基于精细Runge-Kutta混合积分法的车桥耦合振动非迭代求解算法
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摘要
针对结构非线性问题,采用4阶Runge-Kutta法展开精细积分法中响应状态方程的Duhamel项,构造了一种既可以避免迭代又具有较高精度的精细Runge-Kutta混合积分方法,在此基础上提出了适用于车桥耦合振动高效求解的分析框架。车桥耦合系统由车辆、桥梁子系统组成,均采用有限元建模,其中车辆子系统采用部件刚体假定,而桥梁子系统借助于振型叠加法缩减自由度数目;两个子系统内部非线性作用以及系统间的相互作用通过非线性的虚拟力表达。以一节4轴客车匀速通过32m简支梁为研究对象,分别采用分析框架法、Runge-Kutta法进行动力分析。数值结果对比表明:相对于Runge-Kutta法,精细Runge-Kutta混合法能够显著提高计算收敛的积分步长;分析框架可以应用到实际工程中。
A precise Runge-Kutta hybrid integration method with high accuracy was constructed to solve structural nonlinear problems without iteration,Duhamel terms of a state response equation with the precise integration method were expanded in terms of the 4-order Runge-Kutta method.Based on these,an efficient analysis framework for dynamic interaction analysis of coupled vibration of a train-bridge system was proposed.The train-bridge system consisted of a train subsystem and a bridge subsystem,and the models of both subsystems were established by using the finite element method.The rigid component assumption was induced for the train subsystem,while the mode superposition method was applied to the bridge subsystem to reduce its number of DOF.The inner nonlinear effect of both subsystems and the dynamic interaction between two subsystems were taken as virtual forces.A 4-axle vehicle passing through a simply supported beam with a 32m long span at constant speed was taken as a case study.The dynamic analysis of the coupled system was performed by using the proposed framework and Runge-Kutta method,respectively.The comparison of numerical results showed that the calculation efficiency for convergent integral steps of the precise Runge-Kutta hybrid integration method can be greatly larger than that of Runge-Kutta method;and the proposed framework can be applied in practical engineering.
A precise Runge-Kutta hybrid integration method with high accuracy was constructed to solve structural nonlinear problems without iteration,Duhamel terms of a state response equation with the precise integration method were expanded in terms of the 4-order Runge-Kutta method.Based on these,an efficient analysis framework for dynamic interaction analysis of coupled vibration of a train-bridge system was proposed.The train-bridge system consisted of a train subsystem and a bridge subsystem,and the models of both subsystems were established by using the finite element method.The rigid component assumption was induced for the train subsystem,while the mode superposition method was applied to the bridge subsystem to reduce its number of DOF.The inner nonlinear effect of both subsystems and the dynamic interaction between two subsystems were taken as virtual forces.A 4-axle vehicle passing through a simply supported beam with a 32m long span at constant speed was taken as a case study.The dynamic analysis of the coupled system was performed by using the proposed framework and Runge-Kutta method,respectively.The comparison of numerical results showed that the calculation efficiency for convergent integral steps of the precise Runge-Kutta hybrid integration method can be greatly larger than that of Runge-Kutta method;and the proposed framework can be applied in practical engineering.
引文
[1]夏禾,张楠.车辆与结构相互作用[M].北京:科学出版社,2004.
[2]曹雪芹.桥梁结构动力分析[M].中国铁道出版社,1987.
[3]杜宪亭.强地震作用下大跨度桥梁空间动力效应及列车运行安全研究[D].北京交通大学,2011.
[4]翟婉明.车辆-轨道耦合动力学[M].科学出版社,2007.
[5]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,1994,32(2):131-136.ZHONG Wan-xie.On precise time-integration method for structural dynamics[J].Journal of Dalian University of Technology,1994,32(2):131-136.
[6]钟万勰.子域精细积分及偏微分方程数值解[J].计算结构力学及应用,1995,12(3):253-260.ZHONG Wan-xie.Subdomain precise integration method and numerical solution of partial differential equations[J].Computational Structural Mechanics and applications,1995,12(3):253-260.
[7]Wilson E L.Three dimensional static and dynamic analysis of structures:a physical approach with emphasis on earthquake engineering[M].Computers and Structures Inc.,Berkley,California,2002.
[8]吕和祥,于洪洁.精细积分的非线性动力学积分方程及其解法[J].固体力学学报,2001,22(3):303-308.LU He-xiang,YU Hong-jie.An integral equation of non-linear dynamics and its solution method[J].Acta Mechanic Solid Sinica,2001,22(3):303-308.
[9]王海波,余志武.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):105-107,116,173.WANG Hai-bo,YU Zhi-wu.Precise integration multi-step method for nonlinear dynamic equation to avoid calculating inverse of state matrix[J].Journal of Vibration and Shock.2008,27(4):105-107,116,173.
[10]徐萃薇.计算方法引论[M].高等教育出版社,2002.
[11]Xia H,Zhang N.Dynamic analysis of railway bridge under high-speed trains[J].Computers&Structures,2005,83(23):1891-1901.
[2]曹雪芹.桥梁结构动力分析[M].中国铁道出版社,1987.
[3]杜宪亭.强地震作用下大跨度桥梁空间动力效应及列车运行安全研究[D].北京交通大学,2011.
[4]翟婉明.车辆-轨道耦合动力学[M].科学出版社,2007.
[5]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,1994,32(2):131-136.ZHONG Wan-xie.On precise time-integration method for structural dynamics[J].Journal of Dalian University of Technology,1994,32(2):131-136.
[6]钟万勰.子域精细积分及偏微分方程数值解[J].计算结构力学及应用,1995,12(3):253-260.ZHONG Wan-xie.Subdomain precise integration method and numerical solution of partial differential equations[J].Computational Structural Mechanics and applications,1995,12(3):253-260.
[7]Wilson E L.Three dimensional static and dynamic analysis of structures:a physical approach with emphasis on earthquake engineering[M].Computers and Structures Inc.,Berkley,California,2002.
[8]吕和祥,于洪洁.精细积分的非线性动力学积分方程及其解法[J].固体力学学报,2001,22(3):303-308.LU He-xiang,YU Hong-jie.An integral equation of non-linear dynamics and its solution method[J].Acta Mechanic Solid Sinica,2001,22(3):303-308.
[9]王海波,余志武.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):105-107,116,173.WANG Hai-bo,YU Zhi-wu.Precise integration multi-step method for nonlinear dynamic equation to avoid calculating inverse of state matrix[J].Journal of Vibration and Shock.2008,27(4):105-107,116,173.
[10]徐萃薇.计算方法引论[M].高等教育出版社,2002.
[11]Xia H,Zhang N.Dynamic analysis of railway bridge under high-speed trains[J].Computers&Structures,2005,83(23):1891-1901.
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