A broadband bi-stable flow energy harvester based on the wake-galloping phenomenon

Converting flow energy into electricity using piezoelectric galloping flow energy harvesters (FEHs) has received significant attention in the last decade.^1–3 Due to its inherent simplicity and scalability, this concept offers an excellent alternative to the traditional rotary-type turbine, especially for small-scale applications such as those related to powering wireless sensor networks.

As shown in Fig. 1, a piezoelectric wake-galloping FEH consists of a mechanical oscillator coupled to an energy harvesting circuit through a piezoelectric element. The oscillator is placed in the downstream of a bluff body or an obstacle. When an initially laminar fluid flows past the obstacle, it undergoes symmetry breaking in the form of a Von Kármán vortex street shedding from the trailing edge. The periodic nature of the shed vortices induces alternating pressure forces which, in turn, generate a periodic lift on the mechanical oscillator.

Maximum energy transfer from the flow to the oscillator occurs in a frequency bandwidth known as the lock-in region where the vortex shedding frequency is close to the natural frequency of the oscillator. The shedding frequency is a function of the flow speed and the Strouhal number, which depends on the geometry of the bluff body and the Reynolds number of the flow. In the lock-in region, the oscillator can undergo large-amplitude motions that can be transformed into electricity using the piezoelectric transduction element.

Typically, wake-galloping FEHs are designed to possess a linear restoring force.^4,5 As such, they have a very narrow frequency response bandwidth and are not expected to respond well to the broad range of shedding frequencies normally associated with a variable flow speed. To enhance their response bandwidth under varying flow speeds, we exploit designing the harvester such that it has a bi-stable restoring force. A bi-stable potential energy function consists of a two potential wells (stable nodes) separated by a potential barrier (unstable saddle). Consequently, when the harvester interacts with the vortical structures, it can either perform small-amplitude resonant motions within a single potential well (intra-well motion); or large-amplitude non-resonant motions between the two potential wells (inter-well motion). The inter-well dynamics allow the oscillator to couple to the excitation over a wider range of flow speeds.

While this concept has been previously used by many researchers to enhance the performance of vibratory energy harvesters,⁶ it has never been used for wake-galloping FEHs. To illustrate the enhanced performance, we construct an experimental setup following the schematic diagram shown in Fig. 2. A 95 × 12.5 × 0.2 mm³ Stainless Steel beam is laminated onto a microfiber composite (MFC) patch. The cantilever beam is attached to a 25 × 25 × 50 mm³ square-sectioned bluff body at the free end. A rectangular-sectioned obstacle of characteristics width, D = 42.5 mm, is placed at a distance of 114.8 mm in the upstream of the harvester in the wind tunnel to generate the Von Kármán vortex street. The wind tunnel is an open-loop Aerolab Educational Wind Tunnel with test section dimensions of 305 × 305 × 610 mm³ and maximum upstream turbulence levels of 0.2%. The wind velocity was measured using hot-wire sensors both in the downstream (distance of 76.2 mm from the obstacle) and upstream of the obstacle. We noticed that the only sensor which has significantly different velocity reading is the one placed right behind the obstacle. The average of the other sensor readings was used to measure the laminar velocity.

The bi-stable restoring force is created by using two repulsive magnets, A and, B, placed at a distance h as depicted in Fig. 2. The shape of the potential energy function is altered by changing the distance between the two magnets. For a small distance, $h<25 mm$⁠, the repulsive forces between the magnets are large enough to cause the beam to buckle in the Y-direction, which, in turn, produces a bi-stable potential function.

Using the experimental setup depicted in Fig. 3, we compare the response of the linear harvester (without the lower magnet) which has a stiffness of 74.9 N/m and the bi-stable harvester as the wind speed is increased quasi-statically from 0 to $10 m/s$⁠. Both of the steady-state deflection obtained using a laser vibrometer with a resolution of 30μm and root-mean-square (RMS) voltage across a resistive load, $R=100 KΩ$⁠, are recorded.

As shown in Fig. 4(a), the linear harvester exhibits the typical linear lock-in phenomenon where large-amplitude responses occur when the vortex-shedding frequency is close to the natural frequency of the linear harvester. This occurs in a small region approximately between $3 and 5.5 m/s$⁠. As shown in Fig. 4(b), the lock-in phenomenon remains unchanged for the backward sweep illustrating the non-hysteretic nature of the linear harvester.

For the bi-stable system, we notice that the harvester exhibits small-amplitude oscillations up to approximately $1.8 m/s$⁠. Below this speed, the dynamic trajectories remain confined to a single potential well. Near $1.8 m/s$⁠, the harvester jumps to a larger branch of solutions and starts performing large-amplitude periodic inter-well oscillations. These desirable oscillations persist up to approximately $6.2 m/s$⁠, where a jump to the small-amplitude branch of intra-well oscillations occurs. In the region where the bi-stable harvester performs inter-well oscillations, it outperforms the linear harvester. The backward sweep shown in Fig. 5(b) demonstrates similar behavior with the main difference that the jump from small to large-amplitude motion occurs at lower speeds near 5.6 m/s, illustrating the hysteretic behavior of the nonlinear bi-stable harvester.

When inspecting the associated power curves depicted in Figs. 5(a) and 5(b), we clearly observe the enhanced bandwidth associated with the bi-stable response in both directions of the frequency sweep. However, near the lower end of frequencies, the large-amplitude responses associated with the tip deflection are not as prominent. This can be attributed to the fact that the output power is directly proportional to the product of the deflection and the square of the response frequency. As shown in the inset of Fig. 4, when the dynamic trajectories escape a potential well, the effective stiffness of the harvester drops substantially causing the effective frequency to drop. This causes the output power to drop near the lower end of wind velocities. Nevertheless, as the deflection starts to increase with the flow rate, the effective stiffness of the inter-well dynamics increase again due to the hardening nonlinearity. This causes the frequency, and, thereby, the power to increase.

To better understand the response of the bi-stable harvester, we develop a lumped-parameters mathematical model of the system following Fig. 1. In the proposed model, we assume that the dynamics of the piezoelectric beam can be captured by a harmonic oscillator with an effective mass M, a restoring force, $dVdy$⁠, and an effective damping coefficient, C. The dynamics of the oscillator may be used to represent the beam's tip deflection near its first modal frequency as long as the vortex shedding frequency is close to the first modal frequency of the beam, and the first mode is not in an internal resonance with any of the other vibration modes.

When the beam represented by the oscillator is placed in the wake of the bluff body, it undergoes harmonic oscillations in the cross-flow direction. This strains a piezoelectric element of capacitance, C_p, which, in turn, generates a voltage difference, V, across a resistive load, R. The equations governing the dynamics of the system can be written as

The model of the vertical lift force, F_y, depends on the nature of instability which excites the harvester. This, in turn, depends on the type of the flow and the separation distance between the obstacle and the harvester's bluff body.⁷ In our experimental results, shown in Figs. 4 and 5, the separation distance between the two bluff bodies, $x/D=2.7$⁠, is such that a combined wake galloping and galloping instabilities are activated. Specifically, the deflection curve exhibits the typical resonance-type behavior associated with wake galloping for a certain range of wind velocities roughly between 0 and 6 m/s. Subsequently, it exhibits a monotonic but very slow increase in magnitude due to the pure galloping instability. In the adopted model, we decided that this monotonic increase is very small and will only serve to complicate the dynamics. As such, we used a simple uncoupled single-frequency forced model to capture the wake-galloping phenomenon such that $Fy=12ρU2DCF sin(2πStUDt)$⁠.⁸ Here, ρ is the density of the flow per unit length, U is the speed of the mean flow, D is the cross-flow dimension of the bluff body, C_F is an empirical dimensionless lift force coefficient which depends on the geometry and aspect ratio of the bluff body, and S_t is the Strouhal number which can also be determined empirically. For simplicity, in this letter, the Strouhal number (for both systems) is assumed to be invariant with the Reynolds number.

To validate the model, several of the unknown model parameters were experimentally identified. First, the mechanical damping C is measured using the logarithmic decrement method at $0.016 kg/s$⁠. The electromechanical coupling, θ, and the effective capacitance of the MFC layer, C_p, are determined by performing a resistive load sweep at a constant wind velocity and measuring the associated electrical power at each resistance. The electromechanical coupling, θ, is found to be 1.05 × 10⁻⁴ N/V, while the effective capacitance is found to be 84 nF. Finally, the restoring force is obtained experimentally by measuring the force as a function of the tip deflection of the beam using a digital dynamometer. The resulting data is then fit into a polynomial function using a least-square approximation. The best fit yielded a quartic potential function of the form $V=12μy2+14γy4$⁠. For the rectangular obstacle used in the experiment, the lift coefficient, C_F, and Strouhal number, St, can be empirically obtained as $CF=1.21$⁠, and St = 0.13.⁸ 

Figure 6 depicts a comparison between the numerical simulations obtained by solving Equations (1(a)) and (1(b)) using the method of multiple scales⁹ and the experimental measurements for both of the steady-state amplitude and RMS output voltage across a constant resistive load of $100 kΩ$⁠. A quick comparison indicates that there is a good qualitative agreement between the numerical simulations and the experimental data. However, since we neglected the influence of pure galloping in the adopted model, the theoretical results underestimate the experimental data towards the higher end of wind speeds.

The analytical solution graphed in Fig. 6 reveals a more complex picture than initially predicted by the experimental results. Specifically, we notice the presence of multiple co-existing branches of stable (solid lines) and unstable (dashed lines) solutions. The reason that not all of the analytically predicted solutions appear in the experiment stems from two facts. First, some of the available stable solutions have very small basins of attraction, and hence, only a careful selection of initial conditions permits finding these solutions experimentally. Second, in the experiment, the results are limited by the resolution of the wind speed tuner in the wind tunnel, which has a minimum resolution of 0.1 m/s.

To explain the complex behavior of the harvester as observed in Fig. 6, we consider an experiment wherein the wind velocity is quasi-statically increased from zero towards higher wind speed. As shown in Fig. 6(b), the harvester will initially perform small-amplitude intra-well oscillations on the non-resonant branch of solutions, B_n, up to the point, cfA, which represents a cyclic-fold bifurcation in the deflection/voltage versus wind speed parameters space. As a result, the response jumps to one of the adjacent stable periodic solutions. Here, there are two possible adjacent solutions: the small-amplitude intra-well resonant solution, B_r, and the large-orbit inter-well solution, B_L. In the actual experiment, as shown in Fig. 6(a), the response jumps initially to the branch B_r, and continues to follow this branch up to a speed of $1.8 m/s$⁠. At this point, the basin of attraction of the solution, B_r, becomes very small and the harvester's response jumps to the inter-well branch, B_L. Further increase in the wind speed causes the amplitude of the response to increase following the branch B_L up to the point where its basin of attraction becomes too small and the harvester goes back to perform intra-well oscillations on the resonant branch, B_r, beyond U = 5.6 m/s.

Using the qualitatively validated theoretical model, we investigate the influence of the potential shape on the output voltage as shown in Fig. 7(a). The voltage curves reveal that, as the potential shape is varied, the desirable performance criteria, i.e., the minimum wind speed at which the jump to the large-orbit solution occurs and the magnitude of the voltage in the inter-well region has a competing nature. For shallower potential wells and smaller separation distances between the wells, the harvester starts performing large inter-well motions at lower wind speeds, but the resulting inter-well motions are generally smaller. On the other hand, for deeper potential wells and larger separation distances between the wells, the harvester starts performing large inter-well motions at higher wind speeds, but the resulting inter-well motions are generally larger. This demonstrates the presence of an optimal potential shape which can be used to maximize performance for given design parameters.

The authors would like to acknowledge the generous support of the NSF under Grant No. CMMI 1055419.