Considerable Analytical Methods

Abstract

The harmonic balance method (HBM) is a technique used in systems including both linear and nonlinear parts. The fundamental idea of HBM is to decompose the system into two separate subsystems, a linear part and a nonlinear part. The linear part is treated in the frequency domain, and the nonlinear part in the time domain. The interface between the subsystems consists of the Fourier transform pair. Harmonic balance is said to be reached when a chosen number of harmonics N satisfy some predefined convergence criteria. First, an appropriate unknown is chosen to use in the convergence check, which is performed in the frequency domain. Then the equations are rewritten in a suitable form for a convergence loop. One starts with an initial value of the chosen unknown, applies the different linear and nonlinear equations, and finally reaches a new value of the chosen unknown. If the difference between the initial value and the final value of the first N harmonics satisfies the predefined convergence criteria, harmonic balance is reached. Otherwise, an increment of the initial value is calculated by using a generalized Euler method—namely, the Newton–Raphson method.KeywordsPredefined Convergence CriterionHarmonic Balance Method (HBM)Differential Transform Method (DTM)Second-order Analytical ApproximationParameter Expansion MethodThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.