Avoidance of artifacts in harmonic balance solutions for nonlinear dynamical systems
More details
Hide details
Technische Universit¨at Berlin, Chair of Mechatronics and Machine Dynamics, Berlin, Germany
Submission date: 2019-12-20
Final revision date: 2020-01-24
Acceptance date: 2020-01-24
Online publication date: 2020-04-15
Publication date: 2020-04-15
Corresponding author
Lukas Lentz   

Chair of Mechatronics and Machine Dynamics, Technische Universität Berlin, EInsteinufer 5, 10587, Berlin, Germany
Journal of Theoretical and Applied Mechanics 2020;58(2):307-316
Although nonlinear mechanical systems have been the topic of numerous investigations during the last decades, the research on suitable analysis methods is still ongoing. One method that is commonly known and still sees a lot of interest is the Harmonic Balance method. In the basic version of this method, only one harmonic is used to approximate a periodic solution, which allows for fairly easy application. A drawback is that this approach may lead to solutions that are inaccurate or even artifacts which are solutions that possess no physical relevance. In this article, it is demonstrated how an error criterion can be used to access the accuraccy of solutions and how artifacts can be indentified based on this assessment. Subsequently, stability analysis is performed for solutions that possess small errors. The method is applied to an asymmetric Duffing oscillator as well as to a system that consists of two linearly coupled Duffing oscillators. The authors gave a corresponding presentation of their work at PCM-CCM Kraków 2019.
Duffing G., 1918, Forced Oscillations with Variable Natural Frequency and their Technical Significance (in German), Vieweg, Braunschschweig.
Erturk A., Inman D.J., 2011, Broadband piezoelectric power generation on high-energy orbits of the bistable Duffing oscillator with electromechanical coupling, Journal of Sound and Vibration, 330, 2339-2353.
Ferri A., Leamy M., 2009, Error estimates for Harmonic-Balance solutions of nonlinear dynamical systems, 50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, DOI: 10.2514/6.2009-2667.
García-Saldaña J.D., Gasull A., 2013, A theoretical basis for the Harmonic Balance Method, Journal of Differential Equations, 254, 67-80.
Kovacic I., Brennan M.J., 2011, The Duffing Equation, John Wiley & Sons, New York.
Kozłowski J., Parlitz U., Lauterborn W., 1995, Bifurcation analysis of two coupled periodically driven Duffing oscillators, Physical Review E, 51, 1861-1867.
Liu L., Thomas J.P., Dowell E.H., Attar P.J., Hall K.C., 2006, A comparison of classical and high dimensional harmonic balance approaches for a Duffing oscillator, Journal of Computational Physics, 215, 298-320.
Nayfeh A.H., Mook D., 1979, Nonlinear Oscillations, John Wiley & Sons, New York.
Srebro R., 1995, The Duffing oscillator: a model for the dynamics of the neuronal groups comprising the transient evoked potential, Electroencephalography and Clinical Neurophysiology / Evoked Potentials Section, 96, 561-573.
Urabe M., 1965, Galerkin’s procedure for nonlinear periodic systems, Archive for Rational Mechanics and Analysis, 20, 120-152.
Urabe M., Reiter A., 1966, Numerical computation of nonlinear forced oscillations by Galerkin’s procedure, Journal of Mathematical Analysis and Applications, 14, 107-140.
Stokes A., 1972, On the approximation of nonlinear oscillations, Journal of Differential Equations, 12, 535-558.
Van Dooren R., 1988, On the transition from regular to chaotic behaviour in the Duffing oscillator, Journal of Sound and Vibration, 123, 327-339.
Von Wagner U., Lentz L., 2016, On some aspects of the dynamic behavior of the softening Duffing oscillator under harmonic excitation, Archive of Applied Mechanics, 8, 1383-1390.
Von Wagner U., Lentz L., 2018a, On artifact solutions of semi-analytic methods in non-linear dynamics, Archive of Applied Mechanics, 88, 1713-1724.
Von Wagner U., Lentz L., 2019, On the detection of artifacts in Harmonic Balance solutions of nonlinear oscillators, Applied Mathematical Modelling, 65, 408-414.
Journals System - logo
Scroll to top