The Extended Galerkin Method for Approximate Solutions of Nonlinear Vibration Equations

Featured Application This is a novel procedure for solving nonlinear equations of vibrations with asymptotic solutions. It is an extension to the popular Galerkin method by adding an integration of time over one period of vibrations. The method is applicable to a broad class of nonlinear equations as a systematic procedure for approximate solutions. An extension has been made to the popular Galerkin method by integrating the weighted equation of motion over the time of one period of vibrations to eliminate the harmonics from thee deformation function. A set of successive equations of coupled higher-order vibration amplitudes is resulted, and a nonlinear eigenvalue problem is obtained for the frequency-amplitude dependence of nonlinear vibrations with successive displacements. The subsequent solutions of vibration frequencies and deformation are consistent with other successive approximate methods, such as the harmonics balance method. This is an extension of the Galerkin method which has broad applications for asymptotic solutions, particularly for problems in solid mechanics. This extended Galerkin method can also be utilized for the analysis of free and forced nonlinear vibrations of structures as a new technique with significant advantages in calculations.