How to produce the first, second and third modes of vibration in a timpani For a timpani, when hitting it in the center, how do you produce first, second and third modes of vibration? Second mode has a higher frequency than the first, does that imply do you hit the timpani harder?
 A: If you hit in the exact center, then all of the modes involved will be axisymmetric. Hitting the drum creates an initial deformation. That deformation can then be resolved into a superposition of normal modes. That is, the set of deformations can be thought of as a vector space, with the normal modes as a basis. Modes beyond the first one are excited by the initial deformation not being exactly in the "direction" of the first mode; that is, it's not the the space spanned by the first mode. Note that each mode is an eigenvector of the operator describing wave evolution, and thus the "first mode" is unique only up to scalar multiples. So "direction" here is not referring to physical direction, but being a scalar multiple of whatever representative eigenvector  you take for the first eigenvalue.
That is, if your initial hit does not match a scaled version of the first mode, then it will involve other modes, with those modes adding together to create your initial deformation.
Seeing as how it's pretty much impossible to exactly replicate the first mode simply by hitting the drum, any hit will excite modes beyond the first. The first mode involves deformation throughout the drum, while a hit will be highly localized, so you have to have infinitely many modes, with the modes reinforcing in the center of the drum where you hit it, but canceling out everywhere else. It's basically like getting the Fourier transform of a spike, if that means anything to you. Hitting harder will create more of a spike, so I believe that will require more of the further modes.
In practice, it's even more complicated, since you won't be hitting the exact center of the drum, and therefore will be exciting nonaxisymmetric modes as well.
