Rayleigh's quotient for multiple cracked beam and application

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.