Wednesday, 18 April 2012

A lecture on gears

Pay attention at the back, there'll be an exam later. I'm looking at you Middleton.

I've needed to learn a little about gears and gearing theory in order to reproduce the geared facile. This has all been a lot of fun (we may have to disagree on this point, I'm a mathematician and my idea of fun may differ slightly from the norm) and I need to share my findings. I'm not going to go into great details but a Google search will yield many results on this stuff if you are interested.

Cycloidal or Involute?

The first thing to determine is the type of gear profile, there are two contenders: cycloidal or involute. Historically, cycloidal gears were used almost exclusively up until the late 19th century when the new fangled involute profile quickly became the standard. Cycloidal gears are still traditionally used in clock making and have a devoted following. Involute gears have certain benefits, such as a tolerance to incorrect gear spacing, that makes them attractive to industry. It is easy to distinguish between these profiles by eye.

I've been unable to find an absolute date for when the involute profile was introduced, all references I've seen suggest the end of the 19th century. Of course this is the exact time frame when the geared facile was being produced.

Examination of the original gears clearly suggests that they are involute, 
which means that Ellis & Co. were at the cutting edge of the current technology in 1887.

So now that we have established that the gears are involute, we need to calculate the size of the teeth. For gears in a gear train to mesh accurately they need to have some commonality in the tooth size, this is known as the Diametrical Pitch (Pd), which is the ratio of the number of teeth to the pitch diameter. All involute gears with the same Pd will mesh accurately.

All original faciles that I have measured have 37 teeth on the sun gear and 18 on the planet gear. I stand to be corrected on this but the literature of the day backs this up. The final gear ratio in this simple epicyclic train is calculated as follows:

gear ratio = wheel size in inches x (driver + driven)/driven

with a 40" wheel, 37 tooth driven sun gear and 18 tooth driving planet gear we get a gear ratio of 59.46" or the equivalent of a penny farthing with a 59.46" wheel. Ellis and Co. offered the geared facile in three sizes 36", 38" & 40" and claimed gear ratios of 54", 56" & 60" resp. The actual calculated values are 53.51", 56.48" & 59.46" resp, it seems that Ellis & Co. liked to keep things even at the expense of accuracy.

Note that it is very important that at least one of the gears has a prime number of teeth, this ensures that the gears will wear absolutely evenly. With a pulsed power delivery such as a bicycle, if this were not the case, spots of high wear would rapidly occur where the same tooth was repeatedly under peak load.

So how do we calculate the Pd to be used? We have the number of teeth and we also have the centre to centre distance for the gears, from the literature and also from my own measurements I know that the gears ran on a 3" crank.

Centre distance = ((driver teeth + driven teeth)/2)/Pd

gives us a Pd of 9.16666 teeth per inch

which is unfortunate since commercial gear cutters are always sold as whole numbers (and the odd half size). If I use a Pd of 9, I get a centre to centre distance of 3.056", a little too far off for comfort. Fortunately gear cutters in metric countries use an alternative to Pd known as the module.

Module = 25.4/Pd

so we need a module of 25.4/9.1666 = 2.77

metric gear cutters are available in fractional sizes, the closest to 2.77 being 2.75. If I use a module of 2.75 my centres work out to 2.977", less than half the error of using Pd 9 cutters, also remember that tolerance to incorrect spacing that involute gears have?

The last thing to calculate is the outside diameter of the gear blanks, this is given by the following formula:

OD = (teeth + 2)/Pd

this gives sizes of 4.222" for the sun gear and 2.165" for the planet, which match closely to my measurements of the originals.

gear cutters are available in sets of 8, with a different cutter used for a range of teeth

#8 12-13 teeth
#7 14-16
#6 17-20
#5 21-25
#4 26-34
#3 35-54
#2 55-134
#1 135-rack

So I need to buy cutters #3 & #6 for mod 2.75. The only other variable to consider being the pressure angle which I won't go into. As it happens it was cheaper to buy a full set rather than individually.

I had originally intended to cut the teeth myself but now that I have the cutters in my hand, I've realised that my machinery simply isn't massive enough for the job, these are big teeth. Teeth are usually cut at one pass and require an immensely stiff setup to cut the bigger sizes. I'll contact a local gear cutter when I finish the blanks off. I'd rather get a good job done than screw things up.

Has anybody managed to get here yet without falling asleep?

In other news, I've been for a gentle 50km ride at the weekend. My first since the Le Race debacle. The only bits that still hurt are two fingers on my left hand and my right knee. My right knee was OK after the ride, not great but OK. I still haven't looked at the crashed bike yet. For the ride I selected a late 1940's Raleigh Record Ace (RRA), this was for several reasons: it's very comfortable and I still have a pair of shoes for the toe clips that aren't munted from a recent crash.

A late 1940s RRA. Red, of course.

Not disco slippers.

In more other news, Mr. Middleton and partial family have been for a visit. I was unable to provide an earthquake which was a source of some disappointment. However Mr. Middleton was able to console himself by stealing some of my 4130 seamless tubing. I think he is going to use it to make something from this book written by somebody or other.

Also, Tinkerbell wants to come and see my magic cupboard. Actually, she wants help building up a faux pathracer and I happen to have lots of bits that she needs. Sadly, she intends to paint it british racing green!

Seriously though, there is only one colour suitable for bicycles and that's red[1]. We all know that.

[1] Unless it's black.

Oh, and I lied about the exam.

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