Electromechanical Modeling of MEMS-Based Piezoelectric Energy Harvesting Devices for Applications in Domestic Washing Machines

Microelectromechanical system (MEMS)-based piezoelectric energy harvesting (PEH) devices can convert the mechanical vibrations of their surrounding environment into electrical energy for low-power sensors. This electrical energy is amplified when the operation resonant frequency of the PEH device matches with the vibration frequency of its surrounding environment. We present the electromechanical modeling of two MEMS-based PEH devices to transform the mechanical vibrations of domestic washing machines into electrical energy. These devices have resonant structures with a T shape, which are formed by an array of multilayer beams and a ultraviolet (UV)-resin seismic mass. The first layer is a substrate of polyethylene terephthalate (PET), the second and fourth layers are Al and Pt electrodes, and the third layer is piezoelectric material. Two different types of piezoelectric materials (ZnO and PZT-5A) are considered in the designs of PEH devices. The mechanical behavior of each PEH device is obtained using analytical models based on the Rayleigh–Ritz and Macaulay methods, as well as the Euler–Bernoulli beam theory. In addition, finite element method (FEM) models are developed to predict the electromechanical response of the PEH devices. The results of the mechanical behavior of these devices obtained with the analytical models agree well with those of the FEM models. The PEH devices of ZnO and PZT-5A can generate up to 1.97 and 1.35 µW with voltages of 545.32 and 45.10 mV, and load resistances of 151.12 and 1.5 kΩ, respectively. These PEH devices could supply power to internet of things (IoT) sensors of domestic washing machines.


Introduction
In future smart cities, most domestic devices will be connected to internet of things (IoT) to improve their performance and communication with users and other devices. IoT can include everyday objects, such as laptops, mobile phones, washing machines, air conditioners, wearable devices, refrigerators, and other electronic devices [1][2][3][4]. IoT will allow a network between these devices for their remote distance control. To supply these devices, the mechanical vibrations of the surrounding environment The Rayleight-Ritz energy conservation model can be used to obtain the first bending frequency of a single clamped beam [27]. The maximum potential and kinetic energies of this model are given by [28]: where y(x) is the deflection of the beam, E is the Young's modulus of the beam material, L is the beam length, I(x) and A(x) are the bending moment and cross-section area of the beam, f is the bending resonant frequency of the beam, and ρ is the beam density.
Applying the energy conservation law (Pmax = Kmax) and solving for the first bending resonant frequency (f) of a single clamped beam, we obtain:  The Rayleight-Ritz energy conservation model can be used to obtain the first bending frequency of a single clamped beam [27]. The maximum potential and kinetic energies of this model are given by [28]: where y(x) is the deflection of the beam, E is the Young's modulus of the beam material, L is the beam length, I(x) and A(x) are the bending moment and cross-section area of the beam, f is the bending resonant frequency of the beam, and ρ is the beam density. Applying the energy conservation law (P max = K max ) and solving for the first bending resonant frequency (f ) of a single clamped beam, we obtain: Next, the Rayleigh-Ritz method was adapted to obtain the first bending resonant frequency of the PEH devices with a T-shaped structure. For this, we considered a multilayer model with a variable cross-section in the yz plane (see Figure 2) with homogenous and isotropic materials. The geometries of the PEH devices are symmetric with the plane xy. The two electrodes of the PEH devices are negligible Energies 2020, 13, 617 4 of 16 due to their thin thickness (100 nm) in comparison with the substrate PET and piezoelectric layer. In addition, the residual stress of the layers is negligible. Thus, the maximum potential, P max-multilayer , and kinetic, K max-multilayer , energies (two sections (S 1 and S 2 ), two layers for the first section (m = 2) and three layers for the second section (n = 3)) are determined by: Applying the energy conservation law for the multilayer cross-section model (P max-multilayer = K max-multilayer ), we obtained the first resonant bending frequency of the PEH devices: Energies 2020, 13, x FOR PEER REVIEW 4 of 16 Next, the Rayleigh-Ritz method was adapted to obtain the first bending resonant frequency of the PEH devices with a T-shaped structure. For this, we considered a multilayer model with a variable cross-section in the yz plane (see Figure 2) with homogenous and isotropic materials. The geometries of the PEH devices are symmetric with the plane xy. The two electrodes of the PEH devices are negligible due to their thin thickness (100 nm) in comparison with the substrate PET and piezoelectric layer. In addition, the residual stress of the layers is negligible. Thus, the maximum potential, Pmax-multilayer, and kinetic, Kmax-multilayer, energies (two sections (S1 and S2), two layers for the first section (m = 2) and three layers for the second section (n = 3)) are determined by: Applying the energy conservation law for the multilayer cross-section model (Pmax-multilayer = Kmax-multilayer), we obtained the first resonant bending frequency of the PEH devices: The bending stiffness (EIz)sj in the j-th section of the PEH devices can be obtained by [29,30]: The elastic centroid (asj) of each PEH device, in the j-th section, was assumed to be on the xz axis and is defined as [30]:  The bending stiffness (EI z ) sj in the j-th section of the PEH devices can be obtained by [29,30]: The elastic centroid (a sj ) of each PEH device, in the j-th section, was assumed to be on the xz axis and is defined as [30]: Energies 2020, 13, 617 5 of 16 where t isj = h isj − h (i−1)sj is the i-th layer thickness in the j-th section, s j is the j-th section, k is the number of layers on the two different sections (k = m,n, . . . ), A sj is the area on the j-th section, E isj is the Young's modulus of the i-th layer on the j-th section, y sj is the height of the j-th section, h (i−1)sj is the distance from the inferior plane of the first layer to the superior plane of the (i−1)th layer on the j-th section, b isj is the total width of the i-th layer ( b is1 includes the sum of width for each one of the two layers of same material at the same distance), and h 0sj = 0. The deflections, y sj , for both sections of the device structures were obtained by applying the Euler-Bernoulli theory and the Macaulay method as [31,32]: where M sj (x) is the bending moment of the j-th section of each PEH device. The solution of Equations (10) and (11) must consider the boundary conditions of the two sections of each PEH device. Equations (12) and (13) show the boundary conditions for the first and second section of the PEH devices, respectively: To solve Equations (10) and (11), the bending moment functions of the PEH devices are required. This function is determined through the load function, q(x), of the PEH devices, which is obtained using the Macaulay method: where the reaction load (R 0 ) and the bending moment (M 0 ) in the fixed support and the linear weight (w sj ) of the j-th section of the devices are determined as (see Figure 3): where g is the Earth's gravity. The shear stress function, V(x), Equation (18), is obtained by integrating the load function, q(x), with respect to x, considering the integration rules of the Macaulay's functions [32]. Next, the bending moment function, M(x), is determined by integrating the shear stress function, V(x), with respect to x: Energies 2020, 13, 617 The integration constants (C 1 = 0 and C 2 = 0) were calculated using the boundary conditions, in which the shear stress V(0) = R 0 and bending moment M(0) = M 0 . Thus, the bending moment function, M(x), is expressed as: Energies 2020, 13, x FOR PEER REVIEW 6 of 16 The integration constants (C1 = 0 and C2 = 0) were calculated using the boundary conditions, in which the shear stress V(0) = R0 and bending moment M(0) = M0.
Thus, the bending moment function, M(x), is expressed as: The bending moment function for each section of both PEH devices was determined through the Equation (20). The bending moment function for the first section, x ϵ (0,Ls1), and for the second section, x ϵ (Ls1,Ls12), are given by Equations (21) and (22), respectively: The deflection, ysj, into each one of the devices' sections was obtained by substituting the bending moment functions, Ms1 and Ms2, into the Equations (10) and (11), respectively, and using the integration rules of the Macaulay's functions. Furthermore, the boundary conditions (Equations (12) and (13)) were applied to find these deflections: Finally, the deflections of both PEH devices at resonance can be approximated using Qa times the static deflections ( ) [27]: The bending moment function for each section of both PEH devices was determined through the Equation (20). The bending moment function for the first section, x (0,L s1 ), and for the second section, x (L s1 ,L s12 ), are given by Equations (21) and (22), respectively: The deflection, y sj , into each one of the devices' sections was obtained by substituting the bending moment functions, M s1 and M s2 , into the Equations (10) and (11), respectively, and using the integration rules of the Macaulay's functions. Furthermore, the boundary conditions (Equations (12) and (13)) were applied to find these deflections: Energies 2020, 13, 617 Finally, the deflections of both PEH devices at resonance can be approximated using Q a times the static deflections (y s j ) [27]: where Q a is the quality factor of the PEH device due to air damping. The potential and kinetic energies of the devices were estimated using Equations (23), (24), and (8). Next, the first bending resonant frequency of each device was estimated by substituting these energies into Equation (7). Table 1 shows the mechanical properties of the materials of the PEH devices. Table 2 depicts the geometrical parameters of both PEH devices based on the ZnO and PZT-5A layers. Table 3 illustrates the elastic centroid, bending moment, and linear weight for the two sections of both PEH devices.    Considering the proposed analytical modeling, the first bending resonant frequencies of the ZnO and PZT-5A-based PEH devices are 106.17 and 104.89 Hz, respectively.
In addition, the finite element method (FEM) models through software ANSYS ® (ANSYS, Inc., Pittsburgh, USA) were developed to determine the electromechanical behavior of both MEMS-based PEH devices. First, the electromechanical modeling was applied to predict the modal and harmonic response of the PEH devices.
For the modal analysis of both PEH devices, FEM models were developed employing a mesh with hexahedral elements, with five divisions through all layers of each device (see Figure 4). By using a Energies 2020, 13, 617 8 of 16 modal analysis, the first bending resonant frequencies are 108.84 and 108.54 Hz for the FEM models of the ZnO (Figure 5a) and PZT-5A (Figure 5b) PEH devices, respectively. Moreover, three more bending resonant frequencies of the FEM models for each PEH device were determined. Figure 6a,b depicts the second, third, and fourth resonant frequency of the FEM models.
Energies 2020, 13, x FOR PEER REVIEW 8 of 16 PEH devices. First, the electromechanical modeling was applied to predict the modal and harmonic response of the PEH devices.
For the modal analysis of both PEH devices, FEM models were developed employing a mesh with hexahedral elements, with five divisions through all layers of each device (see Figure 4). By using a modal analysis, the first bending resonant frequencies are 108.84 and 108.54 Hz for the FEM models of the ZnO (Figure 5a) and PZT-5A (Figure 5b) PEH devices, respectively. Moreover, three more bending resonant frequencies of the FEM models for each PEH device were determined. Figure 6a and b depicts the second, third, and fourth resonant frequency of the FEM models.  PEH devices. First, the electromechanical modeling was applied to predict the modal and harmonic response of the PEH devices.
For the modal analysis of both PEH devices, FEM models were developed employing a mesh with hexahedral elements, with five divisions through all layers of each device (see Figure 4). By using a modal analysis, the first bending resonant frequencies are 108.84 and 108.54 Hz for the FEM models of the ZnO (Figure 5a) and PZT-5A (Figure 5b) PEH devices, respectively. Moreover, three more bending resonant frequencies of the FEM models for each PEH device were determined. Figure 6a and b depicts the second, third, and fourth resonant frequency of the FEM models.  Energies 2020, 13, 617

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Energies 2020, 13, x FOR PEER REVIEW 9 of 16 (a) (b) Figure 6. Second, third, and fourth vibration mode with a normalized amplitude of PEH devices with ZnO (a) and PZT-5A layers (b) obtained using FEM models.
The first bending resonant frequency of the PEH devices with ZnO and PZT-5A layers obtained through the analytical model had relative errors of 2.45% and 3.36% compared with those of the FEM models.
For the harmonic analysis of the PEH devices, we used the following piezoelectric stress and piezoelectric dielectric matrices [33,34].

6) (26)
For the PZT-5A layer: Figure 6. Second, third, and fourth vibration mode with a normalized amplitude of PEH devices with ZnO (a) and PZT-5A layers (b) obtained using FEM models.
The first bending resonant frequency of the PEH devices with ZnO and PZT-5A layers obtained through the analytical model had relative errors of 2.45% and 3.36% compared with those of the FEM models.
For the harmonic analysis of the PEH devices, we used the following piezoelectric stress and piezoelectric dielectric matrices [33,34].
For the ZnO layer: Energies 2020, 13, 617 10 of 16 For the PZT-5A layer: The Blom model was used to predict the quality factor (Q a ) of the PEH devices due to the air damping. For this, the resonant structure of each device was approximated as a single clamped beam with a substrate (PET) and a proof mass at its end. This quality factor (Q a ) was described as [35]: where f r is first bending frequency, ρ a and µ are the density and viscosity of the air, ρ s is the PET substrate density, and b, h, and L e are the width, thickness, and length of each PEH substrate, respectively. Considering Equations (28)- (30), PEH devices with ZnO and PZT-5A layers have quality factors at an atmospheric pressure of 265.18 and 263.52, respectively.
The damping ratio (ζ) and optimum load resistance (R opt ) of each PEH device can be determined by [27,36]: where W and t p are the width and thickness of the piezoelectric layer, ε 0 is the permittivity constant, and ε 33 is the 3 × 3 element of the piezoelectric matrix.

Results and Discussion
This section reports the results and discussion of the electromechanical behavior of both MEMS-based PEH devices considering piezoelectric layers of ZnO and PZT-5A. Figure 7a and b show the deflections at resonance of both PEH devices with ZnO and PZT-5A layers, which were obtained using the analytical and FEM models. The deflections calculated with the analytical modeling agree well with those of the FEM models. For the PEH devices with ZnO and PZT-5A layers, the relative errors of the analytical modeling are 5.38% and 5.33% in comparison with the FEM models. By using the analytical modeling, the maximum deflections of the ZnO and PZT-5A-based PEH devices achieve values up of 7.43 and 7.68 mm, respectively. On the other hand, the maximum deflections of both devices through FEM models are 7.83 and 8.09 mm, respectively.
The displacements of the PEH devices caused stresses and output voltages in the piezoelectric layers. The maximum normal stresses (x-axis) of the ZnO and PZT-5A layers are 38.45 and 11.74 MPa, respectively. These stresses do not overcome the tensile strengths (412 and 500 MPa) of the ZnO and PZT-5A layers, respectively [37,38]. Furthermore, these devices can operate with the same bending The displacements of the PEH devices caused stresses and output voltages in the piezoelectric layers. The maximum normal stresses (x-axis) of the ZnO and PZT-5A layers are 38.45 and 11.74 MPa, respectively. These stresses do not overcome the tensile strengths (412 and 500 MPa) of the ZnO and PZT-5A layers, respectively [37,38]. Furthermore, these devices can operate with the same bending resonant frequency, with accelerations up to 5.2 and 21.33 m/s 2 for ZnO and PZT-5A layers, respectively. These accelerations generate stresses that do not overcome the ZnO and PZT-5A tensile strengths. The ZnO and PZT-5A-based PEH devices can generate maximum output voltages of 545.32 and 45.10 mV, respectively. Figure 8 shows the generated voltages as a function of the frequency.
The optimum load resistances (151.12 and 1.50 kΩ) reached by the ZnO and PZT-5A-based PEH devices were calculated using Equation (32). These load resistances were utilized to obtain the output power for both PEH devices. Thus, the output power is described as: where rms V is the output voltage.
The output power calculated for ZnO and PZT-5A-based PEH devices is 1.97 and 1.35 µW, respectively. Figure 9 depicts the output powers of the PEH devices with respect to their operation frequency using the FEM models. This output power can supply electrical energy to low-power electronic devices, such as pressure and temperature sensors [39]. The proposed PEH devices can operate at resonance with frequencies caused by vibration sources, such as domestic washing machines. Table 4 indicates the values of the main parameters of the PEH devices obtained through FEM models. The ZnO and PZT-5A-based PEH devices can generate maximum output voltages of 545.32 and 45.10 mV, respectively. Figure 8 shows the generated voltages as a function of the frequency.
The optimum load resistances (151.12 and 1.50 kΩ) reached by the ZnO and PZT-5A-based PEH devices were calculated using Equation (32). These load resistances were utilized to obtain the output power for both PEH devices. Thus, the output power is described as: where V rms is the output voltage. The output power calculated for ZnO and PZT-5A-based PEH devices is 1.97 and 1.35 µW, respectively. Figure 9 depicts the output powers of the PEH devices with respect to their operation frequency using the FEM models. This output power can supply electrical energy to low-power electronic devices, such as pressure and temperature sensors [39]. The proposed PEH devices can operate at resonance with frequencies caused by vibration sources, such as domestic washing machines. Table 4 indicates the values of the main parameters of the PEH devices obtained through FEM models. PEH devices could supply electrical energy to low-power IoT sensors that are exposed to mechanical vibrations to their surrounding environment. For instance, sensors used for monitoring the temperature, pressure, position, humidity, acceleration, and magnetic field. To obtain the maximum output voltage and power of PEH devices, the operation frequency of these devices must be similar to that of the mechanical vibrations' source. An array of PEH devices can be used to increase the supplied power to IoT sensors. In our particular case, we designed two MEMS-based PEH devices to transform the vibrations of domestic washing machines into electrical energy. These devices could be installed in other vibration sources. For this, dimensions of the structure of PEH devices must be modified to achieve operation resonant frequencies approximately equal to those of PEH devices could supply electrical energy to low-power IoT sensors that are exposed to mechanical vibrations to their surrounding environment. For instance, sensors used for monitoring the temperature, pressure, position, humidity, acceleration, and magnetic field. To obtain the maximum output voltage and power of PEH devices, the operation frequency of these devices must be similar to that of the mechanical vibrations' source. An array of PEH devices can be used to increase the supplied power to IoT sensors. In our particular case, we designed two MEMS-based PEH devices to transform the vibrations of domestic washing machines into electrical energy. These devices could be installed in other vibration sources. For this, dimensions of the structure of PEH devices must be modified to achieve operation resonant frequencies approximately equal to those of PEH devices could supply electrical energy to low-power IoT sensors that are exposed to mechanical vibrations to their surrounding environment. For instance, sensors used for monitoring the temperature, pressure, position, humidity, acceleration, and magnetic field. To obtain the maximum output voltage and power of PEH devices, the operation frequency of these devices must be similar to that of the mechanical vibrations' source. An array of PEH devices can be used to increase the supplied power to IoT sensors. In our particular case, we designed two MEMS-based PEH devices to transform the vibrations of domestic washing machines into electrical energy. These devices could be installed in other vibration sources. For this, dimensions of the structure of PEH devices must be modified to Energies 2020, 13, 617 13 of 16 achieve operation resonant frequencies approximately equal to those of the new vibration sources. Therefore, PEH devices could supply electrical energy to different types of sensors.
The operation resonant frequencies of the proposed PEH devices could be altered by changing the thickness of their seismic masses. The PEH devices must be designed to have maximum stresses less than the rupture stresses of their materials. In the FEM models, the thickness of the seismic mass was modified to obtain its effect on the first bending resonant frequency of both PEH devices, as shown in Figure 10a,b. Thus, the thickness of the seismic mass of both PEH devices could be adjusted to change their potential application. the new vibration sources. Therefore, PEH devices could supply electrical energy to different types of sensors. The operation resonant frequencies of the proposed PEH devices could be altered by changing the thickness of their seismic masses. The PEH devices must be designed to have maximum stresses less than the rupture stresses of their materials. In the FEM models, the thickness of the seismic mass was modified to obtain its effect on the first bending resonant frequency of both PEH devices, as shown in Figure 10a and b. Thus, the thickness of the seismic mass of both PEH devices could be adjusted to change their potential application.

Conclusions
The electromechanical modeling of two MEMS-based PEH devices designs with layers of ZnO and PZT-5A was developed. These designs can perform at their first bending resonant frequency,

Conclusions
The electromechanical modeling of two MEMS-based PEH devices designs with layers of ZnO and PZT-5A was developed. These designs can perform at their first bending resonant frequency, with values close to those caused by the vibrations of domestic washing machines. The PEH devices were formed by an array of multilayer beams and a UV-resin seismic mass. These beams included a substrate of polyethylene terephthalate (PET) and two different types of piezoelectric layers (ZnO and PZT-5A). The mechanical behavior of the PEH devices was obtained using analytical models based on the Rayleigh-Ritz and Macaulay methods, as well as the Euler-Bernoulli beam theory. Furthermore, finite element method (FEM) models were generated to estimate the electromechanical behavior of both PEH devices. The results of the mechanical performance of these devices obtained with the analytical models agreed well with those of the FEM models. The PEH devices with ZnO and PZT-5A layers generated 1.97 and 1.35 µW with voltages of 545.32 and 45.10 mV, and load resistances of 151.12 and 1.5 kΩ, respectively. The output voltages of these PEH devices could be supplied to low-power IoT sensors incorporated into domestic washing machines.