= Gear friction work per engagement
{title2=$E_f=\int P_f\,dt$}
<Gear friction work per engagement> is the mechanical <energy> dissipated by sliding during one tooth pair's active engagement. It is the time integral of that pair's <gear sliding-friction loss>:
$$
E_f=\int_{t_a}^{t_r}\mu(t)F_n(t)|v_s(t)|\,dt.
$$
The result has units of <joules>, whereas instantaneous loss has units of <watts>. It excludes other gearbox losses such as support friction. Manufacturer efficiency guidance distinguishes meshing and support losses and emphasizes their dependence on operating conditions.<Technical Data Gearhead by Citizen Micro>[“Gear Efficiency”, meshing/support losses and load- and speed-dependent efficiency]{ref}
For a fixed-axis external <spur gear> pair, let signed contact coordinate $s$ run from $-L_a$ to $L_r$ through the <gear pitch point>. <Gear sliding velocity> has magnitude $(|\omega_1|+|\omega_2|)|s|$. At constant shaft speeds, <gear base-roll speed> gives $ds/dt=u_b>0$ in this chosen direction. If $\mu$ and the force carried by this pair are constant,
$$
E_f=\frac{\mu F_n(|\omega_1|+|\omega_2|)}{u_b}\int_{-L_a}^{L_r}|s|\,ds
=\frac{\mu F_n(|\omega_1|+|\omega_2|)}{2u_b}(L_a^2+L_r^2).
$$
With hypothetical $\mu=0.05$, $F_n=100\,\mathrm N$, angular-speed sum75rad/s, $L_a=5.0586\,\mathrm{mm}$, $L_r=4.5960\,\mathrm{mm}$ and $u_b=0.93969\,\mathrm{m/s}$, this gives about9.32mJ. Doubling both shaft speeds doubles instantaneous <power> but halves <gear contact duration>, leaving this idealized work per engagement unchanged. Real friction and load can themselves change with speed.
<Gear tooth load sharing> is essential when several pairs engage simultaneously. A constant total mesh load is not a constant full load on each active pair. If identical pair events recur at <gear mesh frequency> $f_m$, averaging their correctly modeled losses over many events gives $\overline P=f_mE_f$. The event's force history must already include load sharing; assigning the whole mesh load independently to every overlapping pair would overcount transmitted load and loss.
Back to article page
