具有过阻尼特性的非比例阻尼线性系统的复振型分解法
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摘要
对于具有过阻尼特性的非比例阻尼线性系统,如何应用振型分析方法计算地震反应的问题至今尚无很充分的研究。本文从基于复模态理论的振型分解法一般表达式出发,给出了具有过阻尼特性的非比例阻尼结构动力反应的时程叠加法公式,同时还对具有成对复特征值和实特征值的系统提出了基于反应谱的复完全平方组合(CCQC)法和复平方和开方(CSRSS)法的计算公式。通过对一个典型实例的分析,验证了公式的正确性和有效性。此外,本文阐明了过阻尼体系的基本特性、数值计算方法和基于地震反应谱的峰值反应计算方法,结果表明从具有复特征值的一般线性动力系统发展而来的复振型叠加计算方法对于具有过阻尼特性的系统原则上也是适用的。文中的算例还表明,当非经典阻尼线弹性结构的部分振型出现过阻尼情况时,目前广泛应用于抗震设计规范的强迫解耦法可能导致很大的计算误差。
For non-classically damped linear system with over-critically damping peculiarity,how to solve the seismic response by using complex mode superposition technology is still a problem,and none of the sufficient research is available.Starting from general solution based on the complex mode superposition algorithm,a mode superposition formula in time domain for seismic response of non-classically damped linear system is deduced in this paper.Furthermore,the complex complete quadratic combination(CCQC) formula,and the complex square root of sum of squares(CSRSS) method for calculating maximum seismic responses on the base of design spectra are also provided.The correctness and effectiveness of these formulas are verified by example.The basic characteristics of the non-classically damped linear system with over-critical damping ratios,the numerical integration method of seismic response and the response spectra-based method for calculating peak response are described in detail in this paper.The results showed that the complex mode superposition algorithm derived from general linear dynamic system with complex eigenvalues and eigenvectors is applicable to the system with over-critical damping peculiarity in principle.The numerical example also showed that the forced uncoupling approach,which is popularly used in seismic design code,is likely to result in fairly large errors when over-critical damping ratios present partly in the mode shapes of non-classically damped linear systems.
For non-classically damped linear system with over-critically damping peculiarity,how to solve the seismic response by using complex mode superposition technology is still a problem,and none of the sufficient research is available.Starting from general solution based on the complex mode superposition algorithm,a mode superposition formula in time domain for seismic response of non-classically damped linear system is deduced in this paper.Furthermore,the complex complete quadratic combination(CCQC) formula,and the complex square root of sum of squares(CSRSS) method for calculating maximum seismic responses on the base of design spectra are also provided.The correctness and effectiveness of these formulas are verified by example.The basic characteristics of the non-classically damped linear system with over-critical damping ratios,the numerical integration method of seismic response and the response spectra-based method for calculating peak response are described in detail in this paper.The results showed that the complex mode superposition algorithm derived from general linear dynamic system with complex eigenvalues and eigenvectors is applicable to the system with over-critical damping peculiarity in principle.The numerical example also showed that the forced uncoupling approach,which is popularly used in seismic design code,is likely to result in fairly large errors when over-critical damping ratios present partly in the mode shapes of non-classically damped linear systems.
引文
[1]Foss F K.Co-ordinates which uncouple the linear dynamicsystems[J].Journal of Applied Mechanics,ASME,1958,(24):361-364.
[2]Igusa T,Kiureghian A D,Sackman J L.Modal decompositionmethod for stationary response of non-classically dampedsystems[J].Earthquake Engineering Structure Dynamics,1984,12(1):121-136.
[3]Skinner R I,Robinson W H,McVerry G H.An introduction toseismic isolation[M].John Wiley&Sons Ltd.,1993.
[4]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:地震出版社,1992.
[5]周锡元,董娣,苏幼坡.非正交阻尼线性振动系统的复振型地震响应叠加分析方法[J].土木工程学报,2003,36(5):30-36.
[6]Greco A,Santin A.Comparative study on dynamic analyses ofnon-classically damped linear systems[J].StructuralEngineering and Mechanics,2002,14(6):679-698.
[7]朱位秋.随机振动[M].北京:科学出版社,1992.
[8]Zhou X Y,Yu R F,Dong D.Complex mode superpositionalgorithm for seismic responses of non-classically damped linearMDOF system[J].Journal of Earthquake Engineering,2004,8(4):597-641.
[9]Clough R W,Penzien J.Dynamics of structures[M].Secondedition,McGraw-Hill Inc.,1993.
[10]周锡元,马东辉,俞瑞芳.工程结构中的阻尼与复振型地震响应的完全平方组合[J].土木工程学报,2005,38(1):31-39.
[11]Warburton G B.The dynamic behavior of structures[M].AppendixⅠ,Pergamon Press Ltd.,1976:307-308.
[12]Caughey A K.Classical normal modes in damped lineardynamic systems[J].Journal of Applied Mechanics,ASME,1960,27(6):269-271.
[13]俞瑞芳,周锡元.非比例阻尼弹性结构地震反应强迫解耦方法的理论背景和数值检验[J].工业建筑,2005,35(2):52-56.
[14]桂国庆.非比例阻尼结构体系近似解耦分析中的误差研究[J].工程力学,1994,11(4):40-45.
[15]GB 50011—2001建筑抗震设计规范[S].
[16]Hanson R D,Soong T T.Seismic design with supplementalenergy dissipation devices[R].Earthquake EngineeringResearch Institute,Federal Emergency management Agency,MNO-8,2001.
[2]Igusa T,Kiureghian A D,Sackman J L.Modal decompositionmethod for stationary response of non-classically dampedsystems[J].Earthquake Engineering Structure Dynamics,1984,12(1):121-136.
[3]Skinner R I,Robinson W H,McVerry G H.An introduction toseismic isolation[M].John Wiley&Sons Ltd.,1993.
[4]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:地震出版社,1992.
[5]周锡元,董娣,苏幼坡.非正交阻尼线性振动系统的复振型地震响应叠加分析方法[J].土木工程学报,2003,36(5):30-36.
[6]Greco A,Santin A.Comparative study on dynamic analyses ofnon-classically damped linear systems[J].StructuralEngineering and Mechanics,2002,14(6):679-698.
[7]朱位秋.随机振动[M].北京:科学出版社,1992.
[8]Zhou X Y,Yu R F,Dong D.Complex mode superpositionalgorithm for seismic responses of non-classically damped linearMDOF system[J].Journal of Earthquake Engineering,2004,8(4):597-641.
[9]Clough R W,Penzien J.Dynamics of structures[M].Secondedition,McGraw-Hill Inc.,1993.
[10]周锡元,马东辉,俞瑞芳.工程结构中的阻尼与复振型地震响应的完全平方组合[J].土木工程学报,2005,38(1):31-39.
[11]Warburton G B.The dynamic behavior of structures[M].AppendixⅠ,Pergamon Press Ltd.,1976:307-308.
[12]Caughey A K.Classical normal modes in damped lineardynamic systems[J].Journal of Applied Mechanics,ASME,1960,27(6):269-271.
[13]俞瑞芳,周锡元.非比例阻尼弹性结构地震反应强迫解耦方法的理论背景和数值检验[J].工业建筑,2005,35(2):52-56.
[14]桂国庆.非比例阻尼结构体系近似解耦分析中的误差研究[J].工程力学,1994,11(4):40-45.
[15]GB 50011—2001建筑抗震设计规范[S].
[16]Hanson R D,Soong T T.Seismic design with supplementalenergy dissipation devices[R].Earthquake EngineeringResearch Institute,Federal Emergency management Agency,MNO-8,2001.
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