Mathematical dynamics of electromechanical piezoelectric energy harvesters

This research investigates vibration energy harvesting by modelling several piezoelectric-based structures. The usage of piezoelectric transduction under input vibration environments can be profitable for obtaining electrical energy for powering smart wireless sensor devices for health condition monitoring of rotating machines, structures and defence communication technology. The piezoelectric transduction shows strong prospect in the application of power harvesting because it can be applied at the microelectromechanical system design level in compact configuration with high sensitivity with respect to low input mechanical vibration. In this research work, the important aspects of the continuum thermopiezoelectric system associated with the laws of thermodynamics, Maxwell relations and Legendre transformations have been developed to explore the macroscopic thermopiezoelectric potential equations, the thermopiezoelectric equations of state and energy function forms. The application of the continuum thermopiezoelectric behaviour can be used to further formulate novel analytical methods of the electromechanical cantilevered piezoelectric bimorph beams with the tip mass using the weak and strong forms resulting from Hamiltonian’s principle.The constitutive electromechanical dynamic equations of the piezoelectric bimorph beam under one or two input base excitations can be used to derive the equations of the coupled electromechanical dynamic response of transverse-longitudinal form (CEDRTL), the coupled electromechanical dynamic response of longitudinal form (CEDRL) and the coupled electromechanical dynamic response of transverse form (CEDRT). The derivation of the constitutive electromechanical dynamic equations using the weak form of Hamiltonian’s principle can be further derived using the Ritz method associated with orthonomality whereas the closed form or distributed parameter reduced from strong form of Hamiltonian’s principle, can be further formulated using the convergent eigenfunction series with orthonormality. Laplace transformation can be used to give the solution in terms of the multi-mode transfer functions and multi-mode frequency response functions of dynamic displacement, velocity, electric voltage, current, power and optimal power. Moreover, the broadband multi-electromechanical bimorph beam with multi-resonance can also be explored showing the single- and multi-mode transfer functions and frequency response functions. A parametric case study of the piezoelectric bimorph beam with the tip mass and transverse input excitation is discussed to validate the weak and closed forms of the CEDRTL, under series and parallel connections, using the multi-mode frequency response functions with variable load resistance.A further case study of a broadband multi-electromechanical piezoelectric bimorph beam is also discussed using the weak form of the CEDRT to give the frequency response functions under variable load resistance. Finally, the piezoelectric bimorph beams with and without tip masses under transverse base input excitation are also comprehensively discussed using the weak forms of the CEDRTL and CEDRT models and compared with experimental results for variable load resistance. A piezoelectric bimorph beam with tip mass is investigated to show the close agreement between the CEDRTL model and experimental results using the polar amplitudes from the combined action of simultaneous longitudinal and transverse base input excitation

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

A unified electromechanical finite element dynamic analysis of multiple segmented smart plate energy harvesters: circuit connection patterns

This paper presents the techniques for formulating the multiple segmented smart plate structures with different circuit connection patterns using the electromechanical finite element dynamic analysis. There are three major contributions in the proposed numerical studies. First, the electromechanical discretization has been developed for generalizing the coupled system of Kirchhoff’s smart plate structures with circuit connection patterns. Such constitutive numerical models reduced from the extended Lagrange equations can be used for the physical systems including, but not restricted to, the multiple piezoelectric and electrode segments. Second, the multiple piezoelectric or electrode segments can be arranged electrically in parallel, series, and mixed series–parallel connections with the on–off switching techniques where the electrical outputs of each connection are further connected with the standard AC–DC circuit interfaces. Third, the coupling transformation technique (CTT) has been introduced by modifying the orthonormalized global element matrices into the scalar form equations. As a result, the multimode frequency response function and time-waveform signal response equations are distinctly formulated for each circuit connection. Further parametric numerical case studies are also discussed in this paper. The benefit of using the circuit connection patterns with the on–off switching techniques is that the studies can be used for an adaptive vibration power harveste

Analytical and Experimental Comparisons of Electromechanical Vibration Response of a Piezoelectric Bimorph Beam for Power Harvesting

Power harvesters that extract energy from vibrating systems via piezoelectric transduction show strong potential for powering smart wireless sensor devices in applications of health condition monitoring of rotating machinery and structures. This paper presents an analytical method for modelling an electromechanical piezoelectric bimorph beam with tip mass under two input base transverse and longitudinal excitations. The Euler–Bernoulli beam equations were used to model the piezoelectric bimorph beam. The polarity-electric field of the piezoelectric element is excited by the strain field caused by base input excitation, resulting in electrical charge. The governing electromechanical dynamic equations were derived analytically using the weak form of the Hamiltonian principle to obtain the constitutive equations. Three constitutive electromechanical dynamic equations based on independent coefficients of virtual displacement vectors were formulated and then further modelled using the normalised Ritz eigenfunction series. The electromechanical formulations include both the series and parallel connections of the piezoelectric bimorph. The multi-mode frequency response functions (FRFs) under varying electrical load resistance were formulated using Laplace transformation for the multi-input mechanical vibrations to provide the multi-output dynamic displacement, velocity, voltage, current and power. The experimental and theoretical validations reduced for the single mode system were shown to provide reasonable predictions. The model results from polar base excitation for off-axis input motions were validated with experimental results showing the change to the electrical power frequency response amplitude as a function of excitation angle, with relevance for practical implementation

Electromechanical Piezoelectric Power Harvester Frequency Response Modelling Using Closed-Form Boundary Value Methods

The conversion of mechanical vibration to electrical energy has shown great promise for extending battery life of smart sensor wireless devices for various engineering applications. This paper presents novel analytical models of a piezoelectric bimorph, using the closed-form boundary value (CFBV) method, for predicting the electromechanical power harvester frequency response. The derivations of the coupled electromechanical dynamic response of the transverse-longitudinal (CEDRTL) form based on the CFBV method were developed using the reduced strong form method of the Hamiltonian principle. The equations from CEDRTL can be reduced to give the coupled electromechanical dynamic response of the transverse (CEDRT) form. The electromechanical frequency response functions with variable load resistance were also given in detail using Laplace transformation. The two theoretical studies are compared together and validated with an experimental study. For some cases, when the load resistance approached open circuit, the difference between CEDRTL and CEDRT tended to be more pronounced. Conversely, the CEDRTL and CEDRT models tended to overlap when the load resistance approaches short circuit. Nyquist plots can be used to analyse the shifting frequency and amplitude changes due to variable resistance. Overall, the experimental and CEDRTL model results were very close to each other

Intrinsic Geometries and Properties of Piezo-MEMS Power Harvesters with Tip Mass Offset using New Electromechanical Finite Element Vibration Analysis

Autonomous self-powered wireless sensor devices are inevitable future technology that will potentially become ubiquitous in many sectors such as industry, intelligent infrastructure and biomedical devices. This has spurred a great attention from researchers to develop self-sustained power harvesting devices. For this paper, we present a new numerical technique for modelling the MEMS power harvesters using parametric design optimisation and physical properties for various piezoelectric materials. This technique enables the prediction of optimal power harvesting responses that can be used to identify the performance of piezoelectric materials and particular piezoelectric geometry where this technique can alleviate tedious analytical methods for analysing parametric design optimisation and can assist for analysing piezo-MEMS system response before conducting the micro-fabrication process

Comparative Numerical Studies of Electromechanical Finite Element Vibration Power Harvester Approaches of a Piezoelectric Unimorph

Emerging micro-power harvester research using smart material components shows viable self-powered devices capable of capturing mechanical motion and converting it into useful electrical energy that can be further used to supply electrical voltage into rechargeable power storage via a power management electronic circuit. The micro-power harvesters using piezoelectric materials cover a wide range of applications for powering thin film battery technology and wireless sensor systems that can be used to monitor the health condition of machines and infrastructure and biomedical implant devices. This research focuses on the development of a novel numerical direct method technique with non-orthonormality based on the electromechanical vector transformation for modelling the self-powered cantilevered piezoelectric unimorph beam under input base excitation. The proposed finite element piezoelectric unimorph beam equations were formulated using Hamiltonian’s principle for formulating the global matrices of electromechanical dynamic equations based on the electromechanical vector transformation that can be further employed to derive the electromechanical frequency response functions. This numerical technique was modelled using electromechanical discretisation consisting of mechanical and electrical discretised elements due to the electrode layers covering the surfaces of the piezoelectric structure, giving the single voltage output. The reduced equations are based on the Euler-Bernoulli beam assumption for designing the typical power harvesting device. The proposed finite element models were also compared with orthonormalised electromechanical finite element response techniques, giving accurate results in the frequency domains

Analytical Modeling of Self-Powered Electromechanical Piezoelectric Bimorph Beams with Multidirectional Excitation

Unused mechanical energies can be found in numerous ambient vibration sources in industry including rotating equipment, vehicles, aircraft, piping systems, fluid flow, and even external movement of the human body. A portion of the vibration energy can be recovered using piezoelectric transduction and stored for subsequent smart system utilization for applications including powering wireless sensor devices for health condition monitoring of rotating machines and defence communication technology. The vibration environment in the considered application areas is varied and often changes over time and can have components in three perpendicular directions, simultaneously or singularly. This paper presents the development of analytical methods for modelling of self-powered cantilevered piezoelectric bimorph beams with tip mass under simultaneous longitudinal and transverse input base motions utilizing the weak and strong forms of Hamiltonian’s principle and space- and time-dependent eigenfunction series which were further formulated using orthonormalization. The reduced constitutive electromechanical equations of the cantilevered piezoelectric bimorph were subsequently analysed using Laplace transforms and frequency analysis to give multi-mode frequency response functions (FRFs). The validation between theoretical and experimental results at the single mode of eigenfunction solutions reduced from multi-mode FRFs is also given

An Analytical Method for Vibration Modelling of a Piezoelectric Bimorph Beam for Power Harvesting

This paper presents the development of a mathematical method for modelling a piezoelectric bimorph beam under two input base-transversal and longitudinal excitations. The piezoelectric bimorph beam model was based on the Euler-Bernoulli beam coupled with polarity-electric field for low power harvesting. The piezoelectric bimorph beam with brass centre shim was also coupled to a simple electrical circuit of resistor component. The existence of input base-longitudinal motion can affect the overall strain, polarity and electric field of the cantilevered piezoelectric bimorph, identified to have predominant bending due to input transverse-base motion. The characteristic physical behaviour of the bimorph model for parallel connection can create mode vector configurations of X-poling due to longitudinal extension form and Y-poling due transverse bending form. Conversely, the effect of series connection of the physical bimorph model can create X-poling due to transverse bending and Y-poling due to longitudinal extension forms. A new method of solving the piezoelectric bimorph under two input base-motions using coupling superposition of the elastic-polarity field has been introduced. The governing dynamics equations can be derived analytically using the weak form of Hamiltonian theorem to obtain the constitutive equations. DuBois-Reymond lemma can be used to separate three constitutive dynamic equations based on independent coefficients of the virtual displacement vectors. Furthermore, the solution forms for the three governing dynamics equations were assumed using the three independent normal modes of displacement functions based on the normal modes in the transversal, longitudinal and electric potential mode forms. To this end, the dynamic equations for frequency response, dynamic displacements, accelerations and electric voltage can be further computed analytically according to the suggested formulations