Watch Gear Train Transmission Ratio
Calculates the total step-up velocity ratio from the center wheel (1 revolution per hour) through the third, fourth (seconds), and escape wheels in a horological wheel train.
\[ R_{total} = \prod_{i=1}^{n} \frac{Z_{wheel, i}}{Z_{pinion, i}} = \frac{Z_2 \cdot Z_3 \cdot Z_4 \cdot Z_e}{z_3 \cdot z_4 \cdot z_e \cdot z_{bal}} \]
Plain Text Formula:
Ratio = (Z_center * Z_third * Z_fourth * Z_escape) / (z_third * z_fourth * z_escape)
📐 Variable Definitions & Parameters
| Symbol | Variable Name | Standard Unit | Physical Definition |
|---|---|---|---|
| \[ R_{total} \] | Total Velocity Ratio | dimensionless | Ratio of escape wheel revolutions per hour to mainspring barrel |
| \[ Z_i \] | Wheel Tooth Count | teeth | Driving wheel gear tooth count |
| \[ z_i \] | Pinion Leaf Count | leaves | Driven pinion gear leaf count |
💡 Worked Engineering Example
Problem Statement:
A center wheel (64 teeth) drives a third pinion (8 leaves); third wheel (60 teeth) drives a fourth pinion (8 leaves). Find step-up ratio.
Step-by-Step Solution:
- Step 1 ratio: 64 / 8 = 8
- Step 2 ratio: 60 / 8 = 7.5
- Combined ratio: 8 * 7.5 = 60
🎯 Result: The fourth wheel turns 60 times per hour (1 revolution per minute), driving the seconds hand.