# Damage Detection Using Modal Rotational Mode Shapes Obtained with a Uniform Rate CSLDV Measurement
## 1 Introduction  
Damage detection of structures from changes in their vibration characteristics is widely used in engineering. The basic approach is to compare the modal characters between the damage and the undamaged states of the object structure. In the last three decades, many methods have been developed. The earliest damage detection methods are based on mode frequency changes of structures [1; 2; 3]. However, the mode frequency changes usually reflect the global features and are easily buried by the operational conditions. Afterwards, the detection methods using mode shape changes [1; 2] are proposed to identify the crack or damage of structures. Later on, the change of modal flexibility of a few lower orders of mode shapes [4; 5; 6] were used for damage detection of structures in civil engineering. However, the displacement mode shape can be easily affected by environmental noise and is not sensitive to slight damage. Due to these reasons, methods based on the curvature mode shape [7; 8; 9] or mode strain energy [10; 11; 12] have been further developed and have become the most widely used in practice. This is because the damage indicator based on curvature mode shape or mode strain energy is much more sensitive and can easily localize anomalous features of the structure. In general, the modal strain energy and mode curvature are often applied in combination to damage detection. [PERSON] et al. [13; 14; 15] used the differential quadrature method (DQM) to calculate the modal strain energy to detect the crack of a plate or a circular hollow cylinder structure. [PERSON] et al. [16] proposed a damage severity correction factor (DSCF) based on modal strain energy to reflect the nonlinear relationship between the indicator value and severity of the damage. Also, mode curvatures were measured using a surface-bonded polyvinylidene fluoride (PVDF) piezoelectric strain sensor array to detect the crack of a fiber-reinforced plastic (FRP) sandwich deck panel. However, the quality of mode curvature or modal strain energy achieved in practice depends on the spatial density and the accuracy of displacement mode shapes. [PERSON] et al. [17] proposed an optimal spatial sampling interval method to minimize the effects of noise and truncation errors on the calculation of the curvature mode shape or modal strain energy. [PERSON] et al. [18] developed a novel Laplacian operator and smoothed Teager energy operator (STEO) to eliminate the influence of data noise on mode curvature. Also, [PERSON] et al. [19] formulated a new concept of complex-wavelet mode curvature to reveal and delineate the damage under noisy condition. The mode curvature can be expressed as the second order partial derivatives of the displacement mode shape. The second order differential will greatly amplify the noise effect. Not only that, the spatial resolution of the mode shape also has a great influence [17; 20]. Due to these reasons, measurement noise needs to be strictly controlled and spatial resolution of displacement mode shape should be carefully considered to adapt to various damage detection with these methods. In recent years, the full field vibration measurement techniques, such as continuously scanning laser Doppler vibrometer (CSLDV) [21; 22] and high speed three-dimensional digital image correlation (DIC) [23; 24; 25] techniques, have become mature and widely used. Compared with traditional transducers, the full field techniques can accurately realize noncontact surface measurement and the spatial resolution of mode shape can be greatly improved. For the CSLDV method, there are two ways to scan the structural surface: (a) A uniform rate scan [26], and (b) a sinusoidal rate scan [27]. The demodulation of CSLDV output signals to obtain the operational deflection shape (ODS) is also different according to different scan strategies. For a sinusoidal rate scan method, the displacement mode shape is fitted using lower order polynomial function based on the sideband information of CSLDV output signals.