A finite element based method for estimating natural frequencies of locally damaged homogeneous beams

Vibration-based damage detection from frequency changes requires the calculation of natural frequencies from assumed damage scenarios and conduct a comparison to the actual frequency of the structure. Analytical solutions in obtaining the natural frequency of homogeneous beams are currently limited to beams with uniform cross-sectional area. Changes in cross-sectional area might occur due to damage within the length of the beam. Finite element modeling and analysis is required in these instances, but may not be efficient in terms of computational effort. For the assumed damaged scenarios, there are unlimited number of possible damage combinations for which the natural frequency will be obtained. There is a need for an analytical alternative as a substitute to the finite element method to calculate these frequencies. This study presents an analytical method to estimate the natural frequencies of locally damaged homogeneous beams based on statistical data obtained from finite element modeling and analysis. The method proposes a multiplier function in terms of the extent of area reduction, length, and location of damage in order to estimate the damaged frequency. The function was derived using curve-fitting techniques of data obtained from finite element modeling and analysis of typical beams with assumed damage cases. Examples show that the method is a good alternative to finite element analysis in estimating the natural frequencies of locally damaged homogeneous beams. The method can be used for vibration-based structural health monitoring to predict the damage state of beams given the change in frequency without the computational burden of finite element modeling and analysis.