Measuring gear tooth strength is crucial for engineers to ensure the longevity and reliability of gearboxes and machinery. In this guide, we'll break down the process into simple, easy-to-follow steps.

Understanding Gear Tooth Strength

Gear tooth strength refers to a gear's ability to withstand the stresses placed on it during operation, such as bending and contact stress. The higher the gear tooth strength, the more durable and reliable the gear will be. The strength of a gear tooth is an important property for its performance in a system. In order to measure accurately, the following parameters need to be taken into consideration:

• Contact stress: the amount of pressure created where two gear surfaces meet
• Bending stress: the amount of force applied to gear teeth before they bend
• Surface finish: will affect the surface friction at two gears’ meshing points

Measuring Gear Tooth Strength Using the Lewis Equation

Introduced by Wilford Lewis at the end of the 19th century, the Lewis equation calculates the bending stress of spur gear teeth. Lewis identified a single spur gear tooth as a cantilever beam, with bending load distributed evenly over the tooth’s face width. For simplicity, the original equation dealt only in static loading conditions, ignoring radial force effects and sliding friction.

Versions of the Lewis bending equations are still the basis for most gear design today. There are essentially two equations at play here - the Lewis Equation which determines gear strength, and the Lewis Form Factor which accounts for different tooth pressure angles and the number of teeth.

The Lewis Equation

This standard equation is a quick way to evaluate the pitch and width of a gear required to transmit given force. It uses pre-calculated values of the Lewis Form Factor. Put another way, it states that the maximum allowable load per unit length of the contact surface is proportional to the hardness of the gear (determined using the Brinell or Rockwell hardness tests).

Lewis Gear Strength Formula

W_t = S x F x Y / DP

Where:

• W_t = Maximum transmitted load
• S = Maximum bending tooth stress (taken as ⅓ of the tensile strength)
• F= Face width of the gear
• DP = Diametral pitch (1/module if metric)
• Y= Lewis Factor

Calculating the Lewis Form Factor (Y)

Calculating the Lewis form factor (Y) is the most common way of determining gear tooth strength. This parameter takes into account the tooth geometry, material properties and applied loads to determine the gear's strength. It adjusts the calculated bending stress to a proportional degree to the thickness of the gear tooth and inversely proportional degree to the gear’s tooth height. Calculating the Lewis form factor requires the following measurements:

1. Tooth Dimensions: thickness and height of the gear tooth and the width of its face
2. Pitch Line Velocity: a gear’s rotational speed
3. Transmitted Load: The force applied to the gear tooth.

Once these measurements are found, use the following formula to calculate the Lewis form factor:

Y = K / (Ft x (b / d) ^ 2/3)

Where:

• K = Lewis factor, which varies depending on the gear material and tooth geometry.
• Ft = Transmitted load
• b = Face width of the gear
• d = Pitch diameter of the gear

Lewis Formula Example:

Assuming a gear with the following measurements -

• Tooth thickness: 10 mm
• Tooth height: 20 mm
• Pitch line velocity: 5 m/s
• Ft: Transmitted load: 1000 N
• b: Face width: 50 mm
• d: Pitch diameter: 100 mm

Using the formula Y = K / (Ft X (b / d) ^ 2/3), we can calculate the Lewis form factor:

Y = K / (Ft X (b / d) ^ 2/3)

Y = 0.15 / (1000 N X (50 mm / 100 mm) ^ 2/3)

Y = 0.15 / (1000 N X 0.7937)

Y = 0.000188

US Customary Lewis equation: σ_F=F_tP / bY

SI unit Lewis Equation: σ_F=F_t / bmY

Where:

• σ_F = maximum bending stress
• F_t= tangential load
• P = diametric pitch
• b = face width of gear tooth
• m = module
• Y = Lewis form factor (Y=2xP/3)

BUT, since most gears operate under dynamic loading conditions, the basic Lewis equation was modified:

σ_F=F/bYm(Kv)

Where:

• K_v = velocity factor, determined using Barth's equation - K_v= 6 / (6+V)
• V = velocity of the gear tooth

To take into account contact ratios, stress concentrations and pitch line velocity of dynamic loading, the Lewis equation was updated again:

• Velocity factor (Kv)
• Overload factor (Ko)
• Reliability factor (Kr)
• Load distribution factor (Kx), and
• Gear geometry factor (J)
• J= (Modified Lewis Form Factor)/(Fatigue stress concentration factor)

Resulting in the new modified Lewis Equation:

σ_F=F(Kv)(Ko)(Kx)(Kr)/bmJ

Lewis Form Factor Reference Table:

The following table shows Lewis Form Factor constants for most common 20º pressure angle gears - as well as the previously more common 14.5º pressure angle - from 10-300 teeth.

Barth Equation for Velocity Factor

The Barth Equation is used to complement data learned from the Lewis Equation in the assessment of gear tooth strength. In particular, the Barth Equation for Velocity Factor accounts for dynamic loading effects, whereas the Lewis Equation accounts for gear stress in static conditions. It uses a Barth equation-derived coefficient to account for speed and acceleration and their impact on gear stress.

The formula for the Barth Equation for Velocity Factor is:

Kv=6 / 6+V

Where:

• Kv is the velocity factor
• V is the gear tooth velocity

AGMA Surface Stress Equation

The American Gear Manufacturers Association has refined the basics of the Lewis formula to account for additional real-world factors not well understood during Lewis’ lifetime. These include:

• Geometric factors
• Application factors - variations in load and driver shock
• Dimensional factors - larger or wider teeth are accounted for
• Dynamic factors - a version of the the Barth velocity equation that accounts for gear quality
• Load distribution - a function of load amount and the width of a gear’s teeth
• Idler factors - considers reversed bending in idler gears
• Rim thickness - adjusts and penalizes according to the flexibility of the gear’s rim

Wear Strength Equation - Buckingham Formula

As opposed to the Lewis equation (which assumes a static tooth load), the Buckingham Formula accounts for dynamic pressures of variable loads caused by deflection of gear teeth and machining errors which can create periods of unexpected inertial forces, acceleration and variable loads. Its expected output is the total maximum load on the gear’s teeth, which is dynamic in practice. The formula considers several factors including:

• Applied load
• Gear tooth thickness
• Gear tooth width
• Linear velocity

The Buckingham Wear Strength formula is:

P=K×V^m×Lⁿ×W^p×T^q

Where:

• P is the total maximum load on the gear teeth
• K is a constant
• V is the pitch line velocity, representing the linear velocity of the gear tooth.
• L is the torque or applied load
• W is the width of the gear tooth
• T is the thickness of the gear tooth
• m,n,p,q are dimensionless constants

Consult with a Trusted Gear Manufacturer

Measuring gear tooth strength accurately before designing new components ensures the longevity and reliability of your machinery. By calculating the Lewis form factor, you can determine a gear's strength and make informed decisions about its use in your application. However, there are many factors that affect gear performance, so consulting with a trusted gear manufacturer like WM Berg is recommended. WM Berg’s expertise and experience guide engineers toward precision gear selections that meet the needs of their specific application. We can also design and manufacture custom gearing to meet your specifications, or modify one of our standard gears.

Gear Tooth FAQ