I'm no drum-maker, and I'm definitely not a physicist, but Pearl claims that the reinforcement rings in my walnut solid shell serve a two-fold purpose:Crap re-rings just thought of that. They are there for structure but they will impact sustain I'd bet. Hmmmm that's a tough one-my knee jerk reaction it will increase sustain. But really it depends on shell thickness and size and wood type of re-rings. No matter what I bet it dry out a drum-it prevents deformation, but I bet the sustain will depend on design and wood type of re-ring. So a thin shell is naturally resonate the re-ring might impede motion of shell response so it appears to lose sustain-I think by design and hardness of wood you could overcome that to make it a resonator to return sustain.
The thicker shell is harder to activate so doesn't have a natural sustain, but what energy that does flow it's a lower fundamental long waves travel farther and I think the re-ring stiffness would focus the energy into the re-ring to act like a resonator. The thinner is more tricky my bet because the sustain comes from the shell and the re-ring might impede motion so it would appear to sound like a loss of sustain-but I also think you could design a re-ring that would act like a resonator so sustain maintained and likely this is what generally happens? The dryness likely comes from frequency shift to higher frequency because stiffness and stress increased at that point of re-ring I'd deduce. So generally a re-ring would increase sustain I'd bet but exceptions to rule-and perhaps we can explain why.
I. They preserve the shell's integrity.
II. They lend focus to the shell's tone.
I don't know the exact effect that the reinforcement rings have on sustain. My snare is pretty dry and makes a fairly succinct note, but those traits are also influenced by the shell's dimensions, wood type, thickness, heads, and tuning. As always, a black hole of speculation develops.