Nonstationary extreme response of simple oscillators

• Main Author: Michaelov, George Z
• Format: Dissertation
• Language: English
• Published: ProQuest Dissertations & Theses 01-01-1997
• Subjects: Civil engineering; Mechanical engineering; Mechanics
• ISBN: 0591285029; 9780591285024

To design structures under random loading conditions it is important to characterize the extreme structural response in terms of simple engineering parameters--fractile level, mean, and standard deviation. This study addresses the problem of extreme structural response to nonstationary random excitation by modeling the response and the excitation of a simple dynamic system as evolutionary Gaussian processes. The modulated stationary Gaussian process, a particular evolutionary case, receives a special consideration, and its applications in a number of structural engineering problems are reviewed. The time and the frequency domain characteristics of the nonstationary response of the simple oscillator (single degree-of-freedom dynamic system) under general evolutionary Gaussian excitation are examined. Explicit expressions for these characteristics in the case of modulated white noise excitation are provided. The problem of computing the probability distribution of the extreme value of a random process is examined and four contemporary approaches to its solution are reviewed. These include the relatively simple Poisson and Vanmarke approximations and two more advanced approaches. For all methods considered, the resulting expressions of the probability distribution of the extreme value are derived. The study demonstrates that if simplicity and efficiency for engineering applications are sought, the Poisson and Vanmarke approximations do not have alternatives because the advanced methods require a more significant computational effort. A new method for computing the fractile levels, the mean, and the standard deviation of the extreme value of the nonstationary response process of the simple oscillator is developed. The new method incorporates both Poisson and Vanmarke approximations of the extreme value distribution. Two alternative approaches are introduced. These approaches approximate the probability distribution of the extreme value as a two-parameter Weibull distribution and as a two-parameter Gumbel distribution, respectively. The unknown parameters of both approximations are computed by means of moment equations from the second-order response statistics. Both approximations result in convenient equations from which the fractile levels, the mean value, and the standard deviation of the extreme response are computed. The formulas developed are applied to two nonstationary response problems. The results demonstrate that the new method agrees fairly well with the exact solutions of the equations of both Poisson and Vanmarke approximations.