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Epicyclic Gears 1 Advanced Gear Analysis Epicyclic Gearing Tooth Strength Analysis

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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 Planet Sun

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Epicyclic Gears 3 Epicyclic Gearset Lets 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

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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 = Negative Ring Planet Sun Arm

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Epicyclic Gears 5 The Tabular (Superposition) Method Next, 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 N Ring / N Planet turns clockwise. Since the Planet drives the Sun, the Sun will turn (N Ring / N Planet ) x (-N Planet / N Sun ) = - N Ring / N Sun turns (counter-clockwise). The arm doesnt move. We enter these motions into the second row of the table. 1 Turn Ring Planet Sun Arm

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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: 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.

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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: 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

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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 Teeth 3. Gear = 20 Teeth 4. Ring = 78 Teeth (Fixed) 5. Arm = Output See the Course Materials folder for the solution.

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Epicyclic Gears 9 Epicyclic Tips OUTIN 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.

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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 W t = T/r = 2T/D, where D is the pitch diameter.

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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.

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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

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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 + N 4 /N 3 x (-1) = - N 4 /N 3 rotations. Gear 2 is on the same shaft as Gear 3, so it also turns - N 4 /N 3 rotations. Gear 2 drives Ring 1, which turns + N 2 /N 1 x n 2 = + N 2 /N 1 x (- N 4 /N 3 ) = - N 2 N 4 /N 1 N 3 rotations. The arm was fixed, so it does not turn. The Tabular Method - Example 2

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

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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: N 1 = 100 Teeth Gear 2: N 2 = 40 Teeth Gear 3: N 3 = 20 Teeth Ring 4: N 4 = 78 Teeth The Tabular Method - Example 2

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