Harmonic balance/Galerkin method for non-smooth dynamic systems

Abstract

Models of non-li near systems frequently introduce forces with bounded continuity resulting in nonsmooth (even discontinuous) flow. Examples include systems with clearances, backlash, friction, and impulses. Asymptotic methods require smooth (differentiable) flow and are therefore ill-suited for analyzing non-smooth systems. In these cases, the traditional harmonic balance method may be used to obtain approximate periodic solutions, but the method suffers from extremely slow convergence in general. Generalizations of the traditional harmonic balance method are introduced in this paper that result in superior convergence rates and superior modes of convergence. These improvements derive from the introduction of one or more expansion functions that possesses the same degree of continuity as the exact solution. In particular, forming an infinite series of such functions results in an expansion in the same function space of the exact solution. This expansion converges pointwise to the exact solution and to all derivatives thereof. These improvements are illustrated by example upon re-evaluating a classical single degree-of-freedom model for friction-induced vibration.

A linear complementarity problem formulation for periodic solutions to unilateral contact problems

“…Yet, both of these approaches face some inherent downfalls when nonsmooth nonlinearities-systems exhibiting non-differentiability or discontinuities in the unknowns-are encountered. On the one hand, the harmonic balance method (HBM) is known to produce poor approximations of non-smooth functions with a finite number of harmonics, producing artifacts such as the Gibbs phenomenon [10]. Hence penalty-like approximations of the contact inequalities are introduced [1] to effectively smoothen the nonsmoothness.…”

Multi-resolution Wavelet Analysis for Ferroresonance Phenomena in Power Systems

“…In power systems, the sensitivity analysis is based on the variation of series and parallel capacitances and source voltage. The harmonic balance method is used in order to solve nonlinear differential equations obtained from the model of power systems [7,8]. The analytical solution in a closed form is obtained and used for developing the contour maps of the system for secure and ferroresonance regions.…”

Dynamic analysis of microrobots with Coulomb friction using harmonic balance method

“…Some of these novel approaches are variational iteration method (VIM), homotopy perturbation method (HPM), homotopy analysis method (HAM), and harmonic balance method (HBM) [23][24][25].…”