12
Jul
06

Involute Gears


Involute GearsThe ideal profile for a gear tooth is the involute of a circle. Here is some Mathematica code:
IR = 0.875; OR = 1.125; tmax = Sqrt[OR^2/IR^2 - 1]; theta := Abs[tmax - Mod[t, 2tmax]]; sign := 2Floor[Mod[t/tmax, 2]] - 1;
Rotate[{x_, y_}, theta_] := {x Cos[theta] - y Sin[theta], x Sin[theta] + y Cos[theta]};
ParametricPlot[IR Rotate[{Cos[theta] + theta Sin[theta], sign(Sin[theta] - theta Cos[theta])}, Floor[t/(2tmax)]Pi/5 + 0.109152sign], {t, 0, 20.001tmax}, AspectRatio -> 1];

See also my gear orbit trap.

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