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Stochastic response of bistable vibration energy harvesting system subject to filtered Gaussian white noise
Mechanical Systems and Signal Processing 130 (2019) 201–212
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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp
Stochastic response of bistable vibration energy harvesting system subject to filtered Gaussian white noise Di Liu a,⇑, Yanru Wu a, Yong Xu b,*, Jing Li c a
School of Mathematics, Shanxi University, 030006 Taiyuan, Shanxi, China Department of Applied Mathematics, Northwestern Polytechnical University, 710072 Xi’an, Shaanxi, China c School of Applied Science, Taiyuan University of Science and Technology, 030024 Taiyuan, Shanxi, China b
a r t i c l e
i n f o
Article history: Received 9 January 2019 Received in revised form 28 March 2019 Accepted 2 May 2019 Available online 11 May 2019 Keywords: Energy harvesting Bistable Stochastic response Mean output power Filtered Gaussian white noise
a b s t r a c t Mechanical vibrations have been proved to be a clean and reliable energy source, especially the bistable model can enhance the efficiency of vibration energy harvesting under the case of low-level vibration and attracts more and more attention. In this manuscript, an improved coordinate transformation, based on the equilibrium points of bistable vibration energy harvesting (BVEH) system, is proposed to construct a quasi-conservative stochastic averaging procedure, and this method is applied to the nonlinear BVEH system driven by filtered Gaussian white noise to obtain the dynamic behaviors. Through this transformation, the nonlinear electromechanical coupling BVEH system can be approximated by an equivalent single degree of freedom bistable system, which contains the energydependent frequency functions and the equilibrium points. The analytic expressions of the stationary probability density function of the system state can be obtained by the quasi-conservative stochastic averaging method, and by applying the relationship between the output voltage and the state variables of the system, the mean-square output voltage (MSOV) and the mean output power (MOP) will be obtained. Finally, the variation trends of MSOV and MOP depended on the physical quantities of stochastic BVEH system, such as the excitation intensity and the peak frequency of seismic motion of the filtered Gaussian white noise, the parameters of the vibration system and the electromechanical coupling coefficients, are also analyzed in detail. Corresponding theoretical results are well verified through the direct Monte Carlo simulation. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction With the rapid development of science and technology, excessive exploitation and consumption of energy cause the shortage of energy and the pollution of the environment all over the world, which has attracted increasing attention in recent years. Therefore, full utilization of discarded wastes from industry, agriculture and natural environment, and the development of new and renewable energy sources have significant strategic importance on solving the crisis of energy resource shortage. Among them, the technology of vibration energy harvestering (VEH), which can convert the waste mechanical energy of the environment into electrical energy, has been proven to be an effective means [1,2]. Besides, this technology has the peculiarity of excellent power density, relatively high voltages and low currents, and thus has
⇑ Corresponding author. E-mail addresses: [email protected] (D. Liu), [email protected] (Y. Xu). https://doi.org/10.1016/j.ymssp.2019.05.004 0888-3270/Ó 2019 Elsevier Ltd. All rights reserved.
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been widely used in engineering practice, such as the supply portable or wearable devices, and other wireless sensor networks [3–5]. As we all know, energy harvesting efficiency is an essential standard and important basis to measure the VEH. In earlier studies, the researchers usually adopted the linear model to better understand the influential mechanism of dynamical modeling on energy harvesting efficiency [6–8]. However, some studies showed that the effective energy harvesting for the linear model can only be realized on the special frequency, which means that the efficiency and reliability for VEH technologies are limited and the linear model hindered the extensive use of these technologies [9–11]. To remedy this situation, many researchers improved the models of VEH and developed some novel nonlinear models such as nonlinear monostable, bistable and tristable models et al. [12–18]. At the same time, many theory analysis methods were developed to reveal the run-theory, improvement-mechanism and program-structure of the VEH technologies, which are helpful for the optimization design of these techniques. For example, Triplett and Quinn [19] investigated the response of the nonlinear harvesting system through Poincare-Lindstedt perturbation, and found that the power output is closely related to the nonlinear structure. Stanton et al. [20] found that the bistable vibration energy harvesting (BVEH) technology may increase the capacity of energy harvesting, and this conclusion is confirmed by developing a Melnikov theory. Later, a review paper summarized a portion of BVEH developments to date and highlighted the recent advance of MEMS-scale utility and efficiency metrics in [22] analyzed the influence of fractional order viscoelastic flexible material on the output the BVEH [21]. And then Oumbe electric energy of a tristable energy harvester by using the Krylov–Bogoliubov averaging method. Recently, a method combinated the harmonic balance and the Jacobian matrix were developed to study the analytical solution of tristable energy harvesters, and detailedly analyze on the stability of the system [23]. With the influence of the practical working environment of VEH devices, the VEH system will be inevitably interfered by random forces [24–26]. As the research goes on, researchers have further found that the random forces may seriously affect the vibration behavior of system and therefore, may affect their energy harvesting efficiency, through numerical simulation and experimental methods [27–31]. At the same time, some of the stochastic approximation methods were developed to study the performance of the stochastic nonlinear VEH systems [32–36], especially the bistable VEH systems. These systems can generate a type of vibration from one stable equilibrium position to the other one, which may arouse a large amplitude motion in the broadband range and also increase the output power significantly. For this reason, the study of the dynamic behaviors of the stochastic BVEH has attracted much attention. Daqaq et al. [37] studied the BVEH subject to Gaussian white noise and exponentially correlated noise through the decoupling theory, and then the moment method was used to obtain the stochastic response of the system [38]. Ali et al. [39] analyzed the mean output power (MOP) of the stochastic BVEH system through an equivalent linearization method, and the results were validated by the Monte Carlo (MC) simulation. Then the chaotic behavior and mean-square output voltage (MSOV) of a BVEH system with fractional order nonlinear term and randomly disordered periodic excitation were studied, which was on the basis of random Melnikov theory [40]. Recently, Xu and Li [41] used the standard stochastic averaging method to obtain the stochastic response of nonlinear BVEH system driven by Gaussian white noise. The VEH devices are usually placed in the inside of the mechanical structure, thus the external stochastic force will pass to the VEH devices by the mechanical structure. This means that the systems will suffer a filtered Gaussian white noise disturbance. This force describes a variety of random environmental perturbations and has the characteristic of frequency dependence, including the hydrodynamic forces due to sea-wave kinematics and stationary ground acceleration filtered through soil layers as the typical examples [42]. Compared with the idealized Gaussian white noise, filtered Gaussian white noise may have different effects on the dynamic response of the nonlinear stochastic system because of frequency dependence, which has attracted increasing attention in recent years. Jin [43] used the stochastic averaging method to investigate nonstationary seismic response of a single-degree-of-freedom nonlinear system subject to Kanai-Taijimi excitation. Later, the probabilistic solutions of the responses of the nonlinear multi-degree-of-freedom system were studied when it was excited by filtered Gaussian white noise through the state-space-split method combined with the exponential polynomial closure method [44]. Recently, the filtered Gaussian white noise has been introduced into the research of linear VEH system as a stochastic excitation force. Quaranta et al. [45] proposed a comprehensive method for the electromechanical probabilistic analysis of piezoelectric VEH system subjected to filtered white Gaussian noise at the base. Up to now, only a few researches have been done on the stochastic response of nonlinear VEH system subjected to the filtered Gaussian white noise, especially there is no effective analytical method to research the stochastic response of nonlinear BVEH system under the filtered Gaussian white noise excitation, as well as understand the effect of stochastic excitation on energy harvesting efficiency. In this paper, an improved coordinate transformation, based on the equilibrium points of the BVEH system, is proposed to construst a quasi-conservative stochastic averaging method, which can be used to analyze the stochastic response of the nonlinear BVEH system driven by filtering Gaussian white noise. The paper is organized as follows. Firstly, we establish the basic mathematical model of the BVEH system subject to filtered Gaussian white noise. In Section 3, an improved coordinate transformation, which is depending on the energy-dependence frequency and the equilbrium points of the BVEH system, is used to derive a equialent single degree of freedom system. On this basis, a quasi-conservative stochastic averaging method is used to obtain the probabilistic density function (PDF), MSOV and MOP under the assumption that the BVEH system is quasi-conservative in Section 4. In Section 5, the effects of the excitation, the damping coefficient, natural system frequency, stiffness coefficient, and other parameters on MSOV and MOP of the BVEH system are investigated, and the results are validated by MC. Finally, the conclusions are given in the last section.
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2. The bistable energy harvesters system under filtered white Gaussian noise Consider a class of nonlinear BVEH system, which is modeled as ferromagnetic cantilever beam system coupled to energy harvesting circuit and shown in Fig. 1. The corresponding non-dimensional governing equations can be expressed as
€ þ cX_ x2 X þ bX 3 þ jV ¼ nðtÞ X 0 V_ þ kV X_ ¼ 0
ð1aÞ ð1bÞ
€ denote the displacement, velocity and acceleration of beam, respectively. c represents the damping coefwhere X; X_ and X ficient. x0 denotes natural frequency, b is a positive stiffness coefficient. j represents electromechanical coupling coefficient. V is the output voltage across external resistive load, k denotes the inverse of the product of resistance and capacitance. nðt Þ is filtered Gaussian white noise, which describes the environmental stochastic force and can be modeled as [42]
nðtÞ ¼ 2fg xg u_ g ðt Þ þ x2g ug ðtÞ € g ðt Þ þ 2fg xg u_ g ðt Þ þ x u
2 g u g ðt Þ
ð2aÞ ¼ W ðt Þ
ð2bÞ
where fg and xg represent the damping constant and the peak frequency of seismic motion, respectively, which reflect the local geological site conditions. W ðtÞ is the Gaussian white noise with zero mean and auto-correlation function E½W ðtÞW ðt þ sÞ ¼ S0 dðsÞ, in which dðsÞ is the Dirac’s delta function and S0 is the power spectral density of W ðt Þ. The stochastic acceleration expressed by Eq. (2) is a filtered Gaussian white noise, whose power spectral density function is
SðxÞ ¼ S0
x4g þ 4f2g x2g x2
x2g x2
2
ð3Þ
þ 4f2g x2g x2
Notice that the stationary stochastic model defined by Eqs. (2a) and (2b) can be incorporated into the equation of motion and then the problem is reduced to the case of white noise excitation. This stochastic force also has its physical significance, and is correspond to the model of the mechanical structure and earth as a dynamic system. For this reason, excitation models given by Eq. (2) are often used to describe the stochastic force within the mechanical structure.
Fig. 1. The schematic of the BVEH system subject to filtered Gaussian white noise.
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3. Equivalent stochastic nonlinear bistable energy harvestering system In this section, the BVEH system (1) will be reduced to a new equivalent bistable system without couple through an improved coordinate transformation, the new system will depend on the energy-depedent frequency functions and the equilibrium points of the system. Integrating Eq. (1b), the output voltage can be approximated as [34]
Z V ðt Þ ’
t
X_ ðsÞ exp fkðt sÞgds ¼
0
Z
t
X_ ðt sÞ exp fksgds
ð4Þ
0
From Eq. (4), we find that the coupled output voltage can be approximated by vibration displacement or velocity of the mechanical system. Therefore, the coupled nonlinear BVEH system Eq. (1) will be decoupled and rewritten as
€ þ cX_ x2 X þ bX 3 þ j X 0
Z
t
X_ ðt sÞ exp fksgds ¼ nðt Þ
ð5Þ
0
Carefully observing the nonlinear system (5), we will find that the undamped free vibration of this nonlinear system is still a bistable system, and the three equilibrium points of the system are of the type stable unstable stable, that is xH . Thus the corresponding potential energy U ðX Þ and total energy H of the system (5) are X ¼ ~ xH ð< 0Þ; X ¼ 0 and X ¼ ~
U ðX Þ ¼ H¼
1 2 2 2 X xH ; 4b
1 _2 X þ U ðX Þ 2
ð6Þ
Obviously, the undamped free vibration of system (5) will vibrate around one of the stable equilibrium points ð~ xH or ~ xH Þ when H < U ð0Þ, which means that the system oscillates near the left or right of the equilibrium point in the phase space and the location of oscillation is determined by the initial position. Otherwise, a more complicated motion of BVEH system may happen, across the unstable point from a potential well to the other, and then across back. Correspondently, some motion features of the bistable system for the given total energy H are shown in Fig. 2. In those cases, the corresponding period and frequency of the undamped free vibration of system (5) are calculated by
Z T H ¼ 2T 1=2 ¼ 2
AH
AH
2p xð H Þ ¼ ; TH
1
2
Z AH 2 1 1 pffiffiffiffiffiffi dX ¼ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dX; _X 2 2H 2U ðX Þ AH 1
ð7Þ
where AH1 and AH2 represent the smallest and largest possible diaplacements for a given H, which can be determined through solving the algebraic equation U ðAH Þ H ¼ 0. Now return to the system (5). Under the condition that both the damping and excitation are weak, the total energy H of the original system is a slowly varying function of time. According to the relationship between H and U ð0Þ, the period T H calculated from (7) can be considered as the quasi-period of the original system (1). We introduce a new transformation, which depends on the equilibrium point of the system
Fig. 2. (a) Some characteristics of the conservative bistable system potential energy U given by Eq. (6). (b) phase plane for the system under (a) plotted for different energy levels.
D. Liu et al. / Mechanical Systems and Signal Processing 130 (2019) 201–212
pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi sgn X xH U ðX Þ ¼ H cos h; pffiffiffiffiffiffiffi _X ¼ Y ¼ 2H sin h;
205
ð8Þ
where xH is the equilibrium point of the bistable system (5) and can be determined by
8 > < ~xH ; for H < U ð0Þ and X < 0: ~xH ; for H < U ð0Þ and X > 0: xH ¼ > : 0; for H > U ð0Þ:
ð9Þ
Rt in which U ðX Þ and Hðt Þ are the potential energy and the total energy of the system. hðt Þ ¼ 0 xðHÞds þ /ðtÞ is the total phase, xðHÞ is the quasi-frequency of system (5), which depends on the quasi-period T H , and the relevant calculation formulas are given in Eq. (7). /ðtÞ is the initial phase which changes slowly. Therefore, one can obtain the following expression
hðt sÞ ¼ hðt Þ xðHÞs; pffiffiffiffiffiffiffiffiffiffiffiffiffiffi X_ ðt sÞ ¼ X_ cos ðxðHÞsÞ þ sgnðX x ðHÞÞ 2U ðX Þ sin ðxðHÞsÞ:
ð10Þ
Substituting Eq. (10) into Eq. (4), the output voltage can be rewritten by
V ðt Þ ¼
x2 ðHÞ k X xH þ 2 X_ 2 k þ x ðH Þ k þ x2 ð H Þ 2
ð11Þ
Then the nonlinear electromechanical coupling stochastic BVEH system (1) can be replaced by the following equivalent single degree of freedom bistable system:
€ þ ðc þ N1 ðHÞÞX_ x2 X þ bX 3 þ N 2 ðHÞ X x ¼ nðt Þ X 0 H
ð12Þ
here N 1 ðHÞ ¼ jk= k2 þ x2 ðHÞ and N 2 ðHÞ ¼ jx2 ðHÞ= k2 þ x2 ðHÞ are the equivalent damping coefficient and the equivalent stiffness coefficient of output voltage, respectively. The total mechanical energy of system (12) is defined as
1 x2 N 2 ð H Þ 2 b X2 0 ; 4 b 1 HðtÞ ¼ X_ 2 þ U ðX Þ: 2
U ðX Þ ¼
ð13Þ
Meanwhile, by applying Eq. (9), the stable equilibrium point xH of the mechanical system can be obtained as ~ xH ¼ x20 N 2 ðHÞ =b for H < U ð0Þ. The changing curves of the quasi-frequence and the quasi-period of the BVEH system under different H are shown in Fig. 3. Through the mentioned analysis above, we know that the BVEH system will oscillate in the left or right side of the potential well when H < U ð0Þ, otherwise, the system will oscillate from one potential well to the other. Thus, by solving U ðAH Þ H ¼ 0, the smallest possible position AH1 and the largest possible position AH2 of vibration displacement in different conditions can be expressed as
Fig. 3. The frequence and the period of the BVEH system for the different total energy H.
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AH 1
AH 2
8 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffi x20 þN2 ðHÞ > H > þ 2 ; for H < U ð0Þ and X > b b > > > > ffi < rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffi x20 N2 ðHÞ ¼ 2 Hb ; for H < U ð0Þ and X b > > > ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > rx qffiffiffiffi 2 þN ðH Þ > > 2 H 0 : þ 2 ; for H > U ð0Þ: b b ffi 8 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffi 2 > > x0 Nb 2 ðHÞ 2 Hb ; for H < U ð0Þ and X > > > > > rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < qffiffiffiffi x20 þN2 ðHÞ H ¼ þ 2 ; for H < U ð0Þ and X b b > > > ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi r > > qffiffiffi > x20 þN2 ðHÞ > : þ 2 Hb ; for H > U ð0Þ: b
< 0: > 0:
ð14Þ < 0: > 0:
4. Stochastic averaging of quasi-conservative Substituting Eq. (8) into Eq. (12), we can obtain the following two first-order stochastic differential equations about Hðt Þ and /ðtÞ
Fig. 4. The probabilistic response staatistic of the BVEH system. (a) the joint PDF in analytical result of the displacement and the velocity; (b) the joint PDF in MC result of the displacement and the velocity; (c) the marginal PDF of the displacement; (d) the marginal PDF of the velocity.
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/_ ¼ f 1 ðH; hÞ þ g 1 ðH; hÞnðtÞ; H_ ¼ f ðH; hÞ þ g ðH; hÞnðt Þ;
ð15Þ
1 f 1 ðH; hÞ ¼ ðc þ N1 ðHÞÞ sin h cos h pffiffiffiffiffiffiffi N2 ðHÞxH cos h; 2H pffiffiffiffiffiffiffi 2 f 2 ðH; hÞ ¼ 2Hðc þ N1 ðHÞÞsin h 2HN 2 ðHÞxH sin h; pffiffiffiffiffiffiffi 1 g 2 ðH; hÞ ¼ 2H sin h g 1 ðH; hÞ ¼ pffiffiffiffiffiffiffi cos h; 2H
ð16Þ
2
2
where
Assuming that the correlation times of the filtered Gaussion white noise is shorter than the relaxation time of the energy process, Hðt Þ can be approximated as a Markovian process by stochastic technique [46]. The averaged H process is governed ^ stochastic differential equation by the following Ito
dH ¼ mðHÞdt þ rðHÞdBðtÞ
ð17Þ
Fig. 5. (a) The dependence of the MSOV E½V 2 on the excitation intensity S0 ; (b) The dependence of the MOP E½P on the excitation intensity S0 .
Fig. 6. (a) The dependence of the MSOV E½V 2 on the peak frequency of seismic motion xg of the excitation; (b) The dependence of the MOP E½P on the peak frequency of seismic motion xg of the excitation.
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where mðHÞ and
rðHÞ are averaged drift and diffusion coefficients, which are given as follows:
i E 2 ðt Þ 2 ðt Þ g 1 ðt þ sÞ @g@/ þ g 2 ðt þ sÞ @g@H RðsÞds t D E D E ¼ ðc þ N1 ðHÞÞ X_ 2 N2 ðHÞxH X_ þ pSðxðHÞÞ Z þ1
t Dt E r2 ðHÞ ¼ g 2 ðt þ sÞg 2 ðtÞRðsÞds ¼ 2pSðxðHÞÞ X_ 2
mðHÞ ¼ hf 2 ðH; hÞit þ
DR
þ1 0
h
1
t
t
ð18aÞ ð18bÞ
RT in which h½it ¼ 0 H ½dt=T H is the time-averaging operation. T H is the quasi-period of system, and the relevant calculation formula is given in Eq. (7). One obtains
Z D E 2 AH2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X_ ¼ d 2H 2U ðX Þ; t T H AH 1 Z Z D E 2 AH 2 _ 2 AH2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi XdX ¼ X_ 2 ¼ 2H 2U ðX ÞdX: t T H AH T H AH 1
ð19Þ
1
The exact stationary PDF of the total energy Hðt Þ is
Fig. 7. (a) The dependence of the MSOV E½V 2 on the damping coefficient c; (b) The dependence of the MOP E½P on the damping coefficient c.
Fig. 8. (a) The dependence of the MSOV E½V 2 on the system natural frequence x0 ; (b) The dependence of the MOP E½P on the system frequence x0 .
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pðHÞ ¼
C exp r2 ðHÞ
Z
H
0
2mðxÞ dx r2 ðxÞ
ð20Þ
here C is a constant determined by the condition of normalization. Applying Eq. (13), the joint PDF of displacement X and the velocity X_ is easily obtained as
pðHÞ p X; X_ ¼ j 1_2 : T H H¼2X þUðX Þ
ð21Þ
The correspondingly marginal PDFs of system displacement X and velocity X_ can be obtained respectively as follows
Z
þ1
pðX Þ ¼ p X_ ¼
p X; X_ dX_
1 þ1
Z
p X; X_ dX
ð22aÞ ð22bÞ
1
Applying the relationship between output voltage and state variables (dispalcement and velocity), which is given by Eq. (11), the MSOV can be calculated by
Fig. 9. (a) The dependence of the MSOV E½V 2 on the stiffness coefficient b; (b) The dependence of the MOP E½P on the stiffness coefficient b.
Fig. 10. (a) The dependence of the MSOV E½V 2 on the parameter k; (b) The dependence of the MOP E½P on the parameter k.
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Fig. 11. (a) The dependence of the MSOV E½V 2 on the parameter
h i Z E V2 ¼
þ1
Z
1
þ1
1
j; (b) The dependence of the MOP E½P on the parameter j.
x2 ðHÞ k ðX x ðHÞÞ þ 2 X_ 2 k þ x2 ð H Þ k þ x2 ð H Þ
!2
_ p X; X_ dXdX;
ð23Þ
and the MOP can be obtained through a linear relationship with the MSPV as
h i E½P ¼ jkE V 2 :
ð24Þ
5. Numerical results and stochastic analysis In this section, we show the results obtained by the improved quasi-conservative averaging procedure, which is depended on the energy-dependent frequency and the equilbrium points of the BVEH system, and at the same time use the direct MC numerical simulation method to verify the effectiveness of the proposed method. For simplicity, we choose the parameters of the BVEH system as c ¼ 0:02; x0 ¼ 2:0; b ¼ 3:0; k ¼ 0:05 and j ¼ 0:5, and the parameters of filtered Gaussian white noise as fg ¼ 0:64; xg ¼ 15:0 and S0 ¼ 0:003, unless otherwise mentioned. In addition, the solid lines represent the analytical results by applying our proposed method, while the red stars denote the direct numerical simulation through MC. The joint PDFs of the system displacement and velocity under filtered Gaussian white noise excitation are depicted in Fig. 4. Fig. 4 (a) describes an analytical bimodal joint PDF through our technique, with the comparison of results from the direct MC simu lation is given in Fig. 4 (b), which shows the availability of the presented technique. The marginal PDFs pðX Þ and p X_ of system displacement and velocity are shown in Figs. 4 (c) and (d), respectively. We find that the marginal PDF of system displacement is also bimodal and the probability is equal to zero at location X ¼ 0, which means that the chaotic motion cannot occur and jumps between two potential wells are not frequent, and thus the dominant motion of the system is located in one potential well. To better understand the effects of the filtered Gaussian white noise on MSOV E½V 2 and MOP E½P of the BVEH system, the changes of MSOV and MOP with excitation intensity S0 and the peak frequency of seismic motion xg of the stochastic excitation are shown in Figs. 5 and 6, respectively. It can be seen from Fig. 5 (a), the increase of the noise intensity will increase the MSOV and the corresponding MOP has the same trend. The changes of seismic motion of the stochastic excitation are shown in Fig. 6. We find an opposite trend relative to the effects of the noise intensity, which means that the seismic motion of the stochastic excitation will hinder the production of the MSOV and MOP under the same noise intensity. In Figs. 7–9, we studied the effects of damping coefficient c, the natural frequency x0 and the stiffness coefficient b on MSOV E½V 2 and MOP E½P. One can see from Fig. 7 that the MSOV and MOP will decrease quickly with the increase of damping coefficient, and the decline rate slow down gradually. However, the dependences of the MSOV and MOP show a linear down tend with the increase of x0 and b, which are shown in Figs. 8 and 9. Finally, the effects of the coupling parameters k and j on MSOV and MOP are shown in Figs. 10 and 11, respectively. The increase of the coupling parameter k and j will result in the decrease of the MSOV, but favor the MOP growth. This implies that the coupling parameters benifit the vibration energy scavenging and thus we may improve the energy harvesting capacity through optimizing the coupling.
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6. Conclusions In this manuscript, an improved coordinate transformation, based on the equilibrium points of the BVEH system, is proposed to construct the quasi-conservative stochastic averaging procedure, which is used to analyze the stochastic response of the nonlinear BVEH system driven by filtering Gaussian white noise. First, the nonlinear electromechanical coupling BVEH system can be approximated by an equivalent single degree of freedom system through the new transformation mentioned previously, which contains the energy-dependent frequency functions and the equilibrium points of the bistable system. Next, the quasi-conservative averaging method is used to derive the averaged drift and diffusion coefficients of the total energy process H, which is dependent on the statistical characteristics of filtered Gaussian white noise and the equilibrium points of the systems. Then the analytic expressions of PDFs of the mechanical system state, MSOV and the MOP are obtained. Finally, the effects of the excitation intensity S0 and the peak frequency of seismic motion xg of the stochastic excitation, the parameters of the vibration system including damping coefficient c, the natural frequency x0 and the stiffness coefficient b, and the electromechanical coupling coefficients including the parameters k and j on the MSOV and the MOP of the stochastic BVEH system are also analyzed in detail. The results in this paper have certain significance for improving the energy harvesting capacity by optimizing the stochastic bistable systems. Acknowledgements This work is supported by the National Nature Science Foundation of China (No. 11402139 and 11572247) and the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (No. 2016114). References [1] S. Priya, D.J. Inman, Energy Harvesting Technologies, Springer, US, 2009. [2] A. Erturk, D.J. Inman, Piezoelectric Energy Harvesteing, John Wiley and Sons, 2011. [3] L.M. Swallow, J. 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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp
Stochastic response of bistable vibration energy harvesting system subject to filtered Gaussian white noise Di Liu a,⇑, Yanru Wu a, Yong Xu b,*, Jing Li c a
School of Mathematics, Shanxi University, 030006 Taiyuan, Shanxi, China Department of Applied Mathematics, Northwestern Polytechnical University, 710072 Xi’an, Shaanxi, China c School of Applied Science, Taiyuan University of Science and Technology, 030024 Taiyuan, Shanxi, China b
a r t i c l e
i n f o
Article history: Received 9 January 2019 Received in revised form 28 March 2019 Accepted 2 May 2019 Available online 11 May 2019 Keywords: Energy harvesting Bistable Stochastic response Mean output power Filtered Gaussian white noise
a b s t r a c t Mechanical vibrations have been proved to be a clean and reliable energy source, especially the bistable model can enhance the efficiency of vibration energy harvesting under the case of low-level vibration and attracts more and more attention. In this manuscript, an improved coordinate transformation, based on the equilibrium points of bistable vibration energy harvesting (BVEH) system, is proposed to construct a quasi-conservative stochastic averaging procedure, and this method is applied to the nonlinear BVEH system driven by filtered Gaussian white noise to obtain the dynamic behaviors. Through this transformation, the nonlinear electromechanical coupling BVEH system can be approximated by an equivalent single degree of freedom bistable system, which contains the energydependent frequency functions and the equilibrium points. The analytic expressions of the stationary probability density function of the system state can be obtained by the quasi-conservative stochastic averaging method, and by applying the relationship between the output voltage and the state variables of the system, the mean-square output voltage (MSOV) and the mean output power (MOP) will be obtained. Finally, the variation trends of MSOV and MOP depended on the physical quantities of stochastic BVEH system, such as the excitation intensity and the peak frequency of seismic motion of the filtered Gaussian white noise, the parameters of the vibration system and the electromechanical coupling coefficients, are also analyzed in detail. Corresponding theoretical results are well verified through the direct Monte Carlo simulation. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction With the rapid development of science and technology, excessive exploitation and consumption of energy cause the shortage of energy and the pollution of the environment all over the world, which has attracted increasing attention in recent years. Therefore, full utilization of discarded wastes from industry, agriculture and natural environment, and the development of new and renewable energy sources have significant strategic importance on solving the crisis of energy resource shortage. Among them, the technology of vibration energy harvestering (VEH), which can convert the waste mechanical energy of the environment into electrical energy, has been proven to be an effective means [1,2]. Besides, this technology has the peculiarity of excellent power density, relatively high voltages and low currents, and thus has
⇑ Corresponding author. E-mail addresses: [email protected] (D. Liu), [email protected] (Y. Xu). https://doi.org/10.1016/j.ymssp.2019.05.004 0888-3270/Ó 2019 Elsevier Ltd. All rights reserved.
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been widely used in engineering practice, such as the supply portable or wearable devices, and other wireless sensor networks [3–5]. As we all know, energy harvesting efficiency is an essential standard and important basis to measure the VEH. In earlier studies, the researchers usually adopted the linear model to better understand the influential mechanism of dynamical modeling on energy harvesting efficiency [6–8]. However, some studies showed that the effective energy harvesting for the linear model can only be realized on the special frequency, which means that the efficiency and reliability for VEH technologies are limited and the linear model hindered the extensive use of these technologies [9–11]. To remedy this situation, many researchers improved the models of VEH and developed some novel nonlinear models such as nonlinear monostable, bistable and tristable models et al. [12–18]. At the same time, many theory analysis methods were developed to reveal the run-theory, improvement-mechanism and program-structure of the VEH technologies, which are helpful for the optimization design of these techniques. For example, Triplett and Quinn [19] investigated the response of the nonlinear harvesting system through Poincare-Lindstedt perturbation, and found that the power output is closely related to the nonlinear structure. Stanton et al. [20] found that the bistable vibration energy harvesting (BVEH) technology may increase the capacity of energy harvesting, and this conclusion is confirmed by developing a Melnikov theory. Later, a review paper summarized a portion of BVEH developments to date and highlighted the recent advance of MEMS-scale utility and efficiency metrics in [22] analyzed the influence of fractional order viscoelastic flexible material on the output the BVEH [21]. And then Oumbe electric energy of a tristable energy harvester by using the Krylov–Bogoliubov averaging method. Recently, a method combinated the harmonic balance and the Jacobian matrix were developed to study the analytical solution of tristable energy harvesters, and detailedly analyze on the stability of the system [23]. With the influence of the practical working environment of VEH devices, the VEH system will be inevitably interfered by random forces [24–26]. As the research goes on, researchers have further found that the random forces may seriously affect the vibration behavior of system and therefore, may affect their energy harvesting efficiency, through numerical simulation and experimental methods [27–31]. At the same time, some of the stochastic approximation methods were developed to study the performance of the stochastic nonlinear VEH systems [32–36], especially the bistable VEH systems. These systems can generate a type of vibration from one stable equilibrium position to the other one, which may arouse a large amplitude motion in the broadband range and also increase the output power significantly. For this reason, the study of the dynamic behaviors of the stochastic BVEH has attracted much attention. Daqaq et al. [37] studied the BVEH subject to Gaussian white noise and exponentially correlated noise through the decoupling theory, and then the moment method was used to obtain the stochastic response of the system [38]. Ali et al. [39] analyzed the mean output power (MOP) of the stochastic BVEH system through an equivalent linearization method, and the results were validated by the Monte Carlo (MC) simulation. Then the chaotic behavior and mean-square output voltage (MSOV) of a BVEH system with fractional order nonlinear term and randomly disordered periodic excitation were studied, which was on the basis of random Melnikov theory [40]. Recently, Xu and Li [41] used the standard stochastic averaging method to obtain the stochastic response of nonlinear BVEH system driven by Gaussian white noise. The VEH devices are usually placed in the inside of the mechanical structure, thus the external stochastic force will pass to the VEH devices by the mechanical structure. This means that the systems will suffer a filtered Gaussian white noise disturbance. This force describes a variety of random environmental perturbations and has the characteristic of frequency dependence, including the hydrodynamic forces due to sea-wave kinematics and stationary ground acceleration filtered through soil layers as the typical examples [42]. Compared with the idealized Gaussian white noise, filtered Gaussian white noise may have different effects on the dynamic response of the nonlinear stochastic system because of frequency dependence, which has attracted increasing attention in recent years. Jin [43] used the stochastic averaging method to investigate nonstationary seismic response of a single-degree-of-freedom nonlinear system subject to Kanai-Taijimi excitation. Later, the probabilistic solutions of the responses of the nonlinear multi-degree-of-freedom system were studied when it was excited by filtered Gaussian white noise through the state-space-split method combined with the exponential polynomial closure method [44]. Recently, the filtered Gaussian white noise has been introduced into the research of linear VEH system as a stochastic excitation force. Quaranta et al. [45] proposed a comprehensive method for the electromechanical probabilistic analysis of piezoelectric VEH system subjected to filtered white Gaussian noise at the base. Up to now, only a few researches have been done on the stochastic response of nonlinear VEH system subjected to the filtered Gaussian white noise, especially there is no effective analytical method to research the stochastic response of nonlinear BVEH system under the filtered Gaussian white noise excitation, as well as understand the effect of stochastic excitation on energy harvesting efficiency. In this paper, an improved coordinate transformation, based on the equilibrium points of the BVEH system, is proposed to construst a quasi-conservative stochastic averaging method, which can be used to analyze the stochastic response of the nonlinear BVEH system driven by filtering Gaussian white noise. The paper is organized as follows. Firstly, we establish the basic mathematical model of the BVEH system subject to filtered Gaussian white noise. In Section 3, an improved coordinate transformation, which is depending on the energy-dependence frequency and the equilbrium points of the BVEH system, is used to derive a equialent single degree of freedom system. On this basis, a quasi-conservative stochastic averaging method is used to obtain the probabilistic density function (PDF), MSOV and MOP under the assumption that the BVEH system is quasi-conservative in Section 4. In Section 5, the effects of the excitation, the damping coefficient, natural system frequency, stiffness coefficient, and other parameters on MSOV and MOP of the BVEH system are investigated, and the results are validated by MC. Finally, the conclusions are given in the last section.
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203
2. The bistable energy harvesters system under filtered white Gaussian noise Consider a class of nonlinear BVEH system, which is modeled as ferromagnetic cantilever beam system coupled to energy harvesting circuit and shown in Fig. 1. The corresponding non-dimensional governing equations can be expressed as
€ þ cX_ x2 X þ bX 3 þ jV ¼ nðtÞ X 0 V_ þ kV X_ ¼ 0
ð1aÞ ð1bÞ
€ denote the displacement, velocity and acceleration of beam, respectively. c represents the damping coefwhere X; X_ and X ficient. x0 denotes natural frequency, b is a positive stiffness coefficient. j represents electromechanical coupling coefficient. V is the output voltage across external resistive load, k denotes the inverse of the product of resistance and capacitance. nðt Þ is filtered Gaussian white noise, which describes the environmental stochastic force and can be modeled as [42]
nðtÞ ¼ 2fg xg u_ g ðt Þ þ x2g ug ðtÞ € g ðt Þ þ 2fg xg u_ g ðt Þ þ x u
2 g u g ðt Þ
ð2aÞ ¼ W ðt Þ
ð2bÞ
where fg and xg represent the damping constant and the peak frequency of seismic motion, respectively, which reflect the local geological site conditions. W ðtÞ is the Gaussian white noise with zero mean and auto-correlation function E½W ðtÞW ðt þ sÞ ¼ S0 dðsÞ, in which dðsÞ is the Dirac’s delta function and S0 is the power spectral density of W ðt Þ. The stochastic acceleration expressed by Eq. (2) is a filtered Gaussian white noise, whose power spectral density function is
SðxÞ ¼ S0
x4g þ 4f2g x2g x2
x2g x2
2
ð3Þ
þ 4f2g x2g x2
Notice that the stationary stochastic model defined by Eqs. (2a) and (2b) can be incorporated into the equation of motion and then the problem is reduced to the case of white noise excitation. This stochastic force also has its physical significance, and is correspond to the model of the mechanical structure and earth as a dynamic system. For this reason, excitation models given by Eq. (2) are often used to describe the stochastic force within the mechanical structure.
Fig. 1. The schematic of the BVEH system subject to filtered Gaussian white noise.
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3. Equivalent stochastic nonlinear bistable energy harvestering system In this section, the BVEH system (1) will be reduced to a new equivalent bistable system without couple through an improved coordinate transformation, the new system will depend on the energy-depedent frequency functions and the equilibrium points of the system. Integrating Eq. (1b), the output voltage can be approximated as [34]
Z V ðt Þ ’
t
X_ ðsÞ exp fkðt sÞgds ¼
0
Z
t
X_ ðt sÞ exp fksgds
ð4Þ
0
From Eq. (4), we find that the coupled output voltage can be approximated by vibration displacement or velocity of the mechanical system. Therefore, the coupled nonlinear BVEH system Eq. (1) will be decoupled and rewritten as
€ þ cX_ x2 X þ bX 3 þ j X 0
Z
t
X_ ðt sÞ exp fksgds ¼ nðt Þ
ð5Þ
0
Carefully observing the nonlinear system (5), we will find that the undamped free vibration of this nonlinear system is still a bistable system, and the three equilibrium points of the system are of the type stable unstable stable, that is xH . Thus the corresponding potential energy U ðX Þ and total energy H of the system (5) are X ¼ ~ xH ð< 0Þ; X ¼ 0 and X ¼ ~
U ðX Þ ¼ H¼
1 2 2 2 X xH ; 4b
1 _2 X þ U ðX Þ 2
ð6Þ
Obviously, the undamped free vibration of system (5) will vibrate around one of the stable equilibrium points ð~ xH or ~ xH Þ when H < U ð0Þ, which means that the system oscillates near the left or right of the equilibrium point in the phase space and the location of oscillation is determined by the initial position. Otherwise, a more complicated motion of BVEH system may happen, across the unstable point from a potential well to the other, and then across back. Correspondently, some motion features of the bistable system for the given total energy H are shown in Fig. 2. In those cases, the corresponding period and frequency of the undamped free vibration of system (5) are calculated by
Z T H ¼ 2T 1=2 ¼ 2
AH
AH
2p xð H Þ ¼ ; TH
1
2
Z AH 2 1 1 pffiffiffiffiffiffi dX ¼ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dX; _X 2 2H 2U ðX Þ AH 1
ð7Þ
where AH1 and AH2 represent the smallest and largest possible diaplacements for a given H, which can be determined through solving the algebraic equation U ðAH Þ H ¼ 0. Now return to the system (5). Under the condition that both the damping and excitation are weak, the total energy H of the original system is a slowly varying function of time. According to the relationship between H and U ð0Þ, the period T H calculated from (7) can be considered as the quasi-period of the original system (1). We introduce a new transformation, which depends on the equilibrium point of the system
Fig. 2. (a) Some characteristics of the conservative bistable system potential energy U given by Eq. (6). (b) phase plane for the system under (a) plotted for different energy levels.
D. Liu et al. / Mechanical Systems and Signal Processing 130 (2019) 201–212
pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi sgn X xH U ðX Þ ¼ H cos h; pffiffiffiffiffiffiffi _X ¼ Y ¼ 2H sin h;
205
ð8Þ
where xH is the equilibrium point of the bistable system (5) and can be determined by
8 > < ~xH ; for H < U ð0Þ and X < 0: ~xH ; for H < U ð0Þ and X > 0: xH ¼ > : 0; for H > U ð0Þ:
ð9Þ
Rt in which U ðX Þ and Hðt Þ are the potential energy and the total energy of the system. hðt Þ ¼ 0 xðHÞds þ /ðtÞ is the total phase, xðHÞ is the quasi-frequency of system (5), which depends on the quasi-period T H , and the relevant calculation formulas are given in Eq. (7). /ðtÞ is the initial phase which changes slowly. Therefore, one can obtain the following expression
hðt sÞ ¼ hðt Þ xðHÞs; pffiffiffiffiffiffiffiffiffiffiffiffiffiffi X_ ðt sÞ ¼ X_ cos ðxðHÞsÞ þ sgnðX x ðHÞÞ 2U ðX Þ sin ðxðHÞsÞ:
ð10Þ
Substituting Eq. (10) into Eq. (4), the output voltage can be rewritten by
V ðt Þ ¼
x2 ðHÞ k X xH þ 2 X_ 2 k þ x ðH Þ k þ x2 ð H Þ 2
ð11Þ
Then the nonlinear electromechanical coupling stochastic BVEH system (1) can be replaced by the following equivalent single degree of freedom bistable system:
€ þ ðc þ N1 ðHÞÞX_ x2 X þ bX 3 þ N 2 ðHÞ X x ¼ nðt Þ X 0 H
ð12Þ
here N 1 ðHÞ ¼ jk= k2 þ x2 ðHÞ and N 2 ðHÞ ¼ jx2 ðHÞ= k2 þ x2 ðHÞ are the equivalent damping coefficient and the equivalent stiffness coefficient of output voltage, respectively. The total mechanical energy of system (12) is defined as
1 x2 N 2 ð H Þ 2 b X2 0 ; 4 b 1 HðtÞ ¼ X_ 2 þ U ðX Þ: 2
U ðX Þ ¼
ð13Þ
Meanwhile, by applying Eq. (9), the stable equilibrium point xH of the mechanical system can be obtained as ~ xH ¼ x20 N 2 ðHÞ =b for H < U ð0Þ. The changing curves of the quasi-frequence and the quasi-period of the BVEH system under different H are shown in Fig. 3. Through the mentioned analysis above, we know that the BVEH system will oscillate in the left or right side of the potential well when H < U ð0Þ, otherwise, the system will oscillate from one potential well to the other. Thus, by solving U ðAH Þ H ¼ 0, the smallest possible position AH1 and the largest possible position AH2 of vibration displacement in different conditions can be expressed as
Fig. 3. The frequence and the period of the BVEH system for the different total energy H.
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AH 1
AH 2
8 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffi x20 þN2 ðHÞ > H > þ 2 ; for H < U ð0Þ and X > b b > > > > ffi < rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffi x20 N2 ðHÞ ¼ 2 Hb ; for H < U ð0Þ and X b > > > ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > rx qffiffiffiffi 2 þN ðH Þ > > 2 H 0 : þ 2 ; for H > U ð0Þ: b b ffi 8 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffi 2 > > x0 Nb 2 ðHÞ 2 Hb ; for H < U ð0Þ and X > > > > > rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < qffiffiffiffi x20 þN2 ðHÞ H ¼ þ 2 ; for H < U ð0Þ and X b b > > > ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi r > > qffiffiffi > x20 þN2 ðHÞ > : þ 2 Hb ; for H > U ð0Þ: b
< 0: > 0:
ð14Þ < 0: > 0:
4. Stochastic averaging of quasi-conservative Substituting Eq. (8) into Eq. (12), we can obtain the following two first-order stochastic differential equations about Hðt Þ and /ðtÞ
Fig. 4. The probabilistic response staatistic of the BVEH system. (a) the joint PDF in analytical result of the displacement and the velocity; (b) the joint PDF in MC result of the displacement and the velocity; (c) the marginal PDF of the displacement; (d) the marginal PDF of the velocity.
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/_ ¼ f 1 ðH; hÞ þ g 1 ðH; hÞnðtÞ; H_ ¼ f ðH; hÞ þ g ðH; hÞnðt Þ;
ð15Þ
1 f 1 ðH; hÞ ¼ ðc þ N1 ðHÞÞ sin h cos h pffiffiffiffiffiffiffi N2 ðHÞxH cos h; 2H pffiffiffiffiffiffiffi 2 f 2 ðH; hÞ ¼ 2Hðc þ N1 ðHÞÞsin h 2HN 2 ðHÞxH sin h; pffiffiffiffiffiffiffi 1 g 2 ðH; hÞ ¼ 2H sin h g 1 ðH; hÞ ¼ pffiffiffiffiffiffiffi cos h; 2H
ð16Þ
2
2
where
Assuming that the correlation times of the filtered Gaussion white noise is shorter than the relaxation time of the energy process, Hðt Þ can be approximated as a Markovian process by stochastic technique [46]. The averaged H process is governed ^ stochastic differential equation by the following Ito
dH ¼ mðHÞdt þ rðHÞdBðtÞ
ð17Þ
Fig. 5. (a) The dependence of the MSOV E½V 2 on the excitation intensity S0 ; (b) The dependence of the MOP E½P on the excitation intensity S0 .
Fig. 6. (a) The dependence of the MSOV E½V 2 on the peak frequency of seismic motion xg of the excitation; (b) The dependence of the MOP E½P on the peak frequency of seismic motion xg of the excitation.
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where mðHÞ and
rðHÞ are averaged drift and diffusion coefficients, which are given as follows:
i E 2 ðt Þ 2 ðt Þ g 1 ðt þ sÞ @g@/ þ g 2 ðt þ sÞ @g@H RðsÞds t D E D E ¼ ðc þ N1 ðHÞÞ X_ 2 N2 ðHÞxH X_ þ pSðxðHÞÞ Z þ1
t Dt E r2 ðHÞ ¼ g 2 ðt þ sÞg 2 ðtÞRðsÞds ¼ 2pSðxðHÞÞ X_ 2
mðHÞ ¼ hf 2 ðH; hÞit þ
DR
þ1 0
h
1
t
t
ð18aÞ ð18bÞ
RT in which h½it ¼ 0 H ½dt=T H is the time-averaging operation. T H is the quasi-period of system, and the relevant calculation formula is given in Eq. (7). One obtains
Z D E 2 AH2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X_ ¼ d 2H 2U ðX Þ; t T H AH 1 Z Z D E 2 AH 2 _ 2 AH2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi XdX ¼ X_ 2 ¼ 2H 2U ðX ÞdX: t T H AH T H AH 1
ð19Þ
1
The exact stationary PDF of the total energy Hðt Þ is
Fig. 7. (a) The dependence of the MSOV E½V 2 on the damping coefficient c; (b) The dependence of the MOP E½P on the damping coefficient c.
Fig. 8. (a) The dependence of the MSOV E½V 2 on the system natural frequence x0 ; (b) The dependence of the MOP E½P on the system frequence x0 .
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pðHÞ ¼
C exp r2 ðHÞ
Z
H
0
2mðxÞ dx r2 ðxÞ
ð20Þ
here C is a constant determined by the condition of normalization. Applying Eq. (13), the joint PDF of displacement X and the velocity X_ is easily obtained as
pðHÞ p X; X_ ¼ j 1_2 : T H H¼2X þUðX Þ
ð21Þ
The correspondingly marginal PDFs of system displacement X and velocity X_ can be obtained respectively as follows
Z
þ1
pðX Þ ¼ p X_ ¼
p X; X_ dX_
1 þ1
Z
p X; X_ dX
ð22aÞ ð22bÞ
1
Applying the relationship between output voltage and state variables (dispalcement and velocity), which is given by Eq. (11), the MSOV can be calculated by
Fig. 9. (a) The dependence of the MSOV E½V 2 on the stiffness coefficient b; (b) The dependence of the MOP E½P on the stiffness coefficient b.
Fig. 10. (a) The dependence of the MSOV E½V 2 on the parameter k; (b) The dependence of the MOP E½P on the parameter k.
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Fig. 11. (a) The dependence of the MSOV E½V 2 on the parameter
h i Z E V2 ¼
þ1
Z
1
þ1
1
j; (b) The dependence of the MOP E½P on the parameter j.
x2 ðHÞ k ðX x ðHÞÞ þ 2 X_ 2 k þ x2 ð H Þ k þ x2 ð H Þ
!2
_ p X; X_ dXdX;
ð23Þ
and the MOP can be obtained through a linear relationship with the MSPV as
h i E½P ¼ jkE V 2 :
ð24Þ
5. Numerical results and stochastic analysis In this section, we show the results obtained by the improved quasi-conservative averaging procedure, which is depended on the energy-dependent frequency and the equilbrium points of the BVEH system, and at the same time use the direct MC numerical simulation method to verify the effectiveness of the proposed method. For simplicity, we choose the parameters of the BVEH system as c ¼ 0:02; x0 ¼ 2:0; b ¼ 3:0; k ¼ 0:05 and j ¼ 0:5, and the parameters of filtered Gaussian white noise as fg ¼ 0:64; xg ¼ 15:0 and S0 ¼ 0:003, unless otherwise mentioned. In addition, the solid lines represent the analytical results by applying our proposed method, while the red stars denote the direct numerical simulation through MC. The joint PDFs of the system displacement and velocity under filtered Gaussian white noise excitation are depicted in Fig. 4. Fig. 4 (a) describes an analytical bimodal joint PDF through our technique, with the comparison of results from the direct MC simu lation is given in Fig. 4 (b), which shows the availability of the presented technique. The marginal PDFs pðX Þ and p X_ of system displacement and velocity are shown in Figs. 4 (c) and (d), respectively. We find that the marginal PDF of system displacement is also bimodal and the probability is equal to zero at location X ¼ 0, which means that the chaotic motion cannot occur and jumps between two potential wells are not frequent, and thus the dominant motion of the system is located in one potential well. To better understand the effects of the filtered Gaussian white noise on MSOV E½V 2 and MOP E½P of the BVEH system, the changes of MSOV and MOP with excitation intensity S0 and the peak frequency of seismic motion xg of the stochastic excitation are shown in Figs. 5 and 6, respectively. It can be seen from Fig. 5 (a), the increase of the noise intensity will increase the MSOV and the corresponding MOP has the same trend. The changes of seismic motion of the stochastic excitation are shown in Fig. 6. We find an opposite trend relative to the effects of the noise intensity, which means that the seismic motion of the stochastic excitation will hinder the production of the MSOV and MOP under the same noise intensity. In Figs. 7–9, we studied the effects of damping coefficient c, the natural frequency x0 and the stiffness coefficient b on MSOV E½V 2 and MOP E½P. One can see from Fig. 7 that the MSOV and MOP will decrease quickly with the increase of damping coefficient, and the decline rate slow down gradually. However, the dependences of the MSOV and MOP show a linear down tend with the increase of x0 and b, which are shown in Figs. 8 and 9. Finally, the effects of the coupling parameters k and j on MSOV and MOP are shown in Figs. 10 and 11, respectively. The increase of the coupling parameter k and j will result in the decrease of the MSOV, but favor the MOP growth. This implies that the coupling parameters benifit the vibration energy scavenging and thus we may improve the energy harvesting capacity through optimizing the coupling.
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6. Conclusions In this manuscript, an improved coordinate transformation, based on the equilibrium points of the BVEH system, is proposed to construct the quasi-conservative stochastic averaging procedure, which is used to analyze the stochastic response of the nonlinear BVEH system driven by filtering Gaussian white noise. First, the nonlinear electromechanical coupling BVEH system can be approximated by an equivalent single degree of freedom system through the new transformation mentioned previously, which contains the energy-dependent frequency functions and the equilibrium points of the bistable system. Next, the quasi-conservative averaging method is used to derive the averaged drift and diffusion coefficients of the total energy process H, which is dependent on the statistical characteristics of filtered Gaussian white noise and the equilibrium points of the systems. Then the analytic expressions of PDFs of the mechanical system state, MSOV and the MOP are obtained. Finally, the effects of the excitation intensity S0 and the peak frequency of seismic motion xg of the stochastic excitation, the parameters of the vibration system including damping coefficient c, the natural frequency x0 and the stiffness coefficient b, and the electromechanical coupling coefficients including the parameters k and j on the MSOV and the MOP of the stochastic BVEH system are also analyzed in detail. The results in this paper have certain significance for improving the energy harvesting capacity by optimizing the stochastic bistable systems. Acknowledgements This work is supported by the National Nature Science Foundation of China (No. 11402139 and 11572247) and the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (No. 2016114). References [1] S. Priya, D.J. Inman, Energy Harvesting Technologies, Springer, US, 2009. [2] A. Erturk, D.J. Inman, Piezoelectric Energy Harvesteing, John Wiley and Sons, 2011. [3] L.M. Swallow, J. 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