复阻尼结构动力方程的增维精细积分法
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摘要
为了避开求解复阻尼结构强迫激励动力学方程的积分运算,引入增维精细积分方法。根据复阻尼系统复化对偶原则,将动力学方程和激励波对偶复化为实部和虚部的形式,推导出增维矩阵的精细积分求解过程。结果表明,由于不用求解迭代矩阵H的逆矩阵,避免了矩阵奇异带来的计算解的不稳定性。在计算矩阵仅增加一维的情况下,化积分运算为代数运算,扩大了精细积分法的应用范围。通过对比增维精细积分法和频域法计算结果,二者结果保持较高的一致性。
In order to avoid the integral operation of the forced excitation dynamics equation of complex damped structures,a magnified precise integration method was introduced. According to the dual principle of complex damping system,the dual dynamic equations and the dual excitation waves were divided into real and imaginary parts,and the solving process by using the precise integration of the augmented matrix was derived. The results show that because the inverse matrix of the iteration matrix H doesn't need to be solved,the instability of the computational solution caused by the singularity of matrix is avoided. In the calculation of the matrix,only a one-dimensional integral calculation is increased and the integral operation is transformed into algebraic operations,so,the scope of application of the precise integration method can be expanded. The comparison between the results calculated by using the augmented precise integration method and the frequency domain method shows they are in good consistency.
In order to avoid the integral operation of the forced excitation dynamics equation of complex damped structures,a magnified precise integration method was introduced. According to the dual principle of complex damping system,the dual dynamic equations and the dual excitation waves were divided into real and imaginary parts,and the solving process by using the precise integration of the augmented matrix was derived. The results show that because the inverse matrix of the iteration matrix H doesn't need to be solved,the instability of the computational solution caused by the singularity of matrix is avoided. In the calculation of the matrix,only a one-dimensional integral calculation is increased and the integral operation is transformed into algebraic operations,so,the scope of application of the precise integration method can be expanded. The comparison between the results calculated by using the augmented precise integration method and the frequency domain method shows they are in good consistency.
引文
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[8]张辉东,王元丰.复阻尼模型结构地震时程响应研究[J].工程力学,2010,27(1):109-115.ZHANG Huidong,WANG Yuanfeng.Study on seismic timehistory response of structures with complex damping[J].Engineering Mechanics,2010,27(1):109-115.
[9]潘玉华,王元丰.复阻尼结构动力方程的高斯精细时程积分法[J].工程力学,2012,29(2):16-20.PAN Yuhua,WANG Yuanfeng.Gauss precise time integration of complex damping vibration systems[J].Engineering Mechanics,2012,29(2):16-20.
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[11]钟万勰.暂态历程的精细计算方法[J].计算结构力学及其应用,1995,12(1):1-6.ZHONG Wanxie.Precise computation for transient analysis[J].Computational Structural Mechanics and Applications,1995,12(1):1-6.
[12]MOLER C B,VON LOAN C F.Nineteen dubious ways to compute the exponential of a matrix,twenty-five years later[J].Society for Industrial and Applied Mechanics,2003,45(1):3-49.
[2]GADE S,ZAVERI K,KONSTANTIN-HANSEN H,et al.Complex modulus and damping measurements using resonant and non-resonant methods[C]//SAE Technical Papers,1995.
[3]CORDIOLI J A,BRATTI G,STUMPF C,et al.On the prediction of damping loss factor of fuselage panels with viscoelastic materials using periodic structure theory and finite element method[C]//Proceedings of ISMA,2010:2257-2266.
[4]KOSMATKA J B,LIGUORE S L.Review of methods for analyzing constrained-layer damped structures[J].Journal of Aerospace Engineering,1993,6(3):268-283.
[5]朱镜清.频率相关黏性阻尼理论及有关问题的解[J].振动与冲击,1992,21(4):1-7.ZHU Jingqing.Frequency dependant viscous damping theory and some related problems[J].Journal of Vibration and Shock,1992,21(4):1-7.
[6]何钟怡.复本构理论中的对偶原则[J].固体力学学报,1994,15(2):177-180.HE Zhongyi.The dual principle in theory of complex constitutive equations[J].Acta Mechnica Solid Sinica,1994,15(2):177-180.
[7]王建平,刘宏昭.复阻尼振动系统的对偶原则及其数值方法研究[J].振动工程学报,2004,17(1):62-65.WANG Jianping,LIU Hongzhao.Dual principle of oscillation systems with complex damping and its numerical method[J].Journal of Vibration Engineering,2004,17(1):62-65.
[8]张辉东,王元丰.复阻尼模型结构地震时程响应研究[J].工程力学,2010,27(1):109-115.ZHANG Huidong,WANG Yuanfeng.Study on seismic timehistory response of structures with complex damping[J].Engineering Mechanics,2010,27(1):109-115.
[9]潘玉华,王元丰.复阻尼结构动力方程的高斯精细时程积分法[J].工程力学,2012,29(2):16-20.PAN Yuhua,WANG Yuanfeng.Gauss precise time integration of complex damping vibration systems[J].Engineering Mechanics,2012,29(2):16-20.
[10]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.
[11]钟万勰.暂态历程的精细计算方法[J].计算结构力学及其应用,1995,12(1):1-6.ZHONG Wanxie.Precise computation for transient analysis[J].Computational Structural Mechanics and Applications,1995,12(1):1-6.
[12]MOLER C B,VON LOAN C F.Nineteen dubious ways to compute the exponential of a matrix,twenty-five years later[J].Society for Industrial and Applied Mechanics,2003,45(1):3-49.