Formula & Calculator
Torque Multiplication through Gear Reduction
Calculates the output torque delivered after a gear reduction stage, based on the input torque and the gear ratio.
Interpretation
τ_out = τ_in × N. With a gear ratio N (output/input), torque is multiplied. Used to amplify motor torque at the expense of speed.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| tau_out | Output torque after the gear reduction | N.m |
| tau_in | Input torque from the motor | N.m |
| N | Gear reduction ratio (output speed / input speed) |
What it means
Gear reduction increases the output torque by the gear ratio N (assuming ideal efficiency), while decreasing output speed by the same factor. This is used in robotics to match motor characteristics to the load: a high‑speed, low‑torque motor can be geared to provide high torque at low speed. This formula is used in actuator selection and gearbox design. Understanding torque multiplication is essential for choosing gear ratios to achieve required joint torques and speeds.
Worked example
Torque Multiplication (Gear Reduction) – Two Detailed Examples
Real‑World| Parameter | Value |
|---|---|
| τ_in (N·m) | 5 |
| N | 3 |
| Parameter | Value |
|---|---|
| τ_in | 2 |
| N | 10 |
Common mistakes
- Gear ratio: τ_out = τ_in × N – where N is the gear ratio (output teeth / input teeth).
- Speed reduction: If N > 1, output torque increases, speed decreases (reduction gear).
- Ideal: Assumes 100% efficiency – real gears have losses (use efficiency factor).
- Sign: Both input and output torque directions are related; if gear reverses, sign changes.
- Inertia reflection: The output inertia is reflected to the input multiplied by N² – not covered by this formula.
Applications
Torque multiplication through gear reduction, τ_out = τ_in·N, describes how a gearbox amplifies torque by a factor N (the gear ratio), while reducing speed by the same factor. This is widely used in robotics and automotive drivetrains to increase the torque available from a motor. Engineers use it to match motor characteristics to load requirements, to reduce motor size and cost, and to improve motion control precision. By selecting appropriate gear ratios, they can achieve the desired balance between speed and torque. This formula is essential for designing efficient power transmission systems in robots and vehicles.
- Gearbox design for robotics and automation
- Motor selection and optimisation for torque requirements
- Transmission design in automotive and industrial machinery
- Power train efficiency and speed‑torque trade‑offs
- Simulation of geared mechanical systems
Frequently Asked Questions
A gear reduction with ratio N (output speed = input speed / N) multiplies the torque by the same factor N, ignoring losses: τ_out = N · τ_in. This allows a small motor to drive a heavy load.
Forgetting that the output speed decreases by the same ratio the torque increases, since power is (ideally) conserved. If torque is multiplied by N, speed is divided by N.
Real gears have losses (efficiency η < 1). The actual output torque is τ_out = η·N·τ_in. The power out is η times the power in.
The reflected inertia at the motor shaft is the load inertia divided by N². This reduces the effective inertia seen by the motor, helping acceleration.
Higher gear reduction gives more torque but lower speed. The product of torque and speed (power) is constant (ideally). Choose the gear ratio to match the motor's optimal operating point to the load requirements.
Determine the required output torque and speed. Choose a motor that can provide the power, then select N to transform the motor's torque and speed to match the load.
For a given output torque, higher N reduces the required motor torque and thus the motor current. This can be beneficial for battery‑powered robots.
Spur gears, planetary gears, harmonic drives, and cycloidal drives. Each has different efficiency, backlash, and torque capacity characteristics.
Backlash is the play between gear teeth. It introduces a dead zone, which can affect control accuracy, especially in positioning applications. Low‑backlash gearboxes are preferred for precision.
Robotic joint actuators, conveyor drives, wind turbines, and any system requiring high torque at low speed.