Muhammad R. Hajj

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.

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.

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.

A Model for the Coupled Lift and Drag on a Circular Cylinder

The time-varying coupled lift and drag coefficients acting on a circular cylinder are modeled. Data used for the model are obtained by numerically solving the unsteady Reynolds-Averaged Navier Stokes equations over a wide range of Reynolds numbers. Using spectral moments, we determine the frequency components in the lift and drag coefficients and their phase relations. Using a perturbation technique, we obtain approximate solutions of both the van der Pol and Rayleigh equations. By fitting the amplitude and phase relations, we find that the van der Pol equation is the suitable model for the lift. The Rayleigh equation fails to give the correct phase relation. Because the major frequency in the drag component is twice that of the lift, the drag component is modeled as a quadratic function of the lift. Through analysis with higher-order spectral moments, the correct quadratic relation of the lift that yields the drag is determined. The model and results presented here are a first step in the development of a reduced-order model for vortex-induced vibrations, which includes the motions of the cylinder.

Energy harvesting from a multifrequency response of a tuned bending-torsion system

We investigate the benefits of tuning the frequencies of an energy harvester to extract more energy from a base excitation that comprises three frequency components. The energy harvester is composed of a unimorph cantilever beam with asymmetric tip masses. By adjusting the asymmetry of the tip masses, we can tune this beam–mass structure to harvest energy from multifrequency components of a base excitation. We model the beam using the Euler–Bernoulli beam theory and use the first three global mode shapes of the harvester in a Galerkin procedure to derive a reduced-order model describing its response. We derive an exact analytical solution for the tip deflection, twisting angle, voltage output, and harvested electrical power. Using this solution, we investigate the advantages of harvesting energy from a response that contains multifrequencies in comparison to a response that contains a single frequency by tuning only the fundamental frequency. The advantages of this bending–torsion energy harvester and the effect of its tuning are investigated for different short- and open-circuit configurations. The results show that, through a proper tuning of this bending–torsion harvester, the harvested power can be increased significantly and it can be made to cover a wide range of electrical load resistances.

Wing Kinematics Optimization for Hovering Micro Air Vehicles Using Calculus of Variation

The weight and power constraints imposed on flapping-wing micro air vehicles necessitate optimal design of the flapping kinematics. To date, the approach adopted for kinematics optimization has been to assume specific functions for the Euler angles describing the wing motion with respect to the body. Then, optimization is performed on the parameters of these functions. In another approach, a number of instants over the flapping cycle are selected, and optimization is performed on the magnitude of the Euler angles at these instants. This latter approach provides more freedom for the variations of the Euler angles rather than confining them to certain patterns. Yet, in both approaches, finite-dimensional optimization is adopted and, as such, additional constraints are imposed. Considering that the problem is an infinite-dimensional optimization problem, we use in this work the calculus of variations to obtain true optimality. The combination of the quasi-steady aerodynamics and the calculus of variations ap...

Piezoelectric energy harvesting from hybrid vibrations

The concept of harvesting energy from ambient and galloping vibrations of a bluff body with a triangular cross-section geometry is investigated. A piezoelectric transducer is attached to the transverse degree of freedom of the body in order to convert these vibrations to electrical energy. A coupled nonlinear distributed-parameter model is developed that takes into consideration the galloping force and moment nonlinearities and the base excitation effects. The aerodynamic loads are modeled using the quasi-steady approximation. Linear analysis is performed to determine the effects of the electrical load resistance and wind speed on the global damping and frequency of the harvester as well as on the onset of instability. Then, nonlinear analysis is performed to investigate the impact of the base acceleration, wind speed, and electrical load resistance on the performance of the harvester and the associated nonlinear phenomena that take place. The results show that, depending on the interaction between the base and galloping excitations, and the considered values of the wind speed, base acceleration, and electrical load resistance, different nonlinear phenomena arise while others disappear. Short- and open-circuit configurations for different wind speeds and base accelerations are assessed. The results show that the maximum levels of harvested power are accompanied by a minimum transverse displacement when varying the electrical load resistance.