Non-similar normal oscillations in a strongly non-linear discrete system

Abstract

The free oscillations of a strongly non-linear, discrete oscillator are examined by computing its “non-similar non-linear normal modes.” These are motions represented by curves in the configuration space of the system, and they are not encountered in classical, linear vibration theory or in existing non-linear perturbation techniques. The Mikhlin-Manevich asymptotic methodology is used for solving the singular functional equation describing the non-similar modes and approximate, analytical expressions are derived. For an oscillator with weak coupling stiffness and “mistuning,” both localized and non-localized modes are detected, occurring in small neighborhoods of “degenerate” and “global” similar modes of the “tuned” system. When strong coupling is considered, only non-localized modes were found to exist. An interesting result of this work is the detection of mode localization in the “tuned” periodic system, a result with no counterpart in existing theories on linear mode localization. As a check of the analytical results, numerical integrations of the equations of motion were carried out and the existence of the theoretically predicted non-similar modes was verified.