Identification of multiple cracks in beams under bending

The identification of a transverse crack on a beam is the subject of many investigators. Identifying the crack means to find its position and depth. In many cases there are more than one cracks on a beam. Then the solutions, or the combinations of parameters characterising the cracks are more and the problem becomes more complicated particularly when the crack must be identified using one more parameter, the relative each other angular position.In the present paper the dynamic behaviour of a cracked beam with two transverse surface cracks is studied. Each crack is characterised by its depth, position and relative angle. Both cracks are considered to lie in arbitrary angular positions with respect to the longitudinal axis of the beam and at any distance from the left end. A local compliance matrix of two degrees of freedom, bending in the horizontal and the vertical planes is used to model the rotating transverse crack in the shaft and is calculated based on the available expressions of the stress intensity factors and the associated expressions for the strain energy release rates. The compliance matrix is calculated for the first time at any angle of rotation. Thus, the compliance is given as a function of both the crack depth and the angular location. These expressions are usable, due to the stress intensity function limitations, only for limited regions around the zero angular position of the crack and not for every crack angle. For these cases, B-spline curves are used to interpolate the known points and a function in analytical form is given for every crack depth and angle. It is well known that when a crack exists in a structure, such as a beam, then the natural frequency of vibration decreases. This reduction is studied here for six independent parameters namely the depth, the location, and the rotational angle of each crack. By keeping these six parameters constant, the first three flexural eigenmodes can be computed and plotted.Due to its sensitivity in slope or displacement changes the theory of wavelets is used here to identify the locations of the cracks reducing thus the number of independent parameters. As it is well known the existence of a crack on a beam in bending, creates in the elastic line of the beam a slope discontinuity analog generally to the crack depth and additionally here to the angular position. The wavelet transformation of a vibration mode or of the vibration response of the structure under some circumstances could be used to locate the cracks. If the positions are known, then the depths and the respective angles can be determined. Here the diagrams of the first three eigenvalues versus both the crack depth and the rotational angle, are used to identify the remaining unknown parameters for both cracks.