Developing Variational Vibration Models of Damaged Beams: Toward Intelligent Failure Detection

Inaccuracies in the modeling assumptions about the distributional characteristics of the monitored signatures have been shown to cause frequent false positives in vehicle monitoring systems for high-risk aerospace applications. To enable the development of robust fault detection methods, this work explores the deterministic as well as variational characteristics of failure signatures. Specifically, we explore the combined impact of crack damage and manufacturing variation on the vibrational characteristics of beams. The transverse vibration and associated eigenfrequencies of the beams are considered. Two different approaches are used to model beam vibrations with and without crack damage. The first aproach uses separation of variables. This method enables the direct integration of the resulting ordinary differential equation and determination of different eigenfrequencies and eigenmodes. This approach uses a spring to model localized reduction in stiffness as a result of the cracked beam. The second approach uses a finite difference approach to enable the inclusion of both cracks and manufacturing variation to be considered. The crack model used with the finite difference approach is based on a localized decrease in the Young’s modulus. Using both beam and crack models, Monte Carlo simulations are used to explore the impacts of manufacturing variation on damaged and undamaged beams. Derivations are presented for both models. Conclusions are presented on the choice of modeling techniques to define crack damage, and its impact on the monitored signal, followed by conclusions about the distributional characteristics of the monitored signatures when exposed to random manufacturing variations.