# Damage Detection Using Modal Rotational Mode Shapes Obtained with a Uniform Rate CSLDV Measurement
## 1 Introduction
Hence, the slight changes of mode shapes caused by damage are easily lost during the fitting process, especially for the lower modes. On the contrary, using a uniform rate scan method, the displacement mode shapes can be directly obtained by the demodulation of CSLDV output signals at the corresponding spatial position. Based on that, [PERSON] et al. [28; 29] and [PERSON] et al. [30] used the uniform rate scan CSLDV method to measure the ODS and detect the crack of beam or delamination of composite plate. For the DIC method, mode curvature using measured ODS is derived to detect the crack of sandwich panel and membranes [31; 32]. Actually, advantages of damage detection using the mode curvature and modal strain energy methods are due to the fact that they are based on the second order differential of the displacement mode shape which can significantly amplify the slight discontinuous change caused by the damage. In the same way, the modal rotation, which is expressed as the first order partial derivative of displacement, has the similarity features with mode curvature and therefore can also be applied to damage detection. [PERSON] et al. [33] used the measured modal rotation with DIC method to detect the crack of a beam with the combination of wavelet analysis. [PERSON] et al. [34] proposed a damage detection method for beam structures using the slopes of displacement mode shapes from CLSDV. In addition, [PERSON] et al. [35] proposed a strain based rotational mode shape measurement method for beam structures. The rotational shape can be derived by integrating the strain mode shape. Based on that, the long-gauge strain measurement technique is developed for structural damage detection and health monitoring, especially for long span bridges [36; 37]. The modal rotational mode shape is sensitive to reflect the discontinuous change caused by the damage and is also less sensitive to the noise contamination in the mode shape. Therefore, it is a good balanced parameter suitable for structural damage detection. Another issue for damage detection is that mode shapes of the undamaged state of the structure are hardly obtained in many cases. This causes difficulty in comparing the mode shape changes between the damaged and the undamaged states. [PERSON] and [PERSON] [38] developed a finite difference approximation of Laplace's differential operator to the mode shape from the damaged structure in order to identify and pinpoint structural damage in a beam without a priori knowledge of the undamaged structure. This procedure operates solely on mode shape data of damaged state. Later on, [PERSON] [39] used a modified Laplacian operator on mode shape data to locate and estimate the crack of Euler-Bernoulli beam structures. [PERSON] et al. [40] developed a two-dimensional gapped smoothing method (GSM) to detect the plate damage based on [PERSON]'s method. [PERSON] [41] proposed a two-dimensional surface interpolation method to localize reductions of stiffness in plate-like structures. Most recently, [PERSON] et al. [28; 42] proposed a polynomial function fitted method to the mode shape of the damaged state in order to proximately estimate the mode shape of the undamaged structure. [PERSON] [43] and [PERSON] et al., [44] used the measured mode shape data of the damaged state with principal component analysis (PCA) method to reconstruct the mode shapes of the undamaged structure. The polynomial function fitted ODSs or mode shapes are relatively smooth and can reflect the property of the undamaged structure. In this paper, a damage indicator based on changes of the modal rotational mode shapes between the estimated undamaged and the tested damaged states using a uniform rate CSLDV measurement of the only damaged plate-like structure is put forward. The modal rotational mode shapes are obtained through the derivative of the detailed displacement mode shape of transitional degree-of-freedoms (DOFs) with respect to the orthogonal directions.