A new technique to optimize the use of mode shape derivatives to localize damage in laminated composite plates

Damage localization in laminated composite structures is a very active area of research due to the role that these kind of structures play in the transport industries. The mode shape derivatives, like rotations (first derivative), curvatures (second derivative) and, more recently, third and four derivatives, have been used to localize damage in composite plates. The most used method to compute these derivatives is the application of finite differences. However, finite differences present several well-known problems, such as the error propagation and amplification. The magnitude of the error associated with the computed derivative is not easy to estimate, mainly because the numerical error associated with finite differences depends on the values of derivatives of higher order than the order of the derivative that one wants to compute. A new technique based on the Ritz method to estimate this error is proposed in this paper. The optimal spatial sampling for the numerical differentiation of the mode shapes are defined based on the minimization of the total error. The good performance of the optimal sampling is shown by applying it to the damage localization in a laminated composite plate.