Analysis of forced bilinear oscillators and the application to cracked beam dynamics

TLDR

A closed‑form solution using two square wave functions to model stiffness change is proposed for bilinear oscillators under low‑frequency excitation. The method extends to bilinear forcing functions common in damaged structures, yielding spectral patterns and harmonic magnitudes for a damaged rectangular beam based on crack size and location. Numerical simulations confirm that the solution accurately predicts spectral patterns and harmonic magnitudes for a simply supported beam with a fatigue crack, enabling efficient, computation‑light online diagnostics of damaged structures.

Abstract

A new closed-form solution based on the use of two square wave functions to model stiffness change is proposed for bilinear oscillators under low-frequency excitation. The solution procedure is extended to cases with bilinear forcing function, which usually occur in the dynamics of damaged structures. The proposed solution provides the spectral pattern and the magnitude of each harmonic component for a damaged rectangular beam according to the size and location of the crack. The predicted spectral pattern and the magnitude of each harmonic component for a simply supported beam containing a fatigue crack are validated by numerical simulation; experimental work has not been done. The results show that the proposed solution may be used to predict efficiently spectral patterns of damaged structures, while avoiding lengthy computation. Such a procedure may be developed into an attractive on-line diagnostic tool.