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Math Help - Help with Chords - Gear generating geometry

  1. #1
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    Help with Chords - Gear generating geometry

    Not sure where to start with this question so here goes. If anyone can help or direct me onto other resources I'd much appreciate it !!

    My question relates to gearing and the development of their geometry with basic tools. In particular how an 18th century clockmaker by the name of John Harrison developed his distinct type of gearing known as Chordal Pitch. As the name suggests he based his geometry on a chord and used it to develop the pitch (distance between the teeth) of the driving gear and the pitch of the mating driven gear.

    I'm sure he only had rudimentary tools to work with, so my question is: If you had to accurately scribe two mating gears with only a compass and rule how would you do it ? How would you accurately divide a circle to produce say 60 divisions, then assure the mating gear shares the same chord length (pitch) in a mating gear of say 8 divisions ?

    My math/geometry is abysmal but I can generate the geometry easily using CAD by dividing a circle (driver) then generating a polygon (driven) from 1 division. I'm pretty sure he didn't have a PC so I'm just very curious if anyone can think of a method using basic tools.

    Many thanks in advance.
    Cheers
    Phil
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    Re: Help with Chords - Gear generating geometry

    Quote Originally Posted by inuk101 View Post
    Not sure where to start with this question so here goes. If anyone can help or direct me onto other resources I'd much appreciate it !!

    My question relates to gearing and the development of their geometry with basic tools. In particular how an 18th century clockmaker by the name of John Harrison developed his distinct type of gearing known as Chordal Pitch. As the name suggests he based his geometry on a chord and used it to develop the pitch (distance between the teeth) of the driving gear and the pitch of the mating driven gear.

    I'm sure he only had rudimentary tools to work with, so my question is: If you had to accurately scribe two mating gears with only a compass and rule how would you do it ? How would you accurately divide a circle to produce say 60 divisions, then assure the mating gear shares the same chord length (pitch) in a mating gear of say 8 divisions ?

    My math/geometry is abysmal but I can generate the geometry easily using CAD by dividing a circle (driver) then generating a polygon (driven) from 1 division. I'm pretty sure he didn't have a PC so I'm just very curious if anyone can think of a method using basic tools.

    Many thanks in advance.
    Cheers
    Phil
    1. A remark in advance: I'm not a clockmaker and the only teeth I'm dealing with are my own.

    2. You can produce congruent sectors of a circle by consecutive bisecting the central angle:

    • Starting with quarter circles you can get 8, 16, 32, 64, ... sectors.
    • Strating with a regular hexagon you can get 12, 24, 48, ... sectors


    3. Compare 2 cogwheels with the radii R (large) and r (small). The large cogwheel has 64 teeth, the small one has 8 teeth. Then you know that

    \displaystyle{\frac1{64} \cdot 2\pi R = \frac18 \cdot 2\pi r~\implies~\boxed{\frac Rr = \frac{\frac18}{\frac1{64}} = 8}}

    That means the ratio of the number of teeth correspond to the ratio of the radii.

    4. The active radius of a cogwheel consists of the radius from the center to the base of the teeth plus half of the length of the teeth.

    5. The flank(?) (I mean the side of one tooth where it touches the side of a tooth of the other cogwheel) must be formed as the envelope of the motion of one tooth. (As you may have noticed I definitely reached the limits of my English) See attachment.

    EDIT: Have a look here: Involute gear - Wikipedia, the free encyclopedia
    Attached Thumbnails Attached Thumbnails Help with Chords - Gear generating geometry-zahnflanken.jpg  
    Last edited by earboth; July 28th 2012 at 05:08 AM.
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