Simulation of many acceleration time histories with causality from an observed earthquake motion: Modeling of real part and reproduction of imaginary part by Hilbert transform

Abstract

To conduct the performance design we need many acceleration time histories to evaluate the dynamic nonlinear behavior of concerned structural systems. One practical method is to apply the stochastic process for simulating acceleration time histories. For this purpose we extract the stochastic characteristic of acceleration time history and its uncertainty from an observed earthquake motion. In this research we make a model to simulate a sample real part of Fourier transform of an observed acceleration time history. The imaginary parts of the Fourier transform of a causal acceleration time history can be reproduced from the real part by using the Hilbert transformation. Using this well known fact we develop a method to simulate a causal acceleration time history. First we extract an average trend from the root square of the real part. Dividing the real part by this average trend we obtain a new process, which is named as the standardized real part. After proving that this process have had a stationary characteristic we extract the probabilistic characteristics from this standardized real part process. We then develop a method to simulate a sample process of the standardized real part by using the concept of the autoregressive process. Multiplying the average trend to this sample standardized real part process we can obtain a sample real part process. Appling the concept of the Hilbert transform to this sample real part process we can obtain the imaginary part.