New Approach for the Analysis of Damped Vibrations of Fractional Oscillators

The dynamic behavior of linear and nonlinear mechanical oscillators with constitutive equations involving fractional derivatives defined as a fractional power of the operator of conventional time-derivative is considered. Such a definition of the fractional derivative enables one to analyse approximately vibratory regimes of the oscillator without considering the drift of its position of equilibrium. The assumption of small fractional derivative terms allows one to use the method of multiple time scales whereby a comparative analysis of the solutions obtained for different orders of low-level fractional derivatives and nonlinear elastic terms is possible to be carried out. The interrelationship of the fractional parameter (order of the fractional operator) and nonlinearity manifests itself in full measure when orders of the small fractional derivative term and of the cubic nonlinearity entering in the oscillator's constitutive equation coincide.

[1]  R. Mickens.  Fractional Van Der Pol Equations , 2003 .

[2]  Mohammad Saleh Tavazoei,et al.  Regular oscillations or chaos in a fractional order system with any effective dimension , 2008 .

[3]  A. Hanyga,et al.  Nonlinear differential equations with fractional damping with applications to the 1dof and 2dof pendulum , 2005 .

[4]  Time-domain Analysis Of Structures With Dampers Modelled By Fractional Derivatives , 2001 .

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

[6]  R. Magin.  Fractional Calculus in Bioengineering , 2006 .

[7]  A. Tofighi,et al.  The intrinsic damping of the fractional oscillator , 2003 .

[8]  G. Drăgănescu,et al.  Application of a variational iteration method to linear and nonlinear viscoelastic models with fractional derivatives , 2006 .

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

[10]  Javeed A. Munshi.  Effect of viscoelastic dampers on hysteretic response of reinforced concrete elements , 1997 .

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

[12]  J. Ramos.  Piecewise-linearized methods for oscillators with fractional-power nonlinearities , 2007 .

[13]  Lothar Gaul,et al.  On the numerical evaluation of fractional derivatives in multi-degree-of-freedom systems , 2006, Signal Process..

[14]  James M. Kelly,et al.  EVOLUTIONARY MODEL OF VISCOELASTIC DAMPERS FOR STRUCTURAL ApPLICATIONS , 1997 .

[15]  Juebang Yu,et al.  Chaos in the fractional order periodically forced complex Duffing’s oscillators , 2005 .

[16]  B. Achar,et al.  Damping characteristics of a fractional oscillator , 2004 .

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

[18]  A. M. Abdel-Ghaffar,et al.  Ambient Vibration Studies of Golden Gate Bridge: I. Suspended Structure , 1985 .

[19]  Luis E. Suarez,et al.  Response of Systems With Damping Materials Modeled Using Fractional Calculus , 1995 .

[20]  I. Schäfer,et al.  Description of the Impulse Response in Rods by Fractional Derivatives , 2002 .

[21]  Ji-Huan He.  Approximate analytical solution for seepage flow with fractional derivatives in porous media , 1998 .

[22]  H. Kober.  ON FRACTIONAL INTEGRALS AND DERIVATIVES , 1940 .

[23]  Grzegorz Litak,et al.  Vibration of generalized double well oscillators , 2006, nlin/0610052.

[24]  P. Arena,et al.  Nonlinear Noninteger Order Circuits and Systems — An Introduction , 2000 .

[25]  Analysis of the simple harmonic oscillator with fractional damping , 2003 .

[26]  S. Sakakibara.  Properties of Vibration with Fractional Derivative Damping of Order 1/2. , 1997 .

[27]  Wenchang Tan,et al.  Intermediate processes and critical phenomena: Theory, method and progress of fractional operators and their applications to modern mechanics , 2006 .

[28]  Yuriy A. Rossikhin,et al.  Analysis of the viscoelastic rod dynamics via models involving fractional derivatives or operators of two different orders , 2004 .

[29]  Michael C. Constantinou,et al.  Fractional‐Derivative Maxwell Model for Viscous Dampers , 1991 .

[30]  B. Achar,et al.  Dynamics of the fractional oscillator , 2001 .

[31]  M. Caputo.  Vibrations of an infinite viscoelastic layer with a dissipative memory , 1974 .

[32]  Yuriy A. Rossikhin,et al.  Application of Fractional Calculus for Analysis of Nonlinear Damped Vibrations of Suspension Bridges , 1998 .

[33]  Wenchang Tan,et al.  Intermediate processes and critical phenomena: Theory, method and progress of fractional operators and their applications to modern mechanics , 2006 .

[34]  Leif Kari,et al.  Nonlinear Isolator Dynamics at Finite Deformations: An Effective Hyperelastic, Fractional Derivative, Generalized Friction Model , 2003 .

[35]  J. S. Hwang,et al.  Analytical Modeling of High Damping Rubber Bearings , 1997 .

[36]  G M Zaslavsky,et al.  Chaotic and pseudochaotic attractors of perturbed fractional oscillator. , 2006, Chaos.

[37]  R. Feynman,et al.  RECENT APPLICATIONS OF FRACTIONAL CALCULUS TO SCIENCE AND ENGINEERING , 2003 .

[38]  Marina V. Shitikova,et al.  Analysis of Dynamic Behaviour of Viscoelastic Rods Whose Rheological Models Contain Fractional Derivatives of Two Different Orders , 2001 .

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

[40]  J. A. Tenreiro Machado,et al.  Analysis of the Van der Pol Oscillator Containing Derivatives of Fractional Order , 2007 .

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

[42]  Zhang Wei,et al.  Damping Properties of the Viscoelastic Material Described by Fractional Kelvin-Voigt Model , 1999 .

[43]  Gloria Terenzi,et al.  Dynamics of SDOF Systems with Nonlinear Viscous Damping , 1999 .

[44]  Vasily E. Tarasov,et al.  Dynamics with low-level fractionality , 2005, physics/0511138.

[45]  Rong Deng,et al.  Flexible polyurethane foam modelling and identification of viscoelastic parameters for automotive seating applications , 2003 .

[46]  Ingo Schäfer,et al.  Impulse Responses of Fractional Damped Systems , 2004 .

[47]  Vimal Singh,et al.  Perturbation methods , 1991 .

[48]  Y. Rossikhin,et al.  Damped vibrations of hereditary -elastic systems with weakly singular kernels , 1970 .

[49]  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 .

[50]  M. Giovagnoni,et al.  A fractional derivative model for single-link mechanism vibration , 1992 .

[51]  Anindya Chatterjee,et al.  Galerkin Projections and Finite Elements for Fractional Order Derivatives , 2006 .

[52]  Marina V. Shitikova,et al.  Application of fractional derivatives to the analysis of damped vibrations of viscoelastic single mass systems , 1997 .

[53]  I. Schäfer.  Beschreibung der Dämpfung in Stäben mittels fraktionaler Zeitableitungen , 2000 .

[54]  A. M. Abdel-Ghaffar,et al.  Ambient Vibration Studies of Golden Gate Bridge , 1985 .

[55]  G. Litak,et al.  Vibration of the Duffing oscillator: Effect of fractional damping , 2006, nlin/0601033.

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

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

[58]  Yuriy A. Rossikhin,et al.  Nonlinear free damped vibrations of suspension bridges with uncertain fractional damping , 2008 .

[59]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[60]  Marina V. Shitikova,et al.  APPLICATION OF FRACTIONAL OPERATORS TO THE ANALYSIS OF DAMPED VIBRATIONS OF VISCOELASTIC SINGLE-MASS SYSTEMS , 1997 .

[61]  Ting-Yu Hsu,et al.  A fractional derivative model to include effect of ambient temperature on HDR bearings , 2001 .

[62]  B. Achar,et al.  Response characteristics of a fractional oscillator , 2002 .

[63]  A. Hanyga,et al.  Nonlinear Hamiltonian equations with fractional damping , 2000 .

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