[bloak]

It sounds like you've done plenty of experiments and don't need any theory to make this work, but have you looked at the physics/mathematics of the vibrations? Does the frequency you detect correspond to https://en.wikipedia.org/wiki/Vibration_of_a_circular_membra..., for example? I'm guessing that the interlocking strings are best modelled as a membrane rather than individual strings (https://en.wikipedia.org/wiki/String_vibration) but the racquet head isn't circular, so it's a bit beyond my level of maths to do that properly.

I suppose that to do the maths we'd need to know the spacing of the strings and the mass per unit length of the strings. (And the dimensions of the racquet head, of course, but that must be easy to look up.)

Also, would this work just as well for tennis and squash racquets?

EDIT: Perhaps https://en.wikipedia.org/wiki/Dimensional_analysis would give you the formula, and the constant you could get by experiment, and then the same formula might be applicable to any similarly shaped racquet.

[bloak]

Ah, the string spacing cancels out, doesn't it? I'd expect the frequency to be proportional to sqrt(t/d)/a, where t is the string tension in Newtons, d is the linear density of the strings in kg/m, and a is the diameter of the racquet head (assuming they're all the same shape and measured in the same way, which probably isn't true, unfortunately).

[zhacker]

yeah. Spot on. For the app, I don't take into account the density, but racket head is taken into account. Prominent shapes are oval and isometric, so I have separate co-efficients for those, which were learned using regression. One challenge that I haven't figured out yet is when 2 different types of strings are used. It seems, some advanced players use different strings at different tensions for the horizontal and vertical strings. Like 32x28lbs or other similar configurations.