Abdessattar Abdelkefi

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

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)

Piezoelectric energy harvesting from transverse galloping of bluff bodies

The concept of harvesting energy from transverse galloping oscillations of a bluff body with different cross-section geometries is investigated. The energy is harvested by attaching a piezoelectric transducer to the transverse degree of freedom of the body. The power levels that can be generated from these vibrations and the variations of these levels with the load resistance, cross-section geometry, and freestream velocity are determined. A representative model that accounts for the transverse displacement of the bluff body and harvested voltage is presented. The quasi-steady approximation is used to model the aerodynamic loads. A linear analysis is performed to determine the effects of the electrical load resistance and the cross-section geometry on the onset of galloping, which is due to a Hopf bifurcation. The normal form of this bifurcation is derived to determine the type (supercritical or subcritical) of the instability and to characterize the effects of the linear and nonlinear parameters on the level of harvested power near the bifurcation. The results show that the electrical load resistance and the cross-section geometry affect the onset speed of galloping. The results also show that the maximum levels of harvested power are accompanied with minimum transverse displacement amplitudes for all considered (square, D, and triangular) cross-section geometries, which points to the need for performing a coupled analysis of the system.

Modeling and nonlinear analysis of piezoelectric energy harvesting from transverse galloping

A model for harvesting energy from galloping oscillations of a bar with an equilateral triangle cross-section attached to two cantilever beams is presented. The energy is harvested by attaching piezoelectric sheets to cantilever beams holding the bar. The derived nonlinear distributed-parameter model is validated with previous experimental results. The quasi-steady approximation is used to model the aerodynamic loads. The power levels that can be generated from these vibrations, and the variations of these levels with the load resistance and wind speed, are determined. Linear analysis is performed to validate the onset of galloping speed with experimental measurements. The effects of the electrical load resistance on the onset of galloping are then investigated. The results show that the electrical load resistance affects the onset speed of galloping. A nonlinear analysis is also performed to determine the effects of the electrical load resistance and the nonlinear torsional spring on the level of the harvested power. The results show that maximum levels of harvested power are accompanied by minimum transverse displacement amplitudes. It is also demonstrated that there is an optimum load resistance that maximizes the level of the harvested power.

Design and performance of variable-shaped piezoelectric energy harvesters

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.

Performance enhancement of piezoelectric energy harvesters from wake galloping

Experiments are performed to investigate the effects of wake galloping on the range of flow speeds over which a galloping-based piezoaeroelastic energy harvester can be effectively used. Two different upstream cylinders and a wide range of spacing between the upstream and downstream cylinders are considered. Bifurcation diagrams and type of instability for different setups are determined. The results show a complex relation between the upstream circular cylinder size, the spacing between the two cylinders, the flow speed, and the load resistance on one hand, and the level of harvested power on the other hand.

Theoretical modeling and nonlinear analysis of piezoelectric energy harvesting from vortex-induced vibrations

A nonlinear distributed-parameter model for harvesting energy from vortex-induced vibrations of a piezoelectric cantilever beam with a circular cylinder attached to its end is developed and validated with experimental results. A reduced-order model is derived by using the Euler–Lagrange principle and implementing the Galerkin discretization. A van der Pol wake oscillator is used to model the vortex-induced lift force. A nonlinear analysis is performed to determine the required number of modes in the Galerkin discretization. It is demonstrated that a one- or two-mode approximation in the Galerkin approach is not sufficient to evaluate the performance of the harvester. Based on a five-mode approximation in the Galerkin approach, an identification for the van der Pol wake oscillator coefficients is performed. To design efficient piezoaeroelastic energy harvesters that can generate energy at low freestream velocities, further analysis is performed to investigate the effects of the cylinder’s tip mass, length of the piezoelectric sheet, and electrical load resistance on the synchronization region and performance of the harvester. The results show that depending on the operating freestream velocity, the cylinder’s tip mass, length of the piezoelectric sheet, and electrical load resistance can be optimized to design enhanced piezoaeroelastic energy harvesters from vortex-induced vibrations.

Modeling and performance analysis of cambered wing-based piezoaeroelastic energy harvesters

We investigate the effects of aerodynamic loads on the performance of wing-based piezoaeroelastic energy harvesters. The rigid airfoil consists of pitch and plunge degrees of freedom supported by flexural and torsional springs with a piezoelectric coupling attached to the plunge degree of freedom. The effects of aerodynamic loads are investigated by considering a camber in the airfoil. A two-dimensional unsteady vortex-lattice method (UVLM) is used to model the unsteady aerodynamic loads. An iterative scheme based on Hamming?s fourth-order predictor?corrector method is employed to solve the governing equations simultaneously and interactively. The effects of varying the camber, its location, and the nonlinear torsional spring coefficient are determined. The results show that, for small values of the camber location, the flutter speed changes greatly on increasing the camber of the airfoil. On the other hand, for large values of the camber location, the variation of the flutter speed when changing the camber is very negligible. We demonstrate that the symmetric airfoil case is the best configuration to design enhanced wing-based piezoaeroelastic energy harvesters. Furthermore, the results show that an increase in the camber results in a decrease in the level of the harvested power. For cambered airfoils, we demonstrate that an increase in the camber location leads to an increase in the level of the harvested power. The results show that an increase in the airfoil camber delays the appearance of a secondary Hopf bifurcation.

Performance analysis of galloping-based piezoaeroelastic energy harvesters with different cross-section geometries

The concept of harvesting energy from galloping oscillations of a bluff body with different cross-section geometries attached to a cantilever beam is investigated. To convert these oscillations into electrical power, a piezoelectric transducer is attached to the transverse degree of freedom of the prismatic structure. Modal analysis is performed to determine the exact mode shapes of the structure. A coupled nonlinear distributed-parameter model is developed to determine the effects of the cross-section geometry, load resistance, and wind speed on the level of the harvester power. The quasi-steady approximation is used to model the aerodynamic loads. Linear analysis is performed to investigate the effects of the electrical load resistance and the cross-section geometry on the onset speed of galloping. The results show that the electrical load resistance and the cross-section geometry affect significantly the onset speed of galloping. Nonlinear analysis is performed to determine the effects of the electrical load resistance, cross-section geometry, and wind speed on the system’s outputs and particularly the level of the harvested power. A comparison of the performance of the different cross sections in terms of displacement and harvested power is presented. The results show that different sections are better for harvesting energy over different regions of the flow speed. The results also show that maximum levels of harvested power are accompanied with minimum transverse displacement amplitudes for all considered (square, D, and triangular) cross-section geometries.

Modeling and performance of electromagnetic energy harvesting from galloping oscillations

The modeling and performance of a galloping-based electromagnetic energy harvester are investigated. To convert galloping oscillations into electrical energy, an electromagnetic transducer is used. A set of representative coupled equations that account for the transverse displacement of the bluff body and the induced electromagnetic current are constructed. The galloping force is modeled by using the quasi-steady approximation. The effects of the electrical load resistance on the coupled damping and onset speed of galloping are determined through a linear analysis. It is shown that the electrical load resistance strongly affects the coupled damping and hence the onset speed of galloping of the harvester. For high values of the electrical load resistance, it is demonstrated that the load resistance has a negligible impact on the onset speed of galloping. A nonlinear analysis is then performed to investigate the effects of the electrical load resistance and wind speed on the harvesters outputs. The nonlinear normal form is first derived and validated with numerical predictions in order to characterize the type of instability for various cross-section geometries. The results show that a very good agreement is obtained between the normal form solutions and numerical predictions near Hopf bifurcation. It is also shown that, for well-defined values of wind speeds, both the transverse displacement amplitude and the generated voltage are increasing with the electrical load resistance. On the other hand, there is an optimum value of the electrical load resistance, which varies with the wind speed, at which the levels of the harvested power are maximized.

Comparative modeling of low-frequency piezomagnetoelastic energy harvesters

In order to design efficient low-frequency piezomagnetoelastic energy harvesters, an effective analytical model which considers the effects of the electromechanical coupling and the nonlinear magnetic force is required. In this paper, the harvester consists of a partially covered piezoelectric cantilever beam with a fixed magnet mass at the top of the magnet tip mass. A nonlinear distributed-parameter model based on Euler–Bernoulli beam theory and Galerkin discretization is developed. The used mode shapes in the Galerkin discretization take into account the fact that the magnetic force and the piezoelectric sheet do not cover the whole beam. In addition, we develop an approximated distributed-parameter model that is based on the classical mode shapes of a fully covered piezoelectric cantilever beam in the Galerkin discretization. These distributed-parameter models are compared with a lumped-parameter model and experimental measurements. The results show that the derived distributed-parameter model accurately predicts the experimental measurements and particularly the accompanying softening behavior. On the contrary, it is demonstrated that the approximated distributed-parameter and lumped-parameter models give erroneous predictions of the resonance region, the level of the harvested power, and the softening behavior. In order to investigate the effects of the load resistance and the softening behavior on the performance of the harvester, a parametric study based on the analytically validated model is then performed. The results show that the presence and importance of the softening behavior depends on the electrical load resistance. It is also demonstrated that the presence of the softening behavior plays an important role in the short- and open-circuit configurations.