Closed-form solutions for the time-variant spectral characteristics of non-stationary random processes

Spectral characteristics are important quantities in describing stationary and non-stationary random processes. In this paper, the spectral characteristics for complex-valued random processes are evaluated and closed-form solutions for the time-variant statistics of the response of linear single-degree-of-freedom (SDOF) and both classically and non-classically damped multi-degree-of-freedom (MDOF) systems subjected to modulated Gaussian colored noise are obtained. The time-variant central frequency and bandwidth parameter of the response processes of linear SDOF and MDOF elastic systems subjected to Gaussian colored noise excitation are computed exactly in closed-form. These quantities are useful in problems which require the use of complex modal analysis, such as random vibrations of non-classically damped MDOF linear structures, and in structural reliability applications. Monte Carlo simulation has been used to confirm the validity of the proposed solutions.

[1]  Steen Krenk,et al.  Stochastic Response Analysis , 1983 .

[2]  N. C. Nigam.  Introduction to Random Vibrations , 1983 .

[3]  S. H. Crandall.  First-crossing probabilities of the linear oscillator , 1970 .

[4]  Shahram Sarkani,et al.  Spectral characteristics of nonstationary random processes — a critical review , 1999 .

[5]  Ross B. Corotis,et al.  First Passage of Nonstationary Random Processes , 1972 .

[6]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[7]  Pol D. Spanos,et al.  Oscillator Response to Nonstationary Excitation , 1984 .

[8]  J. Dugundji,et al.  Envelopes and pre-envelopes of real waveforms , 1958, IRE Trans. Inf. Theory.

[9]  P-T. D. Spanos,et al.  Structural reliability under evolutionary seismic excitation , 1983 .

[10]  Joel P. Conte,et al.  Spectral characteristics of non-stationary random processes: Theory and applications to linear structural models , 2008 .

[11]  S. Rice.  Mathematical analysis of random noise , 1944 .

[12]  M. Di Paola.  Transient spectral moments of linear systems , 1985 .

[13]  J. Gary Reid,et al.  Linear System Fundamentals: Continuous and Discrete, Classic and Modern , 1983 .

[14]  Joel P. Conte,et al.  Closed-form solutions for the response of linear systems to fully nonstationary earthquake excitation , 1998 .

[15]  R. Arens,et al.  Complex processes for envelopes of normal noise , 1957, IRE Trans. Inf. Theory.

[16]  G. Solomos,et al.  Barrier Crossing Due to Transient Excitation , 1984 .

[17]  Yu-Kweng Michael Lin.  Probabilistic Theory of Structural Dynamics , 1976 .

[18]  Masanobu Shinozuka,et al.  Simulation of Nonstationary Random Process , 1967 .

[19]  Pol D. Spanos,et al.  Hilbert Transform Generalization of a Classical Random Vibration Integral , 1994 .

[20]  P. Spanos.  Spectral Moments Calculation of Linear System Output , 1983 .

[21]  E. Vanmarcke.  On the Distribution of the First-Passage Time for Normal Stationary Random Processes , 1975 .

[22]  Steen Krenk,et al.  Stationary and Transient Response Statistics , 1982 .

[23]  Steen Krenk,et al.  Stationary and Transient Response Envelopes , 1983 .