1. Principles of Chain Drive and Mechanical Comparison with Belt Drive

In power transmission element design, a Chain Drive is a key positive-engagement transmission system that fuses the advantages of both belt and gear drives. It achieves a reliable, constant velocity ratio transmission without slip, even across relatively long center distances. Compared to belt drives, because it is a geometric engagement drive that does not rely on friction, the initial tension acting on the shaft is very small. This significantly reduces the radial load exerted on the shaft and bearings.

Compared to gear drives, chain drives offer exceptional advantages, such as highly flexible center distance adjustments, ease of multi-shaft transmission, and smooth operation even under harsh, high-temperature, or dusty environments. However, due to the geometric polygonal effect that occurs when sprocket teeth mesh with chain links, velocity fluctuations and minor noise/vibrations can occur. Therefore, managing the minimum number of sprocket teeth is extremely critical for precise, high-speed power transmission.

2. Derivation of the Pitch Circle Diameter (PCD) Formula and Mathematical Foundations of Tension Calculation

Since chain links wrap around a sprocket as chords, the Pitch Circle Diameter (PCD) cannot be calculated using the standard circular relationship π × D; instead, it is derived using triangular geometric relationships. If the chain pitch is P and the number of sprocket teeth is Z, the angle between adjacent teeth is 360° / Z, and half of that angle is 180° / Z. Accordingly, the formula for the radius R from the sprocket center to the link pin is derived as follows:

sin(180° / Z) = (P / 2) / R  =>  R = P / [2 × sin(180° / Z)]

Therefore, the precise formula for the sprocket’s Pitch Circle Diameter (D_p, PCD) is established as:

D_p = P / sin(180° / Z)  [mm]

Based on the derived PCD, the average linear velocity of the driving chain v [m/s] is calculated, and the tension force (F) acting on the tight side of the chain generated by the motor’s transmitted power P_w [kW] is derived as follows:

v = (D_p × π × N_1) / 60,000  [m/s]   (N_1 : Drive RPM)

F = (P_w × 1000) / v  [N]   (Supports conversion where 1 kgf = 9.80665 N)

3. Design Considerations and Lubrication Methods for Roller Chains (KS B 1407 Standards)

To guarantee the design life of roller chains, dynamic frictional wear and the impacts caused by the polygonal effect must be managed. In particular, if the number of teeth on the drive sprocket is too small (less than 15T), the velocity fluctuation rate of the chain rises sharply, and fatigue failure due to link articulation increases dramatically. Therefore, the recommended minimum number of teeth Z_min = 17 (or 21T or more under shock loads) should be strictly followed.

• Calculation of Velocity Fluctuation Rate: The velocity fluctuation rate of the sprocket ε is expressed as ε = (v_max - v_min)/v_max = 1 - cos(180°/Z), which decreases sharply as the number of teeth increases.
• Selection of Lubrication Method: Lubrication methods are specified based on the chain speed. For v < 1.5 m/s, manual or drip-feed lubrication is recommended; for 1.5 m/s < v < 8.0 m/s, oil bath (immersion) lubrication is suitable; and for v > 8.0 m/s, a forced circulation lubrication system using high-pressure nozzles must be adopted to continuously supply a lubricant film inside the link pins (based on KS B 1407 standards).