 Floating-pinion centering stiffness

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 isFor $R_s=-k_{s}x$, $k_s$ 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 obeyWith support reaction $-k_{s}x$, equilibrium givesThe rigid-centering imbalance is $D_{\mathrm{rigid}}=2ke$, soBoth reactions must remain nonnegative for this two-contact model to apply.

For $k=100N/\mu \mathrm{m}$, $e=1\mu \mathrm{m}$ and $\delta =5\mu \mathrm{m}$, choose $k_s=50N/\mu \mathrm{m}$. Displacement is $-0.8\mu \mathrm{m}$ and reactions are $520$ and $480\mathrm{N}$. Raising $k_s$ to $800N/\mu \mathrm{m}$ reduces displacement to $-0.2\mu \mathrm{m}$ 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 obeysIts 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.^[ref]