Nonlinear random vibrations of plates endowed with fractional derivative elements

Abstract This paper deals with the problem of determining the nonlinear response of a plate endowed with fractional derivative elements and exposed to random loads. It shows that an approximate solution of the nonlinear fractional partial differential equation governing the plate vibrations can be obtained via a statistical linearization based approach. The approach is implemented by considering a time-dependent representation of the response involving the eigen-functions of the linear problem. This representation allows deriving a nonlinear fractional ordinary differential equation governing the variation of the time-dependent part of the response, which is linearized in a mean square sense. Then, an iterative procedure provides the response statistics and power spectral density functions. Next, a Boundary Element Method is proposed for conducting relevant Monte Carlo data. The method is developed in conjunction with a Newmark integration scheme for estimating the response in the time domain given spectrum compatible realizations of the excitation. Monte Carlo data and statistical linearization solutions are calculated for square plates with simply supported stress-free edges, but problems involving other boundary conditions can be solved by the proposed approach.

[1]  M. Shitikova,et al.  Application of Fractional Calculus for Dynamic Problems of Solid Mechanics: Novel Trends and Recent Results , 2010 .

[2]  R. Bagley,et al.  On the Fractional Calculus Model of Viscoelastic Behavior , 1986 .

[3]  S. BRODETSKY,et al.  Theory of Plates and Shells , 1941, Nature.

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

[5]  P. Spanos,et al.  Harmonic wavelets based response evolutionary power spectrum determination of linear and non-linear oscillators with fractional derivative elements , 2016 .

[6]  P. Welch.  The use of fast Fourier transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms , 1967 .

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

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

[9]  P. Spanos,et al.  Random vibration and statistical linearization , 1990 .

[10]  G. Failla.  Stationary Response of Beams and Frames with Fractional Dampers through Exact Frequency Response Functions , 2017 .

[11]  Marina V. Shitikova,et al.  The analysis of the impact response of a thin plate via fractional derivative standard linear solid model , 2011 .

[12]  Pol D. Spanos,et al.  Nonlinear Random Vibrations of Beams with Fractional Derivative Elements , 2014 .

[13]  M. Shinozuka,et al.  Simulation of Stochastic Processes by Spectral Representation , 1991 .

[14]  M. Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. U.S. Department of Commerce, National Bureau of Standards , 1965 .

[15]  Wei Xu,et al.  Stationary response of nonlinear system with Caputo-type fractional derivative damping under Gaussian white noise excitation , 2015 .

[16]  Antonina Pirrotta,et al.  Stationary and non-stationary stochastic response of linear fractional viscoelastic systems , 2012 .

[17]  Pol D. Spanos,et al.  Stochastic response determination of nonlinear oscillators with fractional derivatives elements via the Wiener path integral , 2014 .

[18]  Xiaoling Jin,et al.  Response and stability of a SDOF strongly nonlinear stochastic system with light damping modeled by a fractional derivative , 2009 .

[19]  A. Gemant,et al.  A Method of Analyzing Experimental Results Obtained from Elasto‐Viscous Bodies , 1936 .

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

[21]  G. Scimemi,et al.  Finite element method on fractional visco-elastic frames , 2016 .

[22]  I. Podlubny.  Fractional differential equations : an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications , 1999 .

[23]  Thorsten Gerber,et al.  Handbook Of Mathematical Functions , 2016 .

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

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

[26]  K. Miller,et al.  An Introduction to the Fractional Calculus and Fractional Differential Equations , 1993 .

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

[28]  M. Shitikova,et al.  Nonlinear Dynamic Response of a Thin Plate Embedded in a Fractional Viscoelastic Medium under Combinational Internal Resonances , 2014 .

[29]  John T. Katsikadelis,et al.  Non-linear analysis of plates by the Analog Equation Method , 1994 .

[30]  M. Shitikova,et al.  A New Approach for Studying Nonlinear Dynamic Response of a Thin Plate with Internal Resonance in a Fractional Viscoelastic Medium , 2015 .

[31]  Pol D. Spanos,et al.  Response of a non-linear system with restoring forces governed by fractional derivatives—Time domain simulation and statistical linearization solution , 2010 .

[32]  An Accelerated Newmark Scheme for Integrating the Equation of Motion of Nonlinear Systems Comprising Restoring Elements Governed by Fractional Derivatives , 2011 .

[33]  L. G. Jaeger,et al.  Dynamics of structures , 1990 .

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

[35]  M. Shitikova,et al.  A New Approach for Studying Nonlinear Dynamic Response of a Thin Fractionally Damped Plate with 2:1 and 2:1:1 Internal Resonances , 2015 .

[36]  Pol D. Spanos.  Filter approaches to wave kinematics approximation , 1986 .

[37]  P. G. Nutting,et al.  A new general law of deformation , 1921 .