Dear all, I have been a member of NAWCC for many years and have now built a few clocks from scratch but always from plans. Over the years I have tried to understand the complexity of gear trains, but I am despirate to now make my own clock from my own plans and so gear trains are something I will need to have crystal clear in my mind. Please read through my previous threads before asking me to do my own research – I think it will be clear that I read an aweful lot, but I still do not understand how to calculate teeth numbers and ratios! I am quite happy calculating the wheel size and tooth shape given the number of teeth required, but not calculating the number of teeth required for gear ratios. I could explain what I understand and what I don`t, but I think an example is best. I have here a very simple clock. It has: Escape wheel – 30 teeth, pinion 8 leaves Intermediate – 60 teeth, pinion 8 leaves Minute Wheel – 64 teeth, pinion 18 teeth Drive wheel 60 teeth Clearly this gear ration works as it is a running clock! What I can`t see is why and in all honesty, what is trying to be achieved other than a full rotation of the minute wheel once per minute! I have done calculations based on the ratio of gearing and can calculate that the drive arbour turns one rev for 195 revs of the escape wheel….but this doesn`t show me anything! I guess I`m missing the period of the pendulum (double pendulum on this clock) controlling the escape wheel and I should be working back that way? I`d really appreciate any help on this. When deciding on the teeth counts, what exactly are you looking for and how would you work backwards from the clock I have described to show “it works” Any help appreciated. Chris

Hi Chris There are several places you want to think about it. If you want to fit it into a case, you need to think about the pendulum length. Do you want a seconds hand? What source do you want to power it? How many turns will it take to run it for the time required? It is as you think, matching the pendulum to the hands is how you select the total ratio. The size of the pinions helps to optimize the efficiency. More teeth better efficiency. Six is the minimum you could use. 8 or 10 is better. If you have a 1 second beat pendulum and you want a second hand on the escapement wheel, you'll need a 30 tooth escapement. You'll need a ratio of 60:1 to the minute hand. You can choose to do that with one big wheel or more. You have to decide if you want the hands to rotate the same direction ( cw ) or it you don't care. Going from the minute to the hour is a pain because the pitch of the gears will need to be different to make the ratios work. Typically a 4:1 and a 3:1. If you want to make separate hands you can use the same teeth cutters. If on the same shaft you could do the 12:1 as a single gear and then a 1:1 back to the hour. Since it isn't much load, a 5 tooth pinion could be used with a 60 tooth wheel. Tinker Dwight

Hi Tinker, I appreciate your reply. I`m not really concerned with the size and number of leaves on the pinion, how long the pendulum is or if it will fit in a case – quite simply the calculations, how the gear ratio is worked out to calculate that it WILL make a working clock. Even if the example is not practical for a working clock – I want to know how the gearing will “work” and more to the point, what the calculations involved are in making it work. Chris

you might find http://www.atmos-man.com/clockaid.htm useful for calculating gear trains.....the basic formulas are all over the web, try http://www.ppthornton.com/pdf/ppthornton_brochure_2012.pdf for more details...

You need to talk about energy lost in the train. This is not something that is easy to calculate up front. It can be estimated. The quality of your pinions and wheels will determine what you'll need. There are tables and rules of thumb. Other than that, there isn't much to it that you don't already know. You know what the ratio have to be between the minute shaft and the pendulum. How you arrange the gears to achieve that is really up to you. That is the best I can do for you. Jim has some pointers to some stuff that might help. Tinker Dwight

Jim, I'm a little confused by the links, he ppthornton link is calculations for working out wheel measurements once the required wheel is known. The other link is a calculator which I can't see the calculations it is actually doing.,, which is what I want to understand.

Let me ask another question then or in a different way if anyone can help... And I do appreciate all comments. In my original post I listed the wheel in a working clock, I said for example, the intermediate wheel had 60 teeth. If I changed the design to 64 teeth - obviously planting have train differently to fit the new wheel size, other than adjusting the pendulum period, will this still work??

Beats per minute from the full train wheel teeth numbers and pinion numbers and how from this, it decided what pendulum length is needed. For the wheels I listed in my original post, I apparently get a 60 beat per min clock with a one second pendulum, how??

Pendulum length in inches = L beat/sec = B L = 39.1 SrRt( 1 / B ) Each beat on the escapement wheel counts for 2 beats ( once each way ) To calculate the ratio, if it is towards the pendulum, you multiply. If it is towards the minute shaft you divide. In your example 2*(30/8)*(60/8)*64 is the ratio to the minute shaft. 3600 is the end number. This gives a 1 second beat. To know what ratios might work, you need to calculate the various options. If you know the desired total ratio, you can look at the possible selections of wheels and gears. For this, it would be best to setup a spread sheet. You need to consider that you can't make a wheel hit an arbor if the gears are all at the same pitch. As an example: if you had a 8 tooth and a 60 tooth, the radial distance between arbors would be proportional to 8+60. A gear wheel of 68 teeth would be too large to work because the tips of the teeth would be on the middle of the other arbor. In you case, 8+64 is enough greater than 60 to clear. So, ratio and clearance are two factors that you can use in your spread spread sheet to determine what ratios you might want. In the spread sheet, you might want to show different combinations of ratios and what end beat you get with each. Only put down practical ratios. An arbor with a wheel of 68 teeth and a pinion of 8 teeth is not practical to be used with a wheel of 60 teeth ( at least not the same pitch ). In general, it is best to make smaller wheels towards the escapement to ensure you don't have to saw through and arbor. It is possible, though, to have a secondary plate or cock to allow the 68 toothed wheel to work with the 8 toothed pinion and 60 toothed wheel. Just keep in mind that some ratios just can't be made with an integer number of teeth. Practical pinions need to be 6 or 8 to carry loads. 5 could be used at the last stage of a fly where friction is a good thing. 10 is fine. If you put the minute wheel off to the side of the drive train, just realize that this causes an extra reversal of the wheels. A second hand on the escapement would go CCW unless you added another set of gears or idler to reverse it back. Does this help a little? Tinker Dwight

Wow! Thank you... I need to digest the information and I will be back shortly! Very very much appreciate your time and effort in your reply. chris

Here is another way to look at tooth selection: If we factor what you had it would look like this: 2*(5*3*2)*(5*3*2*2)*(2*2*2*2*2*2)/((2*2*2)*(2*2*2))=3600 We can't play with the first two because the escapement to pendulum is always 2. So we have: (5*3*2)*(5*3*2*2)*(2*2*2*2*2*2)/((2*2*2)*(2*2*2))= 1800 To clean things up a little, we can get rid of a lot of 2's: (5*3*2)*(5*3*2*2) = 1800 Now we might say we want one of the pinions to be a 10 tooth. How might we rearrange the numbers to go with one 10 tooth and one 8 tooth. 1: 5*3 = 15, 2*5*3 = 30, 2*2 = 4 2: 5*2 = 10, 3*5 = 15, 3*2*2 = 12 3: 5*3*3 = 45, 5*2 = 10, 2*2 = 4 4: 5*5 = 25, 3*3 = 9, 2*2*2 = 8 5. 5*3*5 = 75, 3*2 = 6, 2*2 = 4 A few more that don't look practical Now we need to select a size for the escapement wheel. I would think 25,30 and 45 are our best options. This leave combo 1,3 and 4. Now lets multiply our pinions to the selections. For # 1 we have 30 tooth escapement and A. 15*10 = 150, 8*4 = 32 or B. 15*8 = 120, 10*4 = 40 For # 3 we have a 45 tooth escapement A. 10*10 = 100, 8*4 = 32 or B. 8*10 = 80, 10*4 = 40 For # 4 we have a 25 tooth escapement A. 10*9 = 90, 8*8 = 64 or B. 8*9 = 72, 10*8 = 80 Now which ones making the wheel size smaller towards the escapement we could do for: center, wheel/pinion, escapement/pinion, #1A 150, 32/10, 30/8 #1B 120, 40/8, 30/10 ( not as good as 1A because pinion getting larger ) #3A 100, 32/10, 45/8 ( won't work 32+8 < 45 unless the pitch of the 45 was smaller. #3B 80, 40/8, 45/10 ( 40+10 > 45 so OK ) #4A 90, 64/10, 25/8 #4B 72, 80/8, 25/10 ( not as good because pinion getting larger ) So, select the wheels you'd like to make. You could do a similar reasoning for one of the pinions being 6 teeth and the other being 8 or 10. This of course all assumed the pitch of your wheels and the escapement was the same. In general, the escapement has a wider pitch so the 45 tooth escapement isn't too practical. Tinker Dwight

I left out one combo that isn't too bad: center to escapement 120, 50/10 24/8 with the escapement being 24 teeth. You can back track the math. Tinker Dwight

(#teeth on center wheel x #teeth on third wheel x #teeth on escape wheel x 2) ÷(# teeth on third wheel PINION x # teeth on escape wheel PINION)=BPH (beats per hour) and there are tables to convert BPH to pendulum length. Generally speaking you do not want to use less than 7 or 8 leaves in a pinion, yes I know less are used in some clocks, the ratio of leaves in a wheel to leaves in a pinion should be about 8:1, I tend to favor escape wheels of 30 teeth, the math and the geometry are easier to deal with than escape wheels with a lot more or less than 30 teeth. It is desirable to make a train geared so as to only require 1 size wheel cutter and one size pinion cutter, unless you have a large selection or are willing to spend a lot of time to make multiples or a lot of $$$$ to buy them.

All good rules of thumb but I think Chris was also looking for how one even knew what ratios would end up working right. Besides rules are sometimes more fun to break. All the one I said would work didn't require more than 2 cutters at most for the 8 and 10 tooth. No reason, one couldn't use all 10 tooth pinions. Tinker Dwight

Generally speaking the more used ratios of clock gearing have been published many times over the last 400 years. The attached scans should provide a starting point for anyone desiring to build their own timepiece. On the whole, pinions with larger tooth counts are preferred over low tooth counts as they have better geometry as to rolling versus sliding friction, but of course that then requires either more arbors in the train and or more teeth on the wheels, and if using one module of cutter then tooth counts in the wheels need to descend while moving up the train, otherwise you may have a bit of an issue with wheels meeting other arbors .....mechanical clocks have been built using only one wheel and at least one design exists using no wheels as we generally speak. Then there is the work of Aaron Dodd Crane who designed, patented, and built some very interesting "gear trains" for all sorts of mechanical devices...

Hi Jim, yes I know all too well about the gearless clock as I built one the other year... http://www.raynerd.co.uk/?page_id=1540 thanks a lot for the attachments. So basically with limited options, there is only realistically the common ratios to choose from!

Tinker: Please can you just explain the issue that you seem to be identifying with 45 ? 40+10 > 45 OK and 32+ 8 <45 not good .... why?? I don`t see where this comes from. I follow it all up to there.

Raynerd, that is some really nice work on the gearless clock, and even better documentation/photos/blog. Commendable on all fronts! The attached pdf is the gear train of something I am working on right (building) at the moment. The are a number of reasons that the gear train consists of 2 different modules as well as using an "extra" 1st wheel in the train...obviously to provide more run time.....the gearing options I selected are also a bit strange but necessary in this mechanism as I am building it. My point in posting this train is to respond to some of your questions of conventional versus non conventional gearing....it can be pretty much what ever you want it to be, as long as ultimately it fits in the overall design you are creating and the train ultimately delivers the BPM or BPH you want. I recall many years ago making a great wheel replacement for a French skeleton clock that was 577 teeth on a diameter of about 8 inches. Why 577 teeth? I have no idea as it would not have been all that easy to index in about 1810 when the clock was created.....point being gearing can be what you want it to be....all the rest is just recommendations based on work done over the centuries where convention generally prevailed over experimentation for most clockmakers.

For simplicity, I'd made the assumption that the pitch or modulus of the escapement was the same as for the other wheels. If so, the diameter is proportional to the number of teeth. the middle wheel would add to the escapements pinion to have a diameter proportional to 38 + 8 or 40. The spacing between the two arbor is 40*K. The escapement wheel of 45 is 45*K. The rim of the escapement would be interfering with the arbor of the middle wheel. One could swap things, making the escapement 32 teeth and the middle wheel 45, as in: 100, 45/10, 32/8 That would work. If you have a wheel someplace that is two small, you can always increase the size of the pinion to make things work out. In the case of 100, 32/10, 45/8, you could have made a 20 tooth pinion, giving: 100, 64/20, 45/8 That would have worked. Since the 32 and 10 both have factors of 2, you could do 3 times 16/5 or 48/15, giving: 100, 48/15, 45/8 48 + 8 is greater than 45 so there is enough space for the escapement wheel between arbors. You only have two rules. The total ratio has to stay 1800 for the one second pendulum and you can't make a wheel larger, higher in the train than will fit between the arbors. It is all about factoring to the prime factors and then arranging them to get what you would like. Of course, you could always use a table like Jim has posted. Using different sized pinions means different addendums for efficient gears ( different tooling ). The one that always is a pain is the hour hand to the minute hand. It is 12:1. That factors to 3 and 4. And the arbors need to have the same centers. This mean two different pitches or do the 12:1 on one wheel, using a second 1:1. To match the pitches, the diameters would need to be the same totals as well. 24:2 and 13:13 would work and then scale up to a useful pinion 72:6 and 39:39, just to use the same pitch. With the one second pendulum, if you wanted the second hand on the escapement wheel it needs to have a 30 tooth escapement. If you used a shorter pendulum, you could move the second hand to the middle wheel. The calculation of the middle would be the same as was done for the minute hand to the pendulum. It has to have prime factor because you can't have half teeth or fractional teeth. Tinker Dwight

The over all ratio worked out nicely with all the prime factor in the numerator. It is possible that the denominator ends up with a factor as well that doesn't easily cancel out. In such cases, the least factored value might leave 7 on the denominator, to get the desired pendulum length. A total ratio of 2314.285714 is just such a total ratio. It would be for a pendulum of 4628.571429 BPH. Still, you have to end up with an integer number of teeth and the diameters of the wheels need to fit between arbors. To get integer teeth, the denominator and numerators need to have prime factor. Tinker Dwight

I'm a new member trying to follow along on this thread. That ratio: 2314.285714:1 would convert to: 575, 350, 395, 218 and 167 tooth wheels and: 50, 96, 191, 44 and 31 tooth pinions. Is this correct? Sorry for the thread intrusion. Best regards, David Malphrus

Yes, that looks to be real close but originally it was a bunch of gears with a ratio of 16200 and a pinion with 7 teeth. You ratios are close enough. Tinker Dwight

Thanks, Dwight. Working with 16200/7 instead of 2314.285714 I get: 460, 540, 490, 288 and 150 and: 80, 92, 90, 98 and 35 --david

You have a lot of values that can be reduced. You can factor out 5s and several 2s and a couple of 3s. That could make the pinions a lot smaller. Tinker Dwight

Thanks on reducing the fractions. Looking at their reduced size, it makes me wonder if Clockmakers strive for hunting tooth arrangements... --david

Occasionally, thought it's been a while, I'll grab the old slide-rule (what's that??). It makes short work of finding an integral solution to a fraction. Johnny

"(what's that??)" I'm old enough to remember them well. All the smart guys had them hanging from their belt in high school (around 1970).

Chris, It looks like you are having lots of fun configuring a clock of your own design. I understand. With all of the good advice from members, did you get one designed? Just a thought if you don't have the resources for multiple wheel and pinion cutters: you can do a lot with standard gears just ordered off of the internet. I built this one without a single traditional clock part (wheel or pinion or anything else).