Piezoaeroelastic Energy Harvesters Research Articles

Nonlinear modeling, characterization, and effectiveness of three-degree-of-freedom piezoaeroelastic energy harvesters

Modeling and performance investigation is conducted on a three-degree-of-freedom piezoaeroelastic system with freeplay and multi-segmented stiffness with and without impact. A piezoelectric transduction mechanism is considered in the plunge degree of freedom and the aerodynamic loading is modeled with the unsteady representation based on the Duhamel formulation, including the stall phenomenon. The typical aeroelastic section model is comprised of a pitching and plunging airfoil with a control surface containing freeplay and multi-segmented stiffness springs in the pitch and control surface motions. Nonlinear characterization is conducted on the response of the energy harvesting system when changing the nonlinear stall coefficient, freeplay gap size in the pitch and control surface springs, and the multi-segmented nonlinear stiffness to simulate impact. Results show that grazing and grazing/sliding bifurcations may be present, and the response is complex with several transitions as the wind speed is increased. Additionally, the presence of freeplay and multi-segmented nonlinearities allow for energy harvesting at speeds smaller than that of the linear flutter velocity. An effective design is determined for a three-degree-of-freedom wing-based energy harvester by selecting the freeplay gap size and multi-segmented nonlinear stiffness in the pitch and control surface and the electrical load resistance.

Performance investigation of piezoaeroelastic energy harvester with trailing-edge flap

This paper presents a piezoaeroelastic energy harvester fitted with trailing-edge flap, for simulating three degrees of freedom plunge-pitch-flap motions and capturing the aeroelastic vibration energy. The flap is fitted with the trailing-edge of the airfoil, and restricted by the torsion spring rod. The governing equations of fluid-structure-electric coupled fields are derived. The influence of the harvester parameters on the output characteristic is investigated numerically. Experimental prototype of the harvester is fabricated, and the numerical results are validated by the experimental method. The results demonstrate that the lesser the pitch stiffness coefficient is and the larger the plunge and flap stiffness coefficients are, the larger aeroelastic vibration and the better harvesting performance can be obtained. The vibration mode changes from flutter-induced vibration to vortex-induced vibration at the smaller flap damping coefficient. The plunge amplitude of 0.057 m and output voltage of 32.18 V are harvested at 18 m/s. The output power can be up to 2.63 mW at 300 kΩ. The plunge amplitude and output voltage obtained numerically are in well agreement with the experimental values. The designed piezoaeroelastic harvester can drive the lower-power microelectromechanical system, which can promote the field application in microelectromechanical systems.

Enhanced performance of piezoaeroelastic energy harvester with rod-shaped attachments

Energy harvesters capture the ambient vibration to persistently drive lower-power devices employed in wireless applications. For enhancing the harvesting performance, a novel piezoelectric aeroelastic (piezoaeroelastic) energy harvester fitted with different rod-shaped attachments is presented. The attachments include square and D-shape rods, and symmetrically fixed on airfoil surface at chord maximum thickness. The performance comparisons of various harvesters are conducted by using simulation and experimental methods. The results demonstrate that square and D-shape rods change flow field characteristic and enhance aerodynamic loads, which reduce flutter velocity and broaden the working bandwidth. The harvesters with square and D-shape rods demonstrate the better output characteristic in both aeroelastic vibration and harvesting performance than without rod. Output voltage of the harvester with square rod is up to 15.14 V at airflow velocity of 16.84 m/s and plunge stiffness coefficient of 278 N/m, and the corresponding output power is 0.765 mW. A maximum enhancement ratio of output power for square rod can reach to 469.7 % at 11.1 m/s. The power density of the harvester with square rod is 4.78 mW/cm3, which demonstrates the superior harvesting performance over others. This work provides a significant guidance for designing the more efficient piezoaeroelastic harvester with square rod.

Effective design and characterization of flutter-based piezoelectric energy harvesters with discontinuous nonlinearities

A comprehensive study on the design and nonlinear characterization of a two-degree of freedom piezoaeroelastic energy harvesting system with freeplay and multi-segmented nonlinearities in the pitch degree of freedom is explored and discussed. The nonlinear governing equations of the considered piezoaeroelastic energy harvesting system are derived and the unsteady representation based on the Duhamel formulation is employed to represent the aerodynamic loads. Nonlinear piezoaeroelastic response analysis is carried out in the presence of freeplay and multi-segmented nonlinearities before and after the linear onset of flutter. Such nonlinearities can be introduced to piezoaeroelastic energy harvesters for performance enhancement through the possible existence of subcritical Hopf bifurcation and aperiodic responses due to the grazing and grazing/sliding bifurcations. It is shown that the existence of discontinuous effects result in the possibility of harvesting energy at lower speeds than the linear onset speed of instability due to the activation of the subcritical Hopf bifurcation. Additionally, the increase of the strength of the multi-segmented nonlinearities leads to the existence of aperiodic responses with the presence of several bifurcations limiting the system's dynamics at low pitch angles with limiting stall issues. It is proved that an effective design with harvesting energy at low wind speeds can be carried out for wing-based energy harvesters by carefully selecting the linear stiffness of the pitch degree of freedom, gap and type of the multi-segmented discontinuity, and electrical load resistance.

Enhanced performance of airfoil-based piezoaeroelastic energy harvester: numerical simulation and experimental verification

This paper performs the detailed simulation analyses of airfoil-based piezoaeroelastic energy harvester with two degrees of freedom self-induced plunge-pitch motions, for exploring the flow field characteristic and enhancing the harvesting performance. The designing of the harvester for achieving flutter is first accomplished, finite element model is then built and simulation analyses are performed, and a prototype of the harvester system is finally fabricated. The obtained simulation results are in good agreement with the experimental and theoretical values. The effects of the key structural parameters of the harvester on flow field, aeroelastic vibration, and harvesting performance are numerically investigated. The results demonstrated that the structural parameters of the harvester determine primarily flutter onset of velocity, and consequently affect the dynamic behavior and harvesting performance. The harvester system takes place flutter and occurs limit cycle oscillations after flutter onset of velocity, which is suitable for harvesting energy. The smaller pitch structural stiffness coefficient and the more moderate both plunge stiffness and pitch damping coefficients are, the better aeroelastic vibration and output performance can be captured. A maximum output voltage of 29.08 V and output power of 3.382 mW can be harvested when the linear and cubic structural stiffness coefficients in pitch are 2.5 N·m and 100 N·m at 14 m/s, respectively, which corresponds to the power density of 21.138 mW/cm3 and demonstrates the superior harvesting performance over others. This work provides a significant guidance for designing the more efficient airfoil-based piezoaeroelastic energy harvester utilized in unmanned aerial vehicles.

A method for investigating aerodynamic load models of piezoaeroelastic energy harvester

This paper performs a comparative analysis of aerodynamic loads and parametric study into a piezoaeroelastic energy harvester, for effectively harvesting aeroelastic vibration. The governing equations of the harvester system based on quasi-steady representation, unsteady formulation, and dynamic stall model are derived theoretically, simulated numerically and validated experimentally. The theoretical results based on the unsteady formulation are in well agreement with the experimental values. A maximum output voltage obtained experimentally is 20.18 V, and corresponds to the voltage density of 126.13 V/cm3 at 15.16 m/s. The nonlinear analyses based on the unsteady formulation are performed to explore the determined relations between the key structural parameters of the harvester and both the aeroelastic vibration and output performance. When the incoming airflow velocity exceeds the critical flutter one, that is 11.32 m/s, the harvester system could take place the bifurcation and occur limit cycle oscillations, and hence aeroelastic vibration and output performance increase rapidly. The cubic nonlinear stiffness coefficient in pitch exerts a greater influence on the aeroelastic vibration and output performance than one in plunge. The smaller linear and cubic nonlinear coefficients in pitch are, the better aeroelastic vibration and output voltage of the harvester can be achieved, which can be enhanced by an order of magnitude by appropriately choosing structural stiffness coefficients in the designed harvester system.

Modeling and analysis of a coupled novel nonlinear magneto-electro-aeroelastic lumped model for a flutter based energy harvesting system

The present paper investigates the modeling and analysis of a novel aeroelastic energy harvesting system using magneto-electro-elastic (MEE) material considering the nonlinear elastic effects of the structure. The main aim of using this group of composite materials, as compared to common piezoelectric materials used in aeroelastic harvesters, is to increase the harvested electrical power. The proposed model to improve energy harvesting is composed of a rigid airfoil connected to an elastic beam, with the elastic beam covered by one or two layers of MEE material and two electrodes connected to the top and bottom of the MEE layers to harvest the electric potential from the generated electric field. An external coil was also provided around the oscillating beam to induce the electrical energy generated by the magnetic field established within the MEE layer. Firstly, simulating the harvesting system as a discrete model considering bending and torsional cubic nonlinear springs, the governing equations are extracted using the constitutive equations of MEE materials, Gauss’s law, and Faraday’s law of induction through Hamilton’s principle. Next, the effects of different design parameters are evaluated to find their optimal values to maximize the generated power as well as the harvester stability. Finally, a comparison is made between the considered MEE harvester and a piezo-aeroelastic (PA) energy harvester, indicating the superiority of the proposed MEE energy harvester.

Piezoaeroelastic energy harvesting based on an airfoil with double plunge degrees of freedom: Modeling and numerical analysis

In this paper, a piezoaeroelastic energy harvester based on an airfoil with double plunge degrees of freedom is proposed to additionally take advantage of the vibrational energy of the airfoil pitch motion. The analytical model of the proposed harvester is built, and an equivalent model using the well-explored pitch–plunge configuration is presented. The nonlinear aerodynamics is calculated based on the dynamic stall model. The dynamic response, average power output, energy harvesting efficiency, and cut-in speed (flutter speed) of the proposed harvester are numerically studied. It is demonstrated that the harvester with double-plunge configuration outperforms its equivalent pitch–plunge counterpart in terms of both the power output and energy harvesting efficiency beyond the flutter boundary. In addition, case studies are performed to reduce the cut-in speed and to enhance the energy harvesting efficiency of the proposed harvester, including the airfoil mass characteristics, the configuration, mass, damping, and stiffness characteristics of the two plunge supporting devices, and the load resistances in the external circuits. It is shown that the cut-in speed is greatly reduced by increasing the airfoil mass while tuning the mass eccentricity. The mass of the first (windward) supporting device should be a bit smaller than that of the second one for an applicable cut-in speed and a high-energy harvesting efficiency. Besides, the decrease of airfoil mass moment of inertia or the damping of the supporting devices is shown to be beneficial for the energy harvesting performance. In addition, the optimal location of the first supporting device is found to be at the airfoil leading edge. Decreasing the distance between the two supporting devices reduces the cut-in speed. The load resistances affect the cut-in speed slightly, and optimal values are found to maximize the energy harvesting efficiency.

A Preliminary Study on Piezo-aeroelastic Energy Harvesting Using a Nonlinear Trailing-Edge Flap

Recently, piezo-aeroelastic energy harvesting has received greater attention. In the present study, a piezo-aeroelastic energy harvester using a nonlinear trailing-edge flap is proposed, and its nonlinear aeroelastic behaviors are investigated. The energy harvester is modeled using a piezo-aeroelastic model of a two-dimensional typical section airfoil with a trailing-edge flap (TEF). A piezo-aeroelastic analysis is carried out using RL and time-integration methods, and the results are verified with the experimental data. The linearizing method using a describing function is used for the frequency domain analysis of the nonlinear piezo-aeroelastic system. From the linear and nonlinear piezo-aeroelastic analysis, the limit cycle oscillation (LCO) characteristics of the proposed energy harvester with the nonlinear TEF are investigated in both the frequency and time domains. Finally, the authors discuss the air speed range for effective piezo-aeroelastic energy harvesting.

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 Identification of Circular Cylinder-based Piezoaeroelastic Energy Harvesters

In this paper, we develop a validated nonlinear distributed-parameter model for harvesting energy from vortex-induced vibrations. The harvester consists of a piezoelectric cantilever beam with a circular cylinder attached to its end. By using the Euler-Lagrange principle and implementing the Galerkin discretization, a reduced-order model is derived. Based on a five-mode approximation in the Galerkin approach, an identification for the van der Pol wake oscillator coefficients is performed. Further analysis is performed to investigate the effects of the cylinder's tip mass 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 can be optimized to design enhanced piezoaeroelastic energy harvesters from vortex-induced vibrations.

Nonlinear analysis and enhancement of wing-based piezoaeroelastic energy harvesters

We investigate the level of harvested power from aeroelastic vibrations for an elastically mounted wing supported by nonlinear springs. The energy is harvested by attaching a piezoelectric transducer to the plunge degree of freedom. The considered wing has a low-aspect ratio and hence three dimensional aerodynamic effects cannot be neglected. To this end, the three dimensional unsteady vortex lattice method for the prediction of the unsteady aerodynamic loads is developed. A strong coupling scheme that is based on Hamming's fourth-order predictor–corrector method and accounts for the interaction between the aerodynamic loads and the motion of the wing is employed. The effects of the electrical load resistance, nonlinear torsional spring and eccentricity between the elastic axis and the gravity axis on the level of the harvested power, pitch and plunge amplitudes are investigated for a range of operating wind speeds. The results show that there is a specific wind speed beyond which the pitch motion does not pick any further energy from the incident flow. As such, the displacement in the plunge direction grows significantly and causes enhanced energy harvesting. The results also show that the nonlinear torsional spring plays an important role in enhancing the level of the harvested power. Furthermore, the harvested power can be increased by an order of magnitude by properly choosing the eccentricity and the load resistance. This analysis is helpful in designing piezoaeroelastic energy harvesters that can operate optimally at specific wind speeds.

Nonlinear dynamics of galloping-based piezoaeroelastic energy harvesters

The normal form is proposed as a tool to analyze the performance and reliability of galloping-based piezoaeroelastic energy harvesters. Two different harvesting systems are considered. The first system consists of a tip mass prismatic structure (isosceles 30° or square cross-section geometry) attached to a multilayered cantilever beam. The only source of nonlinearity in this system is the aerodynamic nonlinearity. The second system consists of an equilateral triangle cross-section bar attached to two cantilever beams. This system is designed to have structural and aerodynamic nonlinearities. The coupled governing equations for the structure’s transverse displacement and the generated voltage are derived and analyzed for both systems. The effects of the electrical load resistance and the type of harvester on the onset speed of galloping are quantified. The results show that the onset speed of galloping is strongly affected by the load resistance for both types of harvesters. The normal form of the dynamic system near the onset of galloping (Hopf bifurcation) is then derived. Based on the nonlinear normal form, it is demonstrated that smaller levels of generated voltage or power are obtained for higher absolute values of the effective nonlinearity. For the first harvesting system, the results show a supercritical Hopf bifurcation for both isosceles 30° or square cross-section geometries. The nonlinear normal form shows that the isosceles triangle section (30°) is more efficient than the square section. For the second harvesting system, the normal form is used to identify the values of the nonlinear torsional spring which changes the harvester’s instability. It is demonstrated that this critical value of the nonlinear torsional spring depends strongly on the load resistance.

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.

Aeroelastic characteristics of linear and nonlinear piezo-aeroelastic energy harvester

In the present study, a piezo-aeroelastic energy harvester using nonlinear aeroelastic behaviors is proposed, and their characteristics and performance are investigated. The energy harvester is modeled by a two-dimensional typical section airfoil. The nondimensional parameters of the harvester are introduced, and the nondimensional piezo-aeroelastic equations are formulated. For the piezo-aeroelastic analysis, the root-locus method and time-integration method are used, and the present method is verified with experimental and analytical results. The iterative method is introduced to calculate the frequency response functions of a nonlinear piezo-aeroelastic energy harvester. The aeroelastic characteristics of a linear piezo-aeroelastic energy harvester and the effects of parameters are investigated. The results show that the linear piezo-aeroelastic energy harvester can be used to generate electricity only at the vicinity of flutter speed. It is assumed that the nonlinear piezo-aeroelastic energy harvester has free play and cubic hardening in pitch. For free play, nonlinear aeroelastic results show that stable limit cycle oscillations are observed in the wide range of air speed below flutter speed when the frequency ratio is 1.3, and unstable limit cycle oscillations are observed at air speeds over flutter speed when the frequency ratio is 0.3. For cubic hardening, unstable limit cycle oscillations are observed at air speeds below flutter speed when the frequency ratio is 0.3, and stable limit cycle oscillations are observed at air speeds over flutter speed when the frequency ratio is 1.3. Finally, the authors discussed how to use these aeroelastic responses for piezo-aeroelastic energy harvesting.

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.

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.

Energy harvesting under combined aerodynamic and base excitations

This paper investigates the transduction of a piezoaeroelastic energy harvester under the combination of vibratory base excitations and aerodynamic loadings. The harvester which consists of a rigid airfoil supported by nonlinear flexural and torsional springs is placed in an incompressible air flow and subjected to a harmonic base excitation in the plunge direction. Under this combined loading, the airfoil undergoes complex motions which strain a piezoelectric element producing a voltage across an electric load. To capture the qualitative behavior of the harvester, a five-dimensional lumped-parameter model which adopts nonlinear quasi-steady aerodynamics is used. A center manifold reduction is implemented to reduce the full model into one nonlinear first-order ordinary differential equation. The normal form of the reduced system is then derived to study slow modulation of the response amplitude and phase near the flutter instability. Below the flutter speed, the response of the harvester is observed to be always periodic with the air flow serving to amplify the influence of the base excitation on the response by reducing the effective stiffness of the system, and hence, increasing the RMS output power. Beyond the flutter speed, two distinct regions are observed. The first occurs when the base excitation is small and/or when the excitation frequency is not close to the frequency of the self-sustained oscillations induced by the flutter instability. In this case, the response of the harvester is two-period quasiperiodic with amplitude modulation due to the presence of two incommensurate frequencies in the response. This amplitude modulation reduces the RMS output power. In the second region, the amplitude of excitation is large enough to eliminate the quasiperiodic response by causing the two frequencies to lock into each other. In this region, the response becomes periodic and the output power increases exhibiting little dependence on the base excitation.

Temperature impact on the performance of galloping-based piezoaeroelastic energy harvesters

The effects of ambient temperature on the level of harvesting energy from galloping oscillations of a bluff body are investigated. A nonlinear-distributed-parameter model is developed to determine variations in the onset speed of galloping and the level of the harvested power when the ambient temperature is varied. The considered harvester consists of a bimorph piezoelectric cantilever beam with a prismatic-structure tip mass. A modal analysis is performed to derive the exact mode shapes and natural frequencies of the beam–structure system and their dependence on temperature variations. The quasi-steady representation is used to model the aerodynamic loads. The linear analysis shows that the temperature and the electrical load resistance affect the onset speed of galloping significantly. The nonlinear analysis shows that temperature variation affects the level of the harvested power.

Sensitivity analysis of piezoaeroelastic energy harvesters

We perform a sensitivity analysis of a piezoaeroelastic energy harvester consisting of a pitching and plunging rigid airfoil supported by flexural and torsional springs with a piezoelectric coupling attached to the plunge degree of freedom. We employ the nonintrusive formulation of the polynomial chaos expansion in terms of the multivariate Hermite polynomials to quantify the effects of variations in the load resistance, the eccentricity (distance between the center of mass and the elastic axis), and the nonlinear coefficients of the springs on the harvested power and the pitch and plunge amplitudes. As a first step, the normal form of the dynamics of the system near the Hopf bifurcation is used to select parameters that ensure a supercritical instability and maximize the generated power. The results show that the harvested power can be mostly affected by the eccentricity. Moreover, decreasing the nonlinear coefficient of the torsional spring results in a decrease in the pitch amplitude and an increase in the plunge amplitude and hence the harvested power. These results give guidance for optimizing and assessing the uncertainty in the performance of piezoaeroelastic energy harvesters.