Nonsmooth Modal Analysis of Piecewise-Linear Impact Oscillators

In linear dynamics, the set of solutions for unforced conservative systems exhibit a vector space structure. Any solution is a linear combination of particular periodic solutions known as modes. Such a strong result no longer holds in the framework of nonlinear dynamics. However, the dynamical behaviour can be captured in the vicinity of equilibrium points. Indeed, there exists two-dimensional surfaces of the state space, which are constituted by a continuum of periodic orbits in the conservative autonomous case. Such manifolds are referred to as nonlinear modes in the vibration community. The latter are usually computed for smooth systems, i.e. systems whose positions are at least twice-differentiable and allows prediction of resonance for slight nonlinearities.