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A floating pinion has mounting freedom that permits a limited position adjustment between its mating meshes. The resulting force balance can improve load division in a split-torque transmission. Floating does not mean unsupported: bearing and shaft constraints still transmit loads, and the allowed displacement must match the intended balancing direction.

An original symmetric scalar model illustrates the mechanism. Let two opposed mesh reactions bewhere $\delta$ sets mean compression, $e$ represents an initial mismatch and $x$ is the pinion's balancing displacement. Assume both contacts remain compressed. Let floating-pinion centering stiffness $k_s$ provide support reaction $-k_{s}x$, with the reaction sign chosen so that equilibrium is $F_2-F_1-k_{s}x=0$. ThenWith negligible centering restraint, $x\to -e$ and the reactions become equal. In the rigid-centering limit, $x\to 0$ and the original imbalance remains.

For stipulated $k=100N/\mu \mathrm{m}$, $e=1\mu \mathrm{m}$ and $\delta =5\mu \mathrm{m}$, rigid centering gives forces $600$ and $400\mathrm{N}$. Free adjustment gives $500$ and $500\mathrm{N}$. At $k_s=200N/\mu \mathrm{m}$, displacement is $-0.5\mu \mathrm{m}$ and forces are $550$ and $450\mathrm{N}$. These describe the chosen scalar model, rather than a complete face-gear support design.

The primary 2002 rig slow-rolled pinions between two face gears, applied loads through hanging masses, pulleys and cables, and used tooth-root strain gauges. Its cantilevered pinion mount allowed a balancing displacement.^[ref] The report's modified fixed-pinion comparison sent nearly all its load to one face-gear mesh, illustrating the effect of constraint in that tested assembly.^[ref]

Finite travel, friction, backlash or loss of one contact can prevent the ideal balancing solution. At low load, the condition that both model forces are nonnegative must be checked; a gear tooth engagement threshold can change the equations. Position freedom can help sharing without guaranteeing identical tooth stress or a perfectly equal torque split at every operating condition.

A gear contact-set consistency check verifies that a proposed solution satisfies force balance and the unilateral contact conditions used to define a gear mesh active-contact set. Solving an equation for selected contacts is insufficient if an omitted tooth should be loaded or an included tooth would require tension.

For the model $F_j=k_j\max (\delta -e_j,0)$, choose a trial active set $\mathcal{A}$ and solveThe check then requires positive $k_j(\delta -e_j)$ for included pairs and $\delta \le e_j$ for excluded pairs. Recomputing every contact with the unilateral law gives the force residualA consistent exact solution has $r_F=0$ together with the inclusion inequalities.

An original example has stiffnesses $100,50,150N/\mu \mathrm{m}$ and gaps $0,2,5\mu \mathrm{m}$. At $F_{\mathrm{applied}}=900\mathrm{N}$, a trial set containing only the first two pairs gives $\delta =20/3\mu \mathrm{m}$. It balances those selected forces but exceeds the third gap by $5/3\mu \mathrm{m}$. The omitted contact would carry $250\mathrm{N}$, giving $r_F=250\mathrm{N}$. Including all three gives the consistent approach $35/6\mu \mathrm{m}$.

Conversely, including all three at applied load $500\mathrm{N}$ gives the un-clipped solution $\delta =4.5\mu \mathrm{m}$ and a third force $-75\mathrm{N}$. Rejecting that tensile contact and solving with the first two gives $\delta =4\mu \mathrm{m}$ and forces $400,100,0\mathrm{N}$.

Numerical tolerances should keep their units and roles clear. A displacement tolerance near a gear tooth engagement threshold is not the same quantity as a force-residual tolerance. At an exact zero-force touch, either a geometric-touch flag or the positive-force active convention may be useful, but the reported convention must be stated. The one-sided gear mesh tangent stiffness can differ across that boundary.

Primary dynamic gear analysis couples tooth stiffness and calculated loads as contact position changes.^[ref] The scalar check above validates internal consistency of its quasistatic model; it does not establish that the model includes every deformation or dynamic effect in a real mesh.

The gear mesh active-contact set identifies the tooth pairs that carry positive compressive normal force in a specified contact model. It is not simply the set of geometrically overlapping teeth from the gear contact ratio. A pair touching at zero force is excluded under the positive-force convention used here.

For unilateral linear contacts,Here $e_j$ is the initial gap, $k_j>0$ the pair stiffness, and $\delta$ the common approach. For a proposed nonempty set $\mathcal{A}$, force balance givesA gear contact-set consistency check then verifies $\delta >e_j$ for included pairs and $\delta \le e_j$ for excluded ones. Solving the linear equation without this check can retain a negative force or omit a contact that should engage.

An original three-pair example uses stiffnesses $100,50,150N/\mu \mathrm{m}$ and gaps $0,2,5\mu \mathrm{m}$. At total load $500\mathrm{N}$, the set $\{1,2\}$ gives approach $(500+50\cdot 2)/150=4\mu \mathrm{m}$ and forces $400,100,0\mathrm{N}$. This set is consistent.

At $900\mathrm{N}$, keeping only those two contacts would give approach $1000/150\simeq 6.667\mu \mathrm{m}$, beyond the third gap. Including all three instead givesand forces approximately $583.33,191.67,125\mathrm{N}$, which sum to $900\mathrm{N}$. The third gear tooth engagement threshold occurs at total load $100\cdot 5+50(5-2)=650\mathrm{N}$.

Within a fixed active set, gear mesh tangent stiffness is the sum of its pair stiffnesses. At a transition, force can remain continuous while that derivative changes. For positive total load the scalar force law is strictly increasing after the first contact, giving a unique approach in this model; the unloaded state need not specify one unique separation.

Real deformation, local contact nonlinearity and dynamics can require more than one common-approach coordinate. A primary NASA analysis coupled tooth stiffness with dynamic-load calculations over contact position and speed.^[ref] The active-set calculation is an explicit quasistatic model, not a substitute for that fuller mechanical description.

The gear mesh tangent stiffness is the local derivative of total mesh force with respect to a specified relative approach, with contact position and other state variables held fixed:It predicts a sufficiently small incremental force $\Delta F\simeq k_{\tan }\Delta \delta$. Unlike a reference-based secant stiffness $F/\delta$, it describes the local response rather than the whole loading interval.

For $F(\delta )={\sum }_j{k}_j\max (\delta -e_j,0)$ away from any gear tooth engagement threshold,Only members of the gear mesh active-contact set contribute. With $k_1=80$, $k_2=120N/\mu \mathrm{m}$ and gaps $0$ and $3\mu \mathrm{m}$, the gear mesh tangent stiffness is $80N/\mu \mathrm{m}$ before the second contact and $200N/\mu \mathrm{m}$ after it. At approach $4.8\mu \mathrm{m}$, an additional $0.1\mu \mathrm{m}$ gives $20\mathrm{N}$ more force within the same branch.

The ideal transition at $3\mu \mathrm{m}$ has distinct left and right derivatives. A centered finite difference with any sufficiently small positive $h$ giveswhere numerical displacements are expressed in micrometres. The value $140$ is the mean of the two branch slopes; it does not establish a unique derivative at the corner, even as $h$ shrinks.

The measured displacement channel also matters. If a contact stiffness $200N/\mu \mathrm{m}$ is in series with support stiffness $300N/\mu \mathrm{m}$, their mechanical compliances add, givingA measurement spanning the whole assembly would therefore not isolate the contact stiffness.

For a perturbation experiment, report the mean load, mesh position, displacement endpoints and whether the increment crosses a contact transition. A primary NASA analysis used a coupled tooth-stiffness/dynamic-load calculation; the derivative above explains a local quasistatic limit, rather than replacing that dynamical model.^[ref]

The gear tooth load fraction ${\eta }_j$ is the fraction of a stated total mesh force carried by tooth pair $j$ at a particular contact position or time:For nonnegative forces resolved in the same direction, $0\le {\eta }_j\le 1$ and ${\sum }_j{\eta }_j=1$. A normal-force fraction must not mix one pair's normal load with another convention's tangential total. At zero total load the ratio is undefined.

An original quasistatic model treats engaged pairs as parallel springs with the same normal approach $\delta$. If $F_j=k_j\delta$ and every pair has zero initial gap, thenTwo pairs with stiffnesses $60$ and $140N/\mu \mathrm{m}$ under total load $1000\mathrm{N}$ approach by $5\mu \mathrm{m}$ and carry $300$ and $700\mathrm{N}$. Their fractions are $0.3$ and $0.7$, rather than one half each. Counting two geometrically overlapping contacts does not establish equal gear tooth load sharing.

An initial gap $e_j$ changes the model toA pair below its gear tooth engagement threshold carries no load in this unilateral model. Changing total load can change both the active pairs and their fractions, even if all the individual stiffnesses remain constant. Gear contact ratio specifies geometric overlap; load fractions require deformation and engagement information.

The fractions also prevent double counting. If total load is $1000\mathrm{N}$, assigning that full load to both of two pairs gives an impossible sum of $2000\mathrm{N}$ within this model. A gear friction work per engagement calculation must use each pair's own $F_j(t)$ before summing its dissipation.

Dynamic motion can alter the quasistatic distribution. A 1986 NASA analysis calculated tooth stiffness and loads as contact position and speed varied, using a coupled stiffness/load iteration.^[ref] Fractions inferred from the simple spring model should therefore be identified as model results, rather than universal operating values.