# Planetary gearing: Getting around the calculations.

The following article is adapted from a chapter in the new edition of Mathematics at Work, published by Industrial Press Inc, New York, NY.

Planetary gearing is a special type of mechanism which provides, in compact form, a means of obtaining a very small or very large change in angular velocity of driven shaft as compared with driving shaft. The distinctive characteristic of planetary gearing is that some of the gears turn on movable centers while others turn on fixed centers. The gears that turn on movable centers are called planet gears and those that turn on fixed centers, sun gears.

The planetary gearing mechanism may consist of external spur gears, of external and internal spur gears, of racks and spur gears, or of bevel gears. That member which receives motion from outside the mechanism is called the driver; that member from which motion is taken outside the mechanism is called the follower; that member which carries one or more bearing pins about which the planet gears rotate is called the train arm. In addition, there is usually one member which is maintained in a fixed position.

The objective in solving planetary gearing problems is usually to determine the number of turns of the driver required to produce one turn of the follower or, vice versa, the number of turns of the follower produced by one turn of the driver. Two methods of solution will be explained and illustrated by application to various problems.

The first method, called the analytical method, is used to accurately determine the ratio between rotational speeds of driver and follower shafts. The second method, called the graphical method, is used to approximately determine the ratio between rotational speeds of driver and follower shafts and is frequently employed to obtain a quick check on the results obtained by the first method.

Analytical Method of Solution. In this method, the revolution of the planet wheel axis or pin through 360 deg in a clockwise direction is taken to be the reference basis for the analysis. The direction of rotation and number of turns made by the driver and by the follower during this single turn of the planet wheel axis are then determined and compared. The step-by-step procedure is as follows:

Step 1. The entire mechanism is considered to be locked together and the whole device is rotated one turn in a clockwise position around that axis about which the planet gear pin is carried.

Step 2. The planet gear pin has now made one revolution in a clockwise direction as required, but the fixed gear has also been rotated the same amount. Hence, the various elements of the mechanism are considered to be again free to turn; the train arm which carries the planet gear pin is now held stationary; and the fixed gear is rotated back in a counter-clockwise direction one turn to its original position.

Step 3. The number of revolutions and direction of rotation of the driver caused by returning the fixed gear to its original position in Step 2 are then noted. Let this be called X.

Step 4. The number of revolutions and direction of rotation of the follower caused by returning the fixed gear to its original position in Step 2 are also noted. Let this be called Y.

Step 5. In Step 1 the driver and follower were both rotated one turn in a clockwise direction. In Step 2 the driver makes X revolutions and the follower Y revolutions. The total number of revolutions made by the driver is 1 [plus or minus] X depending upon whether the driver rotates in a clockwise (plus) or counter-clockwise (minus) direction during Step 2. The total number of revolutions made by the follower is 1 [plus or minus] Y depending upon whether the follower rotates in clockwise (plus) or counter-clockwise (minus) direction during Step 2. The number of turns of the driver for each turn of the follower is, therefore, expressed by the equation:

N= 1 [plus or minus] X/ 1 [plus or minus] Y

Graphical Method of Solution. The graphical method is the same in principle for all planetary gearing problems but varies somewhat in the procedure to be followed depending upon whether the mechanism consists (a) of spur gears having the same circular pitch; (b) of spur gears having different circular pitches; or (c) of bevel gears.

The procedure to be described at this point is applicable to spur gears having the same circular pitch. The modifications of this method required for the other two types of problems will be presented when these problems are specifically discussed.

For the purpose of more clearly outlining this method, reference is made to a typical planetary gearing mechanism having gears of the same circular pitch. This mechanism is diagrammatically shown in the illustration.

Step 1. Draw a line ab tangent to the pitch circle of the driver, if it is a gear, and at the point a where the center line of the train arm intersects this pitch circle. (If the driver is a train arm, this line is drawn through the axis of the outermost bearing pin carried by the train arm and at right angles to the axis of the train arm.)

This line may be of any convenient length and represents the tangential linear velocity of the point from which it is drawn.

If the driver is a gear (as in the illustration) it will usually be found that point a on the pitch circle of the driver coincides with a point on the pitch circle of a planet gear which either itself engages the fixed gear in the train or is keyed to the same shaft as another planet gear which engages the fixed gear. (If the driver is a train arm, this point a will be found to coincide with the axis of a planet gear which engages the fixed gear.) In either case ab can also be taken to represent the linear velocity of point a on that planet gear which engages the fixed gear.

Step 2. Since there is no sliding of the planet gear on the fixed gear, that point c at which their pitch circles are in contact may be considered as having momentarily a linear velocity of zero. If a line is drawn from b to c, then, measuring at right angles to the center line of the train arm, or in other words parallel to ab, the distance to line bc of any point on this planet gear which also coincides with the train arm center line represents the linear velocity of that point.

Step 3. When the follower is the train arm, such a line is now drawn from point p, which is the center of the pin, carried by the train arm, on which the planet gear revolves. This line is drawn parallel to ab until it intersects bc at e. (If the follower is a gear, point p is taken to be the point of tangency between the pitch circles of the planet gear and the follower.) Line pe represents the velocity of point p on the planet gear. It also represents the velocity of coinciding point p on the follower.

Step 4. A line is now drawn from point m, which is the fixed center of the follower and therefore has zero linear velocity, through point e until it intersects line ab at f.

The distance to mf, if it is measured parallel to ab, of any point on the follower which coincides with the axis of the train arm, represents the velocity of the follower at a radius equal to the distance of that point from the axis m. Hence af represents the velocity of the follower at radius am.

Step 5. But ab represents the linear velocity of the driver at radius am. Hence, the value of N which is the number of turns of the driver required to produce one turn of the follower is found by the formula.

N = ab/af

Notation for Planetary Gearing Problems. In the problem that follows, solutions will be worked out by both the analytical and graphical methods. The notation used in these problems will be:

N = number of turns of driver to one of the follower or driven member;

N' = number of turns of follower to one of driver = 1 (division sign here) N;

N1 = number of turns of driver to one complete revolution of planet gear axis;

N2 = number of turns of follower to one complete revolution of planet gear axis;

D = diameter of pitch circle of driver, if driver is a gear (the driver, or the follower, may be the "train arm," and not one of the gears, according to the data of a problem);

D1 = diameter of pitch circle of follower, if follower is a gear;

D2 = diameter of pitch circle of fixed gear;

D3, D4, etc = diameters of pitch circles of planetary gears;

T = number of teeth in driver, if driver is a gear;

T1 = number of teeth in follower, if follower is a gear;

T2 = number of teeth in a fixed gear; and

T3, T4, etc = number of teeth in planetary gears.

Problem-solving

As shown in the illustration, the planetary gear mechanism has a fixed external gear, an internal gear as a driver, a single planet gear, and a train arm acting as follower.

Derivation by the Analytical Method. Step 1. Entire mechanism locked together and rotated once about m.

Step 2. Fixed sun gear D2 rotated back one turn in counter-clockwise direction, with arm held in fixed position.

Step 3. The number of revolutions X made by the driver in Step 2 is found as follows. When the fixed gear D2 is rotated one tuwn in a counter-clockwise direction, planet gear D3 is caused to rotate in a clockwise direction. The number of turns it makes is equal to the pitch diameter of the fixed gear divided by the pitch diameter of the planet gear (or the number of teeth of the fixed gear divided by the number of teeth of the planet gear), thus:

Number of turns of D3 = D2/D3

The number of turns of the driver for each turn of planet gear D3 with which it meshes is equal to the pitch diameter D3 divided by the pitch diameter D (or the number of teeth of the planet gear divided by the number of the number of teeth of the driver).

Number of turns D for one turn D3 = D3/D

Hence, the number of turns X of the driver equals the number of turns of D3 times D3/D, or

X = D2/D3 x D3/D = D2/D

Step 4. Since the follower is the train arm, it makes no revolutions during Step 2, hence Y = 0.

Step 5. It can be seen from the illustration that during Step 2 the driver will rotate in a clockwise or positive direction, therefore

N = 1 + X/l or

N = 1 + D2/D