雙速行星傳動(dòng)地面打磨機(jī)的設(shè)計(jì)含9張CAD圖

雙速行星傳動(dòng)地面打磨機(jī)的設(shè)計(jì)含9張CAD圖,行星,傳動(dòng),地面,打磨,設(shè)計(jì),cad 附錄1：外文翻譯 行星齒輪 介紹 Tamiya行星輪變速箱由一個(gè)約 10500 r/min,3.0V，1.0A的直流電機(jī)運(yùn)行。最大傳動(dòng)比 1:400，輸出速度為26r/min。 四級(jí)行星輪變速箱由兩個(gè) 1: 4 和兩個(gè) 1: 5的傳動(dòng)級(jí)組成，并可以任意選擇組合。 對(duì)于小的機(jī)械應(yīng)用程序這不僅是一個(gè)良好的驅(qū)動(dòng)器，而且還提供了一個(gè)出色檢驗(yàn)的行星齒輪系。 這種齒輪變速箱是一種設(shè)計(jì)非常精心的塑料套件，可在約一個(gè)小時(shí)用很少的工具裝配完成。 參考文獻(xiàn)中給出了裝備資料。 下面讓我們來(lái)開(kāi)始檢驗(yàn)齒輪傳動(dòng)裝置的基本原理和分析行星輪系的技巧。 行星輪系 一對(duì)直齒圓柱齒輪的由節(jié)圓表示在圖表中，它們相切與節(jié)點(diǎn)P點(diǎn)，嚙合齒輪的輪齒齒頂超出了節(jié)圓半徑，在節(jié)圓與齒齒頂之間有一齒頂間隙，。 若節(jié)圓半徑分別為a和b，齒輪軸之間的距離就是 a + b。 為了確保齒輪傳動(dòng)中，一個(gè)節(jié)圓在另一個(gè)節(jié)圓上沒(méi)有滑動(dòng)，必須得有適當(dāng)?shù)男螤畲_保從動(dòng)輪與主動(dòng)輪的運(yùn)動(dòng)一致。 這就意味著接觸線以正常接觸齒廓的形式通過(guò)節(jié)點(diǎn)。這時(shí)，動(dòng)力傳遞脫離高速震動(dòng)達(dá)到可能。 在這里我們不會(huì)進(jìn)一步談?wù)擙X輪輪齒，以及上述有提到的傳動(dòng)裝置的基本原理。 如果一個(gè)齒輪節(jié)圓半徑上有 N 個(gè)齒，這時(shí)在兩個(gè)連續(xù)的齒間的距離，我們稱的齒間距將會(huì)是 2πa/N。如果兩個(gè)齒輪相嚙合，他們之間的齒距必須是相同的。他們之間的節(jié)距通常以2a/N來(lái)表示，我們稱為模數(shù)。 如果你計(jì)算一個(gè)齒輪的齒數(shù)，這時(shí)節(jié)圓直徑的大小是模數(shù)的倍數(shù)，而倍數(shù)則是齒數(shù)。如果你知道兩個(gè)齒輪的節(jié)圓直徑，那么你就能夠得出兩齒輪軸之間的距離。 一對(duì)齒輪的傳動(dòng)比r 驅(qū)動(dòng)輪與從動(dòng)輪之間的角速度之比。 因?yàn)榉侄葓A之間旋轉(zhuǎn)方向的限制條件，r =-a / b =-N 1 /N 2,，因此它們之間的節(jié)圓半徑比與齒數(shù)成正比。齒輪角速度n可以用轉(zhuǎn)/秒，轉(zhuǎn)/分，或者任何類似的單位表示。如果以一齒輪的旋轉(zhuǎn)方向?yàn)檎藭r(shí)另外一個(gè)的方向則為負(fù)。這就是上面的表達(dá)式中的 (-) 標(biāo)志的由于原因。如果其中一個(gè)是內(nèi)齒（齒在齒圈內(nèi)部），這時(shí)傳動(dòng)比為正，因此它們的傳動(dòng)方向一致。 常用漸開(kāi)線齒輪的牙形能夠允許軸線之間一定的變位 ，所以即使它們之間的距離不是很精確也能夠順利的運(yùn)行。齒輪的傳動(dòng)比并不依賴于該軸的精確的間距，而是輪齒或者節(jié)圓諸如此類之間的安裝。稍微增加高于其理論值的距離，能夠使運(yùn)行更容易。因?yàn)槠溆蜗遁^大的齒輪， 在另一方面 齒隙 也增加，它可能不是我們?cè)谀承?yīng)用上所希望的。 一個(gè)行星輪系包含了固定在齒輪軸上的轉(zhuǎn)臂和行星架以及齒輪和旋轉(zhuǎn)的齒輪軸。一個(gè)移動(dòng)的 手臂 或 承運(yùn)人 的有關(guān)該的軸以及齒輪自己可以旋轉(zhuǎn)的齒輪軸。轉(zhuǎn)臂可以是一個(gè)輸入或輸出構(gòu)件而且可被固定固定或可旋轉(zhuǎn)。 最外面的齒輪為內(nèi)齒輪。 一個(gè)簡(jiǎn)單常見(jiàn)的行星輪是如左圖所示的太陽(yáng)-行星輪系。這是三個(gè)行星齒輪輪系用于機(jī)械領(lǐng)域的原因 ； 他們可能被認(rèn)為是在描述該傳動(dòng)裝置的操作之一。 太陽(yáng)輪、 轉(zhuǎn)臂或內(nèi)齒輪可能成為輸入或輸出的鏈接。 如果轉(zhuǎn)臂被固定，就不能旋轉(zhuǎn)，一個(gè)簡(jiǎn)單的三行星輪輪系嗎有n 2 /n 1 =-N 1 /N 2，n 3 /n 2 = + N 2 /N 3，和 n 3 /n 1 =-N 1 /N 3。 這是非常簡(jiǎn)單，不應(yīng)令人困惑。 如果轉(zhuǎn)臂允許移動(dòng)，算出速度比彰顯出了人類的智慧。 嘗試這將顯示該陳述的真實(shí)性 ； 如果你能做到，你應(yīng)得到贊揚(yáng)和聲譽(yù)。 這并不意味這將不可能，只是比較復(fù)雜罷了。 不過(guò)，有一個(gè)非常簡(jiǎn)單的方法獲得所需的結(jié)果。 首先，把這輪系假定認(rèn)為是鎖定的，因此把轉(zhuǎn)臂和所有的作為剛體、。 所有的三個(gè)齒輪和手臂然后有一個(gè)統(tǒng)一的速度比。 行星齒輪任何運(yùn)動(dòng)的特點(diǎn)是可以被第一個(gè)固定支撐轉(zhuǎn)臂和相對(duì)于另外一個(gè)旋轉(zhuǎn)的齒輪實(shí)現(xiàn)，然后鎖定輪系并關(guān)于固定的軸旋轉(zhuǎn)。凈運(yùn)動(dòng)總和或兩個(gè)不同的獨(dú)立的分離運(yùn)動(dòng)來(lái)滿足這問(wèn)題的條件（通常一個(gè)構(gòu)件被固定）。若要進(jìn)行此程序，構(gòu)造的齒輪和轉(zhuǎn)臂臂的角速度列出兩例的每個(gè)表。 鎖定的輪系給定的N1, N2, N3 為齒輪 1、 齒輪 2 和齒輪3。 固定轉(zhuǎn)臂為 0，1，-N 1 /N 2，-N 1 /N 3。 假定我們想知道齒輪1與轉(zhuǎn)臂之間的傳動(dòng)比,當(dāng)齒輪3固定時(shí)， 輪 1 時(shí)齒輪 3 固定的。 第一行乘以常量中，以便在添加第二行時(shí)，齒輪 3 的速度將為零。 此常量為 N 1 /N 3。 現(xiàn)在，做一個(gè)位移，然后另對(duì)應(yīng)于添加這兩行。 我們發(fā)現(xiàn) N 1 /N 3，1 + N 1 /N 3，N 1 /N 3-N 1 /N 2。 第一個(gè)數(shù)字是揮臂速度，第二個(gè)數(shù)字是齒輪1的速度，因此，它們之間的速度比是 N 1 /(N1 + N3) ，再用這個(gè)結(jié)果乘以 N 3。 這就是我們需要的田宮變速器的速度比，在變速器里面，環(huán)齒輪不會(huì)旋轉(zhuǎn)，太陽(yáng)齒輪是輸入端，揮臂速度則是輸出值。這是個(gè)通用過(guò)程，但可以為任何行星齒輪系服務(wù)。 田行星齒輪組件之一有 N 1 = N 2 = 16，N 3 = 48，而另有 N 1 = 12，N 2 = 18，N 3 = 48。 因?yàn)樾行驱X輪必須剛好位于太陽(yáng)和環(huán)齒輪之間，N 3 = 2N 1 + N2 這個(gè)條件必須得到滿足。事實(shí)上，這個(gè)條件得滿足給定齒輪的數(shù)目。 第一個(gè)組件的速度比將是16 /(48 + 16) = 1/4。 第二個(gè)組件的速度比將是12 /(48 + 12) = 1/5。 這兩個(gè)比率如同廣告中介紹的那樣。請(qǐng)注意，太陽(yáng)齒輪和揮臂將向同一個(gè)方向旋轉(zhuǎn)。 通用的求解行星輪系最佳方法是列表法，因?yàn)檫@種方法不包含像公式一樣的隱藏假設(shè)，也不要求應(yīng)用矢量法進(jìn)行計(jì)算。第一步是隔離行星輪系，從行星輪系中分離出齒輪輪系的輸入端和輸出值。 找到輸入速度或轉(zhuǎn)速，使用輸入的行星齒輪輪系。一般情況下，這里有兩個(gè)輸入端，其中之一在簡(jiǎn)單情況下可能為零。現(xiàn)在準(zhǔn)備兩行關(guān)于轉(zhuǎn)速或者角速度的圖表。 第一行對(duì)應(yīng)于圍繞行星軸旋轉(zhuǎn)一次產(chǎn)生的參數(shù)，并由所有1組成。記下第二行，其中假定臂速度為零，使用已知的齒輪比。 你需要的一行是上述兩行組成的一個(gè)線性組合，再加上未知乘數(shù)x和y。把輸入的齒輪值相加, 根據(jù)已知的輸入速度，同時(shí)產(chǎn)生兩個(gè)關(guān)于x和y的兩種線性方程組。現(xiàn)在，把這兩行數(shù)值相加的和乘以其各自的乘數(shù)，就產(chǎn)生了相關(guān)的所有齒輪的速度。最后，借助輸出齒輪傳動(dòng)計(jì)算出輸出速度。參考已經(jīng)采取的正方向，務(wù)必使其旋轉(zhuǎn)方向正確。 田宮齒輪箱工具包 各個(gè)組件從澆口處很好地被切割成單體，就像是用在電子產(chǎn)品中使用的齊平刀加工過(guò)一樣。然后，就可以用一把鋒利的X阿克托刻刀將余下的細(xì)小塑料部件移除。要按照說(shuō)明書(shū)所說(shuō)，小心地除掉所有多余的塑料。 仔細(xì)閱讀說(shuō)明，確保所有事情都按正確的方式運(yùn)行，并位于正確的相對(duì)位置。變速箱組件在輕壓下整體運(yùn)行自如。要注意，棕色部件必須同時(shí)朝正確的相對(duì)方向運(yùn)行。 4 毫米的墊圈由兩個(gè)組件提供，說(shuō)明書(shū)中也有一個(gè)墊圈的全尺寸繪圖。 不過(guò)，較小的墊圈在軸上會(huì)顯得不適合。輸出軸是金屬材質(zhì)。使用較大的長(zhǎng)嘴鉗壓迫E環(huán)使其進(jìn)入墊圈前部的槽。說(shuō)明書(shū)中有一張圖片講述如何執(zhí)行此操作。工具包中有一個(gè)額外的E環(huán)。三個(gè)插針進(jìn)入行星齒輪的傳動(dòng)器，并受到它們的驅(qū)動(dòng)。 現(xiàn)在按照設(shè)計(jì)把變速箱組件堆疊起來(lái)。我使用整個(gè)四個(gè)組件，但要確保把一個(gè)1： 5的部件放在電機(jī)末端的旁邊。因此，我需要長(zhǎng)螺絲刀。橙色的太陽(yáng)齒輪作為最后一個(gè)1:5的部件，務(wù)必把這個(gè)齒輪緊緊地壓進(jìn)電機(jī)軸，壓到它不能滑動(dòng)為止。如果這個(gè)齒輪沒(méi)有放好，電機(jī)加緊鉗將不會(huì)關(guān)閉。通過(guò)該部件自身帶的管子向齒輪注入潤(rùn)滑油，這樣做效果比較好。如果您使用不同的潤(rùn)滑劑，首先從部件上取一塊塑料然后滴上潤(rùn)滑劑進(jìn)行測(cè)試，以確保它和部件能兼容。干石墨潤(rùn)滑油效果也十分不錯(cuò)。在最后一個(gè)組件的所有組成部分上都要涂滿潤(rùn)滑油，因?yàn)檫@個(gè)組件在運(yùn)行時(shí)能達(dá)到最高速度。把電動(dòng)機(jī)放在合適的放置，動(dòng)作要輕但要牢固，晃動(dòng)電動(dòng)機(jī)以便使太陽(yáng)齒輪嚙合。如果太陽(yáng)齒輪沒(méi)有達(dá)到嚙合，電動(dòng)機(jī)的加緊鉗將不會(huì)關(guān)閉?，F(xiàn)在，把電機(jī)終端都布置成一個(gè)垂直的列陣，并按住電動(dòng)鉗。 說(shuō)明的背面顯示如何裝上驅(qū)動(dòng)臂，并對(duì)齒輪箱的使用給出一些提示。齒輪箱上有一個(gè)額外的彈性圓柱銷和兩個(gè)額外的3毫米墊圈。如果有一些小的墊圈，它們可用在機(jī)械螺釘上，以把齒輪箱連接在一起。在輸出端產(chǎn)生的扭矩足夠損壞機(jī)器（最多6千克-厘米），因此，要確保輸出臂可以自由旋轉(zhuǎn)。機(jī)器使用的是擁有變電壓和電流限制標(biāo)準(zhǔn)實(shí)驗(yàn)室直流電源，但也可以使用干電池。對(duì)于D電池來(lái)說(shuō)，1安培的電流都是高負(fù)荷的，因此要提供充足有效的電源供應(yīng)。說(shuō)明書(shū)明確告知不能超過(guò) 4.5V，這是個(gè)好建議。擁有400:1的減量后，無(wú)論輸出負(fù)載怎么樣，電機(jī)都能夠自由運(yùn)行。 齒輪箱在第一次測(cè)試的時(shí)候運(yùn)行良好。經(jīng)秒表檢測(cè)，齒輪箱的輸出轉(zhuǎn)數(shù)維持在47秒20圈，或每分鐘轉(zhuǎn)數(shù)為25.5。這個(gè)數(shù)值符合電動(dòng)機(jī)每分鐘轉(zhuǎn)數(shù)10,200，十分接近設(shè)定規(guī)格。把在序列中的另一個(gè)齒輪箱跟測(cè)試這個(gè)連接起來(lái)也很容易 (各部件都包含進(jìn)去以實(shí)現(xiàn)這一點(diǎn))，并且能達(dá)到大約每小時(shí)4個(gè)轉(zhuǎn)數(shù)。此外，另一個(gè)齒輪箱在四天內(nèi)會(huì)產(chǎn)生一次轉(zhuǎn)數(shù)。這是一個(gè)十分完美的工具，強(qiáng)烈推薦。 其他行星輪系 一個(gè)很著名的行星鏈?zhǔn)峭咛靥?yáng)-行星齒輪，在1781年申請(qǐng)專利，作為曲柄的替代品，使蒸汽引擎的往復(fù)運(yùn)動(dòng)轉(zhuǎn)換成旋轉(zhuǎn)運(yùn)動(dòng)。它由威廉-默多克發(fā)明。在當(dāng)時(shí)，曲柄裝置已獲專利，但是瓦特又不想支付版權(quán)稅。一個(gè)附帶的優(yōu)勢(shì)是輸出的旋轉(zhuǎn)速度增加了1/2。但是，它比曲柄貴得多，并且在曲柄專利過(guò)期后已很少使用。這個(gè)可以觀看維基百科上的動(dòng)畫(huà)。 輸入的是驅(qū)動(dòng)臂，上面裝有具有相同尺寸行星傳動(dòng)齒輪和與其搭配的太陽(yáng)齒輪。為防止行星輪轉(zhuǎn)動(dòng)，行星輪被固定在活塞桿上。雖然出現(xiàn)細(xì)微振蕩，但在每次旋轉(zhuǎn)后都能返回到相同的位置。應(yīng)用表格法來(lái)解釋上述的理論，第一行是1，1，1，其中第一個(gè)數(shù)字指驅(qū)動(dòng)臂，第二個(gè)對(duì)應(yīng)行星齒輪，第三個(gè)對(duì)應(yīng)太陽(yáng)齒輪。 第二行是0、-1，1，在這里面，已經(jīng)逆時(shí)針旋轉(zhuǎn)行星齒輪一周。兩行相加得到1，0，2，這意味著驅(qū)動(dòng)臂的一次轉(zhuǎn)動(dòng) (引擎的一次連擊）傳給太陽(yáng)齒輪兩次轉(zhuǎn)動(dòng)。 我們可以通過(guò)太陽(yáng)和行星齒輪來(lái)闡明另一種分析行星輪系的方法，在這個(gè)方法中我們需要使用速度這個(gè)概念。該方法可能會(huì)比表格方法更令人滿意，并更加清晰地說(shuō)明輪系的工作原理。如上圖所示，A 和O分別是行星齒輪與太陽(yáng)齒輪的中心。A圍繞O的旋轉(zhuǎn)角速度是ω 1，在這里假定是順時(shí)針?lè)较?。如圖位置顯示，A處獲得了一個(gè)向上的速度2ω 1。現(xiàn)在，行星齒輪停止旋轉(zhuǎn)，所以在齒輪上的所有點(diǎn)具有和A相同的速度。這其中包括嚙合節(jié)點(diǎn)P，P也是太陽(yáng)齒輪上的一點(diǎn)，圍繞固定軸O的旋轉(zhuǎn)角速度為ω 2。因此，ω 2=2ω 1，所得結(jié)果跟表格方法計(jì)算出來(lái)的一樣。 左側(cè)圖說(shuō)明了速度法是如何應(yīng)用于上述行星齒輪集的。假定太陽(yáng)齒輪和行星齒輪為相同的直徑（2個(gè)單位）。接下來(lái)，環(huán)形齒輪直徑6。我們先假定太陽(yáng)齒輪是固定的，因此嚙合節(jié)點(diǎn)P也是固定的。A點(diǎn)的速度是驅(qū)動(dòng)臂角速度的兩倍。 由于P點(diǎn)是固定的，所以P '點(diǎn)必須以兩倍于A的速度移動(dòng)，或者四倍于驅(qū)動(dòng)臂的速度移動(dòng)。但是，P'的速度是環(huán)形齒輪角速度的三倍，這樣得出3ωr = 4ωa。如果驅(qū)動(dòng)臂是輸入端，那么速度比就是3: 4，而環(huán)形齒輪是輸入端時(shí)，速度比則是4: 3。 三速自行車輪轂可能包含兩個(gè)這樣的行星輪系，它們由兩個(gè)環(huán)形齒輪連接（其實(shí)，就跟普通的輪系一樣)。輸入端是從后鏈輪齒到一個(gè)輪系的輪臂，而輸出端則是從第二個(gè)輪系的輪臂到輪轂?？梢栽谳嗇S上鎖定一個(gè)或兩個(gè)太陽(yáng)齒輪，要不然就把太陽(yáng)齒輪鎖定在輪臂上而不固定在輪軸上，以使輪系的比例達(dá)到1: 1。三個(gè)齒輪分別是： 高，3: 4，輸出端輪系鎖定；中間，1: 1，兩個(gè)輪系均鎖定；低，4: 3，輸入端輪系鎖定。當(dāng)然，這只是一種可能性，已經(jīng)生產(chǎn)了許多不同的可變輪轂。駛德美愛(ài)馳在1903年推出行星可變輪轂。很受歡迎的AW輪轂擁有上述提及的比率。 鏈?zhǔn)缴禉C(jī)可能會(huì)使用行星輪系。環(huán)形齒輪是固定的，為主要?dú)んw的組成部分。輸入端為太陽(yáng)齒輪，輸出端是從行星搬運(yùn)裝置。太陽(yáng)齒輪和行星齒輪擁有非常不同的直徑以獲得一個(gè)大的減速比。 福特T型車 (1908年-1927) 使用的是反向行星變速器，在這個(gè)裝置中，制動(dòng)帶被應(yīng)用于轉(zhuǎn)載太陽(yáng)齒輪的軸，而制動(dòng)帶選擇的就是傳動(dòng)比。低傳動(dòng)比向前時(shí)為11: 4，而其反向傳動(dòng)比是-4:1，高傳動(dòng)比為1: 1。反向的意思是指，位于行星傳動(dòng)軸上的齒輪驅(qū)動(dòng)軸上的其他齒輪，這些齒輪都跟主軸同心，而主軸上則安裝了制動(dòng)帶。作業(yè)控制裝置其實(shí)就是三個(gè)踏板：低-中性-高，反向，變速器制動(dòng)。應(yīng)用的手動(dòng)閘能夠中和動(dòng)力，以停止左手踏板。前面的火花塞和風(fēng)門(mén)都位于轉(zhuǎn)向柱上。 如上圖所示，汽車試驗(yàn)臺(tái)是一個(gè)錐齒輪行星輪系。在小齒輪的驅(qū)動(dòng)下，內(nèi)齒輪 (冠狀輪)旋轉(zhuǎn)自如，并帶動(dòng)從動(dòng)齒輪。事實(shí)上只奧一個(gè)從動(dòng)齒輪就可以了，但多個(gè)便能提供更好的對(duì)稱性。環(huán)齒輪對(duì)應(yīng)的似乎行星傳動(dòng)裝置，而從動(dòng)齒輪對(duì)應(yīng)的是普通行星鏈上的行星齒輪。從動(dòng)齒輪驅(qū)動(dòng)位于半軸上的側(cè)齒輪，這些側(cè)齒輪對(duì)應(yīng)的是太陽(yáng)齒輪和環(huán)齒輪，也是輸出端的齒輪。當(dāng)在兩個(gè)半軸以相同速度旋轉(zhuǎn)時(shí)，從動(dòng)齒輪不會(huì)旋轉(zhuǎn)。當(dāng)這兩個(gè)半軸旋轉(zhuǎn)速度不同時(shí)，從動(dòng)輪就會(huì)旋轉(zhuǎn)。該試驗(yàn)臺(tái)賦予側(cè)齒輪平等的扭矩（也就是它們是在同等的距離下被從動(dòng)齒輪驅(qū)動(dòng)的），同時(shí)允許他們以不同的速度旋轉(zhuǎn)。如果一個(gè)車輪滑動(dòng)，它就以雙倍速度旋轉(zhuǎn)，而另一個(gè)車輪不旋轉(zhuǎn)。不過(guò)，同樣的（?。┡ぞ貞?yīng)用于兩個(gè)輪子。 使用表格法可以輕松分析角速度。旋轉(zhuǎn)整個(gè)鏈裝置時(shí)，為環(huán)齒輪、從動(dòng)輪、左側(cè)齒輪、右側(cè)齒輪產(chǎn)生的角速度數(shù)值分別是1、0、1、1。把環(huán)齒輪固定時(shí)，分別產(chǎn)生的數(shù)值是0，1，1，-1。如果右側(cè)齒輪是固定的，而環(huán)齒輪旋轉(zhuǎn)一周，我哦們簡(jiǎn)單相加就得到1、1、2、0，這說(shuō)明左側(cè)齒輪已經(jīng)旋轉(zhuǎn)了兩次。速度法當(dāng)然也可以使用。考慮 (equal)從動(dòng)齒輪給側(cè)齒輪施加的力是相同的，這也說(shuō)明扭矩也會(huì)相等。  附錄2：外文原文 Planetary Gears Introduction The Tamiya planetary gearbox is driven by a small DC motor that runs at about 10,500 rpm on 3.0V DC and draws about 1.0A. The maximum speed ratio is 1:400, giving an output speed of about 26 rpm. Four planetary stages are supplied with the gearbox, two 1:4 and two 1:5, and any combination can be selected. Not only is this a good drive for small mechanical applications, it provides an excellent review of epicycle gear trains. The gearbox is a very well-designed plastic kit that can be assembled in about an hour with very few tools. The source for the kit is given in the References. Let's begin by reviewing the fundamentals of gearing, and the trick of analyzing epicyclic gear trains. Epicyclic Gear Trains A pair of spur gears is represented in the diagram by their pitch circles, which are tangent at the pitch point P. The meshing gear teeth extend beyond the pitch circle by the addendum, and the spaces between them have a depth beneath the pitch circle by the dedendum. If the radii of the pitch circles are a and b, the distance between the gear shafts is a + b. In the action of the gears, the pitch circles roll on one another without slipping. To ensure this, the gear teeth must have a proper shape so that when the driving gear moves uniformly, so does the driven gear. This means that the line of pressure, normal to the tooth profiles in contact, passes through the pitch point. Then, the transmission of power will be free of vibration and high speeds are possible. We won't talk further about gear teeth here, having stated this fundamental principle of gearing. If a gear of pitch radius a has N teeth, then the distance between corresponding points on successive teeth will be 2πa/N, a quantity called the circular pitch. If two gears are to mate, the circular pitches must be the same. The pitch is usually stated as the ration 2a/N, called the diametral pitch. If you count the number of teeth on a gear, then the pitch diameter is the number of teeth times the diametral pitch. If you know the pitch diameters of two gears, then you can specify the distance between the shafts. The velocity ratio r of a pair of gears is the ratio of the angular velocity of the driven gear to the angular velocity of the driving gear. By the condition of rolling of pitch circles, r = -a/b = -N1/N2, since pitch radii are proportional to the number of teeth. The angular velocity n of the gears may be given in radians/sec, revolutions per minute (rpm), or any similar units. If we take one direction of rotation as positive, then the other direction is negative. This is the reason for the (-) sign in the above expression. If one of the gears is internal (having teeth on its inner rim), then the velocity ratio is positive, since the gears will rotate in the same direction. The usual involute gears have a tooth shape that is tolerant of variations in the distance between the axes, so the gears will run smoothly if this distance is not quite correct. The velocity ratio of the gears does not depend on the exact spacing of the axes, but is fixed by the number of teeth, or what is the same thing, by the pitch diameters. Slightly increasing the distance above its theoretical value makes the gears run easier, since the clearances are larger. On the other hand, backlash is also increased, which may not be desired in some applications. An epicyclic gear train has gear shafts mounted on a moving arm or carrier that can rotate about the axis, as well as the gears themselves. The arm can be an input element, or an output element, and can be held fixed or allowed to rotate. The outer gear is the ring gear or annulus. A simple but very common epicyclic train is the sun-and-planet epicyclic train, shown in the figure at the left. Three planetary gears are used for mechanical reasons; they may be considered as one in describing the action of the gearing. The sun gear, the arm, or the ring gear may be input or output links. If the arm is fixed, so that it cannot rotate, we have a simple train of three gears. Then, n2/n1 = -N1/N2, n3/n2 = +N2/N3, and n3/n1 = -N1/N3. This is very simple, and should not be confusing. If the arm is allowed to move, figuring out the velocity ratios taxes the human intellect. Attempting this will show the truth of the statement; if you can manage it, you deserve praise and fame. It is by no means impossible, just invoved. However, there is a very easy way to get the desired result. First, just consider the gear train locked, so it moves as a rigid body, arm and all. All three gears and the arm then have a unity velocity ratio. The trick is that any motion of the gear train can carried out by first holding the arm fixed and rotating the gears relative to one another, and then locking the train and rotating it about the fixed axis. The net motion is the sum or difference of multiples of the two separate motions that satisfies the conditions of the problem (usually that one element is held fixed). To carry out this program, construct a table in which the angular velocities of the gears and arm are listed for each, for each of the two cases. The locked train gives 1, 1, 1, 1 for arm, gear 1, gear 2 and gear 3. Arm fixed gives 0, 1, -N1/N2, -N1/N3. Suppose we want the velocity ration between the arm and gear 1, when gear 3 is fixed. Multiply the first row by a constant so that when it is added to the second row, the velocity of gear 3 will be zero. This constant is N1/N3. Now, doing one displacement and then the other corresponds to adding the two rows. We find N1/N3, 1 + N1/N3, N1/N3 - N1/N2. The first number is the arm velocity, the second the velocity of gear 1, so the velocity ratio between them is N1/(N1 + N3), after multiplying through by N3. This is the velocity ratio we need for the Tamiya gearbox, where the ring gear does not rotate, the sun gear is the input, and the arm is the output. The procedure is general, however, and will work for any epicyclic train. One of the Tamiya planetary gear assemblies has N1 = N2 = 16, N3 = 48, while the other has N1 = 12, N2 = 18, N3 = 48. Because the planetary gears must fit between the sun and ring gears, the condition N3 = N1 + 2N2 must be satisfied. It is indeed satisfied for the numbers of teeth given. The velocity ratio of the first set will be 16/(48 + 16) = 1/4. The velocity ratio of the second set will be 12/(48 + 12) = 1/5. Both ratios are as advertised. Note that the sun gear and arm will rotate in the same direction. The best general method for solving epicyclic gear trains is the tabular method, since it does not contain hidden assumptions like formulas, nor require the work of the vector method. The first step is to isolate the epicyclic train, separating the gear trains for inputs and outputs from it. Find the input speeds or turns, using the input gear trains. There are, in general, two inputs, one of which may be zero in simple problems. Now prepare two rows of the table of turns or angular velocities. The first row corresponds to rotating around the epicyclic axis once, and consists of all 1's. Write down the second row assuming that the arm velocity is zero, using the known gear ratios. The row that you want is a linear combination of these two rows, with unknown multipliers x and y. Summing the entries for the input gears gives two simultaneous linear equations for x and y in terms of the known input velocities. Now the sum of the two rows multiplied by their respective multipliers gives the speeds of all the gears of interest. Finally, find the output speed with the aid of the output gear train. Be careful to get the directions of rotation correct, with respect to a direction taken as positive. The Tamiya Gearbox Kit The parts are best cut from the sprues with a flush-cutter of the type used in electronics. The very small bits of plastic remaining can then be removed with a sharp X-acto knife. Carefully remove all excess plastic, as the instructions say. Read the instructions carefully and make sure that things are the right way up and in the correct relative positons. The gearbox units go together easily with light pressure. Note that the brown ones must go together in the correct relative orientation. The 4mm washers are the ones of which two are supplied, and there is also a full-size drawing of one in the instructions. The smaller washers will not fit over the shaft, anyway. The output shaft is metal. Use larger long-nose pliers to press the E-ring into position in its groove in front of the washer. There is a picture showing how to do this. There was an extra E-ring in my kit. The three prongs fit into the carriers for the planetary gears, and are driven by them. Now stack up the gearbox units as desired. I used all four, being sure to put a 1:5 unit on the end next to the motor. Therefore, I needed the long screws. Press the orange sun gear for the last 1:5 unit firmly on the motor shaft as far as it will go. If it is not well-seated, the motor clip will not close. It might be a good idea to put some lubricant on this gear from the tube included with the kit. If you use a different lubricant, test it first on a piece of plastic from the kit to make sure that it is compatible. A dry graphite lubricant would also work quite well. This should spread lubricant on all parts of the last unit, which is the one subject to the highest speeds. Put the motor in place, gently but firmly, wiggling it so that the sun gear meshes. If the sun gear is not meshed, the motor clip will not close. Now put the motor terminals in a vertical column, and press on the motor clamp. The reverse of the instructions show how to attach the drive arm and gives some hints on use of the gearbox. I got an extra spring pin, and two extra 3 mm washers. If you have some small washers, they can be used on the machine screws holding the gearbox together. Enough torque is produced at the output to damage things (up to 6 kg-cm), so make sure the output arm can rotate freely. I used a standard laboratory DC supply with variable voltage and current limiting, but dry cells could be used as well. The current drain of 1 A is high even for D cells, so a power supply is indicated for serious use. The instructions say not to exceed 4.5V, which is good advice. With 400:1 reduction, the motor should run freely whatever the output load. My gearbox ran well the first time it was tested. I timed the output revolutions with a stopwatch, and found 47s for 20 revolutions, or 25.5 rpm. This corresponds to 10,200 rpm at the motor, which is close to specifications. It would be easy to connect another gearbox in series with this one (parts are included to make this possible), and get about 4 revolutions per hour. Still another gearbox would produce about one revolution in four days. This is an excellent kit, and I recommend it highly. Other Epicyclic Trains A very famous epicyclic chain is the Watt sun-and-planet gear, patented in 1781 as an alternative to the crank for converting the reciprocating motion of a steam engine into rotary motion. It was invented by William Murdoch. The crank, at that time, had been patented and Watt did not want to pay royalties. An incidental advantage was a 1:2 increase in the rotative speed of the output. However, it was more expensive than a crank, and was seldom used after the crank patent expired. Watch the animation on Wikipedia. The input is the arm, which carries the planet gear wheel mating with the sun gear wheel of equal size. The planet wheel is prevented from rotating by being fastened to the connecting rod. It oscillates a little, but always returns to the same place on every revolution. Using the tabular method explained above, the first line is 1, 1, 1 where the first number refers to the arm, the second to the planet gear, and the third to the sun gear. The second line is 0, -1, 1, where we have rotated the planet one turn anticlockwise. Adding, we get 1, 0, 2, which means that one revolution of the arm (one double stroke of the engine) gives two revolutions of the sun gear. We can use the sun-and-planet gear to illustrate another method for analyzing epicyclical trains in which we use velocities. This method may be more satisfying than the tabular method and show more clearly how the train works. In the diagram at the right, A and O are the centres of the planet and sun gears, respectively. A rotates about O with angular velocity ω1, which we assume clockwise. At the position shown, this gives A a velocity 2ω1 upward, as shown. Now the planet gear does not rotate, so all points in it move with the same velocity as A. This includes the pitch point P, which is also a point in the sun gear, which rotates about the fixed axis O with angular velocity ω2. Therefore, ω2 = 2ω1, the same result as with the tabular method. The diagram at the left shows how the velocity method is applied to the planetary gear set treated above. The sun and planet gears are assumed to be the same diameter (2 units). The ring gear is then of diameter 6. Let us assume the sun gear is fixed, so that the pitch point P is also fixed. The velocity of point A is twice the angular velocity of the arm. Since P is fixed, P' must move at twice the velocity of A, or four times the velocity of the arm. However, the velocity of P' is three times the angular velocity of the ring gear as well, so that 3ωr = 4ωa. If the arm is the input, the velocity ratio is then 3:4, while if the ring is the input, the velocity ratio is 4:3. A three-speed bicycle hub may contain two of these epicyclical trains, with the ring gears connected (actually, common to the two trains). The input from the rear sprocket is to the arm of one train, while the output to the hub is from the arm of the second train. It is possible to lock one or both of the sun gears to the axle, or else to lock the sun gear to the arm and free of the axle, so that the train gives a 1:1 ratio. The three gears are: high, 3:4, output train locked; middle, 1:1, both trains locked, and low, 4:3 input train locked. Of course, this is just one possibility, and many different variable hubs have been manufactured. The planetary variable hub was introduced by Sturmey-Archer in 1903. The popular AW hub had the ratios mentioned here. Chain hoists may