1. The Fundamental Law of Gearing

Whether using belts, chains, or gears, the fundamental principle relies on the conservation of surface speed. The linear velocity of the belt/chain must be identical at both the driver and driven points (assuming no slip). This leads to the classic inverse relationship:

Where $N$ is rotational speed (RPM) and $D$ is diameter (or number of teeth $T$).
Ratio ($i$) = $N_{in} / N_{out} = D_{out} / D_{in}$.
If $i > 1$, it is a Reducer (Speed down, Torque up).
If $i < 1$, it is an Overdrive (Speed up, Torque down).

2. The Power-Torque Relationship

Power is the rate of doing work. In rotating systems, Power = Torque $\times$ Angular Velocity. A gearbox or belt drive transmits power (minus efficiency losses) but transforms the torque/speed characteristics.

SI Units (Metric): $P (kW) = \frac{T (Nm) \times N (RPM)}{9550}$
$\rightarrow T = \frac{9550 \times P}{N}$

Imperial Units: $P (HP) = \frac{T (ft\cdot lb) \times N (RPM)}{5252}$
$\rightarrow T = \frac{5252 \times P}{N}$

When speed decreases by half, torque doubles (theoretically). This explains why a small motor with a huge reduction ratio can crush rocks.

3. Linear Velocity (Belt Speed)

The speed at which the belt or chain travels over the pulleys is critical.
Calculation: $V = \pi \cdot D \cdot N$
Limits:

• Standard V-Belts: Max ~30 m/s (6000 fpm). Above this, centrifugal force lifts the belt off the pulley, reducing grip and transmitting capability. Dynamic balancing becomes critical.
• Roller Chains: Much lower limits depending on pitch. High speed causes impact noise and rapid wear. Typically < 10-15 m/s.

4. Pitch Diameter vs. Outer Diameter

For accurate engineering, calculations must use the Pitch Diameter (PD), which is the effective diameter where the belt's tensile cord is located.

• V-Belts: The belt rides in the groove. The OD is larger than the PD. Calculations using OD will have a small error (2-5%).
• Chains: The Sprocket PD is purely geometric based on pitch and tooth count: $PD = P / \sin(180/T)$.

This tool assumes "Diameter" input is the effective Pitch Diameter for calculation purposes.

5. System Efficiency

No drive is 100% efficient.
V-Belts: ~95-98% efficient when new, but slip increases with wear.
Synchronous (Timing) Belts: ~98-99% efficient. No slip.
Roller Chains: ~98% efficient if well lubricated.
When calculating output torque, we must multiply by efficiency ($\eta$): $T_{out} = T_{ideal} \times \eta$.

6. Service Factors & Design Power

Simply matching motor HP to load HP is a recipe for failure. Real-world loads have startup shocks, vibrations, and overloads. The Service Factor (SF) is a safety multiplier applied to the Motor Power to determine the Design Power.

A belt or chain drive must be rated for this Design Power, not just the motor nameplate rating.