midterm report of my training holiday

Last night was just the half-time of my stay in Dahab, diving mekka at the Gulf of Aqaba. I am on a strict internet and phone diet with most my communication with home being one-way, by post-card. The internet diet is really good to get most out of my stay here and connect with many people as well as with myself. I am checking email only once per week and today I am using the occasion to also post some impressions here:

• First of all, Dahab is great for windsurfing, so I did in fact come to the right place! Although the rental prices for sail boards are just as expensive as in Berlin, Germany, the service is also as good and includes comfy rides on the rescue boat when you've gone farther than you can make it back. Wind is consistently coming down the gulf from the Nord, although this past week had a lot of quiet days. 
• The Egyptians can be really fun people and it's totally worth learning a few words of Arabic to impress them and put conversations on a more relaxed and friendly level. They'll often start conversations, and when the conversation goes well, will teach me one or two more words. Some people here, unfortunately, have the job of selling absolutely useless things for astronomic prices and dealing with them is just as awful as with any slimy salesman back home. The prime example is when someone desperately tries to sell overpriced drinks at especially touristic places like an oasis or dive site. This really makes being at that place less enjoyable. On the other hand, it is nice to come back to the same shop (bakery, supermarket, restaurant) in Dahab every so often and getting to know the people there better.
• There's a lot of foreigners living here, either they operate a business or they just enjoy the great climate and the cheap cost of living. Almost all dive shops have some foreign staff (and/or owner) which I guess is because diving needs a lot of experience and communication to be safe. The main nations represented here are Germans, Brits, and Russians. It seems that the Russians run most of the sail-renting businesses and also some other shops, restaurants, and –of course– diving schools. Russian signs are as common here as locals who speak a bit of Russian. Most of the local–foreigner communication is done in English, tho.
• The climate in winter is excellent: day temperature just right to wear a T-Shirt outside, without sweating or feeling cold. It's a bit cold for diving, but ok for windsurfing with an optional wet suit when there's a lot of chilly wind.
• In my first week here, I hung out a lot with a nice man from Berlin who I had met at the airport. He used his stay here to plot out his next career moves after completing a second course of studies and to make plans for the next year. I gave him my copy of the “4-hour work-week” to read and we discussed it a bit. He inspired me to do something similar for myself and I quite liked the results. One of the outcomes is my resolution to celebrate my birthday on the fifth of every month, or in other words, have a monthly party to which I invite everybody I like, just to hang out together and to introduce my friends to each other. I will start with a Dahab-themed party on the week-end after coming back home! 
• Besides being a great spot for board sailing, Dahab is the most touristy place I have ever been, which has a lot of advantages and disadvantages. The upside is that there are a lot of cheap places to stay. Right now I have a sea-view room for about 7€ a night, although there's no window, so I need to open the door to actually view the sea, but there's always the sound of waves to take me to sleep at night and wake me up in the morning. There are even camps which offer beds in little rooms for 5€ a night. There are a lot of very interesting people in those camps, among them many seasoned travelers and long-time travelers, which have interesting stories to tell. I myself am more a temporary resident than a traveler. The downside is that tourists are herded like sheep on some places and locals sometimes get bad impressions of foreigners.
• I am trying to live like an expat here, that is, a non-tourist foreigner. I don't go to the restaurants for tourists, avoid souvenir shops and tourist activities, don't go to see the most famous sights. Instead I try to eat where the locals eat, spend my leisure time just hiking or biking around the landscape, and when I do trips then go to any random spot where there are not so many tourists. It as funny to arrive by foot or bicycle at places where tourists are usually brought by bus, jeep, horse, camel or quad-bike. If I pass such a tourist spot while walking or cycling, I do my best to ignore it and move on as quickly as I can.
• Since the Red Sea is know so well for its underwater life, I bought a very simple snorkel so I can enjoy this at my own pace wherever and whenever I feel like it. I have no interest in diving and expect to discover a lot of beautiful things while snorkeling without all the hassle and cost involved with the diving machinery.
• Haggling is said to be important in Egypt, but it doesn't need to be. If you need something, just ask a third person (fellow tourist, camp worker, ...) for a fair price and then buy the thing at approximately that price. With the right people, you'll also get automatic discount for being a regular customer or for speaking a little Arabic. 

Planet of the gears, part one

Fascinated by bicycle hub gears and the vision that this type of gears not only works as part of the hub, but also as part of the chainwheel (for example), I have started to study the subject more deeply. Wikipedia doesn't offer much on the subject. Amazon Germany had a book devoted to epicyclic gears (also known as planetary gears), but it covered mostly aspects that I don't need to understand bicycle gears. Finally I discovered some patents on the subject (all available for free online!) and as I am learning more and more about the subject, I will explain it here in simple terms and with close relation to cycling practice. So here's the first part of what I hope to become a comprehensive and exciting series: the planet of the gears.

In this first part I want to deal with one simple question: what is the transmission ratio of the simplest planetary gear set used in bicycles? The entire complexity of planetary gears (and we'll get to pretty complex arrangements in later posts) can be derived from two formulas which I'll show you after explaining the general setting and the terminology (which I call nomenclatura because I like that word). First of all, I follow the literature by reserving the word “gear” for the cog wheels, that is, physical parts of the gear system. The different settings for transmission ratios which in ordinary English are called first gear, second gear, and so on, will be called first speed, second speed, and so on, so we don't confuse them with the parts of our gearbox.

Now, to the Input and Output parts of a simple planetary gear system. There are three shafts which can be used to transmit a force: the sun gear s, the ring gear r, and the carrier c of the planet gears. If all three are used, the gearing adds two inputs to generate an output or vice versa. To use the gearing as a transmission to translate rotational velocities, one of the three possible shafts will be fixed. If we'd fix the planet carrier, then the planets wouldn't revolve around the sun any more and we had a pretty boring non-planetary transmission. There might be reasons to do this in a practical setting (because it could yield an additional speed), but for the calculation of planetary transmission ratios, we do not need to consider it. We'll either fix the ring gear or the sun gear.

Nomenclatura: Big letters S, P, R denote the number of teeth of the sun gear, planet gears, and ring gear respectively. (The planet carrier itself does not have any teeth.)
Small letters s, c, p, r denote the rotational velocities of those gears and the carrier.

Formula of stationary gears: For two spur gears (that is plain, ordinary cog wheels) with teeth numbers Y and Z and rotational velocities y and z which are engaged, the ratio of rotational speeds is the inverse of the ratio of number of teeth, that is, y/z = Z/Y, or y×Y = z×Z. If one of the gears is a ring gear, a minus has to be thrown into the formula like that: -r×R = z×Z.

Stationary transmission ratio: The weird thing about planetary gears is that gears are not just turning around their shafts but the shafts themselves are moving in space. In order to calculate the transmission ratios of planetary gears, we will first assume that those little planets are not actually moving. We will imagine that we –as the observer– are sitting on the planet carrier and from our relative position the planets do not move (but they still rotate). The transmission ratios observed from this viewpoint are called the stationary transmission ratios.

More Nomenclatura: Superscript x^y denotes rotational speed of shaft x when observed while sitting on shaft y. (This will be an actual superscript as soon as I have found a volunteer who'll edit my blog.)

Formula of translation: z^y = z^x - y^x
Let me explain this formula with a picture: imagine X sits on the curb of a street, y sits on the shoulder of somebody who's walking by towards North, and z sits on the shoulder of a cyclist, also going North. From y's point of view, x is moving southwards and z is moving northwards (assumed it's faster than y) with a speed just slower than seen from x.
As a corollary y^y = y^x - y^x = 0.

Let's apply the formulae: in the stationary case, we observe every velocity from the planet carrier, thus all velocity variables get superscripted with c. The sun gear engages with the planets, thus: s^c×S = p^c×P. And the ring gear engages with the planets, thus -r^c×R = p^c×P. Since we are not interested in the rotational speeds of the planets themselves, we can fuse the two equations to get s^c×S = -r^c×R. Additionally we know c^c = 0.

Now let's look at the case of a fixed sun gear. We want to translate all values x from x^c to x^s, thus we apply x^s = x^c - s^c. Since ring gear and planet carrier are our in- and output, we want to derive the ratio r^s/c^s which equals (r^c - s^c)/(c^c - s^c).
Now we fill in what we know from the stationary case, namely c^c = 0 and r^c = - s^c × S/R.
Thus r^s/c^s = (- s^c × S/R - s^c) / ( - s^c ) = S/R + 1.

I hope that even if you got lost a little in the middle, you'll appreciate the simplicity of the result gear_ratio = r^s/c^s = S/R + 1 which we derived from the simple axioms y×Y = z×Z and z^y = z^x - y^x.

Taking the coarse bounds 0 < S < R, we find that we can use this simple planetary layout to get a ratio 1 < δ(r, c) < 2, that is at most double or half the speed. (How close we can get to 1, that is, what the smallest possible gear step is, depends on some further mechanical parameters.) In the next post we will see, how different usages of this simple gear can be used to build a two-speed gearing (as does the Schlumpf speed-drive) and even a three-speed gearing (as do the three-speed hubs from F&S and Sturmey Archer invented a hundred years ago).


Now the case of a fixed ring gear
s^r = s^c - r^c = s^c + s^c*S/R
c^r = 0 - r^c = + s^c*S/R

δ(s, c) = s^r / c^r = (1 + S/R) / S/R = R/S + 1
rough bounds 0 < S < R
so 2 < δ(s, c) < ∞

Now it is theoretically interesting that we can make gears with ratios from 1 to 2 and from 2 to ∞ and thereby cover the entire possible range (with a small gap at 2, meaning we can't have a transmission ratio of exactly or close to 2).

A ratio of more than double is usually impractical for a single gear step, but this arrangement can be used well in combination with the other one. A simple example is Schlumpf's Mountain Drive which is designed to work in combination with a rear derailer. Since the planetary gear's ratio is so big, the arrangement will spread the available gears further out and avoid gear overlap, that is, more effective gears with less logical gears to shift. We will later see, how two planetary stages can be combined to create a staged gear arrangements whose gears can be shifted in a single sequence with a single shifter and no overlap.

Here are some interesting things to cover in the future:
 - How are multiple gears actually switched?
 - How do the traditional 3, 5, and 7 gear hubs work?
 - How do the newer 4, 8, 11, and 14 gear hubs work?
 - What other improvements can be made to a simple gear box: shifting under load, saving weight, increasing reliability, and much more!