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Constructing transient response probability density of non-linear system through complex fractional moments |
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| Institution: | 1. Department of Engineering Mechanics, Zhejiang University, Hangzhou, China;2. Dipartimento di Ingegneria Civile, Ambientale, Aerospaziale, dei Materiali (DICAM), Univerista di Palermo, Palermo, Italy;1. The State Key Laboratory of Fluid Power Transmission and Control, Department of Engineering Mechanics, Zhejiang University, Hangzhou, China;2. Institute of Fluid Mechanics, China Jiliang University, Hangzhou, China;1. INSA Centre Val de Loire, Université François Rabelais de Tours, LMR EA 2640, Campus de Blois, 3 Rue de la Chocolaterie, CS 23410, 41034 Blois Cedex, France;2. LMA, CNRS, UPR 7051, Centrale Marseille, Aix-Marseille Univ, F-13420 Marseille Cedex 20, France;1. School of Mathematical Sciences, Shanxi University, Taiyuan 030006, China;2. Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an 710072, China;3. School of Applied Science, Taiyuan University of Science and Technology, Taiyuan 030024, China;1. Dipartimento di Ingegneria Civile Ambientale, Aerospaziale, dei Materiali (DICAM), Università degli Studi di Palermo, Viale delle Scienze, Ed. 8, 90128 Palermo, Italy;2. Department of Innovation Engineering, University of Salento, Lecce, Italy |
| Abstract: | The probability density function for transient response of non-linear stochastic system is investigated through the stochastic averaging and Mellin transform. The stochastic averaging based on the generalized harmonic functions is adopted to reduce the system dimension and derive the one-dimensional Itô stochastic differential equation with respect to amplitude response. To solve the Fokker–Plank–Kolmogorov equation governing the amplitude response probability density, the Mellin transform is first implemented to obtain the differential relation of complex fractional moments. Combining the expansion form of transient probability density with respect to complex fractional moments and the differential relations at different transform parameters yields a set of closed-form first-order ordinary differential equations. The complex fractional moments which are determined by the solution of the above equations can be used to directly construct the probability density function of system response. Numerical results for a van der Pol oscillator subject to stochastically external and parametric excitations are given to illustrate the application, the convergence and the precision of the proposed procedure. |
| Keywords: | Probability density function Transient response Non-linear stochastic system Stochastic averaging Complex fractional moment Mellin transform |
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