Timing Belt Pitch Diameter⚠ unverified
Mechanical / Flexible Elements · Compute the pitch diameter of a timing belt pulley
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| teeth | teeth | — | 1.0 | Number of teeth on the pulley, dimensionless |
| pitch | pitch | in | 1.0 | Belt tooth pitch |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | d | in | Pulley pitch diameter, in inches (in) |
The science & history
Understanding the Parameters
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Tooth count $N$ — the pulley must have an integer number of teeth spaced at the belt pitch. Diameter scales linearly with $N$, and the speed ratio of a timing drive is exactly the ratio of tooth counts $N_2/N_1$ — precise and slip-free, the whole point of synchronous belts.
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Belt pitch $p$ — the fixed tooth spacing (a belt-series property, e.g. 5 mm HTD, or imperial fractions). Pulley and belt must share the same pitch to mesh.
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Pitch vs outside diameter — $d = Np/\pi$ is the pitch diameter (where the belt's neutral tension line runs), slightly inside the pulley's outside diameter. Speed and ratio calculations use the pitch diameter.
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Why slip-free matters — because the teeth positively engage, timing belts hold exact angular phase (camshafts, robotics, 3D-printer axes), transmit without the tensioning a friction belt needs, and have a ratio set purely by tooth counts.
Derivation (Approaching a Proof)
The belt teeth are spaced at pitch $p$ along the belt. When wrapped on the pulley, $N$ consecutive teeth must lie on the pitch circle, each occupying an arc of length $p$. So the pitch circumference equals the total tooth spacing:
$$\pi d = N\,p \;\Longrightarrow\; d = \frac{N\,p}{\pi}.$$
Strictly, the teeth are chords of the pitch circle, so the exact relation involves $p = d\sin(\pi/N)$; for the many teeth of a real pulley, $\sin(\pi/N) \approx \pi/N$ and the simple $d = Np/\pi$ is used. It is the belt analog of a gear's $d = N/P_d$ (module × teeth) relation.
Dimensional check. $d = \dfrac{N\,p}{\pi} = \dfrac{(\text{–})\cdot\text{in}}{(\text{–})} = \text{in}$ — a length, as required ($N$ and $\pi$ dimensionless).
History and Development
Toothed (synchronous) belts emerged in the mid-20th century to combine the quiet, lightweight, maintenance- free nature of belts with the positive engagement of gears and chains. The pitch-diameter relation is the foundation of their kinematics; because the ratio is set by integer tooth counts, timing belts are used wherever exact phase or speed is required — automotive cam drives, printers, CNC and robotics axes. Their tooth loading (Timing Belt Tooth Load) and pitch geometry are the two core design quantities.
Related Concepts: Timing Belt Tooth Load, V Belt Speed, Chain Velocity, Belt Power, V-Belt Design
Notes: Pitch diameter (belt neutral line), slightly inside the OD. Slip-free positive drive; speed ratio $= N_2/N_1$ (exact). Exact relation $p = d\sin(\pi/N) \to d = Np/\pi$ for many teeth. Belt/pulley must share the same pitch. Imperial here (pitch in inches).