A novel two-step pseudo-response based adaptive harmonic balance method for dynamic analysis of nonlinear structures

Abstract

Harmonic balance method (HBM) is one of the most popular and powerful methods, which is used to obtain response of nonlinear vibratory systems in frequency domain. The main idea of the method is to express the response of the system in Fourier series and converting the nonlinear differential equations of motion into a set of nonlinear algebraic equations. System response can be obtained by solving this nonlinear equation set in terms of the unknown Fourier coefficients. The accuracy of the solution is greatly affected by the number of harmonics included in the method and it is enhanced as the number of harmonics increases at the expense of computational time; hence, advantage of HBM over time integration method is lost. Therefore, it is desirable to use an adaptive algorithm where the number of harmonics can be optimized in terms of both accuracy and computational effort. In this paper a new adaptive harmonic balance method (AHBM) for the dynamic analysis of nonlinear structures is developed. The new method employs a two-step harmonic selection procedure where the criteria used are based on simple magnitude comparisons that make it easy to understand and program the method. A novel pseudo-response calculation method, which is used at the second harmonic selection step, is developed in order to estimate the response of the nonlinear system with, approximately, no additional computational cost. Due to the two-step harmonic selection procedure, the method eliminates unnecessary harmonics in the response calculation; hence, it is capable of increasing the computational efficiency of HBM significantly. Several case studies are given in order to show the applicability of the proposed adaptive harmonic balance method.