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This document outlines the fundamental calculations required to determine the dimensional properties, forces, and safety factors for a standard involute rack and pinion system. | Symbol | Parameter | Unit (SI) | Unit (Imperial) | | :--- | :--- | :--- | :--- | | $z$ | Number of teeth (Pinion) | - | - | | $m$ | Module | mm | - | | $P$ | Circular Pitch | mm | in | | $d$ | Pitch Diameter | mm | in | | $d_a$ | Tip Diameter (Outer Diameter) | mm | in | | $d_f$ | Root Diameter | mm | in | | $a$ | Center Distance | mm | in | | $v$ | Linear Speed | m/s | ft/min | | $n$ | Rotational Speed | rpm | rpm | | $T$ | Torque | Nm | lb-in | | $F_t$ | Tangential Force | N | lbf | | $F_a$ | Axial Force | N | lbf | | $F_r$ | Radial Force | N | lbf | | $\alpha$ | Pressure Angle | degrees | degrees | 3. Geometric Calculations 3.1 Standard Definitions The Module ($m$) is the ratio of the pitch diameter to the number of teeth. It is the primary metric for gear sizing.