= Idler support reaction
An <idler support reaction> is the <force> exerted by the shaft and bearings that balances the mesh loads and other external loads on an <idler gear>. A vanishing net transmitted <torque> does not imply a vanishing resultant <force>: moments can cancel while forces add. A <free-body diagram> must retain each tooth-contact force and its point of application. The tangential load in each ideal mesh follows from $F_t=|\tau|/r_w$.<Elements of Metric Gear Technology by SDP-SI>[§7, gear forces; tangential transmitted load and pressure-angle separation of mesh forces]{ref}
Consider a steady collinear train with input, idler and output reference radii20,30 and50mm. An input <torque> $2\,\mathrm{N\,m}$ gives a100N tangential mesh component. On the idler, choose contact positions at $x=-30$ and $x=30\,\mathrm{mm}$ relative to its center, with both tangential forces upward. Their moments are $-3$ and $+3\,\mathrm{N\,m}$, but their vector sum is200N upward. For equal20degree pressure angles, the two radial mesh components have magnitude $100\tan20\,{}^{\circ}\approx36.4\,\mathrm N$ and opposite directions, so they cancel. Neglecting other loads, the net support reaction is200N downward.
This is the combined support resultant, not automatically the load on each bearing. For a hypothetical shaft supported at axial positions $z=0$ and $z=100\,\mathrm{mm}$, with the200N upward gear resultant acting at $z=40\,\mathrm{mm}$, <static equilibrium> gives
$$
R_A+R_B=-200\,\mathrm N,\qquad R_B(100\,\mathrm{mm})+200\,\mathrm N(40\,\mathrm{mm})=0.
$$
Thus $R_B=-80\,\mathrm N$ and $R_A=-120\,\mathrm N$. <Idler bearing load sharing> depends on placement even when the total reaction is fixed.
Noncollinear shaft layouts rotate the mesh-force directions, and <helical gears> add <gear axial force>. Acceleration also changes the balance: the sum of <torques> must supply the idler's angular acceleration, and unequal mesh loads need not cancel radially. The zero-net-torque example therefore establishes a steady ideal force balance, rather than a universal bearing-load formula.
Back to article page
