= Floating-pinion balancing mode
A \b[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 is
$$
m\ddot x+c\dot x+Kx=-2ke,\qquad K=2k+k_s.
$$
The static position is $x_{\rm eq}=-2ke/K$. Setting $y=x-x_{\rm eq}$ gives
$$
m\ddot y+c\dot y+Ky=0,\qquad
\omega_n=\sqrt{K/m},\qquad \zeta=\frac{c}{2\sqrt{mK}}.
$$
For $0<\zeta<1$, 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 $k=10^6\,\mathrm{N/m}$, $k_s=0.5\times10^6\,\mathrm{N/m}$, $m=0.25\,\mathrm{kg}$ and $c=250\,\mathrm{N\,s/m}$. It gives $K=2.5\times10^6\,\mathrm{N/m}$, $f_n\simeq503.3\,\mathrm{Hz}$, $\zeta\simeq0.158$ and an envelope reduced to $e^{-1}$ after $2\,\mathrm{ms}$. With mismatch $e=2\,\mathrm{\mu m}$, the equilibrium displacement is $-1.6\,\mathrm{\mu m}$. 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.<Development of Face Gear Technology for Industrial and Aerospace Power Transmission by Gregory F. Heath and others>[Introduction, printed pp. 1–2; test description and Conclusions, printed p. 84]{ref}
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