New model updating method for damped structural systems

In this paper, the following two problems are considered:Problem   I Given a full column rank matrix X∈Rn×k, a diagonal matrix Λ∈Rk×k(k≤n) and matrices Ma∈Rn×n,C0,K0∈Rr×r, find n×nn×n matrices C,K such that MaXΛ2+CXΛ+KX=0, s. t.C([1,r])=C0,K([1,r])=K0, where C([1,r])C([1,r]) and K([1,r])K([1,r]) are, respectively, the r×rr×r leading principal submatrices of CC and KK.Problem   II Given n×nn×n matrices Ca,Ka with Ca([1,r])=C0,Ka([1,r])=K0, find (Cˆ,Kˆ)∈SE, such that ‖Ca−Cˆ‖2+‖Ka−Kˆ‖2=inf(C,M)∈SE(‖Ca−C‖2+‖Ka−K‖2), where SESE is the solution set of Problem I.By applying the theory and methods of the algebraic inverse eigenvalue problems, the solvability condition and the general solution to Problem I are derived. The expression of the solution to Problem II is presented.