Several Inverse Problems Of Vibrating Rod

The inverse problem has been used in many areas, for example particle physics, control design,molecular spectroscopy, structure analysis and so on. In this paper, the inverse problem of vibrating rod is considered as the inverse eigenvalue problem that construct a matrix according some known eigenvalues /eigenvectors.Several inverse problems are discussed as follows:Problemâ… Given the matrix of quality M = diag(m₁,m₂,L,m_n), m_i >0,the eigenpair(λ, X^*)of the rod fixed at one end and free at the other,and another eigenpair (μ,Y^*)(λ>μ)of the rod fixed at two ends.Find the matrix of stiffness J fixed at one end and free at the other,whereProblemâ…¡Given part of physical parameters of the discrete rod k_i , k_i+1,L,k_j, (1 1 = ( x₁, x₂,L ,x_i-1)T ,Y₁ = ( y₁, y₂,L ,y_i-1)T,X₃ = (x_j , x_j+1,L,x_n)T,Y₃ = (y_j , y_j+1,L,y_n)T and M₂ = diag(m_i , m_i+1,L ,m_j-1).Find X₂ =(x_i , x_i+1,L ,x_j-1)T, Y₂ = (y_i , y_i+1,L ,y_j-1)T, M₁ = diag(m₁, m₂,L ,m(i-1)),M₃ = diag(m_j , m_j+1,L,m_n),k₁ , k₂,L ,k(i-1), k_j+1 , k_j+2,L,k_n,such that KX =λMX,KY =μMY with X = (X₁,X₂,X₃)T ,Y=(Y₁,Y₂,Y₃)T,whereProblemâ…¢Given part of physical parameters of the discrete rod k_j+1,k_j+2,L,k_n, m_j+1, m_j+2,L,m_n,λ,μ(λ<μ),X₁ = ( x₁, x₂,L ,x_k)T and Y₁ = ( y₁, y₂,L ,y_k)T .FindX₂ =(x_k+1, x_k+2,L ,x_n)T,Y₂ = (y_k+1 , y_k+2,L ,y_n)T,k₁ , k₂,L ,k_j,m₁ , m₂,L ,m_j suchthat KX =λMX, KY =μMY with X=(?),Y=(?) and makeλbe the biggest eigenvalue andμbe the smallest eigenvalue,whereProblemâ…£Given part of physical parameters of the discrete rod k_j+1,k_j+2,L,k_n,m_j+1, m_j+2,L,m_n,λ,μ(λ<μ),X₁ = ( x₁, x₂,L ,x_k)T and Y₁ = ( y₁, y₂,L ,y_k)T .FindX₂ =(x_k+1, x_k+2,L ,x_n)T,Y₂ = (y_k+1 , y_k+2,L ,y_n)T,k₁ , k₂,L ,k_j,m₁ , m₂,L ,m_j suchthat KX =λMX, KY =μMY with X=(?),Y=(?) and makeλbe the smallest eigenvalue andμbe the second smallest eigenvalue,whereProblemâ…¤Given part of physical parameters of the discrete rod k_j+1,k_j+2,L,k_n,m_j+1, m_j+2,L,m_n,λ,μ(λ<μ),X₁ = ( x₁, x₂,L ,x_k)T and Y₁ = ( y₁, y₂,L ,y_k)T .FindX₂ =(x_k+1, x_k+2,L ,x_n)T,Y₂ = (y_k+1 , y_k+2,L ,y_n)T,k₁ , k₂,L ,k_j,m₁ , m₂,L ,m_j suchthat KX =λMX, KY =μMY with X=(?),Y=(?) and makeλbe the biggest eigenvalue andμbe the second biggest eigenvalue,where Problemâ…¥Given part of physical parameters of the discrete rod k_j+1,k_j+2,L,k_n,m_j+1, m_j+2,L,m_n,λ,μ(λ<μ),X₁ = ( x₁, x₂,L ,x_k)T and Y₁ = ( y₁, y₂,L ,y_k)T .FindX₂ =(x_k+1, x_k+2,L ,x_n)T,Y₂ = (y_k+1 , y_k+2,L ,y_n)T,k₁ , k₂,L ,k_j,m₁ , m₂,L ,m_j suchthat KX =λMX, KY =μMY with X=(?),Y=(?) and makeλbe the biggesteigenvalue andμbe the qth eigenvalue,whereProblemâ…¦Given part of physical parameters of the discrete rod k_j+1,k_j+2,L,k_n,m_j+1, m_j+2,L,m_n,λ,μ(λ<μ),X₁ = ( x₁, x₂,L ,x_k)T and Y₁ = ( y₁, y₂,L ,y_k)T .FindX₂ =(x_k+1, x_k+2,L ,x_n)T,Y₂ = (y_k+1 , y_k+2,L ,y_n)T,k₁ , k₂,L ,k_j,m₁ , m₂,L ,m_j suchthat KX =λMX, KY =μMY with X=(?),Y=(?) and makeλbe the q theigenvalue andμbe the p th eigenvalue,whereThe main results are as follows:1,The sufficent and necessary conditions for the unique solution of Problemâ… are given by using the Jacobi matrix to replace the matrix of stiffness and changing generalized eigenvalue to standard eigenvalue. 2,The solution of Problemâ…¡is given by approximating the system of vibration rod into the system of spring-particle and using matrix block.3,In chapter 3, The sufficent and necessary conditions for the unique solution from Problemâ…¢to Problemâ…¦are given by the discussion of the inverse problem for the orderly defective eigenpairs.