= Overhung idler loading
<Overhung idler loading> occurs when an <idler gear>'s mesh resultant acts outside the axial span of its shaft supports. The overhang creates a lever arm, so the nearer <mechanical bearing> can carry a larger transverse load than the mesh resultant and the farther support can have the opposite reaction direction. This geometry is separate from whether the idler's two transmitted <torques> cancel.<Shafts and Bearings by SDP-SI>[§4.0, case(b), “Overhung Radial Load”, and case(f), external-spur-gear forces]{ref}
Place ideal radial supports A and B at $z=0$ and $z=L$. An upward mesh resultant $P$ at $z=L+e$, with overhang $e>0$, gives <static equilibrium>
$$
R_A+R_B+P=0,\qquad R_BL+P(L+e)=0.
$$
Hence
$$
R_A=P\frac eL,\qquad R_B=-P\left(1+\frac eL\right).
$$
For $P=200\,\mathrm N$, $L=100\,\mathrm{mm}$ and $e=30\,\mathrm{mm}$, A acts upward with60N while B acts downward with260N. Their signed sum balances the applied200N. The larger reaction at B is a lever effect, not an additional external mesh force.
The <overhung shaft bending moment> immediately next to the near support has magnitude $Pe=6\,\mathrm{N\,m}$ in this ideal model. That <bending moment> can exist while the net shaft-axis <torque> on the steady idler is zero. Doubling overhang at fixed load doubles the near-support bending moment and changes the reaction magnitudes to120 and320N.
<Idler bearing load sharing> must therefore include the gear's axial position, not just its pitch radius and speed ratio. Loads in independent transverse directions can be balanced separately and then combined as vectors. A support model with more constraints requires compatible deformations as well as force balance. Shaft bending and support flexibility can also alter the <gear face load distribution>, linking the overhang geometry to tooth loading.
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