Analytical approach to the determination of dynamic characteristics of a cracked beam

A wide spectrum of investigations devoted to the determination of natural frequencies and mode shapes of beams with an open crack are presented in the literature. However, as is well known, an open crack is a fairly crude model of a fatigue crack. The study of the dynamic characteristics of a beam with a closing crack is the main aim of the present paper. The analytical approach which enables one to determine the effect of crack parameters (crack magnitude and location) on different dynamic characteristics of a cantilever Bernoulli} Euler beam with a closing edge transverse crack is performed. Natural frequencies, mode shapes and distortion of time functions describing wave shapes of displacement, acceleration and strain of different cross-sections of a beam are considered as dynamic characteristics to be investigated. The general solution of the problem is derived from the synthesis of particular solutions obtained for the crack-free beam and for the beam with an open crack. The possibility of origination of several modes of vibrations during crack opening is taken into account as well as the peculiarity of strain distribution in the vicinity of a crack. It is shown that analytically predicted relationships between the dynamic characteristics of a cracked beam and crack parameters are well-founded. The analytical approach makes it possible to solve the inverse problem of damage diagnostics with sufficient accuracy for practical purposes.