| With the development of science and technology, the requirement for the engineering structure is higher and higher, and many large complicated structures need to be designed, analyzed, and computed. For acquiring the satisfying result, modification often is required time after time. During the process, we will face a called design sensitivity problem, i.e., to determine the modification methods by calculating the derivatives of the structural modes with respect to design parameters. So the mode derivatives are important for structural dynamics optimal analysis, and control system design. Recently, the derivative data have become necessary for use in structural model modification, system parameter identification, and failure diagnosis. In addition, the calculation of the eigenvector derivatives has an important role in sensitivity analysis. This thesis is focused on the structures with repeated eigenvalues, and computation of eigenvector derivatives with repeated eigenvalues will be deal with by using direct method. Two new methods are presented for computing the eigenvector derivatives. Numerical examples are included to show the two approaches can give the same results that can approach the results that are given by the finite difference, consequently the validity is demonstrated. Considering the real, symmetric eigenvalue problem where K is the structural stiffness matrix, M the mass matrix, whose elements depend continuously on the system parameters, λithe eigenvalue, x ithe eigenvector corresponding to λi, n the total degrees of freedom of the structure, and δijthe Kronecker sign. It is assumed that the eigenvalue problem has m ( m ≤n) repeated eigenvalues λi,i = 1, 2,L,m, that is to say, λ1 = λ2=L=λm=λ, the corresponding eigenvectors are denoted by x1 , x2,L ,xm, when the system parameter p = p0. In the case, the calculation for the derivatives of eigenvalues and eigenvectors is not straightforward. First, we should find the eigenvectors for which derivatives can be defined. Now let X = [ x1 ,x2,L,xm], those adjacent eigenvectors can be expressed in terms of X by an orthogonal transformation: Z = XΓ, where Γin R m×mis orthogonal ΓT Γ=I. For convenience, we denote Fi ≡K?λiM. Substituting z iinto Eq.(1), and differentiating it yields Fi zi′=?Fi′zi i = 1,2,Lm (3) Premultiplying by X Tand noting that X T Fi=0, we get [ X T (K′?λiM′)X?λi′I]γi=0 i = 1,2,Lm (4) where γi is the ith column of Γ. In this paper, we assume that the derivatives of the repeated eigenvalues λi′and the coefficient vector γi can be obtained uniquely from the subeigenvalue problem described by Eq.(4), then we get z i. Next, we solve zi′with direct method. Letwhere v i is a particular solution and the coefficient matrix C is given by the reference [10,11]. Substituting Eq.(5) into Eq.(3) yields Fi vi= ?Fi′zi i = 1,2,Lm (6) Here we will give two new methods to efficiently determine the particular solution v i. 1 Method of adding the borders The objective of this method is to extend the governing equation by using the orthogonal condition, and eliminate the singularity of the coefficient matrix. Premultiplying by z Tjin the Eq.(6) and using z Tj Fi=0, we get z Tj Fi′z i=0 i , j= 1,2,Lm (7) Then the part v iof solution to the following equation Fi vi+ μ1 Mz1+μ2Mz2+L+μmMzm=?Fi′zi i = 1,2,Lm (8) in unknowns v i and μi (i =1,2,L,m)satisfy Eq.(6). Denoting the rest n ? m unknown eigenvectors of the original eigenvalue problem as z m +1 , zm+2,L,zn, and they span an n ? m dimensional subspace V . From Eq.(5), letting vectors v ibelong to V , and we get... |
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