| A new, non-parametric linearization method for nonlinear random vibration analysis is developed. The method employs a discrete representation of the stochastic excitation and concepts from the first-order reliability method, FORM, to compute first-order approximations to stochastic response statistics of interest. The stochastic excitation is discretized and represented in terms of a finite number of standard normal random variables. This representation makes it possible to formulate the tail-probability problem, i.e. the probability that a stochastic response quantity of interest exceeds a specified threshold at a specified point in time, as a time-invariant component reliability problem. Thus, the tail-probability problem can be solved by methods of structural reliability, e.g., FORM, by finding the so called "design point," which is the point of minimum distance from the origin on the limit-state surface in the space of the standard normal random variables. It is shown that for a linear system there exists an inverse relation between the design point of the tail-probability problem and the unit impulse response function of the system. Using this relation, the linear system can be uniquely determined from the knowledge of the design point in terms of its impulse response function.; In the equivalent linearization method developed in this study, the equivalent linear system is defined by matching the design points of the linear and nonlinear responses. Due to this definition, the tail probability of the equivalent linear system is equal to the first-order approximation of the tail probability of the nonlinear system. For this reason, the name Tail-Equivalent Linearization Method (TELM) is assigned to the method. As opposed to the conventional equivalent linearization method (ELM), which is a parametric method, the TELM is a non-parametric linearization method, since the equivalent linear system is defined numerically in terms of its impulse response function without the need to define a parameterized linear system. Having obtained the tail-equivalent linear system (TELS), linear random vibration analysis with the TELS is performed to determine various statistics of the nonlinear response, such as the probability distribution, the mean level-crossing rate and the first-passage probability.; Several important characteristics of the TELS, which have important bearing on the computational aspects of the TELM, are investigated. It is found that the TELS critically depends on the response threshold of interest. Through this dependence, the TELM is able to predict the non-Gaussian distribution of the nonlinear response and accurately predict tail probabilities, which are important in reliability and safety assessment. When the full distribution of the response is of interest, this dependence on the response threshold requires obtaining the design points for a sequence of thresholds. For this purpose, an algorithm to efficiently computing the design points for a sequence of response thresholds is developed. Secondly, the TELS is found to be invariant of the scaling of the excitation. This property makes it possible to estimate response statistics for a sequence of scaled excitations (fragility analysis) with a single determination of the TELS. Thirdly, the TELS is found to be only mildly dependent on the frequency content of broadband excitations. This property allows us to use the TELS obtained for a white-noise excitation to estimate the response statistics for other broadband excitations. Lastly, for nonstationary responses, it is found that the TELS for a selected time point provides fairly reasonable approximation to the first-passage probability. Using this approximation, TELSs determined at a single time point are sufficient for analysis of both stationary and nonstationary responses.; Example applications to single- and multi-degree-of-freedom, non-degrading hysteretic systems illustrate various features of the method. Compar... |
