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Dynamics of a coupled nonlinear energy harvester under colored noise and periodic excitations
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Dynamics of a coupled nonlinear energy harvester under colored noise and periodic excitations Yanxia Zhang , Yanfei Jin , Pengfei Xu PII: DOI: Reference:
S0020-7403(19)33429-0 https://doi.org/10.1016/j.ijmecsci.2020.105418 MS 105418
To appear in:
International Journal of Mechanical Sciences
Received date: Revised date: Accepted date:
10 September 2019 31 December 2019 2 January 2020
Please cite this article as: Yanxia Zhang , Yanfei Jin , Pengfei Xu , Dynamics of a coupled nonlinear energy harvester under colored noise and periodic excitations, International Journal of Mechanical Sciences (2020), doi: https://doi.org/10.1016/j.ijmecsci.2020.105418
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Highlights:
Comparisons of the effects of mono-, bi- and tri-stable potentials are analyzed in a coupled VEH driven by colored noise.
An extended stochastic averaging is applied to deal with the strongly nonlinear tri-stable system under colored noise.
Tri-stable VEH can achieve better SR effect and higher power conversion efficiency compared with bi-stable VEH.
Larger periodic force can enhance SR effect but weaken power conversion efficiency for tri-stable VEH.
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Dynamics of a coupled nonlinear energy harvester under colored noise and periodic excitations Yanxia Zhang, Yanfei Jin, Pengfei Xu Department of Mechanics, Beijing Institute of Technology, Beijing 100081, China
Abstract: Vibration energy harvester (VEH) has proven to be a favorable potential technique to supply continuous energy from ambient vibrations and its performance is greatly influenced by the design of potential structures. Motivated by the enhancement of energy harvesting performance, an electromechanical coupled VEH under colored noise and periodic excitations is investigated theoretically and numerically, and comparisons of the effects of mono-, bi- and tri-stable potentials are analyzed in detail. An uncoupled equivalent system, together with its joint probability density function (PDF), are derived by using the generalized harmonic transformation and the stochastic averaging based on energy-dependent frequency. For three kinds of different potentials, the effects of the periodic excitation, colored noise, potential shape and other crucial system parameters are explored on the dynamical behaviors. Results show that tri-stable VEH outperforms the mono- and bi-stable VEHs in mean harvested power and can achieve better stochastic resonance (SR) effect and higher power conversion efficiency by choosing the optimal system parameters, such as noise intensity, electromechanical coupling coefficient and time constant ratio. Moreover, the periodic excitation can change the topological characteristic of transient PDF, and larger periodic force can enhance SR effect but will weaken the power conversion efficiency. The theoretical results obtained by the proposed stochastic averaging method are well validated by numerical simulations. Keywords: Coupled nonlinear VEH; Stochastic averaging; Colored noise; Periodic excitation; Stochastic resonance.
1. Introduction On account of the growing demand in electric energy and the limitation of battery lifespan, vibration energy harvester (VEH), instead of the traditional chemical energy battery, become a promising potential technique to supply continuous energy because of its ability of harvesting energy from ambient vibrations. In the design of VEH, the electromagnetic transducers [1,2] and piezoelectric transducers [3,4] are the most common ways to realize energy harvesting conversion from vibration to electricity availably. Traditionally, VEHs are designed with linear stiffness type for simplicity. Whereas, linear design suffers from a critical shortcoming in the frequency bandwidth, and a narrow effective bandwidth makes it inefficient to harvest energy from the ambience vibration sources with a wider spectrum. Then, nonlinear stiffness type is introduced into VEHs by taking this point into account, which can extend the effective frequency bandwidth and can enhance energy harvesting performance from ambient vibrations efficiently [5-12]. According to the shape of potential function determined by nonlinear stiffness type, the nonlinear VEHs can be generally divided into three major categories, i.e. mono-stable, bi-stable and tri-stable VEHs. Currently, multitude of researchers have examined the mono-stable and bi-stable nonlinear VEHs [5-9]. Much of research concerning nonlinear mono-stable VEHs [5-7] have manifested that nonlinearity can give rise to extending the range of frequency and result in larger amplitude response compared with linear VEHs. Unfortunately, mono-stable VEHs behave much like linear VEHs especially under a small excitation amplitude. In order to
Corresponding author. E-mail address:[email protected]
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improve the broadband performance efficiently, bi-stable VEH with double-well potential structure is proposed and has exerted a tremendous fascination on researchers because of its favorable capability in harvesting energy over a considerable frequency range in a large-orbit inter-well motion [8-10]. As an example, He et al. [9] concluded that bi-stable VEH, compared with mono-stable one, can produce much more power through investigating the influence of nonlinear stiffness on the average output power of VEHs in the cases of mono-stable and bi-stable potentials. However, due to the existence of a potential barrier between two potential wells, the ideal inter-well motion in the bi-stable VEH cannot be activated if the excitation level is below the required threshold value. To overcome these drawbacks in mono-stable and bi-stable VEHs, the design of the nonlinear VEHs has been further improved to a kind of tri-stable VEH. Tri-stable VEH has been a hot topic in recent studies [11-13], owing to its ability of achieving sustained large-amplitude electric response even for low-lever excitations. For instance, Panyam et al. [11] investigated the response of a tri-stable VEH excited by a harmonic function, and pointed out that tri-stable potential structure is beneficial to extend the vibratory bandwidths. Zhou et al. [12,13] conducted numerical and experimental research on a tri-stable VEH, and revealed that tri-stable VEH, compared with bi-stable VEH, can effectively harvest energy in a larger low frequency range. However, these studies on tri-stable VEHs mainly focused on deterministic systems. Due to the universality and the inevitable of environmental random vibration sources, it is indispensable to study the stochastic dynamical behaviors of the stochastic systems for tri-stable VEHs for purpose further improving the energy harvesting performance. In engineering practical application, noisy stochastic dynamical systems mostly display some interesting nonlinear phenomena, such as stochastic bifurcation, stochastic transition and stochastic resonance (SR) [14-16]. To the authors’ knowledge, current work on the tri-stable VEHs stochastic systems mainly focused on Gaussian white noise excitation [16,17]. Jin et al. [16] discussed the effects of Gaussian white noise and nonlinear stiffness coefficients on the tri-stable VEH stochastic system by using SR theory and concluded that a suitable noise dosage is rewarding to enhance the energy harvesting performance. Nevertheless, since most of noises in nature have a different length of correlation time, Gaussian white noise, unlike colored noise, is just the ideal case for describing the random fluctuations from ambient noisy vibration. Colored noise has the statistical characters of non-zero correlation time which makes it better beneficial to describe the stochastic ambience fluctuations. For example, Liu et al. [18] presented that colored noise has a crucial influence on output power of a coupled mono-stable harvester. Yang et al. [19] studied the power harvested from a bi-stable hybrid harvester under correlated colored noise, and concluded that harvested energy can be enhanced with the suitable choices of noise intensities of colored noise. Zhang et al. [14] conducted a numerical research on a nonlinear tri-stable VEH under colored noise, and concluded that a proper noise intensity of colored noise is exceedingly helpful in enhancing the SR effect. Whereas, to date, the theoretical analysis on colored noise mainly focuses on mono- and bi-stable VEHs. Yet limited works have considered the colored noise in the theoretical research on stochastic dynamics of coupled tri-stable VEHs. Consequently, it is indispensable to pour attention into exploring the influence of colored noise on tri-stable VEHs in practice by considering the characteristic of the stochastic vibration sources in nature. This work is devoted to exploring the stochastic dynamical behaviors of a coupled nonlinear VEH under colored noise and periodic excitations through the extended stochastic averaging method and excavating the superiority of tri-stable VEHs in harvesting energy compared with mono-stable and bi-stable VEHs. It has been proved that stochastic averaging method is an effective and powerful theoretical method in studying stochastic dynamics and has been widely used to stochastic systems currently [20-25]. Due to the complexity of the dynamical behaviors in bi-stable stochastic systems, Zhu et al. [26] developed the stochastic averaging method through introducing the correlation functions of the excitation processes and adopting the variable natural frequency and period of the system corresponding to the different energy levels. Whereas, the dynamical behavior in tri-stable stochastic systems is much more complicated. Although the stochastic averaging method is being employed widely in 3
multitude of researches, it mainly focuses on mono- and bi-stable stochastic dynamical systems. The authors [25] have ever attempted applying the stochastic averaging method of amplitude into tri-stable systems excited only by colored noise, and found that the method has certain limitations on the valid range of theoretical results because the inverse function of triple-well potential does not exist. To date, few method developed is pleasurable and most methods are far from ready to be used in tri-stable stochastic dynamical systems under both noise and periodic excitation. Accordingly, this work, based on the idea of the stochastic averaging method developed by Zhu et al. [26], will extend the method to deal with the case of triple-well potential in the electromechanical coupled VEHs driven by colored noise and periodic force. Motivated by the enhancement of the energy harvesting performance in coupled nonlinear VEHs through exploring the superiority of tri-stable VEH among the three kinds of different potentials, the rest of the paper is organized as follows. Section 2 gives the mathematic model of an electromechanical coupled energy harvester under colored noise and periodic excitation, and shows three kinds of different potentials based on the shape of the potential energy function. Section 3 derives the analytical expression of the joint PDF by the extended stochastic averaging method depending on energy-dependent frequency. Section 4 studies the stochastic dynamics of the electromechanical model from the variations of transient response, steady-state response, mean harvested power, signal-to-noise ratio (SNR) and power conversion efficiency, and exhibits the superiority of tri-stable VEH in energy harvesting. Meanwhile, the theoretical results are verified by the Monte Carlo simulation (MCS). Finally, Section 5 draws some specific conclusions.
2. Electromechanical model A model of the electromechanical coupled VEH is considered, which consists of a mechanical oscillator coupled to an electrical circuit by the mechanism of an electromechanical vibration-to-electricity conversion [7]. The mechanism can be designed as piezoelectric (Fig. 1(a)), or electromagnetic (Fig. 1(b)). The electromechanical coupled equation in piezoelectric VEH and electromagnetic VEH can be expressed as the following general form MX CX
dU ( X ) iYi MX b , dX
Y C pY1 1 1 X (piezoelectric), LY2 R2Y2 2 X (electromagnetic), R1
(1)
where X represents the displacement of the mass M ; the dot denotes a derivative with respect to time t ; X b denotes the base acceleration; C denotes the linear viscous damping coefficient; C p represents the effective capacitance of the piezoelectric element; L denotes the inductance of the harvest coil; i 1, 2 corresponding to the two different cases of piezoelectric VEH and electromagnetic VEH; Y1 represents the inductive voltage in the case of piezoelectric VEH; Y2 represents the inductive current in the case of electromagnetic VEH; 1 and 2 in both cases represent the linear electromechanical coupling coefficient; R1 denotes the equivalent resistance load in piezoelectric VEH, i.e., R1 Rl Rp / (Rl Rp ) , in which Rl is the load resistance, R p is the piezoelectric resistance; R2 denotes the equivalent resistance load in electromagnetic VEH, i.e., R2 Rl Rc , in which Rl is the load resistance, Rc is the coil resistance.
4
Xb
X Xb
dU dX
Colored noise
(a) 1 X
(b)
+ Y1 R p Rl -
Cp
iYi
M
C
Periodic excitation
Rc
Rl'
Y2
L
2X
+ -
Fig. 1. A simplified representation of a coupled VEH with (a) Piezoelectric mechanism and (b) Electromagnetic mechanism.
The potential function U ( X ) of the mechanical oscillator can be expressed as the following general form U(X )
1 1 1 k1 (1 r ) X 2 k2 X 4 k3 X 6 , 2 4 6
(2)
where k1 , k 2 and k 3 denote the linear, cubic and quintic stiffness coefficient, respectively. r is introduced to allow the linear stiffness to be adjusted around its nominal value. To non-dimensionalize Eq. (1), the following transformations are introduced t 0 t , X
1 2
12 k1C p
22 k1 L
X k l2 k l4 X C , Xb b , , 2 c , 3 c , lc lc M 0 k1 k1
(piezoelectric), 1
(electromagnetic), 2
Cp 1 (piezoelectric), Y1 Y (piezoelectric), C p R10 1lc 1
(3)
R2 L (electromagnetic), Y2 Y (electromagnetic), L0 2 lc 2
in which X represents the dimensionless displacement; X b denotes the dimensionless base acceleration;
0 k1 M denotes the natural frequency of the mechanical oscillator; lc k1 k3 represents the length scale; denotes the dimensionless damping coefficient; and denote the dimensionless cubic and quintic stiffness coefficients. 1 and 2 in both cases represent the dimensionless linear electromechanical coupling coefficient; 1 and 2 in both cases represent the dimensionless electromechanical time constants ratio; Y1 and Y2 in both cases represent the dimensionless electric quantity, i.e., the inductive voltage in piezoelectric VEH and
the inductive current in electromagnetic VEH. It should be noted that the introduced transformations for
1 ,1 ,Y1 and 2 , 2 ,Y2 are different in both
cases, but the denoted meaning of each similar parameter is the same because of the homogeneous correlations between the mechanical oscillator and each electrical conversion mechanism. Consequently, for purpose of the dimensionless equation in unified form in both cases, can be used to denote the same correlation, i.e., the dimensionless linear electromechanical coupling coefficient, can be used to denote the dimensionless electromechanical time constants ratio, and Y can be used to denote the dimensionless electric quantity. Thus, the dimensionless electromechanical coupled system can be expressed as the following unified form dU0 ( X ) Y X b , dX Y Y X , X X
with 5
(4)
1 1 1 U 0 ( X ) (1 r ) X 2 X 4 X 6 . 2 4 6
(5)
Here, we consider X b (t ) F sin(t ) , where F sin(t ) is the dimensionless periodic excitation and (t ) is the additive Gaussian colored noise whose statistical properties and spectral density can be given as follows
(t ) 0,
(t ) ( s)
ts exp ,
D
S ( )
D 1 2 2
(6)
,
in which D represents the noise intensity of colored noise and denotes the correlation time of colored noise. Ignoring the mechanical and electrical coupling mechanism, Eq. (4) can be used to investigate the dynamical behaviors of the general nonlinear VEHs (i.e. mono-stable, bi-stable and tri-stable VEHs) excited by colored noise and periodic excitation. Based on the shape of the dimensionless potential energy function, the potential function has three kinds of different potentials as shown in Fig. 2, and the VEHs can be divided into the following three categories: A. Mono-stable when ( r 1 , 0 , 0 ) or ( r 1 , 0 , 2 4(1 r ) ), as shown by the green area A and dashed line in Fig. 2. In such case, the potential function has only one stable equilibrium point X s* 0 and the periodic motion of the particle is just restricted in this stable potential well. B. Bi-stable when r 1 , as shown by the blue area B and asterisk line in Fig. 2. In such case, the potential function has one unstable saddle point X u* 0 and two stable equilibria, i.e. X si * (r 1) (if 0 ) or X si *
2 4(1 r )
2
(if 0 ), where i 1, 2 . As shown in Fig. 3(a), if the total energy H is
greater than zero, the Brownian particle moves from one well to the other, i.e., the periodic motion occurs between two potential wells. Otherwise, the periodic motion is restricted in the right well or the left well depending on the initial condition of the particle. Accordingly, the particle has three kinds of periodic motions, and the periodic motion is determined by the total energy and the initial condition. C. Tri-stable when r 1 , 0 and 2 4(1 r ) , as shown by the red area C and solid line in Fig. 2. In such * case, the potential function has three stable equilibria, i.e. X si
* and two unstable saddle points, i.e. X u
2 4(1 r )
2 4(1 r )
2 , 0 ,
i 1, 2,3
2 . And the particle has four kinds of
periodic motions as shown in Fig. 3(b), jumping among the three wells or restricted in one of three potential wells depending on the total energy and the initial condition.
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Fig. 2. (a) Stable distribution of potential function U 0 ( X ) ; (b) Three kinds of different potentials U 0 ( X ) with different values of
(r , , ) . Green area A and green dashed line: mono-stable; Blue area B and blue asterisk line: bi-stable; Red area C and red solid line: tri-stable.
Fig. 3. Periodic motions with different energy levels in the cases of (a) bi-stable potential, and (b) tri-stable potential.
3. Stochastic Averaging 3.1. The equivalent uncoupled system The dimensionless induced voltage or current equation in Eq. (4) can be integrated at first as the following explicit expression t
Y (t ) C1e t e (t ) X ( )d . 0
(7)
Since the longtime stationary response is only concerned here, the term C1e t can be ignored because of its negligible effect on the system. After using s t , Eq. (7) can be simplified approximately in the following form t
Y (t ) e s X (t s )ds. 0
(8)
Considering the different periodic motions in the multi-stable potential function and introducing the generalized harmonic function, one can get the system displacement and velocity of Eq. (4) as the following form X (t ) A( H (t )) cos ( H (t ))t (t ) X s*i ( H (t )), X (t ) A( H (t )) ( H (t ))sin ( H (t ))t (t ) ,
(9)
where H denotes the total energy; A( H ) represents the energy-dependent amplitude; ( H ) denotes the 7
energy-dependent frequency; X s*i ( H ) are the equilibrium points. For a small s , since the total energy, the frequency, the amplitude, the initial phase and the equilibrium position are slow-varying random processes compared with the state variables of system, X (t s) can be written approximately as
X (t s) X (t )cos ( H (t ))s X (t ) X s*i ( H (t )) sin ( H (t ))s .
(10)
Then, substituting Eq. (10) into Eq. (8), the voltage or current can be gotten by neglecting the exponential decay term Y (t )
2 ( H (t )) X (t ) X s* ( H (t )) 2 X (t ). 2 2 ( H (t )) 2 ( H (t )) i
(11)
Afterwards, substituting Eq. (11) into Eq. (4), the equivalent uncoupled mechanical equation is obtained as dU0 ( X ) 2 ( H ) (12) X 2 2 ( X X s* ) (t ) F sin t. X 2 2 d X ( H ) ( H ) i
The potential function and the total energy function of Eq. (12) can be given as 1 2 ( H ) U ( X ) U0 ( X ) X 2, 2 2 2 (H ) H (X , X )
1 2 X U ( X ). 2
(13) (14)
3.2. Stochastic averaging based on energy-dependent frequency Substituted by two first-order equations associated with system velocity and total energy [27], the equivalent uncoupled system (12) can be reformed as X 2 H 2U ( X ),
(15)
H X f ( X , X ) X (t ) XF sin t ,
(16)
where 2 ( H ) f ( X , X ) 2 X X s* . 2 (H ) 2 2 (H ) i
(17)
The Itoˆ equation of the energy process H (t ) , which is approximate to Markovian process, can be written as d H m( H ) d t ( H ) d B(t ), (18) where B (t ) is a standard Wiener process; m( H ) is the drift coefficient, and ( H ) is diffusion coefficient, which can be calculated by m( H ) X f ( X , X ) t
2 (H )
D XF sin t , t W
2D 2 X , t W
(19) (20)
1 T 1 dt d X represents the time averaging of a quasi-period. The T 0 T X closed-loop integration can be carried out by H U ( A) according to the different energy level in the different
where W 1 2 2 ( H ) , and
t
8
steady states of the potential function U ( X ) in Eq. (13). Especially, if the particle moves in the right potential well 1 T
or
the
left
one
under
bi-stable
1 1 ( ) d X ( ) d X and T T R L R L
d X T
X
* si
or
dX
tri-stable
potential
one
has
1 * * X R d X X L d X 0 . For a given energy TR TL R L
level H (refer to Fig. 3), one can get the energy-dependent period T ( H ) energy-dependent frequency ( H )
function,
dX 2 H 2U ( X )
and the
2 by the iteration method due to the existence of the unknown ( H ) T (H )
in U ( X ) . Figure 4 shows the calculated different energy-dependent frequency for different steady states of potential function.
Fig. 4. The energy-dependent frequency of the displacement and velocity in the unperturbed system in the cases of (a) mono-stable, (b) bi-stable and (c) tri-stable potential function.
Then, the stationary probability density (SPD) of the total energy can be obtained from Eq. (18) as p( H )
C0
2 (H )
2 m( H ) exp 2 d H , ( H )
(21)
where C0 is a normalization constant. It can be proved that d 1 ln T ( H ) X 2 . t dH X2
(22)
t
Considering p( X , X ) p( H ) T ( H ) and Eq. (21), the joint PDF of the equivalent uncoupled system can be derived as p( X , X )
W C0W WF sin t H X exp 2 H (X , X ) 2 0 X 2 D D (H ) D
t t
d H .
(23)
From Eq. (23), the marginal probability density functions are given as p( X )
p( X )
9
p( X , X ) d X ,
(24) p( X , X ) d X .
4. Stochastic Dynamics In this section, the transient response, steady-state response, the mean harvested power, SNR and the power conversion efficiency of a coupled nonlinear VEH are investigated for the goal of enhancing the energy harvesting performance through exploring system dynamical behaviors and optimizing parameters design. Most importantly, comparisons among the mono-, bi- and tri-stable systems in energy harvesting are analyzed in detail to show the superiority of tri-stable VEH. In the following investigation, the stiffness coefficients (r , , ) are chosen as (0,3,3) (mono-stable), (2.5,3, 0) (bi-stable) and (0, 4.2,3) (tri-stable) corresponding to the three kinds of different potentials as displayed in Fig. 2(b), and other system parameters are chosen as 0.02 , 0.2 ,
0.05 , F 0.15 , 0.02 , D 0.005 , 0.5 , unless otherwise mentioned.
4.1. Transient response The effects of the periodic excitation on system response can be characterized by the evolutions of the PDF in one period. The transient inter-well motion of Brownian particle in the potential wells and the transient response affected by periodic force are discussed in this subsection by MCS from the original dimensionless Eq. (4). The generalized potential function can be transformed from Eq. (5) into the following equation 1 1 1 U 0 ( X ) (1 r ) X 2 X 4 X 6 F sin(t ). 2 4 6
(25)
Figure 5 shows the cyclic variations of the generalized potential function by Eq. (25) in the cases of bi-stable potential and tri-stable potential. It can be seen in both subplots that, in one period T ( T 2 ), due to the introduction of the periodic excitation, the potential wells on both sides are tilted asymmetrically upward and downward, raising and lowering the potential barrier periodically. Whereas, the average potential function over one period remains symmetric, as shown in Fig. 2(b). Meanwhile, the transient inter-well transition of the corresponding tri-stable joint PDFs in one period is presented in Fig. 6. One can observe that the stochastic transition is moving toward the right peak at T / 4 , and it is just the opposite at 3T / 4 , where there is only one peak in both cases. However, the two lateral peaks are symmetric and there are three peaks at both T and T / 2 . Thus, the transient PDFs in a period describe the transition motions of a particle in the tri-stable potential as displayed in Fig. 5(b). The effect of the periodic force on the topological characteristic of the transient PDFs at 3T / 4 in the three kinds of different potentials is further explored and presented in Fig. 7. It can be seen in Fig. 7(a) that the inclination of the peak in the case of mono-stable potential increases as F increases. In Fig. 7(b) and 7(c), as F increases, one can see that the particle jumps toward the left potential well gradually, and the bi-/tri-stable transient marginal PDF is changed from two/three peaks to one peak, respectively. It indicates that the amplitude of periodic force can change the topological characteristic of transient PDF, which is similar to the phenomenon of stochastic P-bifurcation. Since the stochastic transition always moves to the local minimum of the potential wells, the introduction of periodic excitation maybe beneficial for particle crossing the potential barrier to harvest much more energy. Accordingly, it is indispensable to further explore the influence of periodic excitation on energy harvesting capacity in the following Section 4.3.
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Fig. 5. The cyclic variation of potential function by Eq. (25) in the cases of (a) bi-stable potential and (b) tri-stable potential.
Fig. 6. The transient joint PDFs in the case of tri-stable potential by MCS from Eq. (4) for fixed F 0.1 at different moments: (a) the initial moment or T , (b) T / 4 , (c) T / 2 , and (d) 3T / 4 .
Fig. 7. The transient marginal PDFs with different F by MCS from Eq. (4) at 3T / 4 in the cases of (a) mono-stable potential, (b) bi-stable potential, and (c) tri-stable potential.
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4.2. Steady-state response and the mean harvested power In this subsection, the steady-state response in the three kinds of different potentials can be obtained by ignoring the periodic excitation, for the reason that there is enough time for the system to reach the local equilibrium point during the period of 1 [28]. The theoretical results of the marginal SPD are determined by Eq. (24), and the numerical results by MCS from original Eq. (4) are also given to verify the precision of the extended stochastic averaging method based on energy-dependent frequency. The marginal SPDs of system displacement and system velocity with different values of noise intensity and correlation time are presented in Fig. 8, where solid lines denote the theoretical results and circle symbols represent MCS results. One can observe that they coincide extremely well. Through analyzing the effect of noise intensity on p ( X ) in the three kinds of different potentials, as displayed in Fig. 8(a)-(c), one can see that the shape structure of SPD remains unchanged and the heights of peaks decreases as noise intensity increases. Meanwhile, the vibration range of system displacement X is broaden. These results indicate that noise intensity of colored noise can weaken the intra-well motion, then accordingly strengthen the inter-well transition and enlarge the range of stochastic response. Whereas, the effect of correlation time is exactly the opposite of noise intensity. Taking the tri-stable potential as an example, Fig. 8(d) manifests the effect of correlation time on the marginal SPD p( X ) . One can see that the increase of correlation time results in the decrease of the range of possible vibrating radius. Consequently, both noise intensity and correlation time have an essential influence on the mean harvested power of the coupled VEH.
Fig. 8. The marginal SPD of system displacement with different values of D in the cases of (a) mono-stable potential, (b) bi-stable potential, and (c) tri-stable potential. (d) The marginal SPD of system velocity with different values of in the case of tri-stable potential. Solid line: stochastic averaging; Circle symbol: MCS.
The mean harvested power is a crucial indicator in assessing the capability of energy harvesters. From Eq. (11), 12
the mean-square voltage or current can be derived as 2
2 (H ) X X s*i ( H ) 2 E Y 2 X p( X , X ) d X d X , 2 2 ( H ) (H ) 2
(27)
Then, the mean harvested power can be given by E P E Y 2 .
(28)
Figure 9(a) depicts the variations of mean harvested power with noise intensity for three kinds of different potentials. It is clear that mean output power E[ P] always keeps increasing continuously as noise intensity D increases for each potential function. Corresponding to the three major classified potential functions, the electromechanical coupled energy harvester can be designed as three kinds of different multi-stable systems, i.e. mono-stable, bi-stable and tri-stable system. Among these systems, it can be seen obviously from Fig. 9(a) that E[ P] in the tri-stable system is higher than that in the bi-stable system and much higher than that in the mono-stable system for any noise intensity of colored noise. That is exactly the result of the different steady-state response in the intra-well and inter-well motions for the three kind of different potentials. It maybe because that the inter-well transition in tri-stable system gives rise to larger jumping range for the particle thereby amplifying the influence of noise intensity on harvested power compared with the mono-/bi-stable systems. Accordingly, the mean harvested power in the tri-stable system is the largest. Therefore, the tri-stable system outperforms the mono-stable and bi-stable systems in harvested power, and noise intensity of colored noise is beneficial to energy harvesting. The influence of correlation time on mean harvested power is also discussed in Fig. 9(b), taking tri-stable system as one example. Obviously, a larger correlation time results in a less harvested power, for the reason that a larger correlation time may reduce the vibration ranges of the particle. Thus, the correlation time of colored noise is not beneficial to energy harvesting in a coupled VEH, which makes it cannot be ignored in the realistic applications. Meanwhile, the theoretical and numerical results, as shown in Fig. 9, present an excellent agreement further validating the extended stochastic averaging method.
Fig. 9. (a) Mean harvested power for three kinds of different potentials corresponding to the parameters values from Fig. 2(b). (b) Mean harvested power with different values of in the case of tri-stable potential. Solid line: stochastic averaging; Circle symbol: MCS.
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4.3. Superiority of tri-stable VEH in energy harvesting The signal-to-noise ratio (SNR) is a crucial quantifier in characterizing the SR phenomenon, which occurs at the maximum of SNR. In this subsection, we focus on the study of SNR numerically from the original Eq. (4) through using the fourth-order Runge–Kutta algorithm, as it is arduous to obtain the identical analytical expression of SNR for the strongly nonlinear systems with three kinds of different potentials. Meanwhile, the influences of system parameters on the power conversion efficiency are also discussed to evaluate the performance of energy harvesting. The definition of SNR can be given by SNR 10 log10 ( Ps / Pn ) d B,
(29)
where Ps denotes the spectrum value at signal frequency in the output power spectrum, and Pn represents the average of the remaining output power spectrum values. The power conversion efficiency % C is defined as the total efficiency to evaluate the power conversion from the ambient noise and the provided periodic mechanical power to the harvested electrical power. That is % Pe Pm 100% ,
where Pe Y 2 , Pm X S () XF sin t
and
1 N
N
(30)
1 lim t d t i 1
tn
tn
n
0
denotes time-average and
ensemble average. In Fig. 10, the effects of noise intensity and periodic excitation on SNR and % are studied for the case of tri-stable potential. Evidently, it can be seen in Fig. 10(a) that SNR decreases first, increases to a maximum later and then keeps decreasing as D further increases for each one periodic excitation. It indicates that there is an optimal noise intensity Dopt to maximize SNR, i.e. SR occurs. Besides, as F increases, the peak of SNR moves toward the upper left gradually. In Fig. 10(b), there is also an optimal noise intensity Dopt to maximize % . Whereas, as F increases, the peak of % moves toward the lower left in a slow manner. These results indicate that larger periodic force can enhance the resonance effect but will weaken the power conversion efficiency for this case study. From viewpoint of occurrence of SR, a suitable dose of noise intensity and periodic force should be adopted to enhance the harvested energy by using SR effect. In both subplots, one can find that the corresponding Dopt decreases with increasing F , for the reason that larger periodic excitation makes it easily for particle to
through the barrier and move among the potential wells. Thus, a smaller optimal noise intensity is needed to induce SR under a larger periodic excitation.
14
Fig. 10. (a) SNR and (b) the power conversion efficiency versus noise intensity for different periodic forces in the case of tri-stable potential.
For the same periodic excitation, the effect of the potential structure on SNR is discussed in Fig. 11. Figures 11(a) and 11(b) depict the influence of the cubic stiffness coefficient on SNR in the cases of bi-stable and tri-stable potentials, respectively. One can see that as increases, both of the peaks in SNR shift to upper left gradually. That is because both of the potential-well depth and the potential-well distance become smaller with increasing cubic stiffness coefficient, then the utilization of colored noise is enhanced leading to higher resonance effect and smaller optimal noise intensity. It indicates that both of the potential-well depth and the potential-well distance have significant influences on the energy harvesting performance for an electromechanical coupled energy harvester. For further study the effect of the potential structure on performance, comparisons of bi-stable and tri-stable potentials under the same potential-well depth and potential-well distance are displayed in Figs. 11(c) and 11(d), respectively. It is clear in both subplots that the tri-stable system has higher superiority in enhancing the output SNR and can achieve better resonance effect than the bi-stable system. The reason may be that, in the presence of the same periodic excitation, the three-inter-well transition in tri-stable system is superior to the two-inter-well transition in bi-stable system in one period, and the synergistic effect of colored noise on the periodic excitation in tri-stable system is also greater than that in bi-stable system when SR occurs. Thus, the tri-stable VEH , compared with the bi-stable VEH, can achieve better SR effect.
15
Fig. 11. SNR versus noise intensity for fixed F 0.15 : (a) in the case of bi-stable potential for different cubic stiffness coefficients; (b) in the case of tri-stable potential for different cubic stiffness coefficients; (c) under the same potential-well depth between bi- and tri-stable potentials; (d) under the same potential-well distance between bi- and tri-stable potentials.
Moreover, comparisons between bi- and tri-stable potentials in the power conversion efficiency are presented in Fig. 12. It is shown that for each potential, the power conversion efficiency % increases invariably first to a maximum and then decreases with the increase of the electromechanical coupling coefficient or the time constant ratio . It indicates that there are an optimal opt and opt that maximize the power conversion efficiency from mechanical power to harvested electrical power. Besides, the value opt of tri-stable system is almost the same with that of bi-stable system, so does the value opt . Whereas, in both subplots, the values % in the case of tri-stable potential are always larger than that in the case of bi-stable potential. The result manifests that tri-stable VEH, compared with bi-stable VEH, can convert much more mechanical energy into harvester electrical energy, and can greatly enhance the energy harvesting performance by choosing the optimal system parameters, such as the electromechanical coupling coefficient and the time constant ratio.
16
Fig. 12. (a) The power conversion efficiency versus the electromechanical coupling coefficient, and (b) the power conversion efficiency versus the time constant ratio in the cases of bi- and tri-stable potentials.
5. Conclusions In this paper, an electromechanical coupled VEH with two different conversion mechanisms under colored noise and periodic excitation is investigated theoretically and numerically for three kinds of different potentials. In particular, the comparisons among mono-, bi- and tri-stable VEHs in dynamical behaviors are analyzed in detail from the viewpoint of enhancing energy harvesting performance. It is found that the tri-stable VEH outperforms the mono- and bi-stable systems in the mean harvested power, SR effect and the power conversion efficiency. The obtained specific conclusions are as the following. 1) The extended stochastic averaging method based on energy-dependent frequency is proposed to deal with the electromechanical coupled tri-stable VEH with colored noise and periodic signal. The theoretical results are well validated by MCS from the original coupled system. 2) The periodic excitation can change the transient topological characteristic and even induce transition in one period. Besides, a larger periodic force can enhance the resonance effect but will weaken the power conversion efficiency. 3) Colored noise has a significant effect on energy harvesting performance, i.e. the inter-well transition and the mean harvested power can be enhanced by decreasing correlation time and increasing noise intensity, and there is an optimal noise intensity to induce SR. 4) There are an optimal time constant ratio and an optimal electromechanical coupling coefficient that maximize the power conversion efficiency from the mechanical power to the harvested electrical power. 5) The potential structure plays a crucial role in improving the performance of the coupled VEH. For the tri-stable potential, all of the noise utilization, SR effect and the power conversion efficiency are enhanced greatly compared with the case of bi-stable potential. The obtained results in this work can provide a significant theoretical guidance in optimizing the design of VEHs and enhancing electric energy harvested from ambient vibrations. Acknowledgements This work has been supported by the National Natural Science Foundation of China Granted Nos. 11832005 and 11772048.
17
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Graphical abstract
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CRediT author statement Yanxia Zhang: Writing- Original draft preparation, Methodology, Software, Data curation, Visualization, Investigation. Yanfei Jin: Conceptualization, Supervision, Validation, Writing- Reviewing and Editing. Pengfei Xu: Software.
Declaration of interests
☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
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Dynamics of a coupled nonlinear energy harvester under colored noise and periodic excitations Yanxia Zhang , Yanfei Jin , Pengfei Xu PII: DOI: Reference:
S0020-7403(19)33429-0 https://doi.org/10.1016/j.ijmecsci.2020.105418 MS 105418
To appear in:
International Journal of Mechanical Sciences
Received date: Revised date: Accepted date:
10 September 2019 31 December 2019 2 January 2020
Please cite this article as: Yanxia Zhang , Yanfei Jin , Pengfei Xu , Dynamics of a coupled nonlinear energy harvester under colored noise and periodic excitations, International Journal of Mechanical Sciences (2020), doi: https://doi.org/10.1016/j.ijmecsci.2020.105418
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Highlights:
Comparisons of the effects of mono-, bi- and tri-stable potentials are analyzed in a coupled VEH driven by colored noise.
An extended stochastic averaging is applied to deal with the strongly nonlinear tri-stable system under colored noise.
Tri-stable VEH can achieve better SR effect and higher power conversion efficiency compared with bi-stable VEH.
Larger periodic force can enhance SR effect but weaken power conversion efficiency for tri-stable VEH.
1
Dynamics of a coupled nonlinear energy harvester under colored noise and periodic excitations Yanxia Zhang, Yanfei Jin, Pengfei Xu Department of Mechanics, Beijing Institute of Technology, Beijing 100081, China
Abstract: Vibration energy harvester (VEH) has proven to be a favorable potential technique to supply continuous energy from ambient vibrations and its performance is greatly influenced by the design of potential structures. Motivated by the enhancement of energy harvesting performance, an electromechanical coupled VEH under colored noise and periodic excitations is investigated theoretically and numerically, and comparisons of the effects of mono-, bi- and tri-stable potentials are analyzed in detail. An uncoupled equivalent system, together with its joint probability density function (PDF), are derived by using the generalized harmonic transformation and the stochastic averaging based on energy-dependent frequency. For three kinds of different potentials, the effects of the periodic excitation, colored noise, potential shape and other crucial system parameters are explored on the dynamical behaviors. Results show that tri-stable VEH outperforms the mono- and bi-stable VEHs in mean harvested power and can achieve better stochastic resonance (SR) effect and higher power conversion efficiency by choosing the optimal system parameters, such as noise intensity, electromechanical coupling coefficient and time constant ratio. Moreover, the periodic excitation can change the topological characteristic of transient PDF, and larger periodic force can enhance SR effect but will weaken the power conversion efficiency. The theoretical results obtained by the proposed stochastic averaging method are well validated by numerical simulations. Keywords: Coupled nonlinear VEH; Stochastic averaging; Colored noise; Periodic excitation; Stochastic resonance.
1. Introduction On account of the growing demand in electric energy and the limitation of battery lifespan, vibration energy harvester (VEH), instead of the traditional chemical energy battery, become a promising potential technique to supply continuous energy because of its ability of harvesting energy from ambient vibrations. In the design of VEH, the electromagnetic transducers [1,2] and piezoelectric transducers [3,4] are the most common ways to realize energy harvesting conversion from vibration to electricity availably. Traditionally, VEHs are designed with linear stiffness type for simplicity. Whereas, linear design suffers from a critical shortcoming in the frequency bandwidth, and a narrow effective bandwidth makes it inefficient to harvest energy from the ambience vibration sources with a wider spectrum. Then, nonlinear stiffness type is introduced into VEHs by taking this point into account, which can extend the effective frequency bandwidth and can enhance energy harvesting performance from ambient vibrations efficiently [5-12]. According to the shape of potential function determined by nonlinear stiffness type, the nonlinear VEHs can be generally divided into three major categories, i.e. mono-stable, bi-stable and tri-stable VEHs. Currently, multitude of researchers have examined the mono-stable and bi-stable nonlinear VEHs [5-9]. Much of research concerning nonlinear mono-stable VEHs [5-7] have manifested that nonlinearity can give rise to extending the range of frequency and result in larger amplitude response compared with linear VEHs. Unfortunately, mono-stable VEHs behave much like linear VEHs especially under a small excitation amplitude. In order to
Corresponding author. E-mail address:[email protected]
2
improve the broadband performance efficiently, bi-stable VEH with double-well potential structure is proposed and has exerted a tremendous fascination on researchers because of its favorable capability in harvesting energy over a considerable frequency range in a large-orbit inter-well motion [8-10]. As an example, He et al. [9] concluded that bi-stable VEH, compared with mono-stable one, can produce much more power through investigating the influence of nonlinear stiffness on the average output power of VEHs in the cases of mono-stable and bi-stable potentials. However, due to the existence of a potential barrier between two potential wells, the ideal inter-well motion in the bi-stable VEH cannot be activated if the excitation level is below the required threshold value. To overcome these drawbacks in mono-stable and bi-stable VEHs, the design of the nonlinear VEHs has been further improved to a kind of tri-stable VEH. Tri-stable VEH has been a hot topic in recent studies [11-13], owing to its ability of achieving sustained large-amplitude electric response even for low-lever excitations. For instance, Panyam et al. [11] investigated the response of a tri-stable VEH excited by a harmonic function, and pointed out that tri-stable potential structure is beneficial to extend the vibratory bandwidths. Zhou et al. [12,13] conducted numerical and experimental research on a tri-stable VEH, and revealed that tri-stable VEH, compared with bi-stable VEH, can effectively harvest energy in a larger low frequency range. However, these studies on tri-stable VEHs mainly focused on deterministic systems. Due to the universality and the inevitable of environmental random vibration sources, it is indispensable to study the stochastic dynamical behaviors of the stochastic systems for tri-stable VEHs for purpose further improving the energy harvesting performance. In engineering practical application, noisy stochastic dynamical systems mostly display some interesting nonlinear phenomena, such as stochastic bifurcation, stochastic transition and stochastic resonance (SR) [14-16]. To the authors’ knowledge, current work on the tri-stable VEHs stochastic systems mainly focused on Gaussian white noise excitation [16,17]. Jin et al. [16] discussed the effects of Gaussian white noise and nonlinear stiffness coefficients on the tri-stable VEH stochastic system by using SR theory and concluded that a suitable noise dosage is rewarding to enhance the energy harvesting performance. Nevertheless, since most of noises in nature have a different length of correlation time, Gaussian white noise, unlike colored noise, is just the ideal case for describing the random fluctuations from ambient noisy vibration. Colored noise has the statistical characters of non-zero correlation time which makes it better beneficial to describe the stochastic ambience fluctuations. For example, Liu et al. [18] presented that colored noise has a crucial influence on output power of a coupled mono-stable harvester. Yang et al. [19] studied the power harvested from a bi-stable hybrid harvester under correlated colored noise, and concluded that harvested energy can be enhanced with the suitable choices of noise intensities of colored noise. Zhang et al. [14] conducted a numerical research on a nonlinear tri-stable VEH under colored noise, and concluded that a proper noise intensity of colored noise is exceedingly helpful in enhancing the SR effect. Whereas, to date, the theoretical analysis on colored noise mainly focuses on mono- and bi-stable VEHs. Yet limited works have considered the colored noise in the theoretical research on stochastic dynamics of coupled tri-stable VEHs. Consequently, it is indispensable to pour attention into exploring the influence of colored noise on tri-stable VEHs in practice by considering the characteristic of the stochastic vibration sources in nature. This work is devoted to exploring the stochastic dynamical behaviors of a coupled nonlinear VEH under colored noise and periodic excitations through the extended stochastic averaging method and excavating the superiority of tri-stable VEHs in harvesting energy compared with mono-stable and bi-stable VEHs. It has been proved that stochastic averaging method is an effective and powerful theoretical method in studying stochastic dynamics and has been widely used to stochastic systems currently [20-25]. Due to the complexity of the dynamical behaviors in bi-stable stochastic systems, Zhu et al. [26] developed the stochastic averaging method through introducing the correlation functions of the excitation processes and adopting the variable natural frequency and period of the system corresponding to the different energy levels. Whereas, the dynamical behavior in tri-stable stochastic systems is much more complicated. Although the stochastic averaging method is being employed widely in 3
multitude of researches, it mainly focuses on mono- and bi-stable stochastic dynamical systems. The authors [25] have ever attempted applying the stochastic averaging method of amplitude into tri-stable systems excited only by colored noise, and found that the method has certain limitations on the valid range of theoretical results because the inverse function of triple-well potential does not exist. To date, few method developed is pleasurable and most methods are far from ready to be used in tri-stable stochastic dynamical systems under both noise and periodic excitation. Accordingly, this work, based on the idea of the stochastic averaging method developed by Zhu et al. [26], will extend the method to deal with the case of triple-well potential in the electromechanical coupled VEHs driven by colored noise and periodic force. Motivated by the enhancement of the energy harvesting performance in coupled nonlinear VEHs through exploring the superiority of tri-stable VEH among the three kinds of different potentials, the rest of the paper is organized as follows. Section 2 gives the mathematic model of an electromechanical coupled energy harvester under colored noise and periodic excitation, and shows three kinds of different potentials based on the shape of the potential energy function. Section 3 derives the analytical expression of the joint PDF by the extended stochastic averaging method depending on energy-dependent frequency. Section 4 studies the stochastic dynamics of the electromechanical model from the variations of transient response, steady-state response, mean harvested power, signal-to-noise ratio (SNR) and power conversion efficiency, and exhibits the superiority of tri-stable VEH in energy harvesting. Meanwhile, the theoretical results are verified by the Monte Carlo simulation (MCS). Finally, Section 5 draws some specific conclusions.
2. Electromechanical model A model of the electromechanical coupled VEH is considered, which consists of a mechanical oscillator coupled to an electrical circuit by the mechanism of an electromechanical vibration-to-electricity conversion [7]. The mechanism can be designed as piezoelectric (Fig. 1(a)), or electromagnetic (Fig. 1(b)). The electromechanical coupled equation in piezoelectric VEH and electromagnetic VEH can be expressed as the following general form MX CX
dU ( X ) iYi MX b , dX
Y C pY1 1 1 X (piezoelectric), LY2 R2Y2 2 X (electromagnetic), R1
(1)
where X represents the displacement of the mass M ; the dot denotes a derivative with respect to time t ; X b denotes the base acceleration; C denotes the linear viscous damping coefficient; C p represents the effective capacitance of the piezoelectric element; L denotes the inductance of the harvest coil; i 1, 2 corresponding to the two different cases of piezoelectric VEH and electromagnetic VEH; Y1 represents the inductive voltage in the case of piezoelectric VEH; Y2 represents the inductive current in the case of electromagnetic VEH; 1 and 2 in both cases represent the linear electromechanical coupling coefficient; R1 denotes the equivalent resistance load in piezoelectric VEH, i.e., R1 Rl Rp / (Rl Rp ) , in which Rl is the load resistance, R p is the piezoelectric resistance; R2 denotes the equivalent resistance load in electromagnetic VEH, i.e., R2 Rl Rc , in which Rl is the load resistance, Rc is the coil resistance.
4
Xb
X Xb
dU dX
Colored noise
(a) 1 X
(b)
+ Y1 R p Rl -
Cp
iYi
M
C
Periodic excitation
Rc
Rl'
Y2
L
2X
+ -
Fig. 1. A simplified representation of a coupled VEH with (a) Piezoelectric mechanism and (b) Electromagnetic mechanism.
The potential function U ( X ) of the mechanical oscillator can be expressed as the following general form U(X )
1 1 1 k1 (1 r ) X 2 k2 X 4 k3 X 6 , 2 4 6
(2)
where k1 , k 2 and k 3 denote the linear, cubic and quintic stiffness coefficient, respectively. r is introduced to allow the linear stiffness to be adjusted around its nominal value. To non-dimensionalize Eq. (1), the following transformations are introduced t 0 t , X
1 2
12 k1C p
22 k1 L
X k l2 k l4 X C , Xb b , , 2 c , 3 c , lc lc M 0 k1 k1
(piezoelectric), 1
(electromagnetic), 2
Cp 1 (piezoelectric), Y1 Y (piezoelectric), C p R10 1lc 1
(3)
R2 L (electromagnetic), Y2 Y (electromagnetic), L0 2 lc 2
in which X represents the dimensionless displacement; X b denotes the dimensionless base acceleration;
0 k1 M denotes the natural frequency of the mechanical oscillator; lc k1 k3 represents the length scale; denotes the dimensionless damping coefficient; and denote the dimensionless cubic and quintic stiffness coefficients. 1 and 2 in both cases represent the dimensionless linear electromechanical coupling coefficient; 1 and 2 in both cases represent the dimensionless electromechanical time constants ratio; Y1 and Y2 in both cases represent the dimensionless electric quantity, i.e., the inductive voltage in piezoelectric VEH and
the inductive current in electromagnetic VEH. It should be noted that the introduced transformations for
1 ,1 ,Y1 and 2 , 2 ,Y2 are different in both
cases, but the denoted meaning of each similar parameter is the same because of the homogeneous correlations between the mechanical oscillator and each electrical conversion mechanism. Consequently, for purpose of the dimensionless equation in unified form in both cases, can be used to denote the same correlation, i.e., the dimensionless linear electromechanical coupling coefficient, can be used to denote the dimensionless electromechanical time constants ratio, and Y can be used to denote the dimensionless electric quantity. Thus, the dimensionless electromechanical coupled system can be expressed as the following unified form dU0 ( X ) Y X b , dX Y Y X , X X
with 5
(4)
1 1 1 U 0 ( X ) (1 r ) X 2 X 4 X 6 . 2 4 6
(5)
Here, we consider X b (t ) F sin(t ) , where F sin(t ) is the dimensionless periodic excitation and (t ) is the additive Gaussian colored noise whose statistical properties and spectral density can be given as follows
(t ) 0,
(t ) ( s)
ts exp ,
D
S ( )
D 1 2 2
(6)
,
in which D represents the noise intensity of colored noise and denotes the correlation time of colored noise. Ignoring the mechanical and electrical coupling mechanism, Eq. (4) can be used to investigate the dynamical behaviors of the general nonlinear VEHs (i.e. mono-stable, bi-stable and tri-stable VEHs) excited by colored noise and periodic excitation. Based on the shape of the dimensionless potential energy function, the potential function has three kinds of different potentials as shown in Fig. 2, and the VEHs can be divided into the following three categories: A. Mono-stable when ( r 1 , 0 , 0 ) or ( r 1 , 0 , 2 4(1 r ) ), as shown by the green area A and dashed line in Fig. 2. In such case, the potential function has only one stable equilibrium point X s* 0 and the periodic motion of the particle is just restricted in this stable potential well. B. Bi-stable when r 1 , as shown by the blue area B and asterisk line in Fig. 2. In such case, the potential function has one unstable saddle point X u* 0 and two stable equilibria, i.e. X si * (r 1) (if 0 ) or X si *
2 4(1 r )
2
(if 0 ), where i 1, 2 . As shown in Fig. 3(a), if the total energy H is
greater than zero, the Brownian particle moves from one well to the other, i.e., the periodic motion occurs between two potential wells. Otherwise, the periodic motion is restricted in the right well or the left well depending on the initial condition of the particle. Accordingly, the particle has three kinds of periodic motions, and the periodic motion is determined by the total energy and the initial condition. C. Tri-stable when r 1 , 0 and 2 4(1 r ) , as shown by the red area C and solid line in Fig. 2. In such * case, the potential function has three stable equilibria, i.e. X si
* and two unstable saddle points, i.e. X u
2 4(1 r )
2 4(1 r )
2 , 0 ,
i 1, 2,3
2 . And the particle has four kinds of
periodic motions as shown in Fig. 3(b), jumping among the three wells or restricted in one of three potential wells depending on the total energy and the initial condition.
6
Fig. 2. (a) Stable distribution of potential function U 0 ( X ) ; (b) Three kinds of different potentials U 0 ( X ) with different values of
(r , , ) . Green area A and green dashed line: mono-stable; Blue area B and blue asterisk line: bi-stable; Red area C and red solid line: tri-stable.
Fig. 3. Periodic motions with different energy levels in the cases of (a) bi-stable potential, and (b) tri-stable potential.
3. Stochastic Averaging 3.1. The equivalent uncoupled system The dimensionless induced voltage or current equation in Eq. (4) can be integrated at first as the following explicit expression t
Y (t ) C1e t e (t ) X ( )d . 0
(7)
Since the longtime stationary response is only concerned here, the term C1e t can be ignored because of its negligible effect on the system. After using s t , Eq. (7) can be simplified approximately in the following form t
Y (t ) e s X (t s )ds. 0
(8)
Considering the different periodic motions in the multi-stable potential function and introducing the generalized harmonic function, one can get the system displacement and velocity of Eq. (4) as the following form X (t ) A( H (t )) cos ( H (t ))t (t ) X s*i ( H (t )), X (t ) A( H (t )) ( H (t ))sin ( H (t ))t (t ) ,
(9)
where H denotes the total energy; A( H ) represents the energy-dependent amplitude; ( H ) denotes the 7
energy-dependent frequency; X s*i ( H ) are the equilibrium points. For a small s , since the total energy, the frequency, the amplitude, the initial phase and the equilibrium position are slow-varying random processes compared with the state variables of system, X (t s) can be written approximately as
X (t s) X (t )cos ( H (t ))s X (t ) X s*i ( H (t )) sin ( H (t ))s .
(10)
Then, substituting Eq. (10) into Eq. (8), the voltage or current can be gotten by neglecting the exponential decay term Y (t )
2 ( H (t )) X (t ) X s* ( H (t )) 2 X (t ). 2 2 ( H (t )) 2 ( H (t )) i
(11)
Afterwards, substituting Eq. (11) into Eq. (4), the equivalent uncoupled mechanical equation is obtained as dU0 ( X ) 2 ( H ) (12) X 2 2 ( X X s* ) (t ) F sin t. X 2 2 d X ( H ) ( H ) i
The potential function and the total energy function of Eq. (12) can be given as 1 2 ( H ) U ( X ) U0 ( X ) X 2, 2 2 2 (H ) H (X , X )
1 2 X U ( X ). 2
(13) (14)
3.2. Stochastic averaging based on energy-dependent frequency Substituted by two first-order equations associated with system velocity and total energy [27], the equivalent uncoupled system (12) can be reformed as X 2 H 2U ( X ),
(15)
H X f ( X , X ) X (t ) XF sin t ,
(16)
where 2 ( H ) f ( X , X ) 2 X X s* . 2 (H ) 2 2 (H ) i
(17)
The Itoˆ equation of the energy process H (t ) , which is approximate to Markovian process, can be written as d H m( H ) d t ( H ) d B(t ), (18) where B (t ) is a standard Wiener process; m( H ) is the drift coefficient, and ( H ) is diffusion coefficient, which can be calculated by m( H ) X f ( X , X ) t
2 (H )
D XF sin t , t W
2D 2 X , t W
(19) (20)
1 T 1 dt d X represents the time averaging of a quasi-period. The T 0 T X closed-loop integration can be carried out by H U ( A) according to the different energy level in the different
where W 1 2 2 ( H ) , and
t
8
steady states of the potential function U ( X ) in Eq. (13). Especially, if the particle moves in the right potential well 1 T
or
the
left
one
under
bi-stable
1 1 ( ) d X ( ) d X and T T R L R L
d X T
X
* si
or
dX
tri-stable
potential
one
has
1 * * X R d X X L d X 0 . For a given energy TR TL R L
level H (refer to Fig. 3), one can get the energy-dependent period T ( H ) energy-dependent frequency ( H )
function,
dX 2 H 2U ( X )
and the
2 by the iteration method due to the existence of the unknown ( H ) T (H )
in U ( X ) . Figure 4 shows the calculated different energy-dependent frequency for different steady states of potential function.
Fig. 4. The energy-dependent frequency of the displacement and velocity in the unperturbed system in the cases of (a) mono-stable, (b) bi-stable and (c) tri-stable potential function.
Then, the stationary probability density (SPD) of the total energy can be obtained from Eq. (18) as p( H )
C0
2 (H )
2 m( H ) exp 2 d H , ( H )
(21)
where C0 is a normalization constant. It can be proved that d 1 ln T ( H ) X 2 . t dH X2
(22)
t
Considering p( X , X ) p( H ) T ( H ) and Eq. (21), the joint PDF of the equivalent uncoupled system can be derived as p( X , X )
W C0W WF sin t H X exp 2 H (X , X ) 2 0 X 2 D D (H ) D
t t
d H .
(23)
From Eq. (23), the marginal probability density functions are given as p( X )
p( X )
9
p( X , X ) d X ,
(24) p( X , X ) d X .
4. Stochastic Dynamics In this section, the transient response, steady-state response, the mean harvested power, SNR and the power conversion efficiency of a coupled nonlinear VEH are investigated for the goal of enhancing the energy harvesting performance through exploring system dynamical behaviors and optimizing parameters design. Most importantly, comparisons among the mono-, bi- and tri-stable systems in energy harvesting are analyzed in detail to show the superiority of tri-stable VEH. In the following investigation, the stiffness coefficients (r , , ) are chosen as (0,3,3) (mono-stable), (2.5,3, 0) (bi-stable) and (0, 4.2,3) (tri-stable) corresponding to the three kinds of different potentials as displayed in Fig. 2(b), and other system parameters are chosen as 0.02 , 0.2 ,
0.05 , F 0.15 , 0.02 , D 0.005 , 0.5 , unless otherwise mentioned.
4.1. Transient response The effects of the periodic excitation on system response can be characterized by the evolutions of the PDF in one period. The transient inter-well motion of Brownian particle in the potential wells and the transient response affected by periodic force are discussed in this subsection by MCS from the original dimensionless Eq. (4). The generalized potential function can be transformed from Eq. (5) into the following equation 1 1 1 U 0 ( X ) (1 r ) X 2 X 4 X 6 F sin(t ). 2 4 6
(25)
Figure 5 shows the cyclic variations of the generalized potential function by Eq. (25) in the cases of bi-stable potential and tri-stable potential. It can be seen in both subplots that, in one period T ( T 2 ), due to the introduction of the periodic excitation, the potential wells on both sides are tilted asymmetrically upward and downward, raising and lowering the potential barrier periodically. Whereas, the average potential function over one period remains symmetric, as shown in Fig. 2(b). Meanwhile, the transient inter-well transition of the corresponding tri-stable joint PDFs in one period is presented in Fig. 6. One can observe that the stochastic transition is moving toward the right peak at T / 4 , and it is just the opposite at 3T / 4 , where there is only one peak in both cases. However, the two lateral peaks are symmetric and there are three peaks at both T and T / 2 . Thus, the transient PDFs in a period describe the transition motions of a particle in the tri-stable potential as displayed in Fig. 5(b). The effect of the periodic force on the topological characteristic of the transient PDFs at 3T / 4 in the three kinds of different potentials is further explored and presented in Fig. 7. It can be seen in Fig. 7(a) that the inclination of the peak in the case of mono-stable potential increases as F increases. In Fig. 7(b) and 7(c), as F increases, one can see that the particle jumps toward the left potential well gradually, and the bi-/tri-stable transient marginal PDF is changed from two/three peaks to one peak, respectively. It indicates that the amplitude of periodic force can change the topological characteristic of transient PDF, which is similar to the phenomenon of stochastic P-bifurcation. Since the stochastic transition always moves to the local minimum of the potential wells, the introduction of periodic excitation maybe beneficial for particle crossing the potential barrier to harvest much more energy. Accordingly, it is indispensable to further explore the influence of periodic excitation on energy harvesting capacity in the following Section 4.3.
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Fig. 5. The cyclic variation of potential function by Eq. (25) in the cases of (a) bi-stable potential and (b) tri-stable potential.
Fig. 6. The transient joint PDFs in the case of tri-stable potential by MCS from Eq. (4) for fixed F 0.1 at different moments: (a) the initial moment or T , (b) T / 4 , (c) T / 2 , and (d) 3T / 4 .
Fig. 7. The transient marginal PDFs with different F by MCS from Eq. (4) at 3T / 4 in the cases of (a) mono-stable potential, (b) bi-stable potential, and (c) tri-stable potential.
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4.2. Steady-state response and the mean harvested power In this subsection, the steady-state response in the three kinds of different potentials can be obtained by ignoring the periodic excitation, for the reason that there is enough time for the system to reach the local equilibrium point during the period of 1 [28]. The theoretical results of the marginal SPD are determined by Eq. (24), and the numerical results by MCS from original Eq. (4) are also given to verify the precision of the extended stochastic averaging method based on energy-dependent frequency. The marginal SPDs of system displacement and system velocity with different values of noise intensity and correlation time are presented in Fig. 8, where solid lines denote the theoretical results and circle symbols represent MCS results. One can observe that they coincide extremely well. Through analyzing the effect of noise intensity on p ( X ) in the three kinds of different potentials, as displayed in Fig. 8(a)-(c), one can see that the shape structure of SPD remains unchanged and the heights of peaks decreases as noise intensity increases. Meanwhile, the vibration range of system displacement X is broaden. These results indicate that noise intensity of colored noise can weaken the intra-well motion, then accordingly strengthen the inter-well transition and enlarge the range of stochastic response. Whereas, the effect of correlation time is exactly the opposite of noise intensity. Taking the tri-stable potential as an example, Fig. 8(d) manifests the effect of correlation time on the marginal SPD p( X ) . One can see that the increase of correlation time results in the decrease of the range of possible vibrating radius. Consequently, both noise intensity and correlation time have an essential influence on the mean harvested power of the coupled VEH.
Fig. 8. The marginal SPD of system displacement with different values of D in the cases of (a) mono-stable potential, (b) bi-stable potential, and (c) tri-stable potential. (d) The marginal SPD of system velocity with different values of in the case of tri-stable potential. Solid line: stochastic averaging; Circle symbol: MCS.
The mean harvested power is a crucial indicator in assessing the capability of energy harvesters. From Eq. (11), 12
the mean-square voltage or current can be derived as 2
2 (H ) X X s*i ( H ) 2 E Y 2 X p( X , X ) d X d X , 2 2 ( H ) (H ) 2
(27)
Then, the mean harvested power can be given by E P E Y 2 .
(28)
Figure 9(a) depicts the variations of mean harvested power with noise intensity for three kinds of different potentials. It is clear that mean output power E[ P] always keeps increasing continuously as noise intensity D increases for each potential function. Corresponding to the three major classified potential functions, the electromechanical coupled energy harvester can be designed as three kinds of different multi-stable systems, i.e. mono-stable, bi-stable and tri-stable system. Among these systems, it can be seen obviously from Fig. 9(a) that E[ P] in the tri-stable system is higher than that in the bi-stable system and much higher than that in the mono-stable system for any noise intensity of colored noise. That is exactly the result of the different steady-state response in the intra-well and inter-well motions for the three kind of different potentials. It maybe because that the inter-well transition in tri-stable system gives rise to larger jumping range for the particle thereby amplifying the influence of noise intensity on harvested power compared with the mono-/bi-stable systems. Accordingly, the mean harvested power in the tri-stable system is the largest. Therefore, the tri-stable system outperforms the mono-stable and bi-stable systems in harvested power, and noise intensity of colored noise is beneficial to energy harvesting. The influence of correlation time on mean harvested power is also discussed in Fig. 9(b), taking tri-stable system as one example. Obviously, a larger correlation time results in a less harvested power, for the reason that a larger correlation time may reduce the vibration ranges of the particle. Thus, the correlation time of colored noise is not beneficial to energy harvesting in a coupled VEH, which makes it cannot be ignored in the realistic applications. Meanwhile, the theoretical and numerical results, as shown in Fig. 9, present an excellent agreement further validating the extended stochastic averaging method.
Fig. 9. (a) Mean harvested power for three kinds of different potentials corresponding to the parameters values from Fig. 2(b). (b) Mean harvested power with different values of in the case of tri-stable potential. Solid line: stochastic averaging; Circle symbol: MCS.
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4.3. Superiority of tri-stable VEH in energy harvesting The signal-to-noise ratio (SNR) is a crucial quantifier in characterizing the SR phenomenon, which occurs at the maximum of SNR. In this subsection, we focus on the study of SNR numerically from the original Eq. (4) through using the fourth-order Runge–Kutta algorithm, as it is arduous to obtain the identical analytical expression of SNR for the strongly nonlinear systems with three kinds of different potentials. Meanwhile, the influences of system parameters on the power conversion efficiency are also discussed to evaluate the performance of energy harvesting. The definition of SNR can be given by SNR 10 log10 ( Ps / Pn ) d B,
(29)
where Ps denotes the spectrum value at signal frequency in the output power spectrum, and Pn represents the average of the remaining output power spectrum values. The power conversion efficiency % C is defined as the total efficiency to evaluate the power conversion from the ambient noise and the provided periodic mechanical power to the harvested electrical power. That is % Pe Pm 100% ,
where Pe Y 2 , Pm X S () XF sin t
and
1 N
N
(30)
1 lim t d t i 1
tn
tn
n
0
denotes time-average and
ensemble average. In Fig. 10, the effects of noise intensity and periodic excitation on SNR and % are studied for the case of tri-stable potential. Evidently, it can be seen in Fig. 10(a) that SNR decreases first, increases to a maximum later and then keeps decreasing as D further increases for each one periodic excitation. It indicates that there is an optimal noise intensity Dopt to maximize SNR, i.e. SR occurs. Besides, as F increases, the peak of SNR moves toward the upper left gradually. In Fig. 10(b), there is also an optimal noise intensity Dopt to maximize % . Whereas, as F increases, the peak of % moves toward the lower left in a slow manner. These results indicate that larger periodic force can enhance the resonance effect but will weaken the power conversion efficiency for this case study. From viewpoint of occurrence of SR, a suitable dose of noise intensity and periodic force should be adopted to enhance the harvested energy by using SR effect. In both subplots, one can find that the corresponding Dopt decreases with increasing F , for the reason that larger periodic excitation makes it easily for particle to
through the barrier and move among the potential wells. Thus, a smaller optimal noise intensity is needed to induce SR under a larger periodic excitation.
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Fig. 10. (a) SNR and (b) the power conversion efficiency versus noise intensity for different periodic forces in the case of tri-stable potential.
For the same periodic excitation, the effect of the potential structure on SNR is discussed in Fig. 11. Figures 11(a) and 11(b) depict the influence of the cubic stiffness coefficient on SNR in the cases of bi-stable and tri-stable potentials, respectively. One can see that as increases, both of the peaks in SNR shift to upper left gradually. That is because both of the potential-well depth and the potential-well distance become smaller with increasing cubic stiffness coefficient, then the utilization of colored noise is enhanced leading to higher resonance effect and smaller optimal noise intensity. It indicates that both of the potential-well depth and the potential-well distance have significant influences on the energy harvesting performance for an electromechanical coupled energy harvester. For further study the effect of the potential structure on performance, comparisons of bi-stable and tri-stable potentials under the same potential-well depth and potential-well distance are displayed in Figs. 11(c) and 11(d), respectively. It is clear in both subplots that the tri-stable system has higher superiority in enhancing the output SNR and can achieve better resonance effect than the bi-stable system. The reason may be that, in the presence of the same periodic excitation, the three-inter-well transition in tri-stable system is superior to the two-inter-well transition in bi-stable system in one period, and the synergistic effect of colored noise on the periodic excitation in tri-stable system is also greater than that in bi-stable system when SR occurs. Thus, the tri-stable VEH , compared with the bi-stable VEH, can achieve better SR effect.
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Fig. 11. SNR versus noise intensity for fixed F 0.15 : (a) in the case of bi-stable potential for different cubic stiffness coefficients; (b) in the case of tri-stable potential for different cubic stiffness coefficients; (c) under the same potential-well depth between bi- and tri-stable potentials; (d) under the same potential-well distance between bi- and tri-stable potentials.
Moreover, comparisons between bi- and tri-stable potentials in the power conversion efficiency are presented in Fig. 12. It is shown that for each potential, the power conversion efficiency % increases invariably first to a maximum and then decreases with the increase of the electromechanical coupling coefficient or the time constant ratio . It indicates that there are an optimal opt and opt that maximize the power conversion efficiency from mechanical power to harvested electrical power. Besides, the value opt of tri-stable system is almost the same with that of bi-stable system, so does the value opt . Whereas, in both subplots, the values % in the case of tri-stable potential are always larger than that in the case of bi-stable potential. The result manifests that tri-stable VEH, compared with bi-stable VEH, can convert much more mechanical energy into harvester electrical energy, and can greatly enhance the energy harvesting performance by choosing the optimal system parameters, such as the electromechanical coupling coefficient and the time constant ratio.
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Fig. 12. (a) The power conversion efficiency versus the electromechanical coupling coefficient, and (b) the power conversion efficiency versus the time constant ratio in the cases of bi- and tri-stable potentials.
5. Conclusions In this paper, an electromechanical coupled VEH with two different conversion mechanisms under colored noise and periodic excitation is investigated theoretically and numerically for three kinds of different potentials. In particular, the comparisons among mono-, bi- and tri-stable VEHs in dynamical behaviors are analyzed in detail from the viewpoint of enhancing energy harvesting performance. It is found that the tri-stable VEH outperforms the mono- and bi-stable systems in the mean harvested power, SR effect and the power conversion efficiency. The obtained specific conclusions are as the following. 1) The extended stochastic averaging method based on energy-dependent frequency is proposed to deal with the electromechanical coupled tri-stable VEH with colored noise and periodic signal. The theoretical results are well validated by MCS from the original coupled system. 2) The periodic excitation can change the transient topological characteristic and even induce transition in one period. Besides, a larger periodic force can enhance the resonance effect but will weaken the power conversion efficiency. 3) Colored noise has a significant effect on energy harvesting performance, i.e. the inter-well transition and the mean harvested power can be enhanced by decreasing correlation time and increasing noise intensity, and there is an optimal noise intensity to induce SR. 4) There are an optimal time constant ratio and an optimal electromechanical coupling coefficient that maximize the power conversion efficiency from the mechanical power to the harvested electrical power. 5) The potential structure plays a crucial role in improving the performance of the coupled VEH. For the tri-stable potential, all of the noise utilization, SR effect and the power conversion efficiency are enhanced greatly compared with the case of bi-stable potential. The obtained results in this work can provide a significant theoretical guidance in optimizing the design of VEHs and enhancing electric energy harvested from ambient vibrations. Acknowledgements This work has been supported by the National Natural Science Foundation of China Granted Nos. 11832005 and 11772048.
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Graphical abstract
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CRediT author statement Yanxia Zhang: Writing- Original draft preparation, Methodology, Software, Data curation, Visualization, Investigation. Yanfei Jin: Conceptualization, Supervision, Validation, Writing- Reviewing and Editing. Pengfei Xu: Software.
Declaration of interests
☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
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