【机械类毕业论文中英文对照文献翻译】行星齿轮
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机械类毕业论文中英文对照文献翻译
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齿轮
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附录1：外文翻译行星齿轮介绍Tamiya行星轮变速箱由一个约 10500 r/min,3.0V，1.0A的直流电机运行。最大传动比 1:400，输出速度为26r/min。 四级行星轮变速箱由两个 1: 4 和两个 1: 5的传动级组成，并可以任意选择组合。 对于小的机械应用程序这不仅是一个良好的驱动器，而且还提供了一个出色检验的行星齿轮系。 这种齿轮变速箱是一种设计非常精心的塑料套件，可在约一个小时用很少的工具装配完成。 参考文献中给出了装备资料。下面让我们来开始检验齿轮传动装置的基本原理和分析行星轮系的技巧。行星轮系一对直齿圆柱齿轮的由节圆表示在图表中，它们相切与节点P点，啮合齿轮的轮齿齿顶超出了节圆半径，在节圆与齿齿顶之间有一齿顶间隙，。 若节圆半径分别为a和b，齿轮轴之间的距离就是 a + b。 为了确保齿轮传动中，一个节圆在另一个节圆上没有滑动，必须得有适当的形状确保从动轮与主动轮的运动一致。 这就意味着接触线以正常接触齿廓的形式通过节点。这时，动力传递脱离高速震动达到可能。 在这里我们不会进一步谈论齿轮轮齿，以及上述有提到的传动装置的基本原理。如果一个齿轮节圆半径上有 N 个齿，这时在两个连续的齿间的距离，我们称的齿间距将会是 2a/N。如果两个齿轮相啮合，他们之间的齿距必须是相同的。他们之间的节距通常以2a/N来表示，我们称为模数。 如果你计算一个齿轮的齿数，这时节圆直径的大小是模数的倍数，而倍数则是齿数。如果你知道两个齿轮的节圆直径，那么你就能够得出两齿轮轴之间的距离。一对齿轮的传动比r 驱动轮与从动轮之间的角速度之比。 因为分度圆之间旋转方向的限制条件，r =-a / b =-N 1 /N 2,，因此它们之间的节圆半径比与齿数成正比。齿轮角速度n可以用转/秒，转/分，或者任何类似的单位表示。如果以一齿轮的旋转方向为正，此时另外一个的方向则为负。这就是上面的表达式中的 (-) 标志的由于原因。如果其中一个是内齿（齿在齿圈内部），这时传动比为正，因此它们的传动方向一致。常用渐开线齿轮的牙形能够允许轴线之间一定的变位 ，所以即使它们之间的距离不是很精确也能够顺利的运行。齿轮的传动比并不依赖于该轴的精确的间距，而是轮齿或者节圆诸如此类之间的安装。稍微增加高于其理论值的距离，能够使运行更容易。因为其游隙较大的齿轮， 在另一方面 齿隙 也增加，它可能不是我们在某些应用上所希望的。一个行星轮系包含了固定在齿轮轴上的转臂和行星架以及齿轮和旋转的齿轮轴。一个移动的 手臂 或 承运人 的有关该的轴以及齿轮自己可以旋转的齿轮轴。转臂可以是一个输入或输出构件而且可被固定固定或可旋转。 最外面的齿轮为内齿轮。 一个简单常见的行星轮是如左图所示的太阳-行星轮系。这是三个行星齿轮轮系用于机械领域的原因 ； 他们可能被认为是在描述该传动装置的操作之一。 太阳轮、 转臂或内齿轮可能成为输入或输出的链接。如果转臂被固定，就不能旋转，一个简单的三行星轮轮系吗有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。 这是非常简单，不应令人困惑。 如果转臂允许移动，算出速度比彰显出了人类的智慧。 尝试这将显示该陈述的真实性 ； 如果你能做到，你应得到赞扬和声誉。 这并不意味这将不可能，只是比较复杂罢了。 不过，有一个非常简单的方法获得所需的结果。 首先，把这轮系假定认为是锁定的，因此把转臂和所有的作为刚体、。 所有的三个齿轮和手臂然后有一个统一的速度比。行星齿轮任何运动的特点是可以被第一个固定支撑转臂和相对于另外一个旋转的齿轮实现，然后锁定轮系并关于固定的轴旋转。净运动总和或两个不同的独立的分离运动来满足这问题的条件（通常一个构件被固定）。若要进行此程序，构造的齿轮和转臂臂的角速度列出两例的每个表。 锁定的轮系给定的N1, N2, N3 为齿轮 1、 齿轮 2 和齿轮3。 固定转臂为 0，1，-N 1 /N 2，-N 1 /N 3。 假定我们想知道齿轮1与转臂之间的传动比,当齿轮3固定时， 轮 1 时齿轮 3 固定的。 第一行乘以常量中，以便在添加第二行时，齿轮 3 的速度将为零。 此常量为 N 1 /N 3。 现在，做一个位移，然后另对应于添加这两行。 我们发现 N 1 /N 3，1 + N 1 /N 3，N 1 /N 3-N 1 /N 2。第一个数字是挥臂速度，第二个数字是齿轮1的速度，因此，它们之间的速度比是 N 1 /(N1 + N3) ，再用这个结果乘以 N 3。 这就是我们需要的田宫变速器的速度比，在变速器里面，环齿轮不会旋转，太阳齿轮是输入端，挥臂速度则是输出值。这是个通用过程，但可以为任何行星齿轮系服务。田行星齿轮组件之一有 N 1 = N 2 = 16，N 3 = 48，而另有 N 1 = 12，N 2 = 18，N 3 = 48。 因为行星齿轮必须刚好位于太阳和环齿轮之间，N 3 = 2N 1 + N2 这个条件必须得到满足。事实上，这个条件得满足给定齿轮的数目。 第一个组件的速度比将是16 /(48 + 16) = 1/4。 第二个组件的速度比将是12 /(48 + 12) = 1/5。 这两个比率如同广告中介绍的那样。请注意，太阳齿轮和挥臂将向同一个方向旋转。通用的求解行星轮系最佳方法是列表法，因为这种方法不包含像公式一样的隐藏假设，也不要求应用矢量法进行计算。第一步是隔离行星轮系，从行星轮系中分离出齿轮轮系的输入端和输出值。 找到输入速度或转速，使用输入的行星齿轮轮系。一般情况下，这里有两个输入端，其中之一在简单情况下可能为零。现在准备两行关于转速或者角速度的图表。 第一行对应于围绕行星轴旋转一次产生的参数，并由所有1组成。记下第二行，其中假定臂速度为零，使用已知的齿轮比。 你需要的一行是上述两行组成的一个线性组合，再加上未知乘数x和y。把输入的齿轮值相加, 根据已知的输入速度，同时产生两个关于x和y的两种线性方程组。现在，把这两行数值相加的和乘以其各自的乘数，就产生了相关的所有齿轮的速度。最后，借助输出齿轮传动计算出输出速度。参考已经采取的正方向，务必使其旋转方向正确。田宫齿轮箱工具包各个组件从浇口处很好地被切割成单体，就像是用在电子产品中使用的齐平刀加工过一样。然后，就可以用一把锋利的X阿克托刻刀将余下的细小塑料部件移除。要按照说明书所说，小心地除掉所有多余的塑料。仔细阅读说明，确保所有事情都按正确的方式运行，并位于正确的相对位置。变速箱组件在轻压下整体运行自如。要注意，棕色部件必须同时朝正确的相对方向运行。 4 毫米的垫圈由两个组件提供，说明书中也有一个垫圈的全尺寸绘图。 不过，较小的垫圈在轴上会显得不适合。输出轴是金属材质。使用较大的长嘴钳压迫E环使其进入垫圈前部的槽。说明书中有一张图片讲述如何执行此操作。工具包中有一个额外的E环。三个插针进入行星齿轮的传动器，并受到它们的驱动。现在按照设计把变速箱组件堆叠起来。我使用整个四个组件，但要确保把一个1： 5的部件放在电机末端的旁边。因此，我需要长螺丝刀。橙色的太阳齿轮作为最后一个1:5的部件，务必把这个齿轮紧紧地压进电机轴，压到它不能滑动为止。如果这个齿轮没有放好，电机加紧钳将不会关闭。通过该部件自身带的管子向齿轮注入润滑油，这样做效果比较好。如果您使用不同的润滑剂，首先从部件上取一块塑料然后滴上润滑剂进行测试，以确保它和部件能兼容。干石墨润滑油效果也十分不错。在最后一个组件的所有组成部分上都要涂满润滑油，因为这个组件在运行时能达到最高速度。把电动机放在合适的放置，动作要轻但要牢固，晃动电动机以便使太阳齿轮啮合。如果太阳齿轮没有达到啮合，电动机的加紧钳将不会关闭。现在，把电机终端都布置成一个垂直的列阵，并按住电动钳。说明的背面显示如何装上驱动臂，并对齿轮箱的使用给出一些提示。齿轮箱上有一个额外的弹性圆柱销和两个额外的3毫米垫圈。如果有一些小的垫圈，它们可用在机械螺钉上，以把齿轮箱连接在一起。在输出端产生的扭矩足够损坏机器（最多6千克-厘米），因此，要确保输出臂可以自由旋转。机器使用的是拥有变电压和电流限制标准实验室直流电源，但也可以使用干电池。对于D电池来说，1安培的电流都是高负荷的，因此要提供充足有效的电源供应。说明书明确告知不能超过 4.5V，这是个好建议。拥有400:1的减量后，无论输出负载怎么样，电机都能够自由运行。齿轮箱在第一次测试的时候运行良好。经秒表检测，齿轮箱的输出转数维持在47秒20圈，或每分钟转数为25.5。这个数值符合电动机每分钟转数10,200，十分接近设定规格。把在序列中的另一个齿轮箱跟测试这个连接起来也很容易 (各部件都包含进去以实现这一点)，并且能达到大约每小时4个转数。此外，另一个齿轮箱在四天内会产生一次转数。这是一个十分完美的工具，强烈推荐。其他行星轮系一个很著名的行星链是瓦特太阳-行星齿轮，在1781年申请专利，作为曲柄的替代品，使蒸汽引擎的往复运动转换成旋转运动。它由威廉-默多克发明。在当时，曲柄装置已获专利，但是瓦特又不想支付版权税。一个附带的优势是输出的旋转速度增加了1/2。但是，它比曲柄贵得多，并且在曲柄专利过期后已很少使用。这个可以观看维基百科上的动画。输入的是驱动臂，上面装有具有相同尺寸行星传动齿轮和与其搭配的太阳齿轮。为防止行星轮转动，行星轮被固定在活塞杆上。虽然出现细微振荡，但在每次旋转后都能返回到相同的位置。应用表格法来解释上述的理论，第一行是1，1，1，其中第一个数字指驱动臂，第二个对应行星齿轮，第三个对应太阳齿轮。 第二行是0、-1，1，在这里面，已经逆时针旋转行星齿轮一周。两行相加得到1，0，2，这意味着驱动臂的一次转动 (引擎的一次连击）传给太阳齿轮两次转动。我们可以通过太阳和行星齿轮来阐明另一种分析行星轮系的方法，在这个方法中我们需要使用速度这个概念。该方法可能会比表格方法更令人满意，并更加清晰地说明轮系的工作原理。如上图所示，A 和O分别是行星齿轮与太阳齿轮的中心。A围绕O的旋转角速度是 1，在这里假定是顺时针方向。如图位置显示，A处获得了一个向上的速度2 1。现在，行星齿轮停止旋转，所以在齿轮上的所有点具有和A相同的速度。这其中包括啮合节点P，P也是太阳齿轮上的一点，围绕固定轴O的旋转角速度为 2。因此， 2=2 1，所得结果跟表格方法计算出来的一样。左侧图说明了速度法是如何应用于上述行星齿轮集的。假定太阳齿轮和行星齿轮为相同的直径（2个单位）。接下来，环形齿轮直径6。我们先假定太阳齿轮是固定的，因此啮合节点P也是固定的。A点的速度是驱动臂角速度的两倍。 由于P点是固定的，所以P 点必须以两倍于A的速度移动，或者四倍于驱动臂的速度移动。但是，P的速度是环形齿轮角速度的三倍，这样得出3r = 4a。如果驱动臂是输入端，那么速度比就是3: 4，而环形齿轮是输入端时，速度比则是4: 3。三速自行车轮毂可能包含两个这样的行星轮系，它们由两个环形齿轮连接（其实，就跟普通的轮系一样)。输入端是从后链轮齿到一个轮系的轮臂，而输出端则是从第二个轮系的轮臂到轮毂。可以在轮轴上锁定一个或两个太阳齿轮，要不然就把太阳齿轮锁定在轮臂上而不固定在轮轴上，以使轮系的比例达到1: 1。三个齿轮分别是： 高，3: 4，输出端轮系锁定；中间，1: 1，两个轮系均锁定；低，4: 3，输入端轮系锁定。当然，这只是一种可能性，已经生产了许多不同的可变轮毂。驶德美爱驰在1903年推出行星可变轮毂。很受欢迎的AW轮毂拥有上述提及的比率。链式升降机可能会使用行星轮系。环形齿轮是固定的，为主要壳体的组成部分。输入端为太阳齿轮，输出端是从行星搬运装置。太阳齿轮和行星齿轮拥有非常不同的直径以获得一个大的减速比。福特T型车 (1908年-1927) 使用的是反向行星变速器，在这个装置中，制动带被应用于转载太阳齿轮的轴，而制动带选择的就是传动比。低传动比向前时为11: 4，而其反向传动比是-4:1，高传动比为1: 1。反向的意思是指，位于行星传动轴上的齿轮驱动轴上的其他齿轮，这些齿轮都跟主轴同心，而主轴上则安装了制动带。作业控制装置其实就是三个踏板：低-中性-高，反向，变速器制动。应用的手动闸能够中和动力，以停止左手踏板。前面的火花塞和风门都位于转向柱上。如上图所示，汽车试验台是一个锥齿轮行星轮系。在小齿轮的驱动下，内齿轮 (冠状轮)旋转自如，并带动从动齿轮。事实上只奥一个从动齿轮就可以了，但多个便能提供更好的对称性。环齿轮对应的似乎行星传动装置，而从动齿轮对应的是普通行星链上的行星齿轮。从动齿轮驱动位于半轴上的侧齿轮，这些侧齿轮对应的是太阳齿轮和环齿轮，也是输出端的齿轮。当在两个半轴以相同速度旋转时，从动齿轮不会旋转。当这两个半轴旋转速度不同时，从动轮就会旋转。该试验台赋予侧齿轮平等的扭矩（也就是它们是在同等的距离下被从动齿轮驱动的），同时允许他们以不同的速度旋转。如果一个车轮滑动，它就以双倍速度旋转，而另一个车轮不旋转。不过，同样的（小）扭矩应用于两个轮子。使用表格法可以轻松分析角速度。旋转整个链装置时，为环齿轮、从动轮、左侧齿轮、右侧齿轮产生的角速度数值分别是1、0、1、1。把环齿轮固定时，分别产生的数值是0，1，1，-1。如果右侧齿轮是固定的，而环齿轮旋转一周，我哦们简单相加就得到1、1、2、0，这说明左侧齿轮已经旋转了两次。速度法当然也可以使用。考虑 (equal)从动齿轮给侧齿轮施加的力是相同的，这也说明扭矩也会相等。附录2：外文原文Planetary GearsIntroductionThe 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.Lets begin by reviewing the fundamentals of gearing, and the trick of analyzing epicyclic gear trains.Epicyclic Gear TrainsA 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 wont 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 2a/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 1s. 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 KitThe 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 TrainsA 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 1, which we assume clockwise. At the position shown, this gives A a velocity 21 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 2. Therefore, 2 = 21, 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 3r = 4a. 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 use epicyclical trains. The ring gear is stationary, part of the main housing. The input is to the sun gear, the output
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