Spectra and autocorrelation functions of a geometrically nonlinear beam

Abstract

The aim of this paper is to explore the Fourier spectra and autocorrelation functions of a geometrically nonlinear elastic beam with supports subjected to harmonic excitation. By using the large deformation theory and considering the deflection caused by both moment and shear as well as the inertia force induced by both displacement and rotation, the equation of motion is derived, which turns out to be a nonlinear fourth‐order partial differential equation. Because of the complexity of the partial differential equation, this paper first employs the Galerkin method to reduce it into a nonlinear second‐order ordinary differential equation. Then the Runge‐Kutta‐Verner numerical method is applied to find the solution. Due to the nonlinearity of the equation, chaotic motion is found to exist in this nonlinear elastic beam system. This paper draws time histories and Poincaré maps to show that chaotic motion indeed exists in the geometrically nonlinear elastic beam. For both regular and chaotic motions, Fourier spectra and autocorrelation functions are then plotted, which can be used as important criteria to diagnose whether the geometrically nonlinear beam will be chaotic in the future because of the significantly different results from those two kinds of motions.