Gear Kinematics and Forces

Concept

Gears are toothed machine elements used to transmit motion and power between rotating shafts. The fundamental law of gearing dictates that the angular velocity ratio between mating gears must remain constant, achieved by the involute tooth profile. Gear kinematics involves understanding parameters like pitch, number of teeth, and pitch diameter. When transmitting torque, the force exerted on the gear tooth can be resolved into a tangential component (the transmitted force) which does the useful work, and a radial component which pushes the gears apart.

Formula & Method

For spur gears, the circular pitch ($p$), diametral pitch ($P$), number of teeth ($N$), and pitch diameter ($d$) are related by:

The gear ratio (or velocity ratio) $m_G$ is:

The transmitted torque $T$ and the tangential force $W_t$ at the pitch circle are related by:

The radial force $W_r$ and the resultant force $W$ are related to the tangential force via the pressure angle $\phi$:

Engineering Meaning

Understanding gear forces is critical for two reasons. First, the tangential force $W_t$ dictates the bending stress at the gear tooth root and the contact stress on the tooth face. Second, both $W_t$ and $W_r$ act as loads on the shaft and the bearings supporting the gear. Accurately determining these forces is the first step in designing the entire gear transmission assembly.

Engineering Check

Ensure that mating gears have the exact same diametral pitch (or module) and pressure angle; otherwise, they will not mesh correctly. When analyzing a multi-stage gear train, carefully track the direction of rotation, as an idler gear will reverse the direction of the output shaft without changing the overall gear ratio.

Explicit Exclusions

This fundamental article does not cover the detailed AGMA stress equations for bending (Lewis equation variations) and pitting resistance (contact stress). It also excludes the kinematics of helical, bevel, and worm gears.