Advanced Gear Analysis

Epicyclic Gears 1Advanced Gear Analysis Epicyclic Gearing Tooth Strength AnalysisEpicyclic Gears 2Epicyclic GearsetAn 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.

INPUTCARRIERRingPlanetSunEpicyclic Gears 3Epicyclic GearsetLets 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.

INPUTOUTPUTINPUT

OUTPUTEpicyclic Gears 4INPUT

OUTPUTThe Tabular (Superposition) MethodTo 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 = PositiveCW = Negative

RingPlanetSunArmEpicyclic Gears 5The 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 doesnt move.We enter these motions into the second row of the table.

1 TurnRingPlanetSunArmEpicyclic Gears 6The Tabular (Superposition) MethodFinally, we sum the motions in the first and second rows of the table.

Now, we can write the relationship:

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 7Planetary GearsetThe 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:

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 .

INPUTOUTPUTRingPlanetSunn = speed; N = # TeethEpicyclic Gears 8Another Epicyclic ExampleAlways 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 9Epicyclic TipsOUTIN 1IN 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 10Gear 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 11The Tabular Method - Example 2First, 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 12First, 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 2Epicyclic Gears 13We 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/N1N3rotations. The arm was fixed, so it does not turn.

The Tabular Method - Example 2Epicyclic Gears 14Then we add the two rows to get the total motionAnd we can write the relationship:

The Tabular Method - Example 2Epicyclic Gears 15We 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 exampleRing 1: N1 = 100 Teeth Gear 2: N2 = 40 TeethGear 3: N3= 20 TeethRing 4: N4 = 78 Teeth

The Tabular Method - Example 2

SunPlanetRingArm

Rotate Whole Assembly CCW+1+1+1+1

SunPlanetRingArm

Rotate Whole Assembly CCW+1+1+1+1

Hold Arm, Rotate Ring CWNRing / NSun- NRing / NPlanet-10

SunPlanetRingArm

Rotate Whole Assembly CCW+1+1+1+1

Hold Arm, Rotate Ring CWNRing / NSun- NRing / NPlanet-10

Total Motion1 + NRing / NSun1 - NRing / NPlanet0+1

Ring 1 Gear 2Gear 3Ring 4Arm 5

Rotate Whole Assembly CCW+1+1+1+1+1

Hold Arm, Rotate Fixed Ring CW

Total Motion

Ring 1 Gear 2Gear 3Ring 4Arm 5

Rotate Whole Assembly CCW+1+1+1+1+1

Hold Arm, Rotate Fixed Ring CW

Total Motion

Ring 1 Gear 2Gear 3Ring 4Arm 5

Rotate Whole Assembly CCW+1+1+1+1+1

Hold Arm, Rotate Ring CW- N2N4/N1N3- N4/N3- N4/N3-10

Ring 1 Gear 2Gear 3Ring 4Arm 5

Rotate Whole Assembly CCW+1+1+1+1+1

Hold Arm, Rotate Ring CW- N2N4/N1N3- N4/N3- N4/N3-10

Total Motion1 - N2N4/N1N31 - N4/N31 - N4/N30+ 1