Involute Function Calculator
Calculate involute function and gear parameters for mechanical design
Common values: 14.5°, 20°, or 25° (typically 20° for modern gears)
Module is the ratio of pitch diameter to number of teeth (typically in mm)
Enter the number of teeth on the gear
How to Use This Calculator
Enter Pressure Angle
Enter the pressure angle in degrees. Common values are 14.5°, 20°, or 25°. Modern gears typically use 20°.
Enter Module
Enter the module value (in mm). The module determines the size of the gear teeth and is standardized (e.g., 1, 1.5, 2, 2.5, 3, 4, 5, etc.).
Enter Number of Teeth
Enter the number of teeth on the gear. This must be a positive integer.
Calculate
Click Calculate to get the involute function value and related gear parameters including pitch diameter, base diameter, addendum, and dedendum.
Formula
inv(α) = tan(α) - α
where α is the pressure angle in radians
Pitch Diameter:
d = m × z
where m = module, z = number of teeth
Base Diameter:
db = d × cos(α)
where d = pitch diameter, α = pressure angle
Addendum:
ha = m
Standard addendum equals the module
Dedendum:
hf = 1.25 × m
Standard dedendum (with clearance)
Example:
For α = 20°, m = 2 mm, z = 30:
- α (radians) = 20° × π/180 = 0.3491 rad
- inv(20°) = tan(0.3491) - 0.3491 = 0.014904
- d = 2 × 30 = 60 mm
- db = 60 × cos(20°) = 56.381 mm
About Involute Function Calculator
The involute function is a fundamental mathematical function in gear design and manufacturing. It describes the shape of gear teeth, which are designed using an involute curve. This function is essential for calculating gear dimensions, contact angles, and ensuring proper meshing between gears.
What is the Involute Function?
The involute function, denoted as inv(α) or inv α, is defined as inv(α) = tan(α) - α, where α is the pressure angle. It's used to calculate the relationship between the pressure angle and the contact point between two meshing gears.
Why Use Involute Gears?
- Constant Velocity Ratio: Involute gears maintain a constant velocity ratio, ensuring smooth transmission
- Tolerance to Center Distance: Gears can work correctly even with slight variations in center distance
- Manufacturing Simplicity: Involute tooth profiles are easier to manufacture than other tooth forms
- Standardization: Most modern gear standards use involute profiles
Key Gear Parameters
- Module (m): The ratio of pitch diameter to number of teeth, determines tooth size
- Pressure Angle (α): The angle between the line of action and the tangent to the pitch circle (typically 20°)
- Pitch Diameter (d): The diameter of the pitch circle where gears make contact
- Base Diameter (db): The diameter of the base circle from which the involute is generated
- Addendum: Height of tooth above the pitch circle
- Dedendum: Depth of tooth below the pitch circle
Applications
The involute function is used in:
- Gear Design: Calculating gear tooth profiles and dimensions
- Gear Manufacturing: CNC programming for gear cutting machines
- Quality Control: Measuring and verifying gear geometry
- CAD Software: Generating accurate gear models
- Mechanical Engineering: Transmission design and analysis
Standard Pressure Angles
Common pressure angles used in gear design:
- 14.5°: Older standard, provides smoother meshing but weaker teeth
- 20°: Modern standard, balances strength and smooth operation
- 25°: Higher strength but requires more precision in manufacturing
Frequently Asked Questions
What is the involute function used for?
The involute function is used in gear design to calculate the relationship between pressure angle and the contact geometry between meshing gears. It's essential for determining gear tooth profiles and ensuring proper gear meshing.
Why is 20° the standard pressure angle?
20° has become the modern standard because it provides a good balance between gear strength and smooth operation. It's stronger than 14.5° (older standard) while still being manufacturable with reasonable precision. Higher angles like 25° are stronger but require tighter manufacturing tolerances.
What is the difference between pitch diameter and base diameter?
Pitch diameter is the theoretical diameter where two gears make contact and is calculated as module × number of teeth. Base diameter is the diameter of the circle from which the involute tooth profile is generated and is always smaller than the pitch diameter (db = d × cos(α)).
Can I use this calculator for helical or bevel gears?
This calculator is designed for spur gears (straight-toothed gears). Helical gears use a normal pressure angle, and bevel gears require additional calculations for the cone geometry. The involute function principle applies, but additional parameters are needed.
What units should I use for the module?
Module is typically specified in millimeters (mm) in metric systems. In the imperial system, gears use diametral pitch instead of module. If you're working in inches, convert to metric first or use a calculator that supports diametral pitch.