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.

Non-linear vibration of Euler-Bernoulli beams

“…In recent years, some promising approximate analytical solutions have been proposed, such as Frequency Amplitude Formulation [13], Variational Iteration [5,6,14,17], Homotopy-Perturbation [3,4,7,24], Parametrized-Perturbation [18], Max-Min [15,19,29], Differential Transform Method [16], Adomian Decomposition Method [22], Energy Balance [23,30], etc.…”

Scrutiny of non-linear differential equations Euler-Bernoulli beam with large rotational deviation by AGM

“…Therefore, most situations of recent studies have been centered on the use of effective analytical methods in order to investigate these kinds of problems. Consequently, many new techniques have been presented in the open literature for homotopy perturbation method (HPM) [11,12], the elliptic Lindstedt-Poincare method (LP) [13,14], Adomian decomposition method [15,16] and the Krylov bogolioubov mitropopolsky method (KBM) [17,18]. Variational iteration method (VIM) has successfully been applied to various engineering applications.…”

Approximate analysis of two-mass–spring systems and buckling of a column

“…An improved He's Energy Balance method for solving nonlinear oscillatory differential equation using a new trial function was presented by Sfahani et al [18]. The problem considered represents the governing equations of the nonlinear, 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 [18].…”