This mechanical engineering calculator computes exact center-to-center axle distances, timing belt pitch lengths, and tooth engagement counts for synchronous belt drives. Designers of 3D printers (such as Voron and CoreXY builds), CNC routers, robotics actuators, and automotive timing assemblies rely on precise center distance sizing to ensure proper belt tensioning and avoid tooth skipping.

### Timing Belt Geometry and Center Distance Equations The calculator solves synchronous closed-loop timing belt geometry:

- **Pitch Length ($L$):** $L = N_b \times p$ (where $N_b$ is belt tooth count and $p$ is tooth pitch in millimeters, such as 2.0mm for GT2). - **Drive Ratio ($R$):** $R = z_2 / z_1$ (where $z_1$ is driver teeth and $z_2$ is driven teeth). - **Center-to-Center Distance ($C$):** Solved iteratively using the standard ASME/ISO synchronous belt approximation: $$b = 2N_b - (z_1 + z_2)$$ $$C \approx \frac{p}{8} \left[ b + \sqrt{b^2 - 8 \left( \frac{z_2 - z_1}{\pi} \right)^2} \right]$$ - **Teeth in Mesh on Smaller Pulley:** Evaluates tooth engagement arc to verify minimum 6-tooth mesh requirements.

### Worked Numeric Example Suppose a robotics engineer designs a GT2 reduction drive for a CoreXY gantry: - **Driver Pulley ($z_1$):** 20 teeth - **Driven Pulley ($z_2$):** 40 teeth (2:1 reduction ratio) - **Belt Tooth Count ($N_b$):** 150 teeth - **Belt Pitch ($p$):** 2.0 mm (GT2 standard)

**Step-by-Step Evaluation:** 1. **Belt Pitch Length:** $150 \times 2.0 = 300.0\text{ mm}$. 2. **Drive Ratio:** $40 / 20 = 2.0\times$. 3. **Intermediate Parameter $b$:** $2(150) - (20 + 40) = 300 - 60 = 240$. 4. **Center Distance ($C$):** Evaluates to approximately **$118.8\text{ mm}$**. 5. **Mesh Verification:** Smaller pulley has $> 8\text{ teeth}$ in mesh, well above the 6-tooth anti-skip safety threshold.

### Client-Side Processing Note All trigonometric and quadratic equations execute locally in client-side JavaScript. No CAD dimensions, 3D printer designs, or motor specifications are transmitted across any network.