【作者】 赵世佳；

【导师】 高文杰；

【作者基本信息】 吉林大学 ， 应用数学， 2010， 硕士

【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.更多还原

【Abstract】 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 ifWhen 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, letBased on the combined approximation approach,whereΨi0=Ψi is known, through the equation (9), we can get the base vectorWrite the generalized eigenvectorΨi of the combination of the base vectorΨiB and the coefficient vector yi where the coefficient vector is LetTherefore, only to solve the s+1 linear equationswe 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.更多还原