基于余弦变换的地震动反应谱计算方法

CALCULATION OF EARTHQUAKE RESPONSE SPECTRA BY COSINE TRANSFORM

针对提高地震动反应谱的计算精度,提出用余弦变换计算地震反应的新方法。给出并证明了基于余弦变换的褶积及微分定理,利用它们,以单自由度系统动力微分方程为出发点,推导出地震反应的余弦变换谱公式,进而通过逆余弦变换求取反应谱的一般表达式。为探讨该方法的计算精度,分别利用精确法和余弦变换法计算了简谐波输入情况下的反应谱,并进行了误差对比分析。研究结果表明:当地震动荷载输入为cos4πt时,精确法计算的位移、速度和加速度反应谱与理论反应谱的均方差分别为0.0012m、0.003m/s、0.16m/s2,而余弦变换法计算的反应谱与理论反应谱的拟合效果非常好,均方差分别为0.0002m、0.002m/s、0.36m/s2,余弦变换法计算的位移和速度反应谱精度分别提高了近6倍和1.5倍,虽然加速度反应谱均方差略大于精确法,但对于长周期部分的计算结果,较之精确法的计算精度来说具有显著的优势。而在对EI Centro地震波的反应谱计算结果中,两种方法所获得的三类反应谱曲线形态相近,进一步证实了余弦变换法计算结果的准确性和可靠性。

A new method of calculating earthquake response based on the cosine transform is proposed in order to improve the calculating accuracy of earthquake response spectra.Convolution and differential theorems of cosine transform are put forward and proved.By employing the transform as well as considering SDOF system＇s dynamic differential equations as a starting point,the formula of cosine-transform-spectrum of earthquake responses is deduced,and then a general expression of response spectrum is obtained.For approaching the calculating accuracy,the response spectra of harmonic conditions is calculated by using exact and cosine transforms respectively.Meanwhile the error analysis is done.It is concluded that the mean square deviation of displacement,velocity and accelerate response spectra are 0.0012m,0.003m/s and 0.16m/s^2 respectively calculated by exact,while the result of cosine transform fit very well with theory,and the mean square deviation are 0.0002m,0.002m/s and 0.36m/s^2,when the load input is cos4πt,which shows the accuracy of displacement and velocity increased nearly 6 and 1.5 fold,although the mean square deviation of acceleration response spectrum is a little greater than that of an exact one;for the long period,the method of cosine transorm is markedly superior to that of an exact one.And in the result of EI Centro,the shape of three type response spectra is similar,further confirmed the accuracy and reliability of the cosine transform method.