I know that modal analysis decouple ODEs so that the modal ODEs can solved individually. The modal superposition method is used to get the total response of the system. Since modal solutions can be superimposed, they must be linear functions. Can someone elaborate more about the linearity of modal solution? How is it linear?
well, modal decomposition essentially linearizes and decouples the given set of equations. Thus, the modal solutions are mutually independent and so, they evolve independent of each other. This is why one can superimpose them! However, the solutions themselves can be nonlinear!
See, you can have a set of coupled equations which on modal decomposition would turn out to be a linear such as, dy1/dx = e^x and dy2/dx = cos x. Now, the answer to the first equation is y1 = e^x and second solution is y2 = sin x. Both solutions are not of "linear" curves, still they will add, provided the the original set of coupled equations (containing y1 and y2) was decomposed into the two differential equations mentioned.
I assume I have correctly understood your question.
As a complement of what has already said: modal analysis can be applied also to partial differential equations. And besides linear modal analysis, there is nonlinear modal analysis. Please find some comments attached.
Your first sentences isn't necessary correct, modal equation can be coupled in general. When this happen, modal equation lost superposition property. So writing an equation in modal form doesn't mean linearity of system, you use method for nonlinear case too. These methods depends on orthogonality of basis function and not superposition property.
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