Macro Fiber Composite Research Articles

Experimental and finite element analysis of PPF controller effectiveness in composite beam vibration suppression

In this paper the problem of vibration reduction is considered. Generally, mechanical vibrations occurring during the operation of a system are undesirable and may have a negative effect on its reliability. A finite element model of a single active blade is developed using the Abaqus software. This structure consists of a multi-layer glass-epoxy composite beam with an embedded macro fiber composite (MFC) piezoelectric actuator. For vibration control the use of a positive position feedback (PPF) controller is proposed. To include the PPF controller in the Abaqus software, a special subroutine is created. The developed control algorithm code makes it possible to solve an additional differential equation by the fourth order RungeKutta method. A numerical dynamic analysis is performed by the implicit procedure. The beam responses with and without controller activation are compared. The control subsystem model also includes the hysteresis phenomenon of the piezoelectric actuator. Numerical findings regarding the PPF controller’s effectiveness are verified experimentally.

Ultrasonic-based tensile force estimation for cylindrical rod at various temperature conditions

This study presents the development of a tensile force estimation technique based on the ultrasonic bulk wave for cylindrical rod in a suspension bridge anchor system. Three piezoelectric macro fiber composites (MFC) transducers are installed on the surface of a cylindrical rod specimen for the generation and sensing of ultrasonic bulk waves. Ultrasonic responses are obtained at various tensile force levels and temperature conditions, and the relationship between the time lag of the ultrasonic response and the tensile force is established. In particular, the temperature effect on the tensile force estimation is theoretically derived and compensated for field applications. The originality of this study includes (1) theoretical derivation and investigation of the relationship between an applied tensile force level and the time lag of an ultrasonic response considering the variations of temperature, density, and elastic modulus, (2) development of a tensile force estimation and temperature compensation technique without temperature measurement, and (3) application to the tensile force estimation of the rod structure commonly used in a suspension bridge anchor system. The performance of the developed tensile force estimation technique was examined based on mock-up experiments of the rod structure of a real suspension bridge anchor system.

Design and development of a high-performance tensile-mode piezoelectric energy harvester based on a three-hinged force-amplification mechanism

When considering durability and reliability, flexible piezoelectric materials, such as PVDF and macro-fiber composite, are preferable to piezoceramics due to the brittleness of piezoceramics. However, flexible piezoelectric materials cannot sustain compressive loads so they need to be operated in either tensile or bending mode. The tensile mode has the advantage of uniform strain distribution over the bending mode. This study proposes a novel tensile-mode piezoelectric energy harvester based on a three-hinged force amplification mechanism. The proposed design consists of a rigid beam and an elastic PVDF film connected to each other via a revolute joint. The assembly is attached to a base via revolute joints with the PVDF film pre-stretched. The PVDF film bears a dynamic tensile load when the harvester is under harmonic excitations. A theoretical model of the proposed harvester is developed and experimentally validated. The simulation and experimental results show that the proposed design exhibits a strong hardening effect due to the nonlinear geometry of the three-hinged mechanism. The effect of preloads and mass distributions are explored to see their impact on the harvesting performance. It is shown that the peak voltage and bandwidth of the harvester decline as the preload increases. By properly tuning the mass distribution, the performance of the harvester can be enhanced. Compared with a bending-mode cantilevered harvester, the voltage output and harvesting bandwidth of the proposed harvester can be improved by 500% and 1250%, respectively.

Study on the Actuating Performance of an Embedded Macro Fiber Composite Considering the Shear Lag Effect.

Macro fiber composite (MFC), which are new ultrathin piezoelectric smart materials, are mostly applied in the fields of shell structure deformation and vibration control. Among others, the application of embedded MFCs in sandwich structures has received wide attention. Currently, its actuating force formula is primarily acquired based on the Bernoulli–Euler Model, which does not consider the shear lag effect and actuating force of MFC ends. To study the actuating performance of an MFC in a sandwich structure, according to its action characteristics, the MFC is divided into upper and lower actuating units without any interaction between to two under the condition of plane strain, and the shear lag effect is considered between the units and the top and bottom of the sandwich structure. The actuating force of the MFC ends is obtained by considering its influence on the bending deformation of the sandwich structure, which deduces the actuating force formula of the embedded MFC. In contrast to ANSYS piezoelectric simulation, the distribution of the MFC interior normal stress is similar to the result from ANSYS piezoelectric simulation, and there is a very small deviation between the MFC end and central normal stress and the result from ANSYS piezoelectric simulation. Taking the end deflection of the sandwich structure with an embedded MFC as an example, the actuating force simulation of the MFC considering the shear lag effect is compared with the ANSYS piezoelectric simulation and actuating force simulation based on the Bernoulli–Euler model. The result indicates that the actuating force simulation of the MFC considering the shear lag effect is closer to the ANSYS piezoelectric simulation, which proves the rationality and necessity of considering the shear lag effect and end actuating force of the MFC.

Dynamic Response of a Composite Fan Blade Excited Instantaneously by Multiple MFC Actuators

The vibration characteristics of a composite fan blade are much more complex than those of a solid titanium fan blade due to the anisotropic material properties and complex excitations coming from unsteady flow and mechanically induced vibration. In this study, the dynamic response measurement of a wide-chord composite fan blade was carried out to study the vibration characteristics using multiple macro fiber composite (MFC) actuators, which can generate complex excitation forces with different frequencies and peak values at different locations. The measured mode shapes and natural frequencies were compared with the finite element simulation results. Based on these results, the responses of the blade under the instantaneous excitation of three MFC actuators with different combinations of several natural frequencies were measured and compared. The results show that the responses of the blade excited by different combinations of MFC actuators with different frequencies were significantly different from those excited by a single MFC actuator. The superposition of different mode shapes may cause the change of the vibration stress state, which indicates that the high cycle fatigue location of the blade under complex excitations may change to an unexpected location. The results will be helpful in understanding the vibration characteristics of the composite blades under complex excitations, and the MFC actuator could be a potential tool in vibration active control.

Lightweight Piezoelectric Bending Beam-Based Energy Harvester for Capturing Energy From Human Knee Motion

Harvesting energy from the human body has attracted substantial attention from industries and academics as it is a promising solution to address the battery life issue. In this article, we developed a bimorph piezoelectric bending beam-based energy harvester to scavenge energy from the human knee motion for powering body-worn electronics such as smartwatches and health monitors. During walking, the human knee will flex and extend to periodically deform piezoelectric macro fiber composites (MFC) bounded to the bending beam, thus generating electricity. The mathematical models of the whole system including the model of the large deflection of the bending beam, movement analysis of the end of the bending beam (slider-crank mechanism), the model of piezoelectric transducers, and measurement of the human knee angle are developed to comprehensively study the harvester. To validate the presented mathematical models, experimental testing on a human body when the subject walks on a treadmill at three different walking speeds is conducted here. The experimental results show good agreement with the simulation results, which indicates that the developed models can characterize the harvester. In addition, the performance of the harvester walking in different contexts is also evaluated. The average power can reach 12.79 mW for level-ground walking, 9.91 mW for stair descending, and 7.70 mW for stair ascending, when walking at a self-selected speed.

Microscopic vibration suppression for a high-speed macro-micro manipulator with parameter perturbation

This paper reports electromechanical dynamics modeling and microscopic vibration suppression for a high-speed macro–micro manipulator with structural flexibility and parameter perturbation. The macro–micro manipulator contains an air-floating macro-motion platform and a macro-fiber-composite (MFC) micromanipulator. The electromechanical dynamics model is derived by combining the assumed mode method and the asymmetric Prandtl-Ishlinskii hysteresis model. Then, a robust control strategy combining a perturbation H-infinity controller with a Kalman filter is proposed to compensate for the hysteresis nonlinearity and suppress the microscopic vibration during and after the macro motion. In particular, an additive perturbation is used to estimate the model uncertainties from the varying end mass and all other unknown dynamics. Also, the Kalman filter is designed to improve the signal-to-noise ratio of the displacement detection based on the process and measurement noises. Several experiments are conducted to validate the effectiveness and feasibility of the proposed dynamic model and robust controller. The elastic vibrations during and after macro motion are significantly suppressed even with the parameter perturbation. Thus, the proposed control strategy can improve the manipulation stability, robustness, and accuracy.

Investigation of damping characteristics onPLArobotic gripper fingers with magnetorheological fluid core and unimorph piezoelectricMFC

AbstractPolylactic acid (PLA) is an environmentally friendly, biodegradable, and low‐cost thermoplastic material suitable for 3D printing. Complex shapes can be easily produced and hence PLA has been used for many applications. This study presents the damping characteristics of two robotic gripper fingers. The first gripper is composed of a PLA surface layer and a magnetorheological (MR) fluid core and the second gripper is composed of a PLA surface layer bonded to a surface of unimorph piezoelectric Macro Fiber Composite (MFC). The magnetic field is applied by using permanent magnets while the magnetic field strength is varied by changing the distance between the permanent magnets. The electric field is applied by using a micro‐amplifier and regulated using data acquisition systems. The natural frequency and damping ratio of the PLA gripper finger are determined by free vibration analysis using LabVIEW software. The damping ratio and vibration amplitude suppression for the PLA grippers was increased as the magnetic field strength changed from 0G to 600G (MR) and as the electric field changed from 0V to 360V (MFC).

Hydrodynamic piezoelectric energy harvesting with topological strong vortex by forced separation

Inspired by the fish tail, we propose a piezoelectric energy harvester based on flow induced vibration with topological strong vortex by forced separation for low-velocity water flow. Observed using computational fluid dynamics, the high surface curvature at the up and down edges of the elliptic cylinder forces the boundary layer to separate and forms strong vortex squeezed on the bluff body. The extended Hamilton’s principle is used to derive the governing equation of the electromechanical coupling system and the Galerkin procedure is introduced to calculate the analytical output of the energy harvester. An experimental study with macro fiber composite in a circulating open U-shaped channel is conducted. The output voltage for the elliptic cylinder with an aspect ratio of 2.5 under the water velocity of 0.45 m/s is 38.4 V, 21.5 times higher than that for the circular cylinder. The hydrodynamic negative and positive damping determined by the shape of the bluff body balance the energy pumped in and dissipated in a period and fix its phase portraits at a stable limit cycle, resulting in the self-excited vibration of the bluff body. Optimal resistance analysis shows that changing parameters of the bluff body such as reducing its mass or increasing its length also improves the energy harvesting efficiency. The results are helpful for the optimal design of energy harvesters at low water velocity.

Active vibration control of smart porous conical shell with elastic boundary under impact loadings using GDQM and IQM

In this paper, the active damping control of the porous metal foam truncated conical shell with smart material macro fiber composite (MFC) is studied. Suppose the shell has arbitrary elastic supported edges, and is subjected to the impact loadings. The elastic supported edges can be achieved by virtual spring technique. Adopting the elastic constraint boundary condition, various classical boundaries can be easily realized by varying the value of the spring stiffnesses. Along the thickness direction, there are three types of porosity distributions of truncated conical shell being considered, and they are nonuniform symmetric, uniform and nonuniform asymmetric distribution, respectively. The sensor layer MFC-d31 and the actuator layer MFC-d33 are applied in the vibration control system. According to the first order shear deformation theory (FSDT) and energy principle, the dynamic equation of the system under electrostatic–mechanical​ coupling is derived. Then it is discretized into ordinary differential equation by generalized differential quadrature method (GDQM). The output charge applied to the sensor can be solved by integral quadrature method (IQM). The convergence and validation of the formulation and numerical calculation are studied by the comparisons between the present solutions and those from the literatures, ANSYS and COMSOL for the truncate conical shell with classical boundary conditions, respectively. Suppose the impact loadings acting on the porous truncated conical shell are step loading and decreasing loading, respectively. The velocity negative feedback approach is employed to diminish the amplitude of the transient response and achieve the active damping control. Then the effects of control gain, external excitation, semi-vertex angle and spring stiffness on the transient response control of the conical shell are studied in detail.

Multi-Mode Shape Control of Active Compliant Aerospace Structures Using Anisotropic Piezocomposite Materials in Antisymmetric Bimorph Configuration

The mission performance of future advanced aerospace structures can be synthetically improved via active shape control utilizing piezoelectric materials. Multiple work modes are required. Bending/twisting mode control receives special attention for many classic aerospace structures, such as active reflector systems, active blades, and compliant morphing wings. Piezoelectric fiber composite (Piezocomposite) material features in-plane anisotropic actuation, which is very suitable for multiple work modes. In this study, two identical macro-fiber composite (MFC) actuators of the F1 type were bonded to the base plate structure in an “antisymmetric angle-ply bimorph configuration” in order to achieve independent bending/twisting shape control. In terms of the finite element model and homogenization strategy, the locations of bimorph MFCs were determined by considering the effect of trade-off control capabilities on the bending and twisting shapes. The modal characteristics were investigated via both experimental and theoretical approaches. The experimental tests implied that the shape control accuracy was heavily reduced due to various uncertainties and nonlinearities, including hysteresis and the creep effect of the actuators, model errors, and external disturbances. A multi-mode feedback control law was designed and the experimental tests indicated that synthetic (independent and coupled) bending/twisting deformations were achieved with improved shape accuracy. This study provides a feasible multi-mode shape control approach with high surface accuracy, especially by employing piezocomposite materials.

Mode-I fracture control of unidirectional laminated composites using MFC actuators

This article presents an experimental and numerical investigation of Mode-I fracture control in aerospace grade unidirectional laminated composites (AS4/914) using the smart material approach. The P1 type macro fiber composites (MFC) actuators are selected for current research because of their higher free strain, high mechanical flexibility, and good blocking force. The opening mode (Mode-I) interlaminar fracture tests have been carried out on DCB specimens having surface bonded MFC actuator patches under the application of electric voltage within a range of 0–1500 V. These fracture control tests have been performed on pre-cracked (PC) and non pre-cracked (NPC) double cantilever beam specimens. The modified pin force model has been adopted for the evaluation of actuation forces in surface bonded MFC actuators. The XIGA-CZM based numerical formulation is utilized for numerical fracture modeling under electromechanical loading conditions. The MFC actuators have significant control over Mode-I fracture energy under the application of peak operating voltage. The numerical model using a modified pin force model in combination with XIGA-CZM based fracture model shows a good agreement with experimental observations.

Intrawell and interwell oscillations for bi-stable cross-ply laminates actuated by MFC under oscillating impulse voltages

The snap-through is a typical geometric nonlinear problem for the bi-stable laminate structures. Using the Macro Fiber Composite (MFC) actuators to actuate the snap-through behavior of the morphing structures has attracted considerable attention. In this paper, the intrawell oscillations and interwell oscillations are studied for the asymmetric four layers cross-ply bi-stable rectangular laminate actuated by the d33 type MFC actuator. It is assumed that the four edges of the bi-stable laminate are free while the geometry center of it is fixed. On the basis of the classical laminated plate theory and the Hamilton’s principle, the equations of motion of the bi-stable laminate which are expressed by curvatures are presented. The stable equilibrium configurations of the bi-stable laminate considering the effects of MFC are given. Under the action of oscillating impulse voltages, the intrawell oscillations and interwell oscillations of the system are analyzed in detail through the frequency-response curves, time response and the phase portraits. The numerical results show that the snap-through behavior is easier to realize for the bi-stable laminate actuated by the oscillating impulse voltages than that actuated by the static voltages. Furthermore, snap-through behavior can be better controlled by changing the amplitude and frequency of the applied voltages in a certain range.

Morphing of bistable variable stiffness composites using distributed MFC actuators

The use of piezoelectrically controlled bistable composite laminates for morphing applications has received increased attention in recent years. So far, most existing investigations have explored the possibility to trigger snap-through using large-sized piezoelectric Macro Fibre Composite (MFC) actuators bonded at the centre of the bistable laminate. However, bonding large-sized MFCs at the centre of the plate leads to flattening of the midsection of the laminate, which can result in the loss of bistability. This study presents an alternative approach of bonding distributed smaller MFCs over the entire surface of the laminate and investigate the resulting stable shapes and the change in snap-through voltages. Self-resetting piezoelectrically controlled active laminates where the MFC patches are distributed over the laminate surface have been investigated. A semi-analytical model using the Rayleigh–Ritz technique is developed to account for the distributed actuation system. Results from the proposed semi-analytical framework are verified using a corresponding finite element model. The bistable shapes, as well as the snap-through and snap-back voltages, are calculated for different distributed MFC configurations and are compared with a single MFC laminate system. Snap-through voltage predictions from the proposed semi-analytical formulation and the finite element model are compared with the experimental results for a single MFC patch available from the literature. Further, the possibility to tailor the snap-through voltages of the proposed laminate-MFC configuration is explored by replacing conventional cross-ply laminates with variable stiffness (VS) laminates generated from curvilinear fibre alignments. A finite element parametric study is performed by tailoring the VS fibre orientation parameters to achieve a bistable laminate-MFC configuration with lower snap-through requirements where the designed laminate is actuated with distributed MFC patches.

Effects of actuator-substrate ratio on hydrodynamic and propulsion performances of underwater oscillating flexible structure actuated by macro fiber composites

Oscillating flexible structures driven by smart actuators are promising thrust-producing components for underwater bionic vehicles. In this study, the hydrodynamic and propulsion performances of an oscillating flexible structure driven by macro fiber composites (MFCs) are investigated at varying actuator-substrate ratios. Numerical and experimental results demonstrate that the underwater resonant frequency increases approximately linearly with increasing actuator-substrate ratio. Moreover, the ratios of the in-air and underwater resonant frequencies vary with differing actuator-substrate ratios, and are mainly determined by the equivalent mass density. The effects of the actuator-substrate ratios on several characteristic parameters related to the propulsion performance are investigated and compared. The results suggest the generated mean thrust of the fully actuated case is the largest, but the power efficiency is the lowest among all the actuator-substrate ratios investigated, because this case has the lowest characteristic velocity and Reynolds number. In contrast, the highest power efficiency and lowest generated thrust are observed in the longest substrate case with the smallest actuator-substrate ratio. An optimal balance between the power efficiency and generated thrust is achieved when the actuator-substrate ratio approaches 0.6, in which case the characteristic parameters of propulsion performance are the highest among all the cases. These findings can provide useful guidelines for the design of efficient underwater bionic devices propelled by flexible fins driven by MFCs and other smart actuators.

The smart morphing winglet driven by the piezoelectric Macro Fiber Composite actuator

AbstractA smart morphing winglet driven by piezoelectric Macro Fiber Composite (MFC) is designed to adjust cant angle autonomously for various flight conditions. The smart morphing winglet is composed of the MFC actuator, DC-DC converter, power supply, winglet part and wing part. A hinge is designed to transfer the bending deformation of intelligent MFC bending actuator to rotation of the winglet structure so as to achieve the adaptive cant angle. Experimental and numerical work are conducted to evaluate the performance of smart morphing winglet. It is demonstrated that the proposed intelligent MFC bending actuator has an excellent bending performance and load resistance. This smart morphing winglet exhibits the excellent characteristic of flexibility on large deformation and lightweight. Moreover, a series of wind tunnel tests are performed, which demonstrate that the winglet driven by intelligent MFC bending actuator produces sufficient deformation in various wind speed. At high wind speed, the cant angle of the winglet can reach 16 degrees, which is still considered to be very useful for improving the aerodynamic performance of the aircraft. The aerodynamic characteristics are investigated by wind tunnel tests with various attack angles. As a result, when the morphing winglet is actuated, the lift-to-drag ratio could vary up to 11.9% and 6.4%, respectively, under wind speeds of 5.4 and 8.5m/s. Meanwhile, different flight phases such as take-off, cruise and landing are considered to improve aerodynamic performance by adjusting the cant angle of winglet. The smart morphing winglet varies the aerofoil autonomously by controlling the low winglet device input voltage to remain optimal aerodynamic performance during the flight process. It demonstrates the feasibility of piezoelectric composites driving intelligent aircraft.

An Alternative Electro-Mechanical Finite Formulation for Functionally Graded Graphene-Reinforced Composite Beams with Macro-Fiber Composite Actuator.

With its extraordinary physical properties, graphene is regarded as one of the most attractive reinforcements to enhance the mechanical characteristics of composite materials. However, the existing models in the literature might meet severe challenges in the interlaminar-stress prediction of thick, functionally graded, graphene-reinforced-composite (FG-GRC)-laminated beams that have been integrated with piezoelectric macro-fiber-composite (MFC) actuators under electro-mechanical loadings. If the transverse shear deformations cannot be accurately described, then the mechanical performance of the FG-GRC-laminated beams with MFC actuators will be significantly impacted by the electro-mechanical coupling effect and the sudden change of the material characteristics at the interfaces. Therefore, a new electro-mechanical coupled-beam model with only four independent displacement variables is proposed in this paper. Employing the Hu–Washizu (HW) variational principle, the precision of the transverse shear stresses in regard to the electro-mechanical coupling effect can be improved. Moreover, the second-order derivatives of the in-plane displacement parameters have been removed from the transverse-shear-stress components, which can greatly simplify the finite-element implementation. Thus, based on the proposed electro-mechanical coupled model, a simple C0-type finite-element formulation is developed for the interlaminar shear-stress analysis of thick FG-GRC-laminated beams with MFC actuators. The 3D elasticity solutions and the results obtained from other models are used to assess the performance of the proposed finite-element formulation. Additionally, comprehensive parametric studies are performed on the influences of the graphene volume fraction, distribution pattern, electro-mechanical loading, boundary conditions, lamination scheme and geometrical parameters of the beams on the deformations and stresses of the FG-GRC-laminated beams with MFC actuators.

Prediction and optimization of poling condition for PZT based-macro fiber composites with interdigitated electrodes

The poling of macro fiber composites (MFCs) as a vital part for manufacturing usable samples, is related with but different from that of their piezoelectric phase owing to the existence of interdigitated electrodes. Herein, beginning with the quantitative relationship between effective electric field and external electric field obtained by finite element analysis, the prediction and optimization of poling condition for lead zirconate titanate (PZT) based-MFCs was presented. The results demonstrate that the predicted values agree with the experimental results, specifically, the poling field range of the investigated MFCs is from 16.7 kV/cm to 33.3 kV/cm. The piezoelectric and dielectric properties of MFCs are improved by adjusting the poling condition up to a suitable value, whereas are degraded due to the interaction of domain alignment and electric field concentration. The optimal poling condition of the MFCs is 26.7 kV/cm for 10 min at room temperature. This research contributes to realizing the personalized-customization of MFCs.

A Hybrid Structure of Piezoelectric Fibers and Soft Materials as a Smart Floatable Open-Water Wave Energy Converter.

An open-water wave energy converter (OWEC) made of a new soft platform has been developed by combining piezoelectric macro-fiber composites (MFCs) and a low-cost elastomer. In the past decades, numerous types of water wave energy conversion platform have been developed and investigated, from buoys to overtopping devices. These harvesters mainly use electromagnetic-based generators, and they have faced challenges such as their enormous size, high deployment and maintenance costs, and negative effects on the environment. These problems hinder their practicality and competitiveness. In this paper, a soft open-water wave energy converter is introduced which integrates piezoelectric MFCs and bubble wrap into an elastomer sheet. The performance of the OWEC was investigated in a wave flume as a floatable structure. The maximum 29.7 µW energy harvested from the small OWEC represents a promising energy conversion performance at low frequencies (<2 Hz). The elastomer was able to protect the MFCs and internal electrical connections without any degradation during the experiment. In addition, the OWEC is a foldable structure, which can reduce the deployment costs in real-world applications. The combination of no maintenance, low fabrication cost, low deployment cost, and moderate energy harvesting capability may advance the OWEC platform to its real-world applications.

Continuous electric field modeling of Macro-Fiber Composites for actuation and energy harvesting

This paper presents a new low-order electric field model for Macro-Fiber Composite devices with interdigitated electrodes. Specifically, the paper proposes a continuous electric field model, where the differential form of Gauss's law is used to obtain spatial terms of the electric field, and the integral form of Gauss's law is used to obtain temporal term of the electric field. By neglecting three-dimensional effects, the method of matched asymptotic expansion is employed to obtain the electric field between two pairs of interdigitated electrodes under quasi-electrostatic conditions. By averaging the solution along with the thickness of piezoceramic fibers and expanding it to the device's entire geometry by Fourier series, the low-order model of the electric field between the interdigitated electrodes is obtained. The matched asymptotic expansion and the low-order model agree well with finite element results. Generalized Hamilton's principle is employed to obtain the coupled electromechanical equations of motion for a beam with the Macro-Fiber Composite device. The Euler-Bernoulli assumptions are used to approximate strain distribution. Under current assumptions, equivalent capacitance and electromechanical coupling terms are obtained without using discontinuous functions (e.g., Heaviside or piecewise functions) for electric field terms. The equivalent capacitance is within 2% of the values reported by the manufacturer. Frequency response functions are obtained for the output voltage, tip displacement, and optimum working load of the energy harvester, and the current and the tip displacement for the actuator by assumed modes solution for a cantilever beam. It is shown that the actuator's current is related to the optimum working load of the energy harvester by Ohm's law. In the limited case, the proposed continuous electric field model reduces to the constant electric field model. When the constant electric field model is compared to the continuous electric field model, it is shown that the constant electric field model overpredicts the equivalent capacitance and electromechanical coupling. The relative difference between the two models for equivalent capacitance is around 41%, and for electromechanical coupling is around 22%. These relative differences are in agreement with the empirical correction factors defined in the earlier constant field models.