Response of Aging Linear Systems to Ergodic Random Input

Response of an aging linear system exposed from a certain age t0 to ergodic random input is analyzed. It is shown that the response, while non‐stationary with respect to time and age, is stationary and ergodic with regard to the birth time τ (the time when the system was built). Consequently, the instantaneous statistical characteristics of all possible response realizations at a chosen age, t, may be determined as the characteristics of the response at age t (and at a fixed exposure age, t0) as the birth time τ is varied, i.e., as the input history is shifted in time against the instant when the system was built. Based on this new idea, the spectral method is generalized for aging systems, using a frequency response function and a spectral density of response that depend on both the current age t and the age t0 when the exposure begins. The relation between the spectral densities of input and response is algebraic, similar to the case of stationary response of nonaging systems. For the special case of no...

[1]  Chris P. Tsokos.  Probability distributions: an introduction to probability theory with applications , 1972 .

[2]  Erik H. Vanmarcke,et al.  Random Fields: Analysis and Synthesis. , 1985 .

[3]  I. Miller.  Probability, Random Variables, and Stochastic Processes , 1966 .

[4]  W. D. Mark.  Spectral analysis of the convolution and filtering of non-stationary stochastic processes , 1970 .

[5]  D. Newland.  An introduction to random vibrations and spectral analysis , 1975 .

[6]  Zdeněk P. Bažant,et al.  Algorithm for Aging Viscoelastic Structures Under Periodic Load , 1984 .

[7]  F. H. Wittmann,et al.  Mathematical Models for Creep and Shrinkage of Concrete , 2022 .

[8]  Bruno O. Shubert,et al.  Random variables and stochastic processes , 1979 .

[9]  H. Saunders,et al.  Book Reviews : AN INTRODUCTION TO RANDOM VIBRATION AND SPECTRAL ANALYSIS D.E. Newland Longman's Inc., New York, NY, 1978 , 1980 .

[10]  Zdeněk P. Bažant,et al.  Spectral Finite Element Analysis of Random Shrinkage in Concrete , 1984 .

[11]  Zdenek P. Bazant,et al.  Spectral Analysis of Random Shrinkage Stresses in Concrete , 1984 .

[12]  P. Zdenllk,et al.  Theory of Creep and Shrinkage in ' Concrete Structures : A · Precis of Recent Developments , 2022 .

[13]  I. Stakgold,et al.  Boundary value problems of mathematical physics , 1987 .

[14]  Zdeněk P. Bažant,et al.  I. – Theory of Creep and Shrinkage in Concrete Structures: A Précis of Recent Developments , 1975 .

[15]  M. B. Priestley,et al.  Power spectral analysis of non-stationary random processes , 1967 .

[16]  Ross B. Corotis,et al.  Time-Dependent Spectral Content of System Response , 1975 .

[17]  John G. Proakis,et al.  Probability, random variables and stochastic processes , 1985, IEEE Trans. Acoust. Speech Signal Process..

[18]  Zden~k P Ba~ant,et al.  Probabilistic Problems in Prediction of Creep and Shrinkage Effects in Structures , 2022 .

[19]  C. Page.  Instantaneous Power Spectra , 1952 .

[20]  A. G. Piersol.  Power spectra measurements for spacecraft vibration data. , 1967 .

[21]  H. J. Stumpf,et al.  Transient Response of a Dynamic System Under Random Excitation , 1961 .

[22]  W. D. Mark,et al.  Random Vibration in Mechanical Systems , 1963 .

[23]  Zdenek P. Bazant,et al.  Random Shrinkage Stresses in Aging Viscoelastic Vessel , 1982 .

[24]  J. K. Hammond,et al.  On the response of single and multidegree of freedom systems to non-stationary random excitations , 1968 .