## 4 Analysis of Marginal Spectrum Change of an Active Lamb Wave
### Application Analysis of Marginal Spectrum Change in Plate Structural Health Monitoring
The marginal spectrum was concentrated in the frequency band 80-120 kHz, which was divided into five continuous bands as follows: 8088 kHz, 88-96 kHz, 96-104 kHz, 104-112 kHz, and 112-120 kHz. The number of corresponding figures of each band was from I to V. The marginal spectrum amplitude in each frequency band is presented in Figure 7a, where \"*\" represents 10 healthy signals and \"\(\Delta\)\" represents 10 damage signals. For both health signals and damage signals, the marginal spectral amplitudes of the last two bands (104-112 kHz and 112-120 kHz, see subfigures VI-V in Figure 7a) were too small, and thus they were not involved in the following analysis. In Figure 7 aI-III, the marginal spectrum amplitude (\"\(\Delta\)\" in the figures) after the damage is different from that before the damage. In subfigures I and II, the corresponding frequency bands were 80-88 kHz and 88-96 kHz, and the marginal spectrum amplitude after damage increased. In subfigure III, the corresponding frequency band was 96-104 kHz, and the marginal spectrum amplitude after damage decreased. Whether it increased or decreased, the change direction of the 10 marginal spectra in the same frequency band was the same after damage, and the change amount was also consistent,  
Figure 6: The first-order IMFs of two decompositions. (**a**) The first-order IMFs obtained by decomposition without zero-phase filtering; (**b**) The first-order IMFs obtained by decomposition with zero-phase filtering. which indicates that the marginal spectral change was not caused by random signals. According to the actual state, in Figure 7a, the change in the marginal spectrum of the active Lamb wave monitoring signal could only be caused by the scattering signals from the plate structure damage. Figure 7: The frequency range around the two excitation frequencies is divided into five frequency bands and the marginal spectrum amplitude of the Lamb wave in each frequency band. “_” is the marginal spectrum amplitude before the plate damage, and “\(\Delta\)” is the marginal spectrum amplitude after the plate damage. Subfigures **(I–V)** relate to the five frequency bands. **(a)** The marginal spectrum of 100 kHz excited Lamb waves; **(b)** The marginal spectrum of 150 kHz excited Lamb waves. The marginal spectrum of the Lamb wave with a center frequency of 150 kHz is presented in Figure 7b. Like the analysis process in Figure 7a, the efficiency frequency band 130-190 kHz was divided into five successive bands, labeled from I to V. Frequency band 178-190 kHz was not involved in the following analysis because the energy was too small. By comparing Figure 7a,b, their marginal spectral energy diffusion directions were different in addition to their different center frequencies. In subfigures I and II, the corresponding frequency bands were 130-142 kHz and 142-154 kHz, and the marginal spectrum amplitude wave decreased after damage. In subfigures III and IV, the corresponding frequency bands were 154-166 kHz and 166-178 kHz, and the marginal spectrum of Lamb wave increased after damage. The change characteristics of the marginal spectrum were like those of Figure 7a. In each subfigure I-IV, the change direction of the marginal spectrum was the same, and the variation amount of the marginal spectrum was consistent. Figure 7b shows that the marginal spectrum changed significantly after damage, and the abnormal of active Lamb wave could be judged to diagnose the plate structure damage. The above marginal spectrum analysis of two different center frequencies shows that the change of the active Lamb wave marginal spectrum after the damage had some characteristics. The marginal spectrum change was not limited to a certain frequency band but rather existed in several continuous frequency bands. In the effective frequency range, the marginal spectrum of the local frequency band increased or decreased after damage, but the whole frequency range did not increase or decrease.