In most cases, a timpanist will hit the drum a few inches in from the edge of the head. This produces the standard, fundamental-dominated sound that you’re probably used to hearing in any timpani-involved setting, especially the orchestra. There are, however, other possibilities.
Playing in the center of the drum will create a very muted, barely pitched “thud”-like sound. No trained percussionist will take this approach unless specifically instructed to by the composer.
The edge brings out many high overtones, sounding to my ears like the timpani’s equivalent of sul pont. The fundamental is still very much present, but with a sharper character. This is also an unorthodox approach and needs to be clarified if it’s a desired sound.
Lastly, it’s possible to get a really interesting high ping-y sound by striking the sides of the drums. It’s a little awkward to play down there, so I’d advise against writing for such a sound on the inside drums. Also, some drum owners might not be comfortable with doing this.
I don’t personally have access to timpani, but I hope to post a video showing these differences in the near future!
For Reference: If you can, take a peak at Elliot Carter’s Eight Pieces for Four Timpani. It’s the best example that I’ve seen of extended technique notation in timpani.