How about finding the principal moments of inertia? That essentially allows you to diagonalize the inertia tensor, which enables you to solve problems like the heavy symmetric spinning top, although you don't have to go through with all of that analysis (requires Euler-Lagrange formulation of mechanics).

Other than that, solving systems of linear ODE's such as you get in linear circuits or mass-spring systems (the vibrational modes you talked about) are still the best examples that I know of.