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The paper presents a method which utilizes substructure normal modes to predict the vibration properties of a cantilever beam with a breathing transverse crack. The two segments of the cantilever beam, separated by the crack, are related to one another by time varying connection matrices representing the interaction forces. The connection matrices are expanded in a Fourier series leading to a classical eigenvalue problem. Subsequently, the initial formulation is extended to avoid interference of the crack interfaces with a time domain formulation. The Lagrange multipliers, used to enforce the exact continuity constraints when the crack is closed, produce the interfaces forces needed for the modelling of interface dry friction.

This paper examines critically the practice of using first-order sensitivities for extrapolation to predict the effect of structural changes. A quality criterion, based on the Cauchy ratio test for convergence of infinite series, is suggested to examine the liability of systems to perform poorly under extrapolation. The efficacy of this approach is demonstrated oh a simple numerical example.