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]

A floating-pinion contact-loss check tests whether the motion predicted by a floating-pinion balancing mode remains compatible with every contact assumed to be compressed. It prevents a two-contact linear solution from being extrapolated into tensile tooth forces.

For the symmetric modelwith $k>0$, mismatch $e$, displacement $x$ and fixed mean approach $\delta >0$, both contacts are compressed only whenAt equality one modeled force is zero. Outside the interval the linear expression makes one force negative; the unilateral model must replace that value by zero and the force-balance equations must be solved again.

Write $x=x_{\mathrm{eq}}+y$ and define the floating-pinion compression marginIf the motion satisfies $|y(t)|\le A(t)$, then $A(t)<M$ is sufficient to retain both contacts. For a steady single sinusoid of amplitude $A$ and fixed $\delta$, the maximum of $|e+x(t)|$ over a complete cycle is $|e+x_{\mathrm{eq}}|+A$, making the condition exact for that stipulated motion.

An original example uses $e=1\mu \mathrm{m}$, $x_{\mathrm{eq}}=-0.5\mu \mathrm{m}$ and $\delta =5\mu \mathrm{m}$. The margin is $4.5\mu \mathrm{m}$. A sinusoidal amplitude $4\mu \mathrm{m}$ retains a minimum compression $0.5\mu \mathrm{m}$; with $k=100N/\mu \mathrm{m}$, the smallest contact force is $50\mathrm{N}$. Amplitude $5\mu \mathrm{m}$ instead makes the unconstrained formula reach $-50\mathrm{N}$, revealing a contact-loss inconsistency rather than a physical tensile contact.

For a damped transient, an amplitude envelope is a sufficient bound but need not be attained at the phase where contact is least compressed. Failure of that sufficient envelope test therefore calls for examining the actual trajectory; it is not automatically proof that a real contact opened.

A controlled dynamic test can compare displacement with load-sensitive observations and check whether the assumed contact set persists. The historical face-gear program examined mounting freedom and torque division under slow rolling; it did not validate every possible dynamic contact assumption.^[ref]

The floating-pinion compression margin is the amount of relative approach remaining before the less compressed contact reaches zero compression in a stated two-contact model. It turns the assumptions of a floating-pinion balancing mode into a quantitative admissibility check.

For $F_1=k(\delta +e+x)$ and $F_2=k(\delta -e-x)$, let equilibrium position be $x_{\mathrm{eq}}$ and defineA positive $M$ means both contacts are compressed at equilibrium. If $x=x_{\mathrm{eq}}+y$ with $|y|\le A$, then $A<M$ guarantees compression throughout the bounded motion. For a steady single sinusoid and fixed $\delta$, this is the exact full-cycle condition; a general transient envelope supplies only a sufficient bound.

An original nominal example has $\delta =5\mu \mathrm{m}$ and $b=e+x_{\mathrm{eq}}=0.5\mu \mathrm{m}$, giving $M=4.5\mu \mathrm{m}$. A displacement amplitude $4.2\mu \mathrm{m}$ passes that nominal check, leaving $0.3\mu \mathrm{m}$ of minimum compression for the sinusoidal model.

A floating-pinion robust contact margin includes bounded parameter uncertainty. If $\delta \in [{\delta }_L,{\delta }_U]$ and $b\in [b_L,b_U]$, a conservative lower margin isFor $\delta =5\pm 0.2\mu \mathrm{m}$ and $b=0.5\pm 0.3\mu \mathrm{m}$, $M_L=4.0\mu \mathrm{m}$. The same amplitude $4.2\mu \mathrm{m}$ is no longer guaranteed admissible, although failure of the bound does not prove contact loss. Amplitude $3.8\mu \mathrm{m}$ leaves at least $0.2\mu \mathrm{m}$ under these independent interval bounds.

In the symmetric static model, $x_{\mathrm{eq}}=-2ke/(2k+k_s)$ givesfor positive mesh stiffness $k$ and nonnegative floating-pinion centering stiffness $k_s$. Mounting freedom can change the margin, but it also changes dynamics; maximizing this one scalar is not a complete gearbox design rule.

A floating-pinion contact-loss check must recompute the contact model once a force reaches zero. The historical face-gear tests provide evidence about mounting freedom and measured torque division under slow rolling, rather than a universal dynamic margin.^[ref]

A floating-pinion robust contact margin is the smallest two-contact compression margin allowed by a stated set of uncertain parameters. It strengthens a nominal floating-pinion compression margin by checking every admissible parameter combination, while retaining the assumptions of the underlying contact model.

Let $\delta$ be the common approach and $b=e+x_{\mathrm{eq}}$ the equilibrium imbalance in the symmetric linear model. If the admissible pairs form a set $\mathcal{U}$, defineFor displacement $y$ from equilibrium with $|y|\le A$, the smaller contact compression is at least $M_{\mathrm{rob}}-A$. Thus $A<M_{\mathrm{rob}}$ guarantees positive compression for both contacts throughout the bounded motion. Equality permits zero compression and does not guarantee maintained compressive contact.

For independent interval bounds, or a bounding rectangle containing all possible pairs,This is exact when every pair in the rectangle is admissible. If the rectangle includes impossible combinations, it remains a conservative lower bound on $M_{\mathrm{rob}}$.

A correlated floating-pinion uncertainty set can materially change the conclusion. In an original example, all quantities are in micrometres andHere $b$ stays positive, so $\delta -|b|=4.5$ for every allowed $u$. The exact joint-set margin is $4.5\mu \mathrm{m}$. Treating the ranges independently instead gives $4.8-0.7=4.1\mu \mathrm{m}$. An amplitude $A=4.3\mu \mathrm{m}$ passes the joint-set guarantee with residual compression $0.2\mu \mathrm{m}$, but fails the rectangular sufficient test. Failure of a conservative test does not itself demonstrate contact loss.

The admissible set must come from defensible parameter bounds, including any shared mounting or thermal effects. A covariance estimate alone supplies no absolute bound for all realizations. If the motion amplitude also depends on uncertain parameters, a direct bound on $\delta -|b|-A$ can preserve correlations that separate worst-case bounds discard.

Floating mounting was physically motivated by face-gear load sharing. NASA's primary report describes mounting freedom and slow-roll torque-division tests; those observations motivate the contact model, but do not validate every dynamic or uncertainty assumption above.^[ref] Once a contact opens, a floating-pinion contact-loss check must use the appropriate changed force law.

A correlated floating-pinion uncertainty set describes combinations of contact-model parameters that can occur together, rather than treating every separate parameter bound as independently attainable. Here correlated means linked by the parameter model; it need not mean a particular statistical correlation coefficient.

For the floating-pinion robust contact margin, write $b=e+x_{\mathrm{eq}}$ andThe set $\mathcal{U}$ can represent shared mounting motion, a common geometric error or another justified constraint. A rectangle based only on separate ranges includes the true set, but can include combinations that its physical cause cannot produce. Minimizing over this larger set gives a conservative lower bound.

An original floating-pinion affine uncertainty bound gives an exact result for bounded shared variables. Supposewith every allowed vector $u$ admissible. Since $\delta -|b|=\min (\delta -b,\delta +b)$, minimizing each affine expression over the box givesThe uncertainty variables are independently bounded here, but the resulting physical parameters can remain linked. Each absolute-value term follows by choosing the sign of $u_j$ that minimizes its coefficient times $u_j$.

For example, in micrometres take $\delta =5+u$, $b=0.5+0.5u$ and $|u|\le 0.2$. The two minima are $4.4$ and $5.2$, so $M_{\mathrm{rob}}=4.4\mu \mathrm{m}$. Separate parameter ranges instead give ${\delta }_L=4.8$ and $b_U=0.6$, hence rectangular bound $4.2\mu \mathrm{m}$. A displacement envelope $A=4.3\mu \mathrm{m}$ leaves guaranteed compression $0.1\mu \mathrm{m}$ under the joint model, although the rectangular test cannot certify it.

A statistical covariance matrix is not itself a hard admissible set: a high-probability region and an absolute bound answer different questions. The physical model also needs to remain valid throughout the selected set. If a contact opens, the linear two-contact force law must be replaced in the floating-pinion contact-loss check.

NASA's face-gear work motivates examining mounting freedom and measured load division, while its slow-roll observations do not establish the particular uncertainty set used in this derivation.^[ref] The value of a joint set is justified constraint information, not an assumption that all uncertainties cancel.

A floating-pinion affine uncertainty bound evaluates the minimum contact-compression margin when the approach and imbalance depend linearly plus a constant on bounded shared parameters. It gives an exact result for a box of uncertainty variables, while preserving the dependence those variables induce between physical quantities.

In the correlated floating-pinion uncertainty set, supposeAll uncertainty vectors in this box are assumed admissible. For the linear two-contact model, the floating-pinion robust contact margin is the infimum of $\delta -|b|$ over that box.

First resolve the absolute value:For any affine expression $q_0+{\sum }_j{q}_j{u}_j$, its minimum is $q_0-{\sum }_j{\rho }_j|q_j|$: choose each bounded variable with the sign opposing its coefficient. Applying this rule to the two branches givesThe branches can attain their minima at different parameter vectors. Taking the smaller of their independently attainable minima is valid; requiring both minima to occur at one vector would impose an unnecessary condition.

For an original example in micrometres, letThe two branch minima are $4.5-0.1-0.3=4.1$ and $5.5-0.3-0.1=5.1$, so $M_{\mathrm{rob}}=4.1\mu \mathrm{m}$. The first is attained at $(u_1,u_2)=(-1,-1)$. Treating the separate physical ranges independently instead gives ${\delta }_L=4.7$ and $\max |b|=0.8$, hence a looser $3.9\mu \mathrm{m}$ bound. A motion amplitude $4.0\mu \mathrm{m}$ leaves guaranteed compression $0.1\mu \mathrm{m}$ in the shared-variable model.

The result is exact for the stated box and affine model. Additional constraints can make it conservative; nonlinear parameter dependence or opening of a contact can invalidate the model itself. A floating-pinion ellipsoidal uncertainty bound addresses a different set shape, and a statistical covariance matrix alone does not certify a hard bound.

NASA's face-gear report supplies the physical motivation for examining mounting freedom and load sharing, not an experimental validation of these chosen uncertainty coefficients.^[ref] A valid guarantee requires defensible parameter bounds and the appropriate floating-pinion contact-loss check.

A floating-pinion ellipsoidal uncertainty bound gives the smallest two-contact compression margin when the uncertain parameter vector lies in a specified ellipsoid. It retains shared parameter effects, as does the floating-pinion affine uncertainty bound, but uses a different admissible-set shape.

Let approach and imbalance be affine functionsand describe the set by $u=u_0+Lz$ with $\Vert z{\Vert }_2\le 1$. Put $\bar{\delta }={\delta }_0+a^{\mathsf{T}}{u}_0$ and $\bar{b}=b_0+c^{\mathsf{T}}{u}_0$. The floating-pinion robust contact margin then satisfies the exact formulaTo prove it, write $\delta -|b|=\min (\delta -b,\delta +b)$. For either linear coefficient vector $q$, the Cauchy-Schwarz inequality givesIf $L^{\mathsf{T}}q\ne 0$, equality is attained at $z=-L^{\mathsf{T}}q/\Vert L^{\mathsf{T}}q{\Vert }_2$. Thus the result supplies both a margin and an attaining direction. The image representation also handles rank-deficient sets without inverting a shape matrix.

For an original example in micrometres, take $u_0=0$, $L$ the identity andOver $u_1^2+u_2^2\le 1$, the two branch minima are $4.5-\sqrt{0.1}$ and $5.5-\sqrt{0.1}$, so the margin is about $4.1838\mu \mathrm{m}$. The first branch attains it at $u=(-1,-3)/\sqrt{10}$. The encompassing box $|u_1|,|u_2|\le 1$ instead gives $4.1\mu \mathrm{m}$ because it admits combinations excluded by the disk.

A covariance-derived ellipse does not automatically certify an absolute engineering bound. Its probability content requires distributional assumptions and an appropriate radius; a hard guarantee requires a defensible admissible set. NASA's face-gear mounting and slow-roll load-sharing observations motivate the physical contact question, not the particular coefficients or set used here.^[ref] The two-contact force law must also remain valid; contact opening requires the floating-pinion contact-loss check.