Mechanical Drives, Roller Chains & Sprocket Kinematics
Roller Chain Sprocket Pitch Diameter & Chordal Ripple Formula
Calculate Exact Polygon Sprocket Pitch Diameter D_p (mm), Chordal Speed Variation Ripple (%), Mean Linear Chain Velocity (m/s), and Chain Strand Tension (N) from kW.
Sprockets are polygons, not circles—dividing N*P by PI() miscalculates pitch diameter AND hides chordal vibration!
Unlike belts wrapping smooth pulleys (D = Circumference / PI()), rigid roller-chain links of pitch P form a regular N-sided polygon around a sprocket! Trigonometry of the half-angle PI() / N gives EXACT Sprocket Pitch Diameter D_p = P / SIN(PI() / N)! Moreover, as each roller seats into a tooth pocket, the effective driving radius oscillates between R_max = D_p / 2 and apothem R_min = (D_p / 2) * COS(PI() / N), inducing Chordal Polygonal Speed Variation Delta_V / V = (1 - COS(PI() / N)) * 100! Below 17 teeth on high-speed drives, chordal action (>1.7% speed ripple every tooth pass) hammers rollers, spikes dynamic noise, and fatigues chain plates!
Verify on an ANSI #60 Chain Drive (Pitch P = 19.05 mm = 3/4 in, B2) with Driver_Teeth N = 19 (B3), Sprocket_RPM = 900.0 RPM (B4), and Transmitted_Power = 15.0 kW (B5): Exact Polygon Pitch Diameter is 19.05 / SIN(PI() / 19) = 115.74 mm (E2, vs 115.22 mm circular approximation!), Chordal Speed Variation stays smooth at (1 - COS(PI() / 19)) * 100 = 1.36% (E3), Mean Chain Velocity equals (19.05 * 19 * 900) / 60,000 = 5.43 m/s (E4), and Working Chain Pull F = 15,000 / v_unrounded = 2,762.81 N (E5)!
| ANSI #60 Sprocket KPI (P = 19.05 mm, N = 19 Teeth, 900 RPM, 15 kW) | Formula | Verified Output |
|---|---|---|
| Polygon Pitch Diameter Dp (mm) | Chain_Pitch_mm / SIN(PI() / Teeth_N) | 115.74 mm (67.62 @ N = 11) |
| Chordal Speed Variation (%) | (1 - COS(PI() / Teeth_N)) * 100 | 1.36% (4.05% @ N = 11!) |
| Mean Linear Chain Velocity (m/s) | (Pitch_mm * Teeth_N * RPM) / 60000 | 5.43 m/s (3.14 @ N = 11) |
| Working Chain Strand Pull (N) | (Power_kW * 1000) / Velocity_m_s | 2,762.81 N |
Exact Polygon Sprocket Pitch Diameter & Chordal Action Formula
Validate Chain_Pitch > 0, integer Sprocket_Teeth N >= 9, RPM > 0, and Power_kW > 0!
=LET(pt_m, B2, nt_s, B3, rp_m, B4, pw_k, B5, IF(OR(pt_m <= 0, nt_s < 9, INT(nt_s) <> nt_s, rp_m <= 0, pw_k <= 0), "Invalid Sprocket Input", ROUND((1 - COS(PI() / nt_s)) * 100, 2)))Undersized 11-Tooth Sprocket Chordal Vibration Spike Check (Set Teeth N = 11)
Shrink Driver_Teeth B3 from 19 down to an undersized 11-tooth sprocket: Pitch Diameter E2 drops to 67.62 mm, Velocity E4 slows to 3.14 m/s, and Chordal Speed Variation E3 nearly triples from 1.36% to 4.05%—inducing severe polygonal whip!
=IF(ISTEXT(E3), E3, ROUND((B5 * 1000) / ((B2 * B3 * B4) / 60000), 2))Try the example
Use the original values or change them to test your understanding.