Stochastic bifurcation analysis of a friction-damped system with impact and fractional derivative damping

Viscoelastic constitutive relationships incorporating fractional derivatives have been previously shown to be extremely useful in describing the frequency dependent behavior of common damping materials. However, the implementation of such models in the analysis of damped mechanical systems is somewhat complicated by the fact that polynomial equations with noninteger order exponents must be solved. This paper develops accurate approximations from which the damping factor and damped natural frequency of such systems may be obtained by evaluating relatively simple algebraic expressions.

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Stochastic analysis of monostable vibration energy harvesters with fractional derivative damping under Gaussian white noise excitation

Nonlinear Dynamics ◽

10.1007/s11071-018-4382-z ◽

2018 ◽

Vol 94 (1) ◽

pp. 639-648 ◽

Cited By ~ 8

Author(s):

Yong-Ge Yang ◽

Wei Xu

Keyword(s):

Fractional Derivative ◽

White Noise ◽

Stochastic Analysis ◽

Gaussian White Noise ◽

Vibration Energy ◽

Energy Harvesters ◽

White Noise Excitation ◽

Fractional Derivative Damping

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STOCHASTIC HOPF BIFURCATION OF QUASI-INTEGRABLE HAMILTONIAN SYSTEMS WITH FRACTIONAL DERIVATIVE DAMPING

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127412500836 ◽

2012 ◽

Vol 22 (04) ◽

pp. 1250083 ◽

Cited By ~ 2

Author(s):

F. HU ◽

W. Q. ZHU ◽

L. C. CHEN

Keyword(s):

Hopf Bifurcation ◽

Fractional Derivative ◽

Hamiltonian Systems ◽

Averaging Method ◽

Stochastic Averaging ◽

Stochastic Averaging Method ◽

Bifurcation Parameter ◽

Integrable Hamiltonian Systems ◽

Fractional Derivative Damping ◽

Stochastic Hopf Bifurcation

The stochastic Hopf bifurcation of multi-degree-of-freedom (MDOF) quasi-integrable Hamiltonian systems with fractional derivative damping is investigated. First, the averaged Itô stochastic differential equations for n motion integrals are obtained by using the stochastic averaging method for quasi-integrable Hamiltonian systems. Then, an expression for the average bifurcation parameter of the averaged system is obtained and a criterion for determining the stochastic Hopf bifurcation of the system by using the average bifurcation parameter is proposed. An example is given to illustrate the proposed procedure in detail and the numerical results show the effect of fractional derivative order on the stochastic Hopf bifurcation.

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Bifurcation Analysis of a Vibro-Impact Viscoelastic Oscillator with Fractional Derivative Element

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127418501705 ◽

2018 ◽

Vol 28 (14) ◽

pp. 1850170 ◽

Cited By ~ 2

Author(s):

Yong-Ge Yang ◽

Wei Xu ◽

YangQuan Chen ◽

Bingchang Zhou

Keyword(s):

Fractional Derivative ◽

Numerical Solutions ◽

Gaussian White Noise ◽

Stochastic Averaging ◽

Stochastic Averaging Method ◽

Stochastic Bifurcation ◽

Damping Force ◽

Impact Oscillator ◽

Viscoelastic Parameters ◽

Viscoelastic Term

To the best of authors’ knowledge, little work has been focused on the noisy vibro-impact systems with fractional derivative element. In this paper, stochastic bifurcation of a vibro-impact oscillator with fractional derivative element and a viscoelastic term under Gaussian white noise excitation is investigated. First, the viscoelastic force is approximately replaced by damping force and stiffness force. Thus the original oscillator is converted to an equivalent oscillator without a viscoelastic term. Second, the nonsmooth transformation is introduced to remove the discontinuity of the vibro-impact oscillator. Third, the stochastic averaging method is utilized to obtain analytical solutions of which the effectiveness can be verified by numerical solutions. We also find that the viscoelastic parameters, fractional coefficient and fractional derivative order can induce stochastic bifurcation.

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