scholarly journals Properties of Vibration with Fractional Derivative Damping of Order 1/2.

1997 ◽  
Vol 40 (3) ◽  
pp. 393-399 ◽  
Author(s):  
Susumu SAKAKIBARA
Keyword(s):  
Fractional Derivative ◽  
Fractional Derivative Damping

Asymptotic Approximations for Systems Incorporating Fractional Derivative Damping

1996 ◽  
Vol 118 (3) ◽  
pp. 572-579 ◽  
Author(s):  
B. S. Liebst ◽  
P. J. Torvik
Keyword(s):  
Fractional Derivative ◽  
Natural Frequency ◽  
Fractional Derivatives ◽  
Asymptotic Approximations ◽  
Damping Factor ◽  
Polynomial Equations ◽  
Algebraic Expressions ◽  
Dependent Behavior ◽  
Fractional Derivative Damping ◽  
Damping Materials

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.

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

2012 ◽  
Vol 22 (04) ◽  
pp. 1250083 ◽  
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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