Non-Stationary Random Earthquake Response Of Linear System

The evaluation has been made for normally used four envelope functions of earthquake ground motion. Being as evaluation standard of the envelope function this paper pointed out that the controlling capability of the parameters involved in functions of the peak position of the envelope with single peak and the shape of the envelope are essential besides simplicity of the formula of envelope function. With the aid of the method for determining damping ratio in structural dynamic experiment the so called stationarity index is used to estimate the gentleness of the peak's slope. By means of this index this paper analyzes the gentleness of the peak's slope for the four normally used envelope functions and some of them have been modified. It is also pointed out that if Kanai power spectrum and Penzien power spectrum is regarded envelope function in time domain, they are powerful and easy in viewpoint of envelope shape controlling ability. Penzien power spectrum is even able to describe the envelope with two peaks like camel. The results of comparative analyses in this paper can be used to select and simulate the envelope of earthquake ground motion. By making use of the time dependent covariance and cross covariance formula which used to calculate the non-stationary response under earthquake random excitation and deduced by Xiyuan Zhou and Hu in 1965, the covariance and cross covariance formula of the four envelope models was deducted and the result of four models compared. Besides, the covariance of a single exponent function deduced according to residue theorem, which provides the correctness of the formula of Zhou and Hu. Finally, the CQC(t) algorithm which used to calculate the non-stationary under earthquake random excitation was derived, the exemplary analysis show that the improvement of accuracy of CQC(t) algorithm is limited for the normal structures which have low period. The meaning of CQC(t) algorithm is the difference in time domain between the peak value location in time axis of input earthquake and model response of structure, which is called delay time. Through the delay time calculating of a series of oscillators with different damp ratio and natural vibration frequency, it is found that it will bring biggish error if the CQC algorithm applied on the structure which has long period, and the time dependent CQC(t) algorithm could solve the problem easily.