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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 floating-pinion balancing mode is a reduced mechanical motion in which a floating pinion shifts between mating meshes and redistributes their reactions. A one-coordinate model can connect the static balance set by floating-pinion centering stiffness with oscillation, damping and the possibility of losing a contact.

For two compressed linear contacts of stiffness $k$, mismatch $e$, support stiffness $k_s$, effective mass $m$ and viscous damping coefficient $c$, the illustrative equation isThe static position is $x_{\mathrm{eq}}=-2ke/K$. Setting $y=x-x_{\mathrm{eq}}$ givesFor $0<\zeta <1$, the oscillation frequency is ${\omega }_d={\omega }_n\sqrt{1-{\zeta }^2}$ and its amplitude envelope decays as $e^{-ct/(2m)}$. These frequencies have angular-frequency units; ordinary frequency is $f_n={\omega }_n/(2\pi )$.

An original example uses $k=10^{6}N/m$, $k_s=0.5\times 10^{6}N/m$, $m=0.25\mathrm{kg}$ and $c=250\mathrm{N}s/m$. It gives $K=2.5\times 10^{6}N/m$, $f_n\simeq 503.3\mathrm{Hz}$, $\zeta \simeq 0.158$ and an envelope reduced to $e^{-1}$ after $2\mathrm{ms}$. With mismatch $e=2\mu \mathrm{m}$, the equilibrium displacement is $-1.6\mu \mathrm{m}$. These are chosen model parameters, not measured gearbox constants.

A floating-pinion contact-loss check must still enforce the assumptions. For reactions $F_1=k(\delta +e+x)$ and $F_2=k(\delta -e-x)$, both contacts remain compressed only while $|e+x|<\delta$. If the oscillation leaves that interval, the two-contact linear equation must change; extrapolating a sinusoid into tensile tooth forces is inconsistent.

A motivating experiment perturbs the pinion slightly about a controlled loaded state, records its displacement and mesh-force proxies, and compares frequency and decay with the model while checking contact continuity. The primary face-gear report used slow rolling to study torque division and mounting freedom; it is evidence for that tested balancing arrangement, not a measurement of the hypothetical resonant frequency above.^[ref]

The floating-pinion compression margin is the amount of relative approach remaining before the less compressed contact reaches zero compression in a stated two-contact model. It turns the assumptions of a floating-pinion balancing mode into a quantitative admissibility check.

For $F_1=k(\delta +e+x)$ and $F_2=k(\delta -e-x)$, let equilibrium position be $x_{\mathrm{eq}}$ and defineA positive $M$ means both contacts are compressed at equilibrium. If $x=x_{\mathrm{eq}}+y$ with $|y|\le A$, then $A<M$ guarantees compression throughout the bounded motion. For a steady single sinusoid and fixed $\delta$, this is the exact full-cycle condition; a general transient envelope supplies only a sufficient bound.

An original nominal example has $\delta =5\mu \mathrm{m}$ and $b=e+x_{\mathrm{eq}}=0.5\mu \mathrm{m}$, giving $M=4.5\mu \mathrm{m}$. A displacement amplitude $4.2\mu \mathrm{m}$ passes that nominal check, leaving $0.3\mu \mathrm{m}$ of minimum compression for the sinusoidal model.

A floating-pinion robust contact margin includes bounded parameter uncertainty. If $\delta \in [{\delta }_L,{\delta }_U]$ and $b\in [b_L,b_U]$, a conservative lower margin isFor $\delta =5\pm 0.2\mu \mathrm{m}$ and $b=0.5\pm 0.3\mu \mathrm{m}$, $M_L=4.0\mu \mathrm{m}$. The same amplitude $4.2\mu \mathrm{m}$ is no longer guaranteed admissible, although failure of the bound does not prove contact loss. Amplitude $3.8\mu \mathrm{m}$ leaves at least $0.2\mu \mathrm{m}$ under these independent interval bounds.

In the symmetric static model, $x_{\mathrm{eq}}=-2ke/(2k+k_s)$ givesfor positive mesh stiffness $k$ and nonnegative floating-pinion centering stiffness $k_s$. Mounting freedom can change the margin, but it also changes dynamics; maximizing this one scalar is not a complete gearbox design rule.

A floating-pinion contact-loss check must recompute the contact model once a force reaches zero. The historical face-gear tests provide evidence about mounting freedom and measured torque division under slow rolling, rather than a universal dynamic margin.^[ref]