 Gear line of action

The gear line of action is the common normal along which ideal tooth contact transmits force. For meshing involute profiles in a transverse plane, it is a straight common tangent to the two gear base circles. The angle it makes with the common tangent to the operating pitch circles is the operating pressure angle.^[ref]

Its intersection with the line of gear centers is the gear pitch point. The actual contact point moves along only a finite portion, the gear path of contact, bounded by usable tooth profiles. Thus a diagram's infinite straight line is a geometric construction, not a claim that teeth can contact everywhere on it.

The geometry explains constant ratio. Let the tooth-normal force magnitude be $F_n$ and the base radii $r_{b1},r_{b2}$. The perpendicular distance from each gear center to the gear line of action is its base radius, so the ideal transmitted moments have magnitudesThese moment arms do not change as the contact point advances along that line. The no-penetration contact condition correspondingly gives $r_{b1}{\omega }_1+r_{b2}{\omega }_2=0$ for an external pair. Compatible profiles have $r_{b2}/r_{b1}=N_2/N_1$, recovering the tooth-count gear ratio.

For example, reference radii $20$ and $40\mathrm{mm}$ at reference gear pressure angle $20{}^{\circ }$ give base radii approximately $18.794$ and $37.588\mathrm{mm}$. A $100\mathrm{N}$ normal force produces moment magnitudes $1.8794$ and $3.7588\mathrm{N}\mathrm{m}$. Their ratio stays two as the point of contact moves.

Ideal normal transmission does not eliminate gear tooth sliding. Relative surface slip generally changes along the gear path of contact and vanishes at the gear pitch point for the ideal external pair. Actual friction associated with that slip adds a tangential contact force; its resultant need not lie exactly on the geometric gear line of action.