Regular and chaotic vibration in a piezoelectric energy harvester with fractional damping

The European Physical Journal Plus, Jun 2015

We examine a vibrational energy harvester consisting of a mechanical resonator with a fractional damping and electrical circuit coupled by a piezoelectric converter. By comparing the bifurcation diagrams and the power output we show that the fractional order of damping changes the system response considerably and affects the power output. Various dynamic responses of the energy harvester are examined using phase trajectory, Fourier spectrum, Multi-scale entropy and 0–1 test. The numerical analysis shows that the fractionally damped energy harvesting system exhibits chaos, and periodic motion, as the fractional order changes. The observed bifurcations strongly influence the power output.

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Regular and chaotic vibration in a piezoelectric energy harvester with fractional damping

Eur. Phys. J. Plus Regular and chaotic vibration in a piezoelectric energy harvester with fractional damping Junyi Cao 2 Arkadiusz Syta 1 Grzegorz Litak 0 1 Shengxi Zhou 2 Daniel J. Inman 4 Yangquan Chen 3 0 Laboratoire de G ́enie Electrique et Ferro ́electricit ́e, Institut National des Sciences Appliqu ́ees de Lyon , 69621 Villeurbanne cedex , France 1 Faculty of Mechanical Engineering, Lublin University of Technology , Nadbystrzycka 36, Lublin 20-618 , Poland 2 State Key Laboratory for Manufacturing Systems Engineering, Xi'an Jiaotong University , Xi'an 710049 , China 3 School of Engineering, University of California , Merced, 5200 North Lake Rd., Merced, CA 95343 , USA 4 Department of Aerospace Engineering, University of Michigan , Ann Arbor, MI 48109-2140 , USA We examine a vibrational energy harvester consisting of a mechanical resonator with a fractional damping and electrical circuit coupled by a piezoelectric converter. By comparing the bifurcation diagrams and the power output we show that the fractional order of damping changes the system response considerably and affects the power output. Various dynamic responses of the energy harvester are examined using phase trajectory, Fourier spectrum, Multi-scale entropy and 0-1 test. The numerical analysis shows that the fractionally damped energy harvesting system exhibits chaos, and periodic motion, as the fractional order changes. The observed bifurcations strongly influence the power output. 1 Introduction of the voltage output. Occurrence of intra- and inter-well oscillations as well as periodic and chaotic vibrations lead to different efficiencies in energy harvesting. Therefore, we identify the types and characteristic properties of given solutions by using nonlinear methods. 2 Mathematical model and equations of motion In the present model we considered the vertical flexible beam with fractional damping (fig. 2) and nonharmonic potential dependent on the magnets orientation angle. The differential equations reads: M x¨(t) + CDαx(t) + Kx(t) − Fm − Θu(t) = Fe −Fm = a0 + a1z(t) + a2z2(t) + . . . + anzn(t), where a0 = 0, a1 = 79.17 [N/m]; a2 = 0; a3 = −2.61 × 105 [N/m3] and an is zero for n > 3. In order to implement the fractional-order operator Dα into numerical simulations we used a polynomial expansion. Namely, for the fractional derivative approximation Dα we used a continued fraction expansion based on Tustin and/or Euler transformation [26–29]. The continued fraction expansion (CFE) based on the Tustin transformation is while CFE based on the Euler transformation is T 1 + z−1 X(z) ≈ X(z) ≈ Qq(z−1) Qq(z−1) where Z{·} is the Z-transform operator, X(z) represents the Z-transform of x(t), X(z) = Z{x[n]} = x[n]z−n and T is the sampling period, and n is the discrete time index. In the numerical simulation, the orders of p and q (in Pp and Qq polynomials) were equal to 10 and the sampling period T was set to 0.002. After the CFE decomposition the algebraic expression (eqs. (4) or (5)) can be transformed back to the time domain using the infinite impulse response (IIR) filter approximated to different fractional operators. 3 Numerical simulations and results Following [23] we use the same set of system parameters. The beam-magnet (fig. 1(a)) system is excited at the base with harmonic excitation. Figure 1(b) shows the restoring force potential V (x) (Fm − Kx = −dV (x)/dx) used in the calculations. The simulations were performed for eqs. (1) and (2) using various values of fractional order α ∈ (0, 1.5]. The results for displacement are summarized in figs. 2(a) and (b). The initial conditions were set for α = 0.01 as (x, x˙ , u, u˙ ) = (0, 0, 0, 0), and increased by 0.01 quasistaticaly during the simulations. Figures 2(a) and (b) show the bifurcation diagram of displacement using the distribution local maxima xmax and the main value of displacement x¯ calculated for 51 cycles, respectively. Note that the single or multiple discrete lines base excitation piezoelectric patches Fig. 2. (a) Bifurcation diagrams of displacement local maxima versus the fractional order α; (b) the mean value x¯ versus the fractional order α. [V1000 e g lta 750 o v f o 500 e c n ira 250 a v Fig. 3. (a) Bifurcation diagrams of displacement local voltage output versus the fractional order α; (b) Variance voltage, Var(u), output against α. 4 Fourier and multi-scale entropy analyses Fig. 4. Voltage open circuit time series and the corresponding phase portrait with delayed coordinate (the time delay Δt was fixed to one quarter of excitation period) for α = 0.35, 0.78, 0.85, 1.00, 1.195, 1.25 for (a)–(f), respectively. The same α values were marked by vertical lines in figs. 2–3. The black stroboscopic points show the cycles of the voltage in terms of Δt (four points per excitation period). of complexity of finite length time series. The concept of multi-scale entropy (MSE) is based on the coarse-graining procedure that uses a coarse-grained time series, as an average of the original data points within nonoverlapping windows by increasing the scale factor τ according to the following formula [31]: −5020 ] 10 V [ eug 0 a t l o v−10 −2020 Fig. 4. Continued. Fig. 5. Schematic picture showing the concept of higher scales. To estimate SampEn(u(τ), m, r) we count the number of vector pairs denoted by u(τ)(i) and u(τ)(j) in the time series of length m and m + 1 having distance d[u(τ)(i), u(τ)(j)] < r. We denote them by Pm and Pm+1, respectively. Finally, we define the sample entropy to be [35] Pm 1 τ τ 20 30 frequency f [Hz] 20 30 frequency f [Hz] × 10−2 2 E S M 1 C 20 30 frequency f [Hz] 5 The 0-1 test The “0-1 test”, invented by Gottwald and Melbourne [36, 37], can be applied for any systems of finite dimension to identify chaotic dynamics, however, it is based on the statistical properties of a single coordinate only. Thus it is suitable to quantify the response where only one parameter is measured in time. The test is related to the universal properties of dynamical systems, such as spectral measures, and can therefore distinguish a chaotic system from a regular one using a single variable. 20 30 frequency f [Hz] 20 30 frequency f [Hz] 20 30 frequency f [Hz] Fig. 6. Continued. A particular advantage of the 0-1 test over the frequency spectrum is that it provides information regarding the dynamics in a single parameter value, similar to the Lyapunov exponent. However, the Lyapunov exponent can be difficult to estimate in any nonsmooth simulated system or measured data [38]. Therefore the 0-1 test can provide a suitable algorithm to identify the chaotic solution [39–44]. To start the analysis, we discretize the investigated time series u(t) → u(i) using the characteristic delay time Δt equal to one quarter of the excitation period 2π/ω (as the points shown in figs. 3 and 4). This roughly indicates the vanishing of the mutual information [39, 45]. Starting from one of the initial map coordinates u(i), for sampling points i = 1, . . . , N , we define new coordinates p(n) and q(n) as p(n) = q(n) = (u(j) − u) (u(j) − u) where n = 1, . . . , N , u denotes the average value of u, σu the corresponding standard deviation, and c ∈ (0, π) is a constant. Note that q(n) is a complementary coordinate in the two-dimensional space. Furthermore, starting from the bounded coordinate u(i) we build a new series p(n) which can be either bounded or unbounded depending on the dynamics of the examined process. Continuing the calculation procedure, the total mean square displacement is defined as 1 N (p(j + n) − p(j))2 + (q(j + n) − q(j))2 . The asymptotic growth of Mc(n) can be easily characterised by the corresponding ratio Kc(n) Kc(n) = In the limit as n → ∞ (in present calculations n = nmax = 270, while N = 2700) we obtain the corresponding values of Kc for a chosen value of c. Note, our choice of the limits on nmax and N (in eqs. (13) and (14)) is consistent with that proposed by Gottwald and Melbourne [40, 41, 46]. N, nmax → ∞ but simultaneously nmax should be about N/10. It is important to note that the parameter c acts like a frequency in a spectral calculation. If c is badly chosen, it could resonate with the excitation frequency or its super- or sub-harmonics. In the 0-1 test, a periodic motion would yield a regular behaviour in the (p, q)- plane with corresponding Mc(n) constant in time. On the other hand, a nonperiodic motion is characterized by an expanding behaviour in the (p, q)-plane with corresponding Mc(n) increasing in time [40]. The disadvantage of the test, its strong dependence on the chosen parameter c, can be overcome by a proposed modification. Gottwald and Melbourne [36, 40] suggested randomly chosen values of c are taken and the median of the corresponding Kc-values are computed. In [40] a covariance formulation is used with Kc = cov(X, Mc) var(X)var(Mc) cov(x, y) = nmax n=1 var(x) = cov(x, x). (x(n) − x)(y(n) − y), where vectors X = [1, 2, . . . , nmax], and Mc = [Mc(1), Mc(2), . . . , Mc(nmax)]. In the above, the covariance cov(x, y) and variance var(x), for arbitrary vectors x and y of nmax elements, and the corresponding averages x and y, respectively, are defined as Note that using eqs. (15) and (16) automatically cancels out the means and eq. (12) can be defined without substraction of u. Finally, the median is taken of the Kc-values in eq. (15), corresponding to 100 equally spaced random values of c ∈ (0, π). Such an average K¯ -value can now be estimated for various orders α. The differences between regular motion (circle like in the (p, q)-plane in figs. 7(b), (d)–(f)) and the chaotic one (random walking like in the (p, q)-plane in figs. 7(a) and (c)) are visible on figs. 7(a)–(f) plotted for c = 1.0. Appart from the pattern, it is also worth noticing the difference in scale, which is about ten times larger in figs. 7(a) and (c) as in figs. 7(a)–(f). The estimated values K¯ = 0.993, −0.005, 0.994, 0.000, −0.005, −0.004 for the cases (a)–(f) (see also figs. 2 and 3), respectively. This is the direct confirmation that the cases, in figs. 7(a) and (c), represent chaotic responses while the rest of cases, in figs. 7(b), (d)–(f), are related to various periodic responses. 6 Summary and conclusions q−30 q−1 −15 −10 −5 strongly and its oscillations are limited to that of a single potential well. For small and medium values of alpha (α < 1) we noticed larger amplitude responses in both the mechanical resonator displacement and voltage output. However the power output significantly depended on the particular solution. For example, the appearance of chaotic vibrations reduced the harvested energy. In most cases, nonlinear systems are characterised by multiple solutions including periodic and nonperiodic, and/or resonant and nonresonant solutions. 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Junyi Cao, Arkadiusz Syta, Grzegorz Litak. Regular and chaotic vibration in a piezoelectric energy harvester with fractional damping, The European Physical Journal Plus, 2015, 103, DOI: 10.1140/epjp/i2015-15103-8