大型非比例阻尼线性系统的地震反应复振型分析方法
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摘要
对于大型非比例阻尼线性系统,当采用基于复振型的振型叠加方法进行动力反应分析时,按照通用算法(例如雅克比法)求解结构全部振型向量的计算工作量很大,甚至是不现实的。本文将经典阻尼系统中行之有效的Lanczos法和子空间迭代法加以推广和改进,给出了一组可对原来的方程进行自由度缩减的实向量基,然后与Foss变换相结合,得到一组实用的复向量基,使之适用于求解复杂非比例阻尼线性系统的任意低阶复振型和相应的复特征值,适用于任意扰力作用下的动力反应分析。理论推导和实例计算表明,本文所给出的复向量基概念清晰,计算效率高,能够适应对具有非比例阻尼特性的大型复杂结构进行动力分析的实际需要,其中包括地震作用下的时程分析和反应谱振型叠加分析。
For large linear non-classically damped system,when mode superposition method based on complex vector is used to analyze seismic responses,the amount of calculating total mode vectors using Jacobi method is fairly large and unpractical indeed.In this paper,Lanczos algorithm and subspace iteration method,which are effective in computing modes of classical damped linear system,are extended and modified to find a set of real vector basis to reduce freedom of primary equation.Combined with Foss transform,a set of applied complex vector basis is derived to analyze the first order complex eigenpairs for dynamic analysis of large non-classically damped linear system under random excitation.From the theoretical analysis and numerical examination,it could be pointed out that the complex vector basis derived in this paper not only has clear physical significance and good computational efficiency,but also is adapted to analyze dynamic responses of large non-classically damped linear system,including seismic responses in time domain and mode superposition based on seismic design spectra in frequency domain.
For large linear non-classically damped system,when mode superposition method based on complex vector is used to analyze seismic responses,the amount of calculating total mode vectors using Jacobi method is fairly large and unpractical indeed.In this paper,Lanczos algorithm and subspace iteration method,which are effective in computing modes of classical damped linear system,are extended and modified to find a set of real vector basis to reduce freedom of primary equation.Combined with Foss transform,a set of applied complex vector basis is derived to analyze the first order complex eigenpairs for dynamic analysis of large non-classically damped linear system under random excitation.From the theoretical analysis and numerical examination,it could be pointed out that the complex vector basis derived in this paper not only has clear physical significance and good computational efficiency,but also is adapted to analyze dynamic responses of large non-classically damped linear system,including seismic responses in time domain and mode superposition based on seismic design spectra in frequency domain.
引文
[1]CAUGHEY T K.Classical normal modes in dampedlinear dynamic systems[J].J Appl Mech,1960,27(2):269-271.
[2]CLOUGH R W,PENZIEN J.Dynamics of Struc-tures[M].Second Edition(McGraw-Hill,Inc.),1993.
[3]HANSTEEN O E,BELL K.On the accuracy of modesuperposition analysis in structural dynamics[J].Earthq Eng Struct Dyn,1979,7:405-411.
[4]CORNWELL R E,CRAIG R R,JOHNSTON C P.On the appliacation of the mode acceleration methodto structural dynamics problem[J].Earthq EngStruct Dyn,1983,11:679-688.
[5]LANCZOS C.An iteration method for the solution ofthe eiegenvalue problem of linear differential and inte-gral operators[J].J Res Natl Bur Standards,1950,45:255-282.
[6]NOUR-HUMID B,CLOUGH R W.Dynamic analysisof structures using Lanczos co-ordinates[J].EarthqEng Struct Dyn,1984,12:365-377.
[7]吴鸿庆,任侠.结构有限元分析[M].北京:中国铁道出版社,2000.(WU H Q,REN X.Structural FiniteElement Analysis[M].Beijing:China Railway Press,2000.(in Chinese))
[8]WILSON E L,YUAN M,DICKENS J M.Dynamic a-nalysis by direct superposition of Ritz vectors[J].Earthq Eng Struct Dyn,1982,10:813-821.
[9]IBRAHIMBEGOVIC A,CHEN H C,WILSON E L,et al.Ritz method for dynamic analysis of large dis-crete linear systems with non-proportional damping[J].Earthq Eng Struct Dyn,1990,19:877-889.
[10]FOSS F K.Co-ordinates which uncouple the lineardynamic systems[J].J of Applied Mechanics,ASME,1958,24:361-364.
[11]周锡元.一般有阻尼线性体系地震反应的振型分解方法[M].北京:中国地震工程研究进展,地震出版社,1992.(ZHOU Xi-yuan.The Modal SuperpositionMethod of Earthquake Response of General DampedLinear System[M].Published in Research Progressof Earthquake Engineering in China,SeismologicalPress,1992.(in Chinese))
[12]ZHOU X Y,YU R F,D ONG D.Complex mode su-perposition algorithm for seismic responses of non-classically damped linear system[J].Journal ofEarthquake Engineering,2004,8(4):597-641.
[13]俞瑞芳,周锡元.非比例阻尼线性系统考虑过阻尼影响的复振型分解法[J].建筑结构学报,2006,27(1):50-59.(YU Rui-fang,ZHOU Xi-yuan.Complex modesuperposition method for non-classically damped sys-tem with over-critical damping peculiarity[J].J ofBuilding Structures,2006,27(1):50-59.(in Chi-nese))
[14]周锡元,俞瑞芳.非比例阻尼线性系统基于规范反应谱的CCQC法[J].工程力学,2006,23(2):10-17.(ZHOU Xi-yuan,YU Rui-fang.CCQC Method forseismic response of non-classically damped linear sys-tem based on code response spectra[J].EngineeringMechanics,2006,23(2):10-17.(in Chinese))
[15]张贤达.矩阵分析与应用[M].北京:清华大学出版社,2004.(ZHANG Xian-da.Matrix Analysis andApplication[M].Beijing:Tinghua University Press,2004.(in Chinese))
[16]LOPEZ O A,TORRES R.The critical angle of seis-mic incidence and the maximum structural response[J].Earth Eng and Struc Dyn,1997,26:881-894.
[17]GB50011-200,建筑抗震设计规[S].北京:建筑工业出版社,2001.(GB50011-2001,Seismic design codefor buildings[S].Beijing:China Architecture andBuilding Press,2001.(in Chinese))
[2]CLOUGH R W,PENZIEN J.Dynamics of Struc-tures[M].Second Edition(McGraw-Hill,Inc.),1993.
[3]HANSTEEN O E,BELL K.On the accuracy of modesuperposition analysis in structural dynamics[J].Earthq Eng Struct Dyn,1979,7:405-411.
[4]CORNWELL R E,CRAIG R R,JOHNSTON C P.On the appliacation of the mode acceleration methodto structural dynamics problem[J].Earthq EngStruct Dyn,1983,11:679-688.
[5]LANCZOS C.An iteration method for the solution ofthe eiegenvalue problem of linear differential and inte-gral operators[J].J Res Natl Bur Standards,1950,45:255-282.
[6]NOUR-HUMID B,CLOUGH R W.Dynamic analysisof structures using Lanczos co-ordinates[J].EarthqEng Struct Dyn,1984,12:365-377.
[7]吴鸿庆,任侠.结构有限元分析[M].北京:中国铁道出版社,2000.(WU H Q,REN X.Structural FiniteElement Analysis[M].Beijing:China Railway Press,2000.(in Chinese))
[8]WILSON E L,YUAN M,DICKENS J M.Dynamic a-nalysis by direct superposition of Ritz vectors[J].Earthq Eng Struct Dyn,1982,10:813-821.
[9]IBRAHIMBEGOVIC A,CHEN H C,WILSON E L,et al.Ritz method for dynamic analysis of large dis-crete linear systems with non-proportional damping[J].Earthq Eng Struct Dyn,1990,19:877-889.
[10]FOSS F K.Co-ordinates which uncouple the lineardynamic systems[J].J of Applied Mechanics,ASME,1958,24:361-364.
[11]周锡元.一般有阻尼线性体系地震反应的振型分解方法[M].北京:中国地震工程研究进展,地震出版社,1992.(ZHOU Xi-yuan.The Modal SuperpositionMethod of Earthquake Response of General DampedLinear System[M].Published in Research Progressof Earthquake Engineering in China,SeismologicalPress,1992.(in Chinese))
[12]ZHOU X Y,YU R F,D ONG D.Complex mode su-perposition algorithm for seismic responses of non-classically damped linear system[J].Journal ofEarthquake Engineering,2004,8(4):597-641.
[13]俞瑞芳,周锡元.非比例阻尼线性系统考虑过阻尼影响的复振型分解法[J].建筑结构学报,2006,27(1):50-59.(YU Rui-fang,ZHOU Xi-yuan.Complex modesuperposition method for non-classically damped sys-tem with over-critical damping peculiarity[J].J ofBuilding Structures,2006,27(1):50-59.(in Chi-nese))
[14]周锡元,俞瑞芳.非比例阻尼线性系统基于规范反应谱的CCQC法[J].工程力学,2006,23(2):10-17.(ZHOU Xi-yuan,YU Rui-fang.CCQC Method forseismic response of non-classically damped linear sys-tem based on code response spectra[J].EngineeringMechanics,2006,23(2):10-17.(in Chinese))
[15]张贤达.矩阵分析与应用[M].北京:清华大学出版社,2004.(ZHANG Xian-da.Matrix Analysis andApplication[M].Beijing:Tinghua University Press,2004.(in Chinese))
[16]LOPEZ O A,TORRES R.The critical angle of seis-mic incidence and the maximum structural response[J].Earth Eng and Struc Dyn,1997,26:881-894.
[17]GB50011-200,建筑抗震设计规[S].北京:建筑工业出版社,2001.(GB50011-2001,Seismic design codefor buildings[S].Beijing:China Architecture andBuilding Press,2001.(in Chinese))
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