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Analytical Verification for Vibration Analysis Technique used in Determination of Cracking in Cantilever Beams

nondestructive experimental technique for determination of structural discontinuities in cantilever beams is presented. The proposed technique is capable of detecting the location of cracks and other possible structural discontinuities on a beam using its first two natural frequencies measured by an accelerometer on a movable mass on the beam at several locations along the beam. To verify the validity of the proposed technique, the vibration response of a cracked cantilever beam with a stationary roving mass was investigated. The beam was modeled as an Euler-Bernoulli beam with a rectangular cross-section. The axial and transverse deformations of the cracked beam were coupled through a stiffness matrix determined using fracture mechanics principles. The developed model was used to determine analytical solutions for the variation of natural frequencies and mode shapes of a cracked cantilever beam versus the position of the roving mass. The analysis indicates that the variation of the natural frequencies versus position of the roving mass can drastically change when the roving mass is close to the position of a discontinuity. Moreover, the effects of the location and depth of the crack, as well as the location and weight of the roving mass, on the natural frequencies and mode shapes of the beam were investigated. The analytical results show that the coupling between the axial and transverse vibrations for moderate values of crack depth or roving mass is weak. Increasing the crack depth, the mass and the rotary inertia increases the coupling effect.

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