Servo & CNC Motion Control, HTD/GT Toothed Timing Belts & Teeth-in-Mesh (TIM)
Toothed Timing Belt Pitch Diameter & Teeth-in-Mesh Formula
Calculate Driver & Driven Timing-Pulley Pitch Diameters (mm), Small-Pulley Engaged Teeth-in-Mesh TIM, and Exact Belt Tooth Count Lp/p across Center Distance C.
Engage fewer than 6 belt teeth on a servo motor timing pulley and elastomeric roots shear off under peak indexing torque!
Fiberglass- or Kevlar-corded **HTD / GT Curvilinear Timing Belts** (`3M`, `5M`, `8M`, `14M` Circular Tooth Pitch `p` mm) lock servo phase without slip across Driver Teeth `Z_1` and Driven Teeth `Z_2`, defining **Pulley Pitch Diameters `d_1 = (p * Z_1) / PI()`** and **`d_2 = (p * Z_2) / PI()`** (`mm`). Unlike friction V-belts, timing-belt torque is carried by shear across simultaneously engaged elastomer teeth on the smaller pulley: **Teeth-in-Mesh `TIM = Z_1 * (theta_1 / 360°)`**, where Small-Pulley Wrap Angle `theta_1 = 180° - 2 * ASIN((d_2 - d_1) / (2 * C)) * (180 / PI())`! Under ISO 5294 / Gates PolyChain & PowerGrip engineering manuals, **`TIM` MUST equal or exceed `6.0` full teeth** (`TIM < 6` forces severe `0.20 * (TIM - 1)` tooth-shear derating or ratcheting jump!), while **Exact Belt Tooth Count `N_b = L_p / p`** guides selection of the nearest catalog belt length!
Verify on an `8M` (`8.0` mm pitch) CNC Ball-Screw Servo Axis—Belt_Tooth_Pitch `p` = 8.0 mm (B2), Servo_Pinion_Teeth `Z_1` = 28 (B3 -> `d_1 = 71.30` mm in E2), Ball-Screw_Pulley_Teeth `Z_2` = 72 (B4 -> `d_2 = 183.35` mm in E3), and Nominal_Center_Distance `C` = 320.0 mm (B5): Engaged Teeth-in-Mesh hits a rock-solid `12.43 >= 6` teeth (E4) and Exact Belt Tooth Count equals `131.2` teeth (`1,049.6` mm -> select standard `1040-8M` `130`-tooth belt! E5)!
| ISO 5294 HTD-8M Timing Belt KPI (p=8 mm, Z2=72: Z1=28/C=320 vs Z1=14/C=190 mm) | Formula | Verified Output |
|---|---|---|
| Driver Pitch Diameter d1 (mm) | (Tooth_Pitch_p_mm * Driver_Z1) / PI() | 71.30 mm (35.65 mm @ Z1=14) |
| Driven Pitch Diameter d2 (mm) | (Tooth_Pitch_p_mm * Driven_Z2) / PI() | 183.35 mm |
| Engaged Teeth-in-Mesh TIM | Z1 * (Small_Wrap_deg / 360) (req >= 6.0!) | 12.43 teeth (5.22 < 6 SHEAR RISK @ Z1=14!) |
| Exact Belt Tooth Count (Lp / p) | (2*C + (PI()/2)*(d2+d1) + (d2-d1)^2/(4*C)) / p | 131.2 teeth (94.1 teeth @ C=190 mm) |
Timing-Pulley Circular Pitch Diameter, Engaged TIM & Belt Tooth-Count Formula
Validate `Tooth_Pitch p > 1` mm, integer `Driver_Z1 >= 10`, `Driven_Z2 >= Z_1`, and non-interfering `Center_Distance C > (d_2 + d_1) / 2`!
=LET(pt_m, B2, zd_r, B3, zv_n, B4, cd_m, B5, d1_m, (pt_m * zd_r) / PI(), d2_m, (pt_m * zv_n) / PI(), lp_m, 2 * cd_m + (PI() / 2) * (d2_m + d1_m) + (((d2_m - d1_m) ^ 2) / (4 * cd_m)), IF(OR(pt_m <= 1, zd_r < 10, zv_n < zd_r, cd_m <= (d2_m + d1_m) / 2), "Invalid Timing Drive", ROUND(lp_m / pt_m, 1)))Undersized 14-Tooth Pinion on Short 190 mm Centers (< 6 Teeth-in-Mesh) Check (Set Z1 B3 = 14, C B5 = 190 mm)
Shrink the motor pinion to `Z_1` B3 = 14 teeth (`d_1 = 35.65` mm in E2) at `C` B5 = 190 mm (`94.1` belt teeth in E5): Engaged Teeth-in-Mesh E4 drops to **`5.22` teeth** (`< 6.0` minimum!)—inviting tooth root peeling and servo encoder following-error jumps under emergency-stop deceleration!
=IF(ISTEXT(E5), E5, LET(d1_m, (B2 * B3) / PI(), d2_m, (B2 * B4) / PI(), wr_d, 180 - 2 * (ASIN((d2_m - d1_m) / (2 * B5)) * (180 / PI())), ROUND((B3 * wr_d) / 360, 2)))Try the example
Use the original values or change them to test your understanding.