Title : The extended galerkin/rayleigh-ritz method for approximate solutions of nonlinear vibration equations
Abstract:
With the wide encounter of nonlinear equations in research activities and practical applications, a search of novel and efficient methods for solving such challenging problems never stopped. In this talk, a novel extension has been made with the popular Galerkin/Rayleigh-Ritz method by integrating the weighted equation of motion or Lagrangian functional over the time of one period of vibrations to eliminate the harmonics arising from the deformation function. A set of successive equations of coupled higher-order vibration amplitudes is obtained, and a nonlinear eigenvalue problem is solved 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 traditional harmonics balance method. This is a novel extension to Galerkin/Rayleigh-Ritz method which has wide applications in approximate solutions particularly for vibration problems in solid mechanics. This extended Galerkin/Rayleigh-Ritz method can also be utilized for the analysis of free and forced nonlinear vibrations of structures as a new technique with significant advantages. Application examples are also presented to a wide array of typical nonlinear problems for simple and efficient solutions and procedure.
Audience Take Away:
- Understanding mathematical description of nonlinear problems.
- Knowing some basic methods for solving typical nonlinear problems.
- Learn to use the extended Galerkin and Rayleigh-Ritz methods
