Harvesting Vibration Energy and Wind Energy by a Bi-stable Harvester: Modeling and Experiments

In realistic environments, there often appears the concurrence of base excitation and blowing wind. Harvesting both vibration energy and wind energy by an unique harvester is attractive. In this paper, we proposed a harvester integrating bi-stability and galloping to realize this aim. The nonlinear dynamical model of the bistable energy harvester under concurrent wind and base excitations is established. The galloping effects on the responses are explored based on the established model, for both harmonic and random excitations. The corresponding experiments are conducted to validate the theoretical prediction. The experimental results are consistent with the simulation results. At a wind speed of U =2 m/s, the bandwidth of large-amplitude inter-well motion of the bi-stable energy harvester is extended by about 18.5%. The critical random excitation level for snap-through is reduced by 58% and the total output voltage at random excitation is increased by 53.4%. Thus, the harvester could scavenge the wind and vibration energies at a high efficiency. These conclusions could be helpful for improving the harvesting efficiency in the real environment.


Introduction
At present, the general solutions for powering wireless sensors are dependent on electrochemical batteries. However, this may bring about some derived problems, such as manual maintenance costs and unavoidable environmental pollution.
Vibration, as a common ambient source, is ubiquitous in the nature. Vibration energy harvesting (VEH), which converts ambient vibration energy to electrical energy, has received a significant amount of attention [1][2]. Many earlier works focused on the scheme exploiting the linear resonance. Unfortunately, the response's amplitudes of such designs will drop significantly if the external exciting frequency does not match with the natural frequency of the structure [3][4][5]. As is well known, the ambient vibration has the feature of weak intensity and broad bandwidth. Therefore, increasing the harvester's adaptability to the broadband environment excitation is significantly important.
As a typical nonlinear system, the bi-stable energy harvester has received much attention because of its broadband property and high voltage output upon the occurrence of snap-through [18][19][20][21][22][23][24]. Stanton et al. [25] proposed a bi-stable energy harvester based on the magnetic repulsion and characterize its complex nonlinear dynamics behaviors. Erturk et al. [26] presented a bi-stable energy harvester with external attraction magnets. Cao et al. [27] carried out numerical and experimental investigations on a bi-stable energy harvester with two external rotational magnets.
The results verified it was preferred to the linear and mono-stable energy harvesters from harmonic base excitation.
Since ambient sources may influence the behavior of energy harvester, some researchers investigate the response of energy harvesting systems under non-harmonic excitation [28,29]. The ambient vibration is usually described as a random process, which has a wide spectrum. Cottone et al. [30] theoretically modeled a bi-stable energy harvester by considering the axial load and experimentally investigated the response under random excitations. As the axial force exceeds the critical buckling load of a beam, the system generated more electrical energy. Zhao and Erturk [31] investigated the relative advantage of mono-stable and bi-stable energy harvester. If only the level of excitation is above the critical value of inter-well oscillation, the bi-stable energy harvester captured more energy. Fang et al. [32] proposed an asymmetric plucking-based bi-stable energy harvester with rotary structure and plectrum, to convert the impulsive excitation to plucking force that helps the harvester jump into the high-energy orbit.
In the nature, the flowing river and air own a huge amount of kinetic energy, transforming which to electric energy has a realistic significance. Galloping energy harvester has attracted a great deal of attention, because it can oscillate dramatically if the speed of wind exceeds the critical onset one [33][34][35][36][37][38][39]. Dai et al. [40,41] developed a distributed-parameter model of an electromagnetic galloping energy harvester, and the effect of magnet displacement and electromagnetic coupling is investigated by a parametric analysis. Zhou et al. [42] proposed a Y-shape bi-stable energy harvester to scavenge the low-speed wind energy.
To improve the harvesting efficiency, the energy harvesters under the hybrid excitations of galloping and base motion are investigated by some researchers [43][44][45][46].
Most researches related to galloping-based energy harvesters considered linear stiffness, and up till now, few works devoted themselves to investigate the interaction between the base excitation and galloping [46,47]. Zhao et al. [48] introduced the impact into the design of aeroelastic energy harvesters, and the results demonstrated a broadened bandwidth for concurrent wind and vibration energy harvesting. In fact, vibration energy harvester in the real environment is inevitably disturbed by the flow of air, but the effect of galloping on the bi-stable energy harvester under random excitation is still not clear. Thus, exploring the effect of galloping on the bi-stable energy harvester will provide new insight and improve energy harvesting performance.
In this paper, the effect of galloping on the bi-stable energy harvesting system has been investigated. The rest of this manuscript is organized as follows. Section 2 provided the process of modeling. In Section 3, numerical simulations are carried out to compare and analyze the response under base excitation solely and combined base excitation and galloping. In Section 4, experimental verification is performed at different excitations to study the effect of galloping on bi-stable energy harvesters.
The result obtained from the experiment accords with the numerical simulation, which demonstrates the positive effect of galloping on a bi-stable energy harvester.
Summary and conclusions are presented in Section 5.

Designing and Modeling
As shown in Figure 1, the bi-stable piezoelectric energy harvester is composed of a steel cantilever beam of length L and a piezoelectric layer (Macro-fiber composites: MFC), which is characterized by length p L , width p where a  is the air density, B L and B D are the length and diameter of the bluff body, respectively. U is the incoming wind speed, i a (i=1, 2,3) are the empirical coefficients, which can be obtained by a polynomial fitting for the total aerodynamical force in the cross-flow direction and the tangential direction of angle of attack. w represents the transverse displacement, and "·" represents the derivative with respect to time t. The total kinetic and potential energies can be given by  The magnetic fields acting on dipole A by dipoles B and C can be given by where 7 0 is the magnetic permeability constant. 2 . and  denote the Euclidean norm and vector gradient operator, respectively. According to the geometrical orientation, the vectors directed from the fixed magnets B and C to the tip magnet A can be expressed as Then, the potential energy of magnetic field can be written as [34] Supposing that the first mode is dominant in the Galerkin expansion, the transverse displacement can be expressed as where q is the generalized time-dependent modal coordinate, and () x  is the first mode shape of the beam and can be given by ( ) 1 cos 2 To derive the dynamical equation of the system, Lagrangian equations are employed, which has the form as follows: is the Lagrangian; F is the nonconservative force, which includes two parts, the galloping excitation galloping F and the damping force  is the natural frequency. Q is the electric charge of piezoelectric layer, and the electric current passing through the resistive load R is expressed by The nonlinear model of the piezoelectric energy harvester subjected to concurrent galloping and base excitations can be derived to give where K is the equivalent stiffness. m F is the nonlinear magnetic force, which can be obtained by the formula as m Ux  .  is the equivalent electro-mechanical coupling coefficient. N represents the coefficients of the first mode of external force due to the base excitation. p C is the capacitance of piezoelectric materials (MFC).
These coefficients can be evaluated by the following formulas: Specifically, the nonlinear magnetic force m F can be approximated by the Taylor series To prove the accuracy of Eq. (11), in Figure 2 (10) can be rewritten as

Simulations and analysis
In this part, the numerical simulation is carried out to investigate the influence of galloping on the response of bi-stable energy harvester under harmonic and random base excitations, respectively.

Harmonic excitations
First, the base excitation is assumed to be a harmonic one, i.e., For comparison, at first no wind is blowing, the structure is excited only by the base motion, the system oscillates around all equilibria and executes a large-amplitude periodic motion, until the excitation frequency reaches 11.4 Hz (Figure 3(a1)). Then, with the frequency increasing, the system returns to oscillate around one equilibrium.
The spectrum from 5 Hz to 11.4 Hz indicates that there are the fundamental frequency and high order harmonic components (see Figure 3(a3)). For the wind speed of U=2 m/s, the system exhibits a broad bandwidth of 7.5 Hz (5 Hz~12.5 Hz), as shown in Figure 3(b1-b3), and there appears a chaotic motion at f=13 Hz. Then, at U=2.5 m/s, as shown in Figure 3(c1-c3), the range of frequency for chaos is extended to 18 Hz, so the voltage generated in this range is enhanced. When the wind speed increases to U=3.6 m/s, both the inter-well chaos response and intra-well response in the high-frequency range evolve into a quasi-periodic response, which can be verified from the diagram of bifurcation, as shown in Figure 3(d1). Thus, it can be concluded that the introduction of wind and increase in wind speed are helpful for extending the bandwidth of inter-well oscillation. Hz and 18 Hz exhibit the intra-well periodic motions, as shown in Figure 4(a1-a4).
As the wind speed increases to 2 m/s, the system executes inter-well chaotic motions, and it can be seen from Figure 4(b) that there is a strange attractor in the phase portraits, and some regular points appear in the Poincaré section. Then, as the frequency increase to 15 Hz, an intra-well period-5 response happens, as shown in    Figure 5(a, b, c), the vibration energy of periodic-1 response is concentrated to the integer multiple of the excitation frequency, while that of periodic-2 response is at one half of the integer multiple of excitation frequency, as  Next, three excitation levels, 0.25 g, 0.35 g and 0.45g are chosen so as to show the influence of base excitation level on the response. When the excitation acceleration is set to Ab=0.25g, as shown in Figure 6(a1-a3), the forward frequency sweeping activates the periodic large-amplitude inter-well oscillations from the beginning of the frequency range, i.e., 5~12 Hz. As the frequency increases further, this largeamplitude oscillation gives way to the intra-well periodic oscillations. As Figure   6(b1-b3) shows, if the base acceleration increases to Ab=0.35g, the inter-well branch of oscillations exists over a wider range of frequencies, compared to that of Ab=0.25g.
When the acceleration is further increased to Ab=0.45g, the input energy could lead the system to cross the potential barrier more easily. Thus, a more wider frequency range of inter-well motion, ranging from 5 Hz to 14 Hz, is formed, and more output is generated, as shown in Figure 6(c1-c3).

Random excitations
In fact, the ambient vibration always demonstrates non-stationary and broadband characteristics. Thus, to show its broadband characteristics more clearly, the proposed energy harvester is excited by random motions. A band-limited random excitation is obtained from a random test, as shown in Figure 7. The frequency of the random excitation is assumed to be uniformly distributed over a bandwidth of 5~50 Hz.    The RMS voltages for these two cases are 3.2 V and 4.8 V, respectively. Then, as D increases to 0.710 -3 g 2 /Hz, the jumping does not happen in the case without wind, i.e., U=0 m/s, and the output voltage is quite small. Then, if a galloping is introduced, e.g., U=2 m/s, the response demonstrates a jump between the adjacent potential wells and produces a large-amplitude displacement. Thus, a high output voltage is generated due to frequently jumping between the adjacent potential wells (see Figure 9(c2, d2)).
The advantage of introducing galloping can be seen clearly from the comparison of RMS output voltages for the system for two cases of U=0 m/s and U=2 m/s, 0 RMS U V   4.7 V for the case without galloping and 2 RMS U V   9.8 V for the case with galloping, as shown in Figure 9(b2, d2). Finally, as D increases to 1.710 -3 g 2 /Hz and 2.510 -3 g 2 /Hz, a nearly regular jump between the adjacent potential wells, or coherence resonance, appears at both U=0 m/s and for U=2 m/s. But their amplitude is different. The galloping effect could lead the system to reach a frequently jumping between two potential wells (see Figure 9(c3, d3, c4, d4)), while for U=0 m/s the jumping seldom happens and the output voltage is relatively small (see Figure 9(a3,   b3, a4, b4)). Therefore, compared to the system without galloping effect, a regular large-amplitude response is produced when it is subjected to the combined base excitation and galloping, and a denser high voltage is generated for the same level random excitation. It should be noted that the positive effects of galloping cannot be measured by peak-peak voltage if the random excitation level is large. Nevertheless, the advantage of hybrid excitation can be characterized clearly from the RMS output voltages for the two situations, 0 RMS U V   10.9 V for the case without galloping and 2 RMS U V   14.2 V for the case with galloping effects, as shown in Figure 9(b4, d4). Figure 9. Simulated time histories of displacements and voltage. (a1, b1) 0.310 -3 g 2 /Hz for U=0 m/s, (a2, b2) 0.910 -3 g 2 /Hz for U=0 m/s, (a3, b3) 1.710 -3 g 2 /Hz for U=0 m/s, (a4, b4) 2.510 -3 g 2 /Hz for U=0 m/s. (c1, d1) 0.310 -3 g 2 /Hz for U=2 m/s, (c2, d2) 0.910 -3 g 2 /Hz for U=2 m/s, (c3, d3) 1.710 -3 g 2 /Hz for U=2 m/s, (c4, d4) 2.510 -3 g 2 /Hz for U=2 m/s.

Experimental findings
To verify the advantage of galloping predicted by the simulation, the validation experiment is conducted. The experimental setup is shown in Figure 10. In the experiment, the harvester consists of a steel cantilever beam, with a D-sectioned prism  In the experiment, the base excitation was designed to sweep from 5 Hz to 25 Hz in both forward and backward directions. Figure 11 shows the experimental transient responses under a sweeping base excitation of 0.25 g, while the wind speeds vary at U=0 m/s, U=2 m/s, U=2.5 m/s and U=3.6 m/s, respectively. In Figure 11, the solid blue line represents the forward sweeping response, while the solid red line represents the backward one. The system responses at four wind speeds all exhibit the hysteretic frequency characteristic, due to which they bend toward the right. At U=0 m/s, i.e., there is no wind, as shown in Figure 11(a1, b1), the forward sweeping frequency activates a large-amplitude oscillation over the range between 5 Hz and 10.4 Hz. Then, as the frequency increases, the high-energy solution will disappear and give way to the low-energy oscillation around the stable equilibrium. As for the backward sweeping, the bandwidth for high-energy branch becomes narrower compared to the one of the forward sweeping, due to the hardening nonlinearity. Next, introducing the wind of U=2 m/s, the bandwidth of inter-well oscillation is extended to a wider range of frequency 6.4 Hz (from 5 Hz to 11.4 Hz), as shown in Figure 11(a2, b2).
Compared to the frequency response at U=0 m/s, the bandwidth of large-amplitude inter-well oscillation at U=2 m/s is extended by about 18.5%. Then, as the wind speed increases to U=2.5 m/s, the chaotic response appears and can keep over a wider range of frequency from 7.5 Hz to 18 Hz, and the system produces a large output voltage.
Finally, as the wind speed reaches U=3.6 m/s, the bandwidth is extended to 12.5 Hz, and the intra-well oscillations in the high-frequency range are turned into the inter-well ones, thereby having a large amplitude. The forward sweeping experiment results are in good agreement with the simulations shown in Figure 3. Experimental results match well with the numerical results in jump down frequency and broadband inter-well band-width. The results demonstrate that as the wind speed increases, the area of less desirable small-amplitude intra-well branch will shrink, while that of the inter-well branch will extend. Therefore, it is experimentally validated that introducing galloping can extend the bandwidth of snap-through motion in the frequency domain such that it can harvest more energy for the wideband excitation. In the random validation experiments, a band-limited random excitation is set in the controller with a bandwidth of 5 Hz~50 Hz. Its intensity is varied from 0.310 -3 g 2 /Hz to 2.510 -3 g 2 /Hz sequently. Figure 12 compares the statistic characteristic of dynamics behaviors for U=0 m/s and U=2 m/s, respectively. In Figure 12(  while that for the case with galloping is 4.6 V. When the excitation increases to D=0.910 -3 g 2 /Hz, the voltage response at U=2 m/s realizes a sharp increase and the RMS voltage reaches 10 V, while the RMS voltage for the case without galloping is only about 5 V. As the excitation increases to the higher level, e.g., D=1.710 -3 g 2 /Hz and D=2.510 -3 g 2 /Hz, a large-amplitude peak-peak voltage can be observed in both cases of U=0 m/s and U=2 m/s. But at U=2 m/s, the large peak-peak voltage appears more frequently. The simulated and measured RMS voltages at 12 excitation levels are listed in Table 2. The maximum error of RMS voltage is -13%, which is mainly caused by the random error in measurement. By summing the voltage under all excitation levels in Table 2, the total output voltage for all random excitation levels can be obtained, which is increased by 53.4%.  The corresponding diagrams of power spectral density are plotted in Figure 13. As shown in Figure 13(a), at D=0.310 -3 g 2 /Hz, the PSDs for both the cases of U=0 m/s and U=2 m/s are relatively low. Then, in Figure 13(b), the excitation increases to D=0.910 -3 g 2 /Hz, since the response at U=2 m/s executes the snap-through motion, the PSD for U=2 m/s is obviously larger than the one for U=0 m/s in the low-frequency range. Furthermore, as the excitation increases to D=1.710 -3 g 2 /Hz, the difference in PSDs for U=0 m/s and U=2 m/s becomes trivial, as shown in Figure   13(c). Finally, as the excitation increases to D=2.510 -3 g 2 /Hz, the galloping effect becomes strong and could lead to the appearance of higher peaks in the low-frequency range, as shown in Figure 13(d).

Conclusions
In this paper, a bi-stable configuration is proposed to harvest both vibration energy and wind energy. From the simulation and experimental analyses, we can see that in this configuration the galloping and vibration effects could enhance each other and reach a high harvesting efficiency. From the analyses and experiments, following conclusions can be drawn.
(1) For the bi-stable energy harvester, if it is under the combined excitation of wind and harmonic vibration, the introduction of galloping could lead the large-amplitude inter-well motion more easily to be triggered in the high-frequency range. The bandwidth for the inter-well nonlinear dynamical behaviors, e.g., the periodic, quasi-periodic or chaotic motion, will be extended with the increase of wind speed.
(2) Under the combined excitation of wind and random vibration, the introduction of galloping could decrease the critical excitation level for snap-through by 58%. Thus, the bi-stable energy harvester can execute snap-through from a low-level ambient excitation.
(3) The validation experiment results exhibit a good agreement with those predicted from theoretical analysis and simulations, confirming the advantage of the bi-stable energy harvester in harvesting wind and vibration energies. As a result, the total output voltage for all random excitation levels can increase by 53.4% averagely.

Figure 1
Con guration of proposed energy harvester subjected to concurrent wind and base excitation.              please see the manuscript le for the full caption