A well-conditioned technique for solving the inverse problem of boundary traction estimation for a constrained vibrating structure

Abstract

 Estimation of the frequency and spatial dependent boundary traction vector from measured vibration responses in a vibrating structure is addressed. This problem, also referred to as the inverse problem, may in some circumstances be ill-conditioned. Here a technique to overcome the ill-conditioning is proposed. A subset of a set of available eigenmodes is chosen such that the problem becomes well-conditioned enough. It is shown that the ill-conditioning originates from the fact that not all eigenmodes are orthogonal over the surface where the traction vector is sought. Consequently, by choosing a set of eigenmodes orthogonal over the surface of interest, the problem becomes well-conditioned. The calculated traction vector is shown to converge to the true one in the sense of a L₂-norm on the boundary of the body. The proposed technique is verified, using numerical simulation of measured responses, with good agreement.