 Pressure distributions with identical resultant force and center of pressure

Pressure distributions with identical resultant force and center of pressure demonstrate that a pressure centroid is not a reconstruction of a pressure field. A normal-load measurement and its two first spatial moments constrain onlyA perturbation $\delta p$ preserving all three integrals leaves the resultant and center of pressure unchanged. It is physically admissible only when the perturbed field remains nonnegative under the nonadhesive pressure model. The polynomial profiles in the parent article give one smooth example; the following construction shows that even a peak-pressure bound is missing.

Take a rectangular region of length $L$ and width $B$, centered about its midpoint, and fix a normal load $N>0$. A uniform field has pressure $N/(LB)$ and centroid $(L/2,B/2)$. Instead place the entire load uniformly on a smaller centered square of side $a\le \min (L,B)$, with zero pressure outside that square. Its pressure isSymmetry fixes the same centroid, and integration still gives total force $N$. Yet its peak is $N/a^2$, which has no upper bound as $a$ shrinks. Smooth symmetric concentrated profiles can provide the same conclusion if a continuous pressure field is required.

For an original numerical example, let $L=0.40\mathrm{m}$, $B=0.20\mathrm{m}$, and $N=800\mathrm{N}$. Uniform pressure is $10\mathrm{kPa}$. Concentrating the same load on a centered $0.05\mathrm{m}$ square gives $320\mathrm{kPa}$ while preserving the measured point $(0.20,0.10)\mathrm{m}$. Both satisfy the stated nonnegative-field moment constraints.

Consequently a force/moment sensor cannot infer local peaks from the centroid alone. Local contact pressure from additional spatial measurements needs more data or justified material/contact assumptions, such as a constitutive model and known geometry. The source's normal-force centroid definition explains exactly which resultant information was retained.^[ref] The examples are inverse-problem constructions; they are not claims that arbitrary pressure maps can be realized by the same elastic surfaces.