¶Kinematics of Gear Trains

A gear train is a mechanical system formed by mounting gears on a frame so the teeth of the gears engage. Gear trains are fundamental in mechanical engineering to transmit torque and adjust the rotational speed between an input power source and an output load.

¶The Gear Ratio

For two meshing gears (a driver and a driven gear), the fundamental kinematic relationship—assuming no slip between the teeth profiles—dictates that their pitch circles roll against each other without slipping. Therefore, their tangential velocities at the point of contact must be equal.

The gear ratio $R$ is defined by the inverse relationship between the number of teeth $N$ and the angular velocity $\omega$: $R = \frac{\omega_{\text{in}}}{\omega_{\text{out}}} = \frac{N_{\text{out}}}{N_{\text{in}}}$

¶Torque Multiplication

By the principle of conservation of energy (power $P = \tau \omega$), the input power must equal the output power in an ideal, frictionless gear system: $\tau_{\text{in}} \omega_{\text{in}} = \tau_{\text{out}} \omega_{\text{out}}$

Substituting the gear ratio yields the torque multiplication equation: $\tau_{\text{out}} = \tau_{\text{in}} \left( \frac{\omega_{\text{in}}}{\omega_{\text{out}}} \right) = \tau_{\text{in}} \left( \frac{N_{\text{out}}}{N_{\text{in}}} \right)$

If $N_{\text{out}} > N_{\text{in}}$, the system is a speed reducer and a torque multiplier. This is critical in robotics and spacecraft mechanisms where electric motors operate at high speeds with low torque, but actuators require low speeds and high torque.

Reference: Shigley, J. E., & Mischke, C. R. (2014). Mechanical Engineering Design (10th ed.). McGraw-Hill.

¶References

• Shigley's Mechanical Engineering Design (Richard G. Budynas, J. Keith Nisbett)
• Machinery's Handbook (31st Edition) (Erik Oberg)
• MIT OCW: 2.007 Design and Manufacturing I