An inverse problem for undamped gyroscopic systems

Linear undamped gyroscopic systems are defined by three real matrices, M>0,K>0, and G(GT=−G); the mass, stiffness, and gyroscopic matrices, respectively. In this paper an inverse problem is considered: given complete information about eigenvalues and eigenvectors, Λ=diag{λ1,λ2,…,λ2n−1,λ2n}∈C2n×2n and X=[x1,x2,…,x2n−1,x2n]∈Cn×2n, where the diagonal elements of ΛΛ are all purely imaginary, XX is of full row rank nn, and both ΛΛ and XX are closed under complex conjugation in the sense that λ2j=λ̄2j−1∈C,x2j=x̄2j−1∈Cn for j=1,…,nj=1,…,n, find M,KM,K and GG such that MXΛ2+GXΛ+KX=0MXΛ2+GXΛ+KX=0. The solvability condition for the inverse problem and a solution to the problem are presented, and the results of the inverse problem are applied to develop a method for model updating.