System Design and Probability Analysis for Up-Shifting of Bicycle Front Sprockets

鏈條與鏈輪是自行車傳動構件中的一套重要機構，傳統鏈輪設計並不關注鏈節空間相對位置與相鄰鏈輪換鏈齒間的相對相角，導致在變速過程中，當鏈條跨越不同鏈輪時，由於撥鏈器在鏈節上的衝壓點瞬間改變產生的鏈條震顫，使得上鏈鏈節跨騎於目標鏈輪輪齒上而滑脫，致使換鏈失敗。本文首先藉由鏈節間的拗折、側偏與扭轉之共軛數學式，導出上鏈時的優化路徑。其次，經由數據分析結果顯示，若是給予上鏈齒適當的齒槽空間切削，增加鏈條側偏能力，在運動學上將可減短上鏈距離及相角，促使鏈輪上鏈點的安排更緊湊，增加鏈輪上鏈點數量，減小鏈輪間隙，使鏈輪塔更形積體化，進而減小撥鏈器所需衝壓力，增進換鏈的效率。 另外，撥鏈器的幾何構型與所需施加之作動力，在一般的鏈輪設計中亦甚少著墨。一般來說，鏈節側偏量不足與撥鏈力過低，經常導致上鏈鏈節跨騎於目標鏈輪輪齒上，使嚙合鏈節掉落回原來鏈輪，導致換檔失敗。事實上，適當的撥鏈方式與給予正確的撥鏈方向，將有助於減輕上鏈時鏈節所需的側向力和鏈條張力、加速鏈條的上鏈過程、延長鏈條和鏈輪的使用壽命、減少變速過程中的振動或噪音，進而提高鏈條的換檔效率。因此，撥鏈器幾何設計的目的即在推導出施加最小作動力時的撥鏈器最佳輪廓。本文以數據分析，證實了最佳軌跡中的所有接觸點在撥鏈器和鏈條之間並非線性排列，透過傳動鏈的準動力平衡差分理論，得出了變速器的優化幾何結構。 然而，在上鏈過程中，自行車鏈條的不穩定擺動就是由撥鏈器引起的。變速器失效也是肇端於撥鏈器與關鍵齒倒角或鏈輪相位角的不匹配。因此，即使透過優化路徑切割出完美倒角，亦不能保證鏈條換檔的成功。因此，本文利用了拉密定理(Lami’s theorem)和瞬時鏈條的靜力平衡，計算出在變速過程中，由撥鏈力引起的鏈條偏擺效應(左右)與弦線運動(上下)，並以此計算整體鏈條定位時，兩者合成效應的聯合機率分析。如此的數據表可以為撥鏈器系統提供效率評估的標準，也可作為鏈條工程設計的參考依據。The relative spatial position of each chain link and the phase angle of engaged tooth are not concerned in traditional sprocket designs for bicycle. As such, the chain shifting might fail regularly. In this study, an optimized path of one chain for up-shifting is derived. Such path is formed by several bended chain links sustain yaw and roll. Empirically, appropriate bending, yaw and torsion angles for tooth chamfer, could be shorten the up-shifting distance and phase angle required, eventually facilitate not only more ascending points compactly arranged on sprocket but larger chain laterodeviation which curtails sprockets clearance and integrates the sprocket unit. Then again, the optimized shifting path promotes efficiency and decreases required movement yielded by derailleur tappet pressure. In addition, some factors such as force applied, force direction, and geometric configuration of a derailleur, which drew scant attention in design of a conventional sprocket. Generally, short yaw of chain link and scarcity of derailleur force induce the chain to straddle on a target sprocket, even make engaged links drop to original one. Thus, gear shifting would be failure. In fact, the proper force applied and direction will contribute to mitigate lateral forces and tensile forces applied on chain links, accelerate the process of chain shifts, extend the service life of chain and sprockets, and reduce vibrations or noises in the process of gear shifting, even improve the efficiency of chain shifts. The purpose of the geometric design for derailleur is to derive an optimal profile that the minimum force applied on. Base on above, all contact points in the optimal trajectory are not linear arrangement between derailleur and chain. In this paper, the optimal geometric configuration of derailleur is formed by employing difference theory of quazi-dynamic equilibrium for transmission chain as the optimal path is derived. However, in the course of up-shifting, unstable swing of bicycle chain is usually caused by derailleur. Moreover, the failure of gear shifting is also arouse by poor match of derailleur with tooth chamfer or phase angle of sprocket. As such, many researchers have devotes themselves on how to cut proper tooth chamfer or various optimized paths for up-shifting of bike chain. However, a perfect tooth chamfer given with an optimized path could not guarantee the success of any gear shifting more, yet. Therefore, this paper employs Lami’s theorem and forces equilibrium for each chain link to propose a probability analysis of positioning for bicycle chain caused by derailleur force and chordal action in the course of gear shifting. Such data could offer a criterion of efficiency evaluation for derailleur system and provide a reference for engineering design also.目次 誌謝 i 中文摘要 ii Abstract iii 目次 iv 表目次 vi 圖目次 vii 第一章 簡介 1 1.1 自行車的發展史 1 1.2 文獻回顧 4 1.3 論文大綱 8 第二章 剛體元件的屬性分析 11 2.1 鏈條 11 2.2 鏈輪 18 2.3 換檔器 21 第三章 換檔路徑的探討 24 3.1 鏈條的自由度 24 3.2 鏈條上鏈路徑的空間模擬 31 3.3 鏈節位置的非線性規畫模型 35 3.4 線性規劃最佳上鏈路徑 37 第四章 自行車鏈條前鏈輪上鏈關鍵齒之齒型設計 41 4.1 上鏈關鍵齒之齒型設計簡介 41 4.2 導鏈齒的切削面與上鏈鏈節的位姿 45 4.3 鏈節對的扭轉緊張模型 51 4.4 鏈節定位假設與上鏈齒的切削 57 4.5 上鏈齒的切削齒廓 61 第五章 自行車前鏈輪上鏈撥鏈片的幾何構形之最佳化設計 72 5.1 換檔器撥鏈片的設計簡介 72 5.2 自行車的負載分析 73 5.3 自行車的傳動與換檔過程 76 5.4 上鏈過程各關鍵點之位置 79 5.5 驅動系統、鏈節位置與撥鏈片的幾何構形 81 5.6 撥鏈片的衝壓輪廓設計 83 第六章 換檔過程中撥鏈力、齒槽深寬與換檔機率的相關性 94 6.1 上鏈過程中路徑機率分析之簡介 94 6.2 上鏈過程中撥鏈偏擺運動和弦線運動的聯合作用 95 6.3 鏈條換檔的橫向擺動 96 6.4 鏈條換檔的複合機率分析 99 6.5 上鏈的機率分析 101 第七章 結論與展望 111 參考文獻 113 附錄-程式碼 11