Electromechanical Frequency Response Functions Research Articles

A general electromechanical model for plates with integrated piezo-patches using spectral-Tchebychev method

This paper presents a general electromechanical model for predicting the dynamics of thin or moderately thick plates with surface-integrated piezo-patches. Using spectral Tchebychev (ST) technique, the boundary value problem governing the electroelastic dynamics of the two dimensional (2D) plate and piezo-patch structure is developed with Mindlin plate theory assumptions. Mass and stiffness contributions of piezo-patch(es) as well as two-way electromechanical coupling behavior are incorporated in the model for both modal analysis and frequency response calculations. To validate the accuracy of the developed solution technique, the modal analysis results are compared against the existing Rayleigh-Ritz solution from the literature as well as the finite-element simulation results for various piezo-patch sizes on thin and moderately thick host plates; and it is shown that the maximum difference in the predicted natural frequencies between the ST and FE results are below 1%. The electromechanical frequency response functions (FRFs) including the vibration response and electrical output of the system under a transverse point force excitation are obtained using the ST model and the results are shown to match perfectly with the finite element (FE) simulations. Additionally, comparisons of the electromechanical FRFs calculated based on Rayleigh-Ritz method from the literature versus the developed framework is presented to highlight that the exclusion of shear deformation terms in the former model leads to an inaccurate estimation of electroelastic behavior for the case of thicker plates with integrated piezo-patches. Finally, the investigated case studies demonstrate that the computational efficiency of the developed method is significantly higher than that of FE simulations.

Combined piezoelectric and flexoelectric effects in resonant dynamics of nanocantilevers

We establish and analyze an analytical framework by accounting for both the piezoelectric and flexoelectric effects in bimorph cantilevers. The focus is placed on the development of governing electroelastodynamic piezoelectric–flexoelectric equations for the problems of resonant energy harvesting, sensing, and actuation. The coupled governing equations are analyzed to obtain closed-form frequency response expressions via modal analysis. The combined piezoelectric–flexoelectric coupling coefficient expression is identified and its size dependence is explored. Specifically, a typical atomistic value of the flexoelectric constant for barium titanate is employed in the model simulations along with its piezoelectric constant from the existing literature. It is shown that the effective electromechanical coupling of a piezoelectric material, such as barium titanate, is significantly enhanced for thickness levels below 100 nm. The electromechanical coupling coefficient of a barium titanate bimorph cantilever increases from the bulk piezoelectric value of 0.065 to the combined piezoelectric–flexoelectric value exceeding 0.3 toward nanometer thickness level. Electromechanical frequency response functions for resonant power generation and dynamic actuation also capture the size-dependent enhancement of the electromechanical coupling. The analytical framework given here can be used for parameter identification and design of nanoscale cantilevers that can be used as energy harvesters, sensors, and actuators.

Modeling and Characterization of a Curved Piezoelectric Energy Harvester for Smart Paver Tiles

Abstract: The transformation of vibrations into low-power electricity has received growing attention over the past couple of decades. The goal in this research field is to power wireless sensors and other small electronic components by harvesting ambient mechanical energy. Piezoelectricity is arguably the most popular transduction mechanism for converting mechanical vibrations into electrical energy. As a promising concept of piezoelectric energy conversion, smart paver tiles are proposed to harness human steps for energy harvesting and a wide range of critical sensing applications ranging from security to healthcare. In this work, we explore the use of a curved piezoelectric transducer (specifically a THUNDER® configuration) for smart paver tiles both analytically and experimentally. Following the description of the smart paver tiles developed, an analytical model is developed and closed-form electromechanical frequency response functions are obtained. Experiments are performed for a range of excitation frequencies (1-10 Hz ), electrical load resistance values, and low force levels. It is shown that the first-order electromechanical model successfully represents the governing dynamics.

Multilayer piezoelectret foam stack for vibration energy harvesting

Electronic devices are high-demand commodities in today’s world, and such devices will continue increasing in popularity. Currently, batteries are implemented to provide power to these devices; however, the need for battery replacement, their cost, and the waste associated with battery disposal present a need for advances in self-powered technology. Energy harvesting technology has great potential to alleviate the drawbacks of batteries. In this work, a novel piezoelectret foam material is investigated for low-level vibration energy harvesting. Specifically, piezoelectret foam assembled in a multilayer stack configuration is explored. Modeling and experimentation of the stack when excited in compression at low frequencies are performed to investigate piezoelectret foam for multilayer energy harvesting. An equivalent circuit model derived from the literature is used to model the piezoelectret stack. Two 20-layer prototype devices and one 40-layer prototype device are fabricated and experimentally tested via harmonic base excitation. Electromechanical frequency response functions between input acceleration and output voltage are measured experimentally. Modeling results are compared to experimental measurements to assess the fidelity of the model near resonance. Finally, energy harvesting experimentation in which the device is subject to harmonic base excitation at the fundamental natural frequency is conducted to determine the ability of the stack to successfully charge a capacitor. For a 20-layer stack excited at 0.5 g, a 100-µF capacitor is charged to 1.45 V in 15 min, and produces a peak power of 0.45 µW. A 40-layer stack is found to charge a 100-µF capacitor to 1.7 V in 15 min when excited at 0.5 g, and produce a peak power of 0.89 µW.

On the energy harvesting via doubly curved piezoelectric panels

This article presents an analytical solution for power output from a doubly curved piezoelectric energy harvester. The energy harvester is made of an elastic core layer coupled with one or two surface-bonded piezoelectric layers. Five mechanical equations of motion together with Gauss’s equation are derived on the basis of first-order shell theory and solved simultaneously for simply supported mechanical boundary conditions. The influence of structural damping is taken into account using Rayleigh damping. The electromechanical frequency response functions that relate the power output and circuit load resistance are identified from the exact solutions. Finally, the performance of the system is analyzed extensively for different parameters such as type of energy harvester including unimorph and bimorph panel in series or parallel connections of piezoelectric layers, circuit load resistance, geometrical parameters, and material properties of core and piezoelectric layers.

Energy harvesting via shallow cylindrical and spherical piezoelectric panels using higher order shear deformation theory

In this article an analytical solution is presented for power output from a piezoelectric shallow shell energy harvester using higher order shear deformation theory (HSDT). The energy harvester is made of an elastic substrate layer coupled with one or two surface bonded piezoelectric layers. Mechanical equations of motion with Gauss’s equation are derived on the basis of HSDT and solved simultaneously for simply-supported mechanical boundary conditions. The electromechanical frequency response functions that relate the power output and circuit load resistance are identified from the exact solutions. Using Rayleigh damping the influence of structural damping is taken into account. Also performance of the system is analyzed extensively for different parameters such as type of energy harvester including unimorph and bimorph panel in series or parallel connections of piezoelectric layers, circuit load resistance, geometrical parameters and material properties of core and piezoelectric layers.

Evaluation of the performance of a lead-free piezoelectric material for energy harvesting

Vibration-based energy harvesting has been explored as an auxiliary power source, which can provide small amounts of energy to power remote sensors installed in inaccessible locations. This paper presents an experimental and analytical study of an energy harvesting device using a lead-free piezoelectric material based on -doped ()() KNL-(NTS)Mo. The harvesting model corresponds to a cantilever beam with a KNL-(NTS)Mo piezoelectric disc attached to it. We analyze the effect of electromechanical coupling and load resistance on the generated electrical power. Electromechanical frequency response functions that relate the voltage output to the translational base acceleration are shown for experimental and analytical results.

Effect of shunted piezoelectric control for tuning piezoelectric power harvesting system responses—analytical techniques

This paper presents new analytical modelling of shunt circuit control responses for tuning electromechanical piezoelectric vibration power harvesting structures with proof mass offset. For this combination, the dynamic closed-form boundary value equations reduced from strong form variational principles were developed using the extended Hamiltonian principle to formulate the new coupled orthonormalized electromechanical power harvesting equations showing combinations of the mechanical system (dynamical behaviour of piezoelectric structure), electromechanical system (electrical piezoelectric response) and electrical system (tuning and harvesting circuits). The reduced equations can be further formulated to give the complete forms of new electromechanical multi-mode frequency response functions and the time waveform of the standard AC–DC circuit interface. The proposed technique can demonstrate self-adaptive harvesting response capabilities for tuning the frequency band and the power amplitude of the harvesting devices. The self-adaptive tuning strategies are demonstrated by modelling the shunt circuit behaviour of the piezoelectric control layer in order to optimize the harvesting piezoelectric layer during operation under input base excitation. In such situations, with proper tuning parameters the system performance can be substantially improved. Moreover, the validation of the closed-form technique is also provided by developing the Ritz method-based weak form analytical approach giving similar results. Finally, the parametric analytical studies have been explored to identify direct and relevant contributions for vibration power harvesting behaviours.

Parametric design-based modal damped vibrational piezoelectric energy harvesters with arbitrary proof mass offset: Numerical and analytical validations

This paper focuses on the primary development of novel numerical and analytical techniques of the modal damped vibration energy harvesters with arbitrary proof mass offset. The key equations of electromechanical finite element discretisation using the extended Lagrangian principle are revealed and simplified to give matrix and scalar forms of the coupled system equations, indicating the most relevant numerical technique for the power harvester research. To evaluate the performance of the numerical study, the analytical closed-form boundary value equations have been developed using the extended Hamiltonian principle. The results from the electromechanical frequency response functions (EFRFs) derived from two theoretical studies show excellent agreement with experimental studies. The benefit of the numerical technique is in providing effective and quick predictions for analysing parametric designs and physical properties of piezoelectric materials. Although analytical technique provides a challenging process for analysing the complex smart structure, it shows complementary study for validating the numerical technique.

Electromechanical finite element modelling for dynamic analysis of a cantilevered piezoelectric energy harvester with tip mass offset under base excitations

A new electromechanical finite element modelling of a vibration power harvester and its validation with experimental studies are presented in this paper. The new contributions for modelling the electromechanical finite element piezoelectric unimorph beam with tip mass offset under base excitation encompass five major solution techniques. These include the electromechanical discretization, kinematic equations, coupled field equations, Lagrangian electromechanical dynamic equations and orthonormalized global matrix and scalar forms of electromechanical finite element dynamic equations. Such techniques have not been rigorously modelled previously by other researchers. There are also benefits to presenting the numerical techniques proposed in this paper. First, the proposed numerical techniques can be used for applications in many different geometrical models, including micro-electro-mechanical system power harvesting devices. Second, applying tip mass offset located after the end of the piezoelectric beam length can result in a very practical design, which avoids direct contact with piezoelectric material because of its brittle nature. Since the surfaces of actual piezoelectric material are covered evenly with thin conducting electrodes for generating single voltage, we introduce the new electromechanical discretization, consisting of the mechanical and electrical discretized elements. Moreover, the reduced electromechanical finite element dynamic equations can be further formulated to obtain the series form of new multimode electromechanical frequency response functions of the displacement, velocity, voltage, current and power, including optimal power harvesting. The normalized numerical strain node and eigenmode shapes are also further formulated using numerical discretization. Finally, the parametric numerical case studies of the piezoelectric unimorph beam under a resistive shunt circuit show good agreement with the experimental studies.

Assumed-modes modeling of piezoelectric energy harvesters: Euler–Bernoulli, Rayleigh, and Timoshenko models with axial deformations

A generalized framework is presented for the electromechanical modeling of base-excited piezoelectric energy harvesters with symmetric and unsymmetric laminates. The electromechanical derivations are given using the assumed-modes method under the Euler–Bernoulli, Rayleigh, and Timoshenko beam assumptions in three sections. The formulations account for an independent axial displacement variable and its electromechanical coupling in all cases. Comparisons are provided against the analytical solution for symmetric laminates and convergence of the assumed-modes solution to the analytical solution with increasing number of modes is shown. Model validations are also presented by comparing the electromechanical frequency response functions derived herein with the experimentally obtained ones in the absence and presence of a tip mass attachment. A discussion is provided for combination of the assumed-modes solution with nonlinear energy harvesting and storage circuitry. The electromechanical assumed-modes formulations can be used for modeling of piezoelectric energy harvesters with moderate thickness as well as those with unsymmetric laminates and varying geometry in the axial direction.