Simulation and analysis of the aeroelastic-galloping-based piezoelectric energy harvester utilizing FEM and CFD

Due to a large oscillation amplitude, galloping can be an admissible scenario to actuate the piezoelectric-based energy harvester. In the case of harvesting energy from galloping vibrations, a prismatic bluff body is attached on the free end of a piezoelectric cantilever beam and the oscillation occurs in a plane normal to the incoming flow. The electrical power then can be extracted from the piezoelectric sheet bonded in the cantilever structure due to the dynamic strain. This study is proposed to develop a theoretical model of a galloping-based piezoelectric energy harvester. A FEM procedure is utilized to determine dynamic characteristics of the structure. Whereas the aerodynamic lift and drag coefficients of the tip bluff body are determined using CDF. The results show that the present method gives precise results of the power generated by harvester. It was found that D-section yields the greatest galloping behavior and hence the maximum power.


Introduction
The lifespan limitation of small batteries is one of the main challenges in the standalone systems/self-power devices application such as a wireless sensor network [1]. Structural vibration is one of the most used mechanical domain which enables to be converted into electrical field by utilizing several well-known energy harvester. Piezoelectric-based vibration energy harvester has been used as favorable transduction mechanism by many researchers due to its simplicity, efficient and wide range operational frequencies. There are many studies have been published aiming the investigation of piezoelectric energy harvester as an electronic devices power supply, either as redundant charging system or primary power supply [2,3,4,5]. There are increasing research interest of aeroelastic-based energy harvesters including vortex-induced-based and galloping-based energy harvesters. Galloping is one of the classical aeroelastic instabilities phenomena which can be characterized as a low-frequency and large-amplitude oscillation. As an aeroelastic phenomenon, galloping is generated by a coupling between the forces of aerodynamic, elastic and inertia. Because causing a large oscillation amplitude, galloping can be an admissible scenario to actuate the piezoelectric-based energy harvester. In the case of harvesting energy from galloping vibrations, a prismatic bluff body is attached on the free end of a piezoelectric cantilever beam and the oscillation occurs in a plane normal to the incoming flow. The electrical power then can be extracted from the piezoelectric sheet which bonded in the cantilever structure due to the dynamic strain. Figure 1 shows a prismatic bluff body tip mass having a D-, square, or triangular -section is mounted at the tip of a flexible cantilever beam. When the bluff body is facing an incoming wind flow, then the occurring of wind-induced vibration causes the box to gallop in the transverse direction, leading to self-excited bending vibrations in the beam. Energy is harvested due to the motion of the piezoelectric sheet bonded in the cantilever beam, whose terminals are connected across a load resistance.  Figure 2 shows a section which is exposed to an air flow with a mean speed U and undergoing both transverse and angular displacements denoted by w and θ, respectively.

Aerodynamic model
The generated lift (L) and drag (D) forces are then formulated as where ρ is the air density , btip is the bluff body section width, ẇ is the transversal velocity, and CL and CD are the lift and drag coefficient, respectively. The aerodynamic analysis is conducted for determining CL and CD with the effective angle of attack of α.
On the tip body the effective angle of attack is given by Projection of the lift and drag components along the vertical direction gives the excitation force Ftip in the downward as and Ltip is length of bluff body.
Assuming small values of the velocity ratio U w /  , the force coefficient can be expanded in powers of the angle of attack as follows  In which EI = EbIb + 2EpIp is the flexural rigidity and ρA = bb(ρbtb + 2ρptp) is the mass per unit length of a bonded Beam-Piezoelectric Element. In the present study suffix 'b' is used for regular beam element, suffix 'p' is used for piezoelectric element. If w indicates the vertical displacements of the beam,   M and   K are global mass and stiffness matrices of the structure, respectively, then the equation of motion for free vibration of entire structure is

Structural Analysis
Solving Equation (7) gives natural frequencies, ωn, and mode shapes, ϕ, of the system.

Equation of motion
The coupled matrix differential equations for a piezoelectric energy harvester, namely electromechanic equation, can be formulated with respect to the generalized coordinate u and the voltage generated by the piezoelectric sheets V as where M and K are matrices of generalized mass and generalized stiffness of the system, respectively. Whereas Θ is the electromechanical coupling coefficient defined as Θ = -E p d31 bp (tp + tb)/2, where d31 is strain coefficient of the piezoelectric. In Equation (9 the equations of motion (Equations (8) and (9)) can be rewritten as     Equation (10) can be solved using standard numerical method to obtain time response of displacement, velocity and electrical charge of harvester. Table 1 shows the geometrical and material properties of cantilever beam and piezoelectric which are applied in this study. A tip mass of 0.08 kg was used for generating galloping. The CFD analysis was performed to obtain the variation of lift and drag coefficients (CL and CD) at angles of attack ranging from -45º to 45º and at Reynolds number of 10800.  The natural frequency of the present energy harvester was found to be 21.90 Hz. Figure  3 shows a comparison of measured and predicted output voltage when air speed 4.47 m/s and load resistance 37 kΩ for a triangular cross section. It can be seen that the trend of the predicted voltage closely approximates that of the measured voltage and the steady-state behavior correlates very well with predictions. Furthermore, this method is applied to predict the characteristic of energy harvester with bluff body of D-and rectangular section. Figures 4a and 5a show samples of the velocity field distribution for Reynolds number of 10800 obtained using CFD to determine CL and CD for Dand rectangular section, respectively. Having been CL and CD determined the aerodynamic coefficient Cy can be calculated. The approximation of Cy then can be conducted utilizing Equation (4). Table  2 shows the constant coefficients for approximate Cy for Dand square section. Meanwhile, the comparison of calculated and approximated of Cy are shown by Figure  4b and 5b. Figure 6 and 7 show the real part of first eigenvalue and the time history of the output voltage plots for the D-and square section for a load resistance of 37 kΩ and an air speed of 4.47 m/s. It can be noted that the galloping onset for D-and square section are 1.85 m/s and 3.05 m/s, respectively.

Conclusions
A piezoelectric energy harvester, based on the concept of aeroelastic galloping was developed in this work. A finite element model was formulated to evaluate the dynamic characteristics of the system. Meanwhile, the lift and drag coefficients of the galloping section are evaluated using CFD. Of the geometries studied, the D-section was found to yield the lowest galloping onset and the greatest galloping behavior and hence the maximum power. Meanwhile, the shortest transient time response was found by the square section of the bluff body. The proposed design is considered to offer a viable solution for a standalone power source for remote wireless sensors.