Response and stability of a SDOF strongly nonlinear stochastic system with light damping modeled by a fractional derivative

A stochastic averaging procedure for a single-degree-of-freedom (SDOF) strongly nonlinear system with light damping modeled by a fractional derivative under Gaussian white noise excitations is developed by using the so-called generalized harmonic functions. The approximate stationary probability density and the largest Lyapunov exponent of the system are obtained from the averaged Ito stochastic differential equation of the system. It is shown that the approximate stationary solutions obtained by using the stochastic averaging procedure agree well with those from the numerical simulation of original systems. The effects of system parameters on the approxiamte stationary probability density and the largest Lyapunov exponent of the system are also discussed.

[1]  Ye Kun,et al.  Stochastic seismic response of structures with added viscoelastic dampers modeled by fractional derivative , 2003 .

[2]  Li Gen-guo,et al.  A numerical method for fractional integral with applications , 2003 .

[3]  Luis E. Suarez,et al.  A Comparison of Numerical Methods Applied to a Fractional Model of Damping Materials , 1999 .

[4]  Peter J. Torvik,et al.  Fractional calculus in the transient analysis of viscoelastically damped structures , 1983 .

[5]  R. Koeller.  Applications of Fractional Calculus to the Theory of Viscoelasticity , 1984 .

[6]  Peter J. Torvik,et al.  Fractional calculus-a di erent approach to the analysis of viscoelastically damped structures , 1983 .

[7]  O. P. Agrawal,et al.  STOCHASTIC ANALYSIS OF DYNAMIC SYSTEMS CONTAINING FRACTIONAL DERIVATIVES , 2001 .

[8]  Anil K. Bajaj,et al.  A Case Study on the Use of Fractional Derivatives: The Low-Frequency Viscoelastic Uni-Directional Behavior of Polyurethane Foam , 2004 .

[9]  A. Chatterjee,et al.  Averaging for Oscillations With Light Fractional Order Damping , 2003 .

[10]  W. Zhu.  Lyapunov exponent and stochastic stability of quasi-non-integrable Hamiltonian systems , 2004 .

[11]  Y. Chen,et al.  Fractional Order Disturbance Observer for Robust Vibration Suppression , 2004 .

[12]  L. Gaul,et al.  Impulse response function of an oscillator with fractional derivative in damping description , 1989 .

[13]  Om P. Agrawal,et al.  Stochastic Analysis of a 1-D System With Fractional Damping of Order 1/2 , 2002 .

[14]  Pol D. Spanos,et al.  Random Vibration of Systems with Frequency-Dependent Parameters or Fractional Derivatives , 1997 .

[15]  Y. K. Cheung,et al.  Averaging Method Using Generalized Harmonic Functions For Strongly Non-Linear Oscillators , 1994 .

[16]  C. Depollier,et al.  Propagation of Transient Acoustic Waves in Layered Porous Media: Fractional Equations for the Scattering Operators , 2004 .

[17]  Om P. Agrawal,et al.  Analytical Solution for Stochastic Response of a Fractionally Damped Beam , 2004 .

[18]  Li Gen-guo,et al.  Dynamical stability of viscoelastic column with fractional derivative constitutive relation , 2001 .

[19]  A. Gemant,et al.  XLV. On fractional differentials , 1938 .

[20]  O. Agrawal,et al.  A Numerical Scheme for Dynamic Systems Containing Fractional Derivatives , 2002 .

[21]  Pankaj Wahi,et al.  Averaging Oscillations with Small Fractional Damping and Delayed Terms , 2004 .

[22]  K. Adolfsson.  Nonlinear Fractional Order Viscoelasticity at Large Strains , 2004 .

[23]  K. B. Oldham,et al.  The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order , 1974 .

[24]  James M. Kelly,et al.  Application of fractional derivatives to seismic analysis of base‐isolated models , 1990 .

[25]  R. Bagley,et al.  A Theoretical Basis for the Application of Fractional Calculus to Viscoelasticity , 1983 .

[26]  Luis E. Suarez,et al.  An Eigenvector Expansion Method for the Solution of Motion Containing Fractional Derivatives , 1997 .

[27]  Om P. Agrawal,et al.  Application of Fractional Derivatives in Thermal Analysis of Disk Brakes , 2004 .

[28]  Yoshiyuki Suzuki,et al.  Response and stability of strongly non-linear oscillators under wide-band random excitation , 2001 .