Spectral element modeling based structure piezoelectric impedance computation and damage identification

Abstract

This paper presents a numerical simulation study on electromechanical impedance technique for structural damage identification. The basic principle of impedance based damage detection is structural impedance will vary with the occurrence and development of structural damage, which can be measured from electromechanical admittance curves acquired from PZT patches. Therefore, structure damage can be identified from the electromechanical admittance measurements. In this study, a model based method that can identify both location and severity of structural damage through the minimization of the deviations between structural impedance curves and numerically computed response is developed. The numerical model is set up using the spectral element method, which is promised to be of high numerical efficiency and computational accuracy in the high frequency range. An optimization procedure is then formulated to estimate the property change of structural elements from the electric admittance measurement of PZT patches. A case study on a pin-pin bar is conducted to investigate the feasibility of the proposed method. The results show that the presented method can accurately identify bar damage location and severity even when the measurements are polluted by 5% noise.

Keywords

PZT / piezoelectric impedance / optimization / spectral element / damage identification

Introduction

Due to deterioration of materials, overloading, severe environment condition and sufficient maintenance, the performance of civil infrastructural systems will degrade during their service life. The techniques to test and diagnosis structural physical condition are thus one recent research focus in civil engineering community. Vibrating based method is one of that promised techniques, which rely on dynamically measured structural signatures (such as modal frequencies, mode shapes and et al.) as the baseline for damage assessment. One key issue of this type of methods is they are so called global methods and not sensitive to structure local micro-crack or initial damage [1]. The piezoelectric impedance-based methods are thus proposed to deal with this type of problems.

The piezoelectric impedance-based method makes use of the dual functions of piezoelectric materials to actuate and sense structural localized high frequency vibration response. Concerning that structural high frequency modes are generally local vibration modes, structural local micro-crack can thus be sensitively captured by measuring high frequency impedance using piezoelectric sensors. Recent research efforts on this topic are put on how to model the electrical impedance of the piezoelectric transducer integrated to various structures [2-8] and how to efficiently model structural impedance in high frequency region. Liang et al. [4] first proposed a mathematical formulation of the electrical impedance of a piezoelectric transducer bonded onto a 1D structure. In their derivations, one end of an axial piezoelectric transducer is attached to a single degree-of-freedom (DOF) mechanical system and the other end is fixed. Giurgiutiu and Zagrai [6] improved the 1D modeling by letting both ends of an axial piezoelectric transducer be attached to a single DOF mechanical system. For structural impedance modeling in high frequency, the spectral element method (SEM) is the most attractive numerical modeling method. The spectral element method uses the frequency dependant shape functions in contrast to the constant polynomial shape functions employed by finite element analysis to construct the element mass and stiffness matrices. Thus it provides accurate dynamic characteristics of a structure and gains very accurate solutions at a high frequency. Recently, dense research works on how to model the cracked beams using SEM and how to identify the damage from the impedance based on this modeling are reported [9-13].

In this study, a numerical study on electro-mechanical impedance technique for structural damage identification is presented. The high frequency structural responses are obtained from a SEM model and the electric admittance curves of PZT patches are simulated using the 1D interaction model. A structural damage identification technique is then presented from the simulated electric admittance curve measurement. A nonlinear optimization approach is presented to locate and quantify damage. Numerical example on a pin-pin bar is conducted to investigate the effectiveness of the proposed method. In the numerical study, the influence of the measurement noise on the results is investigated.

Piezoelectric impedance model

For a PZT patch and host bar structure systems (as shown in Fig. 1), the PZT measured electric admittance curve is the function of dynamic characteristic of PZT and impedance value of the host structure. For 1D consideration, the PZT is seen as a slice for only producing the longitudinal expansion and shrink.

The constitutive equations of the PZT patch are

(1)S_{\mathrm{1}}={\displaystyle \frac{T_{\mathrm{1}}}{{\overline{E}}_p}}+d_{\mathrm{31}}{E}_{\mathrm{3}},

(2)D_{\mathrm{3}}={\overline{ϵ}}_{\mathrm{33}}^{\mathrm{T}}{E}_{\mathrm{3}}+d_{\mathrm{31}}{T}_{\mathrm{1}},

where S_{\mathrm{1}} is the strain; T_{\mathrm{1}} is the stress; {\overline{E}}_{\mathrm{p}}=E_{\mathrm{p}}(\mathrm{1}+i\eta ) is the complex modulus of PZT at zero electric field, and \eta is the mechanical loss factor of PZT; d_{\mathrm{31}} is the piezoelectric constant; D_{\mathrm{3}} is the electric displacement; {\overline{ϵ}}_{\mathrm{33}}^{\mathrm{T}}={ϵ}_{\mathrm{33}}^{\mathrm{T}}(\mathrm{1}-i\delta ) is the complex dielectric constant at zero stress, and \delta is the dielectric loss factor.

The coupled electric admittance of the PZT patch between two electrodes can be described as (Liang et al. [4] and Giurgiutiu and Zagri [6]):

(3)Y(\omega )={\displaystyle \frac{I}{V}}=i\omega {\displaystyle \frac{w_{\mathrm{a}}{l}_{\mathrm{a}}}{h_{\mathrm{a}}}}\left[{\overline{ϵ}}_{\mathrm{33}}^{\mathrm{T}}+\left({\displaystyle \frac{z_{\mathrm{a}}}{z_{\mathrm{s}}+z_{\mathrm{a}}}}\right)d_{\mathrm{31}}^{\mathrm{2}}{\overline{E}}_{\mathrm{p}}\left({\displaystyle \frac{\tan \kappa l_{\mathrm{a}}}{\kappa l_{\mathrm{a}}}}\right)-d_{\mathrm{31}}^{\mathrm{2}}{\overline{E}}_{\mathrm{p}})\right],

(4)z_{\mathrm{a}}={\displaystyle \frac{{\overline{E}}_{\mathrm{p}}{w}_{\mathrm{a}}{h}_{\mathrm{a}}}{i\omega l_{\mathrm{a}}}}·{\displaystyle \frac{\kappa l_{\mathrm{a}}}{\tan \kappa l_{\mathrm{a}}}},

where i is {(-\mathrm{1})}^{-\mathrm{1}/\mathrm{2}}; \omega is the excitation frequency; w_{\mathrm{a}}, h_{\mathrm{a}} and l_{\mathrm{a}} are the width, thickness, and length of the PZT actuator, respectively. \kappa =\sqrt{{\omega }^{\mathrm{2}}\rho /{\overline{E}}_{\mathrm{p}}} is wave number; \rho is the density of the PZT actuator; z_{\mathrm{a}} is the mechanical impedance of the PZT actuator; z_{\mathrm{s}} is the mechanical impedance of the host structure.

From Eq. (3), it is clear that the coupled electric admittance is the function of geometry dimension and capacitance of the PZT material and mechanical impedance of the structure. For a given system, the mechanical impedance of the PZT actuator is a constant and the mechanical impedance of the host structure is the only variable influencing the change of the coupled electric admittance. If the structure is damaged, the parameters, such as mass, stiffness or damping, would be changed. In other words, the coupled electrical admittance would vary. Hence, any changes in electrical admittance curve are attributed to damage or change of structural integrity.

From Eq. (3), structural impedance can be expressed as:

(5)z_{\mathrm{s}}=\left[{\displaystyle \frac{\omega w_{\mathrm{a}}{l}_{\mathrm{a}}{d}_{\mathrm{31}}^{\mathrm{2}}{\overline{E}}_{\mathrm{p}}}{\omega w_{\mathrm{a}}{l}_{\mathrm{a}}\left(d_{\mathrm{31}}^{\mathrm{2}}{\overline{E}}_{\mathrm{p}}-{\overline{ϵ}}_{\mathrm{33}}^T\right)-i{h}_{\mathrm{a}}Y}}·{\displaystyle \frac{\tan \kappa l_{\mathrm{a}}}{\kappa l_{\mathrm{a}}}}-\mathrm{1}\right]z_{\mathrm{a}}.

Equation (5) indicates that structural impedance can be obtained from the measured electric admittance of the PZT patch attached to the host structure.

Generally speaking, the PZT patches attached on a bar can activate three modes of vibration of the host structure (as shown in Fig. 1(b)). To excite the pure axial vibration of the bar, two identical PZT patches should be bonded perfectly and symmetrically onto the top and bottom surfaces of the bar. The dynamic output forces on the host structure can be represented as a pair of axial forces, f_{\mathrm{1}} and f_{\mathrm{2}}, given by

(6)f_{\mathrm{1}}=-f_{\mathrm{2}}=-\mathrm{2}{Z}_{\mathrm{s}}{\dot{u}}_{\mathrm{P}\mathrm{Z}\mathrm{T}}=\mathrm{2}i\omega Z_{\mathrm{s}}(u_{\mathrm{1}}-u_{\mathrm{2}}),

(7)\mathit{f}={\mathit{Z}}_A\mathit{u},

where \mathit{f}={\left[f_1{f}_2\right]}^{\mathrm{T}} is a axial force vector; \mathit{u}={\left[u_1{u}_2\right]}^{\mathrm{T}} is a displacement vector; and the matrix Z_A represents the elemental impedance matrix given by

(8){\mathit{Z}}_A=\mathrm{2}i\omega Z_{\mathrm{s}}\left[\begin{array}{cc}\mathrm{1}& -\mathrm{1}\\ -\mathrm{1}& \mathrm{1}\end{array}\right].

The PZT actuator produces a pair of axial forces for a bar under electric field, the structural impedance of a bar can be derived from Giurgiutiu and Zagrai [6] as:

(9)Z_{\mathrm{s}}={\displaystyle \frac{\rho A}{i\omega }}{\left\{{\displaystyle \sum _{n_u=\mathrm{0}}^{\infty }{\displaystyle \frac{{[U_{n_u}(l_{\mathrm{a}}+x_{\mathrm{a}})-U_{n_u}(x_{\mathrm{a}})]}^{\mathrm{2}}}{{\omega }_{n_u}^{\mathrm{2}}-{\omega }^{\mathrm{2}}}}}\right\}}^{-\mathrm{1}},

where U_{n_u} is the axial mode shape of the bar; {\omega }_{n_u} and {\omega }_{n_w} are nth axial and transverse natural frequency of the bar, respectively.

Spectral element

For the spectral element method, the longitudinal displacement shape function for a bar is

(10)u(x,\omega )=A_{\mathrm{1}}{e}^{-ikx}+A_{\mathrm{2}}{e}^{-ik(L-x)},

where A_{\mathrm{1}} and A_{\mathrm{2}} are constants determined from the boundary conditions; \omega is the radial frequency; k=\omega \sqrt{\rho /E} is the wavenumber.

By substituting values of x equal to zero and L into Eq. (10), the spectral nodal displacements can be expressed in terms of A_m(m = 1, 2) as:

(11)\mathit{y}=\mathit{P}\mathit{A},

where \mathit{y}={\left[u_0{u}_L\right]}^{\mathrm{T}}.

The force-displacement relation of axial vibration can be written as:

(12)N=EA{U}^{\prime }.

Substituting Eq. (10) at xequal to zero and L into Eq. (12), the spectral nodal forces can be expressed in terms of A_m(m = 1, 2) as:

(13)\mathit{F}=\mathit{Q}\mathit{A},

where \mathit{F}={\left[N_0{N}_L\right]}^{\mathrm{T}}.

The spectral force-displacement relation can be obtained by eliminating the coefficients A_m in both Eqs. (11) and (13) as:

(14)\mathit{F}=\mathit{Q}{\mathit{P}}^{-\mathrm{1}}\mathit{y}=\mathit{S}\mathit{y},

where,

(15)\mathit{S}={\displaystyle \frac{EA·ikL}{\mathrm{1}-e^{-\mathrm{2}ikL}}}\left[\begin{array}{ll}\mathrm{1}+e^{-\mathrm{2}ikL}\hfill & -\mathrm{2}{e}^{-ikL}\hfill \\ -\mathrm{2}{e}^{-ikL}\hfill & \mathrm{1}+e^{-\mathrm{2}ikL}\hfill \end{array}\right].

The local stiffness matrix of each element is obtained and then assembled into the global stiffness matrix. The compatibility of the local stiffness matrix is satisfied at the assembly stage by constraining each element displacement to match the displacement at the nodes. Hence,

(16)\mathit{F}=\mathit{S}\mathit{y},

where BoldItalic is the global dynamic stiffness matrix, which is the reciprocal of the transfer function. This matrix is symmetric and banded, as in the case of conventional finite elements, but is frequency dependent.

Damage identification

As shown in Eq. (7), the PZT excitation forces can be expressed as:

(17)\mathit{f}=\mathit{Z}\mathit{u},

where BoldItalic is a displacement vector at the referenced degrees of freedom on which structural impedance is known; \mathit{f} is a PZT excitation force vector; \mathit{Z} is a element structural impedance matrix.

The system displacement vector \mathit{y} can be partitioned as \mathit{y}={[\mathit{u}\mathit{v}]}^{\mathrm{T}}, where \mathit{v} denotes the displacement vector at the degrees of freedom on which structural impedance response is not known. The force vector is partitioned similarly as \mathit{F}={\left[\mathit{f}{\mathit{f}}^{\prime }\right]}^{\mathrm{T}}. The system matrix \mathit{S} can then be expressed in the following partitioned form:

(18)\mathit{S}=\left[\begin{array}{ll}{\mathit{S}}_{\mathbf{1}\mathbf{1}}& {\mathit{S}}_{\mathbf{1}\mathbf{2}}\\ {\mathit{S}}_{\mathbf{2}\mathbf{1}}& {\mathit{S}}_{\mathbf{2}\mathbf{2}}\end{array}\right].

Then, Eq. (16) can be described as:

(19)\left[\begin{array}{ll}{\mathit{S}}_{\mathbf{1}\mathbf{1}}& {\mathit{S}}_{\mathbf{1}\mathbf{2}}\\ {\mathit{S}}_{\mathbf{2}\mathbf{1}}& {\mathit{S}}_{\mathbf{2}\mathbf{2}}\end{array}\right]\left\{\begin{array}{l}\mathit{u}\\ \mathit{v}\end{array}\right\}=\left\{\begin{array}{l}\mathit{f}\\ {\mathit{f}}^{\prime }\end{array}\right\}.

Eliminating the BoldItalic coordinates from Eq. (19) one gets

(20)({\mathit{S}}_{\mathbf{1}\mathbf{1}}-{\mathit{S}}_{\mathbf{1}\mathbf{2}}{\mathit{S}}_{\mathbf{2}\mathbf{2}}^{-\mathbf{1}}{\mathit{S}}_{\mathbf{2}\mathbf{1}})\mathit{u}=\mathit{f}-{\mathit{S}}_{\mathbf{1}\mathbf{2}}{\mathit{S}}_{\mathbf{2}\mathbf{2}}^{-\mathbf{1}}{\mathit{f}}^{\prime }.

If the structure is only subjected to the PZT excitation forces, Eq. (20) can be described by

(21)({\mathit{S}}_{\mathbf{1}\mathbf{1}}-{\mathit{S}}_{\mathbf{1}\mathbf{2}}{\mathit{S}}_{\mathbf{2}\mathbf{2}}^{-\mathbf{1}}{\mathit{S}}_{\mathbf{2}\mathbf{1}})\mathit{u}=\mathit{f}.

Substituting Eq. (17) into Eq. (21) gives

(22)({\mathit{S}}_{\mathbf{1}\mathbf{1}}-{\mathit{S}}_{\mathbf{1}\mathbf{2}}{\mathit{S}}_{\mathbf{2}\mathbf{2}}^{-\mathbf{1}}{\mathit{S}}_{\mathbf{2}\mathbf{1}}-\mathit{Z})\mathit{u}=\mathbf{0}.

For any \mathit{u}, to make Eq. (22) satisfied, the following equation is obtained:

(23){\mathit{S}}_{\mathbf{1}\mathbf{1}}-{\mathit{S}}_{\mathbf{1}\mathbf{2}}{\mathit{S}}_{\mathbf{2}\mathbf{2}}^{-\mathbf{1}}{\mathit{S}}_{\mathbf{2}\mathbf{1}}-\mathit{Z}=\mathbf{0}.

Two sets of variables in Eq. (23), one coming from the numerical analysis and one from the experimental measurement, will be completely equivalent if the numerical model of the structure is correlated with the dynamic behavior of the real structure. The left side of Eq. (23) represents the error function between structural impedance response and its counterpart computed from the spectral element model. The error function for the kth modal frequency point is written as:

(24){\mathit{E}}_k={({\mathit{S}}_{\mathbf{1}\mathbf{1}}-{\mathit{S}}_{\mathbf{1}\mathbf{2}}{\mathit{S}}_{\mathbf{2}\mathbf{2}}^{-\mathbf{1}}{\mathit{S}}_{\mathbf{2}\mathbf{1}}-\mathit{Z})}_{\omega ={\omega }_k}.

Thus, through minimizing the error function and updating system matrices, structure spectral element model resulting from the optimization procedure represents the real damaged structure. The updated quantities of the system matrices show the severity and location of structure damage.

In the spectral element model, a change of wavenumber is analogous to changes in mass and stiffness in model based methods. Therefore, a damaged spectral element can be reconstructed as:

(25){\mathit{s}}_{\mathrm{d}}={\displaystyle \frac{EA·i\alpha kL}{\mathrm{1}-e^{-\mathrm{2}i\alpha kL}}}\left[\begin{array}{ll}\mathrm{1}+e^{-\mathrm{2}i\alpha kL}\hfill & -\mathrm{2}{e}^{-i\alpha kL}\hfill \\ -\mathrm{2}{e}^{-i\alpha kL}\hfill & \mathrm{1}+e^{-\mathrm{2}i\alpha kL}\hfill \end{array}\right],

where \alpha represents the relative change of a damaged wavenumber to an undamaged one.

If multiple PZT patches attached to the host structure are excited one by one, the objective function can be constructed and its value can be minimized to update the system matrices to represent the damaged structure. The objective function can be described as:

(26)J={\displaystyle \sum _{k=\mathrm{1}}^{n\mathrm{f}}{\displaystyle \sum _{l=\mathrm{1}}^{n\mathrm{p}}{\displaystyle \sum _{i=\mathrm{1}}^{n\mathrm{d}}{\displaystyle \sum _{j=\mathrm{1}}^{n\mathrm{d}}\left({\displaystyle \frac{E_{klij}^{\mathrm{2}}}{Z_{klij}^{\mathrm{2}}}}\right)}},}}

where nf represents the number of excitation frequency points; np represents the number of PZT patches; nd represents the number of referenced degrees of freedom. The extent of damage, which means the value of \alpha, may be obtained with the optimal value of the scalars greater than zero.

Numerical study

An example is presented to validate the effectiveness of the damage identification method presented above. The structure is a pin-pin bar as shown in Fig. 2, which undergoes a longitudinal vibration and is modeled with 20 elements. The bar has the dimensions of 200×200×30 mm, with the material properties as follows: Young’s modulus is 210 GPa and mass density is 7800 kg/m³. Each spectral element is assigned one scalar for system matrix. Therefore there are 20 design variables in this problem. The PZT patches are assumed attached to both top and bottom surfaces to excite axial vibration.

Then, Eq. (9) can be re-written as:

(27)Z_{\mathrm{s}}={\displaystyle \frac{\rho A}{i\omega }}{\left\{{\displaystyle \sum _{n_u=\mathrm{0}}^{\infty }{\displaystyle \frac{{[U_{n_u}(l_{\mathrm{a}}+x_{\mathrm{a}})-U_{n_u}(x_{\mathrm{a}})]}^{\mathrm{2}}}{{\omega }_{n_u}^{\mathrm{2}}-{\omega }^{\mathrm{2}}}}}\right\}}^{-\mathrm{1}}.

Equation (27) is used in the present tests to numerically simulate structural impedance curves of the pin-pin bar in both damaged and undamaged states using the first 19 mode shapes of the bar.

The PZT patches are assumed to be mounted on element 5 and 16. The excitation frequency of PZT patches is in the range of 3-10 kHz. Different excitation frequency points were tried because sometimes the objective function did not go to the minimum. For example, if the excitation frequency is close to the natural frequency, structural resonance may occur. Meanwhile, the choice of the initialization value can also influence the optimal value. The identification was assumed to be accurate when the objective function was close to zero.

Damage localization and quantification

Table 1 presents a typical set of the single damage case and the predicted values. The location and extent of damage are simulated to be randomly generated. The accuracy of predictions shown in Table 1 tells that the method clearly identify the location and extent of single structural damage.

At the same time, a typical set of multiple damaged case and the predicted results are presented in Table 2. These cases are also randomly generated. As shown, the method can also identify multiple structural damages.

Effect of measurement noise on damage identification

Generally, the noise affects the measurement results. Therefore, to investigate the robustness of the presented algorithm, random experimental noise is simulated in the numeral studies. A 5% random noise is added to the simulated structural impedance response to express errors in measured results. Structural impedance contaminated with 5% random noise can be described as:

(28){\overline{Z}}_{\mathrm{s}i}=Z_{\mathrm{s}i}(\mathrm{1}+\mathrm{0.05}{r}_i).

where {\overline{Z}}_{\mathrm{s}i} and Z_{\mathrm{s}i} are structural impedance at the ith frequency sampling point with noise and without noise respectively; and r_i is the random number with a mean equal to zero and a variance equal to one.

Table 3 presents a typical set of single damage cases and the identified values with 5% noise. The location and extent of damage are randomly generated. The accuracy of identification as shown in Table 3 is good. The results indicate that the method for single damage identification is robust to disturbance induced by noise.

At the same time, a typical set of the multiple damaged cases with 5% noise and the identified results are presented in Table 4. These cases are also randomly generated. Table 4 presents the locations and extents of structural damage are approximately identified. Comparing the results with 5% noise to the values with no noise, no obvious influence from the added noise can be identified. With the noise, the damage can be still identified clearly, and the robustness of the method to noise is high.

Conclusions

In this paper, a damage identification study using piezoelectric impedance method is presented. The numerical study on a pin-pin bar tells that this model-based method can accurately and efficiently identify both the location and severity of structural damage through minimization of the deviations between PZT electric admittance curve and the baseline response from the numerical model. Moreover, when the measurements are polluted with 5% noise structure damage can still be accurately identified. More detailed discussions on how to select the frequency band for the use of impedance, where and how dense to place the PZT patches on the host structure for best damage identification purpose, how to set the initialization value for the updating will be offered in some coming papers.

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