Rayleigh阻尼系数解法比较及对结构地震反应影响
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摘要
Rayleigh阻尼是一种广泛采用的正交阻尼模型。针对一个直径90m,高15m的穹顶结构,分析比较了四种方法(传统方法、最小二乘法、基于多参考振型的加权最小二乘法和基于结构位移峰值误差优化法)所得Rayleigh阻尼系数对结构地震反应计算精度的影响。四条地震波的分析结果表明:基于结构位移峰值误差的优化方法对于结构位移等以低阶模态控制的动力反应量计算精度最高;传统方法存在选择合适第二阶参考频率的难题;而最小二乘法不是计算Rayleigh阻尼系数的合理方法。当结构的显著贡献模态多且不同动力反应相关显著贡献模态的频率有巨大差异时,Rayleigh阻尼模型将无法构造兼顾低阶模态和高阶模态计算精度的阻尼矩阵,此时需要采用更多阶模态的阻尼比等于精确值的阻尼矩阵构造方法。
Rayleigh damping is a common orthogonal damping model. The effects of Rayleigh damping coefficients on structural seismic response accuracies are discussed for a dome structure with 90m-diameter 15m-height. The following four methods are compared: a traditional method(TM), a least squares method(LS), a weighted least squares method based on multi-reference modes(WLS) and an optimization method based on minimizing the error of structure peak displacements(OM). The numerical results under four seismic excitations indicate that the OM is significantly more accurate than the other methods when significant contribution modes of responses are the first several natural frequencies, such as displacements. In the TM, one should take great care to select the reasonable second reference frequency. It is unreasonable to calculate the Rayleigh damping coefficients for the LS. When there are many significant contribution modes and the difference in natural frequencies of significant contribution modes for different response are great, it is impossible for Rayleigh damping to construct a damping matrix which makes the dynamic responses of the lower and higher modes accuracy at the same time. In this case, it is necessary for complex structures to construct damping matrix that damping ratios in more than two modes are equal to the accurate values.
Rayleigh damping is a common orthogonal damping model. The effects of Rayleigh damping coefficients on structural seismic response accuracies are discussed for a dome structure with 90m-diameter 15m-height. The following four methods are compared: a traditional method(TM), a least squares method(LS), a weighted least squares method based on multi-reference modes(WLS) and an optimization method based on minimizing the error of structure peak displacements(OM). The numerical results under four seismic excitations indicate that the OM is significantly more accurate than the other methods when significant contribution modes of responses are the first several natural frequencies, such as displacements. In the TM, one should take great care to select the reasonable second reference frequency. It is unreasonable to calculate the Rayleigh damping coefficients for the LS. When there are many significant contribution modes and the difference in natural frequencies of significant contribution modes for different response are great, it is impossible for Rayleigh damping to construct a damping matrix which makes the dynamic responses of the lower and higher modes accuracy at the same time. In this case, it is necessary for complex structures to construct damping matrix that damping ratios in more than two modes are equal to the accurate values.
引文
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[16]刘红石.Rayleigh阻尼比例系数的确定[J].噪声与振动控制,1999(6):21―22.Liu Hongshi.Determination of Rayleigh damping scale coefficient[J].Noise and Vibration Control,1999(6):21―22.(in Chinese)
[17]刘红石.相对误差与Rayleigh阻尼比例系数的确定[J].湖南工程学院学报,2001,11(3/4):36―38.Liu Hongshi.Relative errors and determination of Rayleigh damping scale coefficient[J].Journal of Hunan Institute of Engineering,2001,11(3/4):36―38.(in Chinese)
[18]Yang D B,Zhang Y G,Wu J Z.Computation of Rayleigh damping coefficients in seismic time-history analysis of spatial structures[J].Journal of the International Association for Shell and Spatial Structures,2010,51(2):125―135.
[19]潘旦光.直接确定Rayleigh阻尼系数的一种优化方法[J].工程力学,2013,30(9):16―21.Pan Danguang.An optimization method for the direct determination of Rayleigh damping coefficients[J].Engineering Mechanics,2013,30(9):16―21.(in Chinese)
[20]潘旦光.地震反应分析中Rayleigh阻尼系数的优化解[J].工程力学,2013,30(11):15―20,27.Pan Danguang.An optimization solution for Rayleigh damping coefficients in seismic response analysis[J].Engineering Mechanics,2013,30(11):15―20,27.(in Chinese)
[21]GB 50011-2010,建筑抗震设计规范[S].北京:中国建筑工业出版社,2010.GB 50011-2010,Code for Seismic Design of Building[S].Beijing:China Architecture&Building Press,2010.(in Chinese)
[2]Chopra A K.Dynamics of structures:Theory and applications to earthquake engineering[M].New Jersey:Englewood Cliffs,Prentice-Hall,1995:416―420.
[3]Adhikari S.Damping modelling using generalized proportional damping[J].Journal of Sound and Vibration,2006,293(1/2):156―170.
[4]Enrique Luco J.A note on classical damping matrices[J].Earthquake Engineering and Structural Dynamics,2008,37(4):615―626.
[5]Liang Z,Lee G C.Representation of damping matrix[J].Journal of Engineering Mechanics,1991,117(5):1005―1019.
[6]张辉东,王元丰.复阻尼模型结构地震时程响应研究[J].工程力学,2010,27(1):109―115.Zhang Huidong,Wang Yuanfeng.Study on seismic time-history response of structures with complex damping[J].Engineering Mechanics,2010,27(1):109―115.(in Chinese)
[7]Clough R W,Penzien J.Dynamics of structure[M].2nd ed.New York:Mc Graw-Hill Inc,1993:234―245.
[8]董军,邓洪洲,王肇民.结构动力分析阻尼模型研究[J].世界地震工程,2000,16(4):63―69.Dong Jun,Deng Hongzhou,Wang Zhaomin.Studies on the damping models for structural dynamic time history analysis[J].World Information on Earthquake Engineering,2000,16(4):63―69.(in Chinese)
[9]Caughey T K,O’Kelly M E J.Classical normal modes in damped linear dynamic systems[J].Transactions of ASME,Journal of Applied Mechanics,1965,32(3):583―588.
[10]Pant D R,Wijeyewickrema A C,Elgawady M A.Appropriate viscous damping for nonlinear time-history analysis of base-isolated reinforced concrete buildings[J].Earthquake Engineering&Structural Dynamics,2013,42(15):2321―2339.
[11]任德斌,赵文婷.大跨钢拱薄壳穹顶结构地震响应分析[J].工程力学,2014,31(增):166―170Ren Debin,Zhao Wenting.Analysis on seismic response of long-span arch shell steel dome structure[J].Engineering Mechanics,2014,31(Suppl):166―170.(in Chinese)
[12]张喜,楼梦麟,林巧.多维地震下双塔楼结构框架梁行波效应分析[J].工程力学,2012,29(增刊II):133―138.Zhang Xi,Lou Menglin,Lin Qiao.Analysis of traveling wave effects on seismic responses of twin-tower structure under horizontal and vertical seismic excitations[J].Engineering Mechanics,2012,29(Suppl II):133―138.(in Chinese)
[13]周国良,李小军,刘必灯,等.大质量法在多点激励分析中的应用、误差分析与改进[J].工程力学,2011,28(1):48―54.Zhou Guoliang,Li Xiaojun,Liu Bideng,et al.Error analysis and improvements of large mass method used in multi-support seismic excitation analysis[J].Engineering Mechanics,2011,28(1):48―54.(in Chinese)
[14]李田.结构时程动力分析中的阻尼取值研究[J].土木工程学报,1997,30(3):68―73.Li Tian.A study on damping values applied to the time-history dynamic analysis of structures[J].China Civil Engineering Journal,1997,30(3):68―73.(in Chinese)
[15]沈飞,楼梦麟.超高层建筑地震反应分析中高阶振型影响分析[J].工程力学,2012,29(增刊I):23―28.Shen Fei,Lou Menglin.Influence of high modes of high-rise building on its seismic responses[J].Engineering Mechanics,2012,29(Suppl I):23―28.(in Chinese)
[16]刘红石.Rayleigh阻尼比例系数的确定[J].噪声与振动控制,1999(6):21―22.Liu Hongshi.Determination of Rayleigh damping scale coefficient[J].Noise and Vibration Control,1999(6):21―22.(in Chinese)
[17]刘红石.相对误差与Rayleigh阻尼比例系数的确定[J].湖南工程学院学报,2001,11(3/4):36―38.Liu Hongshi.Relative errors and determination of Rayleigh damping scale coefficient[J].Journal of Hunan Institute of Engineering,2001,11(3/4):36―38.(in Chinese)
[18]Yang D B,Zhang Y G,Wu J Z.Computation of Rayleigh damping coefficients in seismic time-history analysis of spatial structures[J].Journal of the International Association for Shell and Spatial Structures,2010,51(2):125―135.
[19]潘旦光.直接确定Rayleigh阻尼系数的一种优化方法[J].工程力学,2013,30(9):16―21.Pan Danguang.An optimization method for the direct determination of Rayleigh damping coefficients[J].Engineering Mechanics,2013,30(9):16―21.(in Chinese)
[20]潘旦光.地震反应分析中Rayleigh阻尼系数的优化解[J].工程力学,2013,30(11):15―20,27.Pan Danguang.An optimization solution for Rayleigh damping coefficients in seismic response analysis[J].Engineering Mechanics,2013,30(11):15―20,27.(in Chinese)
[21]GB 50011-2010,建筑抗震设计规范[S].北京:中国建筑工业出版社,2010.GB 50011-2010,Code for Seismic Design of Building[S].Beijing:China Architecture&Building Press,2010.(in Chinese)
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