- Email: [email protected]
Extending the dynamic range of an energy harvester using nonlinear damping
Journal of Sound and Vibration 333 (2014) 623–629
Contents lists available at ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Rapid Communication
Extending the dynamic range of an energy harvester using nonlinear damping Maryam Ghandchi Tehrani n, Stephen J. Elliott Institute of Sound and Vibration Research, University of Southampton, SO17 1BJ, UK
a r t i c l e i n f o
abstract
Article history: Received 22 March 2013 Received in revised form 28 August 2013 Accepted 24 September 2013 Handling Editor: M.P. Cartmell Available online 1 November 2013
This paper introduces the use of nonlinear damping for extending the dynamic range of vibration energy harvesters. A cubic nonlinear damper is initially considered and the average harvested power and the throw are obtained for different sinusoidal base excitation amplitudes and frequencies, both numerically and analytically. It is demonstrated that when excited at resonance, at an amplitude below its maximum operational limit, the harvested power using a nonlinear damper can be significantly larger than that of a linear energy harvester, therefore expanding its dynamic range. A potential approach to implementing cubic nonlinearity using a shunted electromagnetic device is also presented. & 2013 Elsevier Ltd. All rights reserved.
1. Introduction Energy harvesting from ambient vibration has attracted significant attention in recent years [1,2]. Some interesting applications include low-power wireless sensors [3], harvesting power from human motion [4] and large-scale energy harvesters [5,6]. To increase the range of excitation frequency over which the vibration energy harvester operates [7,8], various nonlinear arrangements have been suggested, particularly using nonlinear springs [9–13]. In some such systems the nonlinear stiffness is accompanied by nonlinear damping. The dynamics of the nanoelectrodynamic resonator described in [14], for example, had both nonlinear stiffness and damping terms. In other applications, however, it is the vibration amplitude that is variable, and efficient harvesting is needed both at high excitation levels and at lower levels. In this paper nonlinear cubic damping, which has previously been used for vibration isolation [15,16], will be investigated as a method of increasing the dynamic range of a power harvester. Depending on the particular practical implementation, it could be that the introduction of nonlinear damping also leads to some nonlinear stiffness. This is not inevitable, however, and the velocity feedback implementation of nonlinear damping by Laalej et al. [15], for vibration isolation, achieved a good approximation to cubic damping with little effect on the device's stiffness. Ruzicke and Derby [17] also outline a model of the behaviour of fluid flow through an orifice as having n-th power damping, but no nonlinear stiffness. Similarly, recent work on the dynamics of small mobile phone loudspeakers [18] has identified significant nonlinear damping but a stiffness that is almost entirely linear. In the initial analysis of the device with cubic damping outlined here, the stiffness of the device is therefore assumed to be linear so as to be able to concentrate on the effect of nonlinear damping.
n
Corresponding author. Tel.: þ44 2380594933. E-mail addresses: [email protected], [email protected] (M. Ghandchi Tehrani), [email protected] (S.J. Elliott).
0022-460X/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jsv.2013.09.035
624
M. Ghandchi Tehrani, S.J. Elliott / Journal of Sound and Vibration 333 (2014) 623–629
Fig. 1. Single degree-of-freedom, base excited, energy harvesting system with a nonlinear damper and an enclosure that restricts the maximum throw.
Nonlinear Coulomb damping, in which the damping opposes the direction of motion but is of constant magnitude, has previously been used to model the damping in electrostatic MEMS devices [3] and in a magnetic levitation device [19]. The objective here, however, is not to model the damping of a particular device, but to design the device to have a particular nonlinearity, in order to achieve better performance. As a first step, we examine the ability of a power harvester with cubic nonlinearity to improve the dynamic range of a linear device. 2. Introductory theory for nonlinear energy harvesting A nonlinear single degree-of-freedom system (spring–mass–damper), as shown in Fig. 1, is subjected to a base excitation, where m is the mass, k is the suspension stiffness, y is the base displacement and z is the relative displacement between the mass and the base. A nonlinear damper is assumed, generating a force f(t), which includes a linear term c1 and a cubic nonlinear term c3. The system is harmonically excited from the base at frequency ω and amplitude Y, and the theory governing the mechanical behaviour of such a system is examined. For this nonlinear mechanical energy harvester, the dynamic equation can be written as follows: mz€ ðtÞ þ c1 z_ ðtÞ þ c3 z_ 3 ðtÞ þkzðtÞ ¼ my€ ðtÞ:
(1)
The harmonic base excitation is assumed to be yðtÞ ¼ Y sin ðωt φÞ:
(2)
where φ is the phase shift necessary to ensure that the response, z, is proportional to sin ðωtÞ, with no phase shift. If non-dimensional variables are used so that z z~ ¼ ; Y
z~ ′ ¼
and defining the non-dimensional parameters as rffiffiffiffiffi k c1 ωn ¼ ; ζ1 ¼ ; m 2mωn
z_ ; ωn Y
ζ3 ¼
z~ ″ ¼
c 3 ωn Y 2 ; 2m
z€ ; ω2n Y
τ ¼ ωn t;
(3)
Ω¼
ω ; ωn
(4)
we obtain the non-dimensional form of Eq. (1) as z~ ″ðτÞ þ 2ζ 1 z~ ′ðτÞ þ2ζ 3 z~ ′3 ðτÞ þ z~ ðτÞ ¼ Ω2 sin ðΩτ φÞ:
(5)
For this particular nonlinear system, it is demonstrated below that the response to a sinusoidal excitation is itself mostly sinusoidal. The method of harmonic balance can thus be used, which is described in detail in [20,21]. We approximate the response at the fundamental frequency Ω as z~ ðτÞ ¼ Z~ sin ðΩτÞ:
(6)
Substituting z~ ðtÞ and its derivatives into Eq. (5) yields Ω2 Z~ sin ðΩτÞ þ 2ζ 1 ΩZ~ cos ðΩτÞ þ 2ζ 3 Ω3 Z~ cos 3 ðΩτÞ þ Z~ sin ðΩτÞ ¼ Ω2 sin ðΩτ φÞ: 3
(7)
Using trigonometry, Eq. (7) can be simplified to 3 1 3 cos ð3ΩτÞ þ cos ðΩτÞ þ Z~ sin ðΩτÞ Ω2 Z~ sin ðΩτÞ þ 2ζ 1 ΩZ~ cos ðΩτÞ þ 2ζ 3 Ω3 Z~ 4 4 ¼ Ω2 ð sin ðΩτÞ cos φ cos ðΩτÞ sin φÞ:
(8)
M. Ghandchi Tehrani, S.J. Elliott / Journal of Sound and Vibration 333 (2014) 623–629
625
Partitioning the Sine terms and ignoring higher order harmonics lead to ð1 Ω2 ÞZ~ ¼ Ω2 cos φ:
(9)
3 3 2ζ 1 ΩZ~ þ ζ 3 Ω3 Z~ ¼ Ω2 sin ϕ: 2
(10)
Similarly partitioning the Cosine terms gives
Squaring Eqs. (9) and (10) and adding them together lead to 9 2 6 ~6 4 2 ζ Ω Z þ 6ζ 1 ζ 3 Ω4 Z~ þ ðð2ζ 1 ΩÞ2 þ ð1 Ω2 Þ2 ÞZ~ Ω4 ¼ 0; 4 3
(11)
From this the normalised throw for the nonlinear harvester is found to be Ω2 Z~ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 ð1 Ω2 Þ2 þð2ζ 1 Ω þð3=2Þζ 3 Ω3 Z~ Þ2
(12)
The equivalent linear damper for the nonlinear harvester, which has the same response at resonance, is thus 3 2 ζ 1ðeqeÞ ¼ ζ 3 Ω2 Z~ : 4
(13)
At resonance, Ω¼1, the throw is obtained by solving the following cubic equation: 3 ~3 ζ Z þ 2ζ 1 Z~ 1 ¼ 0: 2 3
(14)
Eq. (14) has a single real solution, since its discriminant is always negative. Unlike with a nonlinear stiffness [10], the system with nonlinear damping does not exhibit any form of bifurcation or jump phenomena, due to multiple solutions in the response amplitude, which makes its response straightforward to predict in practice. For the linear device, the frequency at which the throw is a maximum can be obtained from 1 Ωmax ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; 1 2ζ 21
(15)
and for the nonlinear device, it can be obtained from solving the cubic equation in Ω2max , 9 4 ζ3 Z~ Ω6max Ω2max þ1 ¼ 0: 8
(16)
both the linear and nonlinear expressions approximate to Ωmax ¼ 1 if the damping is small. If we write zðtÞ ¼ Z sin ðωtÞ, the average power absorbed by the damper, in watts, is the product of the force on the damper and its relative velocity, which is then given by 1 3 P ave ¼ c1 ω2 Z 2 þ c3 ω4 Z 4 ; 2 8
(17)
so that average power, normalised by mω2n Y 2 , can be written as 2 3 4 P~ ave ¼ ζ 1 Ω2 Z~ þ ζ3 Ω4 Z~ : 4
(18)
3. Simulation results Although the analysis above has been conducted for a device with both linear and cubic damping, the initial numerical simulations have been carried out for an entirely nonlinear mechanical energy harvester, with physical parameters of m ¼1 kg, k¼ 4π2 N/m, c1 ¼0 Ns/m and c3 ¼0.052 Ns3/m3, with a maximum throw given by Zmax ¼1 m. Maximum excitation amplitude is obtained from Y max ¼ ð3c3 ωn Z 3max =4 mÞ ¼ 0:246 m. An entirely linear harvester is also considered for comparison, which has the same relative displacement as the nonlinear system at resonance. For such an equivalent linear system, the damping is found to be c1 ¼1.55 Ns/m, so that the linear damping ratio is ζ1 ¼ 0.123. Since the damping ratio is small, the frequency of the maximum throw is almost at resonance: Ω¼1. Fig. 2 shows the relative displacement and the harvested power as a function of input excitation amplitude, when driven at resonance, for both the linear (blue dashed line) and nonlinear (red solid line) systems. Although the two systems have been chosen to have the same throw at resonance, for the maximum excitation amplitude, it is interesting to note that the two systems also produce the same harvested power at the fundamental frequency under these conditions. This result can be verified by substituting the equation for the equivalent linear damping ratio, for an equal on-resonance response, in Eq. (13), into the first term on the right-hand side of Eq. (18), which is then equal to the second term on the right-hand side of Eq. (18), for the harvested power. The maximum power under these conditions is obtained from the first term of Eq. (17) 2 by substituting Z ¼ ðmωn Y max =c1 Þ, to give P max ¼ ðk Y 2max =2c1 Þ which is also equal to P max ¼ 2c21 =3c3 , where c1 and c3 are
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Fig. 2. (a) Relative displacement and (b) for the linear system and nonlinear system with cubic damping, together with the theoretical limit of a highly nonlinear system, the solid line for the nonlinear case is calculated using the harmonic balance method and the dots are the result from time domain simulations (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.).
chosen to give the same relative displacement. For excitation levels below the maximum amplitude, Z max , both the relative displacement and the harvested power are greater for the nonlinear system than they are for the linear system. The power harvested when excited at 20 dB below the maximum amplitude, for example, is about 6.7 dB greater for the nonlinear system than it is for the linear system. Fig. 3 shows the relative displacement and the average harvested power as a function of the excitation frequency when the input excitation amplitude is at the maximum, Y max , and when it is a tenth of this, i.e. 20 dB below the maximum amplitude. Fig. 3 also shows, for the nonlinear system, the peak relative displacement and average harvested power obtained from the steady-state part of numerical time domain simulations of Eq. (1), when excited by sinusoidal base excitation at various frequencies. These results are similar to those obtained using the harmonic balance method, since the responses predicted from the time domain simulations are almost sinusoidal, as shown in Fig. 4. This is because, when the system is driven off resonance, the response is mainly controlled by either the linear mass, at low frequencies, or the linear stiffness, at high frequencies. Near the resonance the analytic and numerical results are very similar, since although the response is controlled by the nonlinear damper, the response at the resonance frequency is much greater than at three times the resonance frequency, so that the harmonics at this frequency, and the higher harmonics, are filtered out to leave a largely sinusoidal response. Although the comparison between the numerical and analytical results is only presented here as a particular case study, this study does appear to be representative for systems of this type.
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Fig. 3. (a) Relative displacement and (b) average harvested power as a function of excitation frequency when Y ¼ Y max , (c) relative displacement and (d) average harvested power when Y ¼ 0:1Y max for the linear and nonlinear system with cubic damping.
The relative displacement and harvested power of the nonlinear device are significantly greater, at resonance, than those of the linear device, when Y is 0:1Y max , since the effective linear damping, in Eq. (13), is much smaller in this case. 4. Implementation with shunted electromechanical device If the inertial mass of the vibration energy harvester drives an electromechanical device, the nonlinear mechanical damping could be implemented using a nonlinear electrical element shunting this device, as shown in Fig. 5. Assuming the electromechanical device is well coupled, then the relative force it produces, fem(t), is proportional to the current, i(t). The voltage it produces, v(t), is also proportional to the relative velocity z_ ðtÞ, so that f em ðtÞ ¼ TiðtÞ;
(19)
vðtÞ ¼ T z_ ðtÞ;
(20)
and
where T is the transduction coefficient. The current in the nonlinear electrical device is now assumed to be proportional to the cube of the voltage across it, so that iðtÞ ¼ Gv3 ðtÞ:
(21)
It is also assumed that the device implementing this characteristic is able to convert the electrical power: PðtÞ ¼ vðtÞiðtÞ ¼ Gv4 ðtÞ ¼ GT 4 z_ 4 ðtÞ;
(22)
which would otherwise be dissipated as heat, into storable electrical energy. Such a device could be realised with a demandset DC/DC switching converter, for example [22]. It is recognised that this is an idealised situation, since in practice mechanical damping and electrical resistance will inevitably be present in the electromechanical device, although the details will depend on the particular design used. Substituting Eq. (20) into Eq. (21), and using Eq. (19), allows the force due to the shunted electromechanical system to be written as f em ðtÞ ¼ GT 4 z_ 3 ðtÞ;
(23)
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M. Ghandchi Tehrani, S.J. Elliott / Journal of Sound and Vibration 333 (2014) 623–629
Fig. 4. The results of time domain simulations, dots, of the system with cubic damping excited at Y ¼ 0:1Y max at normalised frequency of Ω¼ 0.1, 1 and 10, together with the sinusoidal response predicted by the harmonic balance method, solid line.
which provides cubic mechanical damping. An alternative approach would be to vary the value of a linear shunt resistor with the excitation level, so that damping was greater at high excitation levels than at lower levels. This could be achieved by scheduling the shunt resistor on the measured response level or, to give an entirely electrical controller, on the level of voltage generated across the shunt. 5. Conclusions In this paper, nonlinear damping is introduced into a mechanical system for energy harvesting and its performance, including the relative displacement and the average harvested power, is compared with the linear model, using numerical and analytical solutions. The nonlinear harvester can harvest significantly more power at resonance, compared to the linear harvester, when excited below its maximum excitation level, as set by the maximum throw of the device. A potential implementation using an electromechanical device with a nonlinear electrical load is suggested. In practice it is possible that at the same time as using a nonlinear damper, to increase the dynamic range, a nonlinear stiffness will also be included in the design, to increase the frequency range of the harvester, particularly at low levels. The analysis of a system with nonlinear stiffness, as well as nonlinear damping, is likely to be significantly more complicated than that outlined here, because of the bifurcations that are possible in its response [9–13], and is left for future work. Although the simulations here have focused on the case of cubic damping, the results can be readily generalised to the case in which the damping force is proportional to the n-th power of the relative velocity, where n is an odd integer [17]. In this case, the response on resonance is proportional to the driving amplitude to the power of 1/n. The harvested power is then proportional to the input amplitude to the power of (n þ1)/2n. For cubic damping, n is equal to 3, and the slopes of Fig. 2(a) and (b) are thus equal to 1/3 dB/dB and 2/3 dB/dB, respectively, as is observed from these results. As n tends towards infinity, however, so that the damping force would be zero until the relative velocity was unity and then infinitely high for any greater velocity, the slope of Fig. 2(a) would tend to zero and the internal response would be independent of the excitation level, as marked with the black dashed-dotted line, but the slope of the average harvested power would tend to 1/2 dB/dB, however, as shown in Fig. 2(b), defining the theoretical limit for maximum harvested power. The compression in power output with the level achieved with cubic nonlinearity, 1/3 dB/dB, is thus significantly better than that achieved with a linear system, 1 dB/dB, but not very much less than that achieved with the most extreme form of nonlinear damper, which would give 1/2 dB/dB. An alternative approach would be to use a linear damper, whose value varies depending on the
M. Ghandchi Tehrani, S.J. Elliott / Journal of Sound and Vibration 333 (2014) 623–629
629
Fig. 5. An idealised electromechanical system, which couples the relative motion of the inertial mass to a nonlinear electrical device, D, for which the current is proportional to the cube of the voltage across it.
excitation amplitude, to achieve the throw limit. For a linear system at resonance, the relative displacement is Z¼mωnY/c1 and the average harvested power is Pave ¼k2Y2/2c1. If the damping c1 is adjusted with level so that Z is always equal to Z max , corresponding to the dashed-dotted line in Fig. 2(a), then c1 ¼ Ymωn =Z max , and the power harvested is P ave ¼ mω3n YZ max =2, which would again give a slope of 1/2 dB/dB with excitation level, as shown by the dashed-dotted line in Fig. 2(b). Acknowledgements The authors acknowledge the support provided by EPSRC from Grants EP/K005456/1 and EP/K003836/1. The authors would like to thank A. Cammarano, N.D. Sims and P. Green for their comments. References [1] C.B. Williams, R.B. Yates, Analysis of a micro-electric generator for microsystems, Sensors and Actuators A 52 (1996) 8–11. [2] N.G. Stephen, On energy harvesting from ambient vibration, Journal of Sound and Vibration 293 (2006) 409–425. [3] P.D. Mitcheson, T.C. Green, E.M. Yeatman, A.S. Holmes, Architectures for vibration-driven micropower generators, Journal of Microelectromechanical Systems 13 (3) (2004) 429–440. [4] P.D. Mitcheson, E.M. Yeatman, G. Kondala Rao, A.S. Holmes, T.C. Green, Energy harvesting from human and machine motion for wireless electronic devices, Proceedings of the IEEE 96 (9) (2008) 1457–1486. [5] B.S. Hendrickson, S.B. Brown, Harvest of motion, Mechanical Engineering (2008) 56–58. [6] I.L. Cassidy, J.T. Scruggs, S. Behrens, Design of electromagnetic energy harvesters for large-scale structural vibration applications, Proceedings of SPIE (7977), Active and Passive Smart Structures and Integrated Systems, 2011, http://dx.doi.org/10.1117/12.880639. [7] D. Zhu, M.J. Tudor, S.P. Beeby, Strategies for increasing the operating frequency range of vibration energy harvesters: a review, Measurement Science and Technology 21 (2010) 1–29. [8] F. Di Monaco, M. Ghandchi Tehrani, S.J. Elliott, E. Bonisoli, S. Tornincasa, Energy harvesting using semi-active control, Journal of Sound and Vibration 332 (23) (2013) 6033–6043. [9] B.P. Mann, N. Sims, Using nonlinearity to improve the performance of vibration-based energy harvesting devices, Proceedings of the 7th European Conference on Structural Dynamics, Southampton July 2008. [10] R. Ramlan, M.J. Brennan, B.R. Mace, I. Kovacic, Potential benefits of a non-linear stiffness in an energy harvesting device, Nonlinear Dynamics 59 (2010) 545–558. [11] B.P. Mann, N.D. Sims, Energy harvesting from the nonlinear oscillations of magnetic levitation, Journal of Sound and Vibration 319 (1-2) (2009) 515–530. [12] F. Cottone, H. Vocca, L. Gammaitoni, Nonlinear energy harvesting, Physical Review Letters 102 (8) (2009). [13] A. Erturk, D.J. Inman, Broadband piezoelectric power generation on high-energy orbits of the bistable duffing oscillator with electromechanical coupling, Journal of Sound and Vibration 330 (10) (2011) 2339–2353. [14] S.C. Jun, S. Moon, W. Kim, J.H. Cho, J.Y. Kang, Y. Jung, H. Yoon, J. Shin, I. Song, J. Choi, J.H. Choi, M.J. Bae, I.T. Han, S. Lee, J.M. Kim, Nonlinear characteristics in radio frequency nanoelectromechanical resonators, New Journal of Physics 12 (2010) 043023. [15] Z.Q. Lang, X.J. Jing, S.A. Billings, G.R. Tomlinson, Z.K. Peng, Theoretical study of the effects of nonlinear viscous damping on vibration isolation of sdof systems, Journal of Sound and Vibration 323 (2009) 352–365. [16] H. Laalej, Z.Q. Lang, S. Daley, I. Zazas, S.A. Billings, G.R. Tomlinson, Application of nonlinear damping to vibration isolation: an experimental study, Nonlinear Dynamics, 69 (2012) 409–421. [17] E.J. Ruzicka, T.F. Derby, Influence of damping in vibration isolation, Shock Vibration Monograph 7 (1971). [18] W. Klippel, Dominant nonlinearities in micro-speakers, Proceedings of AIA-DAGA 2013, Conference on Acoustics, Merano, Italy, March 2013. [19] P.L. Green, K. Worden, K. Atallah, N.D. Sims., The effect of duffing-type non-linearities and coulomb damping on the response of an energy harvester to random excitations, Journal of Intelligent Material Systems and Structures 23 (2012) 2039–2054. [20] A.H. Nayfeh, D.T. Mook, Nonlinear Oscillations, Wiley-VCH, USA, 2004. [21] K. Worden, G.R. Tomlinson, Nonlinearity in Structural Dynamics, Detection, Identification and Modelling, IOP, Bristol and Phyladelphia, 2001. [22] N. Mohan, T.M. Undeland, W.P. Robbins, Power Electronics – Converters, Applications and Design, John Wiley & Sons, USA, 1995, 148–150 (ISBN 0-471-58408-8).
Contents lists available at ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Rapid Communication
Extending the dynamic range of an energy harvester using nonlinear damping Maryam Ghandchi Tehrani n, Stephen J. Elliott Institute of Sound and Vibration Research, University of Southampton, SO17 1BJ, UK
a r t i c l e i n f o
abstract
Article history: Received 22 March 2013 Received in revised form 28 August 2013 Accepted 24 September 2013 Handling Editor: M.P. Cartmell Available online 1 November 2013
This paper introduces the use of nonlinear damping for extending the dynamic range of vibration energy harvesters. A cubic nonlinear damper is initially considered and the average harvested power and the throw are obtained for different sinusoidal base excitation amplitudes and frequencies, both numerically and analytically. It is demonstrated that when excited at resonance, at an amplitude below its maximum operational limit, the harvested power using a nonlinear damper can be significantly larger than that of a linear energy harvester, therefore expanding its dynamic range. A potential approach to implementing cubic nonlinearity using a shunted electromagnetic device is also presented. & 2013 Elsevier Ltd. All rights reserved.
1. Introduction Energy harvesting from ambient vibration has attracted significant attention in recent years [1,2]. Some interesting applications include low-power wireless sensors [3], harvesting power from human motion [4] and large-scale energy harvesters [5,6]. To increase the range of excitation frequency over which the vibration energy harvester operates [7,8], various nonlinear arrangements have been suggested, particularly using nonlinear springs [9–13]. In some such systems the nonlinear stiffness is accompanied by nonlinear damping. The dynamics of the nanoelectrodynamic resonator described in [14], for example, had both nonlinear stiffness and damping terms. In other applications, however, it is the vibration amplitude that is variable, and efficient harvesting is needed both at high excitation levels and at lower levels. In this paper nonlinear cubic damping, which has previously been used for vibration isolation [15,16], will be investigated as a method of increasing the dynamic range of a power harvester. Depending on the particular practical implementation, it could be that the introduction of nonlinear damping also leads to some nonlinear stiffness. This is not inevitable, however, and the velocity feedback implementation of nonlinear damping by Laalej et al. [15], for vibration isolation, achieved a good approximation to cubic damping with little effect on the device's stiffness. Ruzicke and Derby [17] also outline a model of the behaviour of fluid flow through an orifice as having n-th power damping, but no nonlinear stiffness. Similarly, recent work on the dynamics of small mobile phone loudspeakers [18] has identified significant nonlinear damping but a stiffness that is almost entirely linear. In the initial analysis of the device with cubic damping outlined here, the stiffness of the device is therefore assumed to be linear so as to be able to concentrate on the effect of nonlinear damping.
n
Corresponding author. Tel.: þ44 2380594933. E-mail addresses: [email protected], [email protected] (M. Ghandchi Tehrani), [email protected] (S.J. Elliott).
0022-460X/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jsv.2013.09.035
624
M. Ghandchi Tehrani, S.J. Elliott / Journal of Sound and Vibration 333 (2014) 623–629
Fig. 1. Single degree-of-freedom, base excited, energy harvesting system with a nonlinear damper and an enclosure that restricts the maximum throw.
Nonlinear Coulomb damping, in which the damping opposes the direction of motion but is of constant magnitude, has previously been used to model the damping in electrostatic MEMS devices [3] and in a magnetic levitation device [19]. The objective here, however, is not to model the damping of a particular device, but to design the device to have a particular nonlinearity, in order to achieve better performance. As a first step, we examine the ability of a power harvester with cubic nonlinearity to improve the dynamic range of a linear device. 2. Introductory theory for nonlinear energy harvesting A nonlinear single degree-of-freedom system (spring–mass–damper), as shown in Fig. 1, is subjected to a base excitation, where m is the mass, k is the suspension stiffness, y is the base displacement and z is the relative displacement between the mass and the base. A nonlinear damper is assumed, generating a force f(t), which includes a linear term c1 and a cubic nonlinear term c3. The system is harmonically excited from the base at frequency ω and amplitude Y, and the theory governing the mechanical behaviour of such a system is examined. For this nonlinear mechanical energy harvester, the dynamic equation can be written as follows: mz€ ðtÞ þ c1 z_ ðtÞ þ c3 z_ 3 ðtÞ þkzðtÞ ¼ my€ ðtÞ:
(1)
The harmonic base excitation is assumed to be yðtÞ ¼ Y sin ðωt φÞ:
(2)
where φ is the phase shift necessary to ensure that the response, z, is proportional to sin ðωtÞ, with no phase shift. If non-dimensional variables are used so that z z~ ¼ ; Y
z~ ′ ¼
and defining the non-dimensional parameters as rffiffiffiffiffi k c1 ωn ¼ ; ζ1 ¼ ; m 2mωn
z_ ; ωn Y
ζ3 ¼
z~ ″ ¼
c 3 ωn Y 2 ; 2m
z€ ; ω2n Y
τ ¼ ωn t;
(3)
Ω¼
ω ; ωn
(4)
we obtain the non-dimensional form of Eq. (1) as z~ ″ðτÞ þ 2ζ 1 z~ ′ðτÞ þ2ζ 3 z~ ′3 ðτÞ þ z~ ðτÞ ¼ Ω2 sin ðΩτ φÞ:
(5)
For this particular nonlinear system, it is demonstrated below that the response to a sinusoidal excitation is itself mostly sinusoidal. The method of harmonic balance can thus be used, which is described in detail in [20,21]. We approximate the response at the fundamental frequency Ω as z~ ðτÞ ¼ Z~ sin ðΩτÞ:
(6)
Substituting z~ ðtÞ and its derivatives into Eq. (5) yields Ω2 Z~ sin ðΩτÞ þ 2ζ 1 ΩZ~ cos ðΩτÞ þ 2ζ 3 Ω3 Z~ cos 3 ðΩτÞ þ Z~ sin ðΩτÞ ¼ Ω2 sin ðΩτ φÞ: 3
(7)
Using trigonometry, Eq. (7) can be simplified to 3 1 3 cos ð3ΩτÞ þ cos ðΩτÞ þ Z~ sin ðΩτÞ Ω2 Z~ sin ðΩτÞ þ 2ζ 1 ΩZ~ cos ðΩτÞ þ 2ζ 3 Ω3 Z~ 4 4 ¼ Ω2 ð sin ðΩτÞ cos φ cos ðΩτÞ sin φÞ:
(8)
M. Ghandchi Tehrani, S.J. Elliott / Journal of Sound and Vibration 333 (2014) 623–629
625
Partitioning the Sine terms and ignoring higher order harmonics lead to ð1 Ω2 ÞZ~ ¼ Ω2 cos φ:
(9)
3 3 2ζ 1 ΩZ~ þ ζ 3 Ω3 Z~ ¼ Ω2 sin ϕ: 2
(10)
Similarly partitioning the Cosine terms gives
Squaring Eqs. (9) and (10) and adding them together lead to 9 2 6 ~6 4 2 ζ Ω Z þ 6ζ 1 ζ 3 Ω4 Z~ þ ðð2ζ 1 ΩÞ2 þ ð1 Ω2 Þ2 ÞZ~ Ω4 ¼ 0; 4 3
(11)
From this the normalised throw for the nonlinear harvester is found to be Ω2 Z~ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 ð1 Ω2 Þ2 þð2ζ 1 Ω þð3=2Þζ 3 Ω3 Z~ Þ2
(12)
The equivalent linear damper for the nonlinear harvester, which has the same response at resonance, is thus 3 2 ζ 1ðeqeÞ ¼ ζ 3 Ω2 Z~ : 4
(13)
At resonance, Ω¼1, the throw is obtained by solving the following cubic equation: 3 ~3 ζ Z þ 2ζ 1 Z~ 1 ¼ 0: 2 3
(14)
Eq. (14) has a single real solution, since its discriminant is always negative. Unlike with a nonlinear stiffness [10], the system with nonlinear damping does not exhibit any form of bifurcation or jump phenomena, due to multiple solutions in the response amplitude, which makes its response straightforward to predict in practice. For the linear device, the frequency at which the throw is a maximum can be obtained from 1 Ωmax ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; 1 2ζ 21
(15)
and for the nonlinear device, it can be obtained from solving the cubic equation in Ω2max , 9 4 ζ3 Z~ Ω6max Ω2max þ1 ¼ 0: 8
(16)
both the linear and nonlinear expressions approximate to Ωmax ¼ 1 if the damping is small. If we write zðtÞ ¼ Z sin ðωtÞ, the average power absorbed by the damper, in watts, is the product of the force on the damper and its relative velocity, which is then given by 1 3 P ave ¼ c1 ω2 Z 2 þ c3 ω4 Z 4 ; 2 8
(17)
so that average power, normalised by mω2n Y 2 , can be written as 2 3 4 P~ ave ¼ ζ 1 Ω2 Z~ þ ζ3 Ω4 Z~ : 4
(18)
3. Simulation results Although the analysis above has been conducted for a device with both linear and cubic damping, the initial numerical simulations have been carried out for an entirely nonlinear mechanical energy harvester, with physical parameters of m ¼1 kg, k¼ 4π2 N/m, c1 ¼0 Ns/m and c3 ¼0.052 Ns3/m3, with a maximum throw given by Zmax ¼1 m. Maximum excitation amplitude is obtained from Y max ¼ ð3c3 ωn Z 3max =4 mÞ ¼ 0:246 m. An entirely linear harvester is also considered for comparison, which has the same relative displacement as the nonlinear system at resonance. For such an equivalent linear system, the damping is found to be c1 ¼1.55 Ns/m, so that the linear damping ratio is ζ1 ¼ 0.123. Since the damping ratio is small, the frequency of the maximum throw is almost at resonance: Ω¼1. Fig. 2 shows the relative displacement and the harvested power as a function of input excitation amplitude, when driven at resonance, for both the linear (blue dashed line) and nonlinear (red solid line) systems. Although the two systems have been chosen to have the same throw at resonance, for the maximum excitation amplitude, it is interesting to note that the two systems also produce the same harvested power at the fundamental frequency under these conditions. This result can be verified by substituting the equation for the equivalent linear damping ratio, for an equal on-resonance response, in Eq. (13), into the first term on the right-hand side of Eq. (18), which is then equal to the second term on the right-hand side of Eq. (18), for the harvested power. The maximum power under these conditions is obtained from the first term of Eq. (17) 2 by substituting Z ¼ ðmωn Y max =c1 Þ, to give P max ¼ ðk Y 2max =2c1 Þ which is also equal to P max ¼ 2c21 =3c3 , where c1 and c3 are
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Fig. 2. (a) Relative displacement and (b) for the linear system and nonlinear system with cubic damping, together with the theoretical limit of a highly nonlinear system, the solid line for the nonlinear case is calculated using the harmonic balance method and the dots are the result from time domain simulations (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.).
chosen to give the same relative displacement. For excitation levels below the maximum amplitude, Z max , both the relative displacement and the harvested power are greater for the nonlinear system than they are for the linear system. The power harvested when excited at 20 dB below the maximum amplitude, for example, is about 6.7 dB greater for the nonlinear system than it is for the linear system. Fig. 3 shows the relative displacement and the average harvested power as a function of the excitation frequency when the input excitation amplitude is at the maximum, Y max , and when it is a tenth of this, i.e. 20 dB below the maximum amplitude. Fig. 3 also shows, for the nonlinear system, the peak relative displacement and average harvested power obtained from the steady-state part of numerical time domain simulations of Eq. (1), when excited by sinusoidal base excitation at various frequencies. These results are similar to those obtained using the harmonic balance method, since the responses predicted from the time domain simulations are almost sinusoidal, as shown in Fig. 4. This is because, when the system is driven off resonance, the response is mainly controlled by either the linear mass, at low frequencies, or the linear stiffness, at high frequencies. Near the resonance the analytic and numerical results are very similar, since although the response is controlled by the nonlinear damper, the response at the resonance frequency is much greater than at three times the resonance frequency, so that the harmonics at this frequency, and the higher harmonics, are filtered out to leave a largely sinusoidal response. Although the comparison between the numerical and analytical results is only presented here as a particular case study, this study does appear to be representative for systems of this type.
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Fig. 3. (a) Relative displacement and (b) average harvested power as a function of excitation frequency when Y ¼ Y max , (c) relative displacement and (d) average harvested power when Y ¼ 0:1Y max for the linear and nonlinear system with cubic damping.
The relative displacement and harvested power of the nonlinear device are significantly greater, at resonance, than those of the linear device, when Y is 0:1Y max , since the effective linear damping, in Eq. (13), is much smaller in this case. 4. Implementation with shunted electromechanical device If the inertial mass of the vibration energy harvester drives an electromechanical device, the nonlinear mechanical damping could be implemented using a nonlinear electrical element shunting this device, as shown in Fig. 5. Assuming the electromechanical device is well coupled, then the relative force it produces, fem(t), is proportional to the current, i(t). The voltage it produces, v(t), is also proportional to the relative velocity z_ ðtÞ, so that f em ðtÞ ¼ TiðtÞ;
(19)
vðtÞ ¼ T z_ ðtÞ;
(20)
and
where T is the transduction coefficient. The current in the nonlinear electrical device is now assumed to be proportional to the cube of the voltage across it, so that iðtÞ ¼ Gv3 ðtÞ:
(21)
It is also assumed that the device implementing this characteristic is able to convert the electrical power: PðtÞ ¼ vðtÞiðtÞ ¼ Gv4 ðtÞ ¼ GT 4 z_ 4 ðtÞ;
(22)
which would otherwise be dissipated as heat, into storable electrical energy. Such a device could be realised with a demandset DC/DC switching converter, for example [22]. It is recognised that this is an idealised situation, since in practice mechanical damping and electrical resistance will inevitably be present in the electromechanical device, although the details will depend on the particular design used. Substituting Eq. (20) into Eq. (21), and using Eq. (19), allows the force due to the shunted electromechanical system to be written as f em ðtÞ ¼ GT 4 z_ 3 ðtÞ;
(23)
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Fig. 4. The results of time domain simulations, dots, of the system with cubic damping excited at Y ¼ 0:1Y max at normalised frequency of Ω¼ 0.1, 1 and 10, together with the sinusoidal response predicted by the harmonic balance method, solid line.
which provides cubic mechanical damping. An alternative approach would be to vary the value of a linear shunt resistor with the excitation level, so that damping was greater at high excitation levels than at lower levels. This could be achieved by scheduling the shunt resistor on the measured response level or, to give an entirely electrical controller, on the level of voltage generated across the shunt. 5. Conclusions In this paper, nonlinear damping is introduced into a mechanical system for energy harvesting and its performance, including the relative displacement and the average harvested power, is compared with the linear model, using numerical and analytical solutions. The nonlinear harvester can harvest significantly more power at resonance, compared to the linear harvester, when excited below its maximum excitation level, as set by the maximum throw of the device. A potential implementation using an electromechanical device with a nonlinear electrical load is suggested. In practice it is possible that at the same time as using a nonlinear damper, to increase the dynamic range, a nonlinear stiffness will also be included in the design, to increase the frequency range of the harvester, particularly at low levels. The analysis of a system with nonlinear stiffness, as well as nonlinear damping, is likely to be significantly more complicated than that outlined here, because of the bifurcations that are possible in its response [9–13], and is left for future work. Although the simulations here have focused on the case of cubic damping, the results can be readily generalised to the case in which the damping force is proportional to the n-th power of the relative velocity, where n is an odd integer [17]. In this case, the response on resonance is proportional to the driving amplitude to the power of 1/n. The harvested power is then proportional to the input amplitude to the power of (n þ1)/2n. For cubic damping, n is equal to 3, and the slopes of Fig. 2(a) and (b) are thus equal to 1/3 dB/dB and 2/3 dB/dB, respectively, as is observed from these results. As n tends towards infinity, however, so that the damping force would be zero until the relative velocity was unity and then infinitely high for any greater velocity, the slope of Fig. 2(a) would tend to zero and the internal response would be independent of the excitation level, as marked with the black dashed-dotted line, but the slope of the average harvested power would tend to 1/2 dB/dB, however, as shown in Fig. 2(b), defining the theoretical limit for maximum harvested power. The compression in power output with the level achieved with cubic nonlinearity, 1/3 dB/dB, is thus significantly better than that achieved with a linear system, 1 dB/dB, but not very much less than that achieved with the most extreme form of nonlinear damper, which would give 1/2 dB/dB. An alternative approach would be to use a linear damper, whose value varies depending on the
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Fig. 5. An idealised electromechanical system, which couples the relative motion of the inertial mass to a nonlinear electrical device, D, for which the current is proportional to the cube of the voltage across it.
excitation amplitude, to achieve the throw limit. For a linear system at resonance, the relative displacement is Z¼mωnY/c1 and the average harvested power is Pave ¼k2Y2/2c1. If the damping c1 is adjusted with level so that Z is always equal to Z max , corresponding to the dashed-dotted line in Fig. 2(a), then c1 ¼ Ymωn =Z max , and the power harvested is P ave ¼ mω3n YZ max =2, which would again give a slope of 1/2 dB/dB with excitation level, as shown by the dashed-dotted line in Fig. 2(b). Acknowledgements The authors acknowledge the support provided by EPSRC from Grants EP/K005456/1 and EP/K003836/1. The authors would like to thank A. Cammarano, N.D. Sims and P. Green for their comments. References [1] C.B. Williams, R.B. Yates, Analysis of a micro-electric generator for microsystems, Sensors and Actuators A 52 (1996) 8–11. [2] N.G. Stephen, On energy harvesting from ambient vibration, Journal of Sound and Vibration 293 (2006) 409–425. [3] P.D. Mitcheson, T.C. Green, E.M. Yeatman, A.S. Holmes, Architectures for vibration-driven micropower generators, Journal of Microelectromechanical Systems 13 (3) (2004) 429–440. [4] P.D. Mitcheson, E.M. Yeatman, G. Kondala Rao, A.S. Holmes, T.C. Green, Energy harvesting from human and machine motion for wireless electronic devices, Proceedings of the IEEE 96 (9) (2008) 1457–1486. [5] B.S. Hendrickson, S.B. Brown, Harvest of motion, Mechanical Engineering (2008) 56–58. [6] I.L. Cassidy, J.T. Scruggs, S. Behrens, Design of electromagnetic energy harvesters for large-scale structural vibration applications, Proceedings of SPIE (7977), Active and Passive Smart Structures and Integrated Systems, 2011, http://dx.doi.org/10.1117/12.880639. [7] D. Zhu, M.J. Tudor, S.P. Beeby, Strategies for increasing the operating frequency range of vibration energy harvesters: a review, Measurement Science and Technology 21 (2010) 1–29. [8] F. Di Monaco, M. Ghandchi Tehrani, S.J. Elliott, E. Bonisoli, S. Tornincasa, Energy harvesting using semi-active control, Journal of Sound and Vibration 332 (23) (2013) 6033–6043. [9] B.P. Mann, N. Sims, Using nonlinearity to improve the performance of vibration-based energy harvesting devices, Proceedings of the 7th European Conference on Structural Dynamics, Southampton July 2008. [10] R. Ramlan, M.J. Brennan, B.R. Mace, I. Kovacic, Potential benefits of a non-linear stiffness in an energy harvesting device, Nonlinear Dynamics 59 (2010) 545–558. [11] B.P. Mann, N.D. Sims, Energy harvesting from the nonlinear oscillations of magnetic levitation, Journal of Sound and Vibration 319 (1-2) (2009) 515–530. [12] F. Cottone, H. Vocca, L. Gammaitoni, Nonlinear energy harvesting, Physical Review Letters 102 (8) (2009). [13] A. Erturk, D.J. Inman, Broadband piezoelectric power generation on high-energy orbits of the bistable duffing oscillator with electromechanical coupling, Journal of Sound and Vibration 330 (10) (2011) 2339–2353. [14] S.C. Jun, S. Moon, W. Kim, J.H. Cho, J.Y. Kang, Y. Jung, H. Yoon, J. Shin, I. Song, J. Choi, J.H. Choi, M.J. Bae, I.T. Han, S. Lee, J.M. Kim, Nonlinear characteristics in radio frequency nanoelectromechanical resonators, New Journal of Physics 12 (2010) 043023. [15] Z.Q. Lang, X.J. Jing, S.A. Billings, G.R. Tomlinson, Z.K. Peng, Theoretical study of the effects of nonlinear viscous damping on vibration isolation of sdof systems, Journal of Sound and Vibration 323 (2009) 352–365. [16] H. Laalej, Z.Q. Lang, S. Daley, I. Zazas, S.A. Billings, G.R. Tomlinson, Application of nonlinear damping to vibration isolation: an experimental study, Nonlinear Dynamics, 69 (2012) 409–421. [17] E.J. Ruzicka, T.F. Derby, Influence of damping in vibration isolation, Shock Vibration Monograph 7 (1971). [18] W. Klippel, Dominant nonlinearities in micro-speakers, Proceedings of AIA-DAGA 2013, Conference on Acoustics, Merano, Italy, March 2013. [19] P.L. Green, K. Worden, K. Atallah, N.D. Sims., The effect of duffing-type non-linearities and coulomb damping on the response of an energy harvester to random excitations, Journal of Intelligent Material Systems and Structures 23 (2012) 2039–2054. [20] A.H. Nayfeh, D.T. Mook, Nonlinear Oscillations, Wiley-VCH, USA, 2004. [21] K. Worden, G.R. Tomlinson, Nonlinearity in Structural Dynamics, Detection, Identification and Modelling, IOP, Bristol and Phyladelphia, 2001. [22] N. Mohan, T.M. Undeland, W.P. Robbins, Power Electronics – Converters, Applications and Design, John Wiley & Sons, USA, 1995, 148–150 (ISBN 0-471-58408-8).