The Extended Rayleigh–Ritz Method for Higher Order Approximate Solutions of Nonlinear Vibration Equations

An extension has been made with the popular Rayleigh–Ritz method by integrating the Lagrangian functional of a nonlinear vibration equation of motion over one period of vibrations to eliminate harmonics from the simplification. A set of successive nonlinear equations of coupled higher order amplitudes of deformation is obtained, and a nonlinear eigenvalue problem is presented for the frequency–amplitude dependence of nonlinear vibrations of successive displacements. The subsequent solutions of vibration frequencies and deformation are actually consistent with other successive approximate methods such as the harmonics balance method. This is an extension of the powerful Rayleigh–Ritz method which has broad applications for approximate solutions for vibration problems in solid mechanics. This extended Rayleigh–Ritz method can now be utilized for the analysis of free and forced nonlinear vibrations of structures as a new technique with significant advantages.