= Gear sliding-friction loss
{title2=$P_f=\mu F_n|v_s|$}
<Gear sliding-friction loss> is the rate at which relative tooth-flank sliding dissipates mechanical <energy>. In a simple <Coulomb friction> model with normal contact load $F_n$, <coefficient of friction> $\mu$ and <gear sliding velocity> $v_s$, the nonnegative dissipated <power> is
$$
P_f=\mu F_n|v_s|.
$$
This is a local contact model, rather than the entire gearbox loss. Manufacturer guidance distinguishes gear-meshing losses from losses in rotating supports and warns that overall <gear efficiency> depends on load and speed.<Technical Data Gearhead by Citizen Micro>[“Gear Efficiency”, meshing/support losses and load- and speed-dependent efficiency]{ref}
To derive the sign, let $v_s=v_1-v_2$ along the common tangent. The friction force on surface1 is $f_1=-\mu F_n\operatorname{sgn}(v_s)$ and that on surface2 is $-f_1$. Their combined mechanical <power> is $f_1v_1-f_1v_2=-\mu F_n|v_s|$. The corresponding positive quantity is the rate of mechanical <energy> loss. At zero sliding the ideal Coulomb sliding-loss term vanishes, even though another resistance mechanism can remain.
For a fixed-axis external <spur gear> pair with $|\omega_1|+|\omega_2|=75\,\mathrm{rad/s}$, $|s|=4\,\mathrm{mm}$ gives $|v_s|=0.30\,\mathrm{m/s}$. Hypothetical $\mu=0.05$ and $F_n=100\,\mathrm N$ give $P_f=1.5\,\mathrm W$. Holding these load and friction assumptions fixed while doubling both shaft speeds doubles the instantaneous loss at that geometric point.
<Gear friction work per engagement> integrates this varying loss through the active <gear path of contact>. If one pair carries a constant $F_n$, the contact coordinate runs from $-L_a$ to $L_r$ at constant speed $\dot s=u_b>0$, giving
$$
E_f=\frac{\mu F_n(|\omega_1|+|\omega_2|)}{2u_b}(L_a^2+L_r^2).
$$
For $L_a=5.0586\,\mathrm{mm}$, $L_r=4.5960\,\mathrm{mm}$ and $u_b=0.93969\,\mathrm{m/s}$, the preceding loads and speeds give about $9.32\,\mathrm{mJ}$. Real tooth pairs share varying loads; an overall loss calculation must sum the applicable contacts and integrate their actual loads, sliding and friction model.
Back to article page
