Circle Volvents

Interlocking gear teeth
Author: Claudio Rocchini

A circular involute is a flat geometric curve, a special involute with a circle as an evolute . It is of considerable importance in involute gearing in mechanical engineering , where it occurs as the tooth flank of gears . It is clearly the path of a thread end when the thread is unwound from the circumference of a circle.

properties

The involute of a circle is a spiral with a constant pitch . This property is often wrongly attributed to the Archimedean spiral . The involute of a circle is thus its own parallel curve .

Mathematical representation

The parametric representation of the involute of the unit circle , which starts with the initial slope, is : ${\ displaystyle (1,0)}$ ${\ displaystyle {\ tfrac {\ mathrm {d} y} {\ mathrm {d} x}} | _ {(1,0)} = 0}$ ${\ displaystyle t \ in \ mathbb {R} _ {0} ^ {+}}$

{\ displaystyle x (t) = \ cos (t) + t \ sin (t)}

{\ displaystyle y (t) = \ sin (t) -t \ cos (t)}

The parameter is the length of the unwound piece of thread, i.e. the (unwound) arc length on the underlying unit circle. The following applies to the arc length of the constructed involute ${\ displaystyle t}$

{\ displaystyle s = {\ tfrac {1} {2}} t ^ {2}}

and for their curvature

{\ displaystyle \ kappa = {\ tfrac {1} {t}}}

,

so the parameter is also its radius of curvature at the same time . In polar coordinates it is represented as follows: ${\ displaystyle t}$

{\ displaystyle r (t) = {\ sqrt {1 + t ^ {2}}}}

{\ displaystyle \ varphi (t) = t- \ arctan (t)}

All other geometrically congruent circle involutes emerge from it through rotation around the origin of coordinates and displacement. Furthermore, the curve definition can of course also be continued on all , with all formulas for curve geometry remaining valid except for the arc length, which is too generalized. Geometrically, the original curve is given a further branch, which is created by mirroring itself on the x-axis. ${\ displaystyle t <0}$ ${\ displaystyle s = {\ tfrac {\ operatorname {sgn} (t)} {2}} t ^ {2}}$

literature

Mark Yes. Vygodskij: Higher Mathematics at Hand: Definitions Theorems Examples . Springer, 2013, ISBN, pp. 729–731

Joachim Erven, Dietrich Schwägerl: Mathematics for engineers . Walter de Hruyter, 4th edition 2011, ISBN 9783486707960 , pp. 216-217

Web links

Nora Rebecca Thomas: Geometric Properties of Curves . Staatsexamensarbeit, Uni Main, 2010, pp. 22–28

Individual evidence

^ Volvents in circles. At: mathe.tu-freiberg.de.

[1] Volvents in circles. At: mathe.tu-freiberg.de.