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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]

A floating-pinion robust contact margin is the smallest two-contact compression margin allowed by a stated set of uncertain parameters. It strengthens a nominal floating-pinion compression margin by checking every admissible parameter combination, while retaining the assumptions of the underlying contact model.

Let $\delta$ be the common approach and $b=e+x_{\mathrm{eq}}$ the equilibrium imbalance in the symmetric linear model. If the admissible pairs form a set $\mathcal{U}$, defineFor displacement $y$ from equilibrium with $|y|\le A$, the smaller contact compression is at least $M_{\mathrm{rob}}-A$. Thus $A<M_{\mathrm{rob}}$ guarantees positive compression for both contacts throughout the bounded motion. Equality permits zero compression and does not guarantee maintained compressive contact.

For independent interval bounds, or a bounding rectangle containing all possible pairs,This is exact when every pair in the rectangle is admissible. If the rectangle includes impossible combinations, it remains a conservative lower bound on $M_{\mathrm{rob}}$.

A correlated floating-pinion uncertainty set can materially change the conclusion. In an original example, all quantities are in micrometres andHere $b$ stays positive, so $\delta -|b|=4.5$ for every allowed $u$. The exact joint-set margin is $4.5\mu \mathrm{m}$. Treating the ranges independently instead gives $4.8-0.7=4.1\mu \mathrm{m}$. An amplitude $A=4.3\mu \mathrm{m}$ passes the joint-set guarantee with residual compression $0.2\mu \mathrm{m}$, but fails the rectangular sufficient test. Failure of a conservative test does not itself demonstrate contact loss.

The admissible set must come from defensible parameter bounds, including any shared mounting or thermal effects. A covariance estimate alone supplies no absolute bound for all realizations. If the motion amplitude also depends on uncertain parameters, a direct bound on $\delta -|b|-A$ can preserve correlations that separate worst-case bounds discard.

Floating mounting was physically motivated by face-gear load sharing. NASA's primary report describes mounting freedom and slow-roll torque-division tests; those observations motivate the contact model, but do not validate every dynamic or uncertainty assumption above.^[ref] Once a contact opens, a floating-pinion contact-loss check must use the appropriate changed force law.