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Isn't Poisson's equation basically describing a minimal surface for small z? I'm not saying the badminton racket follows exactly a (discrete) 2D Poisson equati
by jbay808 5y ago
Isn't Poisson's equation basically describing a minimal surface for small z?
I'm not saying the badminton racket follows exactly a (discrete) 2D Poisson equation. But it's certainly related enough to be more than a surface similarly.
The cords are under high tension, which means that any curvature along x (that is, dz^2/dx^2) will result in a net z-axis tension force unless balanced by an oppositely curved cord running in the y direction. Since it's in static equilibrium, there can be no unbalanced forces and so that must be the case. Therefore at each intersection, (d^2/dx^2 + d^2/dy^2)z = 0, which is Poisson's equation in 2D for z height being the function. Approximately, assuming equal tension in x and y, small z, and so on.
- codethief 5y agoI didn't say you're completely wrong, either. :) (In fact, I stressed that both equations are of elliptic type, so solutions might look similar.) One can make a similar heuristic argument, though, as to why the surface you observed should follow the minimal-surface equation. EDIT, as you've updated your comment: > Isn't Poisson's equation basically describing a minimal surface for small z? Small changes in z, I think.
- gpsx 5y agoIf you want to get technical, this isn't a continuous surface but weave of strings. And something that makes this more complex is the location of the string intersections has a static/dynamic friction component that is difficult to model analytically. But I think we accept your approximation if you are looking for numeric results.
- gpsx 5y agoI think that is good intuition. The transverse force is the second derivative of the transverse position, to first order. But then there are higher order effects in it. I think that is where that math breaks down.