Statistical Distribution of Frequency Response in Disordered Periodic Structures

A structure designed to be spatially periodic cannot be exactly periodic. The departure from exact periodicity is known as disorder, and it causes localization of normal modes and additional attenuation of wave motions in the structure not related to damping. The present investigation is directed at a possible adverse effect of disorder, namely, higher structural response near the point at which a dynamic excitation is applied than would occur in a perfectly periodic structure, thereby reducing structure safety and reliability. A systematic procedure is developed herein for the analysis of such an effect for a generic disordered periodic structure. In particular, the probability distribution of structural response is obtained by analysis and by Monte Carlo simulation, which is needed both for fundamental understanding of the effect and for predicting the reliability of a system. It is shown also that, given probability distributions of the disordered parameters of a structure, the mean and standard deviation of structural response can be obtained exactly for a damped randomly disordered periodic structure if the number of disordered cell units is not large and, approximately, if the number is large. Application of the procedure is illustrated by an example, and the results are compared with Monte Carlo simulations.

[1]  P. Taylor,et al.  Spectral Properties of Disordered Chains And Lattices , 1970 .

[2]  Robert E. Kielb,et al.  Flutter and Response of a Mistuned Cascade in Incompressible Flow , 1982 .

[3]  J. Woodhouse,et al.  Vibration isolation from irregularity in a nearly periodic structure: Theory and measurements , 1983 .

[5]  Y. K. Lin,et al.  Propagation of decaying waves in periodic and piecewise periodic structures of finite length , 1989 .

[6]  D. J. Mead,et al.  Coupled flexural-longitudinal wave motion in a periodic beam☆ , 1983 .

[7]  D. J. Mead.  Free wave propagation in periodically supported, infinite beams , 1970 .

[8]  C. Pierre.  Weak and strong vibration localization in disordered structures: A statistical investigation , 1990 .

[9]  E. Dowell,et al.  Localization of vibrations by structural irregularity , 1987 .

[10]  O. Bendiksen,et al.  Vibration Characteristics of Mistuned Shrouded Blade Assemblies , 1986 .

[11]  P. W. Whaley,et al.  Maximum Resonant Response of Mistuned Bladed Disks , 1984 .

[12]  D. J. Mead.  A general theory of harmonic wave propagation in linear periodic systems with multiple coupling , 1973 .

[13]  A. Sinha.  Calculating the statistics of forced response of a mistuned bladed disk assembly , 1986 .

[14]  T. T. Soong,et al.  On the natural frequencies of a disordered linear chain of N degrees of freedom , 1963 .

[15]  D. J. Mead.  Wave propagation and natural modes in periodic systems: II. Multi-coupled systems, with and without damping , 1975 .

[16]  J. N. Yang,et al.  Free Vibration of a Disordered Periodic Beam , 1974 .

[17]  Y. K. Lin.  Stresses in Continuous Skin-Stiffener Panels Under Random Loading , 1962 .

[18]  Christophe Pierre,et al.  Strong Mode Localization in Nearly Periodic Disordered Structures , 1989 .

[19]  Wenhu Huang,et al.  Vibration of Some Structures with Periodic Random Parameters , 1981 .

[20]  Y. K. Lin,et al.  LOCALIZATION OF WAVE PROPAGATION IN DISORDERED PERIODIC STRUCTURES , 1991 .

[21]  B. Kennett,et al.  Seismic Wave Propagation in Stratified Media , 1983 .

[22]  J. N. Yang,et al.  Frequency response functions of a disordered periodic beam , 1975 .

[23]  D. J. Mead.  Wave propagation and natural modes in periodic systems: I. Mono-coupled systems , 1975 .

[24]  Y. K. Lin,et al.  Free vibrations of a continuous beam on elastic supports , 1962 .

[25]  P. Anderson.  Absence of Diffusion in Certain Random Lattices , 1958 .

[26]  Y. K. Lin,et al.  Free Vibration of Continuous Skin-Stringer Panels , 1960 .

[27]  H. Furstenberg.  Noncommuting random products , 1963 .

[28]  L. Brillouin,et al.  Wave Propagation in Periodic Structures , 1946 .

[29]  D. J. Ewins,et al.  Resonant Vibration Levels of a Mistuned Bladed Disk , 1984 .

[30]  T. T. Soong,et al.  On the impulsive admittance and frequency response of a disordered linear chain of N degrees of freedom , 1964 .

[31]  Glen James Kissel,et al.  Localization in disordered periodic structures , 1988 .

[32]  John Ziman,et al.  Models of disorder , 1979 .

[33]  Y. K. Lin,et al.  Dynamics of Beam-Type Periodic Structures , 1969 .

[34]  Earl H. Dowell,et al.  Localized Vibrations of Disordered Multispan Beams: Theory and Experiment , 1987 .

[35]  A. V. Srinivasan,et al.  Fatigue Life Estimates of Mistuned Blades Via a Stochastic Approach , 1980 .

[36]  Oddvar O. Bendiksen,et al.  LOCALIZATION OF VIBRATIONS IN LARGE SPACE REFLECTORS , 1987 .