Development of a Novel Algorithm for a Crack Detection, Localization, and Sizing in a Beam Based on Forced Response Measurements

Abstract

The present work aims at the development of a method for the crack detection, localization and sizing in a beam based on the transverse force and response signals. The Timoshenko beam theory is applied for transverse vibrations of the beam model. The finite element method is used for the cracked beam forced vibration analysis. An open transverse surface crack is considered for the crack model, which contains standard five flexibility coefficients. The effect of the proportionate damping is also included. A harmonic force of known amplitude with sine-sweep frequency is used to dynamically excite the beam, up to few flexible modes, which could be provided with the help of an exciter. In practice, linear degrees of freedom (DOFs) can be measured quite accurately; however, rotational DOFs are difficult to measure accurately. All rotational DOFs, except at crack element, are eliminated by a dynamic condensation scheme; for elimination of rotational DOFs at the crack element, a new condensation scheme is implemented. The algorithm is iterative in nature and starts with a presumption that a crack is present in the beam. For an assumed crack location, flexibility coefficients are estimated with the help of forced responses. The Tikhonov regularization technique is applied in the estimation of bounded crack flexibility coefficients. These crack flexibility coefficients are used to obtain the crack size by minimizing an objective function. With the help of the estimated crack size and measured natural frequency, the crack location is updated. The procedure iterates till the crack size and location get stabilized up to the desired level of accuracy. The algorithm has a potential to detect no crack condition also. The crack flexibility and damping coefficients are estimated as a by-product. Numerical examples, with the simply supported and cantilevered beams, are given to justify the applicability and versatility of the algorithm in practice. With the numerically simulated forced responses, which have the noise contamination and the error in the natural frequency measurements, the estimated crack parameters (i.e., the crack location and size) are in good agreement.