Correlated floating-pinion uncertainty set

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.