- Email: [email protected]
Performance enhancement of wing-based piezoaeroelastic energy harvesting through freeplay nonlinearity
THEORETICAL & APPLIED MECHANICS LETTERS 3, 041001 (2013)
Performance enhancement of wing-based piezoaeroelastic energy harvesting through freeplay nonlinearity Abdessattar Abdelkefi,a) and Muhammad R. Hajj Department of Engineering Science and Mechanics, Virginia Tech, Blacksburg, Virginia 24061, USA
(Received 11 May 2013; accepted 25 June 2013; published online 10 July 2013) Abstract We investigate experimentally how controlled freeplay nonlinearity affects harvesting energy from a wing-based piezoaeroelastic energy harvesting system. This system consisits of a rigid airfoil which is supported by a nonlinear torsional spring (freeplay) in the pitch degree of freedom and a linear flexural spring in the plunge degree of freedom. By attaching a piezoelectric material (PSI-5A4E) to the plunge degree of freedom, we can convert aeroelastic vibrations to electrical energy. The focus of this study is placed on the effects of the freeplay nonlinearity gap on the behavior of the harvester in terms of cut-in speed and level of harvested power. Although the freeplay nonlinearity may result in subcritical Hopf bifurcations (catastrophic for real aircrafts), harvesting energy at low wind speeds is beneficial for designing piezoaeroelastic systems. It is demonstrated that increasing the freeplay nonlinearity gap can decrease the cut-in speed through a subcritical instability and gives the possibility to harvest energy at low wind speeds. The results also demonstrate that an optimum c 2013 value of the load resistance exists, at which the level of the harvested power is maximized. ⃝ The Chinese Society of Theoretical and Applied Mechanics. [doi:10.1063/2.1304101] Keywords energy harvesting, freeplay nonlinearity, piezoelectric material, aeroelasticity, experimental measurements Harvesting energy from ambient1–5 and 6–17 aeroelastic vibrations has been proposed to develop self-powered sensors18 and actuators or to power microelectromechanical systems.19,20 One such system is based on flutter and ensuing limit-cycle oscillations of wing sections. Bryant and Garcia6 showed theoretically and experimentally that we can gather energy from aeroelastic vibrations by attaching an airfoil section to a cantilever. After theoretical and experimental studies, Erturk et al.8 confirmed the effect of piezoelectric power generation on the linear flutter speed. Abdelkefi et al.9–11 investigated theoretically the nonlinear behavior of wing-based piezoaeroelastic energy harvesters with the objective of designing enhanced harvesters that can generate energy at low wind speeds under supercritical Hopf bifurcation conditions. They used cubic representations for the torsional, flexural, and aerodynamic nonlinearities. Nonlinearities associated with the aeroelastic response of wings have structural and aerodynamic sources. Of the structural sources, freeplay nonlinearity, usually associated with moving surfaces, is quite common. It can come up from loose attachments and worn hinges after flying long time. The combination of aerodynamic and structural nonlinearities can vary the system’s response and change its instability from supercritical to subcritical and induce limit-cycle oscillations (LCOs) at wind speeds below the linear flutter speed obtained in Refs. 21–24. This behavior can result in catastrophic damages in real aircraft. However, harvesting energy at low wind speeds can be beneficial for designing piezoaeroelastic systems. Sousa et al.25 presented a) Corresponding
author. Email: [email protected].
modeling and experiments of wing-based piezoaeroelastic energy harvesters having combined nonlinearities (freeplay and cubic) in the nonlinear torsional spring. Effects of varying the freeplay nonlinearity gap at the cut-in speed and the level of harvested power of a wing-based piezoaeroelastic energy harvester are considered in this work. To this end, power harvesting from a rigid airfoil section supported by linear flexural spring in the plunge motion and nonlinear torsional spring in the pitch motion with a piezoelectric coupling attached to the plunge degree of freedom is experimentally investigated. We use a piezoelectric material (PSI-5A4E) that is bonded by two in-plane electrodes with negligible thickness and connected to an electrical load resistance. Four different freeplay nonlinearity gaps in the pitch degree of freedom are considered in order to investigate their effects on the efficiency of the harvester in terms of cut-in speed and harvested power. The experiments were conducted in an open-circuit wind tunnel having a 520 mm × 515 mm test section and with a minimum wind speed of 8.5 m/s. Figures 1 and 2 show the experimental setup used to study the effects of the gap of the freeplay nonlinearity on the performance of a wind-based piezoaeroelastic energy harvester. This experimental setup was previously used by Abdelkefi et al.26 for aeroelastic purposes (identification of concentrated nonlinearity in the pitch degree of freedom). The aim of attaching piezoelectric material to the plunge degree of freedom, as shown in Fig. 2, is to convert aeroelastic oscillations to electrical power. The rigid airfoil consists of an aluminum rigid wing mounted vertically at the 1/4 chord point from the leading edge by using an aluminum shaft. The shaft is connected with bearings to the support of the plunge mechanism, which is a bicantilever beam made of two steel leaf springs (as shown
041001-2 A. Abdelkefi, and M. R. Hajj
Theor. Appl. Mech. Lett. 3, 041001 (2013)
Fig. 1. Experimental setup showing a rigid airfoil in a wind tunnel.
Shaft
Piezoelectric material
Fig. 2. Placement of the piezoelectric material in the plunge mechanism.
in Fig. 2). We insert the steel leaf torsional spring into a slot in the main shaft at the bottom of the wing section, and place the free end of the leaf spring into a support that allows for freeplay variations. A schematic of the pitch freeplay mechanism is presented in Fig. 3. An encoder and an accelerometer were used to measure the rotational and plunge motions, respectively.
Leaf Spring
Shaft
Freeplay variation mechanism
Fig. 3. A schematic of the pitch freeplay mechanism.
To determine the linear flutter speed of the piezoaeroelastic system, the support of the freeplay was closed (zero gap) and initial displacements were applied in the plunge degree of freedom. We noted that, below the speed of 10.9 m/s, all disturbances were damped. Furthermore, LCOs were observed at higher speeds. As such, the linear flutter speed of the piezoaeroelastic system was determined to be 10.9 m/s. This methodology was repeated for different values of the electrical load resistance. It was determined that the electrical load resistance does not affect the linear flutter speed of the harvester. This is expected because the resistive shunt damping27 of the electrical part is negligible compared to the pitch and plunge damping. To study how the freeplay nonlinearity gap in the pitch degree of freedom affects the cut-in speed and the performance of the harvester, we consider four different freeplay gaps, which include a very small gap, ∼0 cm, 0.205 cm, 0.440 cm, and 0.635 cm. Sweep experiments whereby the wind speed was varied between 8.5 m/s and 12.5 m/s were performed. The plotted curves in Figs. 4 and 5 show the bifurcation diagram curves of the RMS generated voltage and RMS pitch angle for different freeplay gaps and when the electrical load resistance is set equal to 104 Ω when increasing and decreasing the wind speed. The plots show bifurcation diagram curves of the average harvested power and RMS plunge amplitude that have the same tendency as the RMS generated voltage curves. Inspecting these curves, we note that when increasing the wind speed from 8.5 m/s to 12.5 m/s, the wing starts to oscillate, for all considered freeplay gaps, at a wind speed of 10.9 m/s which is the linear flutter speed. On the other hand, when decreasing the wind speed from 12.5 m/s to 8.5 m/s, limit-cycle oscillations are observed at speeds that are lower than the linear flutter speed. This is expected because of the freeplay nonlinearity. Furthermore, we note that increasing the freeplay gap is accompanied by a decrease in the cut-in speed. In fact, when the freeplay gap is very small (≈ 0), no oscillations were observed below the linear flutter speed. The cut-in speed is reduced to 10.5 m/s, 9.55 m/s, and less than 8.5 m/s when the freeplay gap is set to 0.205 cm, 0.440 cm, and 0.635 cm, respectively. In contrast to the clear relation between the magnitude of the freeplay nonlinearity and the extent of the region of subcritical instability, the effects of the magnitude of the freeplay nonlinearity on the harvested power in the region above the linear flutter speed are more complex. For wind speed values higher than 10.9 m/s, the generated voltage is the lowest when freeplay gap (0.635 cm) is the largest of the considered four configurations. This is expected because this gap yields the smallest values of the plunge amplitudes and high values of the pitch angle that are associated with the larger freeplay gap. For wind speed values higher than 9.5 m/s, the configuration with freeplay gap of 0.440 cm gives the highest generated voltage and hence the levels of the harvested power are more important. This holds true for the subcritical region between 9.5 m/s and 10.9 m/s.
041001-3 Performance enhancement of wing-based PEH
d ≈ 0 cm d = 0.205 cm (increasing) d = 0.205 cm (decreasing) d = 0.440 cm (increasing) d = 0.440 cm (decreasing) d = 0.635 cm (increasing) d = 0.635 cm (decreasing)
0.3
10 1
U = 11.42 m/s U = 12.59 m/s
0.2
Vrms / V
Vrms / V
Theor. Appl. Mech. Lett. 3, 041001 (2013)
0.1
10 0
10 −1
10 −2
0 8.5
9.0
9.5
10.0 10.5 11.0 U / (m .s-1)
11.5 12.0 12.5
10 3
10 4
10 5
10 6
10 7
R/Ω
Fig. 4. Variation of the root mean square (RMS) generated voltage with increasing and decreasing wind speeds for different freeplay gaps when the electrical load resistance is set equal to 104 Ω.
Fig. 6. Variations of the RMS generated voltage as a function of the electrical load resistance for two different wind speeds and when the freeplay gap is set equal to 0.205 cm. 10 3
d ≈ 0 cm d = 0.205 cm (increasing) d = 0.205 cm (decreasing) d = 0.440 cm (increasing) d = 0.440 cm (decreasing) d = 0.635 cm (increasing) d = 0.635 cm (decreasing)
αrms /( )
8
U = 11.42 m/s U = 12.59 m/s
10
2
Pavg /µW
10
6
10 1 4 2 10 0 10 3
0 8.5
9.0
9.5
10.0
10.5 11.0
11.5 12.0
12.5
10 4
10 5
10 6
10 7
R/Ω
U / (m .s-1)
Fig. 5. Variation of the root mean square (RMS) pitch angle with increasing and decreasing wind speeds for different freeplay gaps when the electrical load resistance is set equal to 104 Ω.
Fig. 7. Variations of the average harvested power as a function of the electrical load resistance for two different wind speeds and when the freeplay gap is set equal to 0.205 cm.
(1)
This result is true for both considered wind speeds. For high values of the electrical load resistance (R > 106 Ω), the generated voltage stabilizes. Inspecting the plotted curves in Fig. 7, we note that an optimum value exists for the electrical load resistance at which the average harvested power is maximized for both wind speeds. This optimum value of the electrical load resistance is near 106 Ω. In summary, in this paper, we investigated experimentally the effects of varying the freeplay nonlinearity gap on the cut-in speed and the level of harvested power of a wing-based piezoaeroelastic energy harvester. The results showed that the right choice of the freeplay gap and the electrical load resistance must be made to enhance the performance of wing-based piezoaeroelastic energy harvesters for specific wind speeds.
It follows from Fig. 6 that the generated voltage increases when the electrical load resistance is increased.
1. I. Kim, H. Jung, B. Lee, et al., Appl. Phys. Lett. 98, 214102 (2011).
To generate energy at the lower wind speeds, one needs to increase the freeplay gap to 0.635 cm. This shows a complex relation between the operating wind speeds, the placement of the harvester, and the choice of the freeplay gap to determine the best performance. Next, we investigate how the electrical load resistance affects the performance of the harvester for two different wind speeds, namely, U = 11.42 m/s and U = 12.59 m/s, and when the freeplay gap is set equal to 0.205 cm, as shown in Figs. 6 and 7. Similar results were obtained for other freeplay gaps. These figures are plotted for the RMS generated voltage and average harvested power which is calculated according to Pavg =
2 Vrms . R
041001-4 A. Abdelkefi, and M. R. Hajj
2. A. F. Arrieta, P. Hagedorn, A. Erturk, et al., Appl. Phys. Lett. 97, 104102 (2010). 3. A. Abdelkefi, F. Najar, A. H. Nayfeh, et al., Smar. Mater. Struc. 20, 115007 (2011). 4. A. Karami and D. J. Inman, Appl. Phys. Lett. 100, 042901 (2012). 5. S. Ben Ayed, A. Abdelkefi, F. Najar, et al., J. Intel. Mater. Syst. Struc., doi: 10.1177/1045389X13489365 (2013). 6. M. Bryant and E. Garcia, Proceedings of SPIE 7288, 728 (2009). 7. S. D. Kwon, Appl. Phys. Lett. 97, 164102 (2010). 8. A. Erturk, W. G. R. Vieira, C. De Marqui, et al., Appl. Phys. Lett. 96, 184103 (2010). 9. A. Abdelkefi, A. H. Nayfeh, and M. R. Hajj, Nonl. Dyn. 67, 925 (2012). 10. A. Abdelkefi, A. H. Nayfeh, and M. R. Hajj, Nonl. Dyn. 68, 531 (2012). 11. A. Abdelkefi, A. H. Nayfeh, and M. R. Hajj, Nonl. Dyn. 68, 519 (2012). 12. A. Abdelkefi, M. R. Hajj, and A. H. Nayfeh, J. Intel. Mater. Syst. Struct. 23, 1523 (2012). 13. A. Abdelkefi, M. R. Hajj, and A. H. Nayfeh, Nonl. Dyn. 70, 1377 (2012). 14. A. Abdelkefi, M. R. Hajj, and A. H. Nayfeh, Nonl. Dyn. 70, 1355 (2012). 15. Y. Yang, L. Zhao, and L. Tang, Appl. Phys. Lett. 102, 064105
Theor. Appl. Mech. Lett. 3, 041001 (2013)
(2013). 16. A. Abdelkefi, M. R. Hajj, and A. H. Nayfeh, Smar. Mater. Struc. 22, 015014 (2013). 17. A. Abdelkefi, Z. Yan, and M. R. Hajj, Smar. Mater. Struc. 22, 055026 (2013). 18. D. J. Inman and B. L. Grisso, Smart Structures and Materials Conference SPIE 6174, 61740T (2006). 19. S. P. Gurav, A. Kasyap, M. Sheplak, et al., Proceedings of the 10th AIAA/ISSSMO Multidisciplinary Analysis and Optimization Conference, 3559 (2004). 20. W. Zhou, W. H. Liao, and W. J. Li, Proc. Smart Structures and Materials Conference. SPIE, 233 (2005). 21. M. D. Conner, D. M. Tang, E. H. Dowell, et al., J. Fluid. Struc. 11, 89 (1997). 22. E. H. Dowell and D. Tang, AIAA J. 40, 1697 (2002). 23. A. Abdelkefi, R. Vasconcellos, F. Marques, et al., J. Soun. Vib. 331, 1898 (2012). 24. R. Vasconcellos, A. Abdelkefi, F. Marques, et al., J. Fluid. Struc. 31, 79 (2012). 25. V. C. Sousa, M de M. Anicezio, C. De Marqui, et al., Sma. Mater. Struc. 20, 094007 (2011). 26. A. Abdelkefi, R. Vasconcellos, A. H. Nayfeh, et al., Nonl. Dyn. 71, 159 (2013). 27. A. Agneni, F. Mastroddi, and G. M. Polli, Comput. Struc. 81, 91 (2003).
Performance enhancement of wing-based piezoaeroelastic energy harvesting through freeplay nonlinearity Abdessattar Abdelkefi,a) and Muhammad R. Hajj Department of Engineering Science and Mechanics, Virginia Tech, Blacksburg, Virginia 24061, USA
(Received 11 May 2013; accepted 25 June 2013; published online 10 July 2013) Abstract We investigate experimentally how controlled freeplay nonlinearity affects harvesting energy from a wing-based piezoaeroelastic energy harvesting system. This system consisits of a rigid airfoil which is supported by a nonlinear torsional spring (freeplay) in the pitch degree of freedom and a linear flexural spring in the plunge degree of freedom. By attaching a piezoelectric material (PSI-5A4E) to the plunge degree of freedom, we can convert aeroelastic vibrations to electrical energy. The focus of this study is placed on the effects of the freeplay nonlinearity gap on the behavior of the harvester in terms of cut-in speed and level of harvested power. Although the freeplay nonlinearity may result in subcritical Hopf bifurcations (catastrophic for real aircrafts), harvesting energy at low wind speeds is beneficial for designing piezoaeroelastic systems. It is demonstrated that increasing the freeplay nonlinearity gap can decrease the cut-in speed through a subcritical instability and gives the possibility to harvest energy at low wind speeds. The results also demonstrate that an optimum c 2013 value of the load resistance exists, at which the level of the harvested power is maximized. ⃝ The Chinese Society of Theoretical and Applied Mechanics. [doi:10.1063/2.1304101] Keywords energy harvesting, freeplay nonlinearity, piezoelectric material, aeroelasticity, experimental measurements Harvesting energy from ambient1–5 and 6–17 aeroelastic vibrations has been proposed to develop self-powered sensors18 and actuators or to power microelectromechanical systems.19,20 One such system is based on flutter and ensuing limit-cycle oscillations of wing sections. Bryant and Garcia6 showed theoretically and experimentally that we can gather energy from aeroelastic vibrations by attaching an airfoil section to a cantilever. After theoretical and experimental studies, Erturk et al.8 confirmed the effect of piezoelectric power generation on the linear flutter speed. Abdelkefi et al.9–11 investigated theoretically the nonlinear behavior of wing-based piezoaeroelastic energy harvesters with the objective of designing enhanced harvesters that can generate energy at low wind speeds under supercritical Hopf bifurcation conditions. They used cubic representations for the torsional, flexural, and aerodynamic nonlinearities. Nonlinearities associated with the aeroelastic response of wings have structural and aerodynamic sources. Of the structural sources, freeplay nonlinearity, usually associated with moving surfaces, is quite common. It can come up from loose attachments and worn hinges after flying long time. The combination of aerodynamic and structural nonlinearities can vary the system’s response and change its instability from supercritical to subcritical and induce limit-cycle oscillations (LCOs) at wind speeds below the linear flutter speed obtained in Refs. 21–24. This behavior can result in catastrophic damages in real aircraft. However, harvesting energy at low wind speeds can be beneficial for designing piezoaeroelastic systems. Sousa et al.25 presented a) Corresponding
author. Email: [email protected].
modeling and experiments of wing-based piezoaeroelastic energy harvesters having combined nonlinearities (freeplay and cubic) in the nonlinear torsional spring. Effects of varying the freeplay nonlinearity gap at the cut-in speed and the level of harvested power of a wing-based piezoaeroelastic energy harvester are considered in this work. To this end, power harvesting from a rigid airfoil section supported by linear flexural spring in the plunge motion and nonlinear torsional spring in the pitch motion with a piezoelectric coupling attached to the plunge degree of freedom is experimentally investigated. We use a piezoelectric material (PSI-5A4E) that is bonded by two in-plane electrodes with negligible thickness and connected to an electrical load resistance. Four different freeplay nonlinearity gaps in the pitch degree of freedom are considered in order to investigate their effects on the efficiency of the harvester in terms of cut-in speed and harvested power. The experiments were conducted in an open-circuit wind tunnel having a 520 mm × 515 mm test section and with a minimum wind speed of 8.5 m/s. Figures 1 and 2 show the experimental setup used to study the effects of the gap of the freeplay nonlinearity on the performance of a wind-based piezoaeroelastic energy harvester. This experimental setup was previously used by Abdelkefi et al.26 for aeroelastic purposes (identification of concentrated nonlinearity in the pitch degree of freedom). The aim of attaching piezoelectric material to the plunge degree of freedom, as shown in Fig. 2, is to convert aeroelastic oscillations to electrical power. The rigid airfoil consists of an aluminum rigid wing mounted vertically at the 1/4 chord point from the leading edge by using an aluminum shaft. The shaft is connected with bearings to the support of the plunge mechanism, which is a bicantilever beam made of two steel leaf springs (as shown
041001-2 A. Abdelkefi, and M. R. Hajj
Theor. Appl. Mech. Lett. 3, 041001 (2013)
Fig. 1. Experimental setup showing a rigid airfoil in a wind tunnel.
Shaft
Piezoelectric material
Fig. 2. Placement of the piezoelectric material in the plunge mechanism.
in Fig. 2). We insert the steel leaf torsional spring into a slot in the main shaft at the bottom of the wing section, and place the free end of the leaf spring into a support that allows for freeplay variations. A schematic of the pitch freeplay mechanism is presented in Fig. 3. An encoder and an accelerometer were used to measure the rotational and plunge motions, respectively.
Leaf Spring
Shaft
Freeplay variation mechanism
Fig. 3. A schematic of the pitch freeplay mechanism.
To determine the linear flutter speed of the piezoaeroelastic system, the support of the freeplay was closed (zero gap) and initial displacements were applied in the plunge degree of freedom. We noted that, below the speed of 10.9 m/s, all disturbances were damped. Furthermore, LCOs were observed at higher speeds. As such, the linear flutter speed of the piezoaeroelastic system was determined to be 10.9 m/s. This methodology was repeated for different values of the electrical load resistance. It was determined that the electrical load resistance does not affect the linear flutter speed of the harvester. This is expected because the resistive shunt damping27 of the electrical part is negligible compared to the pitch and plunge damping. To study how the freeplay nonlinearity gap in the pitch degree of freedom affects the cut-in speed and the performance of the harvester, we consider four different freeplay gaps, which include a very small gap, ∼0 cm, 0.205 cm, 0.440 cm, and 0.635 cm. Sweep experiments whereby the wind speed was varied between 8.5 m/s and 12.5 m/s were performed. The plotted curves in Figs. 4 and 5 show the bifurcation diagram curves of the RMS generated voltage and RMS pitch angle for different freeplay gaps and when the electrical load resistance is set equal to 104 Ω when increasing and decreasing the wind speed. The plots show bifurcation diagram curves of the average harvested power and RMS plunge amplitude that have the same tendency as the RMS generated voltage curves. Inspecting these curves, we note that when increasing the wind speed from 8.5 m/s to 12.5 m/s, the wing starts to oscillate, for all considered freeplay gaps, at a wind speed of 10.9 m/s which is the linear flutter speed. On the other hand, when decreasing the wind speed from 12.5 m/s to 8.5 m/s, limit-cycle oscillations are observed at speeds that are lower than the linear flutter speed. This is expected because of the freeplay nonlinearity. Furthermore, we note that increasing the freeplay gap is accompanied by a decrease in the cut-in speed. In fact, when the freeplay gap is very small (≈ 0), no oscillations were observed below the linear flutter speed. The cut-in speed is reduced to 10.5 m/s, 9.55 m/s, and less than 8.5 m/s when the freeplay gap is set to 0.205 cm, 0.440 cm, and 0.635 cm, respectively. In contrast to the clear relation between the magnitude of the freeplay nonlinearity and the extent of the region of subcritical instability, the effects of the magnitude of the freeplay nonlinearity on the harvested power in the region above the linear flutter speed are more complex. For wind speed values higher than 10.9 m/s, the generated voltage is the lowest when freeplay gap (0.635 cm) is the largest of the considered four configurations. This is expected because this gap yields the smallest values of the plunge amplitudes and high values of the pitch angle that are associated with the larger freeplay gap. For wind speed values higher than 9.5 m/s, the configuration with freeplay gap of 0.440 cm gives the highest generated voltage and hence the levels of the harvested power are more important. This holds true for the subcritical region between 9.5 m/s and 10.9 m/s.
041001-3 Performance enhancement of wing-based PEH
d ≈ 0 cm d = 0.205 cm (increasing) d = 0.205 cm (decreasing) d = 0.440 cm (increasing) d = 0.440 cm (decreasing) d = 0.635 cm (increasing) d = 0.635 cm (decreasing)
0.3
10 1
U = 11.42 m/s U = 12.59 m/s
0.2
Vrms / V
Vrms / V
Theor. Appl. Mech. Lett. 3, 041001 (2013)
0.1
10 0
10 −1
10 −2
0 8.5
9.0
9.5
10.0 10.5 11.0 U / (m .s-1)
11.5 12.0 12.5
10 3
10 4
10 5
10 6
10 7
R/Ω
Fig. 4. Variation of the root mean square (RMS) generated voltage with increasing and decreasing wind speeds for different freeplay gaps when the electrical load resistance is set equal to 104 Ω.
Fig. 6. Variations of the RMS generated voltage as a function of the electrical load resistance for two different wind speeds and when the freeplay gap is set equal to 0.205 cm. 10 3
d ≈ 0 cm d = 0.205 cm (increasing) d = 0.205 cm (decreasing) d = 0.440 cm (increasing) d = 0.440 cm (decreasing) d = 0.635 cm (increasing) d = 0.635 cm (decreasing)
αrms /( )
8
U = 11.42 m/s U = 12.59 m/s
10
2
Pavg /µW
10
6
10 1 4 2 10 0 10 3
0 8.5
9.0
9.5
10.0
10.5 11.0
11.5 12.0
12.5
10 4
10 5
10 6
10 7
R/Ω
U / (m .s-1)
Fig. 5. Variation of the root mean square (RMS) pitch angle with increasing and decreasing wind speeds for different freeplay gaps when the electrical load resistance is set equal to 104 Ω.
Fig. 7. Variations of the average harvested power as a function of the electrical load resistance for two different wind speeds and when the freeplay gap is set equal to 0.205 cm.
(1)
This result is true for both considered wind speeds. For high values of the electrical load resistance (R > 106 Ω), the generated voltage stabilizes. Inspecting the plotted curves in Fig. 7, we note that an optimum value exists for the electrical load resistance at which the average harvested power is maximized for both wind speeds. This optimum value of the electrical load resistance is near 106 Ω. In summary, in this paper, we investigated experimentally the effects of varying the freeplay nonlinearity gap on the cut-in speed and the level of harvested power of a wing-based piezoaeroelastic energy harvester. The results showed that the right choice of the freeplay gap and the electrical load resistance must be made to enhance the performance of wing-based piezoaeroelastic energy harvesters for specific wind speeds.
It follows from Fig. 6 that the generated voltage increases when the electrical load resistance is increased.
1. I. Kim, H. Jung, B. Lee, et al., Appl. Phys. Lett. 98, 214102 (2011).
To generate energy at the lower wind speeds, one needs to increase the freeplay gap to 0.635 cm. This shows a complex relation between the operating wind speeds, the placement of the harvester, and the choice of the freeplay gap to determine the best performance. Next, we investigate how the electrical load resistance affects the performance of the harvester for two different wind speeds, namely, U = 11.42 m/s and U = 12.59 m/s, and when the freeplay gap is set equal to 0.205 cm, as shown in Figs. 6 and 7. Similar results were obtained for other freeplay gaps. These figures are plotted for the RMS generated voltage and average harvested power which is calculated according to Pavg =
2 Vrms . R
041001-4 A. Abdelkefi, and M. R. Hajj
2. A. F. Arrieta, P. Hagedorn, A. Erturk, et al., Appl. Phys. Lett. 97, 104102 (2010). 3. A. Abdelkefi, F. Najar, A. H. Nayfeh, et al., Smar. Mater. Struc. 20, 115007 (2011). 4. A. Karami and D. J. Inman, Appl. Phys. Lett. 100, 042901 (2012). 5. S. Ben Ayed, A. Abdelkefi, F. Najar, et al., J. Intel. Mater. Syst. Struc., doi: 10.1177/1045389X13489365 (2013). 6. M. Bryant and E. Garcia, Proceedings of SPIE 7288, 728 (2009). 7. S. D. Kwon, Appl. Phys. Lett. 97, 164102 (2010). 8. A. Erturk, W. G. R. Vieira, C. De Marqui, et al., Appl. Phys. Lett. 96, 184103 (2010). 9. A. Abdelkefi, A. H. Nayfeh, and M. R. Hajj, Nonl. Dyn. 67, 925 (2012). 10. A. Abdelkefi, A. H. Nayfeh, and M. R. Hajj, Nonl. Dyn. 68, 531 (2012). 11. A. Abdelkefi, A. H. Nayfeh, and M. R. Hajj, Nonl. Dyn. 68, 519 (2012). 12. A. Abdelkefi, M. R. Hajj, and A. H. Nayfeh, J. Intel. Mater. Syst. Struct. 23, 1523 (2012). 13. A. Abdelkefi, M. R. Hajj, and A. H. Nayfeh, Nonl. Dyn. 70, 1377 (2012). 14. A. Abdelkefi, M. R. Hajj, and A. H. Nayfeh, Nonl. Dyn. 70, 1355 (2012). 15. Y. Yang, L. Zhao, and L. Tang, Appl. Phys. Lett. 102, 064105
Theor. Appl. Mech. Lett. 3, 041001 (2013)
(2013). 16. A. Abdelkefi, M. R. Hajj, and A. H. Nayfeh, Smar. Mater. Struc. 22, 015014 (2013). 17. A. Abdelkefi, Z. Yan, and M. R. Hajj, Smar. Mater. Struc. 22, 055026 (2013). 18. D. J. Inman and B. L. Grisso, Smart Structures and Materials Conference SPIE 6174, 61740T (2006). 19. S. P. Gurav, A. Kasyap, M. Sheplak, et al., Proceedings of the 10th AIAA/ISSSMO Multidisciplinary Analysis and Optimization Conference, 3559 (2004). 20. W. Zhou, W. H. Liao, and W. J. Li, Proc. Smart Structures and Materials Conference. SPIE, 233 (2005). 21. M. D. Conner, D. M. Tang, E. H. Dowell, et al., J. Fluid. Struc. 11, 89 (1997). 22. E. H. Dowell and D. Tang, AIAA J. 40, 1697 (2002). 23. A. Abdelkefi, R. Vasconcellos, F. Marques, et al., J. Soun. Vib. 331, 1898 (2012). 24. R. Vasconcellos, A. Abdelkefi, F. Marques, et al., J. Fluid. Struc. 31, 79 (2012). 25. V. C. Sousa, M de M. Anicezio, C. De Marqui, et al., Sma. Mater. Struc. 20, 094007 (2011). 26. A. Abdelkefi, R. Vasconcellos, A. H. Nayfeh, et al., Nonl. Dyn. 71, 159 (2013). 27. A. Agneni, F. Mastroddi, and G. M. Polli, Comput. Struc. 81, 91 (2003).