Gear Parameters and Calculation Formulas

This is a fundamental question in mechanical design. The parameters and calculation formulas for gears form the basis of their design and manufacturing. Below is a systematic introduction, from basic concepts to key parameter calculations.

1. Basic Gear Parameters (Using the Common Involute Spur Gear as an Example)

These are the essential parameters needed to describe a gear, which must typically be specified on a design drawing.

1. Number of Teeth (z)

   • Definition: The total number of teeth on the gear.

   • Explanation: It determines the gear ratio and the size of the gear.

2. Module (m)

   • Definition: A core parameter that determines the size of the gear teeth. It directly represents the ratio of the circular pitch to pi (π).

   • Importance: The module is standardized. A larger module means larger and stronger teeth with higher load-carrying capacity.

   • Calculation Formula: m = p / π, where p is the circular pitch.

3. Pressure Angle (α)

   • Definition: The acute angle between the line of action (common normal to the tooth profiles at the point of contact) and the direction of instantaneous velocity at the pitch point. It usually refers to the pressure angle on the standard pitch circle.

   • Standard Value: The most common is 20°; others include 14.5° and 25°.

   • Influence: The pressure angle affects the gear’s load capacity and tooth shape. A larger pressure angle results in a thicker tooth root and higher strength, but slightly lower transmission efficiency.

4. Addendum Coefficient (hₐ)*

   • Definition: The ratio of the addendum to the module. hₐ = hₐ* × m

   • Standard Value: For a standard full-depth tooth system, hₐ* = 1; for a stub tooth system, hₐ* = 0.8.

5. Dedendum/Clearance Coefficient (c)*

   • Definition: The ratio of the bottom clearance (the gap between the top of one gear’s tooth and the root of the mating gear) to the module. c = c* × m

   • Function: Prevents jamming of the meshing gears and provides space for lubricant.

   • Standard Value: For a standard full-depth tooth system, c* = 0.25.

2. Key Geometric Dimension Calculation Formulas

With the basic parameters above, we can calculate all key dimensions of the gear.

1. Pitch Diameter (d)

   • Definition: The reference circle for design and calculation; the circle where the module and pressure angle are standard.

   • Formula: d = m * z

2. Tip Diameter (dₐ)

   • Definition: The diameter of the circle that passes through the tops of the teeth.

   • Formula: dₐ = d + 2hₐ = m z + 2 m hₐ = m(z + 2)

3. Root Diameter (df)

   • Definition: The diameter of the circle that passes through the bottoms of the tooth spaces.

   • Formula: df = d - 2hf

   • Where Dedendum hf = hₐ + c = (hₐ* + c*) * m

   • Therefore, df = m z - 2 (1 + 0.25) * m = m(z - 2.5)

4. Base Diameter (db)

   • Definition: The circle from which the involute tooth profile is developed.

   • Formula: db = d cos(α) = m z * cos(α)

5. Circular Pitch §

   • Definition: The arc distance along the pitch circle from one tooth flank to the corresponding flank of the adjacent tooth.

   • Formula: p = π * m

6. Tooth Thickness (s) and Space Width (e)

   • Definition: On the pitch circle, the arc length between two opposite flanks of a single tooth is the tooth thickness; the arc length between two flanks of the tooth space is the space width. For standard gears, they are equal.

   • Formula: s = e = p / 2 = π * m / 2

7. Center Distance (a)

   • Definition: The distance between the centers of a pair of mating gears.

   • Formula: a = (d₁ + d₂) / 2 = m(z₁ + z₂) / 2

3. Transmission-Related Parameters and Formulas

1. Gear Ratio (i)

   • Definition: The ratio of the angular velocities (or speeds) of the driving gear to the driven gear. It is also equal to the ratio of the number of teeth of the driven gear to the driving gear.

   • Formula: i = n₁ / n₂ = z₂ / z₁

     • n₁, n₂: Rotational speed of driving and driven gear (rpm)

     • z₁, z₂: Number of teeth of driving and driven gear

2. Contact Ratio (ε)

   • Definition: The average number of tooth pairs in contact during meshing.

   • Importance: A larger contact ratio means smoother transmission and lower load on each tooth pair. It is generally required that ε > 1.2.

   • Formula: (Conceptual) ε = [z₁(tan αₐ₁ - tan α') + z₂(tan αₐ₂ - tan α')] / (2π)

     • Where α' is the operating pressure angle (equal to the standard pressure angle α for standard center distance), and αₐ is the pressure angle at the tip circle. This is a relatively complex formula, usually calculated using tables or software.

4. Parameter Characteristics of Other Gear Types

• Helical Gear:

  • Normal Module (mₙ) and Transverse Module (mₜ): mₙ = mₜ * cos(β)

  • Helix Angle (β): The angle between the tooth trace and the gear axis. Helical gears provide smoother and quieter operation.

  • Calculation Formula: Pitch diameter d = mₜ z = (mₙ / cos(β)) z

  • Center Distance: a = mₙ (z₁ + z₂) / (2 cos(β))

• Bevel Gear:

  • Module at Large End (m): The module at the larger end is taken as the standard value.

  • Pitch Cone Angle (δ): Determines the taper of the gear.

  • Virtual Number of Teeth (zv): zv = z / cos(δ), used for strength calculation and form cutting.

Summary

The key to understanding gear parameters lies in mastering the three most basic concepts: Module (m), Number of Teeth (z), and Pressure Angle (α). These are the foundation for all other dimensional calculations. In practical engineering, based on strength, life, and space requirements, the basic parameters like the module are first determined, and then the formulas above are used to calculate all the geometric dimensions of the gear. For more complex gears (like helical or bevel gears), additional parameters such as the helix angle and cone angle need to be introduced.