Model Updating Using an Incomplete Set of Experimental Modes

Effects of Mode Incompleteness on the Quality of Various Model-Updating Algorithms

“…For N C *2, the perturbation approach returns updated sti!nesses that become increasingly worst as N C increases. Because updating the sti!ness matrix of Figure 1 entails the solution of a least-square problem of size N C ;(3N!2) (see reference [3] for detailed discussion), to render the problem overdetermined requires N C *(3N!2 measured modes. Thus, for N"25, at least nine measured modes are needed to update the sti!nesses to ensure su$cient accuracy based on the heuristics criterion.…”

“…From numerical experiments it was observed that more accurate solutions are obtained when the least-squares system becomes overdetermined. Because updating the mass matrix of Figure 1 involves solving a least-squares problem of size N C ;N (see reference [3] for detailed discussion), to render the problem overdetermined requires N C *(N measured modes. Thus, for N"25, at least "ve measured modes are needed to update the masses to ensure su$cient accuracy.…”

“…The Lagrange multipliers approaches [4,5], the perturbation scheme [6] and the proposed model updating algorithm [3] are applied to the simple system of Figure 1, whose mass matrix is diagonal and whose sti!ness matrix is symmetric and tri-diagonal. The analytical masses and sti!nesses are 2)000 kg and 5)000 N/m, respectively.…”

“…Detailed discussion of every approach is beyond the scope of this note, and interested readers are referred to the survey papers by Mottershead and Friswell [1] and Imregun and Visser [2]. In a recent paper [3], new approaches are developed that use measured natural frequencies and mode shapes to update the analytical mass and sti!ness matrices of a structure. Using the measured modes of vibration of a mass-modi"ed structure and the initial modal survey, the mass matrix of the system can be corrected, after which the sti!ness matrix can be updated by requiring it to satisfy the generalized eigenvalue problem associated with the free vibration of the structure.…”

“…In this technical note, detailed numerical experiments will be performed to compare the results of the mass and sti!ness updating algorithms introduced in reference [3] against the Lagrange multipliers updating formalisms [4,5] and the perturbation updating approach [6]. In addition, error parameters will be introduced to gauge the quality of the updates.…”