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Coupled Van der Pol oscillator with non-integer order connection |
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| Affiliation: | 1. Department of Mathematics, College of Sciences, Yasouj University, Yasouj 75918-74934, Iran;2. Faculty of Mathematical Sciences, University of Tabriz, 29 Bahman St., Tabriz 51665-163, Iran;1. Instituto de Física, Universidade Federal de Goiás, 74.001-970 Goiânia, Goiás, Brazil;2. Instituto de Física, Universidade de São Paulo, 05314-970 São Paulo SP, Brazil;3. Departamento de Física, Universidade Federal da Paraíba, 58051-970 João Pessoa, Paraíba, Brazil;1. Nonlinear Scientific Research Center, Jiangsu University, Zhenjiang 212013, Jiangsu, China;2. Key Laboratory of Measurement and Control for Complex System of Ministry of Education, Research Institute of Automation, Southeast University, Nanjing 210096, Jiangsu, China;1. School of Mathematical Sciences/Institute of Computational Science, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, PR China;2. Department of Mathematics and Computer Science, Anshun University, Anshun, Guizhou 561000, PR China |
| Abstract: | In this paper the dynamics of a system of two oscillators with strong nonlinear connection is considered. The two mass system connected with a spring with pure nonlinear force of any positive rational order (integer or noninteger), on which some additional small nonlinear forces act, is analyzed. The mathematical model of the system contains two coupled second order differential equations of oscillatory type with strong pure nonlinearity and small additional terms. In the paper an analytical solving procedure which introduces the periodical Ateb function is developed. The averaging solution method is adopted to this special function and gives the new type of averaged differential equations.The special attention is given to the steady-state motion of a two-degree-of-freedom Van der Pol oscillator system of positive rational order of nonlinearity. The influence of the order of nonlinearity on the motion of the system is analyzed. using the suggested approximate method three numerical examples are solved. The obtained results are much more accurate than those obtained by the already published methods based on the trigonometric functions. |
| Keywords: | Two-degree-of-freedom system Van der Pol oscillator system Ateb function Steady state motion Nonlinear vibration |
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