Gear Kinematics and Forces

Concept

Gears are toothed machine elements used to transmit motion and power between rotating shafts. The fundamental law of gearing requires that the angular velocity ratio between mating gears remains constant. This is achieved using involute tooth profiles. Gear kinematics involves understanding parameters like pitch, number of teeth, and pitch diameter. When transmitting torque, the force on a gear tooth can be resolved into a tangential (transmitted) component that does useful work, and a radial component that separates the gears.

Formula & Method

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

$p = \frac{\pi d}{N} = \frac{\pi}{P}$

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

$m_G = \frac{N_{gear}}{N_{pinion}} = \frac{d_{gear}}{d_{pinion}} = \frac{\omega_{pinion}}{\omega_{gear}}$

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

$T = W_t \frac{d}{2}$

The radial force $W_r$ and total force $W$ relate to the tangential force via the pressure angle $\phi$:

$W_r = W_t \tan \phi$ $W = \frac{W_t}{\cos \phi}$

Variables & Units

• $N$ = Number of teeth.
• $d$ = Pitch diameter, in meters (m) or millimeters (mm).
• $\omega$ = Angular velocity, in rad/s or rpm.
• $T$ = Transmitted torque, in N·m.
• $W_t$ = Tangential force (transmitted load), in Newtons (N).
• $W_r$ = Radial force, in Newtons (N).
• $\phi$ = Pressure angle, typically $20^\circ$ or $25^\circ$.

Worked Example

Problem: A pinion with 18 teeth and a pitch diameter of $45 \text{ mm}$ rotates at $1000 \text{ rpm}$ and transmits $1.5 \text{ kW}$ of power to a mating gear. The pressure angle is $20^\circ$. Calculate the tangential force ($W_t$) and the radial force ($W_r$) exerted on the gear teeth.

Calculation:

1. Identify the given parameters: $N_p = 18$, $d_p = 0.045 \text{ m}$, $n_p = 1000 \text{ rpm}$, $H = 1500 \text{ W}$, $\phi = 20^\circ$.
2. Calculate the angular velocity in rad/s: $\omega_p = \frac{2 \pi \times 1000}{60} = 104.72 \text{ rad/s}$
3. Calculate the transmitted torque on the pinion: $T = \frac{H}{\omega_p} = \frac{1500}{104.72} = 14.32 \text{ N·m}$
4. Calculate the tangential force: $W_t = \frac{2T}{d_p} = \frac{2(14.32)}{0.045} = 636.4 \text{ N}$
5. Calculate the radial force: $W_r = W_t \tan(20^\circ) = 636.4(0.364) = 231.6 \text{ N}$

Engineering Meaning

Understanding gear forces is critical for two reasons. First, the tangential force $W_t$ determines the bending stress at the root of the gear tooth and the contact stress on the tooth surface. 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 train assembly.

Engineering Check

Ensure that mating gears have the same diametral pitch (or module) and the same pressure angle; otherwise, they will not mesh properly. When analyzing gear trains with multiple stages, track the direction of rotation carefully, as an idler gear will reverse the direction of the output shaft without changing the overall gear ratio.

Explicit Exclusions

This foundational article excludes the detailed AGMA stress formulations for bending (Lewis equation variations) and pitting resistance (contact stresses). It also excludes helical, bevel, and worm gear kinematics.