Impedance-based SHM and Statistical Method for Threshold Level Determination Applied to Aircraft Structures Under Varying Temperature D. S. Rabelo, V. Steffen Jr (*), R.M. Finzi Neto, H.B. Lacerda LMEst – Structural Mechanics Laboratory, INCT (EIE) – National Institute of Science and Technology, Federal University of Uberlandia, School of Mechanical Engineering, Av. João Naves de Ávila, 2121, Uberlândia, MG, 38408-196, Brazil, Phone +55 34 3239 4017, Fax +55 34 3239 4206 (*) Corresponding author:
[email protected] ABSTRACT: The Impedance-based Structural Health Monitoring (ISHM) method has become a promising and attractive tool for damage identification and is considered a nondestructive evaluation technique. However, conventional ISHM studies have mainly focused on structural damage identification, but not so much on statistical modeling approaches in order to determine a threshold for the decision making of the damage detection system. In this study, the ISHM technique is used in a damage detection problem considering temperature variation effects. For this aim, three aluminum 2024-T3 plates were instrumented with small PZT (Lead Zirconate Titanate) patches close to their borders and damage was introduced in the central position of the plates, with temperature ranging from -10 ºC to 60 ºC. This paper proposes a method to statistically determine a threshold for damage detection purposes by using concepts of Statistical Process Control, as well as Confidence Intervals and Normality Tests in order to obtain a diagnosis with a previously determined confidence level. Thus, this work presents a sensitivity evaluation of the ISHM technique as applied to aluminum plates under varying temperature. With the technique proposed, damage threshold levels are determined so that PZT patches placed approximately 280 mm from the damage inserted were able to detect saw cuts of approximately 7 mm long, with 95% confidence intervals inside the temperature range considered. KEYWORDS: Structural Health Monitoring (SHM), Electromechanical Impedance (EMI), Temperature compensation, Threshold determination, Statistical analysis
1. INTRODUCTION Since our societies are heavily dependent upon aerospace, civil and mechanical structures, there are increasing demands to detect structural damage at its earliest possible stage. Strategies for Structural Health Monitoring (SHM) and Nondestructive Evaluation (NDE) technologies are focus of considerable research work over the last twenty years. The forthcoming technologies in SHM are required to be autonomous and to monitor a structure’s integrity in real-time. The main goals of these technologies are to improve life-safety and to reduce maintenance costs by replacing traditional schedule-based maintenance for condition-based maintenance. In the aerospace industry, particularly, besides the ageing of the current aircraft fleet, improvements such as larger capacities and greater use of composite materials motivate the development of SHM systems. Furthermore, additional economic advantages can be gained if the SHM system can reliably prevent unnecessary dismantling of structural components. A variety of damage types are of interest in this sector, with corrosion and cracking in metallic components, also delamination and debonding are subjects of concern for composite components. For commercial aircrafts, fatigue cracks that form around rivets in the fuselage are a major concern. Once a SHM methodology has been established, the full process generally includes the following steps: operational evaluation, data acquisition (DAQ) and networking, feature selection/extraction and finally probabilistic decision-making. [1]. When deployed, the SHM system will have to deal with structures that experience changing operational and environmental conditions. These changing operational and evaluation conditions will produce changes in the measured response and it is imperative that these changes are not interpreted as indications of damage. Varying temperature is a common environmental condition that must be accounted for during the damage detection process [2]. One example of documented success in active local sensing for damage detection using PZT patches is the impedance-based method (ISHM) [3-4]. Advantageous features from these transducers worth mentioning is that they are inexpensive, generally require low power, they are relatively nonintrusive and they have a wide operational frequency range, providing high sensitivity for sensors. The ISHM method monitors variations in mechanical impedance resulting from damage and the mechanical impedance is coupled with the electrical impedance of the PZT acting simultaneously as a sensor and actuator. Although some piezoelectric properties are highly dependent on temperature changes, there are temperature compensation techniques that can considerably compensate for these effects. Temperature variation effects and compensation procedures for these effects have been widely studied [5-9,35-36]. The probabilistic decision making process is one of the main steps of a SHM system. Although the ISHM is commonly used for structural damage identification [10-16], most of the studies make limited use of rigorous statistical models. The statistical models are constructed in order to minimize false diagnoses. False diagnoses fall into either a false positive, when there is an indication of damage when none is present, or a false-negative, when there is no indication of damage when damage is present.
This paper investigates the evaluation of PZT sensors to detect an inserted saw cut into three aluminum 2024-T3 plates, a material commonly used in aircrafts, considering a temperature range from -10 ºC to 60 ºC. The choice of this temperature range does not correspond to the whole temperature range of an aircraft wing surface (-55 to 80o C), however it permits to illustrate the methodology conveyed. The saw cut size was sequentially increased. The temperature compensation technique selected was the effective frequency shift (EFS) through correlation analysis. Sun et al. [5] first proposed the use of cross-correlation between two impedance signatures to compensate for frequency shifts caused by temperature variation. Park et al. [6] compensated for both the frequency and the magnitude shifts through a modified damage metric. Koo et al. [7] modified the method developed by Park et al. [6] to develop EFS to compensate for temperature effects through the maximization of the correlation coefficient. Recent studies have used this technique and a review of different methods is presented in the literature [8-9]. Next, a statistical model was developed to establish threshold indexes according to a pre-defined level of confidence. Thus, this work provides a sensitivity assessment of the ISHM technique considering the rate of success in order to detect the damage inserted for aluminum plates under varying temperature.
2. IMPEDANCE-BASED STRUCTURAL HEALTH MONITORINNG In this section, the principle on which the EMI technique is based is discussed along with the methodology on how structural damage is detected; besides, the issues of temperature variation are briefly discussed. 2.1 Physical principle of the EMI technique The ISHM technique is considered as being a method for non-destructive evaluation [3]. It uses the piezoelectric properties of the PZT patch that is installed in the structure being monitored. The PZT patch is bonded to (or embedded into) a structure and a low electric voltage is applied [19]; a strain is produced in the PZT patch. Then, the response of the mechanical vibration is transmitted to the sensor in the form of an electrical response. In the case a structural modification occurs, such as damage, this is observed in the electric response of the PZT patch. When a small electric voltage is applied to the PZT patch, the piezoelectric effect may be considered approximately linear. With this assumption, the constitutive equations obtained from the Gibbs free energy are as follows [20]: !,!,! !,! !,! 𝑆! = 𝑠!" 𝑇! + 𝑑!" 𝐸! + 𝑑!" 𝐻! + 𝛼!!,! 𝑑𝜃
𝐷! =
!,! 𝑑!" 𝑇!
+
!,!,! 𝜖!" 𝐸!
+
!,! 𝑚!" 𝐻!
+
!,! 𝑝! 𝑑𝜃
(1) (2)
where 𝑠!" is the elastic compliance, 𝑑!" are the piezomagnetic constants, 𝛼! are the thermal expansion coefficients, 𝜖!" is the dielectric permittivity tensor, 𝑚!" are the magnetodielectric constants, 𝑝! are the pyroelectric constants, 𝑇! is the mechanical stress tensor, 𝐸! and 𝐸! are the electric field components, 𝐻! and 𝐻! are the magnetic field components, 𝜃 is the temperature. The superscripts E, H, 𝜃 and T show that the electric field, the magnetic field, the temperature, and the mechanical stress are constant. Given the symmetry of the tensors, from the Einstein notation, the indexes i and j goes from 1 to 6, the indexes k and m from 1 to 3. Equations (1) and (2) describe the converse and direct piezoelectric effect, respectively. The converse effect relates to the PZT’s actuator mode, whereby a strain is activated by an electric field applied. The direct effect relates to the PZT’s sensor mode, whereby a charge is generated in response to a stress field. Although Equations 1 and 2 show several coupling terms, for standard piezoelectric materials it is acceptable to assume that the magnetic effects are negligible [21]. Temperature effects, however, are not particularly negligible since the piezoelectric properties are significantly pyro electric [22], i.e., a thermal change establishes an electric field and vice versa. The mechatronic model that describes the measurement process is shown in Figure 1 for a single-degreeof-freedom (DOF) system.
Figure 1. Mechatronic 1 DOF model illustrating the Impedance-Based Structural Health Monitoring method.
For this system, Liang et al. [23] demonstrated that the PZT’s admittance, 𝑌 𝜔 , which is the inverse of the impedance, can be written as a function of the combined PZT actuator and structure mechanical impedance, as given by Eq. (3): ! 𝑌 𝜔 = 𝑖𝜔𝑎 𝜀!! 1 − 𝑖𝛿 −
𝑍! 𝜔 𝑑! 𝑌 ! 𝑍! 𝜔 + 𝑍! 𝜔 !! !!
(3)
where 𝑍! 𝜔 and 𝑍! 𝜔 are the actuator’s and structure’s mechanical impedances, ! ! respectively. 𝑌!! is the complex Young’s modulus of the PZT with zero electric field, 𝑑!! ! is the piezoelectric coupling constant in the arbitrary x direction, 𝜀!! is the dielectric constant at zero stress, 𝛿 is the dielectric loss tangent of the PZT, and a is a geometric constant of the PZT. Assuming that the mechanical properties of the PZT patch do not vary during the measurement procedure, Eq. (3) shows that the electrical impedance of the PZT patch is directly related to the structure’s impedance. Damage leads to changes
in the structure’s mechanical impedance, thus modifying local dynamic features. Hence, the electrical impedance is measured in order to monitor the so-called health state of the structure, through a comparison with a previous baseline measure, i.e., the pristine condition. The real part of the electrical impedance is more reactive to damage since the imaginary part contains the capacitive portion of the PZT patch, which is more sensitive to temperature variation [7]. Therefore, the ISHM technique usually takes the real part of impedance signatures for damage assessment. Since the goal of the present technique is to obtain high sensitivity to incipient damage, the actuator’s driving frequency range should be in the order of 30 kHz to 250 kHz. It is necessary for the excitation wavelength to be smaller than the characteristic length of the damage to be detected [3]. The effective range for a given structure is determined by trial and error methods or by optimization procedures [24]. As for the sensitive region, Sun et al. [5] suggested that frequencies over 200 kHz should be used to obtain localized damage, while frequencies below 70 kHz should be used to cover larger areas of the monitored structure. Park et al. [3] claim that for a simple PZT a damage located at a radial distance of up to 0.4 m can be identified in composite materials and up to 2 m in bars consisting of a single metal. The curve that represents the impedance response provides a qualitative assessment of the damage. For a quantitative assessment of the failure, a previously defined Damage Metric (DM), or Damage Index (DI) is used [25]. In the present contribution, the damage metric used is the correlation coefficient deviation (CCD). This DM uses two signals, where the first corresponds to the baseline and the second is the test measurement. The CCD is given by Eq. (4): CCD = 1 −
n 1 &( "# Re ( Z1,i ) − Re ( Z1 ) $%"# Re ( Z 2,i ) − Re ( Z 2 ) $% *( ' + ∑ n i=1 )( SZ1 SZ2 ,(
(4)
where Re ( Z1,i ) and Re ( Z 2,i ) are the real part of the impedance from the baseline and test measurement at frequency i, respectively; n is the number of frequency points, Re ( Z1 ) and Re ( Z 2 ) are the average of the baseline and test measurement, respectively; SZ1 and SZ2 are the standard deviations of the baseline and test measurements, respectively. The next topic discusses the effects caused on ISHM systems by temperature variations. 2.1 Temperature Variation Effects The feature selection and extraction step of the SHM process requires a data normalization procedure. Data normalization is referred as the process of separating changes in the features derived from sensor readings that are caused by damage from those changes caused by environmental or operational variations. The ability to perform robust data normalization is one of the biggest challenges faced by SHM when attempting to transfer this technology from research to real world on in situ structures [1].
The experiment performed in this study consists of a reliability assessment of an ISHM system, considering changes in the ambient temperature. Temperature variation plays a major role when a structure is being monitored with PZT patches. In this section, the effect of temperature variation is illustrated in order to show the changes caused in the impedance signatures. In the case of ISHM, the main effects of temperature can be stated as horizontal and vertical shifts, as can be seen in Fig. 2(b) as well as changes in some of the peak amplitudes. Figure 2(a) shows an aluminum plate of dimensions 305 mm x 305 mm x 3 mm, instrumented with a PZT patch type 5H with 15 mm diameter x 1 mm thickness attached on a central position. The tests were performed inside the environmental chamber ESPEC model EPL-4H, whose temperature resolution is 0.5ºC after 30 minutes of settling time to stabilize the internal temperature. In this case, the temperature range utilized was from 0ºC to 50ºC in steps of 10ºC. Each impedance signature was measured with 600 points and the frequency range was from 63 kHz to 66 kHz. By analysing Fig. 2(b), the effects of temperature can be observed predominantly as horizontal shifts due to changes in the resonance frequencies of the system. Vertical shifts can also be seen due to the change in the capacitance of the sensor, as well as changes in some peak amplitudes. In order to avoid false indications of damage, the SHM system requires a data normalization procedure.
(a)
(b)
Figure 2. Temperature variation effects on impedance signatures: (a) Instrumented Al plate of 305 x 305 x 3 mm; (b) Impedance signals shifted with temperature changes.
2.2.1. Data Normalization Procedure – EFS through correlation analysis In order to compensate for temperature changes, the Effective Frequency-Shift (EFS) through correlation analysis technique was implemented. An important feature about this approach is that it is an unsupervised learning machine method, i.e., it does not require data from damage states to perform the diagnostic. For a fixed frequency range, the horizontal and vertical shifts can be considered as being uniforms. However, the presence of damage on a signature is somewhat local and abrupt [5]. This feature allows us to separate the temperature effects. For this purpose, a global average difference from two signals is obtained from Eq. (5):
1 Δ! = 𝑛
!
!
𝑅𝑒 𝑍!,! − !!!
𝑅𝑒 𝑍!,!
(5)
!!!
where Δ! is the vertical shift; Re Z!,! and Re Z!,! are the real parts of the measured impedance of the baseline and test measurement, respectively; n is the number of frequency points selected. Prior to the determination of the horizontal frequency shift, the correlation coefficient between the two impedance signatures is calculated by using Eq. (6): 𝐶𝐶!! !!
1 = 𝑛
!
!!!
𝑅𝑒 𝑍!,! − 𝑅𝑒 𝑍! 𝑅𝑒 𝑍!,! − 𝑅𝑒 𝑍! 𝑆!! 𝑆!!
(6)
where Re Z! and Re Z! are the average of the baseline and test measurement, respectively; S!! and S!! are the standard deviations of the baseline and test measurements, respectively. Afterwards, the frequency shift Δf is obtained through an iterative routine until the maximum correlation coefficient (ideally one) is obtained, [8]. Thus, the compensated signature is obtained by using Eq. 7: 𝑅𝑒 𝑍!"##,! = 𝑅𝑒 𝑍!,!±!! + Δ!
(7)
where Re ( Zcorr,i ) is the real part of the impedance corrected at frequency i, Re ( Z 2,i±Δ f ) is the real part of the test measurement shifted by Δf data points at frequency i ± Δ f . In the frequency axis, the shift will be applied to the right if the test measurement’s temperature is greater than the baseline’s temperature; otherwise, the shift will be oriented to the left. Therefore, it is of the upmost importance to have temperature readings for all measurements performed as close as possible to the sensors in the structure being monitored by an ISHM system. In the end of the process, the damage metrics are updated with the corrected impedance signature. As an illustration, the aluminum plate shown in Fig. 2(a) was tested under a large temperature range. Figures 3(a) and 3(b) show the comparison with the previous measurements performed in the Al plate for the same temperature range. In this case, only one baseline was recorded at -30ºC, and each signature was measured with 600 data points, after 30 minutes of settling time for temperature stabilization.
(a)
(b)
Figure 3. Temperature compensation procedure on an Al plate. (a) – Impedance signatures without compensation; (b) – Impedance signatures with compensation.
Overall, it can be observed that the temperature compensation procedure provides good agreement for the impedance signatures in a wide temperature range. However, since there is some variation on peak amplitudes, a perfect match in practical application will rarely occur, as demonstrated by the resonance peak close to 64 kHz in Fig. 3(b). Despite a good agreement is obtained with temperature compensation techniques, there are limitations regarding the frequency band. This is also true when the desired temperature range is extremely large, which could implicate poor matching in the signatures when the temperature gradient is very large [26]. Therefore, for a structure being designed to work under a given temperature range, the ISHM system would require more than only one baseline in order to provide a more reliable diagnostic.
3. EXPERIMENTAL DESIGN AND SETUP The goal of a NDE reliability demonstration is not to determine the smallest damage size the system can detect – more precisely, it is to determine the largest damage size the system can miss. The probability of false positives for each sensor was determined within the temperature range considered in the tests. Obtaining the false positive rate is an important step, since if this analysis is accompanied by an unacceptable (too high) false positive rate this could increase maintenance costs for the engineering structure. The uncertainty in target decision is caused by both the physical attributes of the targets under test, and the NDE process variables, system settings and test protocol. For the ISHM technique, examples of process variables are the driving frequency bandwidth, the piezoelectric transducers, the measuring device, the sensor bonding procedure, the environmental and operational conditions, etc. The uncertainty caused by the differences between targets is accounted for by using representative specimens with targets of known size. The uncertainty caused by the NDE process is accounted for by a test matrix of different inspections to be performed on the complete set of specimens, [18]. Hence, three specimens (plates) of aluminum 2024-T3 with dimensions 500 x 500 x 1.6 mm and mass of 1.120 kg were instrumented with four PZT patches (diameter of 15 mm x thickness of 0.5 mm type 5H) each. The PZT patches were bonded with an epoxy-
based adhesive (Hysol EA9320NA) and later protected with a sealant. Each specimen was prepared by a different operator according to the draft of Fig. 4(a). In order to introduce damage, a portable milling machine was used with a 0.55 mm diameter cutter made of titanium, so that a through cut was made at the center of each specimen. The impedance-measuring device was the one described in [31], which has a 0.1 Ω resolution. The frequency range was selected by trial and error for each sensor, with a bandwidth of 5Khz and 400 frequency points. The experiments were conducted inside the previously mentioned environmental chamber, where the temperature ranged from -10 ºC to +60 ºC with steps of 5 ºC. At each temperature, 5 measurements were performed and a settling time of 30 minutes was respected for each temperature point. The saw cut lengths were introduced as follows: 3 mm, 5 mm, 7 mm and 10 mm. Although the procedure to insert the cut was manual, a dial indicator from Mitutoyo with 0.001 mm resolution was utilized, in order to check the actual target size. The final target size was a result of the mean of ten repeated measurements by using the dial indicator.
(a)
(b)
(c)
Figure 4. Experimental setup: (a) Draft for bondage of sensors; (b) Instrumented Al panel with 4 bonded PZT patches; (c) Specimens placed inside environmental chamber in the free-free boundary conditions.
The red circle shown in Fig. 4(c) depicts a temperature transducer AD590, from Analog Devices. This temperature sensor was placed close to the specimens, inside the chamber. It was used to assign each impedance signature to its corresponding temperature value for the compensation procedure. Figure 5 depicts the experimental setup for the damaged plate and the impedance analyzer used [31].
(a)
(b)
Figure 5. Experimental setup-2: (a) Zoomed detail of damage (10 mm saw cut) inserted; (b) Illustration of portable impedance meter device;
The test matrix determined during the experiment design is shown in Table 1 and the frequency ranges selected for the twelve PZT transducers are shown: Experiment Baseline recording Run #1 Run #2 Run #3 Run #4 Run #5
Table 1: Test matrix for the experiment Temperature range [ºC] Saw cut length [mm] -5 to +55 0 -10 to +60 -10 to +60 -10 to +60 -10 to +60 -10 to +60
0 3 5 7 10
The baseline recording experiment shown in Table 1 consisted of the first group of measurements performed. A total of seven baselines were recorded at the following temperatures: -5 ºC, 5 ºC, 15 ºC, 25 ºC, 35 ºC, 45 ºC and 55 ºC. The temperature compensation procedure used the baseline that was closer to the test measurement in terms of temperature gradient. Additionally, each experiment run (#1 to #5) consisted of 75 measurements for each transducer since there were a total of 15 temperature points and 5 repetitions for each case. To contribute to the measurement process, an algorithm was used to synchronize the measurements with the experiment’s temperature cycle, considering that each experiment run from Table 1 took approximately 8 hours to be concluded in this automated and continuous measurement process.
Table 2: Frequency range selected for the 12 sensors
Specimen
A
B
C
PZT # 1 2 3 4 5 6 7 8 9 10 11 12
Frequency Range [kHz] 125 – 130 125 – 130 125 – 130 125 – 130 125 – 130 125 – 130 120 – 125 120 – 125 125 – 130 130 – 135 125 – 130 125 – 130
In Table 2, it is important to mention that a different operator prepared each specimen since there could be a variation in the sensitivity since the bonding procedure could influence the electromechanical coupling of each sensor [30]. At the end of the measurement process, a total of 5,048 measurements were performed since there were 12 sensors, 15 temperature points, 5 repetitions, 5 structural health conditions including undamaged as well as additional 548 extra repetitions that were necessary during the experiment.
4. IMPEDANCE SIGNATURE RESULTS AND DAMAGE METRICS The monitoring results of the ISHM system are shown in Figure 5. The effect of temperature is shown in Fig. 6(a), where a total of 15 acquisitions from PZT#6 were isolated: three groups of signatures can be distinguished from this total since 5 measurements were obtained at -10 ºC, -5 ºC and 0 ºC, respectively. The temperature compensation is illustrated in Fig. 6(b), where a compensation routine was implemented, searching for the closest baseline in terms of temperature. In this case, the baseline was recorded at -4.1 ºC. The temperature read by the temperature sensor was slightly different from the target value in the environmental chamber, although for each temperature point a time of 30 minutes was defined as settling time for a stable condition inside the chamber. Figure 6(c) shows the average of the CCD values from PZT#6 for each temperature and structural health condition. UCCD stands for Uncompensated CCD and CCCD stands for Compensated CCD. One can observe that without the temperature compensation procedure, the ISHM system would easily make false decisions. In order to refine the relevant results, Fig. 6(d) shows the CCCD values for each Structural State as obtained from PZT#6.
(a)
(b)
(c)
(d)
Figure 6. Monitoring results: (a) Variations on impedance signals due to temperature change; (b) Results of the temperature compensation procedure; (c) Damage Metrics UCCD and CCCD for PZT#6; (d) CCCD indexes for each structural health state for PZT#6.
In order to simplify the results for the 12 PZTs, an average of each CCCD value was made, resulting in Fig. 7, where the average from all 15 temperature conditions for each PZT was grouped according to its temperature and structural state. It can be seen that with the temperature compensation procedure, the results show a somewhat proportional increasing trend on the damage metric for all PZT sensors as the damage level is increased.
Figure 7. Global results for each PZT averaged throughout the temperature range.
These results show that it is possible to determine through statistics a threshold level that leads the ISHM system to indicate correctly the presence of damage even with temperature variations. The next section contains the statistical analysis procedure used in order to determine a threshold level for each PZT, so that the damage detection system is able to distinguish a healthy state from a faulty state. 5. STATISTICAL ANALYSIS OF DATA AND HIT RATE In SHM, it is highly desirable that a reliable threshold level to be established based on the information acquired with the pristine condition of the structure. After the data is acquired and processed, the next step is the preparation of the raw data for analysis. The first data operation is data editing. This refers to the pre-analysis operations that are designed to detect and eliminate spurious or degraded data signals that might have resulted from acquisitions and recording processes such as excessive noise, signal dropouts, or even from an external cause such as a power supply failure [32]. 5.1. Outlier Detection and Data Cleansing Through Chauvenet’s Criterion Sometimes, one measurement in a series of measurements appears to disagree strikingly from all others. When this happens, the experimenter must decide whether the anomalous measurement resulted from some mistake and should be rejected or was a genuine measurement that should be used with all the others. By doing a data cleansing
procedure, some important statistical parameters and thus the threshold will not be affected by possible outliers. It should be noted that despite the fact that discarding a measurement might sound unjustified, the measurements were repeated many times. If the anomaly shows up more than once, this procedure enables to trace the anomaly’s cause either as a mistake or a real physical effect. All samples analyzed in this study had a minimum size of 75, and each frequency point of the impedance is the average of 2,048 measurements performed by the DAQ hardware [31]. The use of the Chauvenet’s Criterion is shown in the next section as a way to improve the hit rate (or probability of detection) of the ISHM system. The Chauvenet’s criterion was selected in this study due to its simplicity and good performance. 5.1.1. Chauvenet’s Criterion The goal of the Chauvenet’s criterion is to remove values from the sample that have a dispersion relative to the average greater then a standard relative deviation. As normally given, the Chauvenet’s criterion states that if the expected number of measurements at least as deviant as the suspect measurement is less than one-half, then the suspect measurement should be rejected [33]. This criterion presumes that the errors are normally distributed and have constant variance, and specifies that any reading out of a sample of N readings shall be rejected if the magnitude of its deviation dmax from the mean value of the sample is such that the ! probability of occurrence of such deviation exceeds !! . Equation 8 gives the maximum deviation: 𝑑!"# = 0.819 + 0.544× ln 𝑁 − 0.02346×𝑙𝑛 N ! 𝑠
(8)
where 𝑠 is the standard deviation of the series and N is the number of data points. Literature recommendation establishes that Chauvenet’s criterion should not be applied a second time, therefore this recommendation was strictly followed in the present study [34]. Furthermore, the assumption that the sample follows a normal distribution was verified through normality tests. 5.1.2. Normality Tests Among the existent normality tests from the statistics literature, two were chosen to check the samples for normality. In order to be succinct, in this work, results are presented for the PZT#6, which has shown good performance as can be seen in Fig. 6(d) although all other sensors were tested in the same way. The Jarque-Bera (JB) test and the Lilliefors (LF) test were chosen since these tests seem to be statistically more rigorous and fitted to our data samples. The Chi-square goodness-of-fit test was discarded since it presents different results depending on how the sample is binned. The drawback of the Komolgorov-Smirnov test for this case was that this test relies on the information from the population that generated the test sample, whereas in the LF test the basic statistic parameters from the sample are used instead of the population, which in our case is
initially unknown. The JB test uses the statistical moments of Skewness and Kurtosis in order to check if a sample comes from a Gaussian Probability Density Function (PDF). In this work, the MATLAB® codes from the statistics toolbox were used and are briefly described in this section. Both JB and LF tests are two-sided goodness-of-fit tests suitable when a fullyspecified null distribution is unknown and its parameters must be estimated. Equation 9 defines the JB test statistic: 𝐽𝐵 =
𝑁 ! 𝐾! − 3 𝑆! + 6 4
!
(9)
where N is the sample size, Sk is the sample skewness, and Kt is the sample kurtosis. Equation 10 defines the LF test statistic: 𝐿𝐹 = 𝑚𝑎𝑥! 𝑆𝐶𝐷𝐹 𝑥 − 𝐶𝐷𝐹 𝑥
(10)
where SCDF is the empirical cumulative distribution function (CDF) from the sample and CDF is the normal CDF with mean and standard deviation equal to the sample’s parameters. For these two normality tests, sample sizes less than 1,000 and significance levels between 0.001 and 0.50, the test uses a table of critical values computed using a MonteCarlo simulation. Table 03 summarizes the normality test results: Table 3: Normality Test Results for PZT#6 Skewness Kurtosis Statistics 0.6083
2.6546
JB test LF test
H0
p-value[%]
Lim. Val.
0 0
5.45 8.05
4.9982 0.0967
Crit. Val. 5.2745 0.1024
In Table 3, the Skewness and Kurtosis are the statistical moments of the sample analyzed. A simple normality test is to verify that the Skewness of the sample is close to zero, thus indicating symmetry in the PDF, as well as a Kurtosis value of three, indicating the peak shape similar to the classical bell-shape of a Gaussian distribution. H0 corresponds to the null hypothesis “the sample follows a normal distribution” and the result in both cases being 0 means that the null hypothesis should not be rejected. The p value states the result for the hypothesis test. If the p-value is smaller than the significance level, 𝛼, (in this case, 𝛼 = 5%), it means the null hypothesis should be rejected. The two last columns correspond to the limit values and critical values of the normality tests. In order for the null hypothesis to be accepted, the critical value should be higher than the limit value. These results follow the notation used in the MATLAB® statistics toolbox. Therefore, the results indicate that the sample could come from a Gaussian distribution with 95% confidence.
5.2. Threshold Determination with Statistical Process Control and Confidence Intervals As normality has been confirmed, a concept from Statistical Process Control (SPC) was used to determine the Upper Control Limit (UCL) and the Lower Control Limit (LCL) of the damage indexes, since the SPC technique assumes that successive deviations form a normally distributed function. A control chart is shown at Fig. 8 for the damage indexes from PZT#6 in the undamaged cases with only temperature variation.
Figure 8. Control chart for the undamaged case obtained from PZT6. Figure 8 shows that no anomalies were detected. The central line corresponds to the sample mean, which in this case is 1.42%. The other two lines are situated three standard deviations above and below the mean. Between these two limits, according to the normal PDF, 99.73% of all observations shall fit in, if everything is perfectly controlled. Since this is not always the case and we are interested on the upper limit value for the threshold, a methodology in this work is proposed for threshold determination using the concept of confidence interval. A more meaningful procedure for estimating parameters of random variables involves the estimation of an interval, as opposed to a single point value, which will include the parameter being estimated with a known degree of uncertainty. For the case of the mean value estimate, a confidence interval can be established for the mean value 𝜇! based on the sample mean 𝑥 according to Eq. 11: 𝑥−
𝜎! 𝑧!/! 𝑁
≤ 𝜇! < 𝑥 +
𝜎! 𝑧!/! 𝑁
, 𝜈 = 𝑁 − 1
(11)
where 𝜇! and 𝜎! are the population mean and standard deviation, respectively; x is the sample mean, N is the sample size, 𝜈 is the number of DOF, and 𝑧!/! is the standardized variable given by Eq. 12 associated with the significance level 𝛼:
𝑧=
𝑥 − 𝜇! 𝜎!
(12)
Since 𝜎! is unknown, the confidence intervals for the mean 𝜇! and variance 𝜎!! can be determined. For a sample of size N, one can show that Eq. 13 and Eq. 14 [32] give the confidence intervals for the mean and variance values, respectively: 𝑥−
𝑠𝑡!;!/! 𝑁
≤ 𝜇! < 𝑥 +
𝑠𝑡!;! !
𝑁
, 𝜈 = 𝑁 − 1
𝜈𝑠 ! 𝜈𝑠 ! ! ≤ 𝜎 < , 𝜈 = 𝑁 − 1 ! ! ! 𝜒!;!/! 𝜒!;!!!/!
(13) (14)
where s ! is the sample variance, t !;!/! is a student t variable with ν DOF and χ!!;!/! is a chi-square variable with ν DOF. Therefore, these intervals were obtained and the threshold was determined according to Eq. 15: 𝑃𝑍𝑇!!!"#!!"# = 𝜇! !"# + 3 ∗ 𝜎! !"#
(15)
where µ! !"# is the upper limit for the population mean and σ! !"# is the upper limit for the population standard deviation, both obtained choosing a significance level α = 5% applied to Eqs. 13 and 14. It should be noted that the choice of the decision threshold influences both the detectable size and the probability of a false positive. Additionally, the confidence interval obtained with Eq. 13 is generally wider than the one obtained with Eq. 11, although this difference decreases as the sample size increases. However, Eq. 11 provides a confidence interval of the mean with the assumption that the sample variance is equal to the population variance. In this work, Eq. 11 was used to estimate the mean confidence interval since normality has been previously checked. 6. RESULTS AND DISCUSSION A comparison of all (5,048) values of the CCCD damage index with the thresholds of each respective PZT thresholds resulted in a verdict about the system sensitivity evaluation: a damage was found (hit), otherwise, the damage was not found (miss). This comparison was made also for the case in which all specimens were intact, in order to perform a false-positive rate analysis. Tables 4 and 5 show the results of the Hit Rates (HR) as well as probability of false positive (PFP) rates for each PZT in each structural health state, where POD is given by the ratio of the total number of correct diagnosis by the total number of evaluations, which in this case is 75. In order to check the effectiveness of the data cleansing procedure through Chauvenet’s Criterion, Table 4 contains all hit rates without the data cleansing and Table 5 contains all hit rates with the data cleansing procedure.
TABLE 04. HIT RATE RESULTS WITHOUT DATA CLEANSING Sensor # PFP 3mm cut 5mm cut 7mm cut 10mm cut PZT1 0% 49.41% 73.33% 100% 100% PZT2 0% 35.29% 80% 100% 100% PZT3 0% 10.59% 30.67% 100% 100% PZT4 0% 25.88% 100% 100% 100% PZT5 6.67% 22.35% 38.67% 83% 100% PZT6 0% 49.41% 80% 90% 100% PZT7 2,67% 0% 18.67% 44% 76.74% PZT8 2,67% 7.06% 42.67% 57% 65.12% PZT9 0% 51.76% 92% 100% 100% PZT10 0% 0% 66.67% 100% 100% PZT11 0% 48.24% 69.33% 90% 100% PZT12 0% 29.41% 73.33% 100% 100% Sensor # PZT1 PZT2 PZT3 PZT4 PZT5 PZT6 PZT7 PZT8 PZT9 PZT10 PZT11 PZT12
TABLE 05. HIT RATE RESULTS WITH DATA CLEANSING PFP 3mm cut 5mm cut 7mm cut 10mm cut 0% 49.41% 73.33% 100% 100% 0% 37.65% 86.67% 100% 100% 0% 10.59% 30.67% 100% 100% 0% 29.41% 100% 100% 100% 0% 29.41% 73.33% 95% 100% 0% 49.41% 80% 90% 100% 0% 9.41% 77.33% 70% 82.56% 0% 29.41% 46.67% 65% 76.74% 0% 51.76 92% 100% 100% 0% 0% 66.67% 100% 100% 0% 48.25% 69.33% 90% 100% 0% 29.41% 74.67% 100% 100%
Results from Table 05 show that using the data cleansing technique, the major part of sensors could detect a 7 mm crack at a distance of approximately 280 mm, with a Hit Rate higher than 90%, except for sensors #7 and #8; also, the Chauvenet’s Criterion enabled complete removal of undesired false positives. The quantitative result of all measurements can be evaluated in Fig. 9 (without data cleansing) and Fig. 10 (with data cleansing), where at the vertical axis, zero means that the sensor did not perceived damage and one means that the sensor did detect a damage in that measurement. At the horizontal axis, the results are grouped randomly (MATLAB®’s function rand was used) in squares around saw cut lengths in millimeters.
Figure 9 – Illustrative result of 5,048 measurements with the hit/miss verdict (each red point “.” Represents one measurement).
Figure 10 – Illustrative result of 5,048 measurements with the hit/miss verdict after data cleansing. 7. CONCLUSION This paper has presented a statistical method that was designed to determine a threshold level for impedance-based SHM data and perform a sensitivity analysis of the ISHM technique. This method takes into account the concepts of statistical process control and confidence interval from the statistics theory. The decision threshold obtained
with the proposed method was used to perform a sensitivity analysis through the Hit Rate as the damage detection system could successfully detect the introduced damage. The tests were verified by using experimental data as applied to aircraft aluminum panels under varying temperature. In addition, a method to perform data cleansing with the Chauvenet’s criterion was presented to eliminate false-positives, aiming at improving the overall POD results. The experiment performed through this study used the temperature compensation method known as effective frequency shift (EFS) through correlation analysis. This procedure has shown good results for the temperature range analyzed in this work (-10 ºC to 60 ºC) by using a total of seven baseline signatures. The statistical analysis included normality tests as well the verification of the samples from all sensors regarding Gaussian distribution. With the temperature compensation procedure applied, results have shown that the ISHM method can provide a measure of damage level since there was a proportional increasing trend on the damage metrics as the damage level was increased. The statistical analysis with the Hit Rate table has shown that for this inspection of the ISHM system, namely a 7 mm saw cut distant of approximately 280 mm from each sensor could be detected with 95% confidence for 9 out of 12 sensors used in the experiment (see Table 05). It is important to mention that the cuts inserted in the specimens are not the same as fatigue cracks, which are the occurrence of localized and progressive damage due to continuous varying stresses. Since the structural change due to a real fatigue crack is much more significant in the material as compared to a simple saw cut as performed in the present study, considering the crack nucleation and growth, the ISHM should potentially detect a real fatigue crack of smaller sizes. Therefore, real fatigue cracks could be detected with smaller lengths, since this kind of damage produces significant changes in the structural stiffness that are different from the saw cut used in the present work. A dynamic experiment is one of the future research topics to be investigated by the authors. Another topic to be further investigated is the use of the POD Curve according to the standard MIL-1823A [28]. Finally, the authors see this contribution as a necessary step for an effective decisionmaking in the development of a reliable SHM process. Also, the authors have designed a compact impedance analyzer with an associated impedance post processing software, which includes temperature compensation and statistical analysis package aiming at obtaining an onboard/online SHM system. 8. ACKNOWLEDGEMENTS The first author is thankful to FAPEMIG for his PhD scholarship grant number 11302. The authors are also thankful to CNPq, FAPEMIG, and CAPES, Brazilian research agencies, through the INCT-EIE. We thank EMBRAER as well for the interest on this research project. 9. REFERENCES [1] Farrar, CR, Worden, K (2012) Structural Health Monitoring: A Machine Learning Perspective. John Wiley & Sons, Ltd, Chichester, UK.
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