基于Riccati传递矩阵法的线性树形多体系统特征值求解
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摘要
为了提高多体系统传递矩阵法求解线性树形多体系统特征值时的数值稳定性,研究了基于Riccati变换的线性树形多体系统特征值求解方法。建立了元件输入输出端的Riccati传递矩阵递推关系;从树形系统各输入端开始沿传递路径依次求得了各元件联接端的Riccati传递矩阵,并建立了用Riccati传递矩阵表示的系统特征方程;建立了消除系统特征方程极点的方法,从而可以增大求解特征方程时的搜索步长。数值算例计算结果与有限元法计算结果对比验证了该文方法的正确性,与通常多体系统传递矩阵法计算结果对比表明了本文方法具有较高的数值稳定性。
In order to improve the numerical stability in computing the eigenvalues of linear tree multibody systems in the context of transfer matrix method for multibody system( MSTMM),the eigenvalue solving strategy of linear tree multibody systems is studied based on the Riccati transformation. The recursive relations of the Riccati transfer matrices between the input and the output ends of elements is established. Starting from each input end of a tree system,the Riccati transfer matrices of the connection ends of each element are obtained along the transfer path successively. The characteristic equation of the system expressed by Riccati transfer matrix is derived. The searching step can be increased when solving the characteristic equation by proposing a technique to eliminate the poles of the characteristic equation. The proposed method is verified by comparing the results of the numerical example with the results of the finite element method( FEM). And it also proves that theproposed method has better numerical stability relative to the normal MSTMM.
In order to improve the numerical stability in computing the eigenvalues of linear tree multibody systems in the context of transfer matrix method for multibody system( MSTMM),the eigenvalue solving strategy of linear tree multibody systems is studied based on the Riccati transformation. The recursive relations of the Riccati transfer matrices between the input and the output ends of elements is established. Starting from each input end of a tree system,the Riccati transfer matrices of the connection ends of each element are obtained along the transfer path successively. The characteristic equation of the system expressed by Riccati transfer matrix is derived. The searching step can be increased when solving the characteristic equation by proposing a technique to eliminate the poles of the characteristic equation. The proposed method is verified by comparing the results of the numerical example with the results of the finite element method( FEM). And it also proves that theproposed method has better numerical stability relative to the normal MSTMM.
引文
[1]芮筱亭,贠来峰,陆毓琪,等.多体系统传递矩阵法及其应用[M].北京:科学出版社,2008.
[2]Rui Xiaoting,Zhang Jianshu,Zhou Qinbo.Automatic deduction theorem of overall transfer equation of multibody system[J].Advances in Mechanical Engineering,2014,2014(2):1-12.
[3]Rui Xiaoting,Bestle D,Zhang Jianshu,et al.A new version of transfer matrix method for multibody system[J].Multibody System Dynamics,2016,38(2):137-156.
[4]汤华涛,吴新跃.基于有限元的空间变截面梁传递矩阵[J].南京理工大学学报,2014,38(1):78-82.Tang Huatao,Wu Xinyue.Analysis of transfer matrix of space non-uniform beam based on finite element method[J].Journal of Nanjing University of Science and Technology,2014,38(1):78-82.
[5]舒睿洪,王国平,武令伟.含区间参数的柔性弹箭特征值分析[J].兵器装备工程学报,2017(9):48-52.Shu Ruihong,Wang Guoping,Wu Lingwei.Research on eigenvalues with interval parameters of flexible missile[J].Journal of Ordnance Equipment Engineering,2017(9):48-52.
[6]Bestle D,Abbas L,Rui Xiaoting.Recursive eigenvalue search algorithm for transfer matrix method of linear flexible multibody systems[J].Multibody System Dynamics,2014,32(4):429-444.
[7]Chen Gangli,Rui Xiaoting,Yang Fufeng,et al.Study on the natural vibration characteristics of flexible missile with thrust by using Riccati transfer matrix method[J].Journal of Applied Mechanics,2015,83(3):031006-031006-8.
[8]Gu Junjie,Rui Xiaoting,Zhang Jianshu,et al.Riccati transfer matrix method for linear tree multibody systems[J].Journal of Applied Mechanics,2016,84(1):011008-011008-7.
[9]Horner G C,Pilkey W D.The Riccati transfer matrix method[J].Journal of Mechanical Design,1978,1(2):297-302.
[10]Xue H.A combined dynamic finite element-Riccati transfer matrix method for solving non-linear eigenproblems of vibrations[J].Computers&Structures,1994,53(6):1257-1261.
[11]Xue H.A combined finite element-Riccati transfer matrix method in frequency domain for transient structural response[J].Computers&Structures,1997,62(2):215-220.
[12]Stephen N G.On the Riccati transfer matrix method for repetitive structures[J].Mechanics Research Communications,2010,37(7):663-665.
[13]Yu A M,Hao Y.Improved Riccati transfer matrix method for free vibration of non-cylindrical helical springs including warping[J].Shock and Vibration,2012,19(6):1167-1180.
[14]Zheng Y,Xie Z,Li Y,et al.Spatial vibration of rolling mills[J].Journal of Materials Processing Technology,2013,213(4):581-588.
[15]陈淮,杜思义,魏泽丽.基于摄动Riccati传递矩阵的梁桥损伤概率识别方法[J].振动工程学报,2010,23(3):283-289.Chen Huai,Du Siyi,Wei Zeli.A probabilistic damage identification method for beam bridge based on perturbation Riccati transfer matrix[J].Journal of Vibration Engineering,2010,23(3):283-289.
[16]毛文贵,韩旭,刘桂萍.滚动轴承-转子系统RiccatiNewmark加速度传递矩阵法[J].振动与冲击,2015,34(20):80-84.Mao Wengui,Han Xu,Liu Guiping.Riccati transfer matrix method combined with newmark acceleration formulation integration for analyzing sliding bearings and rotor system[J].Journal of Vibration and Shock,2015,34(20):80-84.
[2]Rui Xiaoting,Zhang Jianshu,Zhou Qinbo.Automatic deduction theorem of overall transfer equation of multibody system[J].Advances in Mechanical Engineering,2014,2014(2):1-12.
[3]Rui Xiaoting,Bestle D,Zhang Jianshu,et al.A new version of transfer matrix method for multibody system[J].Multibody System Dynamics,2016,38(2):137-156.
[4]汤华涛,吴新跃.基于有限元的空间变截面梁传递矩阵[J].南京理工大学学报,2014,38(1):78-82.Tang Huatao,Wu Xinyue.Analysis of transfer matrix of space non-uniform beam based on finite element method[J].Journal of Nanjing University of Science and Technology,2014,38(1):78-82.
[5]舒睿洪,王国平,武令伟.含区间参数的柔性弹箭特征值分析[J].兵器装备工程学报,2017(9):48-52.Shu Ruihong,Wang Guoping,Wu Lingwei.Research on eigenvalues with interval parameters of flexible missile[J].Journal of Ordnance Equipment Engineering,2017(9):48-52.
[6]Bestle D,Abbas L,Rui Xiaoting.Recursive eigenvalue search algorithm for transfer matrix method of linear flexible multibody systems[J].Multibody System Dynamics,2014,32(4):429-444.
[7]Chen Gangli,Rui Xiaoting,Yang Fufeng,et al.Study on the natural vibration characteristics of flexible missile with thrust by using Riccati transfer matrix method[J].Journal of Applied Mechanics,2015,83(3):031006-031006-8.
[8]Gu Junjie,Rui Xiaoting,Zhang Jianshu,et al.Riccati transfer matrix method for linear tree multibody systems[J].Journal of Applied Mechanics,2016,84(1):011008-011008-7.
[9]Horner G C,Pilkey W D.The Riccati transfer matrix method[J].Journal of Mechanical Design,1978,1(2):297-302.
[10]Xue H.A combined dynamic finite element-Riccati transfer matrix method for solving non-linear eigenproblems of vibrations[J].Computers&Structures,1994,53(6):1257-1261.
[11]Xue H.A combined finite element-Riccati transfer matrix method in frequency domain for transient structural response[J].Computers&Structures,1997,62(2):215-220.
[12]Stephen N G.On the Riccati transfer matrix method for repetitive structures[J].Mechanics Research Communications,2010,37(7):663-665.
[13]Yu A M,Hao Y.Improved Riccati transfer matrix method for free vibration of non-cylindrical helical springs including warping[J].Shock and Vibration,2012,19(6):1167-1180.
[14]Zheng Y,Xie Z,Li Y,et al.Spatial vibration of rolling mills[J].Journal of Materials Processing Technology,2013,213(4):581-588.
[15]陈淮,杜思义,魏泽丽.基于摄动Riccati传递矩阵的梁桥损伤概率识别方法[J].振动工程学报,2010,23(3):283-289.Chen Huai,Du Siyi,Wei Zeli.A probabilistic damage identification method for beam bridge based on perturbation Riccati transfer matrix[J].Journal of Vibration Engineering,2010,23(3):283-289.
[16]毛文贵,韩旭,刘桂萍.滚动轴承-转子系统RiccatiNewmark加速度传递矩阵法[J].振动与冲击,2015,34(20):80-84.Mao Wengui,Han Xu,Liu Guiping.Riccati transfer matrix method combined with newmark acceleration formulation integration for analyzing sliding bearings and rotor system[J].Journal of Vibration and Shock,2015,34(20):80-84.