The Analysis of Higher Order Nonlinear Vibrations of an Elastic Beam with the Extended Galerkin Method

Motivation The analysis of vibrations of an elastic beam is one of the persistently challenging problems in engineering and mathematics. The differential equation of vibrations of a beam under large deformation, or the typical vibrations of an elastica, is hard to approximate and solve with the formulation in Cartesian coordinates. Objective The higher-order nonlinear vibrations and couplings of modes of beams are analyzed with a novel method for a better understanding of dynamic properties. Method The extended Galerkin method is used to obtain the first- to third-order nonlinear vibration frequencies and mode shapes of a cantilever beam with the exact equation of a beam with larger deflection, and the differential equation is transformed to a nonlinear algebraic equation for an improved analysis and solutions with the aid of linear mode shape functions. Results From a cantilever beam in nonlinear vibrations, the numerical solutions of frequencies and mode shapes show that the approximate solutions are accurate in comparison to the exact solutions for small amplitudes with other methods. Conclusions The effectiveness of the extended Galerkin method in solving nonlinear vibration equations is further demonstrated with a novel procedure and known results.