detection of structural cracks using piezoelectric active sensors
DETECTION OF STRUCTURAL CRACKS USING PIEZOELECTRIC ACTIVE SENSORS Jerome P. Lynch1 (Member, ASCE)
ABSTRACT This study proposes a novel approach to identifying crack damage common to metallic structural elements under extreme cyclic loading. Using two piezoelectric pads surface mounted to a structural element, broadband signals can be used to locally excite the element using one pad and the corresponding system response measured with a second pad. An auto-regressive with exogenous input (ARX) system identification model is then fit to the collected system input-output response data. Complex roots (poles) of the ARX model’s characteristic equation are sensitive to structural damage causing a change in their location on the complex plane. To quantitatively identify the migration of poles in the complex plane, bivariate normal density functions are fit to poles of the same response mode observed during repeated excitation of the element. Certain pole clusters are shown to be more sensitive to damage resulting in statistically significant changes in the fitted normal density function for that cluster. Using the mean of the normal density function, the migration of ARX model poles upon the complex plane are shown to also be well correlated to damage severity. To illustrate the efficacy of the method, a simple aluminum plate with a cut at its midsection is monitored using piezoelectric active sensors commanded by a wireless active sensing unit. Keywords: active sensors, damage detection, wireless sensing, structural health monitoring
INTRODUCTION Structural damage in the form of cracks is common in metallic structures that are exposed to extreme loading scenarios. The 1994 Northridge earthquake, which occurred in the Los Angeles metropolitan region, caused severe damage to many of the region’s civil structures. Structural inspections after the earthquake discovered cracking in the welded column-beam connection of steel moment frames. While some of the steel moment frame structures had discernable permanent drifts, the majority of these structures had no outward signs of distress. Subsequent visual inspection of welded steel moment frame connections revealed the formation 1
Corresponding author: Department of Civil and Environmental Engineering, University of Michigan, Ann Arbor, MI 48109-2125, USA, email: [email protected]
of severe cracks originating from the weld and propagating into the column, with some of the cracks even severing the column completely. While visual inspection is widely used to identify cracks in structural elements, this method is time consuming, labor intensive and expensive. For example, the inspection of steel moment frame connections after the Northridge earthquake cost between a few hundred and one thousand dollars per connection (Hamburger 2000). Today, the field of structural monitoring is rapidly evolving as many new sensing technologies are explored for adoption. For example, wireless communication and mobile computing technologies integrated with sensors have produced monitoring systems with greater functionality and reduced costs, when compared to traditional cable-based alternatives (Lynch 2002). Wireless sensing units installed in civil structures can monitor the behavior of the structure under ambient and forced loads using passive sensors such as accelerometers or strain gages, just to name a few. With computing resources included with the wireless sensing unit, embedded engineering algorithms can be locally executed by the sensing unit in real-time. Algorithms that can interrogate measurement data for indications of structural damage have been of extreme interest in recent years (Doebling et al. 1996). While researchers have proposed many methods for detecting damage in civil structures using global response measurements, these methods assume that only a handful of sensors are installed in a structure. As the cost of wireless sensing continues to decline, opportunities to install monitoring systems with larger sensor densities arise. In these emerging large-scale monitoring systems, some sensors could be installed to monitoring the global behavior of the structure, while others might be intended to monitor the local behavior of critical structural elements or joints. Researchers have recently extended the functionality of wireless sensing units with capabilities to actuate the structural system in which they are installed (Lynch et al. 2004). The intended use of these wireless active sensing units is to command many of the non-destructive evaluation (NDE) technologies that can be used to locally investigate a structure for damage. The wireless active sensing unit can emit acoustic and ultrasonic waves into structural elements using piezoelectric active sensors mounted to the surface of the structure. Active sensors differ from passive sensors in that they are capable of exciting the system in which they are installed, while simultaneously recording the corresponding response to the excitation (Brooks and Iyengar 1998). In recent years, piezoelectric active sensors have been used to detect a variety of damage mechanisms within civil structures (Park et al. 2000, Wu and Chang 2001). Based on the scattering of acoustic and ultrasonic waves emitted into an elastic material by a piezoelectric active sensor, the initiation of cracks in the vicinity of the sensor can be identified (Achenback et al. 1982). In this study, a wireless active sensing unit is employed to emit acoustic waves into a metallic plate to identify the existence of structural damage in the form of cracks. The motivation of the study is to illustrate the potential applicability of using wireless active sensing units to monitor structural elements for the initiation of cracks, such as those formed during the Northridge earthquake in welded steel moment frames. An aluminum plate to which piezoelectric pads have been surface mounted is intentionally cut to simulate crack damage. Using the wireless active sensing unit, the plate is excited with broadband acoustic excitations using one piezoelectric pad, while a second pad is used to record the plate’s corresponding strain response. For each set of input-output responses, autoregressive with exogenous input (ARX) time series models are fit. Roots of the ARX model characteristic equation will be used to
2
identify the occurrence and severity of structural damage within the aluminum plate. ALUMINUM PLATE-PIEZOELECTRIC ACTIVE SENSOR EXPERIMENT A slender aluminum plate (28.6 cm x 6.8 cm x 0.3175 cm) is chosen to serve as the study’s structural element to be monitored using piezoelectric active sensors. Piezoelectrics are a low-cost material that experience strain when a voltage potential is applied across them; similarly, the material outputs a voltage when strained. This unique mechanical-electrical material property has prompted the use of piezoelectric materials as both actuators and sensors in a variety of smart structure applications. Two lead-zirconate-titanate (PZT) piezoelectric pads are mounted to the top surface of the aluminum plate roughly 19 cm apart. One piezoelectric pad will be used in an actuation capacity in order to excite the aluminum plate with low-energy surface acoustic waves. The second piezoelectric pad is intended to act as a sensor that will output a voltage proportional to the strain in the plate surface at the second pad location. Surface acoustic waves traveling across the plate will be sensitive to any cracks that might lie in the vicinity of the two piezoelectric pads. Cracks represent an obstacle in the path of the acoustic wave that will cause the wave to scatter. A prototype wireless active sensing unit proposed by Lynch et al. (2004) is used to issue command signals to the actuation piezoelectric pad, while simultaneously recording the response of the plate using the second piezoelectric pad. The wireless active sensing unit is operated at its maximum sample rate of 40 kHz to record the input-output behavior of the plate. This maximum sample rate will illuminate system modes that are below the Nyquist frequency (20 kHz). The resolution of the wireless active sensing unit actuation interface is 12-bits, while that of the sensing interface is 10-bits. Various excitations are considered for exciting the structural element but ultimately, broadband white noise signals are chosen. Specifically, a set of 12 white noise signals with varying levels of energy are applied to the elastic plate. All of the excitation signals are within the +/- 5V saturation limits of the actuation interface and therefore draw little power from the wireless active sensing unit power supply. With a constant power spectrum, the white noise signals are a valuable tool for exciting all of the piezoelectric-aluminum plate system modes that fall below the Nyquist frequency. The excitation signals are stored in the wireless active sensing unit memory so that they can be repeatedly applied to the plate upon demand. Fig. 1 is a picture of the aluminum plate being monitored by the wireless active sensing unit using piezoelectric active sensors. Also shown is one white noise signal from the excitation set and the corresponding plate response as recorded by the second piezoelectric pad. To simulate cracking of the aluminum plate, a hack saw is used to introduce a cut orthogonal to the plate’s longitudinal axis. The plate is cut in multiple steps resulting in simulated cracks of various lengths. In total, three cuts are made: 1 cm, 2 cm and 3 cm long. For each cut, the 12 white noise excitations stored in the wireless active sensing unit memory bank are identically applied to the aluminum plate. MODELING THE DYNAMIC BEHAVIOR OF THE PLATE SYSTEM Assuming the slender aluminum plate to be elastic, system identification models can be fit to the input-output response time-histories of the system. An autoregressive with exogenous input (ARX) time-series model is selected to serve as a model of the plate’s input-output dynamic
3
Input White Noise with Mean = 0 V and STD = 1.0 V 4 2 0
V
Wireless Active Sensing Unit
−2 −4
0
0.01
0.02
0.03 0.04 0.05 0.06 0.07 Output Signal of Receiving Pad
0.08
0.09
0.1
0
0.01
0.02
0.03
0.08
0.09
0.1
4
V
3 2 1
PZT Pads
0
0.04
0.05 t (sec)
0.06
0.07
FIG. 1. (Left) aluminum plate monitored with a wireless active sensing unit; (right) input white noise and corresponding colored response time-histories behavior. ARX models are essentially linear difference equations that weigh observations of the system output, y, with those of the system input, u, to make a prediction of the system output, ŷ, at time step k:
yˆ (k ) = −a1 y (k − 1) + L − a na y (k − n a ) + b1u (k − 1) + L bnb u (k − nb )
(1)
The weights on the past observations of the system output are a, while those used for the input are b. The number of weights on the input and output, denoted na and nb respectively, are chosen so that only the dynamic behavior of the aluminum plate is captured by the model. If model sizes are selected as too large, the model will capture both the mechanical dynamics of the plate and any electrical noise present in the data acquisition system. For the aluminum plate response data, the appropriate model size is chosen based upon the norm of the residual error of the model using a validation input-output data set separate from the set used to fit the model. As the model order increases, the residual error decreases until a point is reached where the error begins to increase; the final ARX model order is chosen as the point where the error begins to increase. In this study, the optimal ARX model order is determined to be na = 21 and nb = 4. Once an ARX time-series model has been fit to an input-output response pair, the Z-transform is used to derive the system transfer function in the complex z-domain. H ( z) =
b1 z −1 + Lb4 z −4 Y ( z) = U ( z ) 1 + a1 z −1 + L a 21 z − 21
(2)
The denominator of the transfer function is denoted as the polynomial characteristic equation of the linear system. An interesting feature of the characteristic equation is that the complex roots of the equation, also termed poles, embody the frequency, ω, and damping ratio, ξ, of each uncoupled response mode (where T is the time-step of the sampled system):
4
Poles of Undamaged Plate 1
10
14
0.6
18
π/2T
0.2
π/T π/T
−0.2 −0.4 −0.6 −0.8
20
6
9π/10T
2
π/10T
3
π/2T
0.2 0
π/2T
−0.4
5 17 2π/5T
7
15 13
−0.5
11
9
3π/5T
−0.6
π/5T
−0.8
7π/10T 3π/10T 0 Real(z)
12
16
20
−1 −1
1
8
0.1π/5T 0.2 0.3 6 0.4 3π/5T 0.5 4 0.6 0.7 π/10T 0.8 2 0.9
1 3
21 19 9π/10T
7
15 2π/5T
−0.5
π/10T
5
17 13 4π/5T
0.5
10
π/T π/T
−0.2
π/10T
3π/10T 7π/10T
18
0.4
19 9π/10T
14
0.6
3π/5T
21
π/2T
2π/5T
4
1
4π/5T −1 −1
0.8
0.1
8 0.2 π/5T 0.3 0.4 0.5 0.6 0.7 0.8 0.9
9π/10T 16
0.4
Imag(z)
12
2π/5T
4π/5T
Imag(z)
0.8
0
Poles of Damaged Plate − 3cm Cut 1
3π/10T 4π/5T 7π/10T
11 9
3π/5T
π/5T
7π/10T 3π/10T 0 Real(z)
0.5
1
FIG. 2. ARX model characteristic equation root locations on the complex plane for the (left) undamaged and (right) damaged plate (3 cm cut)
−ξω ±ω 1−ξ 2 j T n n
z ROOT = e
(3)
The poles of the system transfer function can be drawn upon the real-imaginary complex plane. Poles within the unit circle are denoted as stable, while those outside the unit circle represent unstable modes of the system. The complex plane can serve as a powerful visualization tool since contours of constant frequency and constant damping can be superimposed upon the plane. For this reason, the complex plane will be used in this study as an expressive domain to analyze if a structure should be characterized as damaged or undamaged. When cracks appear in the aluminum plate, some of the system poles might migrate in the complex plane as a result of subtle changes in the system stiffness and damping. DIAGNOSIS OF DAMAGE IN THE PLATE The piezoelectric-aluminum plate system is first analyzed in an undamaged state with the set of 12 excitations applied. After the excitations are applied to the plate, an ARX model is calculated for each input-output response pair. The ARX model poles are plotted in Fig. 2 for both the undamaged and damaged (3 cm cut) plate. Superimposed on the complex plane are contours of constant damping originating from z = 1 and contours of constant frequency originating from the unit circle boundary. For each ARX model, 21 poles are plotted; the first pole is plotted as a real number representing the first response mode and the remaining poles are plotted as conjugate pole pairs representing the second through the eleventh modes. When the
5
POLE CLUSTER #2 0.3
POLE CLUSTER #6 0.75 0.95
0.25
0.7
Imag(z)
π
Imag(z)
Imag(z)
0.35 0.4 .5 0.3
POLE CLUSTER #12
0.65
/5T 0.9
0.85 0.2
0.1
0.6 0.75
0.8 0.85 0.9 Real(z) POLE CLUSTER #2
0.45
0.5 0.55 Real(z) POLE CLUSTER #6
0. 0.6
−0.25
−0.2 −0.15 Real(z)
FIG. 3. Migration of the estimated bivariate normal distributions for poles on the complex plane as the aluminum plate is cut (arrow indicates migration from no damage to a 1, 2 and 3 cm cut)
poles are plotted for all 12 excitation sources, the poles form 21 clusters that reveal variability in their location upon the complex plane. The variability in the pole locations can be attributed to low levels of electrical noise within the actuation and sensing interfaces of the wireless active sensing unit. In comparing the pole clusters of the undamaged and damaged plate, some of the clusters migrate on the complex plane as a result of the plate damage. To characterize the migration of the pole clusters, bivariate normal probability density functions are first determined using the poles of each pole cluster for the plate in the undamaged and damaged states. Superimposed with each pole cluster in Fig. 2 are elliptical level curves that represent the first standard deviation of the estimated normal density functions. As the crack in the aluminum plate grows, some of the ARX model pole clusters are observed migrating on the complex plane in concert with the cut length. The normal probability density functions for each pole cluster are tracked upon the complex plane as the plate is incrementally damaged. While all of the clusters exhibit some migration, it is observed that pole clusters 2, 6 and 12 exhibit the greatest sensitivity to the growing cut. Fig. 3 presents the migration of the estimated normal density function for each of these damage sensitive pole clusters. For all three clusters, the migration of the cluster mean moves a significant amount with the first introduction of damage in the plate. For the 1 cm long cut, the mean of the pole cluster is outside of the level curve corresponding to the first standard deviation of the normal density function of the undamaged plate; clearly, this is a statistically significant migration. As damage is increased, the means of the pole clusters move an increasing distance away from the corresponding undamaged plate’s cluster means. The absolute difference between the estimated normal density function mean of the damaged and undamaged plate is measured for each cut length. Presented in Fig. 4 for clusters 2, 6 and 12, the absolute difference in the cluster mean increases in tandem with the growing cut length. Also shown in Fig. 4 respectively in dashed and dotted lines are the standard deviations of the estimated normal density function in the real and imaginary coordinates, σRe and σIm, for the undamaged plate clusters. For the first damage state where the cut is only 1 cm, the absolute mean difference for cluster 2 and 6 is approximately
6
POLE CLUSTER #2
POLE CLUSTER #6
0
1 2 Cut Length
3
0.1
|∆ µ|
0.05 0
POLE CLUSTER #12
0.1
|∆ µ|
|∆ µ|
0.1
0.05 0
0
1 2 Cut Length
3
0.05 0
0
1 2 Cut Length
3
FIG. 4. Absolute difference of the pole cluster mean for the damaged and undamaged plate (dotted lines represent the standard deviation of the undamaged plate cluster)
equal to the standard deviations of the estimated normal density function of the undamaged plate. However, as the cut lengthens, the migration of the cluster becomes increasingly significant as shown by Fig. 4. Hence, the cluster mean appears to be a promising indicator of crack damage in the aluminum plate that is strongly correlated to the crack length. CONCLUSIONS This study has explored a new monitoring technique for detecting crack damage in metallic structural elements using piezoelectric active sensors. A wireless active sensing unit was used to collect the input-output response of an aluminum plate excited with broadband white noise excitation signals using piezoelectric pads mounted to the plate surface. Using the characteristic equation roots of ARX models fit to the input-output response data, damage was successfully identified in the aluminum plate. In particular, a set of excitations are applied to the plate in a given structural state (undamaged versus damaged) so that the distribution of ARX model poles upon the complex plane can be statistically described using estimated bivariate normal probability density functions. As damage is introduced into the aluminum plate, changes in the plate stiffness and damping are manifested by migration of pole clusters upon the complex plane. In particular, the mean of the pole clusters corresponding to the second, fourth and seventh response mode (clusters 2, 6 and 12) migrated from the undamaged plate cluster mean in concert with the cut length. The migration of the pole clusters were shown to be statistically significant by comparing the absolute difference of cluster means with the standard deviations of the undamaged plate’s estimated normal probability density functions. Future work is needed to further explore the suitability of using pole clusters in the complex plane for damage detection. Currently, theoretical models of the plate system are being created to validate the proposed methodology and to ascertain ways of identifying a priori those pole clusters most sensitive to structural damage. The wireless active sensing unit used to collect the input-output response data is also being improved so that data can be collected at higher sample rates. High sample rates could illuminate higher order response modes that might exhibit greater sensitivity to structural damage.
7
ACKNOWLEDGMENTS The author would like to express his gratitude to Arvind Sundararajan and Prof. Kincho Law of Stanford University and Drs. Hoon Sohn and Chuck Farrar of Los Alamos National Laboratory for providing assistance in the design and fabrication of the wireless active sensing unit prototype. This research is partially funded by the National Science Foundation (grant number CMS-9988909) and Los Alamos National Laboratory (contract number 75067-001-03). REFERENCES Achenbach, J. D., Gautesen, A. K. and McMaken, H. (1982), Ray Methods for Waves in Elastic Solids, Pitman Advanced Publishing, Boston, USA. Brooks, R. R. and Iyengar, S. (1998), Multi-sensor fusion, Prentice Hall, Upper Saddle River, NJ. Doebling, S. W., Farrar, C. R. and Prime, M. B. (1998), “A Summary Review of Vibration-based Damage Identification Methods,” Shock and Vibration Digest, 30(2), 91-105. Hamburger, R. O. (2000), A Policy Guide to Steel Moment-frame Construction, Federal Emergency Management Agency (FEMA), Report Number 354, Washington D. C., USA. Lynch, J. P. (2002), Decentralization of Wireless Monitoring and Control Technologies for Smart Civil Structures, John A. Blume Earthquake Engineering Center, Technical Report Number 140, Stanford, CA, USA. Lynch, J. P., Sundararajan, A., Law, K. H., Sohn, H. and Farrar, C. (2004), "Piezoelectric Structural Excitation using a Wireless Active Sensing Unit," Proceedings of the 22nd International Modal Analysis Conference (IMAC XXII), Dearborn, MI, USA, January 26-29. Park, G., Cudney, H. H. and Inman, D. J. (2000), “Impedance-based Health monitoring of civil structural components,” Journal of Infrastructure Systems, 6(3), 153-160. Wu, F. and Chang, F. K. (2001), “A Built-in Active Sensing Diagnostic System for Civil Infrastructure Systems,” Proceedings of 8th International Symposium on Smart Structures and Materials, San Diego, CA, March 5-7.
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ABSTRACT This study proposes a novel approach to identifying crack damage common to metallic structural elements under extreme cyclic loading. Using two piezoelectric pads surface mounted to a structural element, broadband signals can be used to locally excite the element using one pad and the corresponding system response measured with a second pad. An auto-regressive with exogenous input (ARX) system identification model is then fit to the collected system input-output response data. Complex roots (poles) of the ARX model’s characteristic equation are sensitive to structural damage causing a change in their location on the complex plane. To quantitatively identify the migration of poles in the complex plane, bivariate normal density functions are fit to poles of the same response mode observed during repeated excitation of the element. Certain pole clusters are shown to be more sensitive to damage resulting in statistically significant changes in the fitted normal density function for that cluster. Using the mean of the normal density function, the migration of ARX model poles upon the complex plane are shown to also be well correlated to damage severity. To illustrate the efficacy of the method, a simple aluminum plate with a cut at its midsection is monitored using piezoelectric active sensors commanded by a wireless active sensing unit. Keywords: active sensors, damage detection, wireless sensing, structural health monitoring
INTRODUCTION Structural damage in the form of cracks is common in metallic structures that are exposed to extreme loading scenarios. The 1994 Northridge earthquake, which occurred in the Los Angeles metropolitan region, caused severe damage to many of the region’s civil structures. Structural inspections after the earthquake discovered cracking in the welded column-beam connection of steel moment frames. While some of the steel moment frame structures had discernable permanent drifts, the majority of these structures had no outward signs of distress. Subsequent visual inspection of welded steel moment frame connections revealed the formation 1
Corresponding author: Department of Civil and Environmental Engineering, University of Michigan, Ann Arbor, MI 48109-2125, USA, email: [email protected]
of severe cracks originating from the weld and propagating into the column, with some of the cracks even severing the column completely. While visual inspection is widely used to identify cracks in structural elements, this method is time consuming, labor intensive and expensive. For example, the inspection of steel moment frame connections after the Northridge earthquake cost between a few hundred and one thousand dollars per connection (Hamburger 2000). Today, the field of structural monitoring is rapidly evolving as many new sensing technologies are explored for adoption. For example, wireless communication and mobile computing technologies integrated with sensors have produced monitoring systems with greater functionality and reduced costs, when compared to traditional cable-based alternatives (Lynch 2002). Wireless sensing units installed in civil structures can monitor the behavior of the structure under ambient and forced loads using passive sensors such as accelerometers or strain gages, just to name a few. With computing resources included with the wireless sensing unit, embedded engineering algorithms can be locally executed by the sensing unit in real-time. Algorithms that can interrogate measurement data for indications of structural damage have been of extreme interest in recent years (Doebling et al. 1996). While researchers have proposed many methods for detecting damage in civil structures using global response measurements, these methods assume that only a handful of sensors are installed in a structure. As the cost of wireless sensing continues to decline, opportunities to install monitoring systems with larger sensor densities arise. In these emerging large-scale monitoring systems, some sensors could be installed to monitoring the global behavior of the structure, while others might be intended to monitor the local behavior of critical structural elements or joints. Researchers have recently extended the functionality of wireless sensing units with capabilities to actuate the structural system in which they are installed (Lynch et al. 2004). The intended use of these wireless active sensing units is to command many of the non-destructive evaluation (NDE) technologies that can be used to locally investigate a structure for damage. The wireless active sensing unit can emit acoustic and ultrasonic waves into structural elements using piezoelectric active sensors mounted to the surface of the structure. Active sensors differ from passive sensors in that they are capable of exciting the system in which they are installed, while simultaneously recording the corresponding response to the excitation (Brooks and Iyengar 1998). In recent years, piezoelectric active sensors have been used to detect a variety of damage mechanisms within civil structures (Park et al. 2000, Wu and Chang 2001). Based on the scattering of acoustic and ultrasonic waves emitted into an elastic material by a piezoelectric active sensor, the initiation of cracks in the vicinity of the sensor can be identified (Achenback et al. 1982). In this study, a wireless active sensing unit is employed to emit acoustic waves into a metallic plate to identify the existence of structural damage in the form of cracks. The motivation of the study is to illustrate the potential applicability of using wireless active sensing units to monitor structural elements for the initiation of cracks, such as those formed during the Northridge earthquake in welded steel moment frames. An aluminum plate to which piezoelectric pads have been surface mounted is intentionally cut to simulate crack damage. Using the wireless active sensing unit, the plate is excited with broadband acoustic excitations using one piezoelectric pad, while a second pad is used to record the plate’s corresponding strain response. For each set of input-output responses, autoregressive with exogenous input (ARX) time series models are fit. Roots of the ARX model characteristic equation will be used to
2
identify the occurrence and severity of structural damage within the aluminum plate. ALUMINUM PLATE-PIEZOELECTRIC ACTIVE SENSOR EXPERIMENT A slender aluminum plate (28.6 cm x 6.8 cm x 0.3175 cm) is chosen to serve as the study’s structural element to be monitored using piezoelectric active sensors. Piezoelectrics are a low-cost material that experience strain when a voltage potential is applied across them; similarly, the material outputs a voltage when strained. This unique mechanical-electrical material property has prompted the use of piezoelectric materials as both actuators and sensors in a variety of smart structure applications. Two lead-zirconate-titanate (PZT) piezoelectric pads are mounted to the top surface of the aluminum plate roughly 19 cm apart. One piezoelectric pad will be used in an actuation capacity in order to excite the aluminum plate with low-energy surface acoustic waves. The second piezoelectric pad is intended to act as a sensor that will output a voltage proportional to the strain in the plate surface at the second pad location. Surface acoustic waves traveling across the plate will be sensitive to any cracks that might lie in the vicinity of the two piezoelectric pads. Cracks represent an obstacle in the path of the acoustic wave that will cause the wave to scatter. A prototype wireless active sensing unit proposed by Lynch et al. (2004) is used to issue command signals to the actuation piezoelectric pad, while simultaneously recording the response of the plate using the second piezoelectric pad. The wireless active sensing unit is operated at its maximum sample rate of 40 kHz to record the input-output behavior of the plate. This maximum sample rate will illuminate system modes that are below the Nyquist frequency (20 kHz). The resolution of the wireless active sensing unit actuation interface is 12-bits, while that of the sensing interface is 10-bits. Various excitations are considered for exciting the structural element but ultimately, broadband white noise signals are chosen. Specifically, a set of 12 white noise signals with varying levels of energy are applied to the elastic plate. All of the excitation signals are within the +/- 5V saturation limits of the actuation interface and therefore draw little power from the wireless active sensing unit power supply. With a constant power spectrum, the white noise signals are a valuable tool for exciting all of the piezoelectric-aluminum plate system modes that fall below the Nyquist frequency. The excitation signals are stored in the wireless active sensing unit memory so that they can be repeatedly applied to the plate upon demand. Fig. 1 is a picture of the aluminum plate being monitored by the wireless active sensing unit using piezoelectric active sensors. Also shown is one white noise signal from the excitation set and the corresponding plate response as recorded by the second piezoelectric pad. To simulate cracking of the aluminum plate, a hack saw is used to introduce a cut orthogonal to the plate’s longitudinal axis. The plate is cut in multiple steps resulting in simulated cracks of various lengths. In total, three cuts are made: 1 cm, 2 cm and 3 cm long. For each cut, the 12 white noise excitations stored in the wireless active sensing unit memory bank are identically applied to the aluminum plate. MODELING THE DYNAMIC BEHAVIOR OF THE PLATE SYSTEM Assuming the slender aluminum plate to be elastic, system identification models can be fit to the input-output response time-histories of the system. An autoregressive with exogenous input (ARX) time-series model is selected to serve as a model of the plate’s input-output dynamic
3
Input White Noise with Mean = 0 V and STD = 1.0 V 4 2 0
V
Wireless Active Sensing Unit
−2 −4
0
0.01
0.02
0.03 0.04 0.05 0.06 0.07 Output Signal of Receiving Pad
0.08
0.09
0.1
0
0.01
0.02
0.03
0.08
0.09
0.1
4
V
3 2 1
PZT Pads
0
0.04
0.05 t (sec)
0.06
0.07
FIG. 1. (Left) aluminum plate monitored with a wireless active sensing unit; (right) input white noise and corresponding colored response time-histories behavior. ARX models are essentially linear difference equations that weigh observations of the system output, y, with those of the system input, u, to make a prediction of the system output, ŷ, at time step k:
yˆ (k ) = −a1 y (k − 1) + L − a na y (k − n a ) + b1u (k − 1) + L bnb u (k − nb )
(1)
The weights on the past observations of the system output are a, while those used for the input are b. The number of weights on the input and output, denoted na and nb respectively, are chosen so that only the dynamic behavior of the aluminum plate is captured by the model. If model sizes are selected as too large, the model will capture both the mechanical dynamics of the plate and any electrical noise present in the data acquisition system. For the aluminum plate response data, the appropriate model size is chosen based upon the norm of the residual error of the model using a validation input-output data set separate from the set used to fit the model. As the model order increases, the residual error decreases until a point is reached where the error begins to increase; the final ARX model order is chosen as the point where the error begins to increase. In this study, the optimal ARX model order is determined to be na = 21 and nb = 4. Once an ARX time-series model has been fit to an input-output response pair, the Z-transform is used to derive the system transfer function in the complex z-domain. H ( z) =
b1 z −1 + Lb4 z −4 Y ( z) = U ( z ) 1 + a1 z −1 + L a 21 z − 21
(2)
The denominator of the transfer function is denoted as the polynomial characteristic equation of the linear system. An interesting feature of the characteristic equation is that the complex roots of the equation, also termed poles, embody the frequency, ω, and damping ratio, ξ, of each uncoupled response mode (where T is the time-step of the sampled system):
4
Poles of Undamaged Plate 1
10
14
0.6
18
π/2T
0.2
π/T π/T
−0.2 −0.4 −0.6 −0.8
20
6
9π/10T
2
π/10T
3
π/2T
0.2 0
π/2T
−0.4
5 17 2π/5T
7
15 13
−0.5
11
9
3π/5T
−0.6
π/5T
−0.8
7π/10T 3π/10T 0 Real(z)
12
16
20
−1 −1
1
8
0.1π/5T 0.2 0.3 6 0.4 3π/5T 0.5 4 0.6 0.7 π/10T 0.8 2 0.9
1 3
21 19 9π/10T
7
15 2π/5T
−0.5
π/10T
5
17 13 4π/5T
0.5
10
π/T π/T
−0.2
π/10T
3π/10T 7π/10T
18
0.4
19 9π/10T
14
0.6
3π/5T
21
π/2T
2π/5T
4
1
4π/5T −1 −1
0.8
0.1
8 0.2 π/5T 0.3 0.4 0.5 0.6 0.7 0.8 0.9
9π/10T 16
0.4
Imag(z)
12
2π/5T
4π/5T
Imag(z)
0.8
0
Poles of Damaged Plate − 3cm Cut 1
3π/10T 4π/5T 7π/10T
11 9
3π/5T
π/5T
7π/10T 3π/10T 0 Real(z)
0.5
1
FIG. 2. ARX model characteristic equation root locations on the complex plane for the (left) undamaged and (right) damaged plate (3 cm cut)
−ξω ±ω 1−ξ 2 j T n n
z ROOT = e
(3)
The poles of the system transfer function can be drawn upon the real-imaginary complex plane. Poles within the unit circle are denoted as stable, while those outside the unit circle represent unstable modes of the system. The complex plane can serve as a powerful visualization tool since contours of constant frequency and constant damping can be superimposed upon the plane. For this reason, the complex plane will be used in this study as an expressive domain to analyze if a structure should be characterized as damaged or undamaged. When cracks appear in the aluminum plate, some of the system poles might migrate in the complex plane as a result of subtle changes in the system stiffness and damping. DIAGNOSIS OF DAMAGE IN THE PLATE The piezoelectric-aluminum plate system is first analyzed in an undamaged state with the set of 12 excitations applied. After the excitations are applied to the plate, an ARX model is calculated for each input-output response pair. The ARX model poles are plotted in Fig. 2 for both the undamaged and damaged (3 cm cut) plate. Superimposed on the complex plane are contours of constant damping originating from z = 1 and contours of constant frequency originating from the unit circle boundary. For each ARX model, 21 poles are plotted; the first pole is plotted as a real number representing the first response mode and the remaining poles are plotted as conjugate pole pairs representing the second through the eleventh modes. When the
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POLE CLUSTER #2 0.3
POLE CLUSTER #6 0.75 0.95
0.25
0.7
Imag(z)
π
Imag(z)
Imag(z)
0.35 0.4 .5 0.3
POLE CLUSTER #12
0.65
/5T 0.9
0.85 0.2
0.1
0.6 0.75
0.8 0.85 0.9 Real(z) POLE CLUSTER #2
0.45
0.5 0.55 Real(z) POLE CLUSTER #6
0. 0.6
−0.25
−0.2 −0.15 Real(z)
FIG. 3. Migration of the estimated bivariate normal distributions for poles on the complex plane as the aluminum plate is cut (arrow indicates migration from no damage to a 1, 2 and 3 cm cut)
poles are plotted for all 12 excitation sources, the poles form 21 clusters that reveal variability in their location upon the complex plane. The variability in the pole locations can be attributed to low levels of electrical noise within the actuation and sensing interfaces of the wireless active sensing unit. In comparing the pole clusters of the undamaged and damaged plate, some of the clusters migrate on the complex plane as a result of the plate damage. To characterize the migration of the pole clusters, bivariate normal probability density functions are first determined using the poles of each pole cluster for the plate in the undamaged and damaged states. Superimposed with each pole cluster in Fig. 2 are elliptical level curves that represent the first standard deviation of the estimated normal density functions. As the crack in the aluminum plate grows, some of the ARX model pole clusters are observed migrating on the complex plane in concert with the cut length. The normal probability density functions for each pole cluster are tracked upon the complex plane as the plate is incrementally damaged. While all of the clusters exhibit some migration, it is observed that pole clusters 2, 6 and 12 exhibit the greatest sensitivity to the growing cut. Fig. 3 presents the migration of the estimated normal density function for each of these damage sensitive pole clusters. For all three clusters, the migration of the cluster mean moves a significant amount with the first introduction of damage in the plate. For the 1 cm long cut, the mean of the pole cluster is outside of the level curve corresponding to the first standard deviation of the normal density function of the undamaged plate; clearly, this is a statistically significant migration. As damage is increased, the means of the pole clusters move an increasing distance away from the corresponding undamaged plate’s cluster means. The absolute difference between the estimated normal density function mean of the damaged and undamaged plate is measured for each cut length. Presented in Fig. 4 for clusters 2, 6 and 12, the absolute difference in the cluster mean increases in tandem with the growing cut length. Also shown in Fig. 4 respectively in dashed and dotted lines are the standard deviations of the estimated normal density function in the real and imaginary coordinates, σRe and σIm, for the undamaged plate clusters. For the first damage state where the cut is only 1 cm, the absolute mean difference for cluster 2 and 6 is approximately
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POLE CLUSTER #2
POLE CLUSTER #6
0
1 2 Cut Length
3
0.1
|∆ µ|
0.05 0
POLE CLUSTER #12
0.1
|∆ µ|
|∆ µ|
0.1
0.05 0
0
1 2 Cut Length
3
0.05 0
0
1 2 Cut Length
3
FIG. 4. Absolute difference of the pole cluster mean for the damaged and undamaged plate (dotted lines represent the standard deviation of the undamaged plate cluster)
equal to the standard deviations of the estimated normal density function of the undamaged plate. However, as the cut lengthens, the migration of the cluster becomes increasingly significant as shown by Fig. 4. Hence, the cluster mean appears to be a promising indicator of crack damage in the aluminum plate that is strongly correlated to the crack length. CONCLUSIONS This study has explored a new monitoring technique for detecting crack damage in metallic structural elements using piezoelectric active sensors. A wireless active sensing unit was used to collect the input-output response of an aluminum plate excited with broadband white noise excitation signals using piezoelectric pads mounted to the plate surface. Using the characteristic equation roots of ARX models fit to the input-output response data, damage was successfully identified in the aluminum plate. In particular, a set of excitations are applied to the plate in a given structural state (undamaged versus damaged) so that the distribution of ARX model poles upon the complex plane can be statistically described using estimated bivariate normal probability density functions. As damage is introduced into the aluminum plate, changes in the plate stiffness and damping are manifested by migration of pole clusters upon the complex plane. In particular, the mean of the pole clusters corresponding to the second, fourth and seventh response mode (clusters 2, 6 and 12) migrated from the undamaged plate cluster mean in concert with the cut length. The migration of the pole clusters were shown to be statistically significant by comparing the absolute difference of cluster means with the standard deviations of the undamaged plate’s estimated normal probability density functions. Future work is needed to further explore the suitability of using pole clusters in the complex plane for damage detection. Currently, theoretical models of the plate system are being created to validate the proposed methodology and to ascertain ways of identifying a priori those pole clusters most sensitive to structural damage. The wireless active sensing unit used to collect the input-output response data is also being improved so that data can be collected at higher sample rates. High sample rates could illuminate higher order response modes that might exhibit greater sensitivity to structural damage.
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ACKNOWLEDGMENTS The author would like to express his gratitude to Arvind Sundararajan and Prof. Kincho Law of Stanford University and Drs. Hoon Sohn and Chuck Farrar of Los Alamos National Laboratory for providing assistance in the design and fabrication of the wireless active sensing unit prototype. This research is partially funded by the National Science Foundation (grant number CMS-9988909) and Los Alamos National Laboratory (contract number 75067-001-03). REFERENCES Achenbach, J. D., Gautesen, A. K. and McMaken, H. (1982), Ray Methods for Waves in Elastic Solids, Pitman Advanced Publishing, Boston, USA. Brooks, R. R. and Iyengar, S. (1998), Multi-sensor fusion, Prentice Hall, Upper Saddle River, NJ. Doebling, S. W., Farrar, C. R. and Prime, M. B. (1998), “A Summary Review of Vibration-based Damage Identification Methods,” Shock and Vibration Digest, 30(2), 91-105. Hamburger, R. O. (2000), A Policy Guide to Steel Moment-frame Construction, Federal Emergency Management Agency (FEMA), Report Number 354, Washington D. C., USA. Lynch, J. P. (2002), Decentralization of Wireless Monitoring and Control Technologies for Smart Civil Structures, John A. Blume Earthquake Engineering Center, Technical Report Number 140, Stanford, CA, USA. Lynch, J. P., Sundararajan, A., Law, K. H., Sohn, H. and Farrar, C. (2004), "Piezoelectric Structural Excitation using a Wireless Active Sensing Unit," Proceedings of the 22nd International Modal Analysis Conference (IMAC XXII), Dearborn, MI, USA, January 26-29. Park, G., Cudney, H. H. and Inman, D. J. (2000), “Impedance-based Health monitoring of civil structural components,” Journal of Infrastructure Systems, 6(3), 153-160. Wu, F. and Chang, F. K. (2001), “A Built-in Active Sensing Diagnostic System for Civil Infrastructure Systems,” Proceedings of 8th International Symposium on Smart Structures and Materials, San Diego, CA, March 5-7.
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