Damage localization in ambient vibration by constructing proportional flexibility matrix

Damage localization approaches based on changes of flexibilities constitute an important technique for damage detection. However, the unavailability of flexibility matrix with output-only data makes flexibility-based approaches not really applicable in the very important cases of ambient vibrations. An algorithm is presented to construct a proportional flexibility matrix (PFM) from a set of arbitrarily scaled tested modal shapes and modal frequencies. The constructed PFM is just within a scalar multiplier to the real flexibility matrix, and the scalar multiplier is theoretically the first modal mass, which is undetermined before the mode is properly scaled. Instead of real flexibilities, the PFMs are incorporated into the damage locating vectors (DLV) method for damage localizations in ambient vibrations. PFMs for the pre- and post-damaged structure need to be comparable before being integrated into the DLV procedure. This requirement is guaranteed when there is at least one reference degree with unchanged mass after damage. Two numerical examples show that a small number of measured modes can produce PFMs with sufficient accuracy to correctly locate the damages by the DLV method from output-only data.