Analytical Relation Between Natural Frequency and Spectral Entropy in Information Theory of Single Degree of Freedom Systems

The eighth (and last) axiom of Structural Health Monitoring (SHM) states that damage increases the complexity of a system. After its introduction, several methods to measure complexity have been explored in the scientific literature. Among these, entropy measures demonstrate to be very straightforward to apply, with a relatively simple concept behind them. Thus, based on this feature, researchers proposed damage identification methodology in several branches of research, such as mechanical engineering, civil engineering, etc. In this work, the authors present an analytical study on the dependency of spectral entropy measures from the modal characteristics of a structural system, showing that an analytical relation exists for what concerns a Single Degree of Freedom (SDoF) system. A comparison between spectral entropy of different physical quantities (displacements, velocity, accelerations, etc.) is discussed for their use in SHM. Finally, different contributions of variation in entropy (i.e., due to external sources and structural properties) are analysed to better understand the causes of these perturbations. The work is of particular interest for cultural heritage structures since many of these are characterised by the presence of protruding elements, most of the time assimilable to SDoF systems (e.g., lanterns, belfries, pinnacles, etc.), when their dynamic is analysed relatively to their base movement.