High-Performance Piezoelectric Energy Harvesters and Their Applications

Pull-in instability as an inherently nonlinear and crucial effect continues to become increasingly important for the design of electrostatic MEMS and NEMS devices and ever more interesting scientifically. This review reports not only the overview of the pull-in phenomenon in electrostatically actuated MEMS and NEMS devices, but also the physical principles that have enabled fundamental insights into the pull-in instability as well as pull-in induced failures. Pull-in governing equations and conditions to characterize and predict the static, dynamic and resonant pull-in behaviors are summarized. Specifically, we have described and discussed on various state-of-the-art approaches for extending the travel range, controlling the pull-in instability and further enhancing the performance of MEMS and NEMS devices with electrostatic actuation and sensing. A number of recent activities and achievements methods for control of torsional electrostatic micromirrors are introduced. The on-going development in pull-in applications that are being used to develop a fundamental understanding of pull-in instability from negative to positive influences is included and highlighted. Future research trends and challenges are further outlined.

Electrostatic pull-in instability in MEMS/NEMS: A review

Aeroelastic energy harvesting: A review

A wide range of microelectromechanical systems (MEMSs) and devices are actuated using electrostatic forces. Multiphysics modeling is required, since coupling among different fields such as solid and fluid mechanics, thermomechanics and electromagnetism is involved. This work presents an overview of models for electrostatically actuated MEMSs. Three-dimensional nonlinear formulations for the coupled electromechanical fluid–structure interaction problem are outlined. Simplified reduced-order models are illustrated along with assumptions that define their range of applicability. Theoretical, numerical and experimental works are classified according to the mechanical model used in the analysis.

https://iopscience.iop.org/article/10.1088/0964-1726/16/6/R01/pdf

Review of modeling electrostatically actuated microelectromechanical systems

A review on nonlocal elastic models for bending, buckling, vibrations, and wave propagation of nanoscale beams

Recently, piezoelectric cantilevered beams have received considerable attention for vibration-to-electric energy conversion. Generally, researchers have investigated a classical piezoelectric cantilever beam with or without a tip mass. In this paper, we propose the use of a unimorph cantilever beam undergoing bending‐torsion vibrations as a new piezoelectric energy harvester. The proposed design consists of a single piezoelectric layer and a couple of asymmetric tip masses; the latter convert part of the base excitation force into a torsion moment. This structure can be tuned to be a broader band energy harvester by adjusting the first two global natural frequencies to be relatively close to each other. We develop a distributed-parameter model of the harvester by using the Euler-beam theory and Hamilton’s principle, thereby obtaining the governing equations of motion and associated boundary conditions. Then, we calculate the exact eigenvalues and associated mode shapes and validate them with a finite element (FE) model. We use these mode shapes in a Galerkin procedure to develop a reduced-order model of the harvester, which we use in turn to obtain closed-form expressions for the displacement, twisting angle, voltage output, and harvested electrical power. These expressions are used to conduct a parametric study for the dynamics of the system to determine the appropriate set of geometric properties that maximizes the harvested electrical power. The results show that, as the asymmetry is increased, the harvester’s performance improves. We found a 30% increase in the harvested power with this design compared to the case of beams undergoing bending only. We also show that the locations of the two masses can be chosen to bring the lowest two global natural frequencies closer to each other, thereby allowing the harvesting of electrical power from multi-frequency excitations. (Some figures in this article are in colour only in the electronic version)

https://iopscience.iop.org/article/10.1088/0964-1726/20/11/115007/pdf

An energy harvester using piezoelectric cantilever beams undergoing coupled bending-torsion vibrations

We investigate the effects of shape variations of a cantilever beam on its performance as an energy harvester. The beam is composed of piezoelectric and metallic layers (unimorph design) with a rigid mass attached to its free end. A reduced-order model based on a one-mode Galerkin approach is derived. Solutions for the tip displacement, generated voltage, and harvested power are then obtained. Linear and quadratic shape variations are considered in order to design piezoelectric energy harvesters that can generate energy at low frequencies and maximize the harvested energy. The results show that the fundamental natural frequency and mode shape are strongly affected when the shape of the beam is varied. The influence of the electrical load resistance and the shape parameters at resonance on the system’s performance is discussed. It is determined that for specific resistance values, the quadratic shape can yield up to two times the energy harvested by a rectangular shape.

Design and performance of variable-shaped piezoelectric energy harvesters:

We model and analyze the deflections and motions of a shaped microbeam in a capacitive-based MEMS device. The model accounts for the system nonlinearities including mid-plane stretching and electrostatic forcing. The differential quadrature method (DQM) is used to discretize the microbeam partial differential equation. It is shown that the use of 11 grid points in the DQM is sufficient to capture the response of the device. It is also observed that, unlike the shooting methods, DQM does not face the problems of system differential equations stiffness and solution sensitivity to the initial guess. The static response to a dc voltage is first determined to investigate the influence of varying the geometric parameters of the device on the range of travel and pull-in voltage. Analytical expressions approximating the range of travel and pull-in voltage, as functions of the capacitor gap size and microbeam width and thickness, are derived. Symmetric and asymmetric spatial distributions of these parameters are considered. For symmetric distribution, an increase (decrease) in the beam width and/or thickness at the middle with respect to those at the endpoints results in an increase (decrease) in the pull-in voltage and a decrease (increase) in the range of travel. An increase (decrease) in the gap size at the middle with respect to those at the endpoints results in an increase (decrease) in the pull-in voltage and an insignificant effect on the range of travel. The dynamic response of the microbeam to a dc voltage is also determined for various distributions of the microbeam width and thickness and the gap size. It is shown that decreasing the microbeam thickness at the middle is the most effective method to reduce the pull-in time.

Modeling and design of variable-geometry electrostatic microactuators

We develop a mathematical model for a resonant gas sensor made up of an microplate electrostatically actuated and attached to the end of a cantilever microbeam. The model considers the microbeam as a continuous medium, the plate as a rigid body, and the electrostatic force as a nonlinear function of the displacement and the voltage applied underneath the microplate. We derive closed-form solutions to the static and eigenvalue problems associated with the microsystem. The Galerkin method is used to discretize the distributed-parameter model and, thus, approximate it by a set of nonlinear ordinary-differential equations that describe the microsystem dynamics. By comparing the exact solution to that associated with the reduced-order model, we show that using the first mode shape alone is sufficient to approximate the static behavior. We employ the Finite Difference Method (FDM) to discretize the orbits of motion and solve the resulting nonlinear algebraic equations for the limit cycles. The stability of these cycles is determined by combining the FDM discretization with Floquet theory. We investigate the basin of attraction of bounded motion for two cases: unforced and damped, and forced and damped systems. In order to detect the lower limit of the forcing at which homoclinic points appear, we conduct a Melnikov analysis. We show the presence of a homoclinic point for a loading case and hence entanglement of the stable and unstable manifolds and non-smoothness of the boundary of the basin of attraction of bounded motion.

Nonlinear dynamics of a resonant gas sensor

We use a discretization technique that combines the differential quadrature method (DQM) and the finite difference method (FDM) for the space and time, respectively, to study the dynamic behavior of a microbeam-based electrostatic microactuator. The adopted mathematical model based on the Euler- Bernoulli beam theory accounts for the system nonlinearities due to mid-plane stretching and electrostatic force. The nonlinear algebraic system obtained by the DQM-FDM is used to investigate the limit-cycle solutions of the microactuator. The stability of these solutions is ascertained usingFloquet theory and/or long- time integration. The method is applied for large excitation amplitudes and large quality factors for primary and secondary resonances of the first mode in case of hardening-type and softening-type behaviors. We show that the combined DQM-FDM technique improves convergence of the dynamic solutions. We identify primary, subharmonic, and superharmonic resonances of the microactuator. We observe the occurrence of dynamic pull-in due to subharmonic and superharmonic resonances as the excitation amplitude is increased. Simultaneous resonances of the first and higher modes are identified for large orbits in both primary and secondary resonances.

Fehmi Najar

Papers

An energy harvester using piezoelectric cantilever beams undergoing coupled bending-torsion vibrations

Design and performance of variable-shaped piezoelectric energy harvesters:

Modeling and design of variable-geometry electrostatic microactuators

Nonlinear dynamics of a resonant gas sensor

Nonlinear Analysis of MEMS Electrostatic Microactuators: Primary and Secondary Resonances of the First Mode