Mathematical Problems in Engineering
Volume 2004 (2004), Issue 2, Pages 145-168
Higher-order Melnikov functions for single-DOF mechanical oscillators: theoretical treatment and applications
1Istituto di Scienza e Tecnica delle Costruzioni, Università Politecnica delle Marche, via Brecce Bianche, Monte D'Ago, Ancona 60131, Italy
2Dipartimento di Ingegneria Strutturale e Geotecnica, Università di Roma “La Sapienza,” via A. Gramsci 53, Roma 00197, Italy
Received 30 October 2003
Copyright © 2004 Stefano Lenci and Giuseppe Rega. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
A Melnikov analysis of single-degree-of-freedom (DOF) oscillators is performed by taking into account the first (classical) and higher-order Melnikov functions, by considering Poincaré sections nonorthogonal to the flux, and by explicitly determining both the distance between perturbed and unperturbed manifolds (one-half Melnikov functions) and the distance between perturbed stable and unstable manifolds (full Melnikov function). The analysis is developed in an abstract framework, and a recursive formula for computing the Melnikov functions is obtained. These results are then applied to various mechanical systems. Softening versus hardening stiffness and homoclinic versus heteroclinic bifurcations are considered, and the influence of higher-order terms is investigated in depth. It is shown that the classical (first-order) Melnikov analysis is practically inaccurate at least for small and large excitation frequencies, in correspondence to degenerate homo/heteroclinic bifurcations, and in the case of generic periodic excitations.