= Floating-pinion contact-loss check
A \b[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 model
$$
F_1=k(\delta+e+x),\qquad F_2=k(\delta-e-x),
$$
with $k>0$, mismatch $e$, displacement $x$ and fixed mean approach $\delta>0$, both contacts are compressed only when
$$
-\delta-e<x<\delta-e,\qquad |e+x|<\delta.
$$
At 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 $x=x_{\rm eq}+y$ and define the <floating-pinion compression margin>
$$
M=\delta-|e+x_{\rm eq}|.
$$
If 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 $|e+x_{\rm eq}|+A$, making the condition exact for that stipulated motion.
An original example uses $e=1\,\mathrm{\mu m}$, $x_{\rm eq}=-0.5\,\mathrm{\mu m}$ and $\delta=5\,\mathrm{\mu m}$. The margin is $4.5\,\mathrm{\mu m}$. A sinusoidal amplitude $4\,\mathrm{\mu m}$ retains a minimum compression $0.5\,\mathrm{\mu m}$; with $k=100\,\mathrm{N/\mu m}$, the smallest contact force is $50\,\mathrm N$. Amplitude $5\,\mathrm{\mu m}$ 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.<Development of Face Gear Technology for Industrial and Aerospace Power Transmission by Gregory F. Heath and others>[Introduction, printed pp. 1–2; Conclusions, printed p. 84, floating versus constrained mounting]{ref}
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