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 epicyclic 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.

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 = -N_{1}/N_{2}, 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, n_{2}/n_{1} = -N_{1}/N_{2}, n_{3}/n_{2} = +N_{2}/N_{3}, and n_{3}/n_{1} = -N_{1}/N_{3}. 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, -N_{1}/N_{2}, -N_{1}/N_{3}. 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 N_{1}/N_{3}. Now, doing one displacement and then the other corresponds to adding the two rows. We find N_{1}/N_{3}, 1 + N_{1}/N_{3}, N_{1}/N_{3} - N_{1}/N_{2}, 0.

The first number is the arm velocity, the second the velocity of gear 1, so the velocity ratio between them is N_{1}/(N_{1} + N_{3}), after multiplying through by N_{3}. 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 N_{1} = N_{2} = 16, N_{3} = 48, while the other has N_{1} = 12, N_{2} = 18, N_{3} = 48. Because the planetary gears must fit between the sun and ring gears, the condition N_{3} = N_{1} + 2N_{2} 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 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.

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 epicyclic 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 ω_{1}, which we assume clockwise. At the position shown, this gives A a velocity 2ω_{1} 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 ω_{2}. Therefore, ω_{2} = 2ω_{1}, 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 usw 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 3ω_{r} = 4ω_{a}. 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 epicyclic 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 ouput 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 epicyclic trains. The ring gear is stationary, part of the main housing. The input is to the sun gear, the output from the planet carrier. The sun and planet gears have very different diameters, to obtain a large reduction ratio.

The Model T Ford (1908-1927) used a reverted epicyclic transmission in which brake bands applied to the shafts carrying sun gears selected the gear ratio. The low gear ratio was 11:4 forward, while the reverse gear ratio was -4:1. The high gear was 1:1. Reverted means that the gears on the planet carrier shaft drove other gears on shafts concentric with the main shaft, where the brake bands were applied. The floor controls were three pedals: low-neutral-high, reverse, transmission brake. The hand brake applied stopped the left-hand pedal at neutral. The spark advance and throttle were on the steering column.

The automotive differential, illustrated at the right, is a bevel-gear epicyclic train. The pinion drives the ring gear (crown wheel) which rotates freely, carrying the idler gears. Only one idler is necessary, but more than one gives better symmetry. The ring gear corresponds to the planet carrier, and the idler gears to the planet gears, of the usual epicyclic chain. The idler gears drive the side gears on the half-axles, which correspond to the sun and ring gears, and are the output gears. When the two half-axles revolve at the same speed, the idlers do not revolve. When the half-axles move at different speeds, the idlers revolve. The differential applies equal torque to the side gears (they are driven at equal distances by the idlers) while allowing them to rotate at different speeds. If one wheel slips, it rotates at double speed while the other wheel does not rotate. The same (small) torque is, nevertheless, applied to both wheels.

The tabular method is easily used to analyze the angular velocities. Rotating the chain as a whole gives 1, 0, 1, 1 for ring, idler, left and right side gears. Holding the ring fixed gives 0, 1, 1, -1. If the right side gear is held fixed and the ring makes one rotation, we simply add to get 1, 1, 2, 0, which says that the left side gear makes two revolutions. The velocity method can also be used, of course. Considering the (equal) forces exerted on the side gears by the idler gears shows that the torques will be equal.

Tamiya Planetary Gearbox Set, Item 72001-1400. Edmund Scientific, Catalog No. C029D, item #D30524-08 ($19.95).

C. Carmichael, ed., *Kent's Mechanical Engineer's Handbook*, 12th ed. (New York: John Wiley and Sons, 1950). Design and Production Volume, p.14-49 to 14-43.

V. L. Doughtie, *Elements of Mechanism*, 6th ed. (New York: John Wiley and Sons, 1947). pp. 299-311.

Epicyclic gear. Wikipedia article on epicyclic trains.

Sun and planet gear. Includes an animation.

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Composed by J. B. Calvert

Created 5 January 2003

Last revised 3 February 2007