The Approximate Analysis of Higher-Order Frequencies of Nonlinear Vibrations of a Cantilever Beam with the Extended Galerkin Method

The nonlinear vibrations of elastic beams with large amplitudes are frequently treated as a typical problem of an elastica. As the continuation of the analysis of the deformation of an elastica, the nonlinear vibration equation of the elastic beam in the rotation angle of the cross-section has been established. Using the deformation function, the nonlinear equation with the inertia effect has been solved by the newly proposed extended Galerkin method (EGM). The solution to the vibration problem of the elastica is compared with earlier approximate solutions including the frequencies and mode shapes obtained by other methods, and the rotation angle and energy of each mode at the high-order frequency are also calculated. This solution procedure provides an alternative technique to the elastica problem by the EGM with possible applications to other nonlinear problems in many fields of science and technology.