Why are normal modes important when analyzing waves or oscillations?

[PhysicsKid0123]

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?

 

[Dr.D]

Your second picture answers one of your questions: Natural frequency for a single degree of freedom system is the square root of the ratio K/M. That how it is calculated, and that's also what it means.

I can't speak to where normal modes (also known as principal modes) are used in physics, but in engineering they are used all the time to analyze complex vibration problems. For a system with N degrees of freedom, there are N natural frequencies and N mode shapes. Any actual motion can be expressed as a combination of these mode shapes, taken in the proper proportions.

This is not obvious, but do not write off modal analysis until you have an opportunity to study it much further. It is an extremely useful approach, both for analysis and for experimental work.