基于组合近似法的亏损振动系统问题研究
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摘要
本文针对线性振动系统首先介绍了亏损现象,阐述了研究振动亏损系统的实际意义、发展状况,以及重分析方法的理论和分类.同时根据特征根几何重数和代数重数的关系给出了亏损系统的定义、特征值亏损的充分必要条件以及特征值λ对应的阻抗矩阵的秩数rank(R(λ))与λ的重数s的关系,进一步运用牛顿第二定律建立了具有n个自由度振动系统运动方程的模型.对于结构修改后的具有粘性阻尼的亏损振动系统,本文利用组合近似方法对特征问题进行重分析计算,将修改后的广义特征向量表示成基向量和系数向量的组合形式,简化了复杂的求解运算.同时还可以根据精度的要求灵活调整基向量的个数.
Vibration is a common and important phenomenon in engineering. Usu-ally, most vibration systems are supposed to be complete. It means the corre-sponding eigenvector sets in the state space are complete, for this situation the reanalysis method has been well developed so far. However, other systems in the practical application appear, for example:pneumatic elastic shaking analysis or structure dynamic problems under the influence of nonconser-vative force. There exist some defective phenomena in the corresponding system of these problems, and this kind of system is called defective system. If only describe the movement in these defective space which is expanded by the eigenvector and superpose these modal, the result of convergence and validity is doubted. That is to say, state analysis theory of complete system is not suitable for the defective system. Shi(Chinese Journal of Theoretical and Applied Mechanics,1989,21(2))supplemented the new linear independence vector based on the incomplete eigenvector system according to the feature of eigenvector system of defective system is not complete and the changed eigenvector system is complete which is called generalized modes. More-over, he put forward the generalized mode theory of the defective system, proved part orthogonality of the generalized mode vector and deducted the expression of frequency response function matrix and impulse response. Yang(Journal of University of Science and Technology of China,1991,21 (3)) changed the state matrix to Jordan normalized form by similarity transfor-mation, treated transformation matrix as coordinate, and made the equation decoupling by changing the original state function coordinate. Furthermore, Chen(Jilin university science journal,2001,31(3)) discussed the computing method of gain matrix, and improved the dynamic characteristics of defective system by assigning the defective multiple eigenvalue for isolated eigenval-ues.
If we want to control the structure of defective system or have a further study, construct compute a modified system. In engineering, people usu-ally need to modify or design the structure for many times so that one can achieve approving property, that is to say we need to have the process of re-vising design-reanalysis-revising design. Reanalysis problem is to have the rapid analysis and calculation of the modified structure. The reanalysis theory and method were developed in the 1970s, it is usually used for the large system which is modified. Since the change of the process is repeated especially for structure optimization problems, sometimes we need to modify dozens times or even hundreds times in order to obtain the optimal design, the computational cost and time problem appear to be important particularly. Large complex structure design changes will be very troublesome and ex-pensive if there is no good algorithm to solve the problems. Therefore, in order to reduce the computational cost, it force people to build rapid anal-ysis and calculation method. According to the parameter modification, the reanalysis method can be divided into accurate reanalysis method and the approximate reanalysis method. Accurate reanalysis method only applies to local small range of structure, including decomposing matrix method, par-allel unit method correction method and the initial force of variation, etc. Differently from the accurate reanalysis method, the approximation method is applicable to the analysis of most moderate modification, thus provides re-sponse approximation solution of structure after modification. The approxi-mate reanalysis problem in the execution of the accuracy and efficiency of the implementation determines the degree of difficulty, which is mainly divided into three types:global multi-point approximation method, local single-point approximation and combined approximation method. Global multi-point ap-proximation has the higher accuracy because most of the information is avail-able. But for large-scale design variables, the calculation amount of global approximation method is very large. Local single-point approximation based on single design point of calculation information, and it is valid only for small changes of design variables. If we give a big modification of the design, the approximation precision will be lower and even meaningless. Combined ap-proximation, this method is put forward by U. Kirsch and it is trying to give the local single-point approximation to global quality, it is the combination of high efficiency of local approximation and high quality of global approx-imation. This method is based on the accurate result of an given point, only to solve a small-scale linear equations. Just in the view of this merit of the combination approximation, we apply this method to the defective system.
This paper firstly presented the vibration system's motion equation with n variance by Newton's second law. where M, C and K are the mass, damping and stiffness matrices respectively; x(t), x(t) and x(t) are the displacement, velocity and acceleration vectors, respectively and F(t) is the external force vector.
Suppose the equation (1) has the solution x=ξeλt, whereλandξare the undetermined eigenvalue and eigenvector respectively. Substituting to the equation (1)gives, where R(λ)=λ2M+λC+K is refered to as a corresponding to the eigenvalueλimpedance matrix.
Theorem 1 The sufficient and necessary condition for eigenvalueλdefective in linear vibration system is thatΦT(2λM+C)Φis a singular matrix, whereΦ= (ξ1,ξ2,…,ξr) is the vibration model base matrix of eigenvalueλ.
Theorem 2 In linear symmetrical vibratory system, the relationship between the rank(R(λ)) of impedance matrix corresponding to eigenvalueλand the multiplicity n of A is:
(1)eigenvalue A is defective if and only if
(2) eigenvalue A is not defective if and only if
When parameters changed,reanalysis needs to be performed on the mod-ified system if we want to control or have a further study of the defective system. This method based on the accurate result of an given point, only to solve a small-scale linear equations, it is suitable for various kinds of design variables and different types of structure. We just based on the character-istics of small amount of calculation algorithm and apply to small structure modification of the eigenvectors of the vibration system problem.
For the vibration system's motion equation with n variance, consider the state matrix A with 2n repeated eigenvalue A, with 2n irrelevant eigenvector nonexistence, we attempt to solve the defective system problem under the condition of small changes of structural parameters.
According to Jordan normal form theory, A characteristic problem is as follows:
whereΨand J are general characteristic vector matrix and Jordan normal form of state matrix respectively. We change the form of equation (5) for applying the combined approximation approach,
where B=(0, e1,…, e2n-1), ei is the unit vector. Expand the equation (6) and write it to the form of characteristic equations,
If the structural parameters have small changes, such that the state matrix has the change△A, based on the equation (5), the changed characteristic problem can be expressed as:
whereΨand J are general eigenvalue vector matrix and Jordan normal form of state matrix A+△A respectively, Based on the above equation, we have
Expand the equation (9) and write it to the form of characteristic equations,
Introduce the mark of combined approximation approach, let
Based on the combined approximation approach,
whereΨi0=Ψi is known, through the equation (9), we can get the base vector
Write the generalized eigenvectorΨi of the combination of the base vectorΨiB and the coefficient vector yi where the coefficient vector is Let
Therefore, only to solve the s+1 linear equations
we have the coefficient vector yi the computation is far less than the original equations. Put the coefficient vector to the equation (13), and repeat the above method for i=2,3,…,2n, we can have the changed general eigenvalue vectorΨi. In the equation (10), using the usual methods for i=1. Summing up the above, we get the changed general eigenvalue vector matrixΨ.
Furthermore, we can also check out whether vectorΨin error range. If it is not in the range, we can use more base vector to improve the precision.
Vibration is a common and important phenomenon in engineering. Usu-ally, most vibration systems are supposed to be complete. It means the corre-sponding eigenvector sets in the state space are complete, for this situation the reanalysis method has been well developed so far. However, other systems in the practical application appear, for example:pneumatic elastic shaking analysis or structure dynamic problems under the influence of nonconser-vative force. There exist some defective phenomena in the corresponding system of these problems, and this kind of system is called defective system. If only describe the movement in these defective space which is expanded by the eigenvector and superpose these modal, the result of convergence and validity is doubted. That is to say, state analysis theory of complete system is not suitable for the defective system. Shi(Chinese Journal of Theoretical and Applied Mechanics,1989,21(2))supplemented the new linear independence vector based on the incomplete eigenvector system according to the feature of eigenvector system of defective system is not complete and the changed eigenvector system is complete which is called generalized modes. More-over, he put forward the generalized mode theory of the defective system, proved part orthogonality of the generalized mode vector and deducted the expression of frequency response function matrix and impulse response. Yang(Journal of University of Science and Technology of China,1991,21 (3)) changed the state matrix to Jordan normalized form by similarity transfor-mation, treated transformation matrix as coordinate, and made the equation decoupling by changing the original state function coordinate. Furthermore, Chen(Jilin university science journal,2001,31(3)) discussed the computing method of gain matrix, and improved the dynamic characteristics of defective system by assigning the defective multiple eigenvalue for isolated eigenval-ues.
If we want to control the structure of defective system or have a further study, construct compute a modified system. In engineering, people usu-ally need to modify or design the structure for many times so that one can achieve approving property, that is to say we need to have the process of re-vising design-reanalysis-revising design. Reanalysis problem is to have the rapid analysis and calculation of the modified structure. The reanalysis theory and method were developed in the 1970s, it is usually used for the large system which is modified. Since the change of the process is repeated especially for structure optimization problems, sometimes we need to modify dozens times or even hundreds times in order to obtain the optimal design, the computational cost and time problem appear to be important particularly. Large complex structure design changes will be very troublesome and ex-pensive if there is no good algorithm to solve the problems. Therefore, in order to reduce the computational cost, it force people to build rapid anal-ysis and calculation method. According to the parameter modification, the reanalysis method can be divided into accurate reanalysis method and the approximate reanalysis method. Accurate reanalysis method only applies to local small range of structure, including decomposing matrix method, par-allel unit method correction method and the initial force of variation, etc. Differently from the accurate reanalysis method, the approximation method is applicable to the analysis of most moderate modification, thus provides re-sponse approximation solution of structure after modification. The approxi-mate reanalysis problem in the execution of the accuracy and efficiency of the implementation determines the degree of difficulty, which is mainly divided into three types:global multi-point approximation method, local single-point approximation and combined approximation method. Global multi-point ap-proximation has the higher accuracy because most of the information is avail-able. But for large-scale design variables, the calculation amount of global approximation method is very large. Local single-point approximation based on single design point of calculation information, and it is valid only for small changes of design variables. If we give a big modification of the design, the approximation precision will be lower and even meaningless. Combined ap-proximation, this method is put forward by U. Kirsch and it is trying to give the local single-point approximation to global quality, it is the combination of high efficiency of local approximation and high quality of global approx-imation. This method is based on the accurate result of an given point, only to solve a small-scale linear equations. Just in the view of this merit of the combination approximation, we apply this method to the defective system.
This paper firstly presented the vibration system's motion equation with n variance by Newton's second law. where M, C and K are the mass, damping and stiffness matrices respectively; x(t), x(t) and x(t) are the displacement, velocity and acceleration vectors, respectively and F(t) is the external force vector.
Suppose the equation (1) has the solution x=ξeλt, whereλandξare the undetermined eigenvalue and eigenvector respectively. Substituting to the equation (1)gives, where R(λ)=λ2M+λC+K is refered to as a corresponding to the eigenvalueλimpedance matrix.
Theorem 1 The sufficient and necessary condition for eigenvalueλdefective in linear vibration system is thatΦT(2λM+C)Φis a singular matrix, whereΦ= (ξ1,ξ2,…,ξr) is the vibration model base matrix of eigenvalueλ.
Theorem 2 In linear symmetrical vibratory system, the relationship between the rank(R(λ)) of impedance matrix corresponding to eigenvalueλand the multiplicity n of A is:
(1)eigenvalue A is defective if and only if
(2) eigenvalue A is not defective if and only if
When parameters changed,reanalysis needs to be performed on the mod-ified system if we want to control or have a further study of the defective system. This method based on the accurate result of an given point, only to solve a small-scale linear equations, it is suitable for various kinds of design variables and different types of structure. We just based on the character-istics of small amount of calculation algorithm and apply to small structure modification of the eigenvectors of the vibration system problem.
For the vibration system's motion equation with n variance, consider the state matrix A with 2n repeated eigenvalue A, with 2n irrelevant eigenvector nonexistence, we attempt to solve the defective system problem under the condition of small changes of structural parameters.
According to Jordan normal form theory, A characteristic problem is as follows:
whereΨand J are general characteristic vector matrix and Jordan normal form of state matrix respectively. We change the form of equation (5) for applying the combined approximation approach,
where B=(0, e1,…, e2n-1), ei is the unit vector. Expand the equation (6) and write it to the form of characteristic equations,
If the structural parameters have small changes, such that the state matrix has the change△A, based on the equation (5), the changed characteristic problem can be expressed as:
whereΨand J are general eigenvalue vector matrix and Jordan normal form of state matrix A+△A respectively, Based on the above equation, we have
Expand the equation (9) and write it to the form of characteristic equations,
Introduce the mark of combined approximation approach, let
Based on the combined approximation approach,
whereΨi0=Ψi is known, through the equation (9), we can get the base vector
Write the generalized eigenvectorΨi of the combination of the base vectorΨiB and the coefficient vector yi where the coefficient vector is Let
Therefore, only to solve the s+1 linear equations
we have the coefficient vector yi the computation is far less than the original equations. Put the coefficient vector to the equation (13), and repeat the above method for i=2,3,…,2n, we can have the changed general eigenvalue vectorΨi. In the equation (10), using the usual methods for i=1. Summing up the above, we get the changed general eigenvalue vector matrixΨ.
Furthermore, we can also check out whether vectorΨin error range. If it is not in the range, we can use more base vector to improve the precision.
引文
[1]徐涛,陈塑寰,张旭莉.广义亏损系统的矩阵摄动[J].吉林工业大学自然科学学报,2000,30(2)：56-61.
[2]UWE PRELLS, Michael I Friswell. A Relationship Between Defective systems and Unit-Rank Modification of Classical Damping [J]. Journal of Vibration and Acoustics.2000,122(2):180-183.
[3]时国勤,诸德超.线性振动亏损系统的广义模态理论[J].力学学报.1989,21(2)：183-192.
[4]孙木楠,方同,马大为,张福祥.亏损系统的时域随机响应分析[J].南京理工大学学报.1999,23(2)：97-100.
[5]陈宇东.亏损系统的特征值配置[J].吉林工业大学自然科学学报.2001,31(3)：36-40.
[6]陈宇东.亏损系统模态控制设计及其在机翼颤振控制中的应用[J].吉林大学学报(工学版)增刊.2006,36：4-7.
[7]陈宇东,陈塑寰,吴伟.亏损系统振动模态的可控可观性[J].计算力学学报.2003,20(6)：697-701.
[8]杨前进,张培强.线性亏损振动系统的模态分析理论[J].中国科学技术大学学报.1991,21(3)：53-61.
[9]陈塑寰,徐涛,韩万芝.线性振动亏损系统的矩阵摄动理论[J].力学学报,1992,24(6)：747-753.
[10]CHEN S H, Yang X W, Lian H D. Comparison of several eigenvalue reanalysis methods for modified structures [J]. Structural Optimization, 2000,20(4):253-259.
[11]KIRSCH U. Combined approximations-a general reanalysis approach for structural optimization[J]. Structural Optimization,2000,20(2):97-106.
[12]LIU X L. Accurate modal perturbation in non-self-adjoint eigenvalue problem[J]. Commun Numer Meth Engng,2001,17:715-725.
[13]CHEN S H. Matrix perturbation theory in structural dynamic[M]. Bei-jing:International academic publisher,1993.
[14]BENNET J M. Triangular factors of modified matrices [J]. Numerical Mathematics,1965,7:217-221.
[15]SOBIESZCZANSKI J. Structural modification by perturbation method[J]. Journal of the Structural Division, ASCE,1968,94(12): 2799-2816.
[16]ARGYRIS J H. The matrix analysis of structures with cutouts and mod-ifications [C]. Proceedings of 9th Int. Congress of Applied Mechanics. Univ. of Brussels,1986.
[17]JIN R, CHEN W, SIMPSON T W. Comparative studies of metamod-elling techniques under multiple modelling criteria[J]. Structural and Multidisciplinary Optimization,2001,23:1-13.
[18]SAKATA S, Ashidaa F, ZAKO M. Structural optimization using Krig-ing approximation[J]. Computer Methods in Applied Mechanics and En-gineering.2003,192(7-8):923-939.
[19]PHANSALKAR S R. Matrix iterative methods for structural reanaly-sis[J]. Journal of Computers & Structures,1974,4:779-800.
[20]BEST G A. Method of structural weight minimization suitable for high speed digital computers[J]. AIAA Journal.1963,1(2):478-479.
[21]FLEURY C. Efficient approximation concepts using second order infor-mation[J]. International Journal of Numerical Methods in Engineering. 1989,28:2041-2058.
[22]FLEURY C. First and second order convex approximation strategies in structural optimization[J]. Structural Optimation,1989,1:3-10.
[23]STARNES J H JR., HAFCTKA R. T. Preliminary design ofcomposite wings for buckling stress and displacement constraints [J]. Journal of Air-craft,1979,16:564-570.
[24]KIRSCH U. Reduced basis approximations of structural displacements for optimal design[J]. AIAA J.1991,29:1751-1758.
[25]KIRSCH U. Efficient reanalysis for topological optimization[J]. Struc-tural Optimization,1993,6:143-150.
[26]KIRSCH U., Liu S. Structural reanalysis for general layout modifica-tions[J]. AIAA J.,1997,35:382-388.
[27]KIRSCH U. Improved stiffness-based first-order approximations for structural optimization[J]. AIAA J.1995,33(1):143-150.
[28]曹宗杰,王明伟.智能结构重频系统反馈控制[J].东北大学学报(自然科学版).2001,22(5)：517-519.
[29]张森文,曹开彬.计算结构动力响应的状态方程直接积分法[J].计算力学学报.2000,17(1)：94-103.
[30]吕和祥,于洪洁,裘春航.动力学方程的积分型直接积分法[J].应用数学和力学.2001,22(2)：151-156.
[31]杨佑发,邹银生.结构动力分析的二阶矩阵扰动振型叠加法[J].重庆大学学报(自然科学版).1999,22(5)：91-94.
[32]陈超核.振型叠加法与Duhamel积分数值解[J].海南大学学报自然科学版.2001,19(4)：311-315.
[33]李海深,杨果林,邹银生.加筋土挡土墙动力特性分析[J].中国公路学报.2004,17(2)：28-32.
[34]陈振藩.线性振动系统的特征值亏损问题[J].航空学报.1986,7(5)：499-503.
[35]陈振藩.拟内积概念与线性振动系统的特征值亏损[J].南京航空航天大学学报.1985,4：11-22.
[36]杨荣,王乐,徐涛,于澜等.具有不完全特征向量的状态向量摄动的快速算法[J].吉林大学学报(工学版).2008,38(5)：1101-1104.
[37]KIRSCH U. Implementation of combined approximations in structural optimization[J]. Computers and Structures.2000,78:449-457.
[38]ADELMAN H M, HAFTKA R T. Sensitivity analysis for discrete struc-tural systems[J]. AIAA J.1986,24(5):823-832.
[2]UWE PRELLS, Michael I Friswell. A Relationship Between Defective systems and Unit-Rank Modification of Classical Damping [J]. Journal of Vibration and Acoustics.2000,122(2):180-183.
[3]时国勤,诸德超.线性振动亏损系统的广义模态理论[J].力学学报.1989,21(2)：183-192.
[4]孙木楠,方同,马大为,张福祥.亏损系统的时域随机响应分析[J].南京理工大学学报.1999,23(2)：97-100.
[5]陈宇东.亏损系统的特征值配置[J].吉林工业大学自然科学学报.2001,31(3)：36-40.
[6]陈宇东.亏损系统模态控制设计及其在机翼颤振控制中的应用[J].吉林大学学报(工学版)增刊.2006,36：4-7.
[7]陈宇东,陈塑寰,吴伟.亏损系统振动模态的可控可观性[J].计算力学学报.2003,20(6)：697-701.
[8]杨前进,张培强.线性亏损振动系统的模态分析理论[J].中国科学技术大学学报.1991,21(3)：53-61.
[9]陈塑寰,徐涛,韩万芝.线性振动亏损系统的矩阵摄动理论[J].力学学报,1992,24(6)：747-753.
[10]CHEN S H, Yang X W, Lian H D. Comparison of several eigenvalue reanalysis methods for modified structures [J]. Structural Optimization, 2000,20(4):253-259.
[11]KIRSCH U. Combined approximations-a general reanalysis approach for structural optimization[J]. Structural Optimization,2000,20(2):97-106.
[12]LIU X L. Accurate modal perturbation in non-self-adjoint eigenvalue problem[J]. Commun Numer Meth Engng,2001,17:715-725.
[13]CHEN S H. Matrix perturbation theory in structural dynamic[M]. Bei-jing:International academic publisher,1993.
[14]BENNET J M. Triangular factors of modified matrices [J]. Numerical Mathematics,1965,7:217-221.
[15]SOBIESZCZANSKI J. Structural modification by perturbation method[J]. Journal of the Structural Division, ASCE,1968,94(12): 2799-2816.
[16]ARGYRIS J H. The matrix analysis of structures with cutouts and mod-ifications [C]. Proceedings of 9th Int. Congress of Applied Mechanics. Univ. of Brussels,1986.
[17]JIN R, CHEN W, SIMPSON T W. Comparative studies of metamod-elling techniques under multiple modelling criteria[J]. Structural and Multidisciplinary Optimization,2001,23:1-13.
[18]SAKATA S, Ashidaa F, ZAKO M. Structural optimization using Krig-ing approximation[J]. Computer Methods in Applied Mechanics and En-gineering.2003,192(7-8):923-939.
[19]PHANSALKAR S R. Matrix iterative methods for structural reanaly-sis[J]. Journal of Computers & Structures,1974,4:779-800.
[20]BEST G A. Method of structural weight minimization suitable for high speed digital computers[J]. AIAA Journal.1963,1(2):478-479.
[21]FLEURY C. Efficient approximation concepts using second order infor-mation[J]. International Journal of Numerical Methods in Engineering. 1989,28:2041-2058.
[22]FLEURY C. First and second order convex approximation strategies in structural optimization[J]. Structural Optimation,1989,1:3-10.
[23]STARNES J H JR., HAFCTKA R. T. Preliminary design ofcomposite wings for buckling stress and displacement constraints [J]. Journal of Air-craft,1979,16:564-570.
[24]KIRSCH U. Reduced basis approximations of structural displacements for optimal design[J]. AIAA J.1991,29:1751-1758.
[25]KIRSCH U. Efficient reanalysis for topological optimization[J]. Struc-tural Optimization,1993,6:143-150.
[26]KIRSCH U., Liu S. Structural reanalysis for general layout modifica-tions[J]. AIAA J.,1997,35:382-388.
[27]KIRSCH U. Improved stiffness-based first-order approximations for structural optimization[J]. AIAA J.1995,33(1):143-150.
[28]曹宗杰,王明伟.智能结构重频系统反馈控制[J].东北大学学报(自然科学版).2001,22(5)：517-519.
[29]张森文,曹开彬.计算结构动力响应的状态方程直接积分法[J].计算力学学报.2000,17(1)：94-103.
[30]吕和祥,于洪洁,裘春航.动力学方程的积分型直接积分法[J].应用数学和力学.2001,22(2)：151-156.
[31]杨佑发,邹银生.结构动力分析的二阶矩阵扰动振型叠加法[J].重庆大学学报(自然科学版).1999,22(5)：91-94.
[32]陈超核.振型叠加法与Duhamel积分数值解[J].海南大学学报自然科学版.2001,19(4)：311-315.
[33]李海深,杨果林,邹银生.加筋土挡土墙动力特性分析[J].中国公路学报.2004,17(2)：28-32.
[34]陈振藩.线性振动系统的特征值亏损问题[J].航空学报.1986,7(5)：499-503.
[35]陈振藩.拟内积概念与线性振动系统的特征值亏损[J].南京航空航天大学学报.1985,4：11-22.
[36]杨荣,王乐,徐涛,于澜等.具有不完全特征向量的状态向量摄动的快速算法[J].吉林大学学报(工学版).2008,38(5)：1101-1104.
[37]KIRSCH U. Implementation of combined approximations in structural optimization[J]. Computers and Structures.2000,78:449-457.
[38]ADELMAN H M, HAFTKA R T. Sensitivity analysis for discrete struc-tural systems[J]. AIAA J.1986,24(5):823-832.