A broadband flow energy harvester induced by the wake of a bluff body

Wake-galloping energy harvesting has been extensively developed to scavenge flow energy from vortex-induced oscillations. Hence, the wake-galloping harvester only has a natural frequency which leads to a very narrow bandwidth. Therefore, it does not operate well under the wide region of shedding frequencies in variable wind speed. To overcome the vital issue, this paper we explored a novel two-degree-of-freedom nonlinear flow energy harvester to collect flow energy induced by the wake of a bluff body. The nonlinear restoring force is realized by using a repulsive magnetic force between two cuboid-shaped permanent magnets, and the electromechanical coupling equations are presented. Based on the method of harmonic balance, the electromechanical governing equations are decoupled, and the first-order harmonic solutions are implemented. The modulation equations are established, and the amplitude–frequency figures of displacement and voltage are depicted with different detuning parameters. The superiority of the presented energy harvester is contrasted with the single-degree-of-freedom linear and nonlinear cases, and the results revealed that the two-degree-of-freedom nonlinear scheme can enhance the bandwidth of flow energy capture. The effect of physical parameters on the scavenged power is discussed. The accuracy and efficiency of the approximate analytical data are examined by numerical simulations.


Introduction
Energy scavenging from environmental vibratory energy for the objective of operating lower powered devices has appeared during the last decade as a feasible technology. Since its innate scalability and simplicity, this method offers an advantage alternative to the traditional batteries for microelectronics facilities such as health-monitoring sensors, medical implants and wireless sensor networks. Converting untapped vibration-based energy into electricity by introducing galloping flow energy harvesters has attracted many investigators' attention in recent years. There are many prominent review papers, such as Truitt and Mahmoodi [1], Wang et al. [2], Abdelkefi [3], McCarthy et al. [4], Nabavi and Zhang [5], Rostami and Armandei [6] and Daqaq et al. [7], reporting the status quo and the development of flow energy harvesting.
The vibration induced in elastic structures by vortex shedding is of practical importance since its possibly destructive main effect on engineering structure. Therefore, vortex-induced vibration energy harvesting has received more and more concerns since its unique dynamical characteristics of self-excited and self-restricted oscillations when the natural frequency of the system is near to the vortex shedding frequency. Traditional devices of wake-galloping harvesters only run linearly, which indicates that the operating lock-in range is narrow. Thus, a small difference between the natural frequency and shedding frequency results in a sharp decrease in the harvesting efficiency. Therefore, in order to capture a broad range of wind speed changes, lots of researchers have been done to proposed broadband wake-galloping energy harvesters, for example, nonlinear magnetic restoring force [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22], linear multi-frequency [23], vortex caused vibration and galloping [24][25][26][27], hybrid piezoelectric and electromagnetic transduction [28,29], concurrent flow-induced vibrations and base excitations [30][31][32][33][34][35] and different-shaped attachments [36][37][38][39][40]. Recently, Chen and Jiang [41] proposed the internal resonance energy harvesting for two-degree-of-freedom nonlinear harvester which can improve the bandwidth of energy harvester and capture more power. Then, two-degree-of-freedom energy harvester has been obtained extensively interesting [42][43][44][45][46][47][48][49][50][51][52]. However, two-degree-of-freedom nonlinear wake-galloping harvesting has not been reported. To emphasize the importance of research in this aspect, this paper develops a two-degree-of-freedom nonlinear system to enhance the captured energy from vortex-induced vibrations.
In the paper, we report an innovative effort to propose high-performance flow energy harvester from vortex-induced vibrations by introducing a two-degree-of-freedom nonlinear system and enabling to broad the bandwidth and scavenge more electricity, which is not reported so far. To illustrate this characteristic, a two-degree-of-freedom nonlinear system from the wake of a bluff body is a e-mail: wajiang@ujs.edu.cn (corresponding author) b e-mail: qbi@ujs.edu.cn designed, as depicted in Fig. 1. For avoiding the impact of wind speed on the auxiliary oscillator, a linear spring oscillator is fixed inside the bluff body as plotted in Fig. 2b.

Basic model
The lumped parameters model of a two-degree-of-freedom nonlinear flow energy harvester from the wake of a bluff body is shown in Fig. 1, and the three-dimensional model is sketched in Fig. 2. As it can be seen, the harvester consists of a steel beam with the related geometry dimension 95 × 12.5 × 0.2 × mm 3 . The beam is pasted with a microfiber composite patch in the fixed end and an auxiliary bluff body at the free end. In the mean time, the bluff body is connected internally to a linear spring oscillator. The nonlinear restoring force is generated by employing two repulsive magnets. In addition, a cuboid obstacle is placed in the upstream of the mechanism to produce the vortex street. The dimensional dynamical equations of the coupling system are modeled by where m 1 denotes an effective mass of the piezoelectric beam, x 1 is the displacement of cantilever beam, and k 1 is the linear stiffness of the harvester, c 1 and c 2 define the structure damping, γ is the nonlinear stiffness coefficient, ρ is the fluid density, C Lo is the lift coefficient, D is the width of the bluff body, S t is the Strouhal number, m 2 denotes the mass of attachment oscillator, k 2 is the stiffness of linear spring, υ is the output voltage of the harvesting, θ is the electromechanical coupling value, C p is the capacitance, R is the resistive load.
It is worth pointing out that the main structure of Fig. 2 has been investigated by Alhadidi and Daqaq [53] and Alhadidi et al. [54]. However, they constructed the system by a single-degree-of-freedom nonlinear oscillator. The model presented herein is a new two-degree-of-freedom nonlinear system where an auxiliary linear spring oscillator m 2 is used to couple with the main structure.
In addition, it is worth reiterating here that the model of the lift force depends on the type of the flow and the separation distance between the bluff body and the obstacle. For different separation layouts, different dynamic types can be evoked. The parameters of the model are based on their experimental results that the separation distance between the bluff body and the obstacle is fixed as x/D = 2.7. As such, a simple uncoupled single-frequency forced model is employed as the right-hand side of Eq. (1) that is utilized to capture the wake-galloping phenomenon. It can be noted that the linear wake-galloping system exhibits the typical lock-in

Air flow
Cp phenomenon where large motion responses arise when the vortex shedding frequency 2π S t U/D is close to the corresponding natural frequency of the linear oscillator.

Approximate analysis
We consider the natural frequencies of linear undamped free mechanical system (1)-(3), written as The first and second natural frequencies can be computed as In the following, the dynamic responses of the two-degree-of-freedom nonlinear flow energy energy harvester are considered. The approximate response is adopted by the method of harmonic balance, which can be written as where ω = 2π S t U D . Taking Eq. (7) into electric Eq. (3), the steady voltage response can be computed as Introducing Eqs. (7) and (8) into Eqs. (1) and (2), simultaneously equating the factors of cos ωt and sin ωt to zero, we have From Eq. (8), we have the relationship of voltage and displacement Furthermore, the output power can be denoted as In the ahead involved analysis, the steady-state displacement response of the two-degree-of-freedom nonlinear flow energy energy harvester can be calculated from Eqs. (9)-(12), the voltage response can be computed from Eq. (13), and the power can be solved by Eq. (14).
The parameter values of the main system are chosen from Alhadidi and Daqaq [53] and Alhadidi et al. [54], and m 1 = 0.0134 kg, k 1 =75 N/m, c 1 = 0.016 N.s/m, γ = 20734 N/m 3 , ρ = 1.225 kg/m 3 , C Lo = 1.21, D = 0.05 m, S t = 0.13, θ = 1.05 × 10 −4 , C p = 84 nF, R = 100 K . In addition, the physical parameter of the auxiliary system is fixed as m 2 = 0.01 kg, c 2 = 0.06 N·s/m, k 2 = 50 N/m. From above parameter values we can calculate the first and second frequencies of Eq. (6) as ω 1 = 48.301 and ω 2 = 109.52. Figure 3 shows the response graphs of displacement and voltage with the wind speed for the two-degree-of-freedom nonlinear flow energy energy harvester, and the response map of displacement and voltage with the corresponding excitation frequency is drawn in Fig. 4. In these figures, solid lines denote stable values and dotted lines depict the unstable cases. It is seen that the curves have typical double peaks and two multi-solution ranges as a result of the two amplitude leaning to the right. Moreover, the frequency-response graphs are slant much to the right, representing a hardening nonlinearity. The bending results in broadband captured performance, which is generally desired for energy harvesting.

Numerical experiments
To assert the accuracy of first-order harmonic term, Eqs.   Figure 7 shows the change rules of both the displacement and the voltage versus the wind speed, and Fig. 8 reports the responses of the displacement and the voltage with the excitation frequency. As can be observed from the motion curves emerging in this figure, there are specific double-jumpings in the pictures for the presented two-degree-of-freedom (TDOF) nonlinear flow energy energy harvester. The curve graph has two spaced multivalued districts so that there are two crest values inflecting to the right, respectively. The double-jumping introduces to wide bandwidth than the traditional single-degree-of-freedom (SDOF) linear and nonlinear energy harvesters. Consequently, we conclude that the presented two-degree-of-freedom nonlinear flow energy energy harvester can capture much electricity than that of the traditional single-degree-of-freedom linear and nonlinear harvesters.

Results and discussion
This section starts with a discussion of the influence of the physical parameter of the two-degree-of-freedom nonlinear flow energy harvester from vortex-induced vibrations, and the corresponding amplitude-frequency curves of displacement and voltage are given.
It is very meaningful to analyze the impact of the auxiliary oscillator on the frequency response curve. Figure 9 shows the frequency response curve of displacement and voltage versus wind speed for different auxiliary mass values (m 2 = 0.005 kg, 0.01 kg and 0.025 kg), and the response track of displacement and voltage with the corresponding excitation frequency is employed in Fig. 10. When the auxiliary mass m 2 is fixed at 0.005 kg, the first and second natural frequencies can be derived as ω 1 = 59.552 and ω 2 = 125.63. While the auxiliary mass m 2 is chosen as 0.025 kg, the first and second natural frequencies can be employed as ω 1 = 33.073 and ω 2 = 101.17. It is noticed that when the auxiliary mass values change, there are still a typical double peaks, and the two amplitude leaning to the right, representing a hardening characteristic. Furthermore, as the auxiliary mass values are decreased, the first peak is rapidly increased, and the path line is bent further to the right, while the second peak is slightly decreased.   12 show the frequency response orbit of the two-degree-of-freedom nonlinear flow energy harvester along with wind speed and excitation frequency for different auxiliary stiffness values (k 2 = 25 N/m, 50 N/m and 75 N/m). When the auxiliary stiffness k 2 is located in 25 N/m, the first and second natural frequencies can be calculated as ω 1 = 41.131 and ω 2 = 90.945. When the auxiliary stiffness k 2 is taken at 75 N/m, the first and second natural frequencies can be addressed as ω 1 = 51.087 and ω 2 = 126.82. It is shown that the natural frequencies increase as the auxiliary stiffness k 2 increases. Moreover, as the auxiliary stiffness values are increased, the first peak is sharply increased, and the orbit line is bent further to the right, while the second peak is slightly decreased.
In the meantime, these figures mean that the response orbit is very sensitive to the auxiliary mass than the auxiliary stiffness. This performance investigations provide a new insight into the usefulness of careful the auxiliary oscillator parameter of the twodegree-of-freedom nonlinear flow energy harvester in order to exploit efficient vortex-induced energy harvesters.
Having addressed the advantageous effect of auxiliary oscillator on the frequency bandwidth, another favorable parameter is the cubic nonlinear coefficients indicated by γ . Figure 13  orbit response also increases as the cubic nonlinearity values are increased. Therefore, large positive cubic nonlinear coefficient is desired for enhanced bandwidth of work effectively for a flow energy harvester. Based on dimensional power Eq. (14), we next discuss the output voltage and power changes of the two-degree-of-freedom nonlinear flow energy harvester related to the circuit parameters. Figure 15 plots the response curve of voltage and power versus resistor with different wind speed. The voltage answering curves illustrate the voltage orbits have apparent double peaks, which are located in different wind speed regions. In addition, the voltage value increases sharply when the resistor is less than 100 K , and then increases slowly after the resistor is greater than 100 K . The power cloud image observes the power augments with the enhancing resistance until the optimal value (about 100 K ) and then descends with the increasing resistance load. Figure 16 plots the changing tendency of the voltage, the power as a result of change in piezoceramic capacitance and wind speed, respectively. As depicted in Fig. 16, the captured voltage increases as the piezoceramic capacitance decreases until around 85 nF, then it leveled off. Simultaneously, the output power curve has a similar trend. Figure 17 draws the variation of the voltage, the power versus electromechanical coupling coefficient and wind speed, respectively. The results indicate that the harvested voltage is rapidly increased as the electromechanical coupling coefficient increases. A similar sharp prosperity of the captured power is also observed.

Conclusions
The paper demonstrated that scheduling two-degree-of-freedom flow energy harvester can be a great boost for the scavenged electric energy. As a simple proof, a piezoelectric cantilever beam flow energy harvester with an auxiliary linear spring oscillator from vortex- induced vibrations is exploited, and the nonlinear restoring force is designed by employing two repulsive magnets. Specifically, the method of harmonic balance is utilized to address the motion response, and the influence of physical parameter on the output voltage and output power is discussed. The captured performance of the presented model and the corresponding single-degree-of-freedom linear and nonlinear resonance cases is compared. The accuracy of approximately harmonic balance solutions was examined by numerical integration. The paper achieves the following results.
(1) A novel two-degree-of-freedom nonlinear flow energy harvester established via vortex-induced vibrations is addressed to enhance the captured bandwidth. (2) The motion responses are analytically observed to predict broadband captured performance, which have significant double peaks, and simultaneously bend to the right with a hardening nonlinearity. (3) The analytical forecast from the method of harmonic balance is verified by the numerical data. (4) The influence of the auxiliary oscillator on the response graphs is employed, and it shows that the response orbit is very sensitive to the auxiliary mass m 2 than the auxiliary stiffness k 2 . The impact of the circuit parameter on the voltage curve and the output power are also discussed, and maximum output power is received with respect to the resistance. (5) The benefit of the designed two-degree-of-freedom nonlinear scheme is contrasted with the single-degree-of-freedom linear and nonlinear cases.