Epicyclic Gears 1

Advanced Gear Analysis

• Epicyclic Gearing

• Tooth Strength Analysis

Epicyclic Gears 2

Epicyclic Gearset

An epicyclic gear set has some gear or gears whose center revolves about some point.

Here is a gearset with a stationary ring gear and three planet gears on a rotating carrier.

The input is at the Sun, and the output is at the planet carrier.

The action is epicyclic, because the centers of the planet gears revolve about the sun gear while the planet gears turn.

Finding the gear ratio is somewhat complicated because the planet gears revolve while they rotate.

INPUT

CARRIER

Ring

PlanetSun

Epicyclic Gears 3

Epicyclic Gearset

Let’s rearrange things to make it simpler:

1) Redraw the planet carrier to show arms rotating about the center.

2) Remove two of the arms to show only one of the planet arms.

INPUT

OUTPUT

INPUT

OUTPUT

Epicyclic Gears 4

INPUT

OUTPUT

The Tabular (Superposition) Method

To account for the combined rotation and revolution of the planet gear, we use a two step process.

First, we lock the whole assembly and rotate it one turn Counter-Clockwise. (Even the Ring, which we know is fixed)

We enter this motion into a table using the convention:

CCW = Positive

CW = NegativeRing

Planet

Sun

Arm

Epicyclic Gears 5

The Tabular (Superposition) MethodNext, we hold the arm fixed, and rotate whichever gear is fixed during operation one turn clockwise.

Here, we will turn the Ring clockwise one turn (-1), holding the arm fixed.

• The Planet will turn NRing / NPlanet turns clockwise.

• Since the Planet drives the Sun, the Sun will turn (NRing / NPlanet) x (-NPlanet / NSun) = - NRing / NSun turns (counter-clockwise).

• The arm doesn’t move.

We enter these motions into the second row of the table.

INPUT1 Turn

Ring

Planet

Sun

Arm

Epicyclic Gears 6

The Tabular (Superposition) Method

Finally, we sum the motions in the first and second rows of the table.

Now, we can write the relationship:

Sun

SunNRingNArm

ArmSun

RingSun

nn

nN

Nn or

1

1

,1

If the Sun has 53 teeth and the Ring 122 teeth, the output to input speed ratio is +1 / 3.3 , with the arm moving the same direction as the Sun.

Epicyclic Gears 7

Planetary Gearset

The configuration shown here, with the input at the Sun and the output at the Ring, is not epicyclic.

It is simply a Sun driving an internal Ring gear through a set of three idlers.

The gear ratio is:

ring

sun

ring

planet

planet

sun

sun

ring

N

N

N

N

N

N

n

n

Where the minus sign comes from the change in direction between the two external gears.

If the Sun has 53 teeth and the Ring 122 teeth, the ratio is -1 / 2.3 .

INPUT

OUTPUT

Ring

Planet

Sun

n = speed; N = # Teeth

Epicyclic Gears 8

Another Epicyclic Example

Always draw this view to get direction of rotation correct.

Here are three different representations of the same gearset:

Given:

1. Ring = 100 Teeth (Input)2. Gear = 40 Teeth3. Gear = 20 Teeth4. Ring = 78 Teeth (Fixed)5. Arm = Output

See the Course Materials folder for the solution.

Epicyclic Gears 9

Epicyclic Tips

OUT IN 1

IN 2

• If you encounter a gear assembly with two inputs, use superposition. Calculate the output due to each input with the other input held fixed, and then sum the results.

• Typically, when an input arm is held fixed, the other output to input relationship will not be epicyclic, but be a simple product of tooth ratios.

• Use the ± sign with tooth ratios to carry the direction information.

Epicyclic Gears 10

Gear Loading • Once you determine the rotational speeds of the gears in a train, the torque and therefore the tooth loading can be determined by assuming a constant power flow through the train.

• Power = Torque x RPM, so Torque = Power / RPM

• If there are n multiples of a component (such as the 3 idlers in the planetary gearset example) , each component will see 1/n times the torque based on the RPM of a single component.

• From the torque, T, compute the tangential force on the

teeth as Wt = T/r = 2T/D , where D is the pitch diameter.

Epicyclic Gears 11

The Tabular Method - Example 2

First, we lock the assembly and rotate everything one turn CCW

Next, we hold the arm fixed, and turn the fixed component (Ring 4) one turn CW, to fill in row two.

Epicyclic Gears 12

First, we lock the assembly and rotate everything one turn CCW

Next, we hold the arm fixed, and turn the fixed component (Ring 4) one turn CW, to fill in row two.

The Tabular Method - Example 2

Epicyclic Gears 13

We enter these motions into row two of the table:

We turn Ring 4 one turn CW (-1).

• Ring 4 drives Gear 3, which turns + N4/N3 x (-1) = - N4/N3 rotations.

• Gear 2 is on the same shaft as Gear 3, so it also turns - N4/N3 rotations.

• Gear 2 drives Ring 1, which turns+ N2/N1 x n2 = + N2/N1 x (- N4/N3) = - N2N4/N1N3

rotations.

• The arm was fixed, so it does not turn.

Ring 1 Gear 2 Gear 3 Ring 4 Arm 5Rotate Whole Assembly

CCW +1 +1 +1 +1 +1

Hold Arm, Rotate RingCW - N2N4/N1N3 - N4/N3 - N4/N3 -1 0

The Tabular Method - Example 2

Epicyclic Gears 14

Then we add the two rows to get the total motion

And we can write the relationship:

1

31

421

31

421

1

,1

RingArm

ArmRing

nn

nNN

NNn

NN

NN

or

Ring 1 Gear 2 Gear 3 Ring 4 Arm 5Rotate Whole Assembly

CCW +1 +1 +1 +1 +1

Hold Arm, Rotate RingCW - N2N4/N1N3 - N4/N3 - N4/N3 -1 0

Total Motion 1 - N2N4/N1N3 1 - N4/N3 1 - N4/N3 0 + 1

The Tabular Method - Example 2

Epicyclic Gears 15

We compute:

The output arm rotates almost twice as fast as the input ring, and in the opposite direction.

• Output direction is dependent on the numbers of teeth on the gears!

For our example

Ring 1: N1 = 100 Teeth Gear 2: N2 = 40 TeethGear 3: N3= 20 TeethRing 4: N4 = 78 Teeth

1

11

1

1

786.1

56.0

1

56.11

1

1

1

1

1

20100

7840

31

42

RingArm

RingRingArm

RingArm

RingArm

nn

nnn

nn

nn

NN

NN

The Tabular Method - Example 2