A hybrid damage detection method using dynamic-reduction transformation matrix and modal force error

• An innovative damage detection method using incomplete modal data is proposed.
• The proposed method just uses translational DOFs.
• The method applies just one mode shape and its corresponding frequency.
• Different examples of 2D and 3D structural model have been studied.
• Results show the good ability of the method in detecting damages in structures.

Condensation methods are commonly used algorithms to fast computation of some low frequencies and corresponding mode shapes of structures by reducing a full rank domain to a restricted one. Hence, these methods are often termed as model reduction techniques. On the other hand, these methods have been used to develop damage detection techniques by several researchers. In this study, a new damage detection method is proposed which uses only one mode shape and its corresponding eigen-value of a structure to conduct damage detection. However, since the number of measured degrees of freedom (DOFs) is most of the time restricted in practice, an iterative hybrid method is developed using dynamic-condensation scheme to carry out damage detection in structures using incomplete modal data. The main characteristic of the proposed method is that measuring the rotational DOFs, which is most of the time very expensive and inaccurate, is not needed. To examine the capability of the proposed method, several examples are studied. In the illustrative example of a cantilever beam, it is assumed that just translational DOFs corresponding to the first mode are available. The capability of the proposed method in damage detection in the cantilever beam and model updating is completely examined in this example. Also a modal assurance criterion is applied to evaluate the robustness of the method in updating the mode shapes. Finally, the ability of the proposed method in damage detection in 2D and 3D frame structures is assessed. Interestingly, the results in most of the cases are exact even with the lack of measured translational DOFs and also demonstrate that the proposed method is completely successful in conducing damage detection in structures.