TY - JOUR
T1 - Principal resonance responses of SDOF systems with small fractional derivative damping under narrow-band random parametric excitation
AU - Liu, Di
AU - Li, Jing
AU - Xu, Yong
PY - 2014/10
Y1 - 2014/10
N2 - The principal resonance responses of nonlinear single-degree-of-freedom (SDOF) systems with lightly fractional derivative damping of order α (0. <. α<. 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.
AB - The principal resonance responses of nonlinear single-degree-of-freedom (SDOF) systems with lightly fractional derivative damping of order α (0. <. α<. 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.
KW - Fractional derivative damping
KW - Multiple scale method
KW - Narrow-band noise
KW - SDOF nonlinear systems
UR - http://www.scopus.com/inward/record.url?scp=84900024224&partnerID=8YFLogxK
U2 - 10.1016/j.cnsns.2014.03.018
DO - 10.1016/j.cnsns.2014.03.018
M3 - 文章
AN - SCOPUS:84900024224
SN - 1007-5704
VL - 19
SP - 3642
EP - 3652
JO - Communications in Nonlinear Science and Numerical Simulation
JF - Communications in Nonlinear Science and Numerical Simulation
IS - 10
ER -