Dynamic Response of Inextensible Beams by Improved Energy Balance Method

Abstract

An improved He's energy balance method (EBM) for solving non-linear oscillatory differential equation using a new trial function is presented. The problem considered represents the governing equations of the non-linear, large-amplitude free vibrations of a slender cantilever beam with a rotationally flexible root and carrying a lumped mass at an intermediate position along its span. Based on the simple EBM, the variational integral of the non-linear conservative system is established, and the Fourier series expansion is employed to address the governing algebraic equations. An alternate procedure for a particular value of the initial condition is then used to estimate the constants. This semi-analytical representation gives excellent approximations to the exact solutions for the whole range of the oscillation amplitude, reducing the respective error of angular frequency in comparison with the simple EBM. Two illustrative examples are considered in order to elucidate the methods described, and to reveal the improvements made by the modified method.