Global Dynamic Analysis of a Typical Bistable Piezoelectric Cantilever Energy Harvesting System

Abstract

This paper focuses on global dynamic behaviors of a bistable piezoelectric cantilever energy harvester with a tip magnet and a single external permanent magnet at the near side. The initial distance between the magnetic tip mass and the external magnet is altered as a key parameter for the enhancement of the energy harvesting performance. To begin with, the dynamical model is established, and the equilibria as well as potential wells of its non-dimensional system are discussed. Three different values of the initial distance are selected to configure double potential wells. Next, the saddle-node bifurcation of periodic solutions in the neighborhood of the nontrivial equilibria is investigated via the method of multiple scales. To verify the validity of the prediction, coexisting attractors and their fractal basins of attraction are presented by employing the cell mapping approach. The best initial distance for vibration energy harvesting is determined. Then, the Melnikov method is utilized to discuss the threshold of the excitation amplitude for homoclinic bifurcation. And the triggered dynamic behaviors are depicted via numerical simulations. The results show that the increase of the excitation amplitude may lead to intra-well period-2 and period-3 attractors, inter-well periodic response, and chaos, which are advantageous for energy harvesting. This study possesses potential value in the optimization of the structural design of piezoelectric energy harvesters.

1. Introduction

With the development of low-power electronic devices, for instance, wireless sensors, MEMS, and implantable medical devices [1], it becomes difficult for traditional batteries with a short lifespan, high cost of maintenance, and the generation of hazardous waste to power these devices. To tackle this problem, researchers have attempted to design energy harvesters for capturing energy from sustainable environmental energy sources and converting the mechanical energy into electric energy. Typical energy harvesting mechanisms include piezoelectric, electromagnetic, and electrostatic ones among which piezoelectricity has the highest energy density, thus harvesting energy the most conveniently [2]. A piezoelectric structure will generate an electrical current if its vibration is excited from a common dynamical load, for instance, air turbulence, human motion, and ocean waves. The traditional structures applied in piezoelectric energy harvesting are the piezoelectric cantilever beams generally having a linear behavior [3]. In previous studies, linear piezoelectric energy harvesting models have been found to be effective in practical applications only if there is a good match between the frequency of the environmental excitation and the resonance frequencies of the energy harvesting devices [4]. Nevertheless, the ambient vibrations of the piezoelectric energy harvesters (PEHs) are often with broadband low frequencies beyond the narrow resonant frequency band, thus resulting in unsatisfactory responses of the energy harvester. This means that to design PEHs, the characteristics of ambient vibration frequencies must be taken into account.

In recent years, to address the issue of low-frequency bandwidth, various nonlinear PEHs have been proposed [5,6,7,8]. Among these, attaching permanent magnets to associated structures is one of the most popular methods to construct PEHs, as permanent magnets can reproduce the nonlinear restoring forces. It was generally accepted that compared with linear PEHs, magnet-based nonlinear ones have a wider frequency bandwidth, and thereby a higher efficiency of energy harvesting [9,10]. The use of permanent magnets in PEHs also helps to obtain multiple equilibrium positions and multiple potential wells of the energy harvesting systems [11]; thus, magnet-based PEHs can be useful in the generation of higher energy outputs over a wide range of frequencies.

A common structure of multi-stable PEHs consists of a cantilever beam with piezoelectric transducers attached to its flat surfaces, a magnetic tip mass attached to its free end, and one [12], two [13,14,15], or three [16] similar external permanent magnets nearby. Multiple potential wells and corresponding coexisting responses of PEHs can be supported by repulsive magnetic forces which can be changed by altering the initial distance between the magnets. Over the past decade, multi-stable configurations of magnets in PEHs have received great attention due to their capacity for achieving high output power under the circumstance of the jump from a stable state to a higher-amplitude one [17,18].

Considering a bistable piezoelectric cantilever PEH with a tip magnet and only one external magnet nearby, Sun and Cao [19] established its lumped parameter model and analyzed the condition of the initial distance between the two magnets for the occurrence of bistability. Numerical simulations of the dynamical system showed that the voltage response had a broadband. To describe nonlinear dynamical behaviors of the same harvester, Tang et al. [20] set up a distributed-parameter model and presented a harmonic analytical solution via applying the harmonic balance method. According to the comparison of optimum resistive loads under different vibration frequencies, it was found that the inter-well and intra-well oscillations bring forth the coexistence of high-energy and low-energy responses, respectively. Barbosa et al. [21] also studied the bistable piezo-magnetoelastic structure subjected to harmonic excitations and presented numerical results for the phenomenon of coexisting inter-well periodic response and chaos which may be beneficial for energy harvesting. For the same bistable piezoelectric energy harvester (BPEH), Shah et al. [22] experimentally verified the maximum initial distance between the tip magnet at the free end of the cantilever beam and the external magnet to maintain bistability and illustrated the function of jump phenomena in enhancing energy harvesting efficiency within the bistable regime.

For the design and performance optimization of PEHs with a magnetic proof mass and two or three external permanent magnets at the near end, the distance between external magnets and the inclination angle of each external magnet have been considered as important structure parameters. Litak et al. [23] undertook an in-depth exploration of the analysis and numerical examples concerning bifurcations of kinetic energy harvester oscillatory systems, and found that bifurcation phenomena could lead to notable alterations in vibration characteristics. Via presenting multiple attractors in the vicinity of tristable wells and their basins of attraction (BAs), Zhu and Shang [24] discussed the mechanism of multi-stability and chaos in order to provide insight into improving the performance of BPEHs. Jung et al. [25] discussed the nonlinear dynamic and energetic characteristics of a BPEH vibrating system with a pair of externally rotatable magnets via numerical simulations and experimental results, and found a spectrum of intricate dynamic behaviors including periodic oscillations, chaos, and jump phenomena. Cao et al. [26] considered the effect of the fractional-order damping in inducing complicated dynamical behaviors of a similar PEH, and numerically depicted that the variation of fractional order of the damping in a certain range can lead to the periodic doubling route from the inter-well periodic motion to chaos and the inverse route, hence contributing to broadband energy harvesting. For a multi-stable PEH configuration consisting of a piezoelectric cantilever beam, an attached tip magnet, and three external permanent magnets located in a line, Ju et al. [27] investigated the transition mechanism from mono-stability to multi-stability by means of static bifurcation analysis, potential well diagrams, and local bifurcations of periodic solutions, which was verified by the experimental results where inter-well motions and the jump phenomenon between intra-well and inter-well motions could be generated. Admittedly, multi-stability is a favorable characteristic for piezoelectric energy harvesting. Nevertheless, without taking into account the disturbance of initial conditions of PEH oscillatory systems, it will be difficult for us to develop a confident estimate of the final response and the output of energy harvesting under realistic conditions. Until now, although sufficient works have discussed the coexistence of multiple responses in multi-stable PEHs in detail, there are only a few reports considering the effect of initial conditions on energy harvesting.

To this end, in this study, we consider a typical bistable piezoelectric energy harvester (BPEH) and discuss the role of the distance between the two magnets in triggering complex dynamics and improving the BPEH performance upon the concept of basins of attraction (BAs). The remaining contents are organized as follows. The governing electromechanical system of the BPEH is constructed, and its equilibria and potential wells are discussed in Section 2. The local bifurcation of the intra-well periodic solution and BAs of periodic attractors in the vicinity of the non-trivial equilibria are presented, and the comparison of dynamics under different values of the distance between the tip magnet and the external one is carried out in Section 3. Section 4 discusses the global bifurcation and the relevant inter-well responses. Finally, the outcomes are summarized in Section 5.

2. Dynamical Model and Unperturbed Dynamics

A bistable piezoelectric cantilever energy harvester is considered whose configuration is displayed in Figure 1 [17,19,20]. Here, the magnetic proof mass A is attached to the tip end of the horizontal beam, and the permanent magnet B with the same size is fixed on the base. The initial horizontal gap between the two magnets, namely, the initial distance, is denoted by d. The cantilever beam consists of a conductive substrate and two piezoelectric layers which are covered by thin electrode layers connecting to a resistive electric load. $X$ is the vertical displacement of magnet A at the moment $t$. As visible in Figure 1, when the fixed end of the cantilever beam is excited in the vertical direction, the beam vibrates, and the subsequent strain energy can be transformed into electrical energy and then stored by the energy harvesting circuits. The magnetic restoring force plays a crucial role in determining the number of potential wells of the system. In a certain range of d, the configuration of the system by the interacting permanent magnets can support two symmetric stable equilibria, $x_{c^{+}}$ and $x_{c^{-}}$ [28].

According to Newton’s Second Law and Kirchhoff’s law, the equation governing the piezo-magnetoelastic structure subjected to the external excitation can be given by [17]

$$m\ddot{X}+c\dot{X}+kX-F_m(X)-{\kappa }_{v}V=-mA\cos \left(\Omega T\right),C_p\dot{V}+\frac{V}{R_L}+{\kappa }_c\dot{X}=0.$$

(1)

where the restoring magnetic force is

$$F_m(X)=\frac{3{\mu }_0{M}_A{V}_A{M}_B{V}_B}{2\pi d^{4}L}\left(X-\left(\frac{1}{L^2}+\frac{5}{d^2}\right)X^3+O(X^5)\right).$$

(2)

In Equations (1) and (2), the variable V represents voltage across the load resistor, the parameter A the amplitude of the external load, and $\Omega$ the frequency of the external excitation. The physical interpretation and values of other parameters are presented on the basis of the same physical properties of BPEH in Ref. [17], as shown in

Table 1. Parameters of system (1).

Parameter	Symbol	Values
Equivalent mass of beam and magnet (g)	$m$	25.9
Equivalent damping of the cantilever beam (N·s/m)	$c$	0.01
Elastic constant of the cantilever beam (N/m)	$k$	774.664
Vacuum permeability (H/m)	${\mu }_0$	$4\pi \times {10}^{-7}$
The magnetization of magnet A (A/m)	$M_A$	$1.1\times {10}^6$
The volume of magnet A (${\mathrm{cm}}^3$)	$V_A$	$2.2$
The magnetization of magnet B (A/m)	$M_B$	$1.1\times {10}^6$
The volume of magnet B (${\mathrm{cm}}^3$)	$V_B$	$2.2$
The horizontal distance between magnet A and the base of the beam (cm)	$L$	7.4
Electromechanical coupling coefficient in relation to voltage (N/V)	${\kappa }_v$	$0.9901$
Equivalent capacitance of piezoelectric ceramics (F)	$C_p$	$2.8\times {10}^{-9}$
Resistive electrical load ($\mathrm{k}\mathsf{\Omega }$)	$R_L$	1000
Electromechanical coupling coefficient in relation to current (A·s/m)	${\kappa }_c$	$0.9901$

.

By introducing the following nondimensional variables

$$x=\frac{X}{L},\hspace{1em}v=\frac{V}{e},\hspace{1em}T={\omega }_{0}t,\hspace{1em}{\omega }_0=\sqrt{\frac{k}{m}},\hspace{1em}e=\frac{mg}{{\kappa }_v},\hspace{1em}\dot{x}=\frac{dx}{dT},\hspace{1em}\dot{v}=\frac{dv}{dT},$$

(3)

into Equation (1), we obtain its nondimensional system

$$\ddot{x}+\xi \dot{x}-px+b{x}^3-\gamma v=-f\cos \left(\omega T\right), \dot{v}+\mu v+\vartheta \dot{x}=0.$$

(4)

where

$$p=\frac{3{\mu }_0{M}_A{V}_A{M}_B{V}_B}{2\pi k{d}^{4}L}-1, b=\frac{3{\mu }_0{M}_A{V}_A{M}_B{V}_{B}L}{2\pi k{d}^4}\left(\frac{1}{L^2}+\frac{5}{d^2}\right), \xi =\frac{c}{2m{\omega }_0}, \gamma =\frac{mg}{kL},f=\frac{mA}{kL}, \mu =\frac{1}{R_L{C}_p{\omega }_0}, \vartheta =\frac{{\kappa }_{c}L}{C_{p}e}.$$

(5)

Note that in Equation (4), the non-dimensional parameters $\xi ,\hspace{0.17em}b,\hspace{0.17em}\gamma ,\hspace{0.17em}f,\hspace{0.17em}\mu$, and $\vartheta$ are positive. Its unperturbed system is a Hamilton system, expressed as

$$\begin{array}{l}\dot{x}=y,\hspace{0.17em}\\ \dot{y}=px-b{x}^3+\gamma v,\hspace{0.17em}\\ \dot{v}=-\mu v-\vartheta y.\end{array}$$

(6)

The equilibrium positions of the nondimensional system (4) can be determined by setting both sides of the above equation to zero. It is apparent that the values of v and y are both zero for the equilibria. The trivial (x, y, v) = (0, 0, 0) is definitely one equilibrium. Whether the system has nontrivial equilibrium depends on the value of the parameter a. For a positive a, there are two nontrivial solutions of the equation $px-b{x}^3=0$, denoted by $\pm x_c$; thus, the unperturbed system has a pair of nontrivial equilibria $P_{\mathrm{1,2}}$($\pm x_c,0,0$). Since the values of parameters ${\mu }_0,\hspace{0.17em}{M}_A,\hspace{0.17em}{V}_A,\hspace{0.17em}{M}_B,\hspace{0.17em}{V}_B,\hspace{0.17em}k$, and L are given, whether p is positive depends on the value of the initial distance d. To get a positive p, 14 mm, 15 mm, and 16 mm are selected for d. On this basis, double potential wells can be configured, and in the Hamilton Equation (6), the Hamiltonian and the function of potential energy can be given by

$$H\left(x,y\right)=\frac{1}{2}{y}^2-\frac{1}{2}p{x}^2+\frac{1}{4}b{x}^4$$

(7)

and

$$V\left(x\right)=-\frac{1}{2}p{x}^2+\frac{1}{4}b{x}^4,$$

(8)

respectively. Altering the values of d, the potential energy is demonstrated in Figure 2. Distinctly, for d = 14 mm, d = 15 mm, or d = 16 mm, the system exhibits three equilibrium points: two centers and one saddle point. The double potential wells are the deepest at d = 14 mm and the shallowest at d = 16 mm. Under the three different values of d, unperturbed orbits are depicted in Figure 3. The nontrivial equilibria are the centers of potential wells, thus being surrounded by homoclinic orbits. In contrast, the trivial equilibrium is a saddle point.

3. Multiple Intra-Well Responses

3.1. Dynamic Responses around the Potential Well Centers

To investigate the jump among coexisted responses and its effect on energy harvesting, periodic solutions and their local bifurcation of the nondimensional system (4) should be discussed first. The method of multiple scales (MMS) is utilized to obtain the approximate-analytic periodic solutions around the potential well centers ($\pm x_c$, 0). Based on the values of system parameters provided in Table 1, it is evident that the values of the nondimensional coefficients $\xi$, $\gamma$, and $\mu$ are small. Additionally, we consider the nondimensional excitation amplitude $f$ to be small. By introducing a small bookkeeping parameter $\epsilon$, we rewrite the parameters and the variable as

$$\xi ={\epsilon }^2\tilde{\xi }, \gamma ={\epsilon }^2\tilde{\gamma }, f={\epsilon }^2\tilde{f}, \mu =\epsilon \tilde{\mu }, x=\pm x_c+\tilde{x}.$$

(9)

Then, Equation (4) can be rescaled as

$$\ddot{\tilde{x}}+{\tilde{\omega }}^2\tilde{x}=3b{x}_c{\tilde{x}}^2-b{\tilde{x}}^3+{\epsilon }^2\tilde{\gamma }v-{\epsilon }^2\tilde{\xi }\dot{\tilde{x}}-{\epsilon }^2\tilde{f}\cos \left(\omega T\right), \dot{v}+\epsilon \tilde{\mu }v+\vartheta \dot{\tilde{x}}=0,$$

(10)

where

$${\tilde{\omega }}^2=-p+3b{x}_c{}^2.$$

(11)

Corresponding to the case of the primary resonance, we can set

$$\omega =\epsilon \sigma +\tilde{\omega }.$$

(12)

Here, $\sigma$ is the so-called detuning parameter. Since the nondimensional displacement x and voltage v in the potential well are both less than 1, we can rewrite x, v, time scales, and the time derivatives of the system (10) as below:

$$\begin{array}{l}\tilde{x}=\epsilon {\tilde{x}}_1+{\epsilon }^2{\tilde{x}}_2+{\epsilon }^3{\tilde{x}}_3, \tilde{v}=\epsilon {\tilde{v}}_1+{\epsilon }^2{\tilde{v}}_2+{\epsilon }^3{\tilde{v}}_3,\\ T_i={\epsilon }^{i}T, D_i=\frac{\partial }{\partial T_i}, \frac{d}{dT}={\displaystyle \sum _{i=0}^n{\epsilon }^i{D}_i\left(i=0,1,2,\cdot \cdot \cdot \right).}\end{array}$$

(13)

By separating the coefficients of ${\epsilon }^1$, ${\epsilon }^2$, and ${\epsilon }^3$ in the system (10), respectively, we have

$${\epsilon }^1:D_0{}^2{\tilde{x}}_1+{\omega }^2{\tilde{x}}_1=0, D_0{v}_1=-\vartheta D_0{\tilde{x}}_1,\mathrm{s}$$

(14)

$${\epsilon }^2:D_0{}^2{\tilde{x}}_2+{\omega }^2{\tilde{x}}_2=-2D_0{D}_1{\tilde{x}}_1+3b{x}_c{\tilde{x}}_1{}^2+2\sigma \omega {\tilde{x}}_1-\tilde{f}\cos \left(\omega T\right), D_0{\tilde{v}}_2=-D_1{\tilde{v}}_1-\vartheta (D_1{\tilde{x}}_1+D_0{\tilde{x}}_2)-\tilde{\mu }{\tilde{v}}_1,$$

(15)

and

$$\begin{array}{l}{\epsilon }^3:D_0{}^2{\tilde{x}}_3+{\omega }^2{\tilde{x}}_3=-(D_1{}^2+2D_0{D}_2){\tilde{x}}_1-b{\tilde{x}}_1{}^3-2D_0{D}_1{\tilde{x}}_2+6b{x}_c{\tilde{x}}_1{\tilde{x}}_2+\tilde{\gamma }{\tilde{v}}_1-\tilde{\xi }{D}_0{\tilde{x}}_1-{\sigma }^2{\tilde{x}}_1+2\sigma \omega {\tilde{x}}_2,\\ \hspace{1em}\hspace{0.17em}{D}_0{\tilde{v}}_3=-D_2{\tilde{v}}_1-D_1{\tilde{v}}_2-\vartheta (D_2{\tilde{x}}_1+D_1{\tilde{x}}_2+D_0{\tilde{x}}_3)-\tilde{\mu }{\tilde{v}}_2.\end{array}$$

(16)

The solution of Equation (14) can be assumed as

$${\tilde{x}}_1=A(T_1,T_2)e^{i\omega T_0}+cc, {\tilde{v}}_1=-\vartheta A(T_1,T_2)e^{i\omega T_0}+cc.$$

(17)

where

$$A=\frac{a(T_1,T_2)}{2}{e}^{i\theta (T_1,T_2)}.$$

(18)

Inserting the solution of Equation (14) (see Equation (15)) in Equation (15), and separating the secular terms there yields

$$D_{1}A=\frac{i\tilde{f}-4i\sigma \omega A}{4\omega }$$

(19)

and the following solution of Equation (15):

$${\tilde{x}}_2=\frac{6bx{}_{c}A\overline{A}}{{\omega }^2}-\frac{bx{}_{c}A^2}{{\omega }^2}{e}^{2i\omega T_0}+cc, {\tilde{v}}_2=\frac{2\vartheta bx{}_{c}A^2}{{\omega }^2}{e}^{2i\omega T_0}-\frac{\tilde{\mu }\vartheta A}{\omega }{e}^{i\omega T_0}+cc.$$

(20)

Similarly, by inserting the solutions of Equations (17) and (20) as well as Equation (19) in Equation (16) and separating the secular terms, we have

$$D_{2}A=-\frac{\tilde{\xi }A}{2}-\frac{15i{b}^2{x}_c^2{A}^2\tilde{A}}{{\omega }^3}+\frac{i\sigma \tilde{f}}{8{\omega }^2}+\frac{3ib{A}^2\tilde{A}}{2\omega }+\frac{i\vartheta \tilde{\gamma }A}{2\omega }.$$

(21)

Let

$$\widehat{a}=\epsilon a.$$

(22)

According to Equations (13), (17) and (20), the approximate-analytic form of the periodic solutions of the nondimensional system (4) surrounding the well centers ($\pm x_c$, 0) is given by

$$x=\pm x_c\pm \frac{3b{x}_c{\widehat{a}}^2}{2{\omega }^2}+\widehat{a}\cos (\omega T+\theta ), v=\frac{\vartheta \widehat{a}(\omega +\tilde{\mu })}{\omega }\cos (\omega T+\theta +\pi ).$$

(23)

It follows from the above equation that the amplitude of the voltage v is $\frac{\vartheta \widehat{a}(\omega +\tilde{\mu })}{\omega }$, proportional to $\widehat{a}$, i.e., the magnitude of the nondimensional displacement x. Since the energy harvesting performance totally depends on the voltage amplitude v, we can concentrate on the variation of $\widehat{a}$ with the system parameters d, f, and ω for discussing the energy harvesting efficiency of the BPEH.

By substituting Equations (18), (19), (21) and (22) into

$$\dot{A}\approx D_{0}A+\epsilon D_{1}A+{\epsilon }^2{D}_{2}A,$$

(24)

and returning the parameters into the original nondimensional ones, we will have

$$\begin{array}{l}\dot{\widehat{a}}=-\frac{\xi }{2}\widehat{a}+\frac{(3\omega -\widehat{\omega })f\sin \theta }{4{\omega }^2},\\ \widehat{a}\dot{\theta }=\frac{\vartheta \gamma \widehat{a}}{2\omega }-(\omega -\widehat{\omega })\widehat{a}+\frac{3b{\widehat{a}}^3({\omega }^2-10b{x}_c^2)}{8{\omega }^3}+\frac{(3\omega -\widehat{\omega })f\cos \theta }{4{\omega }^2}.\end{array}$$

(25)

To obtain the steady-state form of the solution of Equation (4), we can equate $\dot{\widehat{a}}=\dot{\theta }=0$ in the above equation, resulting in

$$\begin{array}{l}2{\omega }^2\xi \widehat{a}=(3\omega -\widehat{\omega })f\sin \theta ,\\ 2\omega \widehat{a}(2{\omega }^2-2\omega \widehat{\omega }-\vartheta \gamma )+\frac{3b{\widehat{a}}^3(10b{x}_c^2-{\omega }^2)}{2\omega }=(3\omega -\widehat{\omega })f\cos \theta .\end{array}$$

(26)

The elimination of the triangulation functions in the above equation yields the following frequency response equation:

$$16{\omega }^6{\xi }^2{\widehat{a}}^2+{(4{\omega }^2\widehat{a}(2{\omega }^2-2\omega \widehat{\omega }-\vartheta \gamma )+3b{\widehat{a}}^3(10b{x}_c^2-{\omega }^2))}^2=4f^2{\omega }^2{(3\omega -\widehat{\omega })}^2.$$

(27)

Its characteristic equation of Equation (25) is

$${(\lambda +\frac{\xi }{2})}^2+\frac{[4{\omega }^2(2{\omega }^2-2\omega \widehat{\omega }-\vartheta \gamma )+3b{\widehat{a}}^2(10b{x}_c^2-{\omega }^2)][4{\omega }^2(2{\omega }^2-2\omega \widehat{\omega }-\vartheta \gamma )+9b{\widehat{a}}^2(10b{x}_c^2-{\omega }^2)]}{64{\omega }^6}=0.$$

(28)

If the equation above has a solution with a positive real part, then the corresponding periodic solution will lose its stability. Therefore, the local bifurcation of the periodic solution will occur once there is a pair of purely imaginary solutions or a trivial one for Equation (28). Owing to no purely imaginary solution in the equation above, the stability of the primary resonance solution within each potential well will be changed only if $\lambda$ = 0, namely,

$$16{\omega }^6{\xi }^2+[4{\omega }^2(2{\omega }^2-2\omega \widehat{\omega }-\vartheta \gamma )+3b{\widehat{a}}^2(10b{x}_c^2-{\omega }^2)][4{\omega }^2(2{\omega }^2-2\omega \widehat{\omega }-\vartheta \gamma )+9b{\widehat{a}}^2(10b{x}_c^2-{\omega }^2)]=0,$$

(29)

which implies the arising of saddle-node (SN) bifurcation of the intra-well periodic solutions.

Based on the analysis above, we can illustrate the variations of the primary resonance responses with the local environmental excitation for three different values of the initial distance d. To begin with, we ought to compare the natural frequencies of the intra-well vibrations under different values of d, denoted by $\tilde{\omega }$. This is because the energy harvesting efficiency will be undesirable if the natural frequency of the BPEH is much higher than the ambient frequencies which are usually low. The variation of the natural frequency with the increase of the distance d is delineated in Figure 4. It shows that $\tilde{\omega }$ monotonically decreases with d. In the comparison of $\tilde{\omega }$ under distances of 14 mm, 15 mm, and 16 mm (see the red star in Figure 4), it seems that the nondimensional natural frequency at d = 14 mm is much higher than 1, i.e. above the blue dash line in Figure 4, thus unsuitable for energy harvesting. Therefore, in the rest of this subsection, we just need to make a comparison for d = 15 mm and d = 16 mm.

Noting the symmetry of intra-well responses surrounding the symmetric well centers, the amplitude variations of the intra-well responses with the nondimensional excitation coefficients at d = 15 mm and d = 16 mm are illustrated in Figure 5, where solid curves and dashed ones denote stable response branches and unstable ones, respectively. It follows from Equation (29) that SN bifurcation points to separate the stable and unstable branches are denoted by small red circles in Figure 5. Numerical results for stable periodic responses of the ordinary differential Equation (4) are simulated via the 4th-order Runge–Kutta method, namely, the software package ODE 45 in MATLAB, represented by the stars in Figure 5. As is evident from Figure 5, the analytical results agree well with the numerical ones. It should be noted that there is some decay between the analytical prediction and numerical examples under the circumstance of a much lower excitation frequency or a higher excitation amplitude ω, which can be ascribed to the limitation of MMS; in other words, the analytic results could be quantitatively precise only under the circumstance of weak nonlinearities, low excitation amplitudes, as well as the excitation frequency close to the resonant frequency. Actually, in the nondimensional BPEH oscillatory system (4), the nonlinear coefficient b for d = 15 mm and d = 16 mm is 179.1 and 121.7, implying a strong nonlinearity. Yet MMS is available, taking into consideration its valid qualitative prediction and the convenience it brings for achieving the approximate-analytical form of the primary resonant solutions.

As illustrated in Figure 5a, the frequency response curves for d = 15 mm and d = 16 mm are totally different. For d = 16 mm, the amplitude of the stable response varies continuously with ω, and its peak is very low. Comparatively, for d = 15 mm, as ω is increased, the periodic solution branches bend to the left, yielding bistable intra-well periodic responses within each potential well. Obviously, for d = 15 mm, varying the frequency ω in the range of [0.7, 0.8], a SN bifurcation point can be observed. On its left side, the response may jump from a lower-amplitude one up to the other higher-amplitude one, which is beneficial for energy harvesting.

Similarly, as is demonstrated in Figure 5b where the nondimensional excitation frequency is 0.8, there are two SN bifurcation points on the branch of d = 15 mm, implying that a jump between bistable periodic branches around each nontrivial equilibrium can result from the rise in f. For a small f lower than its first critical value for the SN bifurcation, we can find a monostable branch with a very low amplitude, implying a single attractor within each potential well. As f becomes higher, a higher-amplitude stable branch will occur, coexisting with the former one. When f exceeds the second critical value for the SN bifurcation, the response will snap through to the higher-amplitude branch owing to the saddle-node bifurcation. To move away from the bistability in the neighborhood of each nontrivial equilibrium, there is a monostable attractor around each potential well center with a much higher amplitude.

By adjusting the initial distance d, we can achieve different stable states in the energy harvesting system, thereby influencing the bandwidth for energy harvesting. From Figure 5, we observe that at d = 15 mm, the system exhibits bistable characteristics. In this configuration, the system can display a high-amplitude response even under a low frequency of the excitation, resulting in a broad bandwidth, thus enabling a better performance of energy collection from ambient vibrations. Conversely, when d increases to 16 mm, the system transitions to a monostable state, leading to an obvious reduction in the bandwidth. This suggests that in the case of d = 15 mm, as a result of SN bifurcation, the jump between two intra-well attractors in the vicinity of each potential well center may be caused by varying the excitation parameters f and ω. Therefore, among the three different values of the initial distance d, 15 mm is the best one for energy harvesting. In the remaining part of this study, we will focus on the global dynamic behaviors of the system (4) at d = 15 mm.

3.2. Coexistence of the Attractors

In the case of multi-stability coexistence, apart from the effect of the excitation on inducing jumps among multiple intra-well vibrations, the disturbance of initial conditions can also have a significant effect on deciding the final dynamic behavior, thus determining the efficiency of energy harvesting. This provides the indispensability of the classification of the attractors’ basins of attraction (BAs). Here, the basin of attraction (BA) means the union of initial conditions leading to the same vibration [24]. Setting $v(0)=0$, the cell mapping method is utilized to depict BAs of the nondimensional dynamical system (4) in the initial-condition plane $-0.15\le x(0)\le 0.15$, $-0.15\le y(0)\le 0.15$ consisting of a $301\times 151$ array of points corresponding to initial conditions. For an attractor marked in one color in the phase plane x-O-y, its BA is denoted by the same hue in the initial-condition plane. The evolution of the coexisting intra-well responses and their BAs with the increase of nondimensional excitation coefficients is detailed in Figure 6 and Figure 7 where the left column depicts the phase maps for the coexisting attractors, and the right one plots their BAs.

For a fixed $f=0.002$, the sequence of the intra-well responses and their BAs with the growth of ω is shown in Figure 6 where $\omega$ ranges from 0.66 to 0.83, lower than 1. For $\omega =0.66$, two intra-well attractors coexist in the vicinity of two different nontrivial equilibria $P_{1,2}(\pm x_c,0)$, denoted by red and black colors, respectively (see Figure 6(a-1)). According to their BAs presented in Figure 6(a-2), the vicinity of each potential well center is single-colored, illustrating that both intra-well attractors have the local stability. In contrast, in the vicinity of the trivial $(0,0)$, their BAs are intermingled, showing that a disturbance of initial conditions can trigger a jump between the two responses. Nevertheless, the two intra-well attractors are symmetric with the same amplitude. According to the analysis in the last section, the voltage amplitude in the BPEH is a proportional function of the amplitude of the final response x. Even though a jump between the two responses can stem from the disturbance of initial states habitually, the energy output will still be stable.

For $\omega$ rising to 0.68, the bistable attractors still coexist, as indicated in Figure 6(b-1). Meanwhile, a pair of higher-amplitude intra-well attractors occur, arising from the SN bifurcation. As evident from Figure 6(b-2), the lower-amplitude attractors are stable in the neighborhood of the potential well centers despite their seriously fractal BA boundaries. Comparatively, the BAs of the new attractors (see the yellow and blue regions, in Figure 6(b-2)) are fractal, rare, and negligible, illustrating a pretty low probability of their occurrence. Since their BAs are out of the vicinity of $(\pm x_c,0)$, they are so-called hidden attractors [29]. It follows from the BAs in Figure 6(b-2) that the energy output is most likely to be low and stable, the same as for $\omega =0.66$.

When $\omega$ is further increased from 0.73 to 0.82 (see Figure 6(c-1–e-2)), the four intra-well attractors still coexist, and the amplitude of the lower-amplitude ones evidently grows. Meanwhile, their BAs are simultaneously eroded by the BAs of the higher-amplitude ones (see Figure 6(c-2,d-2,e-2)), meaning that the occurrence probability of higher-amplitude responses is significantly increased. It also implies that jumps from the lower-amplitude response to a higher-amplitude one can be triggered easily by a tiny change in initial conditions, which definitely helps to obtain a higher energy output.

As the value of $\omega$ grows a bit to 0.83 (see Figure 6(f-1,f-2)), two lower-amplitude attractors disappear; on the other hand, the higher-amplitude ones become locally stable around the potential well centers, suggesting that with the rise in the nondimensional frequency $\omega$, the lower-amplitude intra-well attractors will eventually be replaced by the higher-amplitudes ones, resulting in a better performance of energy output as well as confirming the previous predictions.

Variations of multiple intra-well attractors and their BAs with the nondimensional excitation amplitude f can be seen from Figure 7 where ω = 0.80. To begin with, the inspection of Figure 7(a-1) indicates that for $f=0.001$, two small intra-well periodic attractors are displayed around the potential well centers $(\pm x_c,0)$. The boundary separating their BAs is smooth, as evident from Figure 7(a-2). An arbitrary initial-state point is selected in the neighborhood of each center of the potential well, which will inevitably result in the relevant intra-well attractor. On the contrary, a tiny disturbance of the initial state in the neighborhood of O(0,0) may lead to the jump between the two intra-well attractors. However, it will not influence the energy output as the two attractors are symmetric with the same amplitude.

For $f$ slightly increased to 0.0011, the amplitude of the former attractors grows, and a pair of higher-amplitude attractors occur, as can be seen from the yellow closed loop and the blue one in Figure 7(b-1). According to their discrete and fractal BAs in Figure 7(b-2), they are rare and hidden attractors which can be hard to achieve. As f further increases from 0.0015 to 0.0026 (see Figure 7(c-1–f-2), the amplitude of the two lower-amplitude attractors will grow, and their BAs will be eroded by the BAs of the two higher-amplitude ones. Specifically, for f = 0.0026, the lower-amplitude attractors become rare and hidden (see Figure 7(f-2)), showing that the energy output is most probably to be increased. When f continues to grow (see Figure 7(g-1–i-2), the two lower-amplitude attractors disappear, and the BAs of the remaining two attractors are intermingled and segmented into scatters, revealing that a jump from one response to the other is very easy to achieve. Still, considering the inherent symmetry of the two higher-amplitude responses, the corresponding energy output is insensitive to initial conditions, thereby securing a steady and desirable output in energy harvesting performance.

4. Global Bifurcation and Inter-Well Responses

The main purpose of this section is to discuss global bifurcations of the system (4) and the induced inter-well attractors, as inter-well vibrations can improve the performance of energy harvesting by providing much higher amplitudes and voltages in BPEH.

To obtain the necessary condition for global bifurcations, firstly we need to rewrite the unperturbed orbits in the form of explicit functions of the nondimensional time variable [30]. Based on the unperturbed system and its Hamiltonian, we can express the homoclinic orbits as

$$x_{\mathrm{homo}}^{\pm }=\pm \sqrt{\frac{2p}{b}}\mathrm{sech}(\sqrt{p}T),\hspace{1em}{y}_{\mathrm{homo}}^{\pm }=\mp p\sqrt{\frac{2}{b}}\mathrm{sech}(\sqrt{p}T)\tanh (\sqrt{p}T).$$

(30)

Then, the homologous orbital voltage $v_{\mathrm{homo}}^{\pm }$ can be solved from Equation (6) as

$$v_{\mathrm{homo}}^{\pm }=\pm \sqrt{\frac{2}{b}}\vartheta pq(T)$$

(31)

where

$$q(T)=e^{-\mu T}{\displaystyle \int (e^{\mu T}\mathrm{sech}(\sqrt{p}T)\tanh (\sqrt{p}T))}dT.$$

(32)

Based on the unperturbed system (6), the nondimensional Equation (4) can be rewritten as

$$\begin{array}{l}\dot{x}=y,\\ \dot{y}=px-b{x}^3+(-\xi y+\gamma v-f\cos (\omega T)),\\ \dot{v}=-\mu v-\vartheta y.\end{array}$$

(33)

Inserting homologous orbital expressions (30)–(32) in the Melnikov function of the above system gives rise to

$$\begin{array}{l}M\left(T_0\right)={\displaystyle {\int }_{-\infty }^{+\infty }{y}_{\mathrm{homo}}^{\pm }\left(-\xi y_{\mathrm{homo}}^{\pm }+\gamma v_{\mathrm{homo}}^{\pm }\right)-f\cos \left(\omega (T+T_0)\right)dT}\\ \hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=-\frac{4\xi p\sqrt{p}}{3b}-\frac{2p^2}{b}\gamma \vartheta I_2\mp \pi f\omega \sqrt{\frac{2}{b}}\mathrm{sech}(\frac{\omega \pi }{2\sqrt{p}})\sin (\omega T_0),\end{array}$$

(34)

where

$$I_2={\displaystyle {\int }_{-\infty }^{+\infty }(q(T)\mathrm{sech}(\sqrt{p}T)}\tanh (\sqrt{p}T))dT.$$

(35)

It should be noted that the integral $I_2$ is positive and can be evaluated numerically under given values of the system parameters [30].

The critical condition for homoclinic bifurcation of the system (4) is that there should be a simple root of $T_0$ for the equation $M\left(T_0\right)=0$ satisfying $M^{\prime }\left(T_0\right)\ne 0$. To this end, there should be

$$\frac{4\xi p\sqrt{p}}{3b}+\frac{2p^2}{b}\gamma \vartheta I_2<\pi f\omega \sqrt{\frac{2}{b}}\mathrm{sech}(\frac{\omega \pi }{2\sqrt{p}}).$$

(36)

Otherwise, there would be no real root in the equation $M\left(T_0\right)=0$. Accordingly, only if

$$f>f_0=\frac{\sqrt{2p}p(2\xi +3\sqrt{p}\gamma \vartheta I_2)}{3\pi \omega \sqrt{b}\mathrm{sech}(\frac{\omega \pi }{2\sqrt{p}})},$$

(37)

the nondimensional BPEH oscillatory system (4) will undergo a homoclinic bifurcation. Based on Equations (32), (35) and (37), one can find that the threshold $f_0$ for homoclinic bifurcation is a function of the nondimensional frequency of the external excitation $\omega$. Since the values for the other coefficients in Equation (37) are given, we can obtain the variation of the threshold f₀ for homoclinic bifurcation corresponding to different values of ω (see Figure 8). Theoretical prediction and numerical examples both imply that the excitation-amplitude threshold of homoclinic bifurcation monotonically increases with the nondimensional excitation frequency.

As is well known, homoclinic bifurcation of nonlinear dynamical systems usually leads to complex inter-well dynamical behaviors such as chaos. To find out the dynamic behaviors stemming from the global bifurcation, we now investigate the evolution of global dynamical behaviors and their BAs in the case that $\omega =0.80$ and the excitation amplitude f continues to increase (see Figure 9). For $f=0.0039$, similar as in Figure 7(i-2), there are bistable intra-well attractors whose BAs are both fractal and intermingled with each other. However, the two attractors are period-2 attractors whose phase loops are expanded and nearly touch the homoclinic orbits of the system (4), as shown in Figure 9(a-1).

At $f=0.0040$, besides the two symmetric period-2 responses whose BAs are severely fractal, an inter-well periodic attractor suddenly occurs. Evidence for this is related to the large phase loop and its BA in Figure 9(b-1,b-2), respectively. Owing to its negligible BA at the far side of O(0,0), it is a hidden attractor and a rare attractor. It should be noted that its BA has a smooth boundary, which is totally different from the BAs of the other two attractors. It demonstrates that the ultimate dynamics of the nondimensional BPEH oscillatory system are highly initial-condition sensitive. But considering the symmetry of period-2 attractors and the dominance of their BAs in the initial-state plane, the output voltage of the energy harvesting system most probably relies on the period-2 vibration.

For $f_0$ = 0.0041 (see Figure 9(c-1)), these period-2 responses are extinct; in the meantime, two intra-well period-3 responses occur. Their BAs in Figure 9(c-2) are also intermingled with each other. And the BA of the inter-well attractor expands with a smooth boundary, demonstrating an increasing probability of its occurrence as well as a better energy harvesting performance.

Finally, when $f$ is increased to 0.0042, namely, slightly more than the critical value of f for homoclinic bifurcation, the dynamical behaviors become more complex. The intra-well dynamical behaviors disappear; meanwhile, an inter-well chaos suddenly appears coexisting with the large inter-well periodic attractor, as evident from phase and Poincaré maps a well as spectrum analysis in Figure 9(d-1–d-3), respectively. Noticeably, the BA boundaries of the two inter-well attractors are smooth, as can be identified from Figure 9(d-4). This indicates the initial-state insensitivity of the final dynamical behavior, unlike chaos itself. It follows from Figure 9 that due to the homoclinic bifurcation of the system (4) induced by the increase of f, the system (4) will feature the coexistence of two inter-well responses, which contributes to the increase of energy harvesting output.

5. Discussion

In this work, a typical bistable piezoelectric energy harvester (BPEH) comprised of a piezoelectric cantilever beam and an attached tip magnet under a harmonic tip load as well as a magnetic force from a fixed external magnet is considered. Taking into account the effects of the disturbance of initial conditions on the final dynamical behavior and the energy output of the BPEH, we investigate the bifurcations and global dynamical behaviors of the electromechanical system of the BPEH in detail. Fixing the other physical parameters of the structure, we regard the initial distance between the magnetic proof mass and the fixed permanent magnet as a critical parameter for improving the energy harvesting performance of the BPEH.

To begin with, the nondimensional dynamical system governing the electromechanical system of the BPEH is derived. It follows from the unperturbed system, the Hamiltonian, and the potential-energy function that the construction of the potential wells depends on the value of the initial distance d. To configure symmetric double potential wells, 14 mm, 15 mm, and 16 mm are selected for d. Hence, the system exhibits three equilibrium points where two nontrivial equilibria are potential well centers, and the trivial equilibrium is a saddle point. And the double potential wells are the deepest at d = 14 mm and the shallowest at d = 16 mm.

Next, the method of multiple scales (MMS) is adopted to achieve the approximate-analytical intra-well solutions. The nondimensional dynamical system of the BPEH is found to exhibit a local bifurcation of the periodic solution, namely, a saddle-node (SN) bifurcation, resulting in bistable attractors around each potential well center. Since the energy harvesting efficiency is undesirable in the case of the natural frequency of the BPEH at the far end of the ambient frequencies which are usually low, we compare the natural frequencies of the intra-well vibrations under different values of d and find that 14 mm is unsuitable for the vibrating energy harvesting. According to the comparison between variations of the intra-well response amplitudes with the increase of the excitation coefficients at d = 15 mm and d = 16 mm, it is not difficult to find that d = 15 mm is more beneficial for energy harvesting as the frequency bandwidth is broadened and the response amplitude is significantly increased at d = 15 mm due to the SN bifurcation.

On this basis, for d = 15 mm, a detailed investigation of the coexisting intra-well attractors and BAs is performed. Via the cell-mapping approach, BAs are depicted intuitively. Numerical results further confirm the bistability of the responses within each potential well, in good agreement with the analysis, thus confirming the accuracy of the prediction. This indicates that in the regions that multi-stability coexists, fractal boundaries of BAs as well as hidden attractors occur frequently. It also shows that for the excitation parameters within the small range of the thresholds for SN bifurcation, a jump from one intra-well attractor to the other within the same potential well can be triggered very easily. Since the output-voltage amplitude of the BPEH is proportional to the amplitude of the tip magnet, the jump from a lower-amplitude intra-well response to a higher-amplitude one is beneficial for energy harvesting, while the jump from one attractor to its symmetric one around a different center is unhelpful to harvest vibration energy.

Moreover, the Melnikov method is utilized to derive the necessary condition for homoclinic bifurcation. It suggests that the rise in the excitation amplitude can induce homoclinic bifurcation for which the excitation-amplitude threshold increases monotonically with the excitation frequency. The induced complex dynamical behaviors are displayed using numerical examples in the form of phase maps, basins of attraction, Poincaré maps, and frequency spectra. As depicted, it follows that with the rise in the excitation amplitude, the BAs of intra-well period-2 and period-3 attractors can be broken into scatters. Subsequently, the homoclinic bifurcation caused by the rise in the excitation amplitude triggers the occurrence of inter-well periodic response and chaos, which is favorable for the purpose of energy harvesting.

In previous studies [17,19,20] on this BPEH system, the distance between the tip mass and the external magnet was fixed, and the influence of the disturbance of the initial conditions on the dynamical behavior as well as the efficiency of energy harvesting was neglected. In contrast, we consider the initial distance as a crucial parameter for enhancing the energy harvesting performance. From the viewpoint of global dynamics that takes the variation of initial states into consideration, we determine its optimum value for vibration energy harvesting, which may offer a valuable insight into the design of BPEHs.

We have concentrated on the effect of the initial distance between the tip mass and the external magnet, the ambient excitation, and initial states on global dynamical behaviors of a BPEH oscillatory system for further structural optimization of piezoelectric energy harvesting devices. To assess the energy harvesting efficiency and reliability of BPEHs in practical application, our future studies will include corresponding experiments.