Theorem on the exact nonsimilar steady-state motions of a nonlinear oscillator

In this work the steady-state motions of a nonlinear, discrete, undamped oscillator are examined. This is achieved by using the notion of exact steady state, i.e., a motion where all coordinates of the system oscillate equiperiodically, with a period equal to that of the excitation. Special forcing functions that are periodic but not necessarily harmonic are applied to the system, and its steady response is approximately computed by an asymptotic methodology. For a system with cubic nonlinearity, a general theorem is given on the necessary and sufficient conditions that a excitation should satisfy in order to lead to an exact steady motion. As a result of this theorem, a whole class of admissible periodic functions capable of producing steady motions is identified (in contrast to the linear case, where the only excitation leading to a steady-state motion is the harmonic one). An analytic expression for the modal curve describing the steady motion of the system in the configuration space is derived and numerical simulations of the steady-state motions of a strongly nonlinear oscillator excited by two different forcing functions are presented.