 Floating-pinion balancing mode

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]