SH`Gears`

Author  : Stephan Heiss <Heiss@eicn.ch>, May 1999
Version : 1.0

With this package you can visualize the construction of mechanical involute gears.
You can obtain the shape of gears, with or without deport, and you can see an animation of a rack and gear system.

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<<SH`Gears`

Gear characteristics

({{z, number of cogs}, {α,  contact angle  between cogs, 
                   usually equal 20 °}, {m, modulus,  
                  the diameter on which the contact occurs is d =  
                    m*z.}, {e, deport (see below)}})

Interference

Interference occurs when the foot of the cog is break into by the outgoing rack cog during engagement.
For the standard alpha=20° wheel, one has interference for z < 18.

({{minimalNumberOfCogs[α], Return the minimal number of cogs that a wheel  
                    can have  without interference}})

minimalNumberOfCogs[20°]

An alpha=20° and z<18 wheel is not used in high precision mechanisms because the interference cause a break during the engagement that cause vibrations and overheating.
A common solution is to deport (shift) the rack during the machining of the gear.
This permit for the example the manufacturing of alpha=20° and z=11 gear without interference ( e = +0.4,  see below).
The deport e is given in fraction of the modulus m and is positive when the distance between rack and gear increase.

Rack machining of the gear

A common way to produce gears is to use a milling-cutter with the same shape as the rack.

({{animationMachiningGear[z, α, m, e, n], animation of the gear machining by the 
                      rack. }})

animationMachiningGear[19, 20°, 1, 0, 7];

({{pictureMachiningGear[z, α, m, e, n], picture of the Gear machined by therack. The 
                    picture scale  is adapted for a correct modulus 
                    in milimeters on the printer.}})

pictureMachiningGear[19, 20.°, 5., 0., 20];

({{pictureMachiningCog[z, α, m, e, n], zoom of the above picture.}})

pictureMachiningCog[13, 20.°, 50., 0, 10];

({{pictureInvoluteFamily[z, α, m, e, n], \(Show\ the\ trajectories\ of\ a\ family\ of\ points\ 
                    located\ on\ the\ rack\ during\ the\ machining.
	The\ 
                    envelop\ of\ these\ curve\ is\ the\ shape\ of\ the\ cog\ 
                    and\ is\ a\ circle\ \(involute.\)\)}})

pictureInvoluteFamily[16, 20.°, 50, 0.1, 8];

Shapes

({{shapeCog[z, α, m, e], return the half cog shape (as a list of {x, y} points)}, {ShapeFullCog[z, α, m, e], return the full cog shape}, {shapeCogCenter[z, α, m, e], return the shape of the full cog centered around 
                    the y axis}, {compareDeports[z, α, m, e1, e2], produce a graphical comparaison of two  different
                       deports}, {shapeGear[z, α, m, e], return the shape of the gear}})

A superposition of the machining rack and the computed cog shape

Show[
	pictureMachiningCog[12, 20.°, 30, .3, 10],
	Graphics[{Hue[0.31],Thickness[0.01], Line[shapeCog[12, 20°,30, .3]]}]	
];

On the above picture, one can see a small interference on the foot of the cog.
The interference is supressed by deporting the rack during the machining

pc = shapeFullCog[11, 20°, 1, 0.4];

comparaison of shapes

compareDeports[11, 20°, 30, 0.4, 0];

({{animationRackGear[z, α, m, e, n], generate an n images animation of the rack and
                        gear system}})

A 3 frames animation

animationRackGear[11, 20°, 1, 0.4, 3];

Stephan HeissMel:  Heiss@eicn.ch