= Gear tooth sliding
<Gear tooth sliding> is relative tangential motion of mating tooth surfaces at their contact point. Ideal contact prevents relative normal penetration but does not generally require the tangential surface velocities to agree. For a fixed-axis external <involute> pair, tangential slip vanishes at the <gear pitch point> and changes as the contact moves along the <gear path of contact>.<Elements of Metric Gear Technology by SDP-SI>[§3.1, involute line of action and pitch-circle geometry]{ref}
Let $P$ be the pitch point, let $\mathbf n$ be a <unit vector> along the <gear line of action>, and write a contact location as $Q=P+s\mathbf n$. With a common positive axis $\widehat{\mathbf z}$ and signed <angular velocities> $\omega_1,\omega_2$, rigid-body velocity differences give
$$
\mathbf v_1(Q)-\mathbf v_2(Q)
=(\omega_1-\omega_2)s\,\widehat{\mathbf z}\times\mathbf n.
$$
The velocities at $P$ cancel, leaving this term proportional to distance along the contact line. Since $\omega_2$ has opposite sign in an external pair, the sliding-speed magnitude is $(|\omega_1|+|\omega_2|)|s|$.
For an original example, use $\omega_1=50\,\mathrm{rad/s}$ and $\omega_2=-25\,\mathrm{rad/s}$. At $s=4\,\mathrm{mm}$, <gear sliding velocity> along the common tooth tangent is $0.30\,\mathrm{m/s}$; at $s=-4\,\mathrm{mm}$ it has the opposite sign. At $s=0$ it is zero. Equality of operating pitch-circle speeds must therefore not be used to conclude that the tooth flanks roll without slipping throughout engagement.
In a simplified <Coulomb friction> model with <coefficient of friction> $\mu$ and normal load $F_n$, the dissipated <gear sliding-friction loss> is $P_f=\mu F_n|v_s|$. Hypothetical values $\mu=0.05$, $F_n=100\,\mathrm N$ and $|v_s|=0.30\,\mathrm{m/s}$ give $1.5\,\mathrm W$. This is a local illustrative loss model, not a lubricant specification. Zero ideal sliding at the pitch point also does not eliminate every source of gearbox loss, such as support friction or deformation-related dissipation. Real load and friction can vary through the engagement, so total loss requires integrating the applicable contact model.
Back to article page
