# Rayleigh’s quotient for multiple cracked beam and application

## Rayleigh’s quotient for Euler-Bernoulli multiple cracked beam with different boundary conditions has been derived from the governed equation of free vibration. An appropriate choosing of approximate shape function in terms of mode shape of uncracked beam and specific functions satisfying conditions at cracks and boundaries leads to an explicit expression of natural frequencies through crack parameters that can simplify not only the analysis of natural frequencies of cracked beam but also the crack detection problem. | Vietnam Journal of Mechanics, VAST, Vol. 33, No. 1 (2011), pp. 1 – 12 RAYLEIGH’S QUOTIENT FOR MULTIPLE CRACKED BEAM AND APPLICATION Nguyen Tien Khiem, Tran Thanh Hai Institute of Mechanics, VAST Abstract. Rayleigh’s quotient for Euler-Bernoulli multiple cracked beam with different boundary conditions has been derived from the governed equation of free vibration. An appropriate choosing of approximate shape function in terms of mode shape of uncracked beam and specific functions satisfying conditions at cracks and boundaries leads to an explicit expression of natural frequencies through crack parameters that can simplify not only the analysis of natural frequencies of cracked beam but also the crack detection problem. Numerical analysis of natural frequencies of the cracked beam by using the obtained expression in comparison with the well known methods such as the characteristic equation and finite element method shows their good agreement. The analytical expression of natural frequencies applied to the crack detection problem allows the result of detection to be improved. Key words: Natural frequencies, multiple cracked beam, Rayleigh’s quotient. 1. INTRODUCTION Recently, a significant effort of researchers as well as engineers has been devoted to solve forward and inverse problems in vibration of cracked beam. The most important tool for solving the problems is so-called the characteristic equation relating implicitly natural frequencies with crack parameters. It has been shown in several studies, for instance [1, 2] that for a beam with single crack of extent γ at position xc the characteristic equation has the form F0 (ω) + γF1 (ω, xc) = 0 (1) where F0 and F1 are functions given by boundary conditions of the beam. If the crack extent γ is small, from the equation an explicit expression of natural frequencies through crack parameters can be derived as ∆ωk = ωk − ωk0 = γg(ωk0, xc ) (2) with ωk , ωk0 are frequencies of cracked and uncracked beam respectively.

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