Mitch,
Nice illustration. I obviously love thinking about drums and physics, in fact I am writing curriculum for a science of sound course this summer in hopes that it gets adopted for the 2017-18 school year. Questions like the one you posed are simple on the surface but complex and nuanced when you dig in. Like you said, sustain or resonance isn't exactly what you meant, you were thinking in terms of "bounce" vs "a choked bark". The Newtonian mechanics of it all is pretty complex when you are dealing with coupled oscillators etc. and then you layer on top of that, the human perception of sound phenomenon, and it gets nearly impossible to answer in a "cut and dried" way.
Introductory physics is taught using many simplified and idealized examples, like ignoring air resistance when looking at projectile motion. This is totally necessary because air resistance can't be modeled without the use of mathematics (calculus) that most kids just aren't ready for, and the simplification of the "real world" in this case is not such a big deal because predicting the path of a slow moving projectile can still be pretty accurate even when you neglect air resistance. The unintended side-effect of these simplifications/idealizations is that many students go forth into life with a distorted sense of how complex and nuanced many of the natural phenomenon around them are. If you asked them the drum question they might answer something like..."I don't know the formula for that, but I'm sure there is one." It is one of my greatest challenges as a teacher to get them to understand that nature "is what it is" even though we might wish it were simpler.
Here is one example I heard a professor use and I have been known to repeat...we as humans have done amazing things, put men on the moon, created cell phones blah blah, yet if you tossed a ping pong ball into a creek and wanted to predict exactly where it would be one minute later, even the smartest humans using our most advanced computers could not model that fluid dynamic system accurately enough to say.
BTW- I have not fact checked that, but I'm pretty sure nobody will go to the trouble it would take to prove the statement wrong, so I'm safe.
AL
