Analytic Solution for Nonlinear Multimode Beam Vibration Using a Modified Harmonic Balance Approach and Vieta’s Substitution

Abstract

This paper presents a modified harmonic balance solution method incorporated with Vieta’s substitution technique for nonlinear multimode damped beam vibration. The aim of the modification in the solution procedures is to develop the analytic formulations, which are used to calculate the vibration amplitudes of a nonlinear multimode damped beam without the need of nonlinear equation solver for the nonlinear algebraic equations generated in the harmonic balance processes. The result obtained from the proposed method shows reasonable agreement with that from a previous numerical integration method. In general, the results can show the convergence and prove the accuracy of the proposed method.

Analytic Formula for the Vibration and Sound Radiation of a Nonlinear Duct

“…is implies that the acoustic peak is more important or signi cant when the excitation level is higher. In Figures 7(b) and 7(c), the structural-acoustic peak ratios of the vibration amplitude are almost constant when the excitation level ranges from κ 0.05 to 0.12. ere is an abrupt jump in each curve at around κ 0.12 to 0.15. e abrupt jumps are due to the jump-up phenomenon, which is well known in forced nonlinear panel vibration [34]. en, the structural-acoustic peak ratios of vibration amplitude are monotonically decreasing with the increasing excitation level.…”

“…According to Vieta's substitution [34], equation 17can be simplified using the substitution of c � λ − (G 2 /3G 3 ):…”

“…is motivated the current study to adopt the classical method and Vieta's substitution technique [34] and to develop analytic formulations that would not require an equation solver to compute the nonlinear sound and vibration responses of a rectangular duct. Note that, normally a nonlinear problem would generate a set of nonlinear equations.…”

A modified harmonic balance method for solving forced vibration problems with strong nonlinearity

“…where x is frequency, and A, B, C, and D are constants to be determined. Substituting equation (19) into equation (18) results in JðA; B; C; DÞ ¼…”

“…Consequently, the quest for accurate behavior of the nonlinear dynamical systems led to the development of many analytical approximations. In the literature, several analytical approximate methods are found such as perturbation, 7,8 homotopy analysis, 9,10 homotopy perturbation, 11,12 variational iteration, 13,14 harmonic balance (HBM), [15][16][17][18] etc.…”