Electromechanical Impedance Analysis on Piezoelectric Smart Beam with a Crack Based on Spectral Element Method

Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2015, Article ID 713501, 13 pages http://dx.doi.org/10.1155/2015/713501 Research Article Electromechanical Impedance Analysis on Piezoelectric Smart Beam with a Crack Based on Spectral Element Method Dansheng Wang, Hongyuan Song, and Hongping Zhu School of Civil Engineering and Mechanics, Huazhong University of Science and Technology, Wuhan 430074, China Correspondence should be addressed to Dansheng Wang; Received 11 April 2014; Revised 6 July 2014; Accepted 16 July 2014 Academic Editor: Kui Fu Chen Copyright © 2015 Dansheng Wang 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. An electromechanical impedance (EMI) analysis of a piezoelectric smart beam with a crack is implemented in this paper. Spectral element method (SEM) is used to analyze the EMI response of the piezoelectric smart beam. In this analysis, the spectral element stiffness matrices of different beam segments are derived in this paper. The crack is simulated using spring models, and the EMI signatures of piezoelectric smart beam with and without crack are calculated using SEM, respectively. From the analysis results, it is found that the peak position and amplitude of the EMI signatures have significant changes with the change in crack depth, especially in higher frequency ranges. Different vibration modes of the piezoelectric smart beam are analyzed, and the effect of thickness of the adhesive layer on the admittance is also researched. An experimental study is also implemented to verify the validity of the analysis results using SEM. 1. Introduction The presence of structural damage will result in the changes in the properties of the structure and will also induce the change in the mechanical impedance of the structure. Damage identification can be achieved by comparing the structural mechanical impedances before and after damage. However, structural mechanical impedance is difficult to measure in practice. The coupled electromechanical impedance (EMI) of structure with PZT patches bonded to it is convenient to obtain by using impedance analyzer, even in the high frequency ranges, which is quite sensitive to structural incipient damage. In recent years, the EMI-based technique has been applied to structural health monitoring. It has been successfully used to analyze various engineering structures, including aerospace structures, mechanical structures, and civil structures [1–11]. The EMI technique is based on the direct and converse piezoelectric effects of the PZT patch. In order to analyze the EMI of coupled structures, the model characterizing electromechanical interaction between the host structure and PZT patch should be firstly established. Liang et al. [12] proposed the first EMI model for a PZT patch bonded onto an intact one-dimensional structure. Zhou et al. [13] extended the impedance method to model a twodimensional EMI structure. Bhalla and Soh also presented an improved 2D impedance model to characterize the PZTstructure electroelastic interactions based on the concept of “effective impedance” [14]. Annamdas and Soh have examined the three-dimensional interaction of a PZT transducer or multiple PZT transducers with the host structure based on directional sum impedance formulation [15, 16]. Wang et al. also proposed an embedded 3D electromechanical impedance model for an embedded PZT transducer by considering the interaction between a square PZT patch and a host structure [17]. Spectral element method (SEM) has been proposed to the structural dynamics community as a numerical tool to analyze the dynamic responses of rods, beam, and plates [18]. Spectral element is represented by an exponential interpolation function, which is an exact solution of the wave equation for representative structural element. Therefore, other than classical finite element method, SEM provides dynamic stiffness matrix of a structure, which is more accurate and efficient than the traditional static stiffness matrix, 2 and allows an increased accuracy in modeling the structural dynamic behavior. Subsequently, Palacz and Krawczuk [19] analyzed the dynamic responses of a cracked rod using SEM. The crack is simulated by a rotational spring model, and the node displacement and shape function of the cracked rod are derived. Ostachowicz et al. further summarized the researches on spectral element method and derived the spectral element stiffness matrices of Euler beam, Timoshenko beam, damaged plate, and thin-walled shell structures [20– 23]. Samaratunga et al. presented a new 2D wavelet spectral finite element model for studying wave propagation in thin to moderately thick anisotropic composite laminates [24]. Choi and Inman recently presented modeling of a cableharnessed structure by means of SEM, and a double beam model was formulated. The presented modeling was applied and compared with the conventional FEM to emulate a cableharnessed structure [25]. Some researchers also focused on identifying the damage from the EMI signatures of the cracked beam with surfacebonded PZT patches using SEM. Park et al. implemented damage identification study on a simple one-dimensional structure by combining SEM and EMI techniques [26, 27]. Ritdumrongkul et al. succeeded in quantitative identification of the structure damage based on the SEM and PZT active sensors. In the experiment, a bolted aluminum beam is studied, and the damage is simulated by looseness of the bolt [28]. Wang and Tang further deduced the stiffness matrix of smart Timoshenko beam and successfully analyzed electric admittances of piezoelectric smart beam using the SEM [29]. Combined with EMI method and nonlinear optimization technique, the spectral element model of a simply supported rod was simulated and numerical study on localization and quantitative identification of damage was made by Guo and Sun [30]. Ostachowicz et al. developed their own numerical procedures using SEM in order to calculate damage indexes, which were used both for damage detection and for localization. The proposed methods were applied for structural health monitoring of metallic and fibre reinforced structural elements [31]. Kim and Wang proposed an improved impedance-based damage identification method by incorporating a tunable piezoelectric circuitry with the structure to enrich the impedance measurements. Numerical case study on localizing damage in a fixed-fixed beam using SEM was performed to demonstrate the effectiveness of the new method for structural damage identification [32]. In this paper, a new spectral element model of a cracked Timoshenko beam is proposed. In the model, the crack is modeled accurately by three massless springs, that is, a shear spring, a translational spring, and a rotational spring. According to the relatio (...truncated)