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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。
In order to improve the calculation precision of response spectrum of ground motion, a new method of calculating the seismic response by cosine transform is proposed. Convolution and differential theorems based on cosine transform are given and proved. Using them, the dynamic differential equation of single degree of freedom system is taken as a starting point to derive the cosine transform spectrum formula of seismic response, and then to obtain the generalized response spectrum by inverse cosine transform expression. In order to discuss the calculation precision of this method, the response spectra under the condition of simple harmonic input are calculated respectively by the exact method and the cosine transform method, and the errors are compared and analyzed. The results show that the mean square deviations of displacement, velocity, acceleration response spectrum and theoretical response spectrum calculated by the precision method are 0.0012m, 0.003m / s and 0.16m / s2, respectively, when the local vibration load is input as cos4πt, while the cosine transform The fitting results of response spectrum and theoretical response spectrum are very good, the mean square deviations are 0.0002m, 0.002m / s and 0.36m / s2 respectively. The displacement and velocity response spectrum accuracy calculated by the cosine transform method are respectively increased by nearly 6 times and 1.5 Although the mean square error of the acceleration response spectrum is slightly larger than that of the exact method, the calculation result of the long period has a significant advantage over the precision of the exact method. However, in the calculation of the response spectrum of EI Centro seismic wave, the shape of the three types of response spectra obtained by the two methods are similar, further confirming the accuracy and reliability of the calculation result of the cosine transform method.