙彈簧模型模擬 …………………………………….....………….. 40
3.4 未平衡之問題討論 ……….…………………………………………… 48
3.5 結果比較與分析 ………………………………………....……………. 54
3.5.1 平衡結果的綜合比較 …………………………...…………….….. 54
3.5.2 彈簧連接位置的影響與分析 ………………..……………….….. 56
第四章 介面與程式設計 ………………………………………………..….. 59
4.1 設計構思與簡介 ………………………………………………………. 59
4.2 介面展示與操作說明 ………………………………….……...………. 61
4.2.1 指示介面 ……………………………………………....……….….. 61
4.2.2 單彈簧介面 ………………………………………....………….….. 63
4.2.3 雙彈簧介面 …………………………………………………….….. 65
4.2.4 建議介面 ……………………………………………………….….. 72
第五章 結論與未來展望 ………………………………………………….... 73
5.1 結論 ……………………………………………………….……………. 73
5.2 未來展望 ………………………...………………….……………….… 75
參考文獻 …………………………………………………………………….. 76
附錄A 平衡方程式推導補充說明 …………………………………………. 79
附錄B 建模與彈簧連接之相關參數 ………………………………...……. 85

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