Performance of broadband tristable energy harvesters
MATEC Web of Conferences
Performance of broadband tristable energy harvesters
Shengxi Zhou 2
Grzegorz Litak 0 1
Junyi Cao 3
0 Department of Process Control, AGH University Science and Technology, Department of Process Control , PL Mickiewicza 30, PL-30-059 Krakow , Poland
1 Lublin University of Technology, Faculty of Mechanical Engineering , Nadbystrzycka 36, PL-20-618 Lublin , Poland
2 School of Aeronautics, Northwestern Polytechnical University , Xi'an, 710072 , China
3 School of Mechanical Engineering, Xi'an Jiaotong University , Xi'an, 710049 , China
We analyze dynamical responses of energy harvesters with a mechanical resonator possessing three potential wells. The frequency spectral analysis of simulated systems demonstrates that additional sub-harmonic and super-harmonic resonances appear. These additional solutions make the frequency broadband effect and effectively increase the voltage output. We show the characteristic features of the obtained solutions. Appearance of coexisting periodic solutions and chaotic solutions are also shortly discussed.
Currently, vibration energy harvesting from low-level ambient vibrations has been received
more and more attention [
]. The characteristics and frequency broadband performance of
nonlinear monostable and bistable energy harvesters have been numerically and
experimentally revealed and verif ed [
]. Due to distinct advantages over monostable and bistable
oscillators, a new class of broadband energy harvesters with three potential wells [
2 Modelling and numerical analysis
Figure 1 shows the schematic diagram of the tristable energy harvester (TEH) in this paper.
It is shown that the harvester mainly includes a piezoelectric beam with a tip magnet, two
external magnets, a load resistance and a supporting mechanism. The nonlinear magnetic
force can be produced by the interaction among the tip magnet and two external magnets,
which depends on the relative position of these magnets. Therefore, the equivalent nonlinear
restoring force of the TEH is the vector sum of the linear restoring force of the piezoelectric
beam and the nonlinear magnetic force. It can be found that the TEH has as many as f ve
equilibrium positions (1, 3 and 5 are stable equilibrium positions; 2 and 4 are unstable
equilibrium positions). Under the base acceleration excitation, the nonlinear electromechanical
model of the THE can be described by the following equations:
μ and β are coefficients of the cubic (x3) and quintic terms (x5) of the equivalent nonlinear
restoring force, respectively. The linear term coefficient standing by −x is set to be one for
analysis. In order to make f ve roots of the function Fr(x) (to yield a tristable characteristic),
μ and β should meet the specif c conditions which can be deduced based on Eq. (5). Then,
the potential energy function can be expressed as:
It is noted that all the parameters are dimensionless for numerical analysis, while these
parameters can be theoretically calculated or experimentally measured based on a specif c
energy harvesting device. In this study, the dimensionless system parameters are set to: ξ =
0.8, C p = 0.05, θ = 0.5, R = 1 × 107, μ = 6173.8, β = 8857709.8. As shown in Figure 2,
different from nonlinear monostable and bistable energy harvesters, the TEH has as many as
three potential wells, and two of them are symmetrical and have the same depth in this paper.
When the excitation level is low, the oscillations of the TEH will be limited to one potential
well following low-energy intrawell oscillations. When the excitation level is high enough,
the TEH will overcome their potential barriers and realize high-energy interwell oscillations
(which means that oscillation amplitude is large, and the motions cross all the potential wells).
Therefore, in order to investigate nonlinear characteristics of the TEH, the numerical analysis
should be provided.
The equations of motion, Eqs. (1)-(2), are used in simulations of the system. At f rst,
we perform calculations of non-stationary frequency sweep conditions. Voltage output (Fig.
3a) is ploted against frequency which is increased (Fig. 3b) or decreased with time. One
can see different voltage paths for increasing and decreasing frequencies which are typical
for nonlinear systems. For better clarity, we perform studies of stationary cases (Figs. 4-7),
plotting the phase portrait projected to the corresponding displacement-velocity plane and
voltage response in time and frequency domains.
In Fig. 4, we show the results for low frequency ω = 0.5 with nodal initial conditions.
Although, the low frequency dominates in the response (Figs. 4b-c) the small super-harmonic
frequencies of multiple orders, with respect to the excitation frequencies, are also present.
Finally, in the phase portrait, one can see a number of small loops around local minimum of
the potential and then jump to the other potential well. In the time series, this effect is related
to small corrugation imposed on the leading periodic response with the excitation period.
One should note that this is a nonlinear combination of large and small orbits.
By increasing the excitation frequency, we obtain a single orbit response (ω = 5 in Fig.
5). Note that the shape of the orbit is not circular (Fig. 5a) which is also a typical feature
for nonlinear systems. Consequently, the Fourier spectrum has a small super-harmonic
component of 2ω. In Fig. 6 we plot the corresponding results for ω = 9.5 and the nodal initial
conditions. Interestingly, this response is not periodic. It is presumably chaotic with the
excitation frequency as a leading frequency in the response and chaotic modulation with smeared
frequencies ranges ω ∈ [1.5,3] (see Fig. 6c). However, to tell more about the nature of this
solution, additional tests should be performed. The other interesting feature of this solution
is that the size of the orbit on the phase portrait is fairly small (Fig. 6a). Therefore this
solution is entirely contained in the middle potential well (see Fig. 2). Consequently, the voltage
output is small as well. This solution is not unique in the assumed set of system parameters.
Another solution can be easily found for different initial conditions. In Fig. 7, we show such
a solution for [x, x˙, v] = [0.06, 0, 0]. This solution represents a large orbit as possible to see
in Fig. 7a. The leading response frequency is two times smaller, therefore, the response can
be called as a sub-harmonic 1/2 resonance solution.
We analyzed dynamical responses of energy harvesters with a mechanical resonator
possessing three potential wells. The main advantage of this system is the appearance of additional
solutions absent in the linear system. These are sub-harmonic, super-harmonic and chaotic
solutions. In contrast to the bistable system where dominates odd sub-harmonic solutions
], the considered TEH show the even (namely period 2) sub-harmonic solution with a large
orbit where the symmetry of the attractor is broken by the nonlinear effects of TEH (see Fig.
 S.P. Beeby , M.J. Tudor , N.M. White , Meas. Sci. Technol . 17 , R175 ( 2006 )
 P.D. Mitcheson , E.M. Yeatman , G.K. Rao , A.S. Holmes , T.C. Green , Proceedings of the IEEE 96 , 1457 ( 2008 )
 R.L. Harne , K.W. Wang , Smart Mat. Struct . 22 , 023001 ( 2013 )
 S.P. Pellegrini , N. Tolou , M. Schenk , J.L. Herder , J. Intell . Mater. Syst. Struct . 24 , 1303 ( 2013 )
 J. Twiefel , H. Westermann , J. Intell . Mater. Syst. Struct . 24 , 1291 ( 2013 )
 M.F. Daqaq , R. Masana , A. Erturk , D.D. Quinn , Applied Mechanics Reviews 66 , 040801 ( 2014 )
 A. Syta , G. Litak, M.I. Friswell , S. Adhikari , Eur. Phys. J. B 89 , 99 ( 2016 )
 T. Huguet , A. Badel , A. , M. Lallart, Appl. Phys. Lett . 111 , 173905 ( 2017 )
 M.I. Friswell , S.F. Ali , S. Adhikari , A.W. Lees , O. Bilgen , G. Litak, J. Intell . Mater. Syst. Struct . 23 , 1505 ( 2012 )
 A. Erturk , J. Hoffmann , D.J. Inman , Appl. Phys. Lett . 94 , 254102 ( 2009 )
 F. Cottone , H. Vocca , L. Gammaitoni, Phys. Rev. Lett . 102 , 080601 ( 2009 )
 G Litak , MI Friswell, S Adhikari Appl . Phys. Lett . 96 , 214103 ( 2010 )
 S. Zhou , J. Cao , D.J. Inman , J. Lin , S. Liu , Z. Wang . Appl. Energy 133 , 33 ( 2014 )
 S. Zhou , L. Zuo, Commun. Nonlinear. Sci. 61 , 271 ( 2018 )
 S. Zhou , J. Cao , G. Litak, J. Lin . tm-Tech. Mess. https:// doi.org/10.1515/teme -2017- 0076 ( 2018 )