# How to I design a planetary gear with a gear ratio of 2:1 ( doubling the torque) [closed]

Trying to design a 2:1 gear ratio planetary gear, and I'm not sure how to begin. In the past I've had luck using this site to create an illustrator file that I can extrude in Rhino. This does not generate planetary gears, and I'm also unclear as to how design a planetary gear with a 2:1 ratio.

• Which part of the design is giving you problems? You need to be more specific, otherwise someone might as well just paste a textbook chapter on planetary gear design into the answers. – BarbalatsDilemma Jan 16 '17 at 12:58

(Image from Wikipedia)

When designing planetary gears, there are three different types of gear present to be designed:

• Annulus gear, an internal gear surrounding the other gears (shown above in red)
• Sun gear, an external gear in the middle of the gear set (shown in yellow).
• Planet gears, external gears surrounding the sun gear, usually three or four of them are present (shown in blue).

First of all, it's worth noting about internal gears. The sun and planet gears, being external gears (i.e. teeth point outward), are the exact same as any other kind of spur gear. However, the annulus gear is an internal gear, with the teeth pointing inward. An internal gear may look like a circle with an external gear subtracted from inside of the circle, and it almost is; there is just a slight difference in how far the teeth extend from the pitch circle. Not noting this difference in internal gears can lead to gear teeth undesirably colliding before properly meshing, causing the gear set to jam. Therefore, if you are fine with gears being output as .dxf files, then I suggest using Dr. Rainer Hessmer's gear generator, as it allows you to create both external and internal gears.

When it comes to actually designing these three different types of gears, the following equation, which I'll refer to as the speed rule, should be observed:

$$\omega_C (1+\rho) = \omega_A \rho + \omega_S$$

$\omega_C$ is the rotation speed of the carrier, which is shown in green in the above image. $\omega_A$ is the rotation speed of the annulus gear and $\omega_S$ is the rotation speed of the sun gear. Rotation speeds are all defined to be positive in the same direction of rotation. $\rho$ is the ratio of the number of teeth on the annulus gear to the number of teeth on the sun gear. It is worth noting that $\rho$ cannot be less than or equal to 1.

Often, out of the carrier, the annulus gear and the sun gear, one is selected to be fixed in place ($\omega=0$), one is selected to be the input and one is selected to be the output (it is however possible to have two inputs or two outputs). By fixing one of the rotation speeds, a gear ratio can be determined between the input and output, given as follows:

Fix the carrier: $\omega_C = 0, \frac{\omega_S}{\omega_A} = -\rho$

Fix the annulus: $\omega_A = 0, \frac{\omega_S}{\omega_C} = \rho+1$

Fix the sun: $\omega_S = 0, \frac{\omega_A}{\omega_C} = 1+\frac{1}{\rho}$

It is now a matter of selecting one of the above configurations that give you the appropriate gear ratio. Sometimes it requires a bit of trial and error: stick with a configuration and determine the value of $\rho$. If $\rho$ is less than or very close to one, try another setup.

For your case, with a gear ratio of $2:1$, let's try the following configuration:

Fixed: carrier

Input: sun

Output: annulus

Note that in this configuration, $\frac{\omega_S}{\omega_A} = -\rho$, so the input and output need to be rotation in opposition direction to get a positive value of $\rho$. Let's say the input is spinning in a "negative" direction and the output in a "positive" direction. Therefore:

$$\frac{\omega_S}{\omega_A} = \frac{-2}{1} = -2$$

$$\rho = 2$$

A value of $\rho$ which is greater than 1 is obtained: this configuration seems to work well. Now we enter the second phase of the planetary gear design: assigning the number of teeth.

First of all, the planet gears are generally spaced equally apart from one another. To achieve this, the number of teeth of both the sun gear and the annulus should be a multiple of the number of planets. Otherwise, difficulties may arise when it comes to assembling the gears together. This is a process that may require some trial and error, and it may even required very slightly changing the value of $\rho$ to make it easier to make the teeth number multiples.

In our example let's assume that there are four planet gears. Therefore, it can be seen that a sun gear of 31 (not multiple of 4) teeth is not appropriate, but one of 32 (multiple of 4) teeth is. If the sun gear has 32 teeth, the annulus gear will have $32 \times \rho = 64$ teeth. 64 is also a multiple of 4 and is therefore appropriate.

Once the number of teeth for the sun and annulus gears are selected, the number of teeth for the planet gear is calculated using the following formula:

$$\text{planet teeth} = \frac{\text{annulus teeth} - \text{sun teeth}}{2}$$

Note the division by two: this means that the number of teeth on the annulus and sun gears must both be odd numbers, or both be even numbers. If this turns out not to be the case, doubling the number of teeth on the sun and annulus gears quickly resolves the issue. In our example, the number of teeth on each planet gear is 16.

With the number of teeth selected, now select an appropriate module or diametral pitch, depending on how strong you want the gears, and that is the planetary gear set designed.

So in summary:

• Use speed rule to select appropriate configuration
• Determine ratio of annulus teeth to sun teeth
• Determine number of teeth for each gear
• Select the module/diametral pitch