14 D.-S. Li and X.-H. Li time ( h ) 0 1000 2000 3000 4000 5000 time ( h ) 0 1000 2000 3000 4000 5000 time ( h ) a b c frequency (Hz) 33 33.5 34 34.5 35 35.5 frequency (Hz) frequency (Hz) -1 -0.5 0 0.5 1 33 33.5 34 34.5 35 35.5 0 1000 2000 3000 4000 5000 Fig. 2.6 Filter of frequency changes induced by temperature and damage identification shows the five components of the first nature frequency decomposed by the SGSA. Similar to the analysis of the temperature decomposition, the first two components shown in Fig. 2.5b and c represent the main trend and fluctuation amplitude of the original frequency signal. Compared with the first two components of the temperature in Fig. 2.3, it can be concluded that the first frequency component shown in Fig. 2.5b reflects the frequency variation influenced by the seasonal trend of temperature and structural damage whereas the second component in Fig. 2.5c represents the frequency variation induced by the daily temperature fluctuation. The other frequency components shown in Fig. 2.5d–f can be neglected as noise due to their tiny amplitude. To finally identify structural damage, the first component shown in Fig. 2.5b that includes both temperature and damage influences are further analyzed. If the frequency variation induced by the temperature trend can be filtered out from this component, the remaining part of the frequency changes can be inferred as induced from structural damage. To filter out the frequency change induced by seasonal temperature trend, a cross validation approach is adopted to fit the frequency variation induced only by temperature. The temperature data in Fig. 2.5b are divided into ten segments. The length of first nine segments is set to 500 and the last segment 250. Nine segments of data are then sequentially taken as fitting data and the remaining segment for validation data in turn. A quartic curve is then obtained as shown in red in Fig. 2.6b. The red curve in Fig. 2.6b represents the frequency change induced by temperature and the blue curve shows the frequency change induced both by temperature and damage, which is the same curve as that in Fig. 2.5b. Figure 2.6b shows that the frequency change by temperature trend can be eliminated. Subtracting the blue cure from the red curve in Fig. 2.6b, the blue curve in Fig. 2.5c is obtained, which is the frequency of the beam induced only by the damage. It is easily observed that there is a relatively large change at about the 3000th point (6000 h after Jan 1, 2011). Therefore, it can be deduced that at the 3000th moment, damage occurs in the beam. 2.4 Conclusions A symplectic geometric spectral analysis method is applied to structural damage identification considering the influences of environmental influences. The frequency change induced by seasonal temperature is filtered out through SGSA and damage is identified. It shows that the SGSA method can detect structural damage in the presence of environmental and operational variations successfully. The advantage of this method is that it does not need to directly measure the environment parameters and if feasible to practical damage identification. Acknowledgement The authors appreciate the support by the National Natural Science Foundation of China (Grant No. 51121005, 51578107) and 973 Project (2015CB057704).
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