|
Grootte: 2180
Commentaar:
|
← Versie 25 sinds 2017-03-11 14:06:07 ⇥
Grootte: 0
Commentaar:
|
| Verwijderingen worden op deze manier gemarkeerd. | Toevoegingen worden op deze manier gemarkeerd. |
| Regel 1: | Regel 1: |
| Involute - or evolvent - curves are used as the shape of teeth of gears. For an introduction to more advanced math, see https://en.wikipedia.org/wiki/Involute. <<TableOfContents()>> = What is an involute curve? = The quick and dirty example in the picture says more than words. {{attachment:involute_q_and_d.jpg||width=300}} {{{Involute: a quick and dirty example}}} In plain language. A cord is coiled around a cylinder. At the end of the cord is a pencil. The cord is unrolled while the cord is held taut. The pencil draws a curve. This curve is called an involute. {{attachment:involute_basic_0-4pi.svg}} {{{Involute curve with x- and y-axis, without base circle.}}} = Practical use = The main reason for using gears with involute shapes is that there is no friction between teeth - against common believe. The teeth roll on each other when rotating. It is the gear application that makes a polar calculation based on radii more useful than a Cartesian calculation based on (x,y) - we are only interested in the first small part that starts from the base circle. On the other hand, a polar approach has its limits, see below. = Mathematics = Angles are in radians, 360 degrees equals (2*pi) radians. Overall, see illustration: * The blue curve that is described is the involute. * Points on that curve have coordinates X3 and Y3. * From those points runs a cord A1*R1, tangent to the base circle. * The base circle has a radius R1. In detail: * The first triangle: * Start at point (0,0) * Hypotenuse R1 * The angle is A1 * X1 = R1 * cos (A1) * Y1 = R1 * sin (A1) * The second triangle: * Start at the end of R1 * Hypotenuse is A1*R1 * A1*R1 is perpendicular to R1 * Therefore, the angle is again A1 * X2 = R1 * A1 * sin (A1) * Y2 = R1 * A1 * cos (A1) * About the length of hypotenuse A1*R1: * If A1 goes all the way around from 0 to 2 * pi radians (is 360 degrees) than the length of the cord changes from 0 to 2 * pi * R1. After all, the circumference of a circle equals 2 * pi * R1. * The length of the cord is thus equal to the product of A1 and corner radius R1: A1 * R1. |
