Chapter
Jul 7, 2014
Solutions to Stochastic Dynamical Systems with Fractional Derivative Damping
Authors: Wei Li [email protected], Jun F. Zhao [email protected], Natasa. Trisovic [email protected], and Ying Zhang [email protected]Author Affiliations
Publication: Vulnerability, Uncertainty, and Risk: Quantification, Mitigation, and Management
Abstract
In this paper, the Galerkin method is presented to estimate the approximate stationary and non-stationary responses of nonlinear random dynamical systems with fractional derivative damping. Based on equivalent linearization method, the equation of motion is reverted to a linear equation in terms of envelope and phase process at first. After that, stochastic averaging method is used to get an averaged Ito differential equation with fractional derivative approximated by a periodic function. Finally, the approximate non-stationary probability density function solution of the associated Fokker-Planck equation is obtained by applying the Galerkin method. The application on a Rayleigh oscillator shows that the methods we utilized in this paper are efficient and reliable by comparing the stationary solution between the exact and approximated one.
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© 2014 American Society of Civil Engineers.
History
Published online: Jul 7, 2014
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ASCE Technical Topics:
- Continuum mechanics
- Damping
- Differential equations
- Dynamics (solid mechanics)
- Engineering fundamentals
- Engineering mechanics
- Equations (by type)
- Equations of motion
- Linear functions
- Mathematical functions
- Mathematics
- Nonlinear analysis
- Nonlinear response
- Probability
- Solid mechanics
- Stationary processes
- Stochastic processes
- Structural analysis
- Structural behavior
- Structural dynamics
- Structural engineering
Authors
Affiliations
School of Mathematics and Statistics, Xidian University, Xi'an, Shaanxi, 710071, China. E-mail: [email protected]
Department of Mathematics, Northwestern Polytechnical University, Xi'an Shaanxi, 710072, China. E-mail: [email protected]
University of Belgrade, Faculty of Mechanical Engineering, Department of Mechanics, Kraljice Marije 16, Belgrade, 11000 Serbia. E-mail: [email protected]
Department of Mathematics, Northwestern Polytechnical University, Xi'an Shaanxi, 710072, China. E-mail: [email protected]
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