Torque Multiplication through Gear Reduction

"Gear ratio" is the general term for the ratio of output torque to input torque (equivalently, input speed to output speed). "Gear reduction" specifically refers to the case where the output speed is lower than the input speed (N > 1), which is the common case for motor-to-load transmissions. If N < 1, the gearbox is an overdrive — the output speed is higher and the output torque is lower, as in the top gear of a car. Most gearboxes used in robotics and industry are reduction gearboxes, because motors typically run fast with low torque while loads typically need slow, high torque. The terms are often used interchangeably when the context is clearly reduction.

Choosing a gear ratio is a trade-off between torque and speed. First, determine the required output torque (from the load and acceleration requirements). Second, determine the required output speed (from the motion profile). Third, find a ratio that lets the motor operate within its torque and speed limits at that operating point — the motor's continuous torque must exceed τ_out/(η·N), and its no-load speed must exceed N·ω_out. Fourth, consider inertia matching: the reflected load inertia I_load/N² should be comparable to the motor's own inertia I_motor for best dynamic performance (a ratio between 1:1 and 10:1 is often targeted). Finally, check the gearbox's rated torque, backlash, and efficiency against the requirements. In practice, a few candidate ratios are evaluated against all these criteria, and the best compromise is selected.

Reflected inertia is the effective inertia that a load presents to the motor when viewed through a gearbox. It is I_reflected = I_load / N². This is important because the motor must accelerate both its own inertia and the reflected load inertia. A high gear ratio reduces the reflected load inertia dramatically: a load of 1 kg·m² seen through a 100:1 gearbox appears to the motor as 0.0001 kg·m² — a factor of 10,000 reduction. This makes the load easy to accelerate, allowing the motor to respond quickly and use its full torque capability. The reflected inertia also determines the system's natural frequency and stability: if the reflected load inertia is much larger than the motor inertia, the system may oscillate or be difficult to control. A common rule of thumb is to keep I_load/N² between 1× and 10× the motor inertia.

Real gearboxes lose energy to friction, oil churning, and windage, so the output torque is less than the ideal value: τ_out = η · τ_in · N, where η is the gearbox efficiency. For a single spur gear stage, η is typically 0.97–0.99. For a worm gear, it can be as low as 0.30–0.70, depending on the lead angle. For a multi-stage gearbox, the efficiencies multiply: η_total = η₁ · η₂ · ... · η_k. A three-stage planetary gearbox with 0.97 efficiency per stage has an overall efficiency of about 0.91, so the output torque is about 9% less than the ideal calculation. When sizing a motor, the efficiency must be included, or the motor will be undersized and the system will not meet its performance targets.

No. A gearbox cannot increase both torque and speed simultaneously, because that would violate the conservation of energy: the output power (τ·ω) cannot exceed the input power. If the output torque is multiplied by N, the output speed is divided by N, so τ_out·ω_out = τ_in·ω_in (minus losses). This is the fundamental trade-off of all transmissions. A gearbox can multiply torque (at the cost of speed), multiply speed (at the cost of torque), or change the direction of rotation, but it cannot increase the product τ·ω. The only way to increase both is to increase the input power, which means a larger motor, a higher supply pressure, or a different energy source.

For an ideal gearbox, the torque ratio equals the gear ratio: N = τ_out/τ_in. But for a real gearbox with efficiency η, the torque ratio is η·N, while the speed ratio is exactly N (in the ideal case) or N/η in some formulations. The distinction matters when efficiency is not 1: the gear ratio N is a purely geometric quantity (ratios of teeth or radii) and is the same for torque and speed, while the actual torque multiplication is reduced by efficiency losses. In most engineering contexts, "gear ratio" refers to the geometric ratio N, and the efficiency is applied separately in the torque calculation.