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1 動力與背景 1.1 介紹 最近，對工程師有用的著作中，行星齒輪傳動就像一個簡單明了的運動學解答一樣給予了一個明確的分析。不幸的是沒有一個出版機構愿意出版一個簡單的設計與分析技術。這一技術考慮到了在普通的例子中動力在齒輪傳動中表現(xiàn)。這論文目的是想彌補這樣一個空缺，在大多數(shù)的例子中，能找到全部的速度以及周轉輪系力的解決方法的技術。在這方法發(fā)展后，列線圖表可被用作產生直覺設計裝置，允許設計者通過視覺去分析齒輪傳動的運動形式，而不許不需重復的去解方程來完成。終于,方法設計與解決裝置的出現(xiàn)，在新的一系列雙軸自行車中，把使用行星輪系的實際性當作能源單位來聯(lián)結。 1.2 動力 2002 年研究這項上述方法。很大程度上激發(fā)了被承擔由弗吉尼亞技術人的供給動力的車隊。在多用車早期設計期間進入了每年ASME 競爭, 這表明，對于兩名車手相對不一致輸入動力最有效的辦法是使用行星輪系。概念在設計之后由人力車隊試圖將使用一列行星齒輪象那個被顯示在上圖1 創(chuàng)造近似地會允許兩個車手對腳蹬以同樣速度和近似地會貢獻產品力量的同樣百分比的系統(tǒng)。行星輪系容納在速度和功率輸入上的區(qū)別由二個車手。輪系特點的本質是這份論文焦點。 圖1 圖1: 齒輪傳動被使用在人供給動力的車的Team..s 設計項目中 運用Willis的方法為發(fā)現(xiàn)齒輪傳動的運動學解答, 它被發(fā)現(xiàn)機制被控制了。 那里..代表在齒輪傳動中各個元素的旋轉的速度, 并且R 是齒輪傳動的基本的傳輸比率。 執(zhí)行一個靜態(tài)分析, 扭矩被發(fā)現(xiàn)被控制 那里代表轉矩 在各個元素在傳動, 并且N 代表齒輪傳動中的齒數(shù)。使用這些等式, 它變得明顯, 達到力量均衡的目標以相等和相反輸入速度是不可能的。 如果和被認為是相等和相反, 然后達到力量均衡, T2 和T5 必須并且是相等和對立的。 根據(jù)等式3, 這意味著R 必須是1 。 不幸地, 這采取分母等式1 到零, 可以驅動到無限。什么直覺地似乎一個簡單的問題解決導致了唯一的解答空間。 以最后期限為競爭結束, 設計計劃被摒棄了傾向于一種更加簡單的解答。研究完成在試圖設計一系列齒輪傳動，成為了一個更加寬廣的研究計劃的基礎。 這個項目驅動, 而不是系列齒輪傳動的設計為一個具體目的, 是創(chuàng)造將允許行星齒輪傳動發(fā)展為任一個可能的應用的數(shù)字的一個簡明的設計方法。 由處理行星在最一般的例子中, 這個項目以及允許探索HPV 隊的失敗的原因，設計工程師定義運動學關系在行星齒輪傳動中的三個分支之間沒有首先選擇齒輪的一個物理安排。 1.3背景 一列行星齒輪傳動被定義作為任一列齒輪傳動中，包含至少循軌道運行由轉動關于它自轉和并且關于的軸的一個齒輪, 或載體。 基本行星, 或周轉圓, 齒輪傳動被顯示在表2, 與被簡化的表示法一起被使用為這份論文剩下的人。 基本的傳動包括二個齒輪, 太陽(1) 和行星(2) 齒輪, 并且第三名成員, 此后指行星載體或架(3)。 圖2 圖2: (a) 基本的周轉圓的齒輪傳動和(b) 它的運動學表示法。 因為它很難直接地把轉動傳送到從行星齒輪, 基本的周轉圓的齒輪傳動有些被限制在實際應用。然而，更加有用的是，周轉圓傳動指簡單和復雜行星齒輪傳動, 那里第二個太陽齒輪被使用。 這些齒輪傳動可能會在任何十二個安排被指出的圖3. 是依照由L..vai 最初提出。 圖3 圖3: 簡單和復雜周轉圓的齒輪傳動。 傳動在象限I 和III 被分類作為簡單的周轉圓傳動, 因為行星齒輪是在與兩個太陽齒輪嚙合。那些在象限II 和IV 代表復雜齒輪傳動, 那里行星齒輪部份地是在互相嚙合和部份地在嚙合與二個太陽齒輪。 注意那, 不管安排, 只一個行星載體也許被使用。 當這個圖清楚地顯示周轉圓的齒輪傳動的十二個可能的排列, 記法使用很被難掌握。 為了援助在實際傳動的形象化模擬, 圖4 顯示更低的布置在象限I的一級齒輪傳動。 圖4 圖4: 象限I 的更低布置的周轉圓的齒輪傳動在表3。 行星齒輪傳動第一次出現(xiàn)在古老中國, 是大約2600BC ,。 當磁性指南針它誕生時候, 中國人面對了難題是運用它，橫跨一望無際的戈壁沙漠。 克服這個困難,這機器被發(fā)展了。 這個設備使用了一列相對地復雜行星齒輪傳動，附有驅動的二個輪子維護一個圖在推車上面指向在同樣方向, 不管道路怎樣，都向前運動。這個設備的復雜似乎表明, 中國人使用有差別的驅動相當一段時間的傳動的裝置的誕生之前。 這時，行星齒輪傳動消失從歷史相當一段時間。 這更加可能歸結于缺乏目標, 而不是實際原則的不用。 在裝置以后, 下次出現(xiàn)行星傳動是在什么被命名了安尼可雅機器。1901 年由海綿潛水者發(fā)現(xiàn)在離Antikythera 希臘海島的沿海的附近, 它由學者辨認了作為類型計算器被使用為預言蝕和其它占星術事件。這個特殊設備建于大約82BC, 留下期間行星齒輪傳動通過相對地未被注意的由人類歷史大致2500 年的空白在 行星齒輪的傳動原理在遠東拯救了歐洲的黑暗年代, 由設備的發(fā)現(xiàn)上見證相似與Antikythera 機器由伊朗savant 說出Al Biz4una 名字在第一個世紀廣告晚期。 在巨大新生期間, 行星被獲取的廣泛用途在星盤和時鐘里。機制的用途和發(fā)展在新生過程中一直持續(xù)到當今天。 在這點可以有趣的表明, 從2600BC行星原理成功地被使用了, 在1841機制的衛(wèi)利斯的原則出版之前，任一目的都是為了創(chuàng)造設備的一個分析模型。 1.4 文學回顧 羅伯特?衛(wèi)利斯1857 年的出版物,即機制的原則, 廣泛被看待如同第一出版物單一地致力現(xiàn)在叫動力學領域。 在他的工作中, 衛(wèi)利斯第一次在出版文學里談論分析塑造一周轉圓的行星齒輪傳動。因為這工作純粹地在研究一關于機制的, Willis 提出唯一一種解答為旋轉的速度在齒輪傳動。 在研究這種解答以后, 作者讓剩下的致力周轉圓的齒輪傳動的人去談論機制的應用。當這次討論很好被設想時, 它報道四種齒輪的周轉傳動卓而又模糊的應用, 由于工作年齡的關系。 依照早先的陳述, 這工作研究的僅僅是齒輪傳動的動力學,在機制中任一次關于扭矩的討論都沒有提出。 在他的博士論文關于技術大學的建筑中, 民用和運輸工程學在匈牙利, 周轉圓的齒輪和周轉圓的變速齒輪的理論, Dr Z 。 L..vai 試圖成利用所有早先書面文學關于周轉齒輪傳動并且他稱.周轉傳動的變速齒輪傳動, 哪些看來簡單地都是些多速度傳輸。 在對讀者解釋構成一周轉圓傳動時L..vai第一次確切地指出, 周轉圓傳動有一十二種可能。他還解釋到, 這十二可能清楚地被劃分為有沒有輔助行星或行星對。在第一版中的任一目的都是為了清楚而簡捷的行星傳動的所有可能的排列。 在周轉圓傳動定義以后, L..vai 突然改變了對它的解答的關注。 在簡要地談論解答方法以后由Willis 計劃, 并且通過Kutzbach的圖解方法，作為它適用于沒有輔助行星輪的傳動。最后，他充分談了兩種不同的定義，這定義可能進行適用于Kutzbach方法有輔助行星輪的傳動。再者，他沒提供在這一系統(tǒng)中的解決扭矩的辦法。 麻省理工學院,機械工程Deane Lent教授在1961 年出版了他的著作—— 機制解析和設計。 在這著作中Lent教授再次詳細提出Willis 關于找到有三個與四個齒輪傳動的特別設計方法周轉圓齒輪傳動每一級的轉動速度。當這些技術很好的被寫和簡單應用后, 在這一系統(tǒng)中還是沒有關于扭結的討論。在這出版書中包括了及比所有Willis 談論的更相關的幾種行星齒輪傳動的應用。 約瑟夫?Shigley 和約翰?Uicker 在1980 年出版了他們的動力學文本, 機器和機制的理論。 在這著作中不僅Willis..s 方法學的分析, 而且對周轉圓的齒輪傳動有一個更加完全的定義。他們不僅對這個定義進行了相當數(shù)量的討論,而且他們再次生存了L..vai.的圖象描述行星齒輪傳動的十二可能的變異。然而，最重要地是他們在當前齒輪傳動中提出了扭矩的一個解答技術。不幸地，他們不能把相近的靜態(tài)力量分析作為一般事件; 他們?yōu)橐粋€特別的安排行星輪，通過根據(jù)自由體圖可提出解答。 這個方法相對地簡單, 它限制設計師在早期設計過程中對一對齒輪排布。機制和機械動力學, 哈密爾頓Mabie 和查爾斯?Reinholtz 的出版物,出版了大量和Shigley 和Uicker 一樣的信息。 當動力學的解析和機制的靜態(tài)力量是幾乎相同的, Mabie 和Reinholtz 并且提出一個簡要的部分來考慮在從行星輪系中的流通功率。 當這次討論沒有直接應用這份論文, 它暗示其中使用的方法在齒輪傳動中解決靜態(tài)力量為一般事例。 1981 年約翰?Molnar 出版了他的計算圖。 這著作給出了對計算圖優(yōu)秀介紹, 并且充分談論他們的用途和建筑。 這著作在計算圖的建筑此中被提出是有幫助的。當大多數(shù)這出版物致力于計算圖的再生產，包括問題寬廣的一般類別處理空氣, 水, 并且相關的機械設備, 介紹為新手比足夠的信息提供更多完全地了解對計算圖的建筑和用途為幾乎任一個問題的解答。 2 CAD/CAM 2.1 CAD/CAM導論 縱觀工業(yè)社會的發(fā)展歷史，諸多發(fā)明都被申請為專利，并且新的技術體系也逐步進化。其中比以前任何一項技術能對制造業(yè)產生更迅速、更重大影響的發(fā)明或許就是數(shù)字計算機。計算機在繪圖部門正在被越來越多地應用于設計和工程零部件的詳細說明中。 計算機輔助設計(CAD)就是應用計算機和圖形軟件，在構思到文檔形式的過程中來幫助或改善產品設計。計算機輔助設計通常與一個交互式計算機圖形系統(tǒng)的應用聯(lián)系在一起，稱為計算機輔助設計系統(tǒng)。計算機輔助設計系統(tǒng)是進行產品和零部件的機械設計及幾何建模的強有力的工具。 采用CAD系統(tǒng)支持工程設計有以下優(yōu)點： l 提高生產率 l 提高設計質量 l 統(tǒng)一設計標準 l 創(chuàng)建制造數(shù)據(jù)庫 l 消除手工繪圖的誤差和不相容性 計算機輔助制造（CAM）就是在制造計劃和控制中有效地使用計算機技術。計算機輔助制造是與制造工藝聯(lián)系最緊密的功能，例如工藝過程和生產規(guī)劃、機械加工、進度安排、管理、質量控制和數(shù)字控制（NC）零件加工程序。計算機輔助設計和計算機輔助制造經常結合在一起構成CAD/CAM系統(tǒng)。 這種結合在一起的系統(tǒng)允許在制造一種產品時，從設計階段到計劃階段進行信息傳遞，不再需要手工來輸入零件幾何機構數(shù)據(jù)。在計算機輔助設計期間建立的數(shù)據(jù)庫被儲存起來，然后通過計算機復制造進行進一步的處理，轉變?yōu)椴僮骱涂刂粕a機械、材料處理裝置和進行產品質量自動檢測所必需的數(shù)據(jù)和指令。 2.2 CAD/CAM的基本原理 CAD/CAM基本原理類似于用于在制造業(yè)中判斷任何基于技術的改進原理。它產生于生產力、產品質量和競爭力不斷提高的需求。還有如下一些因素促使一家公司將手工加工方式改造為應用CAD/CAM系統(tǒng)來進行生產： l 不斷增長的生產率 l 更好的產品質量 l 更方便的信息交流 l 在制造過程中共用數(shù)據(jù)庫 l 降低制造樣機的費用 l 加快對用戶的反應 2.3 CAD/CAM的硬件 CAD/CAM系統(tǒng)的硬件部分由以下幾塊組成：（1）一個或多個設計工作站，（2）數(shù)字計算機，（3）繪圖儀、打印機和其他輸出設備，（4）儲存設備。另外，CAD/CAM系統(tǒng)有一個通信接口允許從其他計算機系統(tǒng)或向其他計算機系統(tǒng)傳遞數(shù)據(jù)，因此有利于一些計算機集成。 工作站是CAD系統(tǒng)中計算機和用戶之間的接口。CAD工作站的設計和它的實用特征對用戶輸出的方便性、生產率和質量將產生很重要的影響。工作站必需包括一個圖形顯示終端和一套用戶輸入設備。CAD/CAM系統(tǒng)的應用要求有一臺具有高速中央處理器（CPU）的數(shù)字計算機。它包含主存儲器和邏輯/算術部分。在CAD/CAM中使用最廣泛的輔助存儲介質是硬盤、軟盤或它們兩個的結合。 在CAD系統(tǒng)理典型的輸入/輸出設備如圖10.2所示。輸入設備一半被用來把信息從人或者儲存介質傳遞到一臺能夠執(zhí)行“CAD功能”的計算機中。有兩種基本方法來輸入已經存在的圖形：在圖紙上建模或把圖形數(shù)字化。CAD/CAM的標準輸出設備是陰極射線管顯示器。有兩種主要類型的陰極射線管顯示器：隨機掃描圖形顯示器和光柵掃描顯示器，除陰極射線管顯示器外，還有等離子平板顯示器和液晶顯示器。 2.4 CAD/CAM的軟件 軟件使用戶從一個硬件設備進入一個強有力的設計和制造系統(tǒng)。根據(jù)完成的幾何圖形的維數(shù)，CAD/CAM軟件分為兩大類：二維和三維軟件。在二維空間里描繪對象的CAD設計包稱為二維軟件。早期的系統(tǒng)局限于二維空間。這是一個嚴重的缺陷，因為用二維空間來表示三維的物體本身就容易讓人混淆，而且還存在制造人員自己不能正確讀懂和解釋用來表示三維物體的二維圖形。三維軟件可使零件的三維尺寸---長、寬和高均可見。 CAD/CAM的發(fā)展趨向于用三維來表示圖形。這種表示法接近所描繪的物體和實際形狀和外觀，因此，它們更容易被讀懂和理解。 2.5 CAD/CAM的應用 CAD/CAM的出現(xiàn)對整個制造業(yè)有很大的影響，它能夠將產品開發(fā)標準化、降低設計強度、減少試驗和樣機制造工作，且能夠節(jié)省相當多的成本費用并提高生產率。 CAD/CAM的一些典型應用如下： l 為數(shù)控、計算機數(shù)控和工業(yè)機器人編程； l 在設計鑄造的模具和模型時，可按照預編程序縮小加工余量； l 工具、固定裝置和EDM（電火花機床）電極的設計； l 質量控制和檢測，例如：在CAD/CAM工作站中進行坐標測量機編程； l 工藝計劃與進度安排。 2.6 CAD/CAM的優(yōu)點 使用CAD的原因有很多，最有效的動力就是競爭。為了贏得業(yè)務，公司使用CAD可以創(chuàng)造出更好的設計，并且在設計速度上比競爭對手更快，在成本上花費更少，。通過使用CAD,生產率得到了很大的提高，使用戶能夠很容易地畫多邊形、橢圓、多條平行線和多條平行的曲線。在繪制對稱部分時、復制、旋轉、鏡象這些工具使用起來也是很方便的。很多飛機艙口的樣式就是用CAD程序設計的。用各種不同的顏色填充空白的區(qū)域是藝術和表達的需要。CAD總是提供許多不同類型的字體。能夠將不同的圖形文件格式和掃描材料（照片）導入CAD也是一大優(yōu)點，特別是可以對圖像進行加工、潤飾和加入動畫效果。 CAD系統(tǒng)另一個優(yōu)點是能夠儲存在繪圖中經常用到的實體。常用零件庫可以另外購買或者由繪圖員自己創(chuàng)建。在繪圖中反復使用的一個典型的項目可以在數(shù)秒內檢索并確定它的位置，也可定位在任一角度，以滿足特定的要求。 使用CAD的產品，可以通過插入現(xiàn)有的零件圖到裝配圖中，然后按照要求把他們放在合適的位置來繪制裝配圖。 不同零部件之間的間距能夠在圖中直接測量。如果需要，可以使用裝配圖設計出額外的零部件作為參考。 CAD非常適合文件的快速歸檔。以前，工程師和繪圖員們浪費大約30%的時間去尋找圖紙和其他文檔。用CAD產品可以快速而簡便地編輯圖樣，對以前的東西進行修改，更新零件明細表。 當你用紙繪圖而客戶希望修改圖樣的時候，你就得全部重畫。使用CAD，你可以馬上進行修改，并在幾秒鐘之內打印出新圖，或者通過E-mail和互聯(lián)網(wǎng)立即傳送到世界各個地方。在紙上繪制復雜的幾何圖形時，經常要進行很多測量并且需要確定參考點。在CAD中，這是一個輕而易舉的事情，修改也更容易了。許多CAD程序包含“宏”或者允許用戶定制的附加程序語言。 定制你的CAD系統(tǒng)來你的使它適合你的特定要求，并用它實現(xiàn)你的天才創(chuàng)意，從而使你的CAD系統(tǒng)區(qū)別于你的競爭對手。CAD能夠使企業(yè)完成更出色的設計，而用手工的方式幾乎是不可能，同時排除了概念設計階段的不確定選項。 9 1 MOTIVATION AND BACKGROUND 1.1 Introduction In the current literature available to engineers, planetary gear trains are given a clear treatment as far as a simple kinematic solution. Unfortunately, no publications to date present a simple, concise design and analysis technique that considers both the motion and forces present in a gear train in the general case. This thesis attempts to fill this void by presenting a technique for finding a total speed and force solution to an epicyclic gear train in the most general case possible. After developing this solution, nomographs will be used to create an intuitive design aid, allowing the designer to visualize the performance of a gear train without the need to solve equations repeatedly. Finally, the solution technique and design aids presented will be used to address the practicality of using planetary gear trains as a power coupling element in a new generation of tandem bicycles. 1.2 Motivation The research contained herein was motivated by a design effort undertaken by the Virginia Tech Human Powered Vehicle Team in 2002. During the early design of the multi-rider entry into the annual ASME competition, it was suggested that the most effective method for coupling the relatively inconsistent inputs of two human riders would be to use a planetary gear train. The concept behind the design attempted by the human powered vehicle team was to use a gear train like the one shown in figure 1 to create a system that would allow both riders to pedal at approximately the same speed and contribute approximately the same percentage of the output power. The planetary system accommodates differences in speed and power input by the two riders. The nature of the system behavior is the focus of this thesis. Figure 1 Figure 1: Gear train to be used in the Human Powered Vehicle Team’s design effort Using Willis’s [1] method for finding the kinematic solution of the gear train, it was found that the mechanism was governed by where the ω’s represent rotational speeds of each element in the gear train, and R is the basic transmission ratio of the gear train. Performing a static analysis, the torques were found to be controlled by where the T’s represent torques on each element in the train, and the N’s represent number of teeth in each gear in the train. Using these equations, it became apparent that the goal of achieving power balance at equal and opposite input speeds was impossible. If ω2 and ω5 are assumed to be equal and opposite, then to achieve a power balance, T2 and T5 must also be equal and opposite. According to equation 3, this means R must be 1. Unfortunately, this takes the denominator of equation 1 to zero, which drives ω6 to infinity. What had seemed intuitively a simple problem to solve had led to a singularity in the solution space. With deadlines for competition closing in, the design effort was abandoned in favor of a simpler solution. However, the research done in attempting to design a specific gear train became the foundation of a much broader research project. The drive of this project, rather than the design of a gear train for a specific purpose, is to create a concise design method that will allow development of planetary gear trains for any number of possible applications. By dealing with the planetary in the most general case possible, this project explores the reasons for the failure of the HPV team’s design as well as allowing engineers to define the kinematic relationships between the three branches of the planetary gear train without first selecting a physical arrangement of gears. 1.3 Background A planetary gear train is defined as any gear train containing at least one gear that orbits by rotating about its own axis and also about the axis of an arm, or carrier. The elementary planetary, or epicyclic, gear train is shown in figure 2, along with the simplified representation to be used for the remainder of this thesis. The elementary train consists of two gears, the sun (1) and planet (2) gears, and a third member, hereafter referred to as the planet carrier or arm (3). Figure 2: (a) The elementary epicyclic gear train and (b) its kinematical representation Since it is difficult to directly transmit motion to or from the planet gear, the elementary epicyclic gear train is somewhat limited in practical application. More useful, however, are the epicyclic trains referred to as the simple and complex planetary gear trains, where a second sun gear is used. These gear trains can be realized in any of the twelve arrangements set forth in figure 3, as originally presented by Lévai. The trains in quadrants I and III are classified as simple epicyclic trains, since the planet gears are in mesh with both sun gears. Those in quadrants II and IV represent the complex trains, where the planet gears are partially in mesh with each other and partially in mesh with the two sun gears. Notice that, regardless of arrangement, only one planet carrier may be used. Figure 3 Figure 3: The simple and complex epicyclic gear trains While this figure clearly shows the twelve possible arrangements of the epicyclic gear train, the notation used is difficult to grasp. To aid in the visualization of the actual trains represented, figure 4 shows a gear train of the lower arrangement in quadrant I. Figure 4 Figure 4: Epicyclic gear train of the lower arrangement of quadrant I in figure 3 The planetary gear train first appeared in ancient China, around 2600 BC, in a device referred to as the south pointing chariot. At a time when the magnetic compass was still centuries away from its birth, the Chinese faced the difficult task of navigating across the relatively featureless Gobi Desert. To surmount this difficulty, the south pointing chariot was developed. This device used a relatively complex planetary gear train attached to the two wheels of a cart to maintain a figure atop the cart pointing in the same direction, regardless of the path taken by the cart. The complexity of this device seems to indicate that the Chinese had been using differential drives for quite some time before the birth of the south pointing chariot. At this point, the planetary gear train disappears from history for quite some time. This is more likely due to a lack of writing on the subject, rather than the actual disuse of the principle. After the south pointing chariot, the next appearance of the planetary is in what has been named the Antikythera machine. Discovered by sponge divers off the coast of the Greek island ofAntikythera in 1901, it has been identified by scholars as a type of calculator used for predicting eclipses and other astrological events. This particular device has been dated back to approximately 82 BC, leaving a gap of roughly 2500 years during which the planetary gear train passed relatively unnoticed through human history [8]. The principle of the planetary gear survived Europe’s dark ages in the Far East, evidenced by the discovery of a device similar to the Antikythera machine by an Iranian savant named Al-Biz?na in the late first century AD. During the Great Renaissance, the planetary garnered wide use in astrolabes and clocks. The use and development of the mechanism continued throughout the Renaissance and on until present day. It is interesting to note at this point that, while the planetary has been successfully used since 2600 BC, it was not until the 1841 publication of Willis’s Principles of Mechanism [1] that any attempt was made to create an analytical model of the device. 1.4 Literature Review Robert Willis’s 1857 publication, Principles of Mechanism, is widely regarded as the first publication dedicated solely to the field now called kinematics. In his work, Willis discusses for the first time in published literature the analytical modeling of an epicyclic gear train. As this work is a study purely in mechanism, Willis presents only a solution for the rotational speeds in the gear train. After developing this solution, the author spends the remainder of the work dedicated to epicyclic gear trains in discussing applications of the mechanism. While this discussion is well conceived, it covers four remarkably obscure applications of the epicyclic gear train, owing to the age of the work. As stated previously, this work studied only the pure kinematics of the gear train, without any discussion of the torques present in the mechanism. In his doctoral dissertation for The Technical University of Building, Civil and Transport Engineering in Hungary, Theory of Epicyclic Gears and Epicyclic Change-Speed Gears, Dr Z. Lévai attempts to unify all of the previously written literature on epicyclic trains and what he calls “epicyclic change speed gears”, which appear to simply be multiple speed transmissions. In explaining to the reader exactly what constitutes an epicyclic train Lévai identifies, for the first time, the twelve possible variations on the epicyclic train. It is also stated that these twelve variations can be neatly divided into those with and without auxiliary planets or planet pairs. This is the first publication where any attempt was made to clearly and concisely define all possible arrangements of the planetary train. After defining the epicyclic train, Lévai turns his attention to its solution. After briefly discussing the solution method laid out by Willis, and the graphical method of Kutzbach [5] as it applies to trains without auxiliary planets, he discusses at length two different modifications that can be performed to apply the Kutzbach method to a train with auxiliary planets. Again, he offers no treatment of the torques present in the system. Deane Lent, professor of Mechanical Engineering at Massachusetts Institute of Technology, published his work, Analysis and Design of Mechanisms, in 1961. In this work Lent again presents in detail the methodology of Willis for finding the rotational speeds of each branch of the epicyclic gear train, along with specific methods for the design of three and four gear trains. While these techniques are well written and simple to follow, there is again no discussion of torques present in the system. Also included in this publication are several applications of the planetary gear train, all significantly more relevant than those discussed by Willis. Joseph Shigley and John Uicker published their kinematics text, Theory of Machines and Mechanisms, in 1980. Within this work are not only a treatment of Willis’s methodology, but also a more complete definition of the epicyclic gear train. Not only do they dedicate a significant amount of discussion to this definition, but they also reproduce Lévai’s figure demonstrating the twelve possible variations of the planetary gear train. Most importantly, however, they present a solution technique for the torques present in the gear train. Unfortunately they do not approach the static force analysis for the general case; rather they present the solution in terms of free body diagrams for a specific arrangement of the planetary. While this method is relatively simple, it limits the designer to a single arrangement early in the design process. Mechanisms and the Dynamics of Machinery, the publication of Hamilton Mabie and Charles Reinholtz, presents largely the same information as Shigley and Uicker. While the treatment of the kinematics and static forces of the mechanism are nearly identical, Mabie and Reinholtz also present a brief section considering circulating power flow in controlled planetary gear systems. While this discussion has no direct application to this thesis, it does hint at the methods used herein to solve for the static forces in the gear train for the general case. John Molnar published his Nomographs in 1981. This work presents an excellent introduction to nomographs, as well as discussing at length their use and construction. This work was instrumental in the construction of the nomographs presented herein. While the bulk of this publication is dedicated to the reproduction of nomographs covering the broad general category of problems dealing with air, water, and related mechanical devices, the introduction provides more than enough information for a novice to completely understand the construction and use of nomographs for the solution of nearly any problem. 2 CAD/CAM 2.1 Introduction to CAD/CAM Thoughout the history of our industrial society,many inventions have been patented and whole new technologies have evolved.Perhaps the single development that has impacted manufacturing more quickly and significantly than any previous technology is the digital computer.Computer are being used increasingly for both design and detailing of engineering components in the drawing office. Computer-aided design (CAD) is defined as the application of computers and graphics software to aid or enhance the product design from conceptualization to documentation.CAD is most commonly associated with the use of an interactive computer graphics system,referred to as a CAD system.Computer-aided design systems are powerful tools and are used in the mechanical design and geometric modeling of products and components. There are several good reasons for using a CAD system to support the engineering design function: l To increase the productivity l To improve the quality of the design l To uniform design standards l To eliminate inaccuracies caused by hand-copying of drawings and inconsistency between drawings Computer-aided manufacturing (CAM) is defined as the effective use of computer technology in manufacturing planning and control.CAM is most closely associated with functions in manufacturing engineering,such as process and production planning, machining, scheduling, management, quality control, and numerical control (NC) part programming, Computer-aided design and computer-aided manufacturing are often combined into CAD/CAM systems. This combination allows the transfer of information from the design stage into the stage of planning for the manufacturing of a product,without the need reenter the data on part geometry manually.The database developed during CAD is stored; then it is processed further,by CAM,into the necessary data and instructions for operating and controlling production machinery,material-handling equipment,and automated testing and inspection for product quality. 2.2 Rationale for CAD/CAM The rationale for CAD/CAM is similar to that used to justify any technology-based omprovement in manufacturing.It grows out of a need to continually improve productivity,quality and competitiveness.There are also other reasons why a company might make a conversion from manual processes to CAD/CAM: l Increased productivity l Better quality l Better communication l Common database with manufacturing l Reduced prototype construction costs l Faster response to customers 2.3 CAD/CAM Hardware The hardware part of a CAD/CAM system consists of the following components:(1)oneor more design workstaions,(2)digital computer,(3)plotters,printers and other output devices,and (4)storage devices. The relationship among the components is illustrated in Fig.10.1.In addition,the CAD/CAM system would have a communication interface to permit transmission of data to and from other computer systems,thus enabling some of the benefits of computer integration. The workstation is the interface between computer and user in the CAD system.The design of the CAD workstation and its available features have an important influence on the convenience,productivity,and quality of the user’s output.The workstation must include a graphics display terminal and a set of user input devices.CAD/CAM applications require a digital computer with a high-speed control processing unit(CPU).It contains the main memory and logic/arithmetic section for the system.The most widely used secondary storage medium in CAD/CAM is the hard disk,floppy diskette,or a combination of both. The typical I/O devices used in a CAD system are shown in Fig.10.2. Input devices are generally used to transfer information from a human or storage medium to a computer where “CAD functions”are carried out.There are two basic approaches to input an existing drawing:model the object on a drawing or drawing or digitize the drawing.The standard output device for CAD/CAM is a CRT display.There are two major types of CRT displays:random-scan-line-drawing displays and aster-scan displays.In addition to CRT,there are also plasma panel displays and liquid-crystal displays. 2.4 CAD/CAM Software Software allows the human user to turn a hardware configuration into a powerful design and manufacturing system.CAD/CAM software falls into two broad categories,2-D and 3-D, based on the number of dimensions visible in the finished geometry.CAD packages that represent objects in two dimensions visible in the finished geometrey.CAD packages that represent objects in two dimensions are called 2-D software.Early systems were limited to 2-D.This was a serious shortcoming because 2-D representations of 3-D objects is inherently confusing.Equally problem has been the inability of manufacturing personnel to properly read and interpret complicated 2-D representations of objects.3-D software permits the parts to be viewed with the three-dimensional planes-height,width,and depth-visible.The trend in CAD/CAM is toward 3-D representation of graphic images.Such representations approximate the actual shape and appearance of the object to be produced;therefore,they are easier to read and understand. 2.5 Applications of CAD/CAM The emergence of CAD/CAM has had a major impact on manufacturing,by standardizing product development and by reducing design effort,tryout,and prototype work;it has made possible significantly reduced costs and improved productivity. Some typical applications of CAD/CAM are as follows: l Programming for NC,CNC,and industrial robots; l Design of dies and molds for casting,in which,for example,shrinkage allowances are preprogrammed; l Design of tools and fixtures and EDM(electrical-discharge machining)electrodes; l Quality control and inspection---for instance,coordinate-measuring machines programmed on a CAD/CAM workstation; l Process planning and scheduling. 2.6 Advantage of CAD/CAM There are many reasons for using CAD;the most potent driving force is competition.In order to win business,companies used CAD to produce better designs more quickly and more cheaply than their competitors.Productivity is much improved by a CAD program enabling you to easily draw polygons,ellipses,multiple parallel lines and multiple parallel curves.Copy,rotate and mirror facilities are also very handy when drawing symmetrical parts.Many hatch patterns are supplied with CAD programs. Filling areas in various colors is a requirement in artwork and presentations.Different style fonts for text are always supplies with any CAD programs.The possibility of importing different graphic file formats and scanning of material(photographs)into a CAD program is also an asset especially as the image can be manipulated,retouched and animated. Another advantage of CAD system is its ability to store entities,which are frequently used on drawings.Libraries of regularly used parts can be purchased separately or can be created by the draughtsman.For repetitive use on a drawing,a typical item may be retrieved and positioned in seconds,also oriented at any angle to suit particular circumstances. Using CAD products,assembly drawings can be constructed by inserting existing component drawings into the assembly drawing and positioning them as required. Clearance between different components can be measured directly from the drawing,and if required,additional components designed using assembly as reference. CAD is very suitable for fast doc