Modal Analysis vs. the Solution to the Diff.Eqs.

[Trying2Learn]

TL;DR Summary

Difference between solutions of a system

Hello

The figure below is that of a two body problem in oscillation.

Whether by Free Body Diagrams or Hamilton's Principle, we can obtain the two coupled differential equations that describe the oscillation.

At this point, one could obtain the closed form solution for the displacement.
Another option is to conduct a modal analysis and then assume the solution is based on those two modes.

So, my question is: "what would be the difference in the two solutions?"

And, by that, I do NOT mean: "how do the solutions differ?"

I mean (and forgive me if I do not phrase this properly), how are the two approaches philosophically different?

One (modal analysis) is based on first obtaining the modes and then building a differential solution on them.
The other is a direct attempt, say, to uncouple the differential equations (say, by elevating the order) and then solving each.

But what is the difference, in the two approaches?

And how is it possible to obtain, say, to different solutions, if we know that the solution is possibly unique?

 

[gtoguy97]

The difference between the two approaches is that the modal analysis approach focuses on identifying the modes of oscillation in the system, while the direct approach focuses on finding a closed-form solution. The modal analysis approach is useful for analyzing complex systems and provides insight into the behavior of the system, while the direct approach is useful for obtaining exact numerical solutions. It is possible to obtain multiple solutions from one approach or the other depending on the assumptions used in the analysis. For example, if the assumptions in the modal analysis are incorrect, then the resulting solution will be inaccurate. Similarly, if the assumptions in the direct approach are incorrect, then the resulting solution may not be the correct one.