A damage localisation method based on higher order spatial derivatives of displacement and rotation fields

Abstract

A damage localisation method based on higher order spatial derivatives of displacement and rotation fields Abstract. This paper presents the development of a damage localisation method based on low and higher order spatial derivatives of displacement and rotation fields. The method relies on the Timoshenko beam theory, the Hamilton's principle and the Ritz method, allowing the computation of an approximate analytical solution of natural frequencies, displacement and rotation fields. Since these fields are expressed analytically, through a series expansion, the spatial derivatives are also obtained analytically. Besides the usual curvature difference and damage index, which are based on comparisons of second order spatial derivative of the displacement field of a beam in the undamaged and damaged states, other damage indicators are proposed. These new indicators rely on spatial derivatives of rotation fields. The indicators are applied to the localisation of various cases of damages in beams and a comparison among them is carried out. It is found that, for relatively thick beams, the values of the indicators based on rotation fields spatial derivatives are, in general, lower than those based on displacement fields. It is also observed that these differences are negligible for beams with high length to thickness ratio. IntroductionSeveral methods for structural damage identification, based on vibration characteristics, have been proposed over the past decades [1,2]. The main idea behind them is that whenever there is damage in a structure, its stiffness, mass and damping properties change, therefore changing its dynamic behaviour.The mode shape curvature method, firstly developed by Pandey et al.[3], allows, under certain conditions, the localisation of damage. In this work the curvatures of beams are computed using a second order central finite difference. However, if sparse and noisy displacement fields are used, this differentiation scheme can lead to false localisations of damage. This problem is mainly due to the intrinsic error of the finite difference numerical technique using sparse data and the amplification of noise present in the measurements. A method also based on mode shapes curvatures is the damage index method proposed by Stubbs et al. [4], which relates the second spatial derivative of the undamaged and damaged displacement fields in each segment or element of the structure.These two damage identification methods and similar ones, such as the frequency response functions (FRF) curvatures method [5], have been applied and improved over the years (e.g. [6]-[12]). However, the curvatures are usually computed based on a differentiation of a relatively