非线性动力分析的广义精细积分法
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摘要
针对非线性动力状态方程=H·v+f(v,t),结合广义精细积分法和预估-校正法,提出了用于非线性动力分析的广义精细积分法。在任一时间子域内,对计算过程中待求的vk+j/m(j=1,2,…,m),利用当前时刻的vk进行预估。将离散的非线性项用拉格朗日插值多项式展开并视为外荷载,结合广义精细积分法即可求解非线性系统的动力响应。该方法计算格式统一,易于编程,与四种单步法、一次预-校法及预估校正-辛时间子域法进行数值比较,计算结果表明,该方法具有很高的精度、稳定性及较高的效率。可用于多自由度结构体系的非线性动力反应分析。
Aiming at the nonlinear dynamic state equation = H·v + f( v,t),the generalized precise time domain integration method for nonlinear dynamic analysis was proposed using the generalized precise integration method combined with the predictor-correction one. Firstly,in any time-subdomain,the variable vkat the current time moment was used to pre-estimate the variable to be solved vk + j/m( j = 1,2,…,m) in the process of computation. Then the discrete nonlinear terms were expanded with Lagrange interpolation polynomial and taken as external loads,and the generalized precise integration method was used to directly solve the dynamic response of a nonlinear system. The proposed method was compared with four single-step methods,primary predictor-correction one and the predictor-corrector-symplectic time subdomain one. The numerical example results showed that the proposed method is easy to program and has a uniform computing format,higher accuracy,stability and efficiency; it can be applied in nonlinear dynamic response analysis of multi-DOF structural systems.
Aiming at the nonlinear dynamic state equation = H·v + f( v,t),the generalized precise time domain integration method for nonlinear dynamic analysis was proposed using the generalized precise integration method combined with the predictor-correction one. Firstly,in any time-subdomain,the variable vkat the current time moment was used to pre-estimate the variable to be solved vk + j/m( j = 1,2,…,m) in the process of computation. Then the discrete nonlinear terms were expanded with Lagrange interpolation polynomial and taken as external loads,and the generalized precise integration method was used to directly solve the dynamic response of a nonlinear system. The proposed method was compared with four single-step methods,primary predictor-correction one and the predictor-corrector-symplectic time subdomain one. The numerical example results showed that the proposed method is easy to program and has a uniform computing format,higher accuracy,stability and efficiency; it can be applied in nonlinear dynamic response analysis of multi-DOF structural systems.
引文
[1]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.
[2]LIN Jiahao, SHEN Weiping,WILLIAMS F W. A high precision direct integration scheme for structures subjected to transient dynamic loading[J]. Computer&Structures,1995,6(1):120-130.
[3]赵秋玲.非线性动力学方程的精细积分法[J].力学与实践,1998,20(6):24-26.ZHAO Qiuling. An accurate integration method for solving nonlinear dynamic problems[J]. Mechanics in Engineering,1998,20(6):24-26.
[4]吕和祥,蔡志勤,裘春航.非线性动力问题的一个显式精细积分算法[J].应用力学学报,2001,18(2):34-40.LHexiang,CAI Zhiqin,QIU Chunhang. An explicit precise integration algorithm for nonlinear dynamics problems[J].Chinese Journal of Applied Mechanics, 2001, 18(2):34-40.
[5]张洵安,姜节胜.结构非线性动力方程的精细积分算法[J].应用力学学报,2000,17(4):164-168.ZHANG Xun’an,JIANG Jiesheng. The precise integration algorithm for nonlinear dynamics equations of structures[J].Chinese Journal of Applied Mechanics, 2000, 17(4):164-168.
[6]裘春航,吕和祥,钟万勰.求解非线性动力学方程的分段直接积分法[J].力学学报,2002,34(3):369-378.QIU Chunhang,LHexiang,ZHONG Wanxie. On segmenteddirect-integration method for nonlinear dynamics equations[J]. Acta Mechanica Sinica,2002,34(3):369-378.
[7]王海波,余志武,陈伯望.非线性动力方程的改进分段直接积分法[J].工程力学,2008,25(9):13-17.WANG Haibo, YU Zhiwu, CHEN Bowang. Improved segmented-direct-integration method for nonlinear dynamic equations[J]. Engineering Mechanics, 2008, 25(9):13-17.
[8]李金桥,于建华.非线性动力方程避免状态矩阵求逆的级数解[J].四川大学学报(工程科学版),2004,36(4):26-30.LI Jinqiao,YU Jianhua. Series solution of nonlinear dynamics equations avoiding calculating the inversion of the state matrix[J]. Journal of Sichuan University(Engineering Science),2004,36(4):26-30.
[9]王海波,余志武,陈伯望.非线性动力分析避免状态矩阵求逆的精细积分多步法[J].振动与冲击,2008,27(4):105-108.WANG Haibo,YU Zhiwu,CHEN Bowang. Precise integration multi-step method for nonlinear dynamic equations to avoid calculating inverse of state matrix[J]. Journal of Vibration and Shock,2008,27(4):105-108.
[10]闫海青,唐晨,张皞,等.任意阶显式精细积分多步法的常用形式及其高阶次数值计算[J].计算物理,2004,21(3):333-338.YAN Haiqing,TANG Chen,ZHANG Hao,et al. Common formulae for free-order explicit multistep method of precise time integration and the higher order numerical simulation[J]. Journal of Computational Physics,2004,21(3):333-338.
[11]李炜华,王堉,罗恩.求解非线性结构动力方程的预估校正—辛时间子域法[J].计算力学学报,2014,31(4):453-458.LI Weihua, WANG Yu, LUO En. Predictor-corrector symplectic time-subdomain algorithm for nonlinear dynamic equations[J]. Chinese Journal of Computational Mechanics,2014,31(4):453-458.
[12]范宣华,陈璞,慕文品.结构动力方程的2种精细时程积分[J].西南交通大学学报,2012,47(1):109-114.FAN Xuanhua,CHEN Pu,MU Wenpin. Two precise timeintegration methods for structural dynamic analysis[J].Journal of Southwest Jiaotong University, 2012,47(1):109-114.
[13]江小燕,王建国.非线性动力方程的精细时空有限元方法[J].工程力学,2014,31(1):23-28.JIANG Xiaoyan,WANG Jianguo. The precise space-time finite element method for nonlinear dynamic equation[J].Engineering Mechanics,2014,31(1):23-28.
[14]王海波,陈晋,李少毅.用于非线性动力分析的一种高效精细积分单步法[J].振动与冲击,2017,36(15):158-162.WANG Haibo,CHEN Jin,LI Shaoyi. An efficient precise integration single-step method for the nonlinear dynamic analysis[J]. Journal of Vibration and Shock,2017,36(15):158-162.
[15]富明慧,刘祚秋,林敬华.一种广义精细积分法[J].力学学报,2007,39(5):672-677.FU Minghui,LIU Zuoqiu,LIN Jinghua. A generalized precise time step integration method[J]. Acta Mechanica Sinica,2007,39(5):672-677.
[16]储德文,王元丰.精细直接积分法的积分方法选择[J].工程力学,2002,19(6):115-119.CHU Dewen,WANG Yuanfeng. Integration formula selection for precise direct integration method[J]. Engineering Mechanics,2002,19(6):115-119.
[17]葛根,王洪礼,谭建国.多自由度非线性动力方程的改进增维精细积分法[J].天津大学学报,2009,42(2):113-117.GE Gen,WANG Hongli,TAN Jianguo. Improved incrementdimensional precise integration method for the nonlinear dynamic equation with multi-degree-of-freedom[J]. Journal of Tianjin University,2009,42(2):113-117.
[2]LIN Jiahao, SHEN Weiping,WILLIAMS F W. A high precision direct integration scheme for structures subjected to transient dynamic loading[J]. Computer&Structures,1995,6(1):120-130.
[3]赵秋玲.非线性动力学方程的精细积分法[J].力学与实践,1998,20(6):24-26.ZHAO Qiuling. An accurate integration method for solving nonlinear dynamic problems[J]. Mechanics in Engineering,1998,20(6):24-26.
[4]吕和祥,蔡志勤,裘春航.非线性动力问题的一个显式精细积分算法[J].应用力学学报,2001,18(2):34-40.LHexiang,CAI Zhiqin,QIU Chunhang. An explicit precise integration algorithm for nonlinear dynamics problems[J].Chinese Journal of Applied Mechanics, 2001, 18(2):34-40.
[5]张洵安,姜节胜.结构非线性动力方程的精细积分算法[J].应用力学学报,2000,17(4):164-168.ZHANG Xun’an,JIANG Jiesheng. The precise integration algorithm for nonlinear dynamics equations of structures[J].Chinese Journal of Applied Mechanics, 2000, 17(4):164-168.
[6]裘春航,吕和祥,钟万勰.求解非线性动力学方程的分段直接积分法[J].力学学报,2002,34(3):369-378.QIU Chunhang,LHexiang,ZHONG Wanxie. On segmenteddirect-integration method for nonlinear dynamics equations[J]. Acta Mechanica Sinica,2002,34(3):369-378.
[7]王海波,余志武,陈伯望.非线性动力方程的改进分段直接积分法[J].工程力学,2008,25(9):13-17.WANG Haibo, YU Zhiwu, CHEN Bowang. Improved segmented-direct-integration method for nonlinear dynamic equations[J]. Engineering Mechanics, 2008, 25(9):13-17.
[8]李金桥,于建华.非线性动力方程避免状态矩阵求逆的级数解[J].四川大学学报(工程科学版),2004,36(4):26-30.LI Jinqiao,YU Jianhua. Series solution of nonlinear dynamics equations avoiding calculating the inversion of the state matrix[J]. Journal of Sichuan University(Engineering Science),2004,36(4):26-30.
[9]王海波,余志武,陈伯望.非线性动力分析避免状态矩阵求逆的精细积分多步法[J].振动与冲击,2008,27(4):105-108.WANG Haibo,YU Zhiwu,CHEN Bowang. Precise integration multi-step method for nonlinear dynamic equations to avoid calculating inverse of state matrix[J]. Journal of Vibration and Shock,2008,27(4):105-108.
[10]闫海青,唐晨,张皞,等.任意阶显式精细积分多步法的常用形式及其高阶次数值计算[J].计算物理,2004,21(3):333-338.YAN Haiqing,TANG Chen,ZHANG Hao,et al. Common formulae for free-order explicit multistep method of precise time integration and the higher order numerical simulation[J]. Journal of Computational Physics,2004,21(3):333-338.
[11]李炜华,王堉,罗恩.求解非线性结构动力方程的预估校正—辛时间子域法[J].计算力学学报,2014,31(4):453-458.LI Weihua, WANG Yu, LUO En. Predictor-corrector symplectic time-subdomain algorithm for nonlinear dynamic equations[J]. Chinese Journal of Computational Mechanics,2014,31(4):453-458.
[12]范宣华,陈璞,慕文品.结构动力方程的2种精细时程积分[J].西南交通大学学报,2012,47(1):109-114.FAN Xuanhua,CHEN Pu,MU Wenpin. Two precise timeintegration methods for structural dynamic analysis[J].Journal of Southwest Jiaotong University, 2012,47(1):109-114.
[13]江小燕,王建国.非线性动力方程的精细时空有限元方法[J].工程力学,2014,31(1):23-28.JIANG Xiaoyan,WANG Jianguo. The precise space-time finite element method for nonlinear dynamic equation[J].Engineering Mechanics,2014,31(1):23-28.
[14]王海波,陈晋,李少毅.用于非线性动力分析的一种高效精细积分单步法[J].振动与冲击,2017,36(15):158-162.WANG Haibo,CHEN Jin,LI Shaoyi. An efficient precise integration single-step method for the nonlinear dynamic analysis[J]. Journal of Vibration and Shock,2017,36(15):158-162.
[15]富明慧,刘祚秋,林敬华.一种广义精细积分法[J].力学学报,2007,39(5):672-677.FU Minghui,LIU Zuoqiu,LIN Jinghua. A generalized precise time step integration method[J]. Acta Mechanica Sinica,2007,39(5):672-677.
[16]储德文,王元丰.精细直接积分法的积分方法选择[J].工程力学,2002,19(6):115-119.CHU Dewen,WANG Yuanfeng. Integration formula selection for precise direct integration method[J]. Engineering Mechanics,2002,19(6):115-119.
[17]葛根,王洪礼,谭建国.多自由度非线性动力方程的改进增维精细积分法[J].天津大学学报,2009,42(2):113-117.GE Gen,WANG Hongli,TAN Jianguo. Improved incrementdimensional precise integration method for the nonlinear dynamic equation with multi-degree-of-freedom[J]. Journal of Tianjin University,2009,42(2):113-117.