Analytical estimation of natural frequencies and mode shapes of a beam having two cracks

摘要

In this paper, an analytical estimation based on the Rayleigh's method is extended for a beam having one or two cracks to find natural frequencies and mode shapes in order to overcome weakness of solving eigenvalue problem. Weakness of solving eigenvalue problem to obtain exact natural frequencies and mode shapes is that, an algebraic equation must be solved numerically and then coefficients of trigonometric and hyperbolic terms in mode shapes will be found using matrices obtained from compatibility conditions at each point of cracks and boundary conditions. So this method does not show effects of crack size and location in the explicit form. The advantage of analytical estimation based on the Rayleigh's method over the eigen analysis method is that, the Rayleigh method obtains explicit expression for both natural frequencies and mode shapes in which effect of parameters such as crack size and location on natural frequencies and mode shapes can be investigated analytically. In the analytical estimation method, mode shapes of cracked beam are constructed by adding a polynomial function which shows effect of cracks, to mode shape of undamaged beam. The coefficients of the polynomial function are obtained by using boundary conditions and compatibility equations at the point of cracks. However in this paper it is shown that the accuracy of this estimation decreases when crack depth increases. Therefore, this paper also investigates on the upper limit of crack depth in which natural frequencies have error less than 5% and mode shapes have error less than 7% obtained by analytical estimation in compare to the exact solution. In the literature, it is shown that, analytical estimation based on the Rayleigh's method can predict only the first natural frequency of a simply supported cracked beam with sufficient accuracy, but no more investigation reported for finding the reliable range of crack depth natural mode shapes or higher natural frequencies. So, in this study, upper limit of crack depth has been found for both cases of beam with one or two cracks for first three natural frequencies and mode shapes.