phymath7 said:
does the spring obeys hooke's law while it forms a coupled harmonic oscillator system?
Sort of, yes.
So for clarity let's say you have two oscillating modes an axial mode and a single transverse mode (like p-waves and s-waves in earthquakes) in a simple bar of steel.
Hooke's law only applies in a linear region of strain (think of any very small region of your steel spring) where the material can stretch in one direction, let's say ##\hat{z}##, without a significant change in the spring constant either in that direction or in other orthogonal directions, like ##\hat{x}##. In this case the two oscillating modes can be viewed as uncoupled, they proceed to move in time independent of each other. If you hit it the right way you can excite either or both modes. But after that they will be essentially independent.
This is sort of by definition. Hooke's law applies when Hooke's law applies. If you stretch a spring too far, you can't use it anymore. How do you know what "too far" is? That's when Hooke's law doesn't work anymore. It's an idealized approximation which is simple and often true. But IRL you need to verify that it actually works that way.
But, you asked about coupled modes, where energy can move between modes. This requires some coupling mechanism which isn't defined, so I don't know what the effects are.
Complex structures may have some regions where Hooke's law works and others where it doesn't, it's best to think of this as an approximation of how a small region responds to stress and then figure out how that relates to the whole structure.
The structure may also create coupling between the modes. So, for example if you stretch a coil spring the entire structure will exhibit strain in tension (stretching) and torsion (twisting) because of the way it's constructed in relation to the stress direction you apply. But if you zoom into a small area inside the spring there will be only one strain direction which is a combination of the global tension and torsion directions. A solid metal bar won't couple global tension stress into a torsional strain because it is constructed differently. A tiny section of the coil spring can be viewed as a straight bar which is being pulled off-axis by the neighboring tiny sections it's attached to.
