Involute Function Calculator
Calculate involute function values for gear design and engineering applications
Calculate Involute Function
Range: 0° to 90°
Degrees: 20.000°
Radians: 0.349066 rad
Common angles: 14.5°, 20°, 25°
Involute Function Results
Involute Function Value
Involute Curve Point
x-coordinate: 1.0591
y-coordinate: 0.0140
For unit circle (r = 1)
Mathematical Formula
Involute Function: Inv(α) = tan(α) - α
Engineering Applications
The involute function helps determine gear tooth thickness and spacing for optimal performance.
Example Calculation
Standard 20° Pressure Angle Gear
Input: α = 20° (most common gear pressure angle)
Conversion: 20° = 0.3491 radians
Formula: Inv(α) = tan(α) - α
Calculation: Inv(0.3491) = tan(0.3491) - 0.3491
Result: Inv(20°) = 0.3640 - 0.3491 = 0.0149
Interpretation
This dimensionless value represents the "width" or thickness characteristic of the involute gear tooth profile at a 20° pressure angle.
Engineers use this value to calculate gear tooth spacing and ensure proper meshing between gears.
Common Pressure Angles
20° is the most widely used pressure angle in modern gear design
Gear Design Properties
Pressure Angle
Angle between line of action and perpendicular to line of centers
Involute Angle
Angle related to tooth thickness and gear geometry
Gear Teeth
Involute profile ensures constant velocity ratio
Engineering Tips
Higher pressure angles create stronger, wider teeth
Lower pressure angles provide smoother operation
20° is the industry standard for most applications
Involute function is dimensionless and angle-dependent
Understanding the Involute Function
What is an Involute?
An involute is a curve traced by a point on a taut string as it unwinds from a fixed curve. In gear design, the involute of a circle is used to create the tooth profile because it maintains a constant velocity ratio between meshing gears.
The Involute Function Formula
Inv(α) = tan(α) - α
Where α is the pressure angle in radians. This function relates the pressure angle to the geometry of the involute curve.
Parametric Equations
x = r(cos θ + θ sin θ)
y = r(sin θ - θ cos θ)
Engineering Applications
Gear Design
Primary application in designing involute gears
- • Determines tooth thickness and spacing
- • Ensures constant velocity ratio
- • Minimizes sliding friction
Pressure Angle Selection
Balances strength vs. smoothness
- • 14.5°: Smooth operation, weaker teeth
- • 20°: Standard industrial choice
- • 25°: Higher load capacity
Mathematical Properties
Key characteristics of the involute function
- • Always positive for α > 0
- • Increases monotonically
- • Dimensionless output