The planetary gear train is an important, but complex mechanism. When it comes to analyzing rotation directions and gear speeds, intuition is not enough. We'll look at the superposition method to determine speeds and rotation directions.

For a planetary (a.k.a. epicyclic) gear train there are three possible inputs/outputs: two gears and the planetary gear carrier. Any one of these elements can be fixed, an input, or an output. For example, the basic configuration shown below has a sun gear, a ring gear and a planet gear carrier. The sun gear, ring gear, and carrier can be an input, output, or be fixed.

There are three possible uses of the gear train as well:

- One item is fixed, one input, one output.
- Nothing is fixed, two inputs, one output.
- Nothing is fixed, one input, two outputs.

"Input" and "output" are somewhat arbitrary terms when it comes to analysis. To analyze the gear train when nothing is fixed, we need to know the speed of two items.

In the case of a differential, the input is the drive shaft and the outputs are the axle shafts. However, we would treat them in the opposite sense for analysis. We need to pick speeds of each axle shaft and back-out the drive shaft speed. Alternatively, we would have to know the drive shaft speed and one axle speed to determine the second axle speed.

## Example Planetary Gear Train Analysis

We'll analyze the gear train below for two cases:

- Ring gear (R1) fixed, sun gear (S1) as input, carrier (C1) as output.
- Nothing fixed, sun and carrier as inputs, ring gear as output.

The gears have the following number of teeth:

S1: 22

P1: 12

P2: 20

P3: 20

R1: 94

## Ring Gear Fixed

For superposition, we use this procedure:

- Rotate the entire locked gear train 1 rotation clockwise. Each gear and the carrier rotate 1 revolution.
- With the carrier fixed, rotate the fixed item 1 rotation counterclockwise. Count the rotations of everything else.
- Add the rotations of each item from steps 1 and 2 to get net rotations.
- Calculate the number of rotations relative to the input rotations. This gives relative speeds for each item.

This method is required when the carrier is not fixed. When the carrier is fixed, it's a fixed axis gear train and speeds can easily be calculated by tooth ratios.

For our example, the sun gear is the driver and rotates at 300 RPM clockwise. Since the ring gear is fixed its speed is 0. We're left to find the planet gear speeds and the carrier speed.

Step 1 is easy. Everything rotates 1 revolution.

For step 2, we start by rotating the ring gear back 1 revolution. The carrier does not rotate in step 2 since we fix it for this step.. Then we find the rotation of each remaining gear based on the tooth ratio, except for P1. P1 will rotate the same revolutions as P2 since they share a shaft.

Step 3 simply adds the rotations in steps 1 and 2. Note that this results in the ring gear having a net rotation of 0 since it's fixed in real life.

The speeds are determined from the net rotations and the input speed. The sun gear speed is known to be 300 RPM so it's entered without calculation. R1 is fixed so its speed is entered as 0. The remaining speeds are calculated based on the relative net rotations.

For example, to calculate P1 speed:

P1 speed = S1 speed (S1 net rot/P1 net rot) = 300 (-1.5636/5.7) = 1093.6 CCW

## Nothing Fixed

With nothing fixed we need to alter the procedure.

- Rotate the entire locked gear train v rotations clockwise. v is not known.
- With the carrier fixed, rotate the sun gear w rotations clockwise. w is not known.
- Add the rotations of each item from steps 1 and 2 to get net rotations.
- Using the known input speeds and substitution, solve for v and w.
- Calculate speeds

For our example, the sun gear rotates at 100 RPM clockwise, and the carrier rotates at 50 RPM clockwise. We need to find planet gear speeds and the ring gear speed.

Again step 1 is easy. Everything rotates v revolutions.

For step 2, we start by rotating the sun gear w revolutions. The carrier does not rotate in step 2 since we fix it for this step. Then we find the rotation of each remaining gear based on the tooth ratio, except for P2. P2 will rotate the same revolutions as P1 since they share a shaft.

Step 3 simply adds the rotations in steps 1 and 2.

The sun gear speed is known to be 100 RPM so it's entered without calculation. The carrier speed is known to be 50 RPM so it too is entered without calculation.

To determine speeds, we first need to solve for v and w. Since we know S1 and C1 speeds, we'll use their net rotations.

S1rot = 100 = v + w

C1rot = 50 = v

Therefore, w = 100 - v = 50.

For R1:

R1rot = v + w (NS1NP2NP3 ÷ NP1NP3NR1)

Substituting for v and w:

R1rot = 50 + 50 (22*20*20 ÷ 12*20*94) = 69.5 rotations

The relative speed of S1 and R1 is 100 rotations vs. 69.5 rotations. Therefore the speed of R1 is:

R1 speed = (100 RPM)(69.5 rot ÷ 100 rot) = 69.5 RPM

The planetary gear speeds are calculated in a similar manner.

## The MEboost Planetary Gear Train Tool

As you can see from the examples, calculating speeds is pretty tedious. MEboost has a tool to calculate these speeds very quickly. There are 12 planetary configurations that can be analyzed. To run the tool click the Planetary Gear Train button on the Excel ribbon.

The planetary gear train form will appear. There are several tabs. One is for the analysis and the remaining tabs show the available configurations. For our example, we are analyzing configuration number 10.

After clicking on the analysis tab, we can enter the gear train information for the case where R1 is fixed. Then the speed calculations are made and appear in the results box.

For the case where nothing is fixed we need to enter a second input (C1 rotating at 50 RPM CW).

## Reference

Wilson, Charles; Sadler, Peter; Michels, Walter *Kinematics and Dynamics of Machinery*, Harper Collins, 1983, pp. 467-470 and pp. 482-484.

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