I found some nice videos that describe real-world applications of damped oscillators. First, a video involving a pendulum with adjustable damping, demonstrating underdamped, overdamped, and critically damped behaviors:
Second a really neat video showing how combining two damped oscillators and tuning the damping allows you to stop a structure from swaying too much. If engineers didn’t do this sort of thing, then skyscrapers would be much less pleasant to hang around in.