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Hello there.

I'm in the process of developing a loudspeaker application in which one of the main steps is to calculate the average displacement along "rings" in a loudspeaker cone, in order to project that average displacement into a 2D axisymmetric model. That is, for every point along the edges of an axisymmetric shell model, there exists a three-dimensional ring which, when evaluated, provides the average displacement along itself as a form of projection to the 2D axisymmetric domain. The image below demonstrates what I mean... Averaging occurs in the direction of those arrows, eventually projecting the results in function of the z-coordinate.

Therefore, more exactly, one way of stating this projection is to say that I want to be able to feed into another component, in the z-coordinate, the results of the z-axis displacement evaluated along the rings in the 3D comp, such that for every z-coordinate in the 2D axisymmetric component,

where is the ring of radius 'r' which the cartesian coordinates are defined by

The projection I'm interested in, more exactly, is of the sort

; (a simple circle equation), and

.

It may help to make sense if you remember that this is taking a 3D shell and "revolving" it down to an edge. This can be further simplified and thought of averaging all results in a horizontal plane at height 'z' at all available coordinates and making this averaging operation available in the 2D axisymmetric domain, also along the z coordinate and irrespective to the r coordinate.

What first made sense to me was using the projection I just spoke about earlier, considering how the z coordinate relates to the other two, and mapping it into the secondary z-axis:

However, this produced results that did not make sense at all. They simply were too far from reality. For testing, I simply switched the y-expression in the source map to 'x' or 'y' and, in both cases, it "magically" worked!

I tried to force errors by making the loading condition assymetrical, slanting the geometry at an offset angle, but none could produce innacurate results as far as the average displacement is concerned. The thing is that by looking at the documentation I cannot understand why this works but the approach I came up with doesn't. From my understanding, the one that worked should have weighted more the results where surfaces are perpendicular (or rather parallel?) to the used dimension in the y-direction source map than the other, but results seemed consistent and reliable all the time.

Can anyone try to explain to me why it works?