结构地震反应分析的Newmark精细耦合级数显式方法
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摘要
将Newmark-β法中常平均加速度法的基本假定与精细积分法耦合,对积分项的计算引入指数矩阵的Taylor级数展开式,提出了动力方程的显式耦合级数解,设计了相应的时程积分算法,并应用在结构的地震反应分析中。分析表明,由于该方法是显式的,在质量矩阵为对角矩阵时,不需要计算耦联的方程组,因此可以有效地减少内存占用和机时耗费。该方法的稳定性条件显然满足,其精度可根据Taylor级数展开式的截断阶数进行灵活控制。算例表明该方法对地震作用的有效性和适应性。
An explicit series solution of the structural dynamic equation is put forward by introducing Taylor series expansion.The algorithm is based on the coupled precise time integration method and basic assumptions of constant average acceleration method in Newmark family.And the method is used to analyze the structural seismic response.With the explicit property,the method requires no computation of the coupled equation and can effectively reduce memory occupancy and computer cost when the quality matrix is diagonal.The stable conditions of the proposed method are obviously to be satisfied and its accuracy can be controlled by choosing the number of Taylor series.The example shows the efficiency of the method on earthquake action.
An explicit series solution of the structural dynamic equation is put forward by introducing Taylor series expansion.The algorithm is based on the coupled precise time integration method and basic assumptions of constant average acceleration method in Newmark family.And the method is used to analyze the structural seismic response.With the explicit property,the method requires no computation of the coupled equation and can effectively reduce memory occupancy and computer cost when the quality matrix is diagonal.The stable conditions of the proposed method are obviously to be satisfied and its accuracy can be controlled by choosing the number of Taylor series.The example shows the efficiency of the method on earthquake action.
引文
[1]Newmark N M.A method of computation for structural dynamics[J].Journal of Engineering Mechanics Division,1959,85(3):67―94.
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[5]Fung T C.Complex-time-step Newmark methods with controllable numerical dissipation[J].International Journal for Numerical Methods in Engineering,1998,41(1):65―93.
[6]Fung T C.Weighting parameters for unconditionally stable higher-order accurate time step integration algorithms-Part2[J].International Journal for Numerical Methods in Engineering,1999,45(8):971―1006.
[7]钟万勰.结构动力方程的精细时程积分法[J].大连理工大学学报,1994,34(2):131―136.Zhong Wanxie.On precise time-integration method for structural dynamics[J].Journal of Dalian University of Technology,1994,34(2):131―136.(in Chinese)
[8]汪梦甫,周锡元.结构动力方程的更新精细积分方法[J].力学学报,2004,36(2):191―195.Wang Mengfu,Zhou Xiyuan.Renewal precise time step integration method of structural dynamic analysis[J].Acta Mechanica Sinica,2004,36(2):191―195.(in Chinese)
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[10]Guo Zeying,Li Qingning,Zhang Shoujun.Direct Newmark-precision integration method of seismic analysis for short-leg shear wall structure[C].Proceedings of the Ninth International Symposium on Structural Engineering for Young Experts.Beijing,China,Science Press,2006(1):180―184.
[11]郭泽英,李青宁,张守军.结构地震反应分析的一种新精细积分法[J].工程力学,2007,24(4):35―40.Guo Zeying,Li Qingning,Zhang Shoujun.A new precise integration method for structural seismic response analysis[J].Engineering Mechanics,2007,24(4):35―40.(in Chinese)
[12]李金桥,于建华.非线性动力系统精细积分下的显式级数解[J].四川大学学报(工程科学版),2002,34(2):24―27.Li Jinqiao,Yu Jianhua.Explicit series solution under precise integration for nonlinear dynamics system[J].Journal of Sichuan University(Engineering Science Edition),2002,34(2):24―27.(in Chinese)
[13]孙焕纯,曲乃泗,林家浩.高等计算结构动力学[M].大连:大连理工大学出版社,1992.Sun Huanchun,Qu Naisi,Lin Jiahao.Higher computational structural dynamics[M].Dalian:Dalian University of Technology Press,1992.(in Chinese)
[14]汪梦甫,区达光.精细积分方法的评估与改进[J].计算力学学报,2004,21(6):728―733.Wang Mengfu,Qu Daguang.Assessment and improvement of precise time step integration method[J].Chinese Journal of Computational Mechanics,2004,21(6):728―733.(in Chinese)
[15]Wang M F,Au F T K.Higher-order schemes for time integration dynamic structural analysis[J].Journal of Sound and Vibration,2004,277(3):122―128.
[2]Golley B W.A time-stepping procedure for structural dynamics using Gauss point collocation[J].International Journal for Numerical Methods in Engineering,1996,39(23):3985―3998.
[3]Kajawski J,Gallagher R H.A generalized least-squares family of algorithms for transient dynamic analysis[J].Earthquake Engineering and Structural Dynamics,1989,18(4):539―550.
[4]Tarnow N,Simo J C.How to render second order accurate time stepping algorithms fourth order accurate while retaining the stability and conservation properties[J].Computer Methods in Applied Mechanics and Engineering,1994,115:233―252.
[5]Fung T C.Complex-time-step Newmark methods with controllable numerical dissipation[J].International Journal for Numerical Methods in Engineering,1998,41(1):65―93.
[6]Fung T C.Weighting parameters for unconditionally stable higher-order accurate time step integration algorithms-Part2[J].International Journal for Numerical Methods in Engineering,1999,45(8):971―1006.
[7]钟万勰.结构动力方程的精细时程积分法[J].大连理工大学学报,1994,34(2):131―136.Zhong Wanxie.On precise time-integration method for structural dynamics[J].Journal of Dalian University of Technology,1994,34(2):131―136.(in Chinese)
[8]汪梦甫,周锡元.结构动力方程的更新精细积分方法[J].力学学报,2004,36(2):191―195.Wang Mengfu,Zhou Xiyuan.Renewal precise time step integration method of structural dynamic analysis[J].Acta Mechanica Sinica,2004,36(2):191―195.(in Chinese)
[9]汪梦甫,周锡元.结构动力方程的高斯精细时程积分方法[J].工程力学,2004,21(4):13―16.Wang Mengfu,Zhou Xiyuan.Gauss precise time-integration of structural dynamic analysis[J].Engineering Mechanics,2004,21(4):13―16.(in Chinese)
[10]Guo Zeying,Li Qingning,Zhang Shoujun.Direct Newmark-precision integration method of seismic analysis for short-leg shear wall structure[C].Proceedings of the Ninth International Symposium on Structural Engineering for Young Experts.Beijing,China,Science Press,2006(1):180―184.
[11]郭泽英,李青宁,张守军.结构地震反应分析的一种新精细积分法[J].工程力学,2007,24(4):35―40.Guo Zeying,Li Qingning,Zhang Shoujun.A new precise integration method for structural seismic response analysis[J].Engineering Mechanics,2007,24(4):35―40.(in Chinese)
[12]李金桥,于建华.非线性动力系统精细积分下的显式级数解[J].四川大学学报(工程科学版),2002,34(2):24―27.Li Jinqiao,Yu Jianhua.Explicit series solution under precise integration for nonlinear dynamics system[J].Journal of Sichuan University(Engineering Science Edition),2002,34(2):24―27.(in Chinese)
[13]孙焕纯,曲乃泗,林家浩.高等计算结构动力学[M].大连:大连理工大学出版社,1992.Sun Huanchun,Qu Naisi,Lin Jiahao.Higher computational structural dynamics[M].Dalian:Dalian University of Technology Press,1992.(in Chinese)
[14]汪梦甫,区达光.精细积分方法的评估与改进[J].计算力学学报,2004,21(6):728―733.Wang Mengfu,Qu Daguang.Assessment and improvement of precise time step integration method[J].Chinese Journal of Computational Mechanics,2004,21(6):728―733.(in Chinese)
[15]Wang M F,Au F T K.Higher-order schemes for time integration dynamic structural analysis[J].Journal of Sound and Vibration,2004,277(3):122―128.
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