Detection of Cracks in Beams Using Treed Gaussian Processes

For some years, there has been interest in locating cracks in beams by detecting singularities in mode shape curvatures. Most of the work in the past has depended on the estimation of spatial derivatives (smoothed or otherwise) of the experimentally measured mode shape. This problem is made difficult by the fact that numerical differentiation is notorious for amplifying measurement noise, coupled with that fact that very precise estimates of mode shapes are difficult to obtain. One recent approach, introduced by one of the authors, circumvented the noise issue via a method which did not need numerical differentiation. Briefly, the method applied a Gaussian process regression to the data, using a covariance function that could switch between spatial regions; the switch point—which indicated the crack position—could be determined by a maximum likelihood algorithm. The object of the current paper is to present an alternative approach which uses Treed Gaussian Processes (TGPs). The idea is that separate Gaussian Processes, with standard covariance functions, can be fitted over different spatial regions of the beam, with any switching points learned as part of a decision tree structure. The paper also revisits the idea of using differentiated mode shapes, on the premise that the Gaussian process can ‘see through’ the noise created and perceive the underlying structure.