Lamb Wave Based Structural Damage Detection Using Stationarity Tests
Abstract
:1. Introduction
2. Background
2.1. Basic Concepts
- Trends () representing long term movements in the mean;
- Seasonal effects () and cycles () representing cyclical fluctuations in the series;
- Residuals () representing random or systematic fluctuations.
2.2. Time Series and Stationarity
- For , we have a stationary time series, which appears jagged but always oscillates around the mean, as illustrated in Figure 1;
- For , we have a nonstationary time series, which is smoother and finally explodes, as shown in Figure 2;
- For , we have a random walk time series, which moves up and down; it behaves as a nonstationary time series but slowly, as plotted in Figure 3.
2.3. Unit Root Tests
- If , then , where is a white noise process, and so is a nonstationary random walk time series with a drift that contains a stochastic trend or a unit root (which is represented by the integrated sum ). In this case, shocks have permanent effects, and the time series shows an unpredictable pattern (see Figure 4). It is noted that process in this case is referred to as difference stationary, which is discussed in Section 2.2.
- If , then is a stationary time series around the deterministic linear trend . In this case, shocks have transitory effects, as illustrated in Figure 5. The process in this case is referred to as trend stationary, which is mentioned in Section 2.2.
- If the absolute value of the calculated test statistic is greater than the critical value (in absolute value), then can be rejected (i.e., the time series is stationary).
- If the absolute value of the calculated test statistic is smaller than the critical value (in absolute value), then cannot be rejected (i.e., the time series is nonstationary).
2.4. Common Unit Root (or Stationarity) Tests
- (a)
- Augmented Dickey–Fuller (ADF) test for a unit root
- (b)
- Phillips–Perron (PP) test for a unit root
- (c)
- Kwiatkowski–Phillips–Schmidt–Shin (KPSS) test for stationarity
- (d)
- Leybourne–McCabe (LM) test for stationarity
3. Lamb Wave Based Structural Damage Detection Using Unit Root (or Stationarity) Tests
4. Lamb Wave Experimental Data Contaminated by Varying Temperature
5. Results and Discussion
6. Conclusions
- Both ADF and KPSS tests can detect the damage, while both PP and LM tests are not significant for identifying the damage.
- The ADF test is more stable with the temperature changes than KPSS test. However, the KPSS test can detect damage better than the ADF test.
- Both KPSS and ADF tests can consistently detect damage in the conditions of temperatures varying below 60 °C. However, their t-statistics fluctuate more (or less homogeneous) for temperatures higher than 65 °C.
- Based on these results, both ADF and KPSS tests are suggested to be used together for Lamb wave based structural damage detection in order to enhance the accuracy and reliability of the proposed stationarity-based approach.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Type of Tests | Data at 35 °C | Data at 45 °C | Data at 60 °C | Data at 70 °C |
---|---|---|---|---|
ADF test | 2.81 | 3.83 | 3.28 | 4.42 |
KPSS test | 88.43 | 118.21 | 127.62 | 31.63 |
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Dao, P.B.; Staszewski, W.J. Lamb Wave Based Structural Damage Detection Using Stationarity Tests. Materials 2021, 14, 6823. https://doi.org/10.3390/ma14226823
Dao PB, Staszewski WJ. Lamb Wave Based Structural Damage Detection Using Stationarity Tests. Materials. 2021; 14(22):6823. https://doi.org/10.3390/ma14226823
Chicago/Turabian StyleDao, Phong B., and Wieslaw J. Staszewski. 2021. "Lamb Wave Based Structural Damage Detection Using Stationarity Tests" Materials 14, no. 22: 6823. https://doi.org/10.3390/ma14226823