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Understanding LEGO 4-speed sequential gearboxes
I finally took the time to write down the things I have come to understand with regard to LEGO 4-speed sequential gearboxes. I am receiving many questions about gearboxes and I hope these understandings can help you reason about a gearbox layout while you're building one or trying to design one. I hope this also answers a question I received from @nerdsforprez more than a year ago, which I did not answer yet.
Gearbox layout
Let's take a look at this 4-speed sequential gearbox layout. Black is input, red is output and orange is control.
The main input is divided over a high input (black) with high input ratio and a low input (white) with low input ratio. The high input ratio is 1:1 (via a 12:12 mash) and the low input ratio is 1:2 (via a 8:16 mash). This makes for a combined ratio of (1:1) : (1:2) = 2:1 between the high and low inputs. I will refer to this ratio as the primary ratio. In fact this ratio is the ratio between the two driving rings. Both driving rings have a high output (green) with high output ratio and a low output (yellow) with low output ratio. For both driving rings, the high output ratio is 1:1 * 2:1 = 2:1 (via a 16:16 mash and a 16:8 mash) and the low output ratio is 5:3 * 1:2 = 5:6 (via a 20:12 mash and a 8:16 mash). This makes for a combined ratio of (2:1) : (5:6) = 12:5 between the high and low outputs of each driving ring. I will refer to these ratios as the secondary ratios.
Rotary catch and quadrants
Even though I will explain things in terms of the gearbox layout described above, the first understanding I want to address, applies to practically all 4-speed sequential gearboxes with 2 driving rings. Let's take a look at the rotary catch and driving rings from above and divide the layout into four quadrants. Each quadrant represents one of the four gears of the 4-speed gearbox. When we turn the rotary catch clockwise (seen from the left) with 90-degree steps, it will always make the following path through the four quadrants.
From the path the rotary catch draws, we can see that it toggles from one driving ring to the other driving ring for every 90-degree step. So if we want to obtain a useful gear sequence (either a 1-2-3-4 sequence or a 4-3-2-1 sequence) along that path, we need to tie gears 1 and 3 to one driving ring and gears 2 and 4 to the other driving ring. Otherwise the rotary catch can never 'toggle' between subsequent gears. Now let's take a look at all distributions of the four gears over the four quadrants that meet this requirement.
Starting top-left, this will produce a 1-4-3-2 sequence. Repeating the sequence will give 1-4-3-2-1-4-3-2-etc., which effectively boils down to a 4-3-2-1 sequence.
Starting top-left, this will produce a 1-2-3-4 sequence.
Starting top-left, this will produce a 3-4-1-2 sequence. Repeating the sequence will give 3-4-1-2-3-4-1-2-etc., which effectively boils down to a 1-2-3-4 sequence.
Starting top-left, this will produce a 3-2-1-4 sequence. Repeating the sequence will give 3-2-1-4-3-2-1-4-etc., which effectively boils down to a 4-3-2-1 sequence.
Starting top-left, this will produce a 2-3-4-1 sequence. Repeating the sequence will give 2-3-4-1-2-3-4-1-etc., which effectively boils down to a 1-2-3-4 sequence.
Starting top-left, this will produce a 2-1-4-3 sequence. Repeating the sequence will give 2-1-4-3-2-1-4-3-etc., which effectively boils down to a 4-3-2-1 sequence.
Starting top-left, this will produce a 4-3-2-1 sequence.
Starting top-left, this will produce a 4-1-2-3 sequence. Repeating the sequence will give 4-1-2-3-4-1-2-3-etc., which effectively boils down to a 1-2-3-4 sequence.
Surprisingly, every distribution that meets the requirement, will produce either a 1-2-3-4 sequence or a 4-3-2-1 sequence. What this tells us, is that it's enough to tie gears 1 and 3 to one driving ring and gears 2 and 4 to the other driving ring, to obtain a useful gear sequence. Nothing else matters! Primary ratio vs. secondary ratios
The next understanding I want to address, concerns the relation between the primary ratio (the ratio between the high and low input) and the secondary ratios (the ratios between the high and low outputs of both driving rings). We have already seen that in the gearbox layout at hand, the high and low output ratios are the same for both driving rings.
One thing we can say about 4-speed gearboxes in general, is that the ratios between gears 1 and 3 and between gears 2 and 4 need to make a bigger difference than the ratios between gear 1 and 2 and between 3 and 4. Now when we take into account that gears 1 and 3 need to be tied to one driving ring and gears 2 and 4 need to be tied to the other driving ring, and we use the same high and low output ratios for both driving rings, we can say that the secondary ratios, which constitute the ratios between gears 1 and 3 and between gears 2 and 4, need to be bigger than the primary ratio, which constitutes the ratios between gears 1 and 2 and between gears 3 and 4. The gearbox discussed in the beginning of this post has a primary ratio of 2:1 and secondary ratios of 12:5, so it meets the above requirement. Check!
Swapping and reversing
If we go back to the distributions we listed above, we can see that half of them generate a 1-2-3-4 sequence and half of them generate a 4-3-2-1 sequence. When we study them more thoroughly, we can see that all 1-2-3-4 distributions have a horizontally flipped counterpart with a 4-3-2-1 sequence. In other words, if we flip the distribution horizontally, we reverse the gear sequence.
Example: Swapping 1-3 with 2-4 in a 4-3-2-1 sequence produces a 3-4-1-2 sequence. Repeating the sequence will give 3-4-1-2-3-4-1-2-etc., which effectively boils down to 1-2-3-4.
Example: Swapping 1-3 with 4-2 in a 1-2-3-4 sequence produces a 4-3-2-1 sequence.
What this tells us, is that when we mirror the gearbox layout left-to-right (top-down in the quadrants), which boils down to swapping the high and low inputs, the effect is that we reverse the gear sequence.
Practical value: If you find yourself in a situation where you want to swap the upshifting and downshifting directions, simply swap the high and low inputs, like in the image above.
Finally, if we take one more look at the gear distributions above, we can see that when we swap gears 1 and 3 or gears 2 and 4 in any distribution, we get a distribution with the reversed order. 1-2-3-4 will produce 4-3-2-1 and 4-3-2-1 will produce 1-2-3-4. When we swap both gears 1 and 3, and gears 2 and 4, we reverse the order twice and get again the same order.
Example: Swapping 1 and 3 in a 1-2-3-4 sequence produces a 3-2-1-4 sequence. Repeating the sequence will produce 3-2-1-4-3-2-1-4, which effectively boils down to a 4-3-2-1 sequence.
Example: Swapping 2 and 4 in a 1-2-3-4 sequence produces a 1-4-3-2 sequence. Repeating the sequence will produce 1-4-3-2-1-4-3-2, which effectively boils down to a 4-3-2-1 sequence.
Example: Swapping 1 and 3, and 2 and 4 in a 1-2-3-4 sequence produces a 3-4-1-2 sequence. Repeating the sequence produces 3-4-1-2-3-4-1-2, which effectively boils down to a 1-2-3-4 sequence.
What this tells us, is that when we mirror one side of the gearbox front-to-back (swap the high and low outputs of one driving ring), we will reverse the gear sequence. When we mirror both sides front-to-back (swap the high and low outputs of both driving rings), we won't affect the gear sequence.
Practical value: If it's more convenient for the rest of your build to mirror your gearbox layout front-to-back, like in the image above, you can do so without any consequences. If it's more convenient to mirror only the left side or the right side of your gearbox layout, you need to also swap the upshifting and downshifting directions. If you want to inspect the gearbox used in this post in 3D, here it is in Stud.io format and here in LDD format.
Gearbox layout
Let's take a look at this 4-speed sequential gearbox layout. Black is input, red is output and orange is control.
The main input is divided over a high input (black) with high input ratio and a low input (white) with low input ratio. The high input ratio is 1:1 (via a 12:12 mash) and the low input ratio is 1:2 (via a 8:16 mash). This makes for a combined ratio of (1:1) : (1:2) = 2:1 between the high and low inputs. I will refer to this ratio as the primary ratio. In fact this ratio is the ratio between the two driving rings. Both driving rings have a high output (green) with high output ratio and a low output (yellow) with low output ratio. For both driving rings, the high output ratio is 1:1 * 2:1 = 2:1 (via a 16:16 mash and a 16:8 mash) and the low output ratio is 5:3 * 1:2 = 5:6 (via a 20:12 mash and a 8:16 mash). This makes for a combined ratio of (2:1) : (5:6) = 12:5 between the high and low outputs of each driving ring. I will refer to these ratios as the secondary ratios.
Rotary catch and quadrants
Even though I will explain things in terms of the gearbox layout described above, the first understanding I want to address, applies to practically all 4-speed sequential gearboxes with 2 driving rings. Let's take a look at the rotary catch and driving rings from above and divide the layout into four quadrants. Each quadrant represents one of the four gears of the 4-speed gearbox. When we turn the rotary catch clockwise (seen from the left) with 90-degree steps, it will always make the following path through the four quadrants.
From the path the rotary catch draws, we can see that it toggles from one driving ring to the other driving ring for every 90-degree step. So if we want to obtain a useful gear sequence (either a 1-2-3-4 sequence or a 4-3-2-1 sequence) along that path, we need to tie gears 1 and 3 to one driving ring and gears 2 and 4 to the other driving ring. Otherwise the rotary catch can never 'toggle' between subsequent gears. Now let's take a look at all distributions of the four gears over the four quadrants that meet this requirement.
Starting top-left, this will produce a 1-4-3-2 sequence. Repeating the sequence will give 1-4-3-2-1-4-3-2-etc., which effectively boils down to a 4-3-2-1 sequence.
Starting top-left, this will produce a 1-2-3-4 sequence.
Starting top-left, this will produce a 3-4-1-2 sequence. Repeating the sequence will give 3-4-1-2-3-4-1-2-etc., which effectively boils down to a 1-2-3-4 sequence.
Starting top-left, this will produce a 3-2-1-4 sequence. Repeating the sequence will give 3-2-1-4-3-2-1-4-etc., which effectively boils down to a 4-3-2-1 sequence.
Starting top-left, this will produce a 2-3-4-1 sequence. Repeating the sequence will give 2-3-4-1-2-3-4-1-etc., which effectively boils down to a 1-2-3-4 sequence.
Starting top-left, this will produce a 2-1-4-3 sequence. Repeating the sequence will give 2-1-4-3-2-1-4-3-etc., which effectively boils down to a 4-3-2-1 sequence.
Starting top-left, this will produce a 4-3-2-1 sequence.
Starting top-left, this will produce a 4-1-2-3 sequence. Repeating the sequence will give 4-1-2-3-4-1-2-3-etc., which effectively boils down to a 1-2-3-4 sequence.
Surprisingly, every distribution that meets the requirement, will produce either a 1-2-3-4 sequence or a 4-3-2-1 sequence. What this tells us, is that it's enough to tie gears 1 and 3 to one driving ring and gears 2 and 4 to the other driving ring, to obtain a useful gear sequence. Nothing else matters! Primary ratio vs. secondary ratios
The next understanding I want to address, concerns the relation between the primary ratio (the ratio between the high and low input) and the secondary ratios (the ratios between the high and low outputs of both driving rings). We have already seen that in the gearbox layout at hand, the high and low output ratios are the same for both driving rings.
One thing we can say about 4-speed gearboxes in general, is that the ratios between gears 1 and 3 and between gears 2 and 4 need to make a bigger difference than the ratios between gear 1 and 2 and between 3 and 4. Now when we take into account that gears 1 and 3 need to be tied to one driving ring and gears 2 and 4 need to be tied to the other driving ring, and we use the same high and low output ratios for both driving rings, we can say that the secondary ratios, which constitute the ratios between gears 1 and 3 and between gears 2 and 4, need to be bigger than the primary ratio, which constitutes the ratios between gears 1 and 2 and between gears 3 and 4. The gearbox discussed in the beginning of this post has a primary ratio of 2:1 and secondary ratios of 12:5, so it meets the above requirement. Check!
Swapping and reversing
If we go back to the distributions we listed above, we can see that half of them generate a 1-2-3-4 sequence and half of them generate a 4-3-2-1 sequence. When we study them more thoroughly, we can see that all 1-2-3-4 distributions have a horizontally flipped counterpart with a 4-3-2-1 sequence. In other words, if we flip the distribution horizontally, we reverse the gear sequence.
Example: Swapping 1-3 with 2-4 in a 4-3-2-1 sequence produces a 3-4-1-2 sequence. Repeating the sequence will give 3-4-1-2-3-4-1-2-etc., which effectively boils down to 1-2-3-4.
Example: Swapping 1-3 with 4-2 in a 1-2-3-4 sequence produces a 4-3-2-1 sequence.
What this tells us, is that when we mirror the gearbox layout left-to-right (top-down in the quadrants), which boils down to swapping the high and low inputs, the effect is that we reverse the gear sequence.
Practical value: If you find yourself in a situation where you want to swap the upshifting and downshifting directions, simply swap the high and low inputs, like in the image above.
Finally, if we take one more look at the gear distributions above, we can see that when we swap gears 1 and 3 or gears 2 and 4 in any distribution, we get a distribution with the reversed order. 1-2-3-4 will produce 4-3-2-1 and 4-3-2-1 will produce 1-2-3-4. When we swap both gears 1 and 3, and gears 2 and 4, we reverse the order twice and get again the same order.
Example: Swapping 1 and 3 in a 1-2-3-4 sequence produces a 3-2-1-4 sequence. Repeating the sequence will produce 3-2-1-4-3-2-1-4, which effectively boils down to a 4-3-2-1 sequence.
Example: Swapping 2 and 4 in a 1-2-3-4 sequence produces a 1-4-3-2 sequence. Repeating the sequence will produce 1-4-3-2-1-4-3-2, which effectively boils down to a 4-3-2-1 sequence.
Example: Swapping 1 and 3, and 2 and 4 in a 1-2-3-4 sequence produces a 3-4-1-2 sequence. Repeating the sequence produces 3-4-1-2-3-4-1-2, which effectively boils down to a 1-2-3-4 sequence.
What this tells us, is that when we mirror one side of the gearbox front-to-back (swap the high and low outputs of one driving ring), we will reverse the gear sequence. When we mirror both sides front-to-back (swap the high and low outputs of both driving rings), we won't affect the gear sequence.
Practical value: If it's more convenient for the rest of your build to mirror your gearbox layout front-to-back, like in the image above, you can do so without any consequences. If it's more convenient to mirror only the left side or the right side of your gearbox layout, you need to also swap the upshifting and downshifting directions. If you want to inspect the gearbox used in this post in 3D, here it is in Stud.io format and here in LDD format.