Inverse Vibration Problems Of Spring-mass Systems

The inverse eigenvalue problem concerns the construction of matrices under certain constraints which is from known complete or partial information of eigenvalues and eigenvectors,its purpose is to construct the matrix with given spectral data and specific structure matrices.Inverse eigenvalue problems have important applications in many fields,and they have becomed one of the most flourish research topics in numerical algebra.Based on the inverse eigenvalue problems of Jacobi matrices,this thesis studied the applications of its theories and algorithms in structure dynamics.we proposed and solved the inverse problems of undamped spring-mass vibration successfully.In this paper,we mainly consider spring-mass vibration inverse problems as follows:??P Given positive real numbers ?,?1,?2,c,and non-zero vectors x?Rn,Y1?Rp,Y1?Rn-p.Find stiffness matrix K and mass matrix M,such that(Kn-?Mn)x=0,(Kp-?1Mp)Y1=0,(Kp+1,n-?2Mp+1,n)Y2=0,where Kp,Mp are the p order leading principal submatrices of K,M,Kp+1,n,Mp+1,n are the lower right principal submatrices of K,M,positive numbers ?,?1,?2 are the max generalized eigenvalues of(Kn,Mn),(Kp,Mp),(Kp+1,n,Mp+1,n),and the sum of the elastic coefficients is c.In this paper,we study the problem P in three states:fixed-fixed,fixed-free and free-free.The necessary conditions under which the problem is solvable are obtained respectively,the necessary and sufficient conditions under which the problem is uniquely solvable are also obtained respectively.The corresponding numerical algorithms are given to calculate the solution of problem P in three states,numerical expeiments show that the algorithms are feasible and effective.The research in this paper enriches the research results of the spring-mass vibration inverse problems and Jacobi matrices inverse problems,and has certain theoretical and practical significance.