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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]

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]

A gear torque-path load fraction specifies how a stated total transmitted torque is divided among paths of a split-torque transmission. With all branch torques referred to the same shaft coordinate,For positive total torque and nonnegative contributions, the fractions lie between zero and one and sum to one. If a path carries reverse or circulating power, the signed ratio requires a different interpretation; it need not lie in that interval.

The common coordinate is essential. Two branches carrying $10\mathrm{N}\mathrm{m}$ at $100$ and $200\mathrm{rad}{\mathrm{s}}^{-1}$ carry powers $1000$ and $2000\mathrm{W}$. Their equal raw torques do not imply equal power shares. If ideal gearing brings both contributions to a common output speed $50\mathrm{rad}{\mathrm{s}}^{-1}$, their output torques are $20$ and $40\mathrm{N}\mathrm{m}$, giving fractions $1/3$ and $2/3$.

For parallel elastic paths with the same twist coordinate and zero gaps, ${\tau }_j=K_j\theta$ gives ${\lambda }_j=K_j/{\sum }_{\ell }{K}_{\ell }$. In that model a stiffness ratio $2:3$ gives fractions $0.4$ and $0.6$, regardless of the total torque while the same linear branches remain active. Gaps and a floating pinion can change that allocation.

A sensor ratio also needs a measurement model. If two channels obey $s_1=a_1{\tau }_1$ and $s_2=a_2{\tau }_2$, thenEqual torques with sensitivities $a_1=2$ and $a_2=1$ give raw signals in the ratio $2:1$; the raw signal fraction $2/3$ is not the torque fraction $1/2$. Cross-talk and changing tooth-load distributions may require a fuller calibration model.

The primary face-gear program reported approximately $48\%/52\%$ division at the input pinions and a different split between its idlers. The report also explains why tooth strain was not an invariant direct proxy for transmitted load.^[ref] Report the reference shaft, sign convention, operating condition and measurement uncertainty with a torque-path fraction.

Gregory F. Heath is a mechanical engineer whose documented research includes rotorcraft transmissions and face gear technology. His2010 author biography describes him as an associate technical fellow at Boeing Rotorcraft-Mesa and a technical integrator for an advanced rotorcraft-drive program.^[ref]

Heath coauthored Development of Face Gear Technology for Industrial and Aerospace Power Transmission by Gregory F. Heath and others with Robert R. Filler and Jie Tan. Their2002 report documents a250-horsepower proof-of-concept split-torque transmission using concentric face gears. The tests examined whether floating input pinions could divide the load nearly evenly between parallel transmission paths.^[ref]

The mechanical question is more demanding than obtaining the desired gear ratio. If two parallel paths have unequal geometry or effective stiffness, one can carry more torque than intended. In an ideal equal-speed example, dividing a100N·m input into50N·m in each path also divides input power equally. A60/40N·m split preserves the total transmitted torque but raises the larger path's load20% above the intended50N·m share. A floating pinion supplies freedom intended to improve load balance; the report's apparatus and measurements tested how that mechanism behaved in a manufactured transmission.

Heath also coauthored the later gear-pitting detection study with Paula J. Dempsey, David G. Lewicki and Perumal Shanthakumaran, linking transmission development with experimental assessment of damage indicators.^[ref]

Jie Tan coauthored Development of Face Gear Technology for Industrial and Aerospace Power Transmission by Gregory F. Heath and others with Gregory F. Heath and Robert R. Filler. Its2002 title page identifies Tan with The Boeing Company in Mesa, Arizona. The work investigated concentric face gears and floating pinions as a means of distributing transmitted torque between parallel paths.^[ref]

The report is useful because its apparatus revealed that different load divisions in one split-torque transmission need not agree. While input-pinion mesh division was close to48/52, the idlers were reported near57/43. The authors related the idler imbalance to how backlash had been set using strain indications, and discussed the limitations of inferring the split from those indications alone.^[ref]

In an illustrative100-unit load normalization, a57-unit path is14% above the intended50-unit equal share. A geometric gear ratio can remain correct despite that unequal internal loading. Tan's coauthored study therefore provides a concrete connection between transmission geometry, adjustment and experimentally inferred forces, rather than treating the desired speed ratio as a complete description of mechanical performance.

A split-torque transmission divides transmitted load among two or more mechanical paths and subsequently combines their contributions. Its purpose is to let several meshes carry a total load that would otherwise pass through one path. The gear torque-path load fraction depends on compatibility, stiffness, gaps and mounting; a symmetric layout alone does not prove equal sharing.

For an original idealization, refer both branch torques to the same shaft coordinate and let their common elastic twist be $\theta$. With torsional stiffnesses $K_1,K_2$ and no initial gaps,Values $K_1=40000$ and $K_2=60000\mathrm{N}m/rad$ under total torque $1000\mathrm{N}\mathrm{m}$ give $\theta =0.01\mathrm{rad}$ and branch torques $400$ and $600\mathrm{N}\mathrm{m}$. If each branch were limited to $600\mathrm{N}\mathrm{m}$, this sharing would limit total load to $1000\mathrm{N}\mathrm{m}$; equal sharing would permit $1200\mathrm{N}\mathrm{m}$ within the same simplified capacity assumptions.

Parallel-path gear efficiency also differs from the product rule for serial stages. If input power fractions are $a_j$ with ${\sum }_j{a}_j=1$, and each branch delivers fraction ${\eta }_j$ of its input power to the common output boundary, thenFor fractions $0.4,0.6$ and efficiencies $0.96,0.94$, the result is $0.948$. Multiplying the two branch efficiencies would describe a different arrangement of power boundaries.

The primary face-gear program reported input-pinion torque division about $48\%/52\%$, while its two idlers shared about $57\%/43\%$. These were measured results with stated calibration limitations, not exact consequences of the layout.^[ref]

A meaningful load-sharing test measures both paths over relevant load and position ranges. Strain gauges need an appropriate measurement calibration because different tooth load distributions can produce different strain responses for the same transmitted load. Floating pinions provide one mechanism for reducing imbalance, but the complete assembly and its constraints still determine the operating split.