Formula & Calculator
Belt and Pulley Speed Ratio
Relates the rotational speeds and diameters of two pulleys connected by a belt, based on conservation of belt surface speed.
Interpretation
For a belt‑and‑pulley system, the product of speed and diameter is constant: N₁·D₁ = N₂·D₂. This ensures that the belt linear speed is the same on both pulleys. It is used to change rotational speed and torque.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| N1, N2 | Rotational speeds of pulley 1 and pulley 2 | RPM |
| D1, D2 | Diameters of pulley 1 and pulley 2 | mm |
What it means
In a belt drive, the linear speed of the belt must be identical on both the driver and driven pulleys. This leads to the relationship N₁ D₁ = N₂ D₂, where N is rotational speed (e.g., rpm) and D is the pulley diameter. This formula assumes no slip between the belt and pulley. It shows that the speed ratio is inversely proportional to the diameter ratio. If the driven pulley is larger, it turns slower but produces higher torque. This principle is used in various machines, such as conveyor systems, automotive accessories (alternator, water pump), and industrial drives. The belt tension and friction determine the torque transmission capacity. In practice, slip may occur, which affects the speed ratio. The formula is also applicable to chain drives and gear trains (with teeth count instead of diameters). Proper selection of pulley sizes is crucial for achieving desired output speeds and torque. This relationship is fundamental in mechanical power transmission design.
Worked example
Belt and Pulley – Two Examples
Real‑World| Parameter | Value |
|---|---|
| N1 | 1500 RPM |
| D1 | 100 mm |
| D2 | 200 mm |
| Parameter | Value |
|---|---|
| N1 | 500 RPM |
| D1 | 200 mm |
| D2 | 100 mm |
Common mistakes
- Speed N in rpm: Not rad/s – use rpm consistently.
- Diameter D: Use the pitch diameter of the pulley (or outer diameter if belt thickness ignored).
- Belt slip: The formula assumes no slip; in reality, slip reduces the speed ratio.
- Direction of rotation: The product equality holds regardless of direction, but ensure you assign N₁ and N₂ correctly.
- Units: Both diameters in same units (mm, m, etc.) – the ratio cancels.
Applications
In a belt and pulley system, the product of speed and diameter is constant, ensuring the belt travels at the same linear speed. This principle is used to transmit power between shafts at different speeds and distances. It is common in machinery, such as in conveyors, fans, compressors, and textile equipment. The speed ratio allows engineers to select appropriate pulley diameters to achieve desired output speeds. Belt drives are favoured for their simplicity, low cost, and overload protection. By applying this relationship, engineers can design efficient power transmission systems that accommodate space constraints and speed requirements.
- Power transmission between shafts in machinery
- Conveyor belt and material handling systems
- Automotive accessory drives (alternator, water pump)
- HVAC fan and blower drives
- Textile and printing machinery
Frequently Asked Questions
For a belt drive (or chain drive), the speed ratio is given by N₁·D₁ = N₂·D₂, where N is the rotational speed (RPM) and D is the diameter of the pulley. This is based on the fact that the belt speed is the same on both pulleys: v = π·D·N. Thus, N₂/N₁ = D₁/D₂.
RPM and diameter must be in consistent units (e.g., inches, mm). The formula applies to flat belts, V‑belts, and timing belts (assuming no slip). It is used in industrial drive systems, automotive engines (alternator, water pump), and conveyor systems.
- Applying it to gear trains – for gears, the ratio is based on tooth counts, not diameters.
- Ignoring belt slip – real belts have some slip (1‑2%), which changes the output speed slightly.
- Using the wrong diameter – for V‑belts, use the pitch diameter, not the outer diameter.
- Confusing driver and driven – ensure you know which is input (driver) and which is output (driven).
From N₁·D₁ = N₂·D₂, we get N₂ = N₁ · (D₁/D₂). For example, if the driver is 10 cm and runs at 1000 RPM, and the driven is 20 cm, the driven speed is 1000 × (10/20) = 500 RPM.
If the driver pulley is smaller than the driven pulley (D₁ < D₂), the driven pulley rotates slower (N₂ < N₁). This is a speed reduction, but torque increases (T₂ = T₁ × (D₂/D₁)). This is used to multiply torque in machinery.
If D₁ > D₂, the driven pulley rotates faster (N₂ > N₁). This increases speed but reduces torque. This is used in applications like a supercharger or a high‑speed spindle.
The formula assumes the belt runs on the pitch circle. For thick belts, the effective diameter is the pitch diameter (which includes half the belt thickness). For flat belts, the neutral axis is used. The difference is usually small and often neglected, but for precision drives (like timing belts), the pitch diameter is used.
Both use the same speed ratio formula (N₁/N₂ = D₂/D₁ for belts, or N₁/N₂ = N₂_teeth/N₁_teeth for chains). However, chains do not slip (positive engagement), so the ratio is exact. Belts can slip under high torque, so the actual output speed may be slightly lower than calculated.
The belt length is approximately:
L = 2·C + π/2·(D₁ + D₂) + (D₁ − D₂)²/(4·C), where C is the center distance. This formula is used for selecting the correct belt length. The exact length depends on the belt type and wrap angle.
The torque is limited by the friction between the belt and the pulley. The maximum tension difference (T₁ − T₂) is given by the capstan equation: T₁/T₂ = e^(μ·θ), where μ is the coefficient of friction and θ is the wrap angle. The torque is T = (T₁ − T₂)·D/2. This sets the power capacity of the drive.