In the coupled harmonic oscillation problem you can find the normal frequencies, modes and coordinates fairly easily. But unless you have very specific initial conditions the actual motion will be a complicated superposition of these modes. So what good is finding these results?
Are most physical oscillating systems subject to very specific initial conditions?
In dynamic loading of structures it is a good thing to be able to predict the superposition of harmonic oscillations.
For example, if a water tower was subjected to an earthquake, we might want to know how the resonating frequency of the tower relates to the frequency of the earthquake’s shaking. If they get into a harmony together the tower will shake itself to death.
re: initial conditions are fairly straightforward for structures problems. So I’m not sure about your second question.
If that’s not what you are asking then please excuse me. 
Yeah, that makes sense. Suppose you had a triatomic molecule and you wanted to get it highly excited, you’d expose it to radiation that matched one of its eigenfrequencies.
This is probably also how you get a population inversion in a lasing material. In other words you impose the initial conditions to match what you want to accomplish. Thanks.
I’m happy that I was able to give you insight. You always ask the coolest questions! Usually I’m not able to answer them.
The little I’ve remebered about structural dynamics is because it’s a favorite subject of my Dad’s, but I’m a hydrology/hydraulics person. Speaking of eigenfrequencies ~ he did his Ph.D. dissertation on eigenvectors and eigenvalues. 