Waveform fractal dimension for mode shape-based damage identification of beam-type structures

Mode shape-based structural damage identification has been a research focus during the last couple of decades. Most of the existing methods need a numerical or measured baseline mode shape serving as a reference to identify damage, and this requirement extremely limits the practicability of the methods. Recently, waveform fractal dimension such as Katz’s waveform fractal dimension (KWD) has been explored and applied to mode shape for crack identification without a baseline requirement. In this study, different from the popular KWD, an approximate waveform capacity dimension (AWCD) is formulated first, from which an AWCD-based modal abnormality algorithm (AWCD-MAA) is systematically established. Then, the basic characteristics of AWCD-MAA on abnormality detection of mode shapes, e.g., crack localization, crack quantification, noise immunity, etc., are investigated based on an analytical crack model of cantilever beams using linear elastic fracture mechanics. In particular, from the perspective of isomorphism, a mathematical solution on the use of applying waveform fractal dimension to higher mode shapes for crack identification is originally proposed, from which the inherent deficiency of waveform fractal dimension to identify crack when implemented to higher mode shapes is overcome. The applicability and effectiveness of the AWCD-MAA is validated by an experimental program on damage identification of a cracked composite cantilever beam using smart piezoelectric sensors/actuators (i.e., Piezoelectric lead–zirconate–titanate (PZT) and polyvinylidene fluoride (PVDF)). The proposed AWCD-MAA provides a novel, viable method for crack identification of beam-type structures without baseline requirement, and it largely expands the scope of fractal in structural health monitoring applications.