Nonlinear steady state vibrations of beams made of the fractional Zener material using an exponential version of the harmonic balance method

Abstract

This paper presents the application of an exponential version of the harmonic balance method to the analysis of steady state vibration of geometrically nonlinear systems. A detailed description of the method and of the corresponding numerical procedure is provided. The von Karman theory is used to describe the effects of geometric nonlinearity. The material of the beams is modelled with the help of the Zener model using the fractional calculus. The problem is solved using an exponential version of the harmonic balance method. In the above-mentioned version, the complex calculus is used in contrast to the ordinary harmonic balance method, where the steady state vibrations are described with the help of the trigonometric functions. It significantly simplifies derivation of the amplitude equations. Moreover, the exponential version of the harmonic balance method is more elegant in comparison with the ordinary one. A detailed derivation of the amplitude equations is presented. The modified continuation method is proposed to solve the nonlinear amplitude equations and to determine the response curves. Moreover, the results of the exemplary calculation are presented and compared with known results in order to justify the efficiency and the correctness of the proposed approach.