Analysis of Strongly Nonlinear Systems by Using HBM-AFT Method and Its Comparison with the Five-Order Runge–Kutta Method: Application to Duffing Oscillator and Disc Brake Model

Non-linear dynamic problems are difficult to analyze since no general methods exist to deal with them, which namely depend on several factors such as the nature of the problem, the type of non-linearity or the type of solution to be sought. While some methods are used to study weak nonlinearities, o...

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Bibliographic Details
Published in:	International journal of applied and computational mathematics Vol. 6; no. 2
Main Authors:	Ghorbel, Ahmed, Zghal, Becem, Abdennadher, Moez, Walha, Lassâad, Haddar, Mohamed
Format:	Journal Article
Language:	English
Published:	New Delhi Springer India 01-04-2020 Springer Nature B.V
Subjects:	Algorithms Applications of Mathematics Applied mathematics Computational mathematics Computational Science and Engineering Disc brakes Duffing oscillators Harmonic balance method Linearity Mathematical and Computational Physics Mathematical Modeling and Industrial Mathematics Mathematics Mathematics and Statistics Nonlinear analysis Nonlinear systems Nonlinearity Nuclear Energy Numerical methods Operations Research/Decision Theory Original Paper Runge-Kutta method Theoretical Alternating frequency time method Disc brake model Runge–Kutta method Duffing oscillator Harmonic balance method
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Summary:	Non-linear dynamic problems are difficult to analyze since no general methods exist to deal with them, which namely depend on several factors such as the nature of the problem, the type of non-linearity or the type of solution to be sought. While some methods are used to study weak nonlinearities, others can more easily solve strong nonlinearities. Therefore, the aim of the present paper is to find a method to solve the strong nonlinearities and to present the theoretical aspects of the harmonic balance method (HBM) in the form of algorithms to facilitate programming. Indeed, the first objective is to present the two continuation techniques based on the Newton–Raphson algorithm and the Alternating Frequency Time method (AFT). As for the second goal, it pertains to the proposition of a method for calculating the response curve by combining HBM and AFT with arc length continuation to solve the systems with strong non-linearities. For illustration, two applications were investigated, namely a Duffing model and a developed model describing a disc brake with non-linear friction. A comparison of the performances of the two continuation techniques was analyzed to demonstrate their advantages and disadvantages. Finally, the HBM–AFT was applied to study the Duffing oscillator and the disk brake model. The obtained results have shown a good agreement with those found in the literature for the Duffing model and with the Runge–Kutta numerical method of the order 5 for the brake model.
ISSN:	2349-5103 2199-5796
DOI:	10.1007/s40819-020-0803-z