An explicit expression of natural frequencies through crack parameters is derived for multiple cracked beams with simply supported boundaries using the Rayleigh quotient. The obtained expression provides not only a simple tool for calculating natural frequencies of multiple cracked beams, but also allows employing the so-called crack scanning method for detecting multiple cracks in simply supported beams from measured natural frequencies. A numerical example demonstrates that the crack scanning method, in combination with the Rayleigh quotient, enables consistent identification of cracks with 1% relative depth.
Abstract. A new approach is proposed for calculating natural frequencies and crack detection in a stepped cantilever beam with arbitrary number of cracks. This is based an explicit expression of the natural frequencies in term of crack parameter derived in the form similar to the so-called Rayleigh quotient for vibrating beam. The obtained simple relationship between natural frequencies and crack parameters enables not only accurate calculating the natural frequencies but also to develop an efficient procedure for detecting multiple cracks from given natural frequencies. The proposed technique called crack scanning method is illustrated and validated by numerical results.
A new approach is proposed for calculating natural frequencies and crack detection in a stepped cantilever beam with arbitrary number of cracks. This is based an explicit expression of the natural frequencies in term of crack parameter derived in the form similar to the so-called Rayleigh quotient for vibrating beam. The obtained simple relationship between natural frequencies and crack parameters enables not only accurate calculating the natural frequencies but also to develop an efficient procedure for detecting multiple cracks from given natural frequencies. The proposed technique called crack scanning method is illustrated and validated by numerical results.
This paper analyses free vibrations of framed nanostructures made of Functionally Graded Material (FGM) based on the Nonlocal Elastic Theory (NET) and the Dynamic Stiffness Method (DSM). FGM characteristics vary nonlinearly throughout the height of the beam element. The NET considers the nonlocal parameter that perfectly captured the size effect of nanostructures. However, the NET makes nonlocal paradoxes in the bending and vibration behaviour of framed nanostructures with the free ends. To overcome these phenomena, the nanostructure is modelled according to the Euler–Bernoulli beam theory and the variational-consistent nonlocal boundary conditions have been derived. The exact solutions of differential equations of motion and variational-consistent nonlocal boundary conditions are found using the DSM. The influences of the nonlocal, material, geometry parameters and Pasternak’s foundation on the free vibration are then analyzed. It is shown that the study can be applied to other FGMs as well as more complicated framed structures.
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