## 3 Study on Instantaneous Phase of an Active Lamb Wave Monitoring Signal
### Analysis of the Instantaneous Phase of the Monitoring Signal  
An experimental setup was built for the active Lamb wave based monitoring, and an epoxy glass fiber plate was used as the object of monitoring. The damage types of the composite plate include matrix fracture, fiber breakage, and internal delamination [30]. The damage monitoring method proposed in this paper focuses on damage with an explicit damage boundary that can easily scatter Lamb waves, such as a matrix fracture, fiber breakage, and crack. In the following experiment, a through-hole was used to simulate the scattered waves of these damages. Both the actuator and the sensor were PZTs with a diameter of 8 mm and a thickness of 0.5 mm. The material of PZT was lead zirconate titanate. The epoxy glass fiber plate used in the experiment was 2 mm thick and the center frequency of excitation signal was 100 kHz and 150 kHz. Thus, there were two fundamental modes of Lamb wave propagated in the plate. If the distance between the actor and the sensor was short, then the waves of these two modes would mix, and if the distance was long, then the attenuation of the waves would be large. According to the experimental conditions, the propagating distance of the wave was suggested to be set at 15-60 cm to ensure that the sensing signal was single-mode and had enough energy. The experimental setup is presented in Figure 4a. The center distance between the actuator (PZT1) and sensor (PZT2) was 40 cm. The damage in the plate was a through-hole with a diameter of about 6 mm, which was approximately at a distance of 20 cm and 25 cm from PZT1 and PZT2, respectively. Equation (6) was used as the mathematical expression of the excitation signal (with a center frequency of 100 kHz and 150 kHz and five peaks). The actuator PZT1 was excited once, and the monitoring system collected the transmitted Lamb wave by sensor PZT2, performing the one-time signal data collection. To reduce the impact of random noise on results, each monitoring signal denoted the average value of 20 single successive received signals. To ensure the reliability of the analysis results, 10 monitoring signals were respectively collected under the condition of both plate health and damage. Using 10 signals was a strategy for experimental data analysis. The purpose was to observe whether the instantaneous parameter changes of the signal caused by the damage were ubiquitous in the signal and whether the difference of instantaneous parameters between the same state signals could conceal the instantaneous parameter changes of the signal caused by damage. The response of the plate structure under excitation by the signal with the center frequency of respectively 100 kHz and 150 kHz is presented in Figure 4b, wherein the solid line denotes the response when there was not any damage,and the dashed line denotes the response when there was damage. Following the analysis in Section 2.2, only the Lamb waves labeled as Wave packet 1 in Figure 4b were used to study the instantaneous phase and the marginal spectrum. The instantaneous phase analysis is shown in Figure 4c,d with a center frequency of 100 kHz and 150 kHz, respectively. In Figure 4c,d, subfigure I presents the instantaneous phase of 10 monitoring signals after damage and 10 monitoring signals before damage; subfigure II denotes the locally enlarged view of subfigure I; subfigure III shows the information for the monitoring wave. As shown in Figure 4c,dII, the instantaneous phase of the 10 damage signals presented the same change concerning the instantaneous phase of 10 health signals, which indicates that the change was caused by a specific cause rather than a random signal. This change indicates that the monitoring signal was mixed with the scattering wave, and then it can be inferred that there was damage in the propagation path of the Lamb wave in a certain direction, which means that the plate structure was damaged. The damage diagnosis result obtained from the instantaneous phase change was consistent with the actual situation of the plate.