基于余弦变换的地震动反应谱计算方法
|
摘要
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/s2 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/s2,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.
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/s2 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/s2,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.
引文
[1]Biot M A.Theory of vibration of buildings during earthquakes[J].Zeitschrift für Angewandte Mathematik und Mechanik,1934,14(4):213―223.
[2]Biot M A.A mechanical analyzer for the prediction of earthquake stresses[J].Bull Seismol Soc Amer,1941,31:151―171.
[3]Biot M A.Analytical and experimental methods in enginering seismology[J].ASCE Transactions,1942,108:365―408.
[4]Housner G W.Characteristic of strong motion earthquakes[J].Bull Seismol Soc Amer,1947,37(1):19―31.
[5]Housner G W.The behavior of inverted pendulum structures during earthquakes[J].Bull Seismol Soc Amer,1963,53(2):403―418.
[6]Housner G W,Jennings P C.Generation of artificial earthquake[J].ASCE,1964,90(1):113―150.
[7]谢礼立,于双久.强震观测与分析原理[M].北京:地震出版社,1982.Xie Lili,Yu Shuangjiu.The theory of strong earthquake motion measurement and analysis[M].Beijing:Seismological Press,1982.(in Chinese)
[8]大崎顺彦.地震动的谱分析入门[M].吕敏申,谢礼立,译.北京:地震出版社,1980.Ohsaki Y.The approach of spectra analysis of ground motion[M].Translated by Lu Minshen,Xie Lili.Beijing:Seismological Press,1980.(in Chinese)
[9]Nigam N C,Jennings P C.Calculation of response spectra from strong-motion earthquake records[J].Bull Seismol Soc Amer,1969,59(2):909―922.
[10]朱敏,朱镜清.反应谱计算的三角插值解析公式法[J].世界地震工程,2001,17(3):62―64.Zhu Min,Zhu Jingqing.Calculation of responses spectra based on trigonometric interpolation[J].World Information on Earthquake Engineering,2001,17(3):62―64.(in Chinese)
[11]陈国兴,庄海洋.基于抛物线内插的反应谱计算公式及其精度分析[J].防灾减灾工程学报,2003,23(3):56―61.Chen Guoxing,Zhuang Haiyang.Computation formulas of the earthquake response spectra based on parabolic interpolation method and the analysis of its precision[J].Journal of Disaster Prevention and Mitigation Engineering,2003,23(3):56―61.(in Chinese)
[12]Kanamori H,Maechling P,Hauksson E.Continuous monitoring of strong motion parameters[J].Bull Seismol Soc Amer,1999,89(1):311―316.
[13]Lee V W.Efficient algorithm for computing displacement velocity and acceleration responses of an oscillator to arbitrary ground motion[J].Soil Dynamics and Earthquake Engineering,1990,9(6):288―300.
[14]Newmark N M.A method of computation for structural dynamics[J].ASCE,1959,85(3):67―94.
[15]Ahmed N,Natarjan T,Rao K R,Discrete cosine transform[J].IEEE Trans Compute,1974,23(1):90―93.
[16]Andreas S S,Stefan B J,Samuel D S.Transform methods for seismic data compression[J].IEEE Transaction on Geoscience and Remote Sensing,1991,29(3):407―415.
[17]Artyom M Grigoryan.An algorithm for calculation of the discrete cosine transform by paired transform[J].IEEE Transactions on Signal Processing,2005,53(1):265―273.
[18]Wang Zhongde.A fast algorithm for the discrete sine transform implemented by the fast cosine transform[J].IEEE Transactions on Signal Processing,1982,30(5):814―815.
[19]Cvetkovic Z,Popvic M V.New fast algorithms for the computation discrete cosine and sine transform[J].IEEE Transactions on Signal Processing,1992,40(8):2083―2086.
[20]Rao K R,Yip P.Discrete cosine transform:Algorithms,advantages and applications.New York:Academic Press,1990.
[21]Dinstein I,Rose K,Heiman A.Variable block-size transform image coder[J].IEEE Transcomm,1990,38(11):2073―2078.
[22]黄文胜,赵焕东.余弦变换三维物体轮廓术的改进[J].光学仪器,2000,22(6):8―13.Huang Wensheng,Zhao Huandong.Improved cosine transform profilometry for the automatic measurement of 3-D objcet shapes[J].Optical Instruments,2000,22(6):8―13.(in Chinese)
[23]《数学手册》编写组.数学手册[M].北京:人民教育出版社,1979.《Mathematics Handbook》compile group.Mathematics handbook[M].Beijing:People’s Education Press,1979.(in Chinese)
[24]Jain A K.A fast Karhunen-Loeve transform for a class of stochastic processes[J].IEEE Transcomm,1976,24(6):1023―1029.
[2]Biot M A.A mechanical analyzer for the prediction of earthquake stresses[J].Bull Seismol Soc Amer,1941,31:151―171.
[3]Biot M A.Analytical and experimental methods in enginering seismology[J].ASCE Transactions,1942,108:365―408.
[4]Housner G W.Characteristic of strong motion earthquakes[J].Bull Seismol Soc Amer,1947,37(1):19―31.
[5]Housner G W.The behavior of inverted pendulum structures during earthquakes[J].Bull Seismol Soc Amer,1963,53(2):403―418.
[6]Housner G W,Jennings P C.Generation of artificial earthquake[J].ASCE,1964,90(1):113―150.
[7]谢礼立,于双久.强震观测与分析原理[M].北京:地震出版社,1982.Xie Lili,Yu Shuangjiu.The theory of strong earthquake motion measurement and analysis[M].Beijing:Seismological Press,1982.(in Chinese)
[8]大崎顺彦.地震动的谱分析入门[M].吕敏申,谢礼立,译.北京:地震出版社,1980.Ohsaki Y.The approach of spectra analysis of ground motion[M].Translated by Lu Minshen,Xie Lili.Beijing:Seismological Press,1980.(in Chinese)
[9]Nigam N C,Jennings P C.Calculation of response spectra from strong-motion earthquake records[J].Bull Seismol Soc Amer,1969,59(2):909―922.
[10]朱敏,朱镜清.反应谱计算的三角插值解析公式法[J].世界地震工程,2001,17(3):62―64.Zhu Min,Zhu Jingqing.Calculation of responses spectra based on trigonometric interpolation[J].World Information on Earthquake Engineering,2001,17(3):62―64.(in Chinese)
[11]陈国兴,庄海洋.基于抛物线内插的反应谱计算公式及其精度分析[J].防灾减灾工程学报,2003,23(3):56―61.Chen Guoxing,Zhuang Haiyang.Computation formulas of the earthquake response spectra based on parabolic interpolation method and the analysis of its precision[J].Journal of Disaster Prevention and Mitigation Engineering,2003,23(3):56―61.(in Chinese)
[12]Kanamori H,Maechling P,Hauksson E.Continuous monitoring of strong motion parameters[J].Bull Seismol Soc Amer,1999,89(1):311―316.
[13]Lee V W.Efficient algorithm for computing displacement velocity and acceleration responses of an oscillator to arbitrary ground motion[J].Soil Dynamics and Earthquake Engineering,1990,9(6):288―300.
[14]Newmark N M.A method of computation for structural dynamics[J].ASCE,1959,85(3):67―94.
[15]Ahmed N,Natarjan T,Rao K R,Discrete cosine transform[J].IEEE Trans Compute,1974,23(1):90―93.
[16]Andreas S S,Stefan B J,Samuel D S.Transform methods for seismic data compression[J].IEEE Transaction on Geoscience and Remote Sensing,1991,29(3):407―415.
[17]Artyom M Grigoryan.An algorithm for calculation of the discrete cosine transform by paired transform[J].IEEE Transactions on Signal Processing,2005,53(1):265―273.
[18]Wang Zhongde.A fast algorithm for the discrete sine transform implemented by the fast cosine transform[J].IEEE Transactions on Signal Processing,1982,30(5):814―815.
[19]Cvetkovic Z,Popvic M V.New fast algorithms for the computation discrete cosine and sine transform[J].IEEE Transactions on Signal Processing,1992,40(8):2083―2086.
[20]Rao K R,Yip P.Discrete cosine transform:Algorithms,advantages and applications.New York:Academic Press,1990.
[21]Dinstein I,Rose K,Heiman A.Variable block-size transform image coder[J].IEEE Transcomm,1990,38(11):2073―2078.
[22]黄文胜,赵焕东.余弦变换三维物体轮廓术的改进[J].光学仪器,2000,22(6):8―13.Huang Wensheng,Zhao Huandong.Improved cosine transform profilometry for the automatic measurement of 3-D objcet shapes[J].Optical Instruments,2000,22(6):8―13.(in Chinese)
[23]《数学手册》编写组.数学手册[M].北京:人民教育出版社,1979.《Mathematics Handbook》compile group.Mathematics handbook[M].Beijing:People’s Education Press,1979.(in Chinese)
[24]Jain A K.A fast Karhunen-Loeve transform for a class of stochastic processes[J].IEEE Transcomm,1976,24(6):1023―1029.
版权所有：© 2023 中国地质图书馆 中国地质调查局地学文献中心