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| 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. | 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. However, this page is more practical. <<TableOfContents()>> = What is the involute curve of a circle? = {{attachment:involute_q_and_d.jpg||width=250}} {{{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 involute of a circle - you can make involutes based on other shapes too. {{attachment:involute_basic_0-4pi.svg||width=200}} {{{Involute curve with x- and y-axis, without base circle.}}} = Practical use = |
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| 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. | 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 = {{attachment:involute_with_base.svg||width=250}} {{{Involute with base circle}}} Angles are in radians, 360 degrees equals (2 * pi) radians. Overall, see illustration: * The blue curve is the involute of a circle. * 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. {{attachment:involute_math.svg||width=650}} {{{Mathematics of involute}}} In detail: * The first triangle: * Start at point (0,0) * Hypotenuse is 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 (equals 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 angle A1 and radius R1, hence {{{A1 * R1}}}. * Cartesian coordinates: * Points on the involute curve have Cartesian coordinates (X3,Y3). * {{{X3 = X1 + X2 = R1 * cos (A1) + R1 * A1 * sin (A1)}}} * {{{Y3 = Y1 - Y2 = R1 * sin (A1) - R1 * A1 * cos (A1)}}} * Polar coordinates: * Points on the involute curve have polar coordinates (R2,A2). * About R2: * R1 and A1*R1 are always perpendicular. * Radius R2 can be calculated using the Pythagorean theorem. * {{{R2 = sqrt (R1^2 + (A1 * R1)^2) = R1 * sqrt (1 + A1^2)}}}. * About A2: * This can be calculated by using asin, acos or atan. * Unfortunately, these functions do not go all the way around with one argument supplied. For example, {{{A2 = acos (X3 / R2)}}} and that is only true for {{{0 < A2 < pi}}}. See example. * Fortunately, many environments offer a solution when X- and Y values are both entered as arguments. For example, !LibreOffice Calc has a formula {{{ATAN2(X-value;Y-value)}}} and LISP for CAD offers {{{(ATAN Y-value X-value)}}}. {{attachment:involute_ang_acos.svg||width=250}} {{{Wrongly drawn by using acos(X3/R2) for angle A2}}} * Additional information: * For a gear application, A1 is not interesting while R1 and R2 are important. * Polar part R2 is defined as a relation of R1 and A1, as stated before: {{{R2 = R1 * sqrt (1 + A1^2)}}}. * If we solve this equation in order to find A1, the answer becomes: * {{{A1 = 2 * sqrt ((R2^2 - R1^2) / 2)}}} |
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. However, this page is more practical.
What is the involute curve of a circle?
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 involute of a circle - you can make involutes based on other shapes too.
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
Involute with base circle
Angles are in radians, 360 degrees equals (2 * pi) radians.
Overall, see illustration:
- The blue curve is the involute of a circle.
- 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.
Mathematics of involute
In detail:
- The first triangle:
- Start at point (0,0)
- Hypotenuse is 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 (equals 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 angle A1 and radius R1, hence A1 * R1.
- Cartesian coordinates:
- Points on the involute curve have Cartesian coordinates (X3,Y3).
X3 = X1 + X2 = R1 * cos (A1) + R1 * A1 * sin (A1)
Y3 = Y1 - Y2 = R1 * sin (A1) - R1 * A1 * cos (A1)
- Polar coordinates:
- Points on the involute curve have polar coordinates (R2,A2).
- About R2:
- R1 and A1*R1 are always perpendicular.
- Radius R2 can be calculated using the Pythagorean theorem.
R2 = sqrt (R1^2 + (A1 * R1)^2) = R1 * sqrt (1 + A1^2).
- About A2:
- This can be calculated by using asin, acos or atan.
Unfortunately, these functions do not go all the way around with one argument supplied. For example, A2 = acos (X3 / R2) and that is only true for 0 < A2 < pi. See example.
Fortunately, many environments offer a solution when X- and Y values are both entered as arguments. For example, LibreOffice Calc has a formula ATAN2(X-value;Y-value) and LISP for CAD offers (ATAN Y-value X-value).
Wrongly drawn by using acos(X3/R2) for angle A2
- Additional information:
- For a gear application, A1 is not interesting while R1 and R2 are important.
Polar part R2 is defined as a relation of R1 and A1, as stated before: R2 = R1 * sqrt (1 + A1^2).
- If we solve this equation in order to find A1, the answer becomes:
A1 = 2 * sqrt ((R2^2 - R1^2) / 2)
- For a gear application, A1 is not interesting while R1 and R2 are important.