Discrete singular convolution and spectral finite element method for predicting electromechanical impedance applied on rectangular plates

Abstract

The impedance-based structural health monitoring using piezoelectric wafer-active sensor has been increasingly developed for aerospace, civil, and mechanical structures. Using electromechanical coupling effects of piezoelectric wafer-active sensor, impedance of piezoelectric wafer-active sensor can detect any change in a structure. Piezoelectric wafer-active sensor is embedded and bounded to the structure in order to monitor the structure in a preferred frequency range. This article presented a general model to predict the impedance of piezoelectric wafer-active sensor bounded to a plate structure and also developed a general model for the influence of interaction of piezoelectric wafer-active sensor and plate on the structural response in impedance-based structural health monitoring. To obtain equations of vibrations, in the first step, potential and kinetic energies of fully free Kirchhoff and Mindlin plates with piezoelectric wafer-active sensor are derived. Numerical solutions for structural vibration of the plate developed using discrete singular convolution methods and applied based on Rayleigh–Ritz method in very high frequencies. After calculating mass and stiffness matrices of structure and piezoelectric wafer-active sensor, impedance of piezoelectric wafer-active sensor was determined using piezoelectric constitutive equations. Also, a three-dimensional spectral finite element method was used to model impedance of plate. An experimental setup was used for modal analysis to obtain low natural frequencies and calculate the impedance of piezoelectric wafer-active sensor in high frequencies. Finally, the comparison of numerical results of three-dimensional spectral element method and experimental results verified this model.

Keywords

Complex impedance structural health monitoring Kirchhoff plate Mindlin plate discrete singular convolution experimental setup

Introduction

Over the past few decades, light-weighted and inexpensive piezoelectric wafer-active sensors (PWASs) have been used for impedance-based structural health monitoring (ISHM) to detect damage and fault in aerospace, civil, and mechanical structures (Giurgiutiu and Zagrai, 2005; Lopes et al., 2000; Park et al., 2003, 2005; Park and Inman, 2005; Peairs et al., 2004; Sepehry et al., 2011, 2014; Zagrai and Giurgiutiu, 2001). In the ISHM method, PWAS is embedded and bonded to the structure, and then, it is actuated using a harmonic signal in a preferred frequency range to generate the impedance of PWAS. This impedance signature is related to mechanical impedance of the structure (Giurgiutiu and Zagrai, 2001). Therefore, PWAS can detect any change in the structure using impedance variation, and it can be used as a damage sensor. For real-time structural health monitoring (SHM) application, these signals can be compared with each other to determine the damage during the service structures. In an ISHM model, vibration interaction between structure and PWAS is used to drive impedance signature of PWAS.

A continuous model of a beam and PWAS was developed by Giurgiutiu (2007) for calculation of impedance of PWAS. A model based on the energy method was presented by Sepehry et al. (2010) for an Euler–Bernoulli beam considering PWAS attached to the beam. Yang and Miao (2010) modeled the effect of external vibration on PWAS impedance for two-dimensional (2D) system. Bhalla and Soh (2004) studied the 2D interaction of PWAS with structure based on mechanical impedance to determine the electrical impedance of PWAS. Also, the effect of temperature on the impedance of a fully clamped plate was investigated by Sepehry et al. (2014). Adhesive effect of impedance signature for circular disk has been studied in Rugina et al. (2015).

Traditionally, the ISHM models used transverse vibration to calculate the impedance of PWAS. However, for real application, in-plane vibration should be considered in high-frequency analysis. For solving the continuous plate model and approximating the solution of boundary value problem, Rayleigh–Ritz method was applied to discretize the system. In Rayleigh–Ritz method, a set of shape functions which satisfy at least the geometrical boundary conditions was assumed to model vibration modes. With the choice of proper shape functions, accuracy of the Rayleigh–Ritz method was guaranteed. Discrete singular convolution (DSC) algorithm is a local method to solve the partial differential equations (Baltacıoğlu et al., 2011; Civalek, 2006, 2007a, 2007b, 2008b, 2013; Demir et al., 2010; Wei et al., 2001; Zhu and Wang, 2011). At first, a potential method to solve the partial differential equations using DSC algorithm was proposed by Wei (1999). Different works were examined by numerical solution including quantum problem (Wei, 2000) and mechanical vibration (Wei, 2001b). The ISHM is a high-frequency method for structural monitoring of damage. A few works used vibration modeling in high-frequency analysis with different algorithm (Lim et al., 2005; Wan and Wei, 2000). In these methods, DSC algorithm is a good, robust, and efficient method to solve the vibration problems in high frequency (Civalek and Gürses, 2009; Lim et al., 2005; Seçgin and Sarıgül, 2009; Wan and Wei, 2000; Zhao et al., 2002).

Spectral element method (SEM) has been used in several applications (Mohamed and Masson, 2010; Patera, 1984; Rowlatt and Phillips, 2016). Patera (1984) developed the SEM, which potentially increased the low-convergence rate problem of finite element method (FEM). Spectral finite elements can be used for domain decomposition in complex domain methods for the solution of boundary value problems which combine the generality and flexibility of the FEM (Kim et al., 2015; Maday and Patera, 1989). The spectral finite element method (SFEM) has three main advantages over the low-order FEM: first, it has higher accuracy provided by the spectral basis; second, the mass matrix will be diagonal using Gauss–Lobatto quadrature rule in elemental Lagrangian bases; and finally, it provides exponential rate of convergence for smooth solution problems (Valenciano and Chaplain, 2005). For evaluating the integrals appearing after the weak formulation, the Gauss–Lobatto quadrature rule was used. Normally, Legendre polynomials and the Gauss–Lobatto–Legendre (GLL) quadrature rule were used in SFEM for finite domains.

The main purpose of this study is to develop a general model to investigate the influence of interaction of PWAS and plate on the structural response in ISHM. This article presents Kirchhoff and Mindlin plates equations to model a fully free plate. The numerical results of dynamic model are calculated based on Rayleigh–Ritz method in high frequencies. DSC method have been developed and applied for calculation of mass and stiffness matrices based on Rayleigh–Ritz method. After that, the related equations of PWAS were implemented to find impedance of PWAS. Then, three-dimensional (3D) SFEM was used to model plate and PWAS interaction. Then, impedance of PWAS was calculated. A modal analysis setup was used for comparing low natural frequency for experiment, DSC, and SFEM. Also, this setup could be used for determination of Rayleigh damping coefficient. Finally, an experimental setup was used for comparing the experimental, DSC (Kirchhoff and Mindlin plates), and SFEM data of impedance signal in high frequencies.

Governing equations

The impedance model of PWAS is useful because of the following reasons:

Understanding the effect of mechanical vibration on the response of electrical impedance of PWAS;

Optimization of PWAS dimensions and location on structure;

Reducing the cost and time of tests in SHM.

Modeling of Kirchhoff plate

Stress and displacement field

The plate with attached PWAS is shown in Figure 1. The mechanical displacement field will be determined as (Sepehry et al., 2014)

\begin{matrix} u_{0} (x, y, z, t) = u (x, y, t) - z \frac{\partial w}{\partial x} (x, y, t) \\ v_{0} (x, y, z, t) = v (x, y, t) - z \frac{\partial w}{\partial y} (x, y, t) \\ w_{0} (x, y, z, t) = w (x, y, t) \end{matrix}

(1)

where $u_{0} (x, y, z, t)$ , $v_{0} (x, y, z, t)$ , and $w_{0} (x, y, z, t)$ are the displacements of an arbitrary point of the plate along the x, y, and z directions, respectively, and u and w are the displacements of the mid-surface of the plate.

Using PWAS constitutive equations, we have (Sepehry et al., 2014)

\begin{matrix} ε = S^{E} σ + d^{t} E \\ D = d σ + ε^{T} E \end{matrix}

(2)

where $σ$ is the stress, $ε$ is the strain, $E$ is the electrical field, $ε^{T}$ is the permittivity, $d$ is a matrix for the direct piezoelectric effect, S is a matrix of compliance coefficients, and D is the vector of electric displacement. Also, superscripts E, T, and t indicate a zero, or constant, for electric field, stress field, and the transpose, respectively. Equation (2) can be defined as (Sepehry et al., 2014)

\begin{matrix} σ = C^{E} ε - e^{t} E \\ D = e ε + ε^{E} E \end{matrix}

(3)

In which stress and electrical displacement are defined as a function of strain and electrical field. Also, new constants can be defined as follows

C^{E} = (S^{E})^{- 1}, e^{t} = (S^{E})^{- 1} d^{t}, ε^{E} = (ε^{T} - d e^{t})

(4)

Using plate assumption, only normal and shear strain in x and y planes can be considered, so we have (Sepehry et al., 2014)

C^{E} = \frac{E_{p}}{(1 - υ_{p}^{2})} [\begin{array}{l} 1 & υ_{p} & 0 \\ υ_{p} & 1 & 0 \\ 0 & 0 & \frac{1 - υ_{p}}{2} \end{array}], e^{t} = [\begin{array}{l} 0 & 0 & e_{31} \\ 0 & 0 & e_{31} \\ 0 & 0 & 0 \end{array}], ε^{E} = [\begin{array}{l} ε_{11}^{E} & 0 & 0 \\ 0 & ε_{22}^{E} & 0 \\ 0 & 0 & ε_{33}^{E} \end{array}]

(5)

where $E_{p}$ is Young’s modulus of PWAS and $υ_{p}$ is Poisson’s ratio. Therefore, using equations (3) and (5), stress field can be defined as (Sepehry et al., 2014)

\begin{matrix} σ_{x x} = \frac{E_{p}}{(1 - υ_{p}^{2})} (ε_{x x} + υ ε_{y y}) - e_{31} E_{z} \\ σ_{y y} = \frac{E_{p}}{(1 - υ_{p}^{2})} (ε_{y y} + υ ε_{x x}) - e_{31} E_{z} \\ σ_{x y} = \frac{E_{p} (1 - υ_{p})}{(1 - υ_{p}^{2})} (ε_{x y}) \end{matrix}

(6)

Using equation (3) and assuming the electrical field is only in z direction, electrical displacement can be calculated as follows (Sepehry et al., 2014)

D_{z} = e_{31} (\frac{\partial u}{\partial x} - z \frac{\partial^{2} w}{\partial x^{2}}) + e_{31} (\frac{\partial v}{\partial y} - z \frac{\partial^{2} w}{\partial y^{2}}) + ε_{33}^{E} E_{z}

(7)

The stress field of the plate can be obtained by elimination of electrical field of first PWAS constitutive equations.

Kinetic and potential energies

Kinetic and potential energies of PWAS and plate could be calculated by Sepehry et al. (2014)

T_{P W A S} = \frac{1}{2} \int_{A_{p}} ρ_{p} h_{p} ({\overset{\cdot}{u}}^{2} + {\overset{\cdot}{v}}^{2} + {\overset{\cdot}{w}}^{2}) d A_{p}

(8)

T_{p l a t e} = \frac{1}{2} \int_{A} ρ h ({\overset{\cdot}{u}}^{2} + {\overset{\cdot}{v}}^{2} + {\overset{\cdot}{w}}^{2}) d A

(9)

The potential energies of PWAS and plate using stress and electrical displacement fields were defined as (Sepehry et al., 2014)

U_{P W A S} = \frac{1}{2} \int_{V_{p i e z o}} (σ_{x x} ε_{x x} + σ_{x y} ε_{x y} + σ_{y y} ε_{y y} - D_{z} E_{z}) d v

(10)

U_{p l a t e} = \frac{1}{2} \int_{V_{p l a t e}} (σ_{x x} ε_{x x} + σ_{x y} ε_{x y} + σ_{y y} ε_{y y} + σ_{y y} ε_{y y}) d v

(11)

The resulting electric field is assumed to be uniform over the PWAS, so we have

E_{z} = \frac{V (t)}{h_{p}}

(12)

System discretization

Since the system has continuous mechanical characteristics, a partial differential equation describes its motion, and Rayleigh–Ritz method can be used to discretize the system equation. Finally, the Lagrange equation is used to obtain the system differential equation of motion. In Rayleigh–Ritz method, system response is assumed as follows

\begin{matrix} u (x, y, t) = \sum_{i = 1}^{N_{u}} \sum_{j = 1}^{N_{u}} ϕ_{i j}^{u} (x, y) q_{i j}^{u} (t) \\ v (x, y, t) = \sum_{i = 1}^{N_{u}} \sum_{j = 1}^{N_{v}} ϕ_{i j}^{v} (x, y) q_{i j}^{v} (t) \\ w (x, y, t) = \sum_{i = 1}^{N_{w}} \sum_{j = 1}^{N_{w}} ϕ_{i j}^{w} (x, y) q_{i j}^{w} (t) \end{matrix}

(13)

where $ϕ_{i j} (x, y)$ is the modal shape function which must satisfy boundary conditions and $q_{i j}$ is the modal function. The superscripts u, v, and w indicate modal shape and modal function for mechanical displacements of $u (x, y, t)$ , $v (x, y, t)$ , and $w (x, y, t)$ , respectively. Also, $N_{u}$ , $N_{v}$ , and $N_{u}$ are number of these modal shapes and modal functions in mechanical displacement.

The kinetic and potential energies of the system are function of modal functions; therefore, the differential equation of the system can be obtained using Lagrange equation.

Electrical response of PWAS

Recalling equation (7), which represents the electrical displacement and integration of $D_{z}$ over the electrodes’ area $A_{p}$ , yields the total charge $Q (t)$ as

Q (t) = \int_{A_{p}} D_{z} d A_{p}

(14)

Finally, the impedance, $Z (i ω)$ , defined as the ratio of the voltage to current can be calculated by Sepehry et al. (2014)

Z (i ω) = \frac{V (i ω)}{I (i ω)} = {([\begin{matrix} - e_{31} h_{p} φ^{w} & e_{31} φ^{u} & e_{31} φ^{v} \end{matrix}] {(T - M ω^{2} + C i ω + K)}^{- 1} N + \frac{ε_{33}^{E} (x_{p 2} - x_{p 1}) (y_{p 2} - y_{p 1})}{h_{p}})}^{- 1}

(15)

where M , C , and K are the mass, damping, and stiffness matrices, respectively, and N is the force and moment vector created by PWAS actuator.

This equation shows that the peaks in real part of impedance are natural frequencies of structure. Also, the real part of impedance is a nonlinear combination of mechanical model of structure and capacitance of PWAS. For structure with Rayleigh damping without PWAS, amplitude of natural frequency decreases with the increase in frequency amplitude. But in the real part of impedance of PWAS, this damping does not result in similar effect on the amplitude of the real part of impedance. This is because of nonlinear combination of mechanical model of structure and capacitance of PWAS as shown in equation (15).

Modeling of Mindlin plate

Stress and displacement field

The mechanical displacement field of Mindlin plate will be determined by Hou et al. (2004)

u_{0} (x, y, z, t) = u (x, y, t) - z θ ψ (x, y, t), v_{0} (x, y, z, t) = v (x, y, t) - z θ (x, y, t), w_{0} (x, y, z, t) = w (x, y, t)

(16)

where $u_{0} (x, y, z, t)$ , $v_{0} (x, y, z, t)$ , and $w_{0} (x, y, z, t)$ are the displacements of an arbitrary point of the plate along the x, y, and z directions, respectively; moreover, u and w are the displacements of the mid-surface of the plate; and $θ_{0}$ and $ψ_{0}$ are the rotations of cross sections along x and y directions, respectively.

According to equation (3) and using Mindlin plate assumption, we have

C = \frac{E_{p}}{1 - υ_{p}^{2}} [\begin{matrix} 1 υ_{p} & 0 & 0 & 0 \\ υ_{p} & 1 & 0 & 0 & 0 \\ 0 & 0 & \frac{1 - υ_{p}}{2} & 0 & 0 \\ 0 & 0 & 0 & k \frac{1 - υ_{p}}{2} & 0 \\ 0 & 0 & 0 & 0 & k \frac{1 - υ_{p}}{2} \end{matrix}], ε^{E} = [\begin{array}{l} ε_{11}^{E} & 0 & 0 \\ 0 & ε_{22}^{E} & 0 \\ 0 & 0 & ε_{33}^{E} \end{array}], e^{t} = [\begin{matrix} 0 & 0 & 0 & 0 & e_{31} \\ 0 & 0 & 0 & 0 & e_{31} \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{matrix}]

(17)

where $E_{p}$ is Young’s modulus of PWAS, $υ_{p}$ is Poisson’s ratio, and $k = π^{2} / 12$ is shear correction factor. Therefore, using equations (3) and (17), stress field can be defined as (Hou et al., 2004)

\begin{matrix} σ_{x x} = \frac{E}{1 - υ^{2}} (ε_{x x} + υ ε_{y y}) σ_{y z} = k \frac{E}{1 - υ^{2}} (\frac{1 - υ}{2}) ε_{y z} \\ σ_{y y} = \frac{E}{1 - υ^{2}} (ε_{y y} + υ ε_{x x}) σ_{x y} = \frac{E}{1 - υ^{2}} (\frac{1 - υ}{2}) ε_{x y} \\ σ_{x z} = k \frac{E}{1 - υ^{2}} (\frac{1 - υ}{2}) ε_{x z} \end{matrix}

(18)

Using equation (3) and assuming the electrical field is only in z direction, electrical displacement can be calculated as follows

D_{z} = e_{31} (\frac{\partial u}{\partial x} - z \frac{\partial ψ}{\partial x}) + e_{31} (\frac{\partial v}{\partial y} - z \frac{\partial θ}{\partial y}) + ε_{33}^{E} E_{z}

(19)

The stress field of the plate can be obtained by elimination of electrical field of first PWAS constitutive equations.

Kinetic and potential energies

Kinetic and potential energies of PWAS and Mindlin plate could be calculated by

T_{P W A S} = \frac{1}{2} \int_{x_{1}}^{x_{2}} \int_{y_{1}}^{y_{2}} \int_{h / 2}^{(h / 2) + h_{p}} ρ {{(\frac{\partial u}{\partial t})}^{2} + {(\frac{\partial v}{\partial t})}^{2} + {(\frac{\partial w}{\partial t})}^{2} + z^{2} {(\frac{\partial θ}{\partial t})}^{2} + {(\frac{\partial ψ}{\partial t})}^{2}} d x d y d z

(20)

T_{p l a t e} = \frac{1}{2} \int_{x_{1}}^{x_{2}} \int_{y_{1}}^{y_{2}} \int_{h / 2}^{(h / 2) + h_{p}} ρ {{(\frac{\partial u}{\partial t})}^{2} + {(\frac{\partial v}{\partial t})}^{2} + {(\frac{\partial w}{\partial t})}^{2} + z^{2} {(\frac{\partial θ}{\partial t})}^{2} + {(\frac{\partial ψ}{\partial t})}^{2}} d x d y d z

(21)

The potential energies of plate and PWAS using stress and electrical displacement fields are defined as

U_{P W A S} = \frac{1}{2} \int_{x_{1}}^{x_{2}} \int_{y_{1}}^{y_{2}} \int_{h / 2}^{(h / 2) + h_{p}} {ε_{x x}^{2} + 2 υ ε_{y y} ε_{x x} + ε_{y y}^{2} + k (\frac{1 - υ}{2}) ε_{x z}^{2} + \frac{1 - υ}{2} k ε_{y z}^{2} + \frac{1 - υ}{2} ε_{x y}^{2} - D_{3} E_{3}} d x d y d z

(22)

U_{p l a t e} = \frac{1}{2} \int_{0}^{a} \int_{0}^{b} \int_{- h / 2}^{h / 2} {ε_{x x}^{2} + 2 υ ε_{y y} ε_{x x} + ε_{y y}^{2} + k (\frac{1 - υ}{2}) ε_{x z}^{2} + \frac{1 - υ}{2} k ε_{y z}^{2} + \frac{1 - υ}{2} ε_{x y}^{2}} d x d y d z (23)

(23)

System discretization

Using Rayleigh–Ritz method, system response is assumed as follows

\begin{matrix} u (x, y, t) = \sum_{i = 1}^{N_{u}} \sum_{j = 1}^{N_{u}} φ_{i j}^{u} (x, y) q_{i j}^{u} (t) \\ v (x, y, t) = \sum_{i = 1}^{N_{v}} \sum_{j = 1}^{N_{v}} φ_{i j}^{v} (x, y) q_{i j}^{v} (t) \\ w (x, y, t) = \sum_{i = 1}^{N_{w}} \sum_{j = 1}^{N_{w}} φ_{i j}^{w} (x, y) q_{i j}^{w} (t) \\ θ (x, y, t) = \sum_{i = 1}^{N_{θ}} \sum_{j = 1}^{N_{θ}} φ_{i j}^{θ} (x, y) q_{i j}^{θ} (t) \\ ψ (x, y, t) = \sum_{i = 1}^{N_{ψ}} \sum_{j = 1}^{N_{ψ}} φ_{i j}^{ψ} (x, y) q_{i j}^{ψ} (t) \end{matrix}

(24)

where $φ_{i j} (x, y)$ is the modal shape function which must satisfy boundary conditions and $q_{i j}$ is the modal function; the superscripts $θ$ , $ψ$ , u, v, and w indicate modal shape and modal function for mechanical displacements of $θ (x, y, t)$ , $ψ (x, y, t)$ , $u (x, y, t)$ , $v (x, y, t)$ , and $w (x, y, t)$ , respectively; and $N_{θ}$ , $N_{ψ}$ , $N_{u}$ , $N_{v}$ , and $N_{w}$ are number of these modal shapes and modal functions in mechanical displacement.

Electrical response of PWAS

The impedance, $Z (i ω)$ , defined as the ratio of the voltage to current can be calculated similar to equation (15).

The mass, damping, and stiffness matrices and force and moment vectors created by PWAS actuator for Mindlin plate are shown in Appendix 2.

Generation of characteristic function using DSC

In this section, three different characteristic functions based on DSC method have been developed and applied for determination of plate mode shapes. The DSC algorithm is a general method to solve singular convolution problem. Let T and $η (x)$ be the distribution and an element of the space of the test function, respectively. A singular convolution can be defined by Wei (2001a)

F (t) = (T * η) (t) = \int_{- \infty}^{\infty} T (t - x) η (x) d x

(25)

where $T (t - x)$ is a singular kernel. In this article, only singular kernel of delta type is considered. A singular kernel type can be defined by Hou et al. (2004) as

T (x) = δ^{(n)} (x), (n = 0, 1, 2, \dots)

(26)

For $n \geq 1$ , these kernels can be used for numerical solution of partial differential equations. With a sufficiently smooth approximation, a DSC can be considered as (Hou et al., 2004)

F_{α} (t) = \sum_{k} T_{α} (t - x_{k}) f (x_{k})

(27)

where $F_{α} (t)$ is an approximation for $F (t)$ and $x_{k}$ is an appropriate set of discrete points on which the DSC equation (18) is well defined. In this equation, the original test function $η (x)$ was replaced by $f (x)$ . Gauss’ delta sequence kernel can be considered as follows

δ_{σ} (x) = \frac{1}{\sqrt{2 π σ}} e^{- x^{2} / 2 σ^{2}}

(28)

According to equation (13), DSC-Ritz algorithm can be applied to this equation in which

X_{i}^{u, v, w, θ, ψ} = δ_{i σ} B_{x}^{u, v, w, θ, ψ}, Y_{i}^{u, v, w, θ, ψ} = δ_{i σ} B_{y}^{u, v, w, θ, ψ}

(29)

where $B_{x}^{u, v, w, θ, ψ}$ and $B_{y}^{u, v, w, θ, ψ}$ are the boundary functions and must satisfy geometric boundary conditions for transverse and in-plane vibrations. For fully free plate, boundary functions can be written for in-plane vibration as follows

B_{x}^{u} = B_{y}^{u} = B_{x}^{v} = B_{y}^{v} = 1

(30)

And for transverse vibration as

B_{x}^{w} = B_{y}^{w} = B_{x}^{θ} = B_{y}^{θ} = B_{x}^{ψ} = B_{y}^{ψ} = 1

(31)

In equation (20), $δ_{i σ}$ is a DSC delta kernel and for Gauss’ kernel can be given by Wei et al. (2002) and Xiang et al. (2010)

δ_{i σ} (x) = e^{- {(x - x_{i})}^{2} / 2 σ^{2}}

(32)

where $x_{i}$ is a grid-point coordinate along x-axis in equation (32). The parameter σ is chosen as $σ = r Δ$ , where $Δ = 2 / (N - 1)$ is the grid spacing and $r$ is an adjustable parameter. The parameter $r$ gives more accurate results for the interval $2 \leq r \leq 3$ in applied mechanics (Civalek, 2008a; Wei, 2001b). Similar to Sepehry et al. (2014), r = 2.2 for transverse and in-plane vibration is chosen.

3D SEM

Similar to FEM, SFEM decomposes the whole domain into small elements to solve differential equations. Regarding to solution accuracy, the spectral element has benefits over the finite element. Higher order interpolation function of SFEM has more accuracy than the linear interpolation function of the FEM for an element. Also, a diagonal mass matrix using nodal quadrature rule in GLL point was obtained in SFEM. For plate with low thickness, a hybrid SEM could be used. In this method, only two points were used in the thickness direction (Ha and Chang 2009), and Gauss quadrature rule was applied for accurate calculation of stiffness matrix. The hybrid SFEM leads to faster results and low memory requirement.

Inside each element, the model can be heterogeneous

x (ξ) = \sum_{a = 1}^{n_{a}} N_{a} (ξ) x_{a}

(33)

Points within the reference cube are denoted by the vector $ξ = (ξ, η, ζ)$ , where $- 1 \leq ξ \leq 1$ , $- 1 \leq η \leq 1$ , and $- 1 \leq ζ \leq 1$ . The geometry of the spectral elements is controlled by $n_{a} = 27$ points, $x_{a}$ , as shown in Figure 2.

Lagrange polynomial is given as

h_{α} (ξ) = \underset{m \neq j}{Π} \frac{(ξ - ξ_{m})}{(ξ_{j} - ξ_{m})}

(34)

n + 1 GLL points are the roots of Lagrange polynomial

(1 - ξ^{2}) {P'}_{n} (ξ) = 0

(35)

And we have

h_{α} (ξ_{β}) = δ_{α β}

(36)

3D displacements can be defined as

\begin{matrix} u (ξ, η, ζ) = \sum_{i} \sum_{j} \sum_{k} h_{i} (ξ) h_{j} (η) h_{k} (ζ) u_{i j k} \\ v (ξ, η, ζ) = \sum_{i} \sum_{j} \sum_{k} h_{i} (ξ) h_{j} (η) h_{k} (ζ) v_{i j k} \\ w (ξ, η, ζ) = \sum_{i} \sum_{j} \sum_{k} h_{i} (ξ) h_{j} (η) h_{k} (ζ) w_{i j k} \end{matrix}

(37)

where $u_{i j k}$ , $v_{i j k}$ , and $w_{i j k}$ are the nodal displacements, and i, j, and k denote the index of nodes in the ξ, η, and ζ directions, respectively.

In each element, the H matrix, which relates displacement fields to nodal displacements, can be expressed as (Ha and Chang, 2009)

\begin{matrix} u = Hd \\ d = {\begin{matrix} d_{111} & d_{112} & \begin{matrix} d_{121} & \dots & d_{n_{1} n_{2} 2} \end{matrix} \end{matrix}} \\ d_{i j k} = {\begin{matrix} u_{i j k} & v_{i j k} & w_{i j k} \end{matrix}} \\ H = {\begin{matrix} H_{111} & H_{112} & \begin{matrix} H_{121} & \begin{matrix} \dots & H_{n_{1} n_{2} 2} \end{matrix} \end{matrix} \end{matrix}} \\ H_{ijk} = [\begin{matrix} h_{i} (ξ) h_{j} (η) h_{k} (ζ) & 0 & 0 \\ 0 & h_{i} (ξ) h_{j} (η) h_{k} (ζ) & 0 \\ 0 & 0 & h_{i} (ξ) h_{j} (η) h_{k} (ζ) \end{matrix}] \end{matrix}

(38)

In both structural and PZT (lead zirconate titanate) elements, the B _u matrix, which relates nodal displacements to strain field, is represented by Ha and Chang (2009)

\begin{matrix} ε = B_{u} d \\ ε = [\begin{matrix} ε_{x x} & ε_{y y} & \begin{matrix} ε_{z z} & ε_{y z} & \begin{matrix} ε_{z x} & ε_{x y} \end{matrix} \end{matrix} \end{matrix}] \\ B_{u} = [\begin{matrix} B_{111} & B_{112} & \begin{matrix} B_{121} & \dots & B_{n_{1} n_{2} 2} \end{matrix} \end{matrix}] \\ B_{i j k} = [\begin{matrix} \frac{\partial h_{i} (ξ)}{\partial x} h_{j} (η) h_{k} (ζ) & 0 & 0 \\ 0 & h_{i} (ξ) \frac{\partial h_{j} (η)}{\partial y} h_{k} (ζ) & 0 \\ 0 & 0 & h_{i} (ξ) h_{j} (η) \frac{\partial h_{k} (ζ)}{\partial z} \\ 0 & h_{i} (ξ) h_{j} (η) \frac{\partial h_{k} (ζ)}{\partial z} & h_{i} (ξ) \frac{\partial h_{j} (η)}{\partial y} h_{k} (ζ) \\ h_{i} (ξ) h_{j} (η) \frac{\partial h_{k} (ζ)}{\partial z} & 0 & \frac{\partial h_{i} (ξ)}{\partial x} h_{j} (η) h_{k} (ζ) \\ h_{i} (ξ) \frac{\partial h_{j} (η)}{\partial y} h_{k} (ζ) & \frac{\partial h_{i} (ξ)}{\partial x} h_{j} (η) h_{k} (ζ) & 0 \end{matrix}] \end{matrix}

(39)

It is assumed that the electric potential for the PWAS layer is distributed linearly through the thickness (Ge et al., 2014). So, the electric voltage $V_{p}$ in the PWAS can be calculated by

E_{z} = - \frac{\partial V_{p}}{\partial z} = - \frac{V}{h_{p}} = - [B_{V}] V

(40)

The differential equation of global system is given by

{\begin{matrix} M \overset{\cdot\cdot}{q} + C \overset{\cdot}{q} + K q = - K_{u V} V_{a} \\ K_{V V} V_{s} = - K_{V u} q \end{matrix}

(41)

where $V_{a}$ and $V_{s}$ are the actuator voltage and sensor voltage, respectively.

In which mass and stiffness matrix element can be defined for plate and PWAS

\begin{matrix} [M^{e}] = \underset{V}{\int \int \int} ρ {[N]}^{T} [N] d V \\ [K^{e}] = \underset{V}{\int \int \int} {[B]}^{T} [D] [B] d V \end{matrix}

(42)

where $ρ$ and $[D]$ are the density and elastic matrix, respectively, and C is the Rayleigh damping and defined as

C = α M + β K

(43)

According to piezoelectric constitutive equations, we have the following equation for PWAS element

\begin{matrix} \begin{matrix} σ = [C] ε + {[e]}^{T} E \\ D = [e] ε + [ε^{S}] E \end{matrix} \\ [K_{u V}^{e}] = {[K_{u V}^{e}]}^{T} = \underset{V}{\int \int \int} {[B_{u}]}^{T} {[e]}^{T} [B_{V}] d V \\ [K_{V V}^{e}] = \underset{V}{\int \int \int} {[B_{v}]}^{T} [ε^{S}] [B_{V}] d V \end{matrix}

(44)

For PWAS impedance modeling, charge of PWAS could be calculated as follows

Q (t) = \int \int_{A} D \cdot d A

(45)

where A is the PWAS area. Electrical impedance could be given as follows

Z (ω) = \frac{V (ω)}{j ω Q (ω)}

(46)

Numerical and experimental implementation

Material and geometry of plate and PWAS

Tables 1 to 5 show the material and geometrical properties of aluminum beam, plate, and PWAS PSI-5H4E used in numerical and experimental implementation. Numerical integrations are calculated by Gaussian quadrature with appropriate number of polynomials. For all simulations after convergence study similar to Sepehry et al. (2014), $N_{w} = 35 and N_{u} = N_{v} = 20$ are the set for Kirchhoff plate and $N_{w} = 40, N_{u} = N_{v} = 20$ and $N_{θ} = N_{ψ} = 25$ are the set for Mindlin plate DSC characteristic function. Also, for SFEM simulation, we set aluminum and PWAS elements as 5 × 5 × 1.

Table 1.

Material properties of aluminum plate, beam, and PWAS.

	Aluminum	PWAS
$E_{11} (G P a)$	70	60
$E_{22} (G P a)$	70	60
$E_{33} (G P a)$	70	60
$G_{13} (G P a)$	26.92	23
$G_{23} (G P a)$	26.92	23
$G_{12} (G P a)$	26.92	23
ν	0.3	0.3
$ρ (kg m^{- 3})$	2700	7750
$d_{31} = d_{32} ({CN}^{- 1})$	—	$- 320 \times 10^{- 12}$
$d_{33} = d_{15} = d_{24} ({CN}^{- 1})$	—	$650 \times 10^{- 12}$
$ε_{31}^{s} = ε_{32}^{s} (F m^{- 1})$	—	$3860 \times 8.85 \times 10^{- 12}$
$ε_{33}^{s} (F m^{- 1})$	—	$3860 \times 8.85 \times 10^{- 12}$
α	$10^{- 3}$	$10^{- 3}$
β	$8 \times 10^{- 8}$	$8 \times 10^{- 8}$

PWAS: piezoelectric wafer-active sensor.

Table 2.

Geometrical properties of aluminum beam.

Property	Symbol	Value
Length	$a$	$337 mm$
Width	$b$	$20 mm$
Thickness	$h$	$2 mm$

Table 3.

Geometrical properties of PWAS PSI-5H4E on beam.

Property	Symbol	Value
Length	$L_{p}$	$10 mm$
Width	$b_{p}$	$20 mm$
Thickness	$h_{p}$	$0.267 mm$
Distance of PWAS first edge from origin in x-axis	$x_{p 1}$	$110 mm$
Distance of PWAS end edge from origin in x-axis	$x_{p 2}$	$x_{p 2} = x_{p 1} + L_{p}$

PWAS: piezoelectric wafer-active sensor.

Table 4.

Geometrical properties of aluminum plate.

Property	Symbol	Value
Length	$a$	$50 cm$
Width	$b$	$50 cm$
Thickness	$h$	$0.8 mm$

Table 5.

Geometrical properties of PWAS PSI-5H4E on plate.

Property	Symbol	Value
Length	$L_{p}$	$2 cm$
Width	$b_{p}$	$1.5 cm$
Thickness	$h_{p}$	$0.267 mm$
Distance of PWAS first edge from origin in x-axis	$x_{p 1}$	$192.5 mm$
Distance of PWAS end edge from origin in x-axis	$x_{p 2}$	$x_{p 2} = x_{p 1} + L_{p}$
Distance of PWAS first edge from origin in y-axis	$y_{p 1}$	$240 mm$
Distance of PWAS end edge from origin in y-axis	$y_{p 2}$	$y_{p 2} = x_{p 1} + b_{p}$

PWAS: piezoelectric wafer-active sensor.

Experimental setup

For calculation of low natural frequency and Rayleigh damping coefficient, a modal analysis using impact hammer (PCB Model 086C02), accelerometer (B & K 4366), and charge amplifiers TE5858 as shown in Figure 3 is used.

For impedance method, a chirp signal is applied to PWAS, and then, the output current is measured. The oscilloscope used in this setup is Pico Scope 4424 and arbitrary function generator is Agilent function generator (AWG) 33220A as shown in Figure 4. Also, experimental setup for beam structure is shown in Figure 5.

The adhesive used to bond the PWAS transducers to the aluminum plate is Cyanoacrylate glues CN-Y epoxy. A copper tape conductive adhesive is used to access the bottom electrode of PWAS. Then, the electrodes of the transducers are soldered by two 20-cm-long electrical wires.

Results and discussions

In this section, the experimental results of electromechanical impedance were compared with DSC and SFEM for fully free plate and beam. The results of investigation in frequency range of 10–20 kHz for beam and 3D SFEM are shown in Figure 6. From this figure, it is clear that there is a good agreement between 3D SFEM and experimental results.

Also, the frequency range of 10–14 kHz was investigated. Results of low natural frequency using DSC, modal analysis, and SFEM are presented in Table 6. It can be seen that there is a good agreement between Kirchhoff plate, SFEM, and modal analysis in low frequencies. Also, modal analysis is used for determination of Rayleigh damping coefficient.

Table 6.

Five first natural frequencies of plate (Hz).

Natural frequency	Experimental	SFEM	DSC
1	6.094	6.01	5.17
2	11.6	11.3	11
3	15.19	15.8	17.4
4	26.15	26.5	27.3
5	29.5	28.75	27.3

SFEM: spectral finite element method; DSC: discrete singular convolution.

From this table, it is found that there is a good agreement between experimental, SFEM, and Kirchhoff plate results.

Convergence study of natural frequencies in real part of impedance using DSC method for Kirchhoff plate is shown in Table 7. Figure 7 shows the real part of impedance signal for experimental results, SFEM, and DSC for frequency range of 10–14 kHz. Number of degree of freedom (DOF) for SFEM is 23,673. The real part of impedance at this frequency range can be better visualized by focusing on 10–10.18 kHz as shown in Figure 8. Also, in this figure, Mindlin plate assumption is presented. For better displaying the data, DSC results is amplified by factor of 10. This is because of difference between Kirchhoff plate and experimental result. In all, 16 numbers of first natural frequencies of plate in frequency range 10–14 kHz is presented in Table 8. The third column shows the natural frequency of Kirchhoff plate method. According to this column, only, five natural frequencies are obtained using Kirchhoff method. The fourth column shows the percentage of error between the SFEM and experimental results. Figures 10 and 11 show the imaginary and phase angle of impedance for frequency range 10–10.18 kHz, respectively.

Table 7.

Convergence study of natural frequency using DSC in real part of impedance in 10–14 kHz (Hz).

$N_{w} = 30, N_{u} = N_{v} = 17$	$N_{w} = 33, N_{u} = N_{v} = 19$	$N_{w} = 35, N_{u} = N_{v} = 20$
10,059.791	10,049.900	10,048.317
10,194.532	10,189.939	10,180.059
10,230.633	10,220.845	10,211.072
10,292.421	10,278.994	10,274.008
10,353.373	10,341.395	10,333.126

DSC: discrete singular convolution.

Table 8.

16 first natural frequencies of plate in frequency range 10–14 kHz (Hz).

Experimental	SFEM	Kirchhoff	% Error, SFEM/Experimental
10,029.488	10,026.422	—	0.5908
10,050.238	10,039.911	10,048.317	0.41306
10,072.158	10,072.747	—	0.10258
10,082.216	10,087.490	—	0.18694
10,121.728	10,114.374	—	0.30619
10,140.472	10,144.298	—	0.28617
10,148.363	10,155.588	—	0.35735
10,174.035	10,175.260	10,180.059	0.74881
10,191.796	10,209.109	10,211.072	0.55205
10,268.229	10,230.512	—	0.59502
10,314.901	10,268.688	10,274.008	0.34189
10,331.431	10,305.054	10,333.126	0.33341
10,370.921	10,344.504	—	0.61199
10,391.099	10,363.162	—	0.41738
10,405.832	10,390.345	—	0.36929
10,426.334	10,401.769	—	0.42947

SFEM: spectral finite element method.

Table 8 and Figures 8 to 11 show a good agreement between experimental results and SFEM. However, the result of Mindlin plate has good agreement with experimental results but the result of Kirchhoff plate is not close to the experimental results. This is because of two reasons: first, the wavelength is low in high frequency, so 2D modeling cannot present a good model for high frequency and 3D modeling should be used. This is because this difference will be more with increasing dimension of the plate. Second, low-order model of the plate similar to the Kirchhoff model cannot represent the structure properly in high frequency. So, a higher order model of the plate should be applied in this frequency range. Another problem with spectral modeling such as Rayleigh–Ritz in high frequency (even for higher order models) is singularity in mass and stiffness matrices. To overcome this problem, digits number should be set very high. This solution significantly reduces the speed of run-time simulation (Table 9). Although Mindlin plate result is in good agreement with experimental result in frequency range 10–10.18 kHz, simulation time is very low because of very large and non-sparse matrices. For showing efficiency of proposed SFEM in very high frequencies, the experimental and SFEM results of real part of impedance for frequency range 90–100 kHz is shown in Figure 11. Number of DOF for this simulation is 242,892. A total of 16 numbers of first natural frequencies of plate in frequency range 90–100 kHz is exhibited in Table 10. The third column shows the percentage of error between the SFEM and experimental results. Table 10 and Figure 11 show good agreements between the experimental and SFEM for this very high-frequency range. Since 3D SFEM is similar to FEM, this method can be used as a general approach for modeling the PWAS impedance and complex geometry structure.

Table 9.

Simulation time computing real part of impedance in 10–14 kHz (hr).

Method	Simulation time (hr)
Kirchhoff plate	34.2536
Mindlin plate	59.1858
3D SFEM	3.1669

SFEM: spectral finite element method.

Table 10.

16 first natural frequencies of plate in frequency range 90–100 kHz (Hz).

Experimental	SFEM	% Error SFEM/Experimental
90,511.662	90,919.662	0.4488
90,652.176	91,008.51	0.3915
90,772.081	91,284.312	0.5611
90,997.034	91,313.704	0.3468
91,036.403	91,352.287	0.3458
91,213.566	91,416.572	0.2221
91,262.546	91,548.868	0.3122
91,291.868	91,694.379	0.4395
91,355.207	91,859.504	0.549
91,434.778	91,964.204	0.5757
91,659.373	92,277.238	0.6696
91,771.818	92,499.628	0.7868
91,961.647	92,639.647	0.7319
92,278.328	92,681.328	0.4348
92,488.047	93,096.706	0.6538
92,565.497	93,259.67	0.7443

SFEM: spectral finite element method.

Conclusion

The main purpose of this study was to develop a general model to investigate the influence of variation of PWAS and plate on structural response in ISHM. Rayleigh–Ritz method was used to discretize dynamic model. In this method, a characteristic function is needed to satisfy boundary conditions. DSC was used as characteristic function of Kirchhoff plate in high frequency. Also, 3D SFEM was used for modeling the interactions between plate and PWAS. Two experimental setups were used for comparing experimental results with SFEM and Kirchhoff plate data. Modal analysis setup was used to measuring the low natural frequency and Rayleigh damping coefficient. Another setup was used for high-frequency measuring of PWAS impedance. Finally, the experimental results of impedance were compared with Kirchhoff plate, Mindlin plate, and SFEM. It was observed that there is a good agreement between Kirchhoff, SFEM, and experiment data in low frequency. However, in the high frequency, Kirchhoff results were not close to the experimental results but SFEM results showed very good agreement with the experimental results. However, the results of Mindlin plate were in a good agreement with experimental data in frequency range 10–10.18 kHz. This is because of the higher model of Mindlin compared to Kirchhoff plate. Rayleigh–Ritz method needs very high RAM and has very low simulation results. So, for large-dimension plate and high-frequency impedance, SFEM was proposed and compared to experimental results. It was observed that the difference between natural frequencies obtained by 3D SFEM and experimental results was lower than 0.7% in very high frequency. Finally, it was found that this 3D SFEM as an element discretization can be used for general case of structure with complex geometry.

Footnotes

Appendix 1

Appendix 2

The mass and stiffness matrices for Mindlin plate in equation (47) is defined as

(47)

\begin{matrix} M = M_{p l a t e} + M_{P W A S} \\ K = K_{p l a t e} + K_{P W A S} \end{matrix}

where $K_{P W A S}, K_{p l a t e}, M_{P W A S}$ and $M_{p l a t e}$ are the PWAS, plate stiffness, PWAS, and plate mass, respectively, which are defined as

(48)

\begin{matrix} M_{p l a t e} \\ = [\begin{matrix} M_{p l a t e}^{u u} & 0 & 0 & 0 & 0 \\ 0 & M_{p l a t e}^{v v} & 0 & 0 & 0 \\ 0 & 0 & M_{p l a t e}^{w w} & 0 & 0 \\ 0 & 0 & 0 & M_{p l a t e}^{φ φ} & 0 \\ 0 & 0 & 0 & 0 & M_{p l a t e}^{θ θ} \end{matrix}], \\ M_{P W A S} [\begin{matrix} M_{P W A S}^{u u} & 0 & 0 & M_{P W A S}^{u φ} & 0 \\ 0 & M_{P W A S}^{v v} & 0 & 0 & M_{P W A S}^{v θ} \\ 0 & 0 & M_{P W A S}^{w w} & 0 & 0 \\ M_{P W A S}^{φ u} & 0 & 0 & M_{P W A S}^{φ φ} & 0 \\ 0 & M_{P W A S}^{θ v} & 0 & 0 & M_{P W A S}^{θ θ} \end{matrix}] \\ K_{p l a t e} \\ = [\begin{matrix} K_{p l a t e}^{u u} & K_{p l a t e}^{u v} & 0 & 0 & 0 \\ K_{p l a t e}^{v u} & K_{p l a t e}^{v v} & 0 & 0 & 0 \\ 0 & 0 & K_{p l a t e}^{w w} & K_{p l a t e}^{w φ} & K_{p l a t e}^{w θ} \\ 0 & 0 & K_{p l a t e}^{φ w} & K_{p l a t e}^{φ φ} & K_{p l a t e}^{φ θ} \\ 0 & 0 & K_{p l a t e}^{θ w} & K_{p l a t e}^{θ φ} & K_{p l a t e}^{θ θ} \end{matrix}], \\ K_{p l a t e} [\begin{matrix} K_{P W A S}^{u u} & K_{P W A S}^{u v} & K_{P W A S}^{u w} & K_{P W A S}^{u φ} & K_{P W A S}^{u θ} \\ K_{P W A S}^{v u} & K_{P W A S}^{v v} & K_{P W A S}^{v w} & K_{P W A S}^{v φ} & K_{P W A S}^{v θ} \\ K_{P W A S}^{w u} & K_{P W A S}^{w v} & K_{P W A S}^{w w} & K_{P W A S}^{w φ} & K_{P W A S}^{w θ} \\ K_{P W A S}^{φ u} & K_{P W A S}^{φ v} & K_{P W A S}^{φ w} & K_{P W A S}^{φ φ} & K_{P W A S}^{φ θ} \\ K_{P W A S}^{θ u} & K_{P W A S}^{θ v} & K_{P W A S}^{θ w} & K_{P W A S}^{θ φ} & K_{P W A S}^{θ θ} \end{matrix}] \end{matrix}

where

And moment vector is defend as

(50)

N = [\begin{matrix} N^{u u} & N^{v v} & N^{w w} & N^{ψ ψ} & N^{θ θ} \end{matrix}]

where

(51)

\begin{matrix} N^{u u} = \int_{x_{1}}^{x_{2}} \int_{y_{1}}^{y_{2}} [2 e_{31} (\frac{\partial φ_{i j}^{u}}{\partial x})] d x d y, N^{v v} = \int_{x_{1}}^{x_{2}} \int_{y_{1}}^{y_{2}} [2 e_{31} (\frac{\partial φ_{i j}^{v}}{\partial x})] d x d y \\ N^{ψ ψ} = \int_{x_{1}}^{x_{2}} \int_{y_{1}}^{y_{2}} [2 e_{31} h_{p} (\frac{\partial φ_{i j}^{ψ}}{\partial x})] d x d y, N^{θ θ} = \int_{x_{1}}^{x_{2}} \int_{y_{1}}^{y_{2}} [2 e_{31} h_{p} (\frac{\partial φ_{i j}^{θ}}{\partial x})] d x d y \\ N^{w w} = 0 \end{matrix}

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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