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 L 2 -norm on the boundary of the body. The proposed technique is verified, using numerical simulation of measured responses, with good agreement.

Dynamic Traction Vector Field Estimation in a Structure Using Hybrid Strain Analysis

“…Together with the method presented in reference [11] the method proposed here may be used as a tool for calculating the dynamic boundary traction vector acting on, for example, interfaces between structural parts. This is a good starting point for deriving physical models describing damping at joints and interfaces between di!erent parts.…”

“…The test structure is the same as in [11], i.e., an aluminium plate having dimensions 0)520;0)300;0)0042 m (x;y;z). The plate is attached to two supports consisting of Plexiglas (PMMA) and having dimensions 0)01;0)300;0)01 m (x;y;z), as depicted in and "xed to the ground along the other edge (z"!0)01 m).…”

“…In several applications, when solving the inverse problem, it is often reported, (see for example references [4}10]) that the problem equations become ill-conditioned. In Sehlstedt [11], a well-conditioned technique for obtaining the spatial and frequency-dependent boundary traction vector is proposed. However, when using the method proposed here for obtaining the boundary traction vector, for example, at interfaces between structural parts, there is no problem with ill-conditioning; moreover, the dynamic stress tensor and needed traction vectors are obtained, not just on the boundary but, in the whole structure.…”

Boundary traction vector estimation in a vibrating structure using series expansion of the stress tensor

“…In Sehlstedt [16], it was shown that the ill-conditioned system of equations, that often arises when solving the inverse problem, using modal based techniques, is caused by the well-known fact that modes, orthogonal in a volume, are in general not orthogonal in a subspace of the volume, such as, e.g., a boundary. One way to overcome the ill-conditioning, which was proposed in [16], is to choose a subset of the set of available modes which are orthogonal over the boundary in question.…”

“…One way to overcome the ill-conditioning, which was proposed in [16], is to choose a subset of the set of available modes which are orthogonal over the boundary in question.…”