ENERGY HARVESTING FROM TRANSVERSE GALLOPING ENHANCED BY PARAMETRIC EXCITATION

Transcription

1 ENERGY HARVESTING FROM TRANSVERSE GALLOPING ENHANCED BY PARAMETRIC EXCITATION Guilherme R. Franzini Rebeca C. S. Santos Celso P. Pesce Offshore Mechanics Laboratory (LMO), Escola Politécnica (EPUSP), University of São Paulo, Brazil. ABSTRACT The objective of this paper is to numerically investigate the effects of parametric excitation on the energy harvesting from transverse galloping of a square prism. The rigid square prism is elastically supported and is free to oscillate in the transverse direction with respect to the free-stream velocity. A polynomial fit is used for modeling the fluid force. Energy can be harvested from the system by means of a piezoelectric coupling and a linear dashpot. Curves of standard-deviations of electric tension, prism displacement and time-averaged powers harvested at the electric resistance and at the dashpot are presented as functions of the non-dimensional free-stream velocity. The analysis reveals that the parametric excitation in the principal Mathieu s instability may enhance the electric tension and the power that can be harvested from the system. Additionally, the paper presents maps of dissipated power as functions of the parameters that govern the parametric excitation for a given non-dimensional free-stream velocity, showing how the energy harvesting from both the dashpot and the electric resistance can be enhanced. Keywords: Transverse galloping, energy harvesting, piezoelectricity, parametric excitation. INTRODUCTION Fluid-structure interaction phenomena usually involve self-excited oscillations. Such flow-induced vibrations (FIV) can be either resonant (such as, for example, vortex-induced vibrations - VIV) or non-resonant (flutter and galloping). It is very common investigations that focus on the mitigation of FIV. However, there is another class of investigations that aims to collect part of the kinetic energy from the flow, converting it into another class of energy, such as potential or electrical energy. It is important to emphasize that most of the investigations on the electromechanical energy conversion from FIV focuses on low-power devices. Consider firstly energy harvesting from VIV. One of the pioneer devices is VIVACE (Vortex-Induced Vibrations Aquatic Clean Energy). Details concerning this device can be found in [] and [2]. For another kind of vibrating systems, cables, numerical studies described in [3] and [4] investigated the influence of the structural damping parameter and the ratio between the structural natural frequency and the vortex shedding frequency for elastically supported rigid cylinders. Additionally, the authors investigated the effects of distributed energy harvesters on flexible cylinders. Energy harvesting from flutter and galloping were also investigated both from the experimental point-ofview (for example, [5]) or through numerical simulations (see [6], [7]). For all, two common mechanisms of energy conversion are electromagnetic effects (see [8]) and piezoelectricity (references [9], [0], []). Another class of interesting dynamic problems is the parametric excitation, which may occur if one or more parameters of the equation of motion depends explicitly on time. A particular category of parametric excitation problems is composed by systems in which the stiffness depends harmonically on time, leading to Mathieu s equation. Depending of the values of frequency and amplitude of the stiffness variation, the trivial solution may become unstable, giving rising to oscillatory ones. A common tool to check the stability of the undamped Mathieu s equation is the Strutt s diagram. The textbooks [2] and [3] provide excellent didactic material on such a theme. Parametric excitation may occur in slender offshore structures such as TLP s tethers and risers. In such structures, the geometric stiffness is dominant. The motion of the floating units causes tension modulation and, hence, geometric stiffness modulation to the structure. Depending on the amplitude and, mainly, the frequency of the tension modulation, lateral motions may be observed. Corresponding author. gfranzini@usp.br

2 Examples of studies on parametric instability in risers and TLP s tethers can be found in [4], [5], [6] and [7]. Furthermore, reference [8] studies the combined effects of parametric excitation and flow-induced vibration, showing an enhancement of oscillation amplitudes. This enhancement may be useful for energy harvesting from FIV. In this brief literature review, energy harvesting from FIV as well as some aspects of oscillations induced by parametric excitation, particularly in the offshore engineering scenario, were mentioned. However, at least to the authors knowledge, there is an investigation lack regarding the simultaneous presence of parametric excitation and energy harvesting from FIV. It is important to highlight that the parametric excitation may be externally imposed (for example through an active system) or naturally driven by environmental conditions such as, for example, the case of Mathieu s instability of risers and tethers, caused by the wave induced motion of the floating platform. However, despite its practical importance, this paper does not present any consideration regarding the power necessary to externally trigger Mathieu s instability. The objective of this paper is to numerically investigate the influence of parametric excitation on piezoelectric energy harvesting from the transverse galloping of a square prism. Oscillation amplitudes, electric tension harvested, power dissipated at the electrical resistance and at the dashpot are investigated, at two excitation frequency ratios with respect to the natural frequency of the system ω n. The dimensions of the prism and the free-stream velocity range were chosen to be compatible with University of São Paulo experimental facilities. The following sections present the mathematical model and discussion on some results. Finally, final remarks are presented. MATHEMATICAL MODEL This investigation focuses on the transverse galloping of a square prism under structural parametric excitation. Energy is harvested from the structure by means of a lumped piezoelectric device. Fig. sketches the problem. The prism has mass m s, length L and square cross-section side D. The electric load resistance is R and the prism is subjected to a free-stream velocity U. The linear stiffness of the system oscillates as k(t) = k + k cosωt and the linear structural damping constant is c. Consider firstly the problem of pure parametric excitation, i.e., U = 0. As described, for example, in [4], [5], [6] and [7] the fluid force on immersed struc- FIGURE : SKETCH OF THE PROBLEM. tures subjected to parametric excitation can be decomposed into two terms: an inertial term and a viscous one. The corresponding equation of motion may then be written as: m s ÿ +cẏ +( k + k cosωt)y = m a ÿ 2 ρc DDL ẏ ẏ () being m a = C a m d the added mass (m d is the prism displaced mass, C a is the added mass coefficient) and C D the drag coefficient. Notice that the added mass term can be incorporated into the left-hand side of Eqn. and the total mass m then defined. The standard problem (i.e, under no parametric excitation) of transverse galloping is usually modeled under a quasi-steady approach. As described, for example, in [9], the equation of motion of a square prism is given by: mÿ + cẏ + ky = F y = 2 ρu DLC 2 y ( ) ẏ k C y = A k (2) U 3 k= As pointed out in [20], such a quasi-steady approach may be said valid if: U r = U ω n D > 20 = 3.8 (3) 2π where U is the free-stream velocity, ω n is the natural frequency in rad/s and D is the dimension of the crosssection. For the combined problem of transverse galloping under parametric excitation, an ad-hoc model is proposed

3 by superposing the effects of the mentioned phenomena. Such an ad-hoc proposition has to be confirmed through deeper experimental and theoretical investigations. Thus, the equation of motion of the combined problem reads: TABLE : PARAMETERS USED IN THE SIMULA- TIONS. Parameter Value ρ 000kg/m 3 mÿ + cẏ + ( k + k cosωt)y + 2 ρc DDL ẏ ẏ = = 2 ρu 2 DLC y (4) The piezoelectric coupling can be considered following the procedure described, for example, in [0]. This approach consists in including an additional equation, coupling the voltage measured across the load resistance, V, with the prism velocity, ẏ. The piezohydroelastic system is then modeled as: mÿ + cẏ + ( k + k cosωt)y + 2 ρc DDL ẏ ẏ θv = = 2 ρu 2 DLC y (5) C P V + V + θẏ = 0 (6) R where C P is the capacitance of the piezoelectric system, R the electrical resistance and θ the electromechanical coupling term. The system composed by Eqns. 5 and 6 was simulated in MATLAB R R202b using a fourth-fifth Runge- Kutta scheme (ode45 function) with adaptive time-step. The transverse force coefficient series, given by Eqn. 2, was truncated in 3 terms, so not able to capture hysteretic behaviors, see [20]. RESULTS AND DISCUSSION The parameters used in the simulations are given in Tab.. The dimensions proposed are proper for further experimental investigations at USP facilities. The properties of the piezoelectric component are the same employed in [0]. The drag coefficient was selected as C D = 2.2 and the added mass coefficient was considered as that of a square, assuming C a =.5 (see [2]). The values of A k (k =,...,3) used for the evaluation of the lift coefficient are the same presented in [9]. Trivial linear stability analysis of Eqn. 2 indicates that the critical velocity for the onset of self-excited oscillations is U c = 4mζ ω n /ρdla. Considering the parameters presented in Tab., the critical free-stream velocity is U c = 0.002m/s. D L m s ω n = k m+ma ζ s = 20mm 500mm kg π 2 rad/s c 2mω n 0.0 k R C P θ U k 2 5kΩ F N/V 0. < U < 0.3m/s U r 3.8 < U < 9.55 A 2.69 A 2 0 A C D 2.2 C a.5 Fig. 2 presents the standard deviation (y(t)/d) and V (t) as functions of the non-dimensional free-stream velocity U r = U ω n D for three conditions: the standard galloping (or no parametric excitation case, see Eqn. 2) and the cases with parametric excitation with Ω : ω n = : and Ω : ω n = 2 :, being this latter condition also defined in the literature as principal Mathieu s instability condition. Considering U r < 5, the condition with Ω : ω n = 2 : leads to a increase in both (y(t)/d) and V (t). For U r > 5, the Ω : ω n = : condition presents higher values of oscillation amplitudes, reaching.3 diameters at U r 9.5. Considering the conditions tested, the electrical voltage varies from 50mV to 200mV. It is important to emphasize that [0] obtained similar values of electrical voltage from the VIV of a circular cylinder for the same values of R,θ and C p. Curiously, the voltage obtained through the load resistance is higher for Ω : ω n = 2 : than is under no

4 (y/d) fd/fn Ω : ω n = 2 : Ω : ω n = : Ω : ω n = 2 : Ω : ω n = : 0 (a) (y/d). FIGURE 3: OSCILLATION FREQUENCY V [mv] 00 Ω : ωn = 2 : Ω : ωn = : 50 (b) V. V [mv] CD = 2.2;Ω : ωn = 2 : CD = 0;Ω : ωn = 2 : no parametric excitation 50 FIGURE 2: STANDARD-DEVIATION of y(t)/d AND V (t). FIGURE 4: DEPENDENCE OF V ON THE MEAN DRAG COEFFICIENT. parametric excitation for U r < 6, despite the lower values of (y(t)/d) between U r = 5 and U r = 6. This fact may be explained by calculating the dominant oscillation frequency as a function of U r. As can be seen in Fig. 3, the dominant oscillation frequency decreases as U r increases under no excitation and for the : parametric excitation case. On the other hand, the dominant frequency is constant f d f n = ω n /2π for the Ω : ω n = 2 : case. The case Ω : ω n = 2 : is, obviously, a Mathieu s resonant response. On the other hand, the other two cases are non-resonant ones, dominated by galloping, where the dominant frequency is expected to decrease as the oscillation amplitude increases. Notice that θ couples the structural and electric circuit dynamics through the velocity ẏ, which is proportional to f d. Notice, however, that all the simulations herein pre- sented make use of quadratic hydrodynamic damping coefficient C D = 2.2. This value is typical of static simulations or experiments. A sensitivity analysis regarding the coefficient C D is certainly of interest. Fig. 4 presents an initial discussion on this subject, showing the variation of V as functions of U r for different conditions. As expected, V increases as C D decreases, at Ω : ω n = 2 :, for the whole range of U r. Additionally, the same plot reveals that the parametric excitation at Ω : ω n = 2 : with C D = 0 leads to electric voltage larger than that obtained under no parametric excitation for U r < 7.5. Indeed, the influence of C D in the response to parametric excitation is interesting and shall be investigated in a further work. Now, we focus on the power dissipated in the dashpot (P D ) and the electrical power dissipated at the electrical resistance (P el ) and their dependence on the excitation frequency Ω. As well known, these quantities can be

5 evaluated as: It is interesting to evaluate the effects of the parametric excitation on the power harvested. For this, two coefficients γ and γ 2 can be defined as: P D (t) = F D (t)ẏ(t) = c(ẏ(t)) 2 (7) P el (t) = (V (t))2 R (8) where the signal in Eqn. 7 indicates power harvested from the system. Fig. 5 presents the variation of P D and P el as functions of U r. Despite the lower values of (y(t)/d) for U r < 5, Fig. 5(a) shows that the parametric excitation at Ω : ω n = 2 : enhances the power dissipated at the dashpot in comparison to the no parametric excitation case. Once again, this result may be related to the larger oscillation frequency in the case Ω : ω n = 2 :. The electrical power at the resistance is presented in Fig. 5(b). PD[mW] Pel[mW] Ω : ωn = 2 : Ω : ωn = : (a) P D. Ω : ω n = 2 : Ω : ω n = : 0 (b) P el. FIGURE 5: AVERAGED VALUES P D AND P el. γ = P D P 0 D γ 2 = P el P 0 el (9) (0) where () indicates time-averaged values and the the superscript () 0 refers to the power harvested without parametric excitation. Fig. 6 presents the variation of γ and γ 2 with U r. Considering firstly the Ω : ω n = : case, the behavior of both curves are qualitatively similar, being almost invariant with U r and close to 0.9. On the other hand, the parametric excitation at Ω : ω n = 2 : enhances the power dissipated at the dashpot for U r < 8.5 and the electrical power for U r < 6. Notice that the parametric excitation at the principal Mathieu s instability condition enhances the energy harvesting by circa 30% at U r = Now, we investigate the influence of the parameters that govern the parametric excitation ( k and Ω) on the energy harvesting, at a constant non-dimensional velocity U r = 5. Such investigation is carried out by plotting the power dissipated at the dashpot and at the electrical resistance as functions of k/ k and n = Ω/ω n. The colored maps corresponding to P D and P el are presented in Fig. 7. Close to the principal Mathieu s instability, there is a marked increase in the power harvested at the dashpot and at the electrical resistance (Figs. 7(a) and 7(b) respectively). In addition to the principal Mathieu s instability, it can be noticed an enhancement of the power harvested at n. However, the amplitude of stiffness modulation necessary to trigger the parametric excitation and, consequently, to increase the power harvested is larger than that for the principal Mathieu s instability. Finally, it is investigated the energy harvesting efficiency. Before the presentation of the results, some comments should be made. In [3] and [4], the efficiency is defined as the ratio between the time-averaged power dissipated at the dashpot in one oscillation period T and the flux of kinetic energy density measured across a section of dimensions L and D, labeled φ K. Notice that this definition does not discuss the way kinetic energy would be converted into electric energy. In [6], the efficiency is computed by taking the ratio between the power imparted by the flow to the body and φ K. A comprehensive discus-

6 .35.3 Ω : ωn = 2 : Ω : ωn = : γ U r = U /ω nd.3.25 Ω : ωn = 2 : Ω : ωn = : (a) P D. γ U r = U /ω nd FIGURE 6: γ AND γ 2 COEFFICIENTS AS FUNC- TIONS OF NON-DIMENSIONAL FREE-STREAM VELOCITY. (b) P el. FIGURE 7: POST-CRITICAL VALUES OF P el AND P D AT U r = 5. sion regarding different definitions of efficiency is certainly of interest, but is left for a further paper. We then compute two values of energy harvesting efficiency. η D and η el are defined, respectively, as the ratio between the time-averaged power dissipated at the dashpot and at the electrical resistance and φ K. Mathematically, we have: η D = η el = P D 2 ρu 3 DL P el 2 ρu 3 DL () (2) In these definitions, the parametric excitation power delivered to the system is not taken into account. Implicitly, this mean that the excitation energy is supposed to be freely delivered by another source, as is the case of wave induced floating body motions driving risers or tethers stiffness. Fig. 8(a) presents the variation of η D as a function of U r. For the chosen set of parameters and the range of reduced velocities simulated, it can be noticed an increase in both η D and η el for U r < 6. For the sake of comparison with previous findings, [4] reports a maximum efficiency of energy harvesting from VIV as η = 23%. The authors also point out that the efficiency of the VIVACE device is close to 22%. However, it is important to emphasize that, contrary to transverse galloping, VIV occurs at much lower velocities (typically 0.64 < U r <.59). As φ k has a cubic dependence on the free-stream velocity, the flux of kinetic energy across the cross-section is much larger in galloping than in VIV. The electrical efficiency is much lower than those associated to the dashpot, as can be seen in Fig. 8(b). In fact, φ K = 0.35W at the maximum free-stream velocity simulated, being this value much higher than the corresponding electrical power at Ω : ω n = 2 : (0.006mW - see Fig. 5(b)). The power dissipated at the dashpot is one order of magnitude larger than that measured at the electrical resistance.

7 ηd[%] ηel[%] Ω : ωn = 2 : Ω : ωn = : 0.02 U r = U /ω nd (a) η D. Ω : ωn = 2 : Ω : ωn = : subjected to parametric excitation may be larger than one, indicating that the parametric excitation may enhance energy harvesting. Additionally, a map of dissipated power was plotted as functions of the parameters k and Ω. The region close to the principal Mathieu s instability is characterized by a marked increase of the power that can be harvested at the electrical resistance and at the dashpot. The energy harvesting can also be increased at Ω : ω n = :, but the amplitude of stiffness modulation k must be significantly larger than that in the principal Mathieu s instability region. Further work may include a deeper discussion on the effects of the parameters that govern the mathematical model in the energy harvesting. The mathematical model shall be re-written in non-dimensional form and extensive investigations shall be carried out, regarding the influence of the parameters on electric voltage, power dissipated at the dashpot and measured at the electrical resistance on the energy harvesting efficiency U r = U /ω nd (b) η el. FIGURE 8: ENERGY HARVESTING EFFICIENCY. FINAL REMARKS This paper addressed the effects of parametric excitation on the energy harvested from the transverse galloping of a square prism. The interest for the combined aspect of parametric excitation and energy harvesting lies on the fact that parametric excitation may lead to selfexcitated oscillations and, then, might enhance the energy harvested from a fluid-structure problem also characterized by self-excitated oscillations. We consider energy harvesting from both the dashpot and the electrical resistance in the piezoelectric system. The equations of motion were numerically integrated and time-histories of electric voltage, prism displacement and powers dissipated at the dashpot and at the electric resistance were analyzed. It was found that the parametric excitation with Ω : ω n = 2 : enhances the standard-deviations of both the power dissipated at the dashpot and at the electrical resistance, in part of the free-stream velocity range. Therefore, the ratio between the power harvested at Ω : ω = 2 : (principal Mathieu s instability condition) and those not ACKNOWLEDGMENT First and third authors acknowledge CNPq for financial support, grant numbers 30595/205-0 and /204-5 respectively. Second author acknowledges a scholarship from University of São Paulo through NAP-OceanoS - Research Nucleous Sustainable Ocean ( english.php). We also acknowledge FINEP, project number 87/0. REFERENCES [] Bernitsas, M. M., Raghavan, K., Ben-Simos, Y., and Garcia, E. M. H., VIVACE (Vortex Induced Vibration Aquatic Clean Energy): A new concept in generation of clean and renewable energy from fluid flow. In Proceedings of OMAE International Conference on Offshore Mechanics and Artic Engineering. [2] Bernitsas, M. M., Ben-Simos, Y., Raghavan, K., and Garcia, E. M. H., The VIVACE converter: model tests at high damping and Reynolds number around 0 5. In Proceedings of OMAE International Conference on Offshore Mechanics and Artic Engineering. [3] Grouthier, C., Michelin, S., and de Langre, E., 202. Optimal energy harvesting by vortex-induced vibrations in cables. In Proceedings of the 0th FIV International Conference on Flow-Induced Vibrations Conference (& Flow-Induced Noise).

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