The First Passage Failure of SDOF Strongly Nonlinear Stochastic System with Fractional Derivative Damping

The first passage failure of single-degree-of-freedom (SDOF) strongly nonlinear stochastic system with fractional derivative damping is studied. The stochastic averaging method for SDOF strongly nonlinear stochastic system with fractional derivative damping under white noise excitations using the generalized harmonic function is introduced. The averaged Itô equation for Hamiltonian is obtained by using the stochastic averaging method. Then, a backward Kolmogorov equation governing the conditional reliability function and a Pontryagin equation governing the mean of the first passage time are established. The conditional reliability function, and the conditional probability density and mean first passage time are obtained from solving these equations together with suitable initial condition and boundary conditions. Finally, two examples are worked out in detail and the solutions are confirmed by using Monte Carlo simulation of original systems.

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

[2]  O. Agrawal.  AN ANALYTICAL SCHEME FOR STOCHASTIC DYNAMIC SYSTEMS CONTAINING FRACTIONAL DERIVATIVES , 1999 .

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

[4]  Ingo Schäfer,et al.  Fractional Calculus via Functional Calculus: Theory and Applications , 2002 .

[5]  W. Zhu,et al.  First-Passage Failure of Quasi-Integrable Hamiltonian Systems , 2002 .

[6]  T. Soong,et al.  MODELING OF VISCOELASTIC DAMPERS FOR STRUCTURAL ApPLICATIONS , 1995 .

[7]  Weiqiu Zhu,et al.  First-passage failure and its feedback minimization of quasi-partially integrable Hamiltonian systems , 2003 .

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

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

[10]  J. Roberts,et al.  First-passage time for randomly excited non-linear oscillators , 1986 .

[11]  M. Shitikova,et al.  Analysis of free non-linear vibrations of a viscoelastic plate under the conditions of different internal resonances , 2006 .

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

[13]  Chunbiao Gan,et al.  First-passage failure of quasi-non-integrable-Hamiltonian systems , 2001 .

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

[15]  Francesco Mainardi,et al.  Fractional relaxation in anelastic solids , 1994 .

[16]  Jerzy T. Sawicki,et al.  Nonlinear Vibrations of Fractionally Damped Systems , 1998 .

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

[18]  F. Rüdinger,et al.  Tuned mass damper with fractional derivative damping , 2006 .

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

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

[21]  M. Shitikova,et al.  Applications of Fractional Calculus to Dynamic Problems of Linear and Nonlinear Hereditary Mechanics of Solids , 1997 .

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

[23]  Peter J. Torvik,et al.  FRACTIONAL CALCULUS - A DIFFERENT APPROACH TO THE FINITE ELEMENT ANALYSIS OF VISCOELASTICALLY DAMPED STRUCTURES. , 1981 .

[24]  Juhn-Horng Chen,et al.  Chaotic dynamics of the fractionally damped van der Pol equation , 2008 .

[25]  Nicos Makris,et al.  Spring‐viscous damper systems for combined seismic and vibration isolation , 1992 .

[26]  Marina V. Shitikova,et al.  Analysis of nonlinear free vibrations of suspension bridges , 1995 .

[27]  W. Feller.  Diffusion processes in one dimension , 1954 .

[28]  J. Roberts,et al.  First Passage Probability for Nonlinear Oscillators , 1976 .

[29]  T. Pritz,et al.  ANALYSIS OF FOUR-PARAMETER FRACTIONAL DERIVATIVE MODEL OF REAL SOLID MATERIALS , 1996 .

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

[31]  G. Solomos,et al.  Barrier Crossing Due to Transient Excitation , 1984 .

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

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

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

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

[36]  James M. Kelly,et al.  Visco-hyperelastic model for filled rubbers used in vibration isolation , 1997 .

[37]  Y. K. Lin,et al.  On Statistics of First-Passage Failure , 1994 .

[38]  O. P. Agrawal,et al.  A New Lagrangian and a New Lagrange Equation of Motion for Fractionally Damped Systems , 2001 .

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

[40]  Hsien-Keng Chen,et al.  Chaotic dynamics of the fractionally damped Duffing equation , 2007 .

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

[42]  Wei Zhang,et al.  Fractional Calculus Approach to Dynamic Problems of Viscoelastic Materials , 1999 .

[43]  Marina V. Shitikova,et al.  Analysis of nonlinear vibrations of a two-degree-of-freedom mechanical system with damping modelled by a fractional derivative , 2000 .

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