Dynamic response of hysteretic systems to Poisson-distributed pulse trains

Abstract A single-degree-of-freedom hysteretic system subjected to a specific non-Gaussian random excitation in the form of a Poisson-distributed train of random pulses is considered. The hysteretic behaviour is described by the Bouc-Wen model of smooth hysteresis. The total hysteretic energy dissipation is assumed as a cumulative damage indicator. The state variables of the system together with the damage indicator form in that case a Poisson-driven, non-diffusive Markov vector process. Two equivalent systems are introduced by substituting the original non-analytical, non-algebraic non-linearities by equivalent linear and cubic forms in the pertinent state variables. Equations for mean responses are obtained by direct averaging of the governing equations, whereas the equations for second- and higher-order joint central moments are derived from the equivalent systems with the help of a generalized Ito's differential rule. Appearing in the equations for mean responses and for equivalent coefficients, expectations of non-algebraic functions of state variables are performed with respect to a non-Gaussian joint probability density function assumed in the form of a generalized, bivariate Gram-Charlier expansion. In the case of equivalent cubic non-linearities, the equations for moments form an infinite hierarchy which is truncated with the help of a cumulant-neglect closure technique. Mean values and variances of the response variables are evaluated by the numerical integration of equations for moments for two equivalent systems. For the sake of comparison the excitation process is also substituted by a Gaussian white noise and the usual equivalent linearization technique, combined with the Gaussian closure, is applied. The cases of non-zero-mean as well as zero-mean excitation processes are included. The case of general pulses is dealt with by a suitable augmentation of the state vector. The accuracy of the analytical techniques developed is verified against Monte Carlo simulations.

[1]  M. K. Kaul,et al.  Stochastic seismic analysis of yielding offshore towers , 1974 .

[2]  鈴木 祥之.  Seismic reliability analysis of hysteretic structures based on stochastic differential equations , 1986 .

[3]  Thomas T. Baber,et al.  Modal analysis for random vibration of hysteretic frames , 1986 .

[4]  Leon E. Borgman,et al.  RANDOM HYDRODYNAMIC FORCES ON OBJECTS , 1967 .

[5]  Y. Wen.  Equivalent Linearization for Hysteretic Systems Under Random Excitation , 1980 .

[6]  Shay Assaf,et al.  Approximate analysis of non-linear stochastic systems , 1976 .

[7]  Søren Nielsen,et al.  Dynamic response of non-linear systems to poisson-distributed random impulses , 1992 .

[8]  S. H. Crandall.  Non-gaussian closure for random vibration of non-linear oscillators , 1980 .

[9]  P. Thoft-Christensen,et al.  Response analysis of hysteretic multi‐storey frames under earthquake excitation , 1989 .

[10]  J. Roberts,et al.  System response to random impulses , 1972 .

[11]  J. Roberts.  The yielding behaviour of a randomly excited elasto-plastic structure , 1980 .

[12]  Y. K. Lin.  Application of Nonstationary Shot Noise in the Study of System Response to a Class of Nonstationary Excitations , 1963 .

[13]  P-T. D. Spanos,et al.  Formulation of Stochastic Linearization for Symmetric or Asymmetric M.D.O.F. Nonlinear Systems , 1980 .

[14]  Philip Protter,et al.  ℋp stability of solutions of stochastic differential equations , 1978 .

[15]  Thomas T. Baber.  Nonzero mean random vibration of hysteretic systems , 1984 .

[16]  E. Renshaw,et al.  STOCHASTIC DIFFERENTIAL EQUATIONS , 1974 .

[17]  T. T. Baber,et al.  Stochastic response of multistorey yielding frames , 1982 .

[18]  H. W. Liepmann,et al.  On the Application of Statistical Concepts to the Buffeting Problem , 1952 .

[19]  Y. Wen.  Method for Random Vibration of Hysteretic Systems , 1976 .

[20]  J. Karl Hedrick,et al.  Improved Statistical Linearization for Analysis and Control of Nonlinear Stochastic Systems: Part I: An Extended Statistical Linearization Technique , 1981 .

[21]  C. Tung,et al.  RANDOM RESPONSE OF HIGHWAY BRIDGES TO VEHICLE LOADS , 1967 .

[22]  Donald L. Snyder,et al.  Random point processes , 1975 .

[23]  A. Tylikowski,et al.  Vibration of a non-linear single degree of freedom system due to poissonian impulse excitation , 1986 .

[24]  J. Roberts,et al.  The Response of an Oscillator With Bilinear Hysteresis to Stationary Random Excitation , 1978 .

[25]  Thomas K. Caughey,et al.  Random Excitation of a System With Bilinear Hysteresis , 1960 .

[26]  P. Spanos,et al.  Hysteretic structural vibrations under random load , 1979 .

[27]  J. Roberts,et al.  The response of linear vibratory systems to random impulses , 1965 .

[28]  R. N. Iyengar,et al.  Study of the Random Vibration of Nonlinear Systems by the Gaussian Closure Technique , 1978 .

[29]  P. Thoft-Christensen,et al.  Stochastic response of hysteretic systems , 1990 .

[30]  J. Roberts.  On the response of a simple oscillator to random impulses , 1966 .