How to Design a Simpson Planetary Gearset

Hi! I’m a high school mechanical engineering student who was recently introduced to the world of gears and planetary gearsets. While learning about such things, I noticed something missing from my school curriculum: designing a fully functioning gearbox from the ground up.

That realization sent me down a rabbit hole. I studied planetary gear systems in depth and eventually decided to design a model of a simple 3-speed plus reverse "automatic" gear train. When choosing a layout, I narrowed it down to either a general compound planetary gearset or a Simpson planetary gearset. I chose the Simpson gearset because of its compact and somewhat niche design.

One challenge I ran into was figuring out how to calculate gear tooth counts based on desired gear ratios. I couldn’t find a clear, easy to understand method online, so I developed my own approach. It may or may not be the industry-standard method, but it works. So, in this Instructable, I’ll walk you through exactly how I designed my Simpson planetary gearset from scratch.

NOTE: This is not an Instructable on how gears themselves work or even on how a single planetary gearset works. This Instructable assumes that all readers have basic knowledge about gear terms like module, pitch circle, etc. as well as the different parts of a planetary gearset and how they interact with one another.

1. Pen/Pencil/Marker and paper
2. Calculator
3. Basic algebra knowledge
4. Willis Equation (This link is very helpful for learning how to derive it and how to apply it, and I will be referencing it a lot in Step 3)
5. 3D printer (or access to one) and filament
6. Ball bearings (sizes below if using my design):
7. 5×10×4 mm (10x)
8. 8×16×5 mm (4x)
9. 12×28×8 mm (3x)
10. M2.5×16 mm nuts and bolts (5x, if using my design)
11. Glue (superglue/quick drying glue is best)
12. Needle Nose Pliers
13. Foam Board/Cardboard

The schematic above (Source: Wikipedia) is that of a very straightforward Simpson Gearset, and my design will be closely modeled after it.

Key aspects of the Simpson Gearset:

1. 3 speed + reverse gear train
2. Two planetary gearsets that share a common sun gear
3. Two connections to the input shaft via the common sun gear (from a housing that wraps around everything else) and the ring gear in gearset 1
4. Two brakes that hold either the common sun gear or the planet carrier in gearset 2 stationary
5. Output is derived from the planet carrier in gearset 1 and the ring gear in gearset 2, which are permanently connected to each other and the output shaft

How to get each gear ratio (C1 is Clutch One, C2 is Clutch Two, B1 is Brake One, B2 is Brake Two):

1. First gear: Connect C2 and B2 (Input: Ring of gearset 1; Stationary: Planet Carrier of gearset 2). Power flows through both gearsets.
2. Second Gear: Connect C2 and B1 (Input: Ring of gearset 1; Stationary: Sun Gear). Since the output comes from the Planet Carrier of gearset 1, you can see that this gear is "contained" in gearset 1.
3. Third Gear: Connect C1 and C2 (Input: Ring of gearset 1 and Sun Gear; Stationary: Nothing). Having two inputs rotating at the same speed will force everything else to follow, including the output, resulting in a 1:1 gear ratio.
4. Reverse Gear: Connect C1 and B2 (Input: Sun Gear; Stationary: Planet Carrier of gearset 2). Since the output comes from the Ring Gear of gearset 2, you can see that this gear is "contained" in gearset 2.

This step is very straightforward.

In a Simpson gearset:

1. Third gear is always 1:1
2. Reverse ratio is determined by the 1st and 2nd gear ratios

Because of this, you only need to choose 1st and 2nd gear ratios, and the rest of the system follows naturally.

General guidelines:

1. 1st gear should be greater than 2:1
2. 2nd gear should fall between 1:1 and 2:1

For my design, I chose:

1. 1st gear: 2.8:1
2. 2nd gear: 1.6:1

I felt that these values provided a good balance between torque multiplication and smooth progression between gears.

This is a fun one! Here's the method that I used to get my tooth counts:

Few things to note:

1. Since there are two planetary gearsets, I’ll refer to them by location:
2. Input-side gearset → subscript I
3. Output-side gearset → subscript O
4. The sun gear is common to both gearsets and will simply be labeled S.
5. Notation:
6. ω = rotational speed (units don’t matter because they are consistent and derived from the gear ratios)
7. τ = gear tooth count
8. Planet gear notation:
9. Tooth count → τ_P(I/O)
10. Rotational speed → from the carrier, ω_C(I/O)​​
11. All tooth counts include a variable x, representing a common multiplier. This ensures all tooth counts are whole numbers.

Begin by listing all known values and unknowns at the top of your page (Picture 1).

USING SECOND GEAR:

We know that this gear ratio is formed using only the input side gearset. Sun is stationary, Ring is input, Planet Carrier is output. Using the "Fixed sun gear" equation:

1. 1.6 = 1+ τ_S / τ_RI
2. τ_S = 3x
3. τ_RI = 5x
4. τ_PI = (τ_RI - τ_S) / 2 = 1x (This ensures the planet gears fit symmetrically between the sun and ring)
5. Fill in those values at the top of the page (Picture 2; as of now, x can be any natural number)

USING FIRST GEAR:

First gear is more complex:

1. Input comes from the input-side ring gear
2. The output-side carrier is held
3. Power flows through both gearsets

To use the Willis equation, we need:

1. Tooth counts for the sun gear and both ring gears
2. Rotational speeds for sun, both rings, and both carriers

Known values:

1. ω_CO = 0
2. ω_CI = ω_RO ​​= output speed = 1 (2.8 : 1)
3. τ_S = 3x
4. τ_RI = 5x
5. ω_RI = input speed = 2.8 (2.8 : 1 gear ratio)

The unknowns are τ_RO​​ and ω_S. However, because the sun gear is shared, its rotational speed is the same in both gearsets. By solving the Willis equation for ω_S, we can equate the input-side and output-side expressions and solve for τ_RO. (It doesn't matter what ω_S actually is; all we are using it to do is relate both gearsets to each other)

1. Willis equation: ω_R ⋅ τ_R = ω_C ⋅ (τ_R + τ_S) – τ_S ⋅ ω_S
2. Solved for ω_S: ω_S = (ω_C ⋅ (τ_R + τ_S) – ω_R ⋅ τ_R) / τ_S
3. Refer to picture 3 where I plug in the values and solve
4. τ_RO = 6x
5. τ_PO = (τ_RO - τ_S) / 2 = 1.5x
6. Fill in those values at the top of the page (Picture 3; now x must be a positive multiple of 2 to cancel out the 1.5)

CALCULATING REVERSE GEAR:

We know that the reverse gear is formed using only the output side gearset. Planet Carrier is stationary, Sun is input, Ring is output. Using the "Fixed carrier" equation:

1. R = -τ_RO / τ_S
2. R = -2 (Reverse ratio of 2:1)
3. Fill in this value at the top of the page (Picture 4; everything has been solved!)

For my gearset, I made "x" equal to 10. Based on your calculations, you can choose any value of x that cancels out all decimals and results in a reasonable number of teeth for all gears. My gear teeth counts came out to:

Common:

1. Sun --> 30

Input side:

1. Planet --> 10
2. Ring --> 50

Output side:

1. Planet --> 15
2. Ring --> 60

Since I didn't want my model to take too much space when mounted on the foam board, I decided on a gear module of 1 mm for my gears. This selection is up to you as well.

Time to model the gearbox! You can use my files, or make your own, but the general parts you need are:

1. 2x Sun Gear (surrounding geometry may be identical)
2. Sun gear housing (where the stationary brake or input clutch will be applied)
3. 2x Ring Gear (most likely not going to be identical; input ring gear must have a clutch system and output ring gear must connect to the output shaft)
4. Input clutches for input ring and sun (take a look at mine, I made these "dog gears" that are keyed onto the input shaft and each other)
5. Planet gears as denoted by the equation (τ_{S +} τ_R) mod (# of planet gears) = 0 (30+50 mod 5 = 0 and 30+60 mod 5 = 0 so I decided on 5 planet gears per gearset for my design)
6. Planet carriers (input side carrier should connect to output side ring gear by passing through everything in between; output side planet carrier should extend out and around the ring gear so it is where the brake will be applied)
7. Input and output shafts and a power system (crank, motor and motor mount, etc.)
8. At least 2x Stands
9. Optional 2x Brake mechanism (I just used my hand for my model)

If you use my design, you need:

1. 2x Stands
2. 2x SunGear
3. 1x SunGear Housing (2 parts)
4. 1x RingGear Input Side
5. 1x Ring Input Clutch
6. 1x Sun Input Clutch
7. 1x Input Shaft (2 parts: shaft and crank)
8. 1x Output Shaft (2 parts: shaft and "end plate")
9. 5x PlanetGear Input Side
10. 1x PlanetCarrier Input Side (2 parts)
11. 1x RingGear Output Side
12. 5x PlanetGear Output Side
13. 1x PlanetCarrier Output Side (2 parts)

The only setting I would recommend is use 0.1 or 0.2 mm layer height (make the first layer 0.2mm regardless) Print each part in an orientation that requires the least amount of support material. For the gears, print the sun and planet gears with the gear at the bottom, and print the ring gears so that the gear layers are at the top and the back side of each ring gear is at the bottom. Other than that, it is up to your discretion what settings you want to use.

I will explain my process for assembling the gearbox here (again, only if you use my design):

NOTE: If you need to sand any parts to fit in the holes, make sure to glue the parts to each other. This gearbox isn't meant to be taken apart after it is fully assembled.

1. Fit the ball bearings as follows:
2. 5×10×4 mm --> 5 in each planet carrier where the planet gears are supposed to go
3. 8×16×5 mm --> 1 in the Sun Gear Housing, 1 in the Output Side Carrier, and 1 in each of the Sun Gears (press each bearing in from the side with the actual gear until they are flush with the bottom surface)
4. 12×28×8 mm --> 1 in each of the stands, and then 1 in the back side of the Input Side Ring Gear
5. Put the 5 bolts through the holes in the Sun gear that will be used for the input side gearset. This will take a little bit of forcing through because they don't fit very easily.
6. Fit the planet gears into their respective carriers and the carriers around their respective suns. Glue the front and back of each carrier to each other.
7. Fit the loose ends of the bolts through the flat part of the sun gear housing and through the output side sun gear. Use pliers to carefully get the nuts on and tighten them. The "rod" of the input side carrier should go through everything, and be held in place by the 4 medium ball bearings.
8. Slide the input clutches onto each other and onto the input shaft. Slide the end of the input shaft with the square extrusion through one of the stands until it fits entirely through the ball bearing. Then fit the crank onto the input shaft.
9. Fit the output shaft through the other stand until it is flush with one side. Then color the notch in the little part that connects to the output shaft with a pencil or a marker, and fit the two together.
10. Put the free end of each shaft into their respective ring gears, then fit the ring gears onto the assembly. For the output side ring, make sure it also fits into the little square extrusion that is at the end of the input side planet carrier.
11. Glue the remaining part of the sun gear housing to the part that is already assembled.
12. Finally, glue the two stands to a piece of foam board or cardboard. Now you're done!

After all the hard work, it's finally time to see how the gears work in real life.

Neutral: Disconnect the two clutches.

1st gear: Connect the smaller clutch and hold the output side planet carrier stationary.

2st gear: Connect the smaller clutch and hold the sun gear housing stationary.

3rd gear: Connect both clutches.

Reverse Gear: Connect the larger clutch and hold the output side planet carrier stationary.