1 Introduction

Simulation of the performance of PEG energy harvesting devices can be carried out on the base of finite element modeling in computer software such as ACELAN , ANSYS, etc. Experimental and numerical studies of PEG of stack and bimorph types are discussed in [1,2,3,4]. The ACELAN software implemented the possibility of calculating the process of polarization of piezoelectric elements based on the theory developed in [5]. The use of inhomogeneously polarized piezoelements in the bimorph type of PEG significantly increases its output characteristics [6]. In [7,8,9,10], a method for inhomogeneous polarization of rectangular bimorph piezoelements was developed and an applied theory of calculating its cylindrical bending vibrations for piecewise constant polarization was constructed. In this paper we consider axisymmetric vibrations of a bimorph piezoelement in the form of a multilayer circular disk with piezoceramic layers are partially covered with electrodes and are inhomogeneously polarized, with circular and annular regions with a thickness polarization (they are covered with electrodes along one flat face) and an electrically-connected annular section between them with radial polarization (Fig. 26.1).

Fig. 26.1
figure 1

a Piezoelectric element top view, axial section, b polarization scheme

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1.1 Research Purpose

The efficiency of PEG with the shape of a bimorph circular plate with axisymmetric bending vibrations is studied. The piezoceramic layer of the transducer has an inhomogeneous polarization.

1.2 Research Scope

At this study, we consider the following frameworks of the problem:

  1. (i)

    Inhomogeneous polarization creation technology and PEG design;

  2. (ii)

    Applied theory of flexural axisymmetric vibrations of PEG;

  3. (iii)

    Finite element modeling of PEG with an inhomogeneously polarized piezoelement;

  4. (iv)

    Numerical and analytical calculation of the output characteristics of the power harvesting device;

  5. (v)

    Optimization of the structural characteristics of PEG for obtaining its greatest efficiency.

2 Research Method

2.1 Continuous Formulation of the Problem

The mathematical model of flexural steady vibrations of a bimorph circular plate can be reduced to the boundary problem of electro-elasticity, which consists of a system of differential equations [11, 12].

We assume that the following constitutive equations are satisfied (piezoelectric medium is Ω j  = Ω pk ):

$$ \rho_{pk} {\ddot{\mathbf{u}}} + \alpha_{dj} \rho_{j} {\dot{\mathbf{u}}} - \nabla \cdot {\varvec{\upsigma}} = {\mathbf{f}}_{j} ;\quad \nabla \cdot {\mathbf{D}} = 0, $$
(26.1)
$$ {\varvec{\upsigma}} = {\mathbf{c}}_{j}^{E} \cdot \cdot ({\varvec{\upvarepsilon}} + \beta_{dj} {\dot{\varvec{\upvarepsilon}}}) - {\mathbf{e}}_{j}^{T} \cdot {\mathbf{E}};\quad {\mathbf{D}} + \varsigma_{d} {\dot{\mathbf{D}}} = {\mathbf{e}}_{j} \cdot \cdot ({\varvec{\upvarepsilon}} + \varsigma_{d} {\dot{\varvec{\upvarepsilon}}}) + {\boldsymbol{\backepsilon }}_{j}^{S} \cdot {\mathbf{E}}, $$
(26.2)
$$ {\varvec{\upvarepsilon}} = (\nabla {\mathbf{u}} + \nabla {\mathbf{u}}^{T} )/2;\quad {\mathbf{E}} = - \nabla \varphi , $$
(26.3)

where ρ(x, t) is the continuous function of coordinates (density); u(x) is the displacement vector-function; σ is the stress tensor, f are the mass forces vector; D is the electric induction vector; c E j is the elastic constant tensor; e j is the tensor of piezoelectric constants ; ε is the strain tensor; E is the electric field vector; φ(x) is the electric potential function; э S j is the dielectric permittivity tensor; α dj , β dj , ζ d are non-negative damping coefficients, and the other symbols are the standard designations for theory of electroelasticity with the exception of index “j”, corresponding to area Ω j . (For elastic media Ω j  = Ωe, the piezomodules ej are equal to zero.)

For the media Ω j  = Ω em with pure elastic properties, only stress fields would be considered. Similar (26.1)–(26.3) and constitutive relationships are used, neglecting electric fields and piezoelectric connectivity effects. Equations (26.1)–(26.3) are added to the mechanical and electrical boundary conditions, as well as the initial conditions in the case of non-stationary problem. Numerical modeling of devices that can be described with (26.1)–(26.3) are performed using finite element method.

The boundary conditions are divided into mechanical and electrical. In particular, for an electrode \( (S_{e} ) \), included in external circuit, besides the condition of constancy of the electric potential, which presents itself unknown function in this case, it is necessary to add the condition , defining electric current passes through this electrode:

$$ \int\limits_{{S_{e} }} {\dot{D}_{m} ds} = I. $$
(26.4)

In addition to previous equations, all material properties are handled as functions of coordinates:

$$ \rho_{k} = \rho_{pk} (x);\quad{\mathbf{c}}_{j}^{E} = {\mathbf{c}}_{j}^{E} (x);\quad{\varvec{\upvarepsilon}}_{{\mathbf{a}} {j} }^{S} = {\varvec{\upvarepsilon}}_{{\mathbf{a}} {j} }^{S} (x);\quad{\mathbf{e}}_{j}^{T} = {\mathbf{e}}_{j}^{T} (x), $$
(26.5)

where we have

$$ g = g^{i} + \left| P \right|(g^{a} - g^{i} ),\,{\text{for tensors}}\,{\mathbf{c}}_{j}^{E} \,{\text{and}}\, {\boldsymbol{\backepsilon }}_{j}^{S} ;\,{\text{and}}\,g = \left| P \right|g^{a} ,\,{\text{for tensor}}\,{\mathbf{e}}_{j}^{T} . $$
(26.6)

Here g are corresponding tensor components, i points isotropic state, a points anisotropic state. Tensor of piezoconstants \( {\mathbf{e}}_{j}^{T} \) will be zero for isotropic bodies. In previous paper [4], we presented specific modules of ACELAN software for describing, presenting and modeling the non-homogeneous polarization.

The matrices of elastic constants , piezoelectric constant and permittivity have the corresponding forms for regions with transverse polarization, respectively:

$$ \left[ {\begin{array}{*{20}c} {c_{11} } & {c_{12} } & {c_{13} } & 0 & 0 & 0 \\ {c_{12} } & {c_{11} } & {c_{13} } & 0 & 0 & 0 \\ {c_{13} } & {c_{13} } & {c_{33} } & 0 & 0 & 0 \\ 0 & 0 & 0 & {c_{44} } & 0 & 0 \\ 0 & 0 & 0 & 0 & {c_{44} } & 0 \\ 0 & 0 & 0 & 0 & 0 & {\frac{1}{2}c_{11} - \frac{1}{2}c_{12} } \\ \end{array} } \right];\quad\left[ {\begin{array}{*{20}c} 0 & 0 & {e_{31} } \\ 0 & 0 & {e_{31} } \\ 0 & 0 & {e_{33} } \\ 0 & {e_{15} } & 0 \\ {e_{15} } & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } \right];\quad\left[ {\begin{array}{*{20}c} {g_{11} } & 0 & 0 \\ 0 & {g_{11} } & 0 \\ 0 & 0 & {g_{33} } \\ \end{array} } \right]. $$
(26.7)

For regions with polarization in radial direction, we have another forms:

$$ \left[ {\begin{array}{*{20}c} {c_{33} } & {c_{13} } & {c_{13} } & 0 & 0 & 0 \\ {c_{13} } & {c_{11} } & {c_{12} } & 0 & 0 & 0 \\ {c_{13} } & {c_{12} } & {c_{11} } & 0 & 0 & 0 \\ 0 & 0 & 0 & {\frac{1}{2}c_{11} - \frac{1}{2}c_{12} } & 0 & 0 \\ 0 & 0 & 0 & 0 & {c_{44} } & 0 \\ 0 & 0 & 0 & 0 & 0 & {c_{44} } \\ \end{array} } \right];\quad \left[ {\begin{array}{*{20}c} {e_{33} } & 0 & 0 \\ {e_{31} } & 0 & 0 \\ {e_{31} } & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & {e_{15} } \\ 0 & {e_{15} } & 0 \\ \end{array} } \right];\quad \left[ {\begin{array}{*{20}c} {g_{33} } & 0 & 0 \\ 0 & {g_{11} } & 0 \\ 0 & 0 & {g_{11} } \\ \end{array} } \right]. $$
(26.8)

2.2 Technology of Piezoelements Polarization

Polarization of the piezoelement is performed in two stages using technological electrodes on the lower surface , which are removed after the second stage. Calculation of the residual polarization field is conducted in the finite element software ACELAN [7]. Figure 26.2 shows half of the axial section of the upper piezoceramic disk and two polarization stages: the first stage involves transverse polarization of the regions, covered by the electrodes (Fig. 26.2a), and the second stage involves polarization in radial direction of the section between them (Fig. 26.2b).

Fig. 26.2
figure 2

First (a) and second (b) stages of piezoelement polarization

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The lower piezoceramic disk is polarized in a similar way to the circuit, shown in Fig. 26.2 (\( - V_{0} \) is replaced on \( + V_{0} \)).

2.3 Applied Theory of Flexural Axisymmetric Vibrations of PEG

Under the assumption of Kirchhoff-Love hypotheses for mechanical quantities and a single normal, the angle of inclination of the normal \( \vartheta \), radial and circular deformation, expressed through the deflection \( U_{z} = U_{z} (r) \) of the median surface , takes the form, respectively:

$$ \vartheta = - \frac{{dU_{z} (r)}}{dr},\quad \varepsilon_{r} = - \frac{{d^{2} U_{z} (r)}}{{dr^{2} }}z,\quad \varepsilon_{t} = - \frac{1}{r}\frac{{dU_{z} (r)}}{dr}z. $$
(26.9)

Electrical potential distribution in parts with transverse polarization is found as

$$ \varphi (r,\theta ,z) = V_{p} \frac{z}{2h}\left( {1 - \frac{2z}{h}} \right) + V_{m} \frac{z}{2h}\left( {1 + \frac{2z}{h}} \right) +\Phi (r)\left( {1 - \frac{{4z^{2} }}{{h^{2} }}} \right) . $$
(26.10)

where h is the plate thickness; \( V_{m} ,V{}_{p} \) are the electrical potentials on the upper and lower electrodes, respectively.

In a ring with polarization in radial direction, we have:

$$ \varphi (r,\theta ,z) =\Phi (r) . $$
(26.11)

System of differential equations for areas with transverse polarization for a two-layer piezoactive bimorph for deflection \( U_{z} = U_{z} (r) \) and the electric potential of the mid-surface \( \Phi =\Phi (r) \) will take the forms:

$$ \frac{{d^{4} U_{z} }}{{dr^{4} }}D + 2D\frac{1}{r}\frac{{d^{3} U_{z} }}{{dr^{3} }} - D\frac{1}{{r^{2} }}\frac{{d^{2} U_{z} }}{{dr^{2} }} + D\frac{1}{{r^{3} }}\frac{{d^{2} U_{z} }}{{dr^{2} }} - F\frac{{d^{2}\Phi }}{{dr^{2} }} - F\frac{1}{r}\frac{{d\Phi }}{dr} - \omega^{2} \rho hU_{z} = p(r); $$
(26.12)
$$ \frac{{d^{2}\Phi }}{{dr^{2} }} - 3G\frac{{d^{2} U_{z} }}{{dr^{2} }} - G\frac{1}{r}\frac{{dU_{z} }}{dr} + H( - V_{m} + V{}_{p} - 4\Phi ) = 0, $$
(26.13)

where \( H = 3\frac{{c_{33} g_{33} + e_{33}^{2} }}{{h^{2} g_{n} c_{33} }} \), \( F = \frac{2h}{3}\frac{{(c_{13} e_{33} - c_{33} e_{31} )}}{{c_{33} }} \), \( G = \frac{{c_{13} e_{33} - c_{33} e_{31} }}{{2g_{11} c_{33} }} \), \( \omega \) is the angular frequency of vibration.

Internal mechanical (\( M_{r} \) is the moment, \( Q_{r} \) is the transverse force) and electrical (\( \varphi \) is the electric potential, \( D_{r} \) is the electric induction) factors are expressed as

$$ M_{r} = - D\left( {\frac{{d^{2} U_{z} }}{{dr^{2} }} + \frac{1}{r}\frac{{dU_{z} }}{dr}} \right) - \frac{{h^{3} }}{18}( - V_{m} + V{}_{p} - 4\Phi )H;\quad \quad Q_{r} = - D\left( {\frac{{d^{3} U_{z} }}{{dr^{3} }} + \frac{1}{r}\frac{{d^{2} U_{z} }}{{dr^{2} }} - \frac{1}{r}\frac{{dU_{z} }}{dr}} \right) - J\frac{{d\Phi }}{dr}; $$
(26.14)
$$ \varphi = \frac{1}{12}( - V_{m} + V{}_{p} + 8\Phi );\quad D_{r} = - \frac{2}{3}g_{11} \frac{{d\Phi }}{dr}, $$
(26.15)

where \( J = \frac{2}{3}h\frac{{(c_{33} e_{15} - c_{13} e_{33} + c_{33} e_{31} )}}{{c_{33} }} \).

For a region with longitudinal polarization, the system of differential equations corresponding to the system (26.11), (26.12) has the form:

$$ D_{1} \frac{{d^{4} U_{z} }}{{dr^{4} }} + 2D_{1} \frac{1}{r}\frac{{d^{3} U_{z} }}{{dr^{3} }} - D_{2} \frac{1}{{r^{2} }}\frac{{d^{2} U_{z} }}{{dr^{2} }} + D_{2} \frac{1}{{r^{3} }}\frac{{dU_{z} }}{dr} - F_{1} \frac{{d^{3}\Phi }}{{dr^{3} }} - F_{2} \frac{1}{r}\frac{{d^{2}\Phi }}{{dr^{2} }} - \omega^{2} \rho hU_{z} = p(r); $$
(26.16)
$$ K\frac{{d^{2}\Phi }}{{dr^{2} }} + L\frac{{d^{3} U_{z} }}{{dr^{3} }} - N\frac{1}{r}\frac{{d^{2} U_{z} }}{{dr^{2} }} + N\frac{1}{{r^{2} }}\frac{{dU_{z} }}{dr} = 0; $$
(26.17)

where

$$ D_{1} = \frac{{h^{3} }}{12}\frac{{(c_{11} c_{33} - c_{13}^{2} )}}{{c_{11} }};\quad D_{2} = \frac{{h^{3} }}{12}\frac{{(c_{11}^{2} - c_{12}^{2} )}}{{c_{11} }};\quad F_{1} = \frac{{h^{2} }}{4}\frac{{(c_{11} e_{33} - c_{13} e_{31} )}}{{c_{11} }};\quad N = \frac{h}{4}\frac{{(c_{12} e_{31} - c_{11} e_{31} )}}{{c_{11} }};\quad F_{2} = \frac{{h^{2} }}{4}\frac{{(2c_{11} e_{33} - c_{13} e_{31} + c_{12} e_{31} - 2c_{13} e_{31} )}}{{c_{11} }};\quad K = \frac{{(c_{11} g_{33} + g_{31}^{2} )}}{{c_{11} }};\quad L = \frac{h}{4}\frac{{(c_{11} e_{33} - c_{13} e_{31} )}}{{c_{11} }}. $$

Internal mechanical and electrical factors are expressed as

$$ \begin{aligned} M_{r} & = - D_{1} \frac{{d^{2} U_{z} }}{{dr^{2} }} - D_{3} \frac{1}{r}\frac{{dU_{z} }}{dr} + F_{1} \frac{d}{dr}\Phi ; \\ Q_{r} & = - D_{1} \frac{{d^{3} U_{z} }}{{dr^{3} }} - D_{1} \frac{1}{r}\frac{{d^{2} U_{z} }}{{dr^{2} }} + D_{2} \frac{1}{{r^{2} }}\frac{{dU_{z} }}{dr} + F_{1} \frac{{d^{2}\Phi }}{{dr^{2} }} + F_{3} \frac{1}{r}\frac{{d\Phi }}{dr}; \\ \end{aligned} $$
(26.18)
$$ \varphi =\Phi (r);\quad D_{r} = \frac{1}{h}F_{1} \frac{{d^{2} U_{z} }}{{dr^{2} }} + N\frac{1}{r}\frac{{dU_{z} }}{dr} - K\frac{d}{dr}\Phi , $$
(26.19)

where

$$ D_{3} = - \frac{{h^{3} }}{12}\frac{{(c_{11} c_{13} - c_{12} c_{13} )}}{{c_{11} }},F_{3} = \frac{{h^{2} }}{4}\frac{{(c_{11} e_{33} - c_{13} e_{31} - c_{11} e_{31} + c_{12} e_{31} )}}{{c_{11} }}. $$

The boundary conditions and congruence conditions have the forms:

  • for \( r = 0 \):

  • \( \vartheta = 0 \) and in the absence of concentrated force or inertial mass, we have

  • \( Q_{r} = 0 \) from (26.14) and \( D_{r} = 0 \) from (26.13);

  • for \( r = r_{i} ,\,i = 1,2 \), we have congruence of parts with different polarizations.

The conditions for continuity of deflections \( U_{z} \), angles \( \vartheta \), moments and transverse forces, we obtain from (26.14) and (26.18), electric potential and electrical induction are found from (26.15) and (26.19).

3 Results and Discussion

3.1 Static Load

Next, the efficiency of a uniformly and inhomogeneously polarized PEG is compared. The thickness of the piezolayers (from PZT -4) is equal to 0.2 mm, the radius is 19 mm, the thickness of the steel plate , on which they are pasted, equals 0.1 mm. The load is a uniform pressure with intensity of 1 kPa. For piecewise constant polarization, the dimensions of the electrodes vary.

Table 26.1 presents the results of calculation of deflection and output potential for PEG with homogeneous polarization (No. 1) and for different electrode sizes (Nos. 2–4).

Table 26.1 Static load
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3.2 Steady-State Vibrations

Steady-state vibrations of the PEG at the first antiresonance frequency are considered. The shape of the vibrations, with the distribution of the axial displacement, is shown in Fig. 26.3a, the distribution of the electrical potential for homogeneous polarization (No. 1 in Table 26.2) is shown in Fig. 26.3b, and for the inhomogeneous polarization (No. 3 in Table 26.2), it is shown in Fig. 26.3c.

Fig. 26.3
figure 3

Distributions of deflection (a) and electric potential (b) at homogeneous polarization, (c) inhomogeneous polarization

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Table 26.2 Harmonic analysis
Full size table

Table 26.2 shows the calculation results for the first antiresonance frequency: the deflection at the center and the output potential for PEG with homogeneous polarization No. 1 and for different electrode sizes Nos. 2–4.

4 Conclusion

An applied theory of axisymmetric vibrations of PEG with a piecewise constant polarization is constructed. The results of the calculations show the significant efficiency of PEG with regions of polarization in radial direction, when generating an output potential in comparison with PEG with homogeneous polarization of piezoceramic layers. It is shown that the output potential increases with the growth of the region with polarization in radial direction. There is a limiting size of this section, which can be efficiently polarized. So, it is possible to make several such sections of a given size to create an effective device.