Construction of orthogonal projector for the damage identification by measured substructural flexibility

• An orthogonal projector is constructed for the free–free structure.
• It is accurate to extract the free–free stiffness and flexibility.
• It is helpful for the flexibility-based model updating.

The substructuring methods have been popularly used in model updating, system identification and damage assessment. In the substructuring methods, the global structure is divided into free–free substructures. The independent substructures move freely, and their stiffness matrix is singular and rank-deficient. The flexibility matrix of the free–free substructure, which is associated with the inverse of the stiffness matrix, is not easy to be determined. This study expands on the previous research of the substructuring methods by taking a deeper look at the analysis of a free–free substructure. An orthogonal projector is formulated to add/remove the rigid body components from the generalized stiffness and flexibility matrices of a free–free substructure, and thus make the substructural flexibility useful to model updating or damage identification. The orthogonal projector is derived both for the full and partial measured flexibility, and it can remove all rigid body components regardless its participation factor. The accuracy of the proposed method in extraction of the free–free flexibility and in damage identification is verified by an experimental beam. The properties addressed in this paper are not limited to be used for the analysis of a free–free substructure in many substructuring methods, and they are promising to be generalized to a range of analysis relevant to a free–free structure.