Analytical predictions of natural frequencies of cracked simply supported beams with a stationary roving mass

Natural frequencies of a damaged simply supported beam with a stationary roving mass are studied theoretically. The transverse deflection of the cracked beam is constructed by adding a polynomial function, which represents the effects of a crack, to the polynomial function which represents the response of the intact beam [J. Fernández-Sáez, L. Rubio, C. Navarro, Approximate calculation of the fundamental frequencies for bending vibrations of cracked beams, Journal of Sound and Vibration 225 (1999) 345–352]. By means of the boundary and kinematics conditions, approximate closed-form analytical expressions are derived for the natural frequencies of an arbitrary mode of transverse vibration of a cracked simply supported beam with a roving mass using the Rayleigh's method. The natural frequencies change due to the roving of the mass along the cracked beam. Therefore the roving mass can provide additional spatial information for damage detection of the beam. That is, the roving mass can be used to probe the dynamic characteristics of the beam by roving the mass from one end of the beam to the other. The presence of a crack causes the local stiffness of the beam to decrease which, in turn, causes a marked decrease in natural frequency of the beam when the roving mass is located in the vicinity of the crack. The magnitude of the roving mass used varied between 0% and 50% of the mass of the beam. The predicted frequencies are shown to compare very well with those obtained using the finite element method and the experimental results. Finally, the effects of crack depth, crack location and roving mass on the natural frequency of the beam are investigated. It is shown that the natural frequencies of the cracked beam decrease as the crack depth increases and as the roving mass is traversed closer to the crack location.