The Post Where I Update Everything Ever, Part II: Wubba Wubba Drive

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First, these videos must be watched before I can explain what the hell is going on:

Anyways, the arrangement of the gears in this contraption is known as a compound planetary geartrain. The idea is not unlike a harmonic drive – the difference in tooth counts between two ring gears is exploited by a rolling constrained element inside. However, unlike the harmonic drive, or the chainarmonic drive, there is no deformation element – in the case of the harmonic drive it’s the strain in the flexible spline, and in the chainarmonic drive, well, it’s a fucking chain.

Either way, the method has been in use for quite some time to get a ridiculous gear ratio in a small space – it has historically been seen in winches, bicycle hub gears, turboshaft engine output gearboxen, and apparently also automotive power mirror adjusters as the Wikipedia image shows. It’s also the secret of how the 2.007 kit motors, the BO-P5 and BO-P6, drop a 40:1 and 120:1 respectively in two stages of gearing, if you even count the output ring gear as a “stage”. Normal planetary gearboxes are hard pressed to get 7 or 8:1 in one stage, and that is with the sun gears being tiny and comparatively fragile.

But what is the actual gear ratio of a compound planetary geartrain of the type shown in the Wiki table of planetary gearbox arrangements, row 4 column 3? I’ll drop it here in case Wikipedia ever disappears or that page gets troll edited:

where blue indicates a driving element, green and cyan indicating small and large idler gears respectively, and red indicates a driven (output) element. Black indicates a fixed element. As can be observed from the diagram, this is the classical ‘compound planetary’ arrangement of a fixed ring gear, rotating ring gear, sun gear, and two idlers of different tooth counts.

…or is it? I spent a while staring at this trying to figure out why the blue carrier stick was…. well, blue. I decided that this must have been a graphical error – if the carrier were forced to rotate with the sun gear at the same speed, there would never be dependent rotation of the planets at all. However, there was another arrangement which lacked a graphical error and seemed more plausible at the time, and it is row 5, column 4:

This is a strange illustration for sure. I’m not sure why the wikipedia editors chose to make a strange arrangement of bevel gears which nobody in their right mind would actually try to build – I’ll chalk this one up to Wikipedia being crowdsourced. But topologically speaking, this is equivalent to the former diagram. The blue input sun interfaces directly with the cyan idler which is constrained to the fixed ring gear, and mechanically locked to the green idler, which drives the output gear. So why was one including an extra factor of 1 + (z_fixed / z_driving)? That expression, 1 + (z_fixed/z_driving) is exactly the ratio of a single stage conventional planetary with carrier output.

I wasn’t sure which one was wrong or right, so I decided to try out both in Inventor using rotational mates and the model of the first (104:1) gearbox. The result clearly showed that it is in fact the former, which really says that the diagram is wrong because it indicates the blue carrier stick as being rotationally driven. In actuality, the carrier just holds the idler gears and should ideally not experience torque. What this experiment told me was that if I used a conventional sun gear input and fixed ring for the first stage, the compound planetary ratio of ( [z_small_idler * z_fixed] / [z_large_idler * z_output] ) is multiplied by a further conventional planetary gear ratio of 1 + (z_fixed / z_driving) …even though it superficially looks like division – the compound ratio is 1 – (something close to 1), resulting in a very small number, further multiplied by a 1 / (large number, the conventional planetary gear ratio) resulting in a number smaller still.

This means either I am misunderstanding the distinctions between the two diagrams, or that they’re wrong. I’m not quite confident enough to say the latter, but regardless they’re really, really wack, bro.

Empirically, then, the ratio is confirmed to be of the form 1-(z_small_idler * z_fixed)/(z_large_idler * z_output ) /  1+( z_fixed / z_driving) for tooth count z.

I now present converse method of determining the gear ratio from Shane Colton, who can put up with infinitely more math and basic mechanical physics than I can:

This is pretty much all going up because we determined that about a year from now that entire conversation about compound planetary gearbox design will have been forgotten and therefore have to be rediscovered. Now, I definitely don’t want to go through all of that again, so I will say that the speed equation on the bottom left, of the same form as the Wikipedia equation, can be algebraically transformed into the torque expression in the top right (appropriately inverted to change to speed instead of torque). It’s missing something, though, namely anything involving the fixed ring gear.

Right away, it can be seen that R5, the diameter of the fixed ring gear, can be eliminated as a design input, reducing the system to four degrees of freedom. This is because the ring gear is assumed to be as large as the sun plus a plant – in other words, R1 (sun) + 2 * R4 (first stage, larger planet)… From a pure radius equivalency perspective, this makes sense. If these were pitch circles of gears, then the ring gear would have to be exactly one sun radius plus one planet diameter.

Therefore we define the Soft Rule as follows: The ring gear pitch diameter must be one sun pitch radius plus one planet pitch diameter. From a tooth count perspective, this means the ring gear tooth count must equal the sun gear tooth count plus 2 * first stage planet gear tooth count – or concisely, Z_s + 2 * Z_p1 = Z_r where p1 is the first stage planet (previously called z_large_idler). If soft rule is obeyed, then you only select the sun tooth count, planet tooth counts, and output ring tooth count: four degrees of freedom. Why is it called “soft rule”? Because…. yeah.

The speed ratio with 4 degrees of freedom is then 1-(Z_p2 / Z_p1) / 2*(Z_r2 / Z_s) where r2 is the second stage output ring gear.

However, there is another dependency which can be eliminated in order to simplify ratio selection even further. Let the radius of the output ring (R2) equal the radius of the sun (R1) plus the radius of each planet (R3 + R4). From a tooth count perspective this is Z_r2 = Z_s + Z_p1 + Z_p2. Geometrically, this is saying that the difference in tooth count between the ring gears is the same as the difference in tooth count between the planets, which must be true if the planets ride on the same idler axle.

If this constraint is allowed, then the speed equation above reduces further to 1-(Z_p2 / Z_p1) /  (2 + [2*(Z_p1 + Z_p2)/Z_s1]) and only has three degrees of freedom: the sun and two planets.

We now have three different expressions for the speed ratio of a compound planetary gearbox with conventional first stage planetary input:

  1. 1-(Z_p2 * Z_r1)/(Z_p1 * Z_r2 ) /  (1+[ Z_r1 / Z_s])
  2. 1-(Z_p2 / Z_p1) / 2*(Z_r2 / Z_s) for dependent first stage ring gear
  3. 1-(Z_p2 / Z_p1) /  (2 + [2*(Z_p1 + Z_p2)/Z_s1]) for dependent first stage and output ring gears.

With this example of Wubba Wubba Drive, I can check if they are all equivalent:

In this example:

Z_s = 12,

Z_p1 = 20,

Z_p2 = 16,

Z_r1 = 52,

Z_r2 = 48.

For the first expression, the result is (1 – (16*52)/(20*48)) / (1 + (52/12)) =  0.025 or 40:1

For the second expression, the result is (1 – (16/20) / 2*(48 / 12) = 0.025, or also 40:1.

Finally, for the third expression, the result is (1 – (16/20) / (2 + (2*(16 + 20)/12)) = 0.025.

So all you really need to pick, if there are no other hard constraints on the ring gear size, the tooth counts of only the sun gear and the two planet gears. However, this is still all assuming just one planetary gear. Clearly, any weird combination of teeth can be done with just one planet, but generally planetary geartains have multiple (minimum 2, usually 3 or 5). In this case, the gear tooth counts would be limited to those which make sense and do not force the teeth to “elastically compensate”. This includes the usual design constraints such as the ring and sun must all share a common denominator that is the number of planets (for a rotationally symmetric box… you can totally fudge it like the Tamiya planetary gearbox’s second stage (yellow gears) and have the gears not totally symmetric – can you unsee it now?!). For two-planet systems, the constraint is pretty loose.

It also turns out that this compound planetary gearbox layout is not the most efficient design possible. The whole concept of relative/differential gearing, and harmonic drives, relies on wedging action and leverage, so they are as a whole not the most efficient possible power transmission method for a given ratio. If the planets are small relative to the ring gears, there is significantly more wedging action under load and the associated loss due to increased sliding friction. This somewhat old school but excellent writeup (Thanks Dale!) explains how efficiency for such a gearbox can be calculated. In fact, the example design doesn’t have a sun gear at all – the power is input directly to the planetary carrier, there is only one planet gear set, and a counterweight on the carrier keeps the system in balance.

This makes me think that 1. The weird bevel gear thing on Wikipedia isn’t crazy after all, just drawn strangely, and 2. The first diagram with the blue carrier stick is entirely wrong, especially because

…these two are right next to eachother, and they have the same gear ratio. The fuck is going on?

Anyways, what will I use Wubba Wubba Drive for? Well, it’s already been partially given away by the giant steel serrated knocker thing hanging off the second model. But, before I give more details on what contraption it’s going on, I’m going to design the rest of it first. Stay tuned for more wubbing.

The Post Where I Update Everything Ever, Part I: Watching the Seedlings Grow

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…because not only have I not had time (huh?) to really work on anything in depth, but…

Okay, so it really is because my day to day business is taken up by watering the seedlings assisting in teaching MIT’s renowned 2.007 Design & Manufacturing I course (to atone for my sins), partaking primarily in the logistics of the class and also in part acting as a design advisor and tutor. That task is not one which is very atomic. Students can come and hunt you down to ask questions and consult for design advice almost any time you venture through the shop, and there are lots of little overhead things you do if you are in charge of an entire lab section which take a nontrivial amount of time – things like ordering parts, office hours, Arduino crash course study sessions, grading… Oh god the grading.

Because of that, there haven’t been much in terms of new developments, at least not at the speed I am usually known for. The hell is this “not updating my site in a month” thing? Usually if I don’t throw up something in a week and a half, something has gone horribly wrong, such as being hit by a bus angry quadrotor. To remedy this, I’m just going to dump everything that’s been going on.

Here’s some updates from the Electric Vehicle Section front, which in the grand scope of growing young seedling engineers, is like my personal victory garden or something. Remember the one-powerpoint-slide “billboard for the class:

The format of the section itself is purely lab-oriented – students are still obligated to complete the class homework and exams, but this replaces the robot kit with free reign on Hobbyking and McMaster, among other places. I’m quickly realizing that this model is not exactly sustainable – while it was simple enough to handle the 4 or 5 students of years past, there’s 11 this year. Ultimately, this shows the conflict between class resources and ‘customizability’ where the students gets more flexible design and execution privileges – a “kit” based EV class would be no fun in my opinion, yet this level of freedom is a little much for both me, and the less experienced members of the class, to handle effectively. It’s not that it’s not working, but it is alot of work – conveying the message of using analysis and CAD in designing systems while actually building the system itself is a Herculean task that 2.007 tries to tackle to varying degrees of success every year. A better balance might be a more limited but immediately available set of parts and materials – 5 different motors, a bag of sprockets, and a pile of aluminum, for example. It’s something I have to think about if I want to take this effort EVEN MOAR SRS BIDNESS than it is this year.

Enough philosophizing – here’s Deathscooter-Adrian lecturing to the students early in the term about vehicle design…or how not to design your vehicle in his particular case.

Some labs start with a “mini lecture” which covers little details of EVs that the main 2.007 lectures cannot cover, such as application of DC motor theory specifically to designing drivetrains.

The students are provided “$300” in materials and parts budget and the option of beheading a Razor A3 type scooter, like most of the MIT and Georgia Tech Electric Scooter Teams have done.

What it really translates into is that major parts such as the motor, controller, drivetrain parts, and frame materials should fit under $300 in nominal costs – you can perform an infinite amount of nickel and diming if you count every screw or steel rod turned shafting. A budget is proposed and submitted before ordering begins – I’m the order aggregator, sending out compiled McMaster, Hobbyking, SDP, Monster Scooter Parts, etc. orders every Monday and Wednesday.

 Now, you definitely didn’t have to build just a scooter or even stay solo – two students decided to team up and combine their budgets to create a go-kart competitor to tinykart.

One major breakthrough for this year is a battery donation by A123Systems. These are A123’s new ALM12V7 12 volt lead-acid replacement modules that are essentially 8 A123 26650 type cells in a box that have been trained to pretend to be lead-acid. They have onboard charge regulation and balancing circuitry, and feature a built-in 30A (to be jacked up to 40A for the class) fuse and output protection. These are so much better than

1. Lead acid batteries, from 2010

2. Trying to get newbies to solder their own A123 packs from our pile of cells last year, which was a white-hair generating experience for all involved.

They are also of a “difficult” form factor – Fitting a house brick inside your vehicle is one of those design constraints that make the course interesting.

So shoutouts to A123. Ya’ll should… uhhh… go do commercial/industrial business with them or something, because they still won’t sell directly to end users and hobbyists. Hey, I love them and all, but I’m also not going to hold back in expressing my pet peeve of snooty American companies refusing to sell directly to the market as every Chinese shady parts house has learned to do in the past few years. Ever wonder why there are no American-made random small vehicle parts?

Here’s a more recent picture of the team effort go-kart, now named Melonkart officially, because when you have more money, you can afford bigger motors.

building shop

Now, all this is actually happening while I am still putting together a fab shop. Along with the other graduate students and Ph.D students of our group, we’ve been commissioning….

Mobile “fabrication cell” workstations with mini shop tools, and…

…electronic workstations with the latest in nice things. This table is alot messier now…

Mmmm….infrastructure budget.

Anyways, that’s the latest from the “Charles actually does legit things” front. Several of the students have actually picked up the habit of blogging about their projects:

At the end of the semester, there will be games.