Why are they important? I've been learning about them quite a bit and have no idea why they are significant. What is the motivation for their discovery, their use, or even their mention in physics for that matter. All I really know is that when you look/have solutions with the same angular frequencies and if they are in phase and plug them in for the differential equation of motion (I'll upload a pic for example), you're able to find resonant frequencies or natural frequencies. But why is that the approach? Why not have different frequencies or different phases. What would the equations be telling you then? Why can't we consider both? This further leads me to question what are natural (resonant frequencies exactly)? I know about the story of that bridge that collapsed because the wind hit the bridge at the same frequency as the bridge and eventually went chaotic until it collapsed. But I'm still not sure what exactly they mean in a different context other than that. Are natural frequencies just this kind hidden attribute of all objects? If so, what determines natural frequencies? Is it length, density, shape, etc. What does this frequency tells us about the object?