Roller chain looks like the simplest power transmission element on the planet — pin, bushing, roller, side plates, repeat — but the geometry hides two behaviors that catch designers off guard: chain elongation and the chordal effect.

Pitch is everything. A chain's pitch is the center-to-center distance between adjacent pins. ANSI standard pitches (in eighths of an inch) give you the size code: #40 chain has 4/8" = 0.500" pitch, #60 has 0.750", #80 has 1.000". A sprocket must have the same pitch — you can't mix. The tooth count and pitch together set the pitch diameter: PD = P / sin(180°/N), where P is pitch and N is tooth count.

The chordal effect (polygon action). A chain doesn't wrap a sprocket as a smooth circle — it wraps as a polygon. As each link engages, the effective radius oscillates between the pitch radius (pin at tooth tip) and a smaller chord radius (pin between teeth). The result: even at constant sprocket RPM, chain linear velocity pulsates. The variation is roughly:

• 6 teeth: ±13.4% speed variation (violently rough)
• 12 teeth: ±3.4%
• 19 teeth: ±1.4%
• 25+ teeth: negligible

Rule of thumb: never use fewer than 17 teeth on the driver for smooth, quiet operation. High-speed drives want 21+.

Chain elongation, not stretch. Chains don't stretch elastically in service — the pins and bushings wear, and each tiny bit of material loss adds up across every link. A chain with 100 links losing 0.001" per joint gets 0.100" longer. Once elongation exceeds 3% of pitch (some say 1.5% for hardened sprockets), the chain rides up the sprocket teeth, contact shifts to the tooth tips, and both chain and sprocket destroy each other fast. That's why you replace chain and sprockets together — a worn sprocket eats a new chain in weeks.

Real-world example: A motorcycle final drive uses #520 chain (5/8" pitch, 0.25" width) with a 15T front and 45T rear sprocket — a 3:1 reduction. The 15T front violates the "17 teeth minimum" for smoothness, which is exactly why motorcycle chains are noisy and why racers going to smaller front sprockets for more torque accept measurably faster chain wear. Odd tooth counts (15, 45) are also chosen deliberately so the same chain link doesn't meet the same sprocket tooth every revolution — spreading wear evenly.

Quick sizing rule: Wrap angle on the smaller sprocket should exceed 120°, meaning the center distance should be 30–50× the chain pitch for typical drives.