I don't think you could point to any one person or group and say, "They discovered mechanical resonance." It's something that's always been around and people have used it to their advantage for centuries, if not millennia.
I've done more than a few vibration tests looking for the kind of resonances where the response is detrimental to whatever's being tested in a variety of machines or devices. A typical approach for a resonance search is to use some kind of measured, controlled input, such as a shaker table, and record the motion (usually with an accelerometer) of both the table and the DUT (device under test). The first step is usually a "sine sweep" where the shaker is operating over a range of frequencies, sweeping up and then down (or vice versa) and the data is examined for regions where the response is high relative to the excitation. In the end, what you're trying to identify are those frequencies where the DUT has a sensitivity that are also present in the environment which the device is supposed to operate.