Let's look at two practical examples of harmonic analysis. A spring-mass damper system is probably the simplest example to demonstrate this. When we pull a mass in the spring's axial direction for a distance and let it move freely, its path shows a sinusoidal behavior with time. Check out the lecture video to find out how it behaves when it's under harmonic loading.
Another example is the resonance analysis of a recreational drone arm. When the drone blade is rotating, it is likely that a disturbance can introduce a sinusoidal twisting force. If that twisting force frequency matches the natural frequency of the drone arm, it may damage the drone arm structure. Check out the lecture video and handout for more information.