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Floating-pinion centering stiffness describes the local restoring-force response of the mounting to displacement along a specified balancing coordinate. If mounting reaction is $R_s(x)$, a convenient positive restoring convention isFor $R_s=-k_{s}x$, $k_s$ has force-per-length units. A rotational coordinate would require a torque-per-angle stiffness instead; the coordinate and displacement endpoints must be stated before comparing values.

In an original symmetric model of a floating pinion, two compressed mesh reactions obeyWith support reaction $-k_{s}x$, equilibrium givesThe rigid-centering imbalance is $D_{\mathrm{rigid}}=2ke$, soBoth reactions must remain nonnegative for this two-contact model to apply.

For $k=100N/\mu \mathrm{m}$, $e=1\mu \mathrm{m}$ and $\delta =5\mu \mathrm{m}$, choose $k_s=50N/\mu \mathrm{m}$. Displacement is $-0.8\mu \mathrm{m}$ and reactions are $520$ and $480\mathrm{N}$. Raising $k_s$ to $800N/\mu \mathrm{m}$ reduces displacement to $-0.2\mu \mathrm{m}$ but gives forces $580$ and $420\mathrm{N}$. Stronger centering keeps the location closer to its reference while preserving more of the original imbalance.

Dynamic response is another question. With an effective mass $m$ and viscous coefficient $c$, the same linearized floating-pinion balancing mode obeysIts undamped natural frequency is $\sqrt{(2k+k_s)/m}$ in angular-frequency units. A favorable static split does not by itself establish favorable dynamic behavior, and the toy mode does not include every shaft or housing motion.

A stiffness measurement should separate mounting reaction from the mesh reactions or report the combined assembly stiffness explicitly. The primary 2002 face-gear program used a cantilevered mounting to permit pinion balancing and examined a constrained comparison; it did not establish one universal centering stiffness for all transmissions.^[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]

A floating-pinion contact-loss check tests whether the motion predicted by a floating-pinion balancing mode remains compatible with every contact assumed to be compressed. It prevents a two-contact linear solution from being extrapolated into tensile tooth forces.

For the symmetric modelwith $k>0$, mismatch $e$, displacement $x$ and fixed mean approach $\delta >0$, both contacts are compressed only whenAt equality one modeled force is zero. Outside the interval the linear expression makes one force negative; the unilateral model must replace that value by zero and the force-balance equations must be solved again.

Write $x=x_{\mathrm{eq}}+y$ and define the floating-pinion compression marginIf the motion satisfies $|y(t)|\le A(t)$, then $A(t)<M$ is sufficient to retain both contacts. For a steady single sinusoid of amplitude $A$ and fixed $\delta$, the maximum of $|e+x(t)|$ over a complete cycle is $|e+x_{\mathrm{eq}}|+A$, making the condition exact for that stipulated motion.

An original example uses $e=1\mu \mathrm{m}$, $x_{\mathrm{eq}}=-0.5\mu \mathrm{m}$ and $\delta =5\mu \mathrm{m}$. The margin is $4.5\mu \mathrm{m}$. A sinusoidal amplitude $4\mu \mathrm{m}$ retains a minimum compression $0.5\mu \mathrm{m}$; with $k=100N/\mu \mathrm{m}$, the smallest contact force is $50\mathrm{N}$. Amplitude $5\mu \mathrm{m}$ instead makes the unconstrained formula reach $-50\mathrm{N}$, revealing a contact-loss inconsistency rather than a physical tensile contact.

For a damped transient, an amplitude envelope is a sufficient bound but need not be attained at the phase where contact is least compressed. Failure of that sufficient envelope test therefore calls for examining the actual trajectory; it is not automatically proof that a real contact opened.

A controlled dynamic test can compare displacement with load-sensitive observations and check whether the assumed contact set persists. The historical face-gear program examined mounting freedom and torque division under slow rolling; it did not validate every possible dynamic contact assumption.^[ref]