= Floating-pinion centering stiffness
\b[Floating-pinion centering stiffness] describes the local restoring-force response of the mounting to displacement along a specified balancing coordinate. If mounting reaction is $R_s(x)$, a convenient positive restoring convention is
$$
k_s=-\frac{dR_s}{dx}.
$$
For $R_s=-k_sx$, $k_s$ has force-per-length units. A rotational coordinate would require a torque-per-angle stiffness instead; the coordinate and displacement endpoints must be stated before comparing values.
In an original symmetric model of a <floating pinion>, two compressed mesh reactions obey
$$
F_1=k(\delta+e+x),\qquad F_2=k(\delta-e-x).
$$
With support reaction $-k_sx$, equilibrium gives
$$
x=-\frac{2ke}{2k+k_s},\qquad
D=F_1-F_2=\frac{2ke\,k_s}{2k+k_s}.
$$
The rigid-centering imbalance is $D_{\rm rigid}=2ke$, so
$$
\frac{D}{D_{\rm rigid}}=\frac{k_s}{2k+k_s}.
$$
Both reactions must remain nonnegative for this two-contact model to apply.
For $k=100\,\mathrm{N/\mu m}$, $e=1\,\mathrm{\mu m}$ and $\delta=5\,\mathrm{\mu m}$, choose $k_s=50\,\mathrm{N/\mu m}$. Displacement is $-0.8\,\mathrm{\mu m}$ and reactions are $520$ and $480\,\mathrm N$. Raising $k_s$ to $800\,\mathrm{N/\mu m}$ reduces displacement to $-0.2\,\mathrm{\mu m}$ but gives forces $580$ and $420\,\mathrm N$. Stronger centering keeps the location closer to its reference while preserving more of the original imbalance.
Dynamic response is another question. With an effective mass $m$ and viscous coefficient $c$, the same linearized <floating-pinion balancing mode> obeys
$$
m\ddot x+c\dot x+(2k+k_s)x=-2ke.
$$
Its undamped natural frequency is $\sqrt{(2k+k_s)/m}$ in angular-frequency units. A favorable static split does not by itself establish favorable dynamic behavior, and the toy mode does not include every shaft or housing motion.
A stiffness measurement should separate mounting reaction from the mesh reactions or report the combined assembly stiffness explicitly. The primary 2002 face-gear program used a cantilevered mounting to permit pinion balancing and examined a constrained comparison; it did not establish one universal centering stiffness for all transmissions.<Development of Face Gear Technology for Industrial and Aerospace Power Transmission by Gregory F. Heath and others>[Introduction, printed p. 1; Conclusions, printed p. 84, floating mounting and modified fixed-pinion comparison]{ref}
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