 Floating-pinion contact-loss check

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]