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Gear Ratios to Gear Tooth Count

  1. Mar 15, 2015 #1
    Hello Ladies and Gents,

    I am look for some help with designing the gear trains for an orrery (a mechanical model of the solar system) I am planning. I need some advice on how to go about calculating the specification of the gears in the mechanism which I will machine. I will use cycloidal teeth

    I am struggling to understand the relationship between the gear ratio; which I understand to be the "ratio of the angular velocity of the input gear verses the angular velocity of the output gear" and the number of teeth a gear must possess to honour that ratio.

    I can’t find a clear explanation of how to determine the appropriate number of teeth a gear must possess to honour a specific ratio. All of the examples on the Forum have a known number of teeth on one or more of the gears in the train.

    For example:

    As in the diagram below; I need the gear that drives the moon (GEAR A) to rotate 13.36838 times per 1 revolution it makes of the fixed ring gear – but how many teeth do I need to reach this outcome. I can guess that 12 teeth on the moon gear and 156 teeth on the fixed ring gear will cause 13.0 revolutions of the moon gear per 1 revolution the arm makes of the fixed ring gear…

    But this does not get me to the 13.36383 revolutions I need and nor do I find this method repeatable for more complex/less favourable ratios.

    What is/is there a formulaic method to calculate exact gear ratio into teeth counts?

    Many thanks for assistance rendered.
  2. jcsd
  3. Mar 15, 2015 #2


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    The determination of tooth numbers to exactly develop a specified ratio is a tricky subject not deal with in many places at all. It is really quite an art. I can direct you to one (and only one) book that I know of that deals directly with this subject. It is
    Mechanics of Machines
    by S. Doughty
    available from Lulu in a paper back version. This topic is dealt with in some detail on pp. 200 - 209 of that text.
  4. Mar 16, 2015 #3


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    you know that 156/12 = 13

    you need to solve x/y = 13.36838
    where x and y are the smallest integer solution.

    Unfortunately, your moon gear will require fifty thousand teeth...

  5. Mar 16, 2015 #4
    Thanks for the replies chaps. I see from the rationalization that it is very hard in practice to make accurate systems that are practically machinable. I take it people have always just dealt with this error in clockwork mechanism or designed units of measure that divide rationally into one another.

    As a thought, and your feedback welcome - would using pulleys and belts be a more accurate way to achieve finer scale measurements (albeit in low torque applications). I'm thinking that the circumference of the pulley could be an infinitely finer scale (as fine as the resolution of ones equipment) and would allow for non interger ratios that would otherwise restrict one when using toothed gears?

    Many thanks also for the reference to Doughty. Will follow that up as no doubt a useful text.

  6. Mar 16, 2015 #5


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    This article describes how an Orrery can be constructed from a Meccano set (or an Erector set for US readers).

    http://www.sis.org.uk/bulletin/94/meccano.pdf [Broken]

    The details of construction give the specification of the gears required to simulate the orbits of the planets (there are quite a few gears involved).
    Last edited by a moderator: May 7, 2017
  7. Mar 16, 2015 #6
    The solution proposed by Billy_Joule is based on the assumption of a single gear pair. A much better approach is to several compound pairs, which allows you to come much closer with realistic tooth numbers. It may still not be possible to exactly hit the required ratio, but it should be possible to be quite close.
  8. Mar 16, 2015 #7

    jack action

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    If we try the solutions proposed by Dr. D and billy_joule, you can define a set of «correcting» gears, with an exact ratio of 13/12, for example. That means that your gear ratio needed is now 13.36838 / (13/12) = 12.340043. Putting 12.34 in Wolfram, you get a ratio of 617/50.

    So combining both ratios, we get 617/50 * 13/12 = 13.3683333333 or a -0.000349 % error.

    You play around with different sets of «correcting» gears (14/13, 15/14, 16/15, ...) and when you get to 20/19, you find out that you need a ratio of 12,699961 (or 12.7) which can be obtained with the ratio 127/10. 127/10 * 20/19 = 13.36842105... which has an even slightly less error (+0.000307 %) than the previous set. But more importantly, 127/10 = 254/20 and 254/20 * 20/19 = 254/19 ! So only one gear set needed !

    The fun is to play around with these numbers until you have the precision desired. You can also add multiple sets of «correcting» gears.
  9. Mar 16, 2015 #8


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    Jack, are you proposing a single stage with 254 teeth on the gear? I don't think that will work out too well. Please let me know who is going to cut that gear for you.
  10. Mar 16, 2015 #9
    The principal of correcting ratios is very cleaver. I think 254 teeth might be pushing it a bit. Can this principal be applied to compound gear trains and if so would you be kind enough to show a worker example?

    How would one add multiple sets of correcting gears As you suggest?
    Last edited: Mar 16, 2015
  11. Mar 16, 2015 #10

    jack action

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  12. Mar 16, 2015 #11
    Hi Jack,

    So when you developed the 127/10 which is a ratio of 12.7 and multiplied it by 20/19 which is 1.05263 to get 13.3634 what does this do to the gear train in real terms?

    Is there now a 127 toothed gear meshed to a 10 toothed gear and then a 20 tooth gear on the same shaft as the 10 toothed gear, the 20 tooth gear then meshes a 19 tooth gear on a separate shaft?

  13. Mar 16, 2015 #12

    jack action

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    Yes, exactly.

    Here's one with 3 gear sets, each with a ratio of 3:1 (or 1:3, depending on which end is driven). Total gear ratio is 3 * 3 * 3 = 27:1 (or 1:27):

  14. Mar 17, 2015 #13
    Thanks again Jack, unfortunately this is sinking in quite slowly with me and I have another question related to the above example where we have the 127:10 and 20:19 compound gear chain giving us the desired 13.36 ratio.

    Let's imagine for whatever practical reason that a 127 tooth gear is to large (it likey is not, but I'm bound to run into this problem soon or later). So is it possible to turn the 127:10 into a compound gear train that can drive the 20:19 at same ratio of 12.70 (that being 127/10 driving 20/19 giving the overall 13.36)? The output gear of the complete compound train (if correctly calculated) should be turning 13.36 times faster than the input or vice versa depending on the driven-driver - correct?

    If the above statement is correct the I can go about making a compound gear train with a 12.7 :1 ratio, but 127/10 has no (useful) commonly divisible values to give whole integers - I think only 1 is common.

    So to make a ratio of 12.70 do I apply the same method as we discussed early (correction gear)?

    127/10 = 12.7

    12.7/(e.g.13/12)= 11.72 = 293/25

    (293/25)* (13/12) = 12.7 (ended up with a larger gear than I started with)

    Neither 293/25 nor 127/10 can be simplified so with no common divisible factors I can't see how to get a smaller compound drive train to reduce the practical size of the 127 toothed gear.

    I'm probably confusing myself here but any lead from here would be great,
  15. Mar 17, 2015 #14

    jack action

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    First, I'm no expert in gear making. It is just a fun math problem for me to solve. You should take a new look at the first video I posted about the wooden gears for a clock, it is very instructive.

    Now here is how you can apply it to your problem.

    To get the smallest ratios you have to use the roots. For example, for 2 gear sets, each set must have a ratio of 3.6562795... [= (13.36838)^(1/2)]. For 3 gear sets, each set must have a ratio of 2.373338... [= (13.36838)^(1/3)]. And so on. Note that the more gear sets you have, the smaller is your «average» ratio, so less difference in the number of teeth between the driver and the driven gears.

    But all of these won't give round numbers since you don't start with one to begin with. So let's say you are satisfied with a ratio of approx 3.6562795. Choose the first «round» ratio below that, say, 3.5. This is 7/2 = 14/4 = 28/8 = 56/16, depending on what you think your minimum number of teeth should be.

    Now the second gear set should be 13.36838 / 3.5 = 3.819537. Again not a round number. Again we take a lower ratio than this one, say, 3.6666666. This is 11/3 = 22/6 = 44/12. The gear set ratio that is left is 13.36838 / 3.5 / (11/3) = 1.04169.... This is our «correcting» gear set ratio. Multiply that ratio with different number of teeth (12, 13, 14, etc.) and you find out that when multiplied by 24 it gives 25.00061 teeth needed to achieved that ratio.

    So 56/16 * 44/12 * 25/24 = 13.3680555555... or an error of -0.0024 %. If that is not enough precision for you, you can always play around with other ratios.
  16. Mar 17, 2015 #15
    Thanks very much for taking the time to reply in such detail jack, you're obviously a very proficient mathematician - I on the other hand am a bit slow with picking it up especially concept that are Not as formulaic in approachs as perhaps any area such as mechanics. Let me work thought the example above and try it with some other numbers and let you know how I go. I hope you won't mind if I drop back with some questions on this
  17. Mar 17, 2015 #16


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    I don't have a good understanding of the application for this gear train, so let me simply make a couple of general comments.

    1. In making wooden gears, you can be somewhat sloppy; why not? You cannot form the wooden teeth with great precision anyway. If you want real precision in the resulting motion, you cannot play quite as fast and loose with the center distance as Matthias does in his video. The more you open up the center distance, the lower the contact ratio goes. A contact ratio significantly greater than 1.0 is necessary simply to assure that there is always at least one tooth pair engage. Without that assurance, there will be points in the motion when there are no tooth pairs engaged. Also, lowering the contact ratio will make for a more noisy gear set (but for wooden gears, who cares?).

    2. According to the AGMA, to avoid undercutting on the teeth, the pinion must have at least 18 teeth for a 20 deg pressure angle and 12 teeth for a 25 deg pressure angle. Thus a 10 tooth pinion is assured to undercut. This will (1) weaken the tooth, and (2) result in non-involute action at points in the motion.

    Will wooden gears work for this application, or do you need precision cut metal gears? I don't know, but you need to be aware of these potential pitfalls.
  18. Mar 17, 2015 #17
    Great additional information OldEng63, that really helps me with the finer design considerations. I'll just summarise a couple of things for you information and fire off some questions that result. So firstly;

    I'm planning to CNC the gears on a Sherline lathe/mill system out of 3.175mm (or 1/8" in old money) thick brass sheet so, obviously hoping for very little play and an accurate mechanism.

    I'm aiming to use a cycloidal tooth profile, which research has lead me to believe can tolerate a higher pressure angle that results from smaller circumference/less teeth - does that fit with your knowledge?

    I wasn't planning to producing cycloidal gears with less than 18 teeth (although I think I can get away with a 12 tooth cycloidal); Where less than 18 teeth are requiring I was going to fabricate a lantern gear out of two brass end plates and thin steel bar stock (~<1mm dia) connecting them - I think the lantern gear can cope with both the increased pressure angle and otherwise reduced contact ratio???

    The centre to centre distance and resulting contact ratio is not something I've studied in detail yet, but I understand this to refer to a ratio that represents the average number of teeth of two meshed gears in contact at any one time. So essentially the more gear teeth are meshed the smoother and more reliable the systems is?

    So could it be said that having two gears meshed with 25 teeth and 24 teeth would produce a higher contact ratio that 10 teeth meshed with 150 teeth?
  19. Mar 17, 2015 #18
    One of the fundamental requirements for a satisfactory gear pair is that the angular velocity ratio be constant. If this is not met, with the driving gear turning at constant speed, the driven gear with speed up and slow down repeatedly as the teeth go through engagement. This makes for noise and for torsional vibration in the gear train.

    The spoke and lantern pair does not have a constant velocity ratio.

    Cycloidal gears have a constant velocity ratio only when mounted on exactly the correct center distance. This is a difficult criterion to meet, and gives you no leeway in the shaft center distance assignment. Cycloidal gears are virtually unheard of outside the watch and clock industry, and for good reason. Involute gears have many advantages, and I would certainly urge you to go with the involute profile.

    It seems that you are demonstrating why gear train design is usually left to folks who have studied it in considerable detail and know a lot about the subject.
  20. Mar 18, 2015 #19

    jack action

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    Before you go too deep into gear design, remember that most of the theory out there is for making gears that can transmit large loads and/or high-precision positioning and/or minimal wear and/or minimal noise. But if you are making a simple toy (no offense), where only relative positioning of two shafts is of importance, you can be a lot looser on the criteria. IMHO, you could simply used cage gear and it would be fine. Otherwise, it can get crazy and if you are not math-oriented, you won't like it. Here's an equation for contact ratio (http://psas.pdx.edu/lv2cguidance/contact_ratio/ [Broken]):

    http://psas.pdx.edu/lv2cguidance/contact_ratio/fcc0c10c2303e33528223039a2453320.png [Broken]

    For more info on gear design, I like http://www.thegears.org/spur-and-helical-gears-for-parallel-shafts-design-theory/ [Broken] and you got some good stuff here too.
    Last edited by a moderator: May 7, 2017
  21. Mar 18, 2015 #20
    Or, for that matter, for a toy perhaps just use frictions wheels and forget about tooth forms altogether.
  22. Mar 18, 2015 #21
    Jack I’ve been working through the methods you set out in your post yesterday and this technique works amazingly well. I’ve been able to identify multiply combinations of favorable gear sets to drive the motion of the inner planets and moon of earth to the desired accuracy. I had to set up a matrix of sorts in Excel to help me spot the factorization and also speed the selection of the correcting gear set ratio, but it’s done the trick! Really great practical advice and well explained, thanks so much.

    Dr D thanks also for the insight into gear tooth profiles and associated parameters. I am aware of the limited contemporary use of cycloidal gear profiles, save horology, but remember a orrery is essentially an astronomical clock, albeit a relative one. I was drawn to the cycloidal gears on the basis of the larger reductions they can facilitate compared with involuted gears; similarly I understand a cycloidal gear is less likely to undercut for a smaller given number of teeth (?). Also I’m seeing that cycloidal gears are more adept at functioning without lubrication and typically run with less friction. As Jack also points out there will not be significant amounts for force being transmitted through the gear train. I’m planning to centre gears on 1/8” steel shafts with milling to 0.001” so hoping centre to centre distance should not cause me strife. The issue with centre distance only causes problems when there is play in the gear mount right? So if the addendum and dedendum arcs are correct (allowing enough clearance - the BS 987 (see below) suggests using 95% of the theoretical addendum height to account for machining tolerance and centre to centre diameter shift) there should be a constant velocity ratio between cycloid all gear pairs?

    Never the less I’ve been working on ascertaining the contact ratios and my understanding thus far is that as long as the contact ratio is > 1 in the worst case, then the angular velocity should be constant between the two gears (?).I’ve been basing my calculations and design of cycloidal gears on an article by Hugh Sparks (2013) which draws from the British Standard for Cycloidal gears (BS:978). Sparks does a very good job of explaining the design consideration and associated calculations of colloidal gears in plain terms - very useful piece of work.
  23. Mar 19, 2015 #22


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    There are two parts to the center distance problem:

    1) There is the error due to machining inaccuracies -- this can be managed by careful machine work.

    2) There is the situation where we would like to deliberately open up center distance in order to achieve desired shaft center line positions -- this can be easily done with involute gears, but is not tolerated at all by cycloidal gears.
  24. May 10, 2015 #23
    Sorry to arrive late to the party.
    I did not find this particular blog back in March... I was certainly looking for any questions to try and see if there was sufficient interest in this topic. Plus during February, March and April I was busy fine tuning my proposed gear calculator. This was my first major project to implement while learning JavaScript.

    Unfortunately there is no formulaic method to convert an arbitrary ratio to gear train teeth numbers. Mathematically, the process of determining gears is labour intensive. It belongs to the class of problems of NP-hard computational complexity. That is, no shortcut equation exists that can churn out solutions.

    Around 150 years ago French clockmakers formalized the finding of arbitrary fractions by using a Stern-Brocot tree. This is a structured approach to finding fractions which are close to any desired number eliminating hours of guesswork. Another related method is the use of continued fractions.

    The subsequent factoring process of such fractions into usable gear tooth counts would then be applied. There were no clever techniques to speed that part up.

    It seems that even in today's scholarly machinist's textbooks, this is as far as they go. The factoring part - choosing the gear teeth numbers is ultimately a trial and error exercise after a number of candidate rationals have been found.

    I was determined to find a solution that's practical on today's computers. By "practical" I mean that it should do the calculations reasonably quickly for 2, 3, or even 4-stage gear trains.

    I believe I have achieved a workable gear train calculator. Possibly the first of its kind on the internet.

    See Gear Train Calculator
    For the example for ratio 13.36838 it recommends gears 31:71,22:59,17:37 which gives a ratio accurate to 6 decimal places: 13.36838019665344
    Mark111's original post did suggest a planetary gear arrangement... so the ratio might have to be modified as the arm's rotation adds another full rotation to the planet gear as I understand it.

    Although late to this particular thread, I hope that it can be useful for gear problems others may encounter. It's free. I'm not selling or pushing anything. Optimizations, algorithms, "difficult" problems is my hobby.

    I'm just a retired dinosaur programmer that's putting his skills to good use instead of evil. :)
  25. May 10, 2015 #24
    That is a very interesting program, and you are most certainly correct about the difficulty of the problem. Have you (or do you plan) to publish your approach? I would be very interested in knowing more details about how you solved this problem.
  26. May 10, 2015 #25
    Thanks for your interest.
    I could certainly go into how I "solved" the problem.
    Not sure if this is the appropriate thread to do that in... Maybe a new thread? Or on my site I could put up a "theory" page.
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