Stationary response of Duffing oscillator with hardening stiffness and fractional derivative |
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| Institution: | 1. Faculty of Sciences and Technologie, University of Khemis Miliana, 44000, Algeria;2. UMAB University of Mostaganem, 27000, Algeria;3. Department of Mathematics, Faculty of Science and Information Technology, Al-Zaytoonah University of Jordan, P.O. Box 130 Amman 11733, Jordan;4. Laboratory of Pure and Applied Mathematics, Faculty of SEI, UMAB, University of Mostaganem, 27000, Algeria;1. Department of Mathematics, Taiyuan Normal University, Jinzhong 030619, Shanxi, China;2. College of Mechanical Engineering, Taiyuan University of Science and Technology, Taiyuan 030024, Shanxi, China;3. Institute of Advanced Forming and Intelligent Equipment, Taiyuan University of Technology, Taiyuan 030024, Shanxi, China;4. Key Laboratory for Engineering & Computational Science (Taiyuan Normal University), Shanxi Provincial Department of Education, Jinzhong 030619, Shanxi, China |
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| Abstract: | The stationary response of Duffing oscillator with hardening stiffness and fractional derivative under Gaussian white noise excitation is studied. First, the term associated with fractional derivative is separated into the equivalent quasi-linear dissipative force and quasi-linear restoring force by using the generalized harmonic balance technique, and the original system is replaced by an equivalent nonlinear stochastic system without fractional derivative. Then, the stochastic averaging method of energy envelope is applied to the equivalent nonlinear stochastic system to yield the averaged Itô equation of energy envelope, from which the corresponding Fokker–Planck–Kolmogorov (FPK) equation is established and solved to obtain the stationary probability densities of the energy envelope and the amplitude envelope. The accuracy of the analytical results is validated by those from the Monte Carlo simulation of original system. |
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