4.7 Article

Principal resonance responses of SDOF systems with small fractional derivative damping under narrow-band random parametric excitation

Journal

Publisher

ELSEVIER SCIENCE BV
DOI: 10.1016/j.cnsns.2014.03.018

Keywords

SDOF nonlinear systems; Fractional derivative damping; Narrow-band noise; Multiple scale method

Funding

  1. National Natural Science Foundation of China [11372247, 11202160, 61303020]
  2. Program for New Century Excellent Talents in University
  3. Shaanxi Project for Young New Star in Science Technology
  4. NPU Foundation for Fundamental Research and New Faculties and Research Area Project
  5. Youth Science and Technology Research Fund of Shanxi Province
  6. Program of TYUST for Doctors' scientific research [20122019]
  7. Shanxi University [010951801005]

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The principal resonance responses of nonlinear single-degree-of-freedom (SDOF) systems with lightly fractional derivative damping of order alpha (0 < alpha < 1) subject to the narrow-band random parametric excitation are investigated. The method of multiple scales is developed to derive two first order stochastic differential equation of amplitude and phase, and then to examine the influences of fractional order and intensity of random excitation on the first-order and second-order moment. As an example, the stochastic Duffing oscillator with fractional derivative damping is considered. The effects of detuning frequency parameter, the intensity of random excitation and the fractional order derivative damping on stability are studied through the largest Lyapunov exponent. The corresponding theoretical results are well verified through direct numerical simulations. In addition, the phenomenon of stochastic jump is analyzed for parametric principal resonance responses via finite differential method. The stochastic jump phenomena indicates that the most probable motion is around the larger non-trivial branch of the amplitude response when the intensity of excitation is very small, and the probable motion of amplitude responses will move from the larger non-trivial branch to trivial branch with the increasing of the intensity of excitation. Such stochastic jump can be considered as bifurcation. (C) 2014 Elsevier B.V. All rights reserved.

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