Distributed Sensor Networks for Health Monitoring of Civil Infrastructures
View this Special IssueResearch Article  Open Access
Feasibility Verification of Mountable PZTInterface for Impedance Monitoring in TendonAnchorage
Abstract
This study has been motivated to numerically evaluate the performance of the mountable PZTinterface for impedance monitoring in tendonanchorage. Firstly, electromechanical impedance monitoring and feature classification methods are outlined. Secondly, a structural model of tendonanchorage subsystem with mountable PZTinterface is designed for impedance monitoring. Finally, the feasibility of the mountable PZTinterface is numerically examined. A finite element (FE) model is designed for the labscaled tendonanchorage. The FE model of the PZTinterface is tuned as its impedance signatures meet the experimental test results at the same frequency ranges and also with identical patterns. Equivalent model properties of the FE model corresponding to prestress forces inflicted on the labtested structure are identified from the finetuning practice.
1. Introduction
The fundamental of the impedancebased method is to utilize the electromechanical (EM) impedance of a coupled PZTstructure system to detect the change in structural characteristics at local critical region. This monitoring technique was first proposed by Liang et al. [1]. Since then, many researchers have improved the method and applied it into various damage detection problems [2–7]. Kim et al. [8, 9] applied the impedancebased method for prestressloss monitoring in tendonanchorage connection by detecting the change in the pattern of the impedance responses at the anchorage.
Although the method has shown the excellent performance in detecting prestressloss in tendonanchorage, it still has several limitations for practical applications. Firstly, the effective frequency band which is sensitive to the variation of prestress force could be very significant, even above 800 kHz [9]. In this case, a high performance impedance analyzer is needed to get signals from such a high frequency range. Therefore, it is almost impossible to adopt a wireless impedance device which has a measurable frequency range of 10 kHz~100 kHz [10, 11]. Secondly, the impedance frequency bands which are sensitive to prestressloss should be predetermined before employing the impedancebased method for the damage detection job. Generally, the effective frequency range is varied dependent on target structures and usually determined by trial and error. This causes difficulty when dealing with real structures.
To overcome the abovementioned limitations, Nguyen and Kim [12] designed a fixedtype PZTinterface for monitoring prestressloss in tendonanchorage subsystems. The implementation of the PZTinterface was successful as sensitively indicating various prestresslosses in the tendonanchorage. Also, the sensitive frequency range to prestressloss was reduced to below 100 kHz. However, this design of PZTinterface must be installed during the construction of the tendonanchorage connection. So it is impossible to apply it into existing cablesupported structures. Also, the presence of the aluminum PZTinterface between anchor and bearing plate may cause the loss of the bearing capacity of the connection.
To deal with these drawbacks of the fixedtype PZTinterface, a mountable PZTinterface was proposed by Huynh and Kim [13], as shown in Figure 1, for impedance monitoring of tension forces from existing tendonanchorages. This interface can be easily bonded on and also detached from the anchorage surface as it is a clampedclamped beam embedded with a PZT patch. Although the new PZTinterface was successfully tested for tensionloss monitoring from a labscaled tendonanchorage, its feasibility should be verified by numerical simulation. Since the PZTinterface’s geometry should be fitted dependent on host structures, the numerical simulation (as well as verification) is the key to the design of appropriate PZTinterfaces corresponding to the target’s specifications.
(a)
(b)
In the numerical simulation of the electromechanical impedance, the coupling interaction between the PZT sensor and the host structure is an important issue. Several simplified impedance models have been proposed for describing the PZT sensorstructure interaction. Liang et al. [1] developed a 1dof springmassdamper system via combining the electromechanical analysis of a PZT sensor and the structural dynamic analysis of the host structure. To consider the multidirectional effects caused by the PZT sensor’s vibration, Zhou et al. [14] and Bhalla and Soh [15] introduced twodimensional impedance models from which the planar coupled vibration was taken into account. By adopting the effective impedance concept, Zuo et al. [16] have developed a threedimensional PZTstructure model considering the coupled vibration along the radial direction and along the thickness direction of the PZT sensor. The performance evaluation of the theoretical impedance models has been validated by experimental tests conducted on beamlike structures [1, 17] and thin plates [14, 15] with bonded PZT actuators. However, the direct application of the socalled “impedance model” is limited when dealing with host structures with complex characteristics (i.e., geometry, material properties, and boundary conditions). In such cases, appropriate modeling techniques are needed to fully capture the coupled vibration of the PZT sensor and the host structure.
This study has been motivated to numerically evaluate the performance of the mountable PZTinterface for impedance monitoring in tendonanchorage. Firstly, electromechanical impedance monitoring and feature classification methods are outlined. Secondly, a structural model of the anchorage’s components is designed from the force equilibrium state. In the model, the tension force is represented by interfacial structural parameters of the tendonanchorage connection. A mountable PZTinterface is designed for tensionloss monitoring from postinstallation into existing anchorages. Finally, the feasibility of the mountable PZTinterface is numerically examined. A finite element (FE) model is designed for the labscaled tendonanchorage. The FE model of the PZTinterface is tuned as its impedance signatures meet the experimental test results (presented by [13]) at the same frequency ranges and also with identical patterns. Equivalent model properties of the FE model corresponding to prestress forces inflicted on the labtested structure are identified from the finetuning practice.
2. Theory of Impedance Monitoring
2.1. Electromechanical Impedance
To monitor structural change, a piezoelectric material (e.g., PZT) is surfacebonded to structure at the examined region. As shown in Figure 2, the interaction between the PZT and the structure is simply explained by the 1D freebody diagram of PZTstructure system. Due to the inverse piezoelectric effect, an input harmonic voltage induces a deformation of PZT. Because the PZT is bonded to the structure, a force against that deformation is induced into the structure and the PZT as well.
For 1dof system, structural mechanical impedance of the host structure is obtained by the ratio of force to velocity as follows [1]: where and are the damping coefficient and the mass of the structure, respectively; is the angular natural frequency of the structure; and is the angular frequency of the excitation voltage. As shown in (1), structural mechanical impedance is a function of mass, damping, and stiffness (i.e., stiffness is introduced from natural frequency, ). Thus, the change in structural parameters can be represented by the change in structural mechanical impedance.
In practice, the electric current is measured and then it is utilized to calculate electromechanical impedance as follows [1]:where is the complex Young’s modulus of the PZT patch at zero electric field; is the complex dielectric constant at zero stress; is the piezoelectric coupling constant in direction at zero stress; is the wave number where is the mass density of the PZT patch; and , , and are the width, length, and thickness of the piezoelectric transducer, respectively. The parameters and are structural damping loss factor and dielectric loss factor of piezoelectric material, respectively.
As shown in (2), the electromechanical impedance, , is a combining function of the mechanical impedance of the host structure, , and that of the piezoelectric patch, . Therefore, the change in structural parameters (, , and ) can be represented by the change in electromechanical impedance.
An important issue for impedance monitoring is to decide the target frequency range. The frequency range should be selected appropriately in order to realize structural change in electromechanical impedance. Generally, if the excitation frequency is not identical to the natural frequency of the host structure (i.e., ), the impedance of the structure is very large compared with the mechanical impedance of the PZT sensor (i.e., ). As a result, the term is neglected in (2). On the other hand, if the PZT sensor is excited by a frequency matching with the natural frequency of the structure (i.e., ), structural mechanical impedance takes only the term of damping coefficient (i.e., ). Consequently, the structural impedance for that frequency is comparable with the mechanical impedance of the PZT, and electromechanical impedance is expressed as follows: In (3), the contribution of structural mechanical impedance to electromechanical impedance is the damping coefficient , which causes resonant responses in electromechanical impedance signature. It means that electromechanical impedance at resonance represents not only modal damping, but also the natural frequency of the structure. Therefore, the structural change could be identified sensitively by the change in electromechanical impedance at the resonant frequency.
2.2. Quantification of Electromechanical Impedance
To quantify the change in electromechanical impedance, the root mean square deviation (RMSD) index is utilized. RMSD index is calculated as [18] where and are the impedances measured at two events for the th frequency, respectively, and denotes the number of frequency points in the sweep. Ideally, the RMSD is equal to 0 if the two events are identical (e.g., there is no structural change). Otherwise, the RMSD is larger than 0.
The correlation coefficient deviation (CCD) index can also be used to quantify the change of the whole impedance signatures [3]. The CCD index is calculated as follows: in which is the expectation operation; signifies the real parts of the electromechanical impedances of the th frequency; signifies the mean values of impedance signatures (real part); and signifies the standard deviation values of impedance signatures. Note that the asterisk () denotes the structural change. The CCD index is equal to 0 if there is no structural change. Otherwise, the CCD is larger than 0.
3. Mountable PZTInterface for TendonAnchorage Connection
3.1. Structural Model of TendonAnchorage
A tendonanchorage subsystem can be modeled by a series of components such as bearing plate, anchor block, and tendon and contact forces in equilibrium condition, as shown in Figure 3(a). The prestress force is modeled by tendon force acting on the anchor block and transformed to contact pressure and bearing stress in the interface of anchor block and bearing plate. According to the contact mechanism [19], the interaction in the contact interface can be simplified by damping coefficients and spring stiffness parameters, as shown in Figure 3(b). On the other hand, the variation of interfacial stiffness and damping parameters are associated with the variation of contact pressures [20]. Hence, the variation of tendon force can be treated as the variation of those structural parameters at the contact interface.
(a)
(b)
3.2. Structural Model of TendonAnchorage with Mountable PZTInterface
A fixedtype PZTinterface was proposed to overcome the difficulties in monitoring the high frequency response corresponding to the change in the tendonanchorage system [12]. The interface is a thin plate equipped with a PZT patch. It reduces the frequency range enough to deal within 10–100 kHz measurement. However, the fixedtype design of this interface has drawbacks since it interferes with the bearing capacity of the tendonanchorage and also it needs to be installed during the construction.
As an alternative measuring technique, a mountable PZTinterface has been newly modeled in order to overcome the limitations of the fixedtype interface in practice, as shown in Figure 4(a). The mountable interface is a clampedclamped beam with a PZT patch bonded on its middle. As tightly bonded to the surface of the bearing plate, the clamped interfacing makes the stress fields of the PZTinterface almost equivalent to those of the bearing plate on its surface, while the top and bottom midsurfaces are freely deformed along with the deformation in the bearing plate. Therefore, any change in structural parameters due to cable forceloss would lead to the change in the EM impedance response of PZTinterface (Figure 4(b)).
(a)
(b)
(c)
On the basis of the coupling interaction between the PZT sensor and the host structure [1, 17], a 1dof structural model using a series of massspringdampness is proposed as shown in Figure 4(c). The clampedclamped PZTinterface is modeled as 1dof system with its mass (), stiffness (), and damping parameters (), as shown in Figure 4(c). Also, the tendonanchorage is modeled as bearing plate’s mass (), bearing plate’s stiffness () and damping (), contact stiffness (), and contact damping (). The series model is selected for the simplification since the bearing plate is target structure and the contact parameters are considered as boundary conditions. The equivalent structural stiffness of the PZTinterface on the bearing plate () is simplified as follows: where is the total contact stiffness () and is a factor representing the ratio of the bearing plate’s stiffness to the total contact stiffness as Suppose is the ratio of the PZTinterface’s stiffness to the equivalent contact stiffness; the equivalent structural stiffness of the PZTinterface can be rewritten as For the simplified model of the tendonanchorage without the mountable PZTinterface (), the bearing plate is the target structure and its equivalent structural stiffness is derived on the basis of (8) as follows:
Since the bearing plate is clamped between two surfaces, its equivalent stiffness would be very large due to the contact pressure and the stress field acting at the interface. Substituting (9) into (8) leads the ratio of the equivalent stiffness with the PZTinterface () to that of the bearing plate () as follows: As the stiffness of the PZTinterface is much smaller than the contact stiffness (i.e., ), (10) is simplified as and it leads . Assume the mass of the PZTinterface is comparable with that of the bearing plate, natural frequencies of the PZTinterface subsystem would be quite reduced as compared to those of the tendonanchorage model.
For simplified structural models, structural parameters are needed to be determined to obtain the consistent experimental results. The structural parameters could be identified from the finetuning process by matching the calculated resonant response of the simplified model and the measured resonant response of the experimental model [20, 21]. In the present study, the 1dof structural model of the PZTinterface was proposed firstly to explain how to represent the loss of cable force from the change in the electromechanical impedance (Figure 4(c)) and secondly to prove that the sensitive frequency band could be reduced when the PZTinterface device is implemented into tendonanchorage (see (10)).
4. Numerical Verification of Mountable PZTInterface for Cable Force Monitoring
4.1. FE Modeling for Numerical Simulation of TendonAnchorage
In order to numerically evaluate the applicability of the PZTinterface device for cable force monitoring using impedancebased technique, FE model of tendonanchorage connection was established by the commercially available FE package, COMSOL Multiphysics [22]. Figure 5 shows the geometry of the FE model that was in accordance with the labscale model test presented by Huynh and Kim [13]. It should be noted that the bonding layers between the PZT sensor and PZTinterface with the mother structure were not simulated in the present work. In FE modeling, the bearing plate, anchor head, and the PZTinterface were discretized by the elastic solid elements in 3D, as shown in Figure 6(a). The properties of the steel anchorage and the aluminum interface specified in the FE model are listed in Table 1. The PZT patch is added by the piezoelectric material, PZT5A type as shown in Table 2, which could deal with both mechanical and electrical fields. To acquire the electromechanical impedance, a harmonic excitation voltage with an amplitude of 1 V was simulated to the top surface of the PZT patch, and the bottom one was set as the ground electrode.


(a)
(b)
According to the foregoing analytical model of tendonanchorage connection, the variation of tendon force can be treated as the variation of structural parameters at the contact interface. Therefore, the cable force can be monitored via the interfacial structural stiffness (socalled contact stiffness). As shown in Figure 6(a), spring elements in the , , and axes were added to the contact surface of the bearing plate to represent the contact stiffness caused by cable force. It is assumed that the structural parameters are uniformly distributed on the contact boundary surface. As given in Table 3, four cases of spring stiffness (i.e., C0, C1, C2, and C3) were considered to investigate the change in the piezoelectric impedance response of PZTinterface. By assuming that the variation of cable force mainly causes the variation of spring constant in the direction normal to the contacting surface, only the variation of is simulated in the FE analysis.

As shown in Figure 7, the real part of the electromechanical impedance computed by the numerical simulation (case C0 in Table 3) is compared to the corresponding experimental result (for tension force 49.05 kN). It is worth noting that the spring constants specified in the C0 scenario were obtained by using trialanderror method, and the identification of the contact stiffness is beyond the scope of this paper. As observed in Figure 7, the resonance occurred within 10 kHz~100 kHz, and both resonant and nonresonant regions of the numerical impedance signatures show good agreement with those of the experimental result [13].
The difference in the resonant frequencies between the FE modeling and the labscaled testing is very small, about 0.25% and 0.55% for the first and the second resonant vibrations, respectively. It indicates that the FE simulation can provide reasonable impedance responses for cable force monitoring via interfacial stiffness monitoring. As shown in Figure 6(b), the deformation contour of the anchorage excited by PZT patch was generated at the resonant frequency of 19.52 kHz. It is also observed that the maximum displacement of the PZTinterface is about 2.35 × 10^{−8} m due to the resonance event.
4.2. Contact Stiffness versus Impedance Response
Figure 8 shows real impedance signatures of the PZTinterface obtained by the FE modeling in the wide frequency range 10 kHz–100 kHz and two sensitive frequency ranges of 15–25 kHz and 75–95 kHz for four contact stiffness scenarios (as listed in Table 3). Similar to the experiment, the impedance signatures were recorded with 901 interval points for the frequency band of 10–100 kHz and 501 points for both frequency bands of 15–25 kHz and 75–95 kHz. As observed in Figures 8(b) and 8(c), the impedance signatures in the resonant bands of 15–25 kHz and 75–95 kHz are varied when decreasing the contact stiffness . The frequency range of 15–25 kHz is more sensitive than that of 75–95 kHz. The resonant frequencies tend to decrease when decreasing the spring stiffness . The change in the resonant frequency due to the contact stiffnessloss is summarized in Table 3. The frequencyshift is very small, 0.81% and 0.19% for the first and the second resonant frequencies, respectively. Obviously, the pattern of the numerical impedance variation due to the change in contact stiffness is wellmatched with the experimental observations [13]. Furthermore, the impedance response via the PZTinterface was sensitive to the change in contact stiffness even when the examined frequency range was 10 kHz~100 kHz.
(a) Frequency range 10–100 kHz
(b) Frequency range 15–25 kHz
(c) Frequency range 75–95 kHz
For contact stiffness monitoring, both RMSD values (see (4)) and CCD indices (see (5)) are examined. Figures 9 and 10, respectively, show the change in RMSD and CCD indices associated with the decrement of contact stiffness for the wide frequency band (i.e., 10–100 kHz) and the narrow ones (i.e., 15–25 kHz and 75–95 kHz). As observed in the figures, RMSD and CCD indices increase linearly with the loss of interfacial stiffness. It is also observed that RMSD and CCD indices of the wide frequency range 10–100 kHz show a good indication of the contact stiffnessloss as the sensitive frequency range 15–25 kHz. In comparison, of RMSD and CCD approaches, RMSD indices are varied much higher than the CCD indices due to the contact stiffness variation for all considered frequency ranges (i.e., 10–100 kHz, 15–25 kHz, and 75–95 kHz). These numerical results are consistent with the experimental results obtained from the labscaled test structure described previously. From the feature analysis of the impedance signatures, the RMSD is found to be the sensitive indication of the contact stiffnessloss as the prestress force was reduced.
(a) Frequency range 10–100 kHz
(b) Frequency range 15–25 kHz
(c) Frequency range 75–95 kHz
(a) Frequency range 10–100 kHz
(b) Frequency range 15–25 kHz
(c) Frequency range 75–95 kHz
5. Summary and Conclusions
In this study, the feasibility of the mountable PZTinterface was numerically analyzed for impedance monitoring in the tendonanchorage connection. Firstly, electromechanical impedance monitoring and feature classification methods were outlined. Secondly, a structural model of tendonanchorage subsystem with mountable PZTinterface was designed for impedance monitoring. Finally, the feasibility of the mountable PZTinterface was numerically examined. A finite element (FE) model was designed for the labscaled tendonanchorage. The FE model of the PZTinterface was tuned as its impedance signatures met the experimental test results presented by Huynh and Kim [13]. Equivalent model properties of the FE model corresponding to prestress forces inflicted on the labtested structure were identified from the finetuning practice.
The interfacial stiffness which represents the prestress force in the tendonanchorage subsystem was successfully monitored by the electromechanical impedance response of the PZTinterface. The feature analysis of the impedance signatures shows that the RMSD changed to make the sensitive indication of the contact stiffnessloss as the prestress force was reduced. The impedance response via the PZTinterface was sensitive to the change in contact stiffness even when the examined frequency range was 10 kHz~100 kHz. It is noted that two resonance frequencies occurred near 20 kHz and 82 kHz by implementing the mountable interface device. Since the PZTinterface is mobile and adaptable to be applied in any existing cableanchorage connections, this numerical evaluation is promising for designing appropriate devices for practical applications. As for the future work, the bonding layer should be considered in the numerical simulation to examine its effect on the electromechanical impedance of the PZTinterface.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This research was supported by a grant from a Strategic Research Project (Development of Smart Prestressing and Monitoring Technologies for Prestressed Concrete Bridges) funded by the Korea Institute of Construction Technology. The graduate student, Mr. ThanhCanh Huynh, involved in this research was also partially supported by the Brain Korea 21 Plus (BK21 Plus) Program of Korean Government.
References
 C. Liang, F. P. Sun, and C. A. Rogers, “Coupled electromechanical analysis of adaptive material systems—determination of the actuator power consumption and system energy transfer,” Journal of Intelligent Material Systems and Structures, vol. 5, no. 1, pp. 12–20, 1994. View at: Publisher Site  Google Scholar
 G. Park, H. H. Cudney, and D. J. Inman, “Feasibility of using impedancebased damage assessment for pipeline structures,” Earthquake Engineering and Structural Dynamics, vol. 30, no. 10, pp. 1463–1474, 2001. View at: Publisher Site  Google Scholar
 A. N. Zagrai and V. Giurgiutiu, “Electromechanical impedance method for crack detection in thin plates,” Journal of Intelligent Material Systems and Structures, vol. 12, no. 10, pp. 709–718, 2001. View at: Publisher Site  Google Scholar
 S. Bhalla and C. K. Soh, “Structural impedance based damage diagnosis by piezotransducers,” Earthquake Engineering & Structural Dynamics, vol. 32, no. 12, pp. 1897–1916, 2003. View at: Publisher Site  Google Scholar
 Y. Yang, Y. Hu, and Y. Lu, “Sensitivity of PZT impedance sensors for damage detection of concrete structures,” Sensors, vol. 8, no. 1, pp. 327–346, 2008. View at: Publisher Site  Google Scholar
 J.T. Kim, W.B. Na, J.H. Park, and D.S. Hong, “Hybrid health monitoring of structural joints using modal parameters and EMI signatures,” in Smart Structures and Materials 2006—Sensors and Smart Structures Technologies for Civil, Mechanical, and Aerospace Systems, vol. 6174 of Proceedings of the SPIE, San Diego, Calif, USA, March 2006. View at: Publisher Site  Google Scholar
 D. L. Mascarenas, Development of an impedancebased wirelesssensor node for monitoring of bolted joint preload [M.S. thesis], Department of Structural Engineering, University of California, San Diego, Calif, USA, 2006.
 J. T. Kim, J. H. Park, D. S. Hong, H. M. Cho, W. B. Na, and J. H. Yi, “Vibration and impedance monitoring for prestressloss prediction in PSC girder bridges,” Smart Structures and Systems, vol. 5, no. 1, pp. 81–94, 2009. View at: Publisher Site  Google Scholar
 J.T. Kim, J.H. Park, D.S. Hong, and W.S. Park, “Hybrid health monitoring of prestressed concrete girder bridges by sequential vibrationimpedance approaches,” Engineering Structures, vol. 32, no. 1, pp. 115–128, 2010. View at: Publisher Site  Google Scholar
 D. L. Mascarenas, M. D. Todd, G. Park, and C. R. Farrar, “Development of an impedancebased wireless sensor node for structural health monitoring,” Smart Materials and Structures, vol. 16, no. 6, pp. 2137–2145, 2007. View at: Publisher Site  Google Scholar
 J.H. Park, J.T. Kim, D.S. Hong, D. Mascarenas, and J. Peter Lynch, “Autonomous smart sensor nodes for global and local damage detection of prestressed concrete bridges based on accelerations and impedance measurements,” Smart Structures and Systems, vol. 6, no. 56, pp. 711–730, 2010. View at: Publisher Site  Google Scholar
 K.D. Nguyen and J.T. Kim, “Smart PZTinterface for wireless impedancebased prestressloss monitoring in tendonanchorage connection,” Smart Structures and Systems, vol. 9, no. 6, pp. 489–504, 2012. View at: Publisher Site  Google Scholar
 T.C. Huynh and J.T. Kim, “Impedancebased cable force monitoring in tendonanchorage using portable PZTinterface technique,” Mathematical Problems in Engineering, vol. 2014, Article ID 784731, 11 pages, 2014. View at: Publisher Site  Google Scholar
 S.W. Zhou, C. Liang, and C. A. Rogers, “An impedancebased system modeling approach for induced strain actuatordriven structures,” Journal of Vibration and Acoustics, vol. 118, no. 3, pp. 323–331, 1996. View at: Publisher Site  Google Scholar
 S. Bhalla and C. K. Soh, “Structural health monitoring by piezoimpedance transducers. I: modeling,” Journal of Aerospace Engineering, vol. 17, no. 4, pp. 154–165, 2004. View at: Publisher Site  Google Scholar
 C. Zuo, X. Feng, and J. Zhou, “A threedimensional model of the effective electromechanical impedance for an embedded PZT transducer,” Mathematical Problems in Engineering, vol. 2013, Article ID 218026, 10 pages, 2013. View at: Publisher Site  Google Scholar
 V. Giurgiutiu and A. N. Zagrai, “Embedded selfsensing piezoelectric active sensors for online structural identification,” Journal of Vibration and Acoustics, vol. 124, no. 1, pp. 116–125, 2002. View at: Publisher Site  Google Scholar
 F. P. Sun, Z. Chaudhry, C. Liang, and C. A. Rogers, “Truss structure integrity identification using PZT sensoractuator,” Journal of Intelligent Material Systems and Structures, vol. 6, no. 1, pp. 134–139, 1995. View at: Publisher Site  Google Scholar
 K. L. Johnson, Contact Mechanics, Cambridge University Press, Cambridge, UK, 1985.
 S. Ritdumrongkul, M. Abe, Y. Fujino, and T. Miyashita, “Quantitative health monitoring of bolted joints using a piezoceramic actuatorsensor,” Smart Materials and Structures, vol. 13, no. 1, pp. 20–29, 2004. View at: Publisher Site  Google Scholar
 V. Giurgiutiu and C. A. Rogers, “Modal expansion modeling of the electromechanical (E/M) impedance response of 1D structures,” in Proceedings of the European COST F3 Conference on System Identification & Structural Health Monitoring, Universidad Politecnica de Madrid, Madrid, Spain, June 2000. View at: Google Scholar
 COMSOL, Inc., 2013, /http://www.comsol.com.
 Efunda Inc., 2010, http://www.efunda.com.
Copyright
Copyright © 2015 ThanhCanh Huynh et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.