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

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.