In gearbox sizing, it’s important to determine the required input speed in rpm and ensure that it doesn’t exceed the gearbox’s maximum speed capability. But the linear velocity of the gear teeth, known as pitch line velocity, also plays a significant role in gearbox performance.

Pitch line velocity is measured at the pitch line of the gear, which is midway along the length of the gear teeth. For circular gears, the pitch line is more correctly referred to as the pitch circle, which is an imaginary circle that rolls without slipping when aligned with the pitch circle of the mating gear.

Strictly speaking, “pitch line” is the correct term when referring to a linear gear rack, and “pitch circle” is the correct term when referring to a circular gear. However, the term “pitch line” is often used when discussing the equivalent linear velocity of a circular gear – i.e. “pitch line velocity.”

Pitch line velocity is a function of the gear’s pitch diameter and its rotational speed:

Where:

PLV = pitch line velocity (m/s)

d_p = pitch diameter (m)

ω = rotational speed (rpm)

Pitch line velocity is important for gear design and selection for several reasons. First, the American Gear Manufacturers Association standard 9005-D94, Industrial Gear Lubrication, specifies that a gear’s pitch line velocity is one of the primary criteria for selecting gear lubrication. Pitch line velocity determines the contact time between gear teeth, which has a significant impact on the required oil viscosity. High pitch line velocities are usually accompanied by light loads and short contact times, making low-viscosity oils suitable. However, low pitch line velocities are associated with high loads and long contact times, which make high-viscosity, or even EP-rated, oils necessary.

In addition to lubrication considerations, pitch line velocity also affects the load capacity and service life of gear teeth. The ability of gears to transmit the required torque for the desired operating life depends on the ability of the gear teeth to withstand bending stress. Tooth bending stress is determined according to the Lewis formula:

Where:

σ = tooth bending stress (MPa)

W_t = tangential force on tooth (N)

P = diametrical pitch (mm^-1)

F = face width (mm)

Y = Lewis Form Factor

But as gear teeth come into initial contact, they experience greater stresses, based on the velocity of the gear. In order to account for these stresses, a velocity factor, K_v, was developed. The velocity factor depends on both the pitch line velocity of the gear and the quality of the gear (Q_v), and can be obtained from AGMA charts such as the one below.

This velocity factor K_v is used to modify the Lewis equation:

Thus, the higher the pitch line velocity, the greater the bending stress on the gear teeth.

Note: The AGMA has developed an equation for bending stress that replaces the Lewis Form Factor with a geometry factor, J, and includes factors for other conditions that affect gear service life, such as overload, load distribution, and mounting.

Feature image credit: Philip J. O’Keefe, PE