Characteristics of a tri-stable piezoelectric vibration energy harvester by considering geometric nonlinearity and gravitation effects

Mechanical Systems and Signal Processing 138 (2020) 106571

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Characteristics of a tri-stable piezoelectric vibration energy harvester by considering geometric nonlinearity and gravitation effects Guangqing Wang a,⇑, Zexiang Zhao a, Wei-Hsin Liao b, Jiangping Tan a, Yang Ju a, Yin Li a a b

School of Information and Electronic Engineering, Zhejiang Gongshang University, Hangzhou 310018, China Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong, Shatin, NT, Hong Kong, China

a r t i c l e

i n f o

Article history: Received 22 October 2019 Received in revised form 9 December 2019 Accepted 14 December 2019

Keywords: Tri-stable piezoelectric vibration energy harvester Nonlinear distributed-parameter model Geometric nonlinearity Gravitation effect Asymmetric potential wells Energy harvesting enhancement

a b s t r a c t Piezoelectric energy harvester (PEH) with tri-stable potential wells has attracted considerable attention and has been studied in recent years. However, the influence mechanism of the geometric nonlinearity (GNL) and the gravitation effect (GE) on the equilibrium solution bifurcation, potential energy function, and dynamic characteristics of such device remains uninvestigated. In this paper, a global nonlinear distributed-parameter model by considering the GNL and GE of the tri-stable PEH is originally established. The effects of the GNL and GE on the equilibrium solution bifurcations and potential energy function in the parameter’s space are firstly identified. It is found that the tri-stable PEH has a potential energy function with asymmetric potential wells that are not observed in the previous works. The influence mechanism of the asymmetric potential wells caused by the GE and GNL on the dynamic responses and the effective bandwidth of the tri-stable PEH are numerically analyzed. A prototype of the tri-stable PEH is manufactured and is subsequently used to validate the theories by experiments. The results indicates that the derived model by considering the GNL and GE has high accuracy, and the tri-stable PEH with asymmetric wells can not only enhance the dynamic response outputting at lower base excitation but also broaden the effective bandwidth. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction Sustainable self-powered technologies for converting vibration energy from various ambient sources into electrical energy are most prospective technologies and have become a hot topic in recent years [1–3]. Among these technologies, vibration-based piezoelectric energy harvester (PEH) has received considerable attentions because of its superior ability to convert various vibration energies into electrical energy. It is expected to be a potential alternative for solving the needs for chemistry batteries in low power electronic devices due to the advantages of consistent power generation and significant cost saving [4,5]. Early configurations for PEH were designed based on linear, single-frequency resonant harvesters and generally have been a piezoelectric cantilever beam with a tip mass. Such PEH has a narrow bandwidth that can only output a peak value under resonance [6]. A small shift of the exciting frequency from the resonant frequency of the PEH can lead to significant reduction in energy harvesting efficiency. This makes such PEH unsatisfied the practical application requirements ⇑ Corresponding author. E-mail address: [email protected] (G. Wang). https://doi.org/10.1016/j.ymssp.2019.106571 0888-3270/Ó 2019 Elsevier Ltd. All rights reserved.

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G. Wang et al. / Mechanical Systems and Signal Processing 138 (2020) 106571

as the base vibrations usually occur randomly over a wide frequency bandwidth [7]. Efforts have been put on various ways to improve the output performances of such PEH with active or adaptive strategies [1,8–12]. Among these strategies, nonlinear techniques has been proved to be more suitable for energy harvesting from base vibrations due to the less sensitivity to variations in the frequency characteristics of the operating environment [13,14]. Nonlinear PEHs have been developed in recent years and they can be generally classified into three main categories, namely, mono-, bi- and tri-stable PEHs. Compared to the mono- and bi-stable PEHs, tri-stable PEH has attracted much interest due to its shallower and wider potential wells that is beneficial to enhance the energy harvesting ability. A tri-stable PEH can be realized with the magnetic field coupling, and the most common configuration is generally composed of a piezoelectric cantilever beam with a tip magnet and two external magnets. Up to now, several tri-stable PEHs have been developed to indicate the wide applications of superior energy harvesting ability [15–20]. Zhou et al. [15] modeled and investigated a broadband tri-stable PEH induced by a magnetic field their results indicated that the tri-stable PEH has shallower potential wells to produce higher energy output over wider frequency ranges. Similar results have also been obtained by Kim and Seok [16], Zhu et al. [17], and Masana and Daqaq [18]. Zhou and Zou [19] used the harmonic balance method to study the nonlinear dynamic mechanism of asymmetric tri-stable PEH. The results showed that the orbit height for large-amplitude interwell oscillations greatly depends on the potential barrier. Xu and Jin [20] studied the stochastic resonance phenomenon of an asymmetric tri-stable system. Wang et al. [21] presented an improved magnetic force model and used it to analyze the nonlinear dynamic characteristics of a tri-stable PEH. Their results showed that the proposed model has high accuracy even the magnet distance is small, and the tri-stale PEH exhibits better output performance in a wide frequency range than the bistable PEH. The above research has demonstrated that tri-stable PEH has more unique dynamic behaviors and excellent energy harvesting ability. However, there are still rarely nonlinear analyses into the dynamic characteristics as well as the potential energy of the tri-stable PEH by considering the GNL of the piezoelectric cantilever beam and the GE of the tip magnet. Up to now, most of the dynamic models of the tri-stable PEH are derived based on the assumption that the oscillating amplitude is small enough to obey the linear Euler-Bernoulli beam theory that the GNL of the cantilever beam can be ignored [12–23]. It seems unreasonable, especially when the tri-stable PEH oscillates in a large-amplitude interwell motion. As the tri-stable PEH is generally expected to oscillate in a large-amplitude interwell motion to enhance the energy harvesting output, this results in a very large deformation in the cantilever beam of the tri-stable PEH. Consequently, the GNL is inevitably generated into the piezoelectric cantilever beam [21]. In addition, the influences of the GE of the tip magnet on the equilibrium solution bifurcations, potential wells and dynamic characteristics of the tri-stable PEH should also be considered. From the above research, an important conclusion has been drawn that the tri-stable PEH possesses three stable equilibriums and three symmetric potential wells. However, this conclusion holds only for exceptionally small mass of the tip magnet respects to the equivalent mass of the cantilever beam. If the mass of the tip magnet is not small enough to be neglected, the gravitation effects of the tip magnet should be considered both in the potential energy analysis and dynamic responses. Moreover, the center position of the equilibrium solutions will also shift with the variation of the GE, leading to various bifurcation behaviors of the equilibrium solution. Not only that, the combined effects of GNL and GE on the tri-stable PEH can lead to rich dynamic phenomena and complicated bifurcation behaviors. All these have been neglected in most of the previous works [12–23]. Therefore, further investigation on the characteristics of the tri-stable PEH by considering the GNL and GE is necessary, which can provide more physical insights into the underlying of nonlinear characteristics and bifurcation mechanism of the tri-stable PEH. Motivated by the above discussions, we developed a global nonlinear distributed-parameter model of the tri-stable PEH by considering the GNL and GE. This paper is organized as follows: in Section 2, a global nonlinear distributedparameter model of the tri-stable PEH by considered the GNL and GE is originally developed with the modified Hamilton’s principle; Section 3 analyses the static characteristics, such as the equilibrium solution bifurcation and the potential wells, of the tri-stable PEH based on the derived model; in Section 4, some numerical simulations are performed to analyze the influence of the GNL and GE on the nonlinear dynamic responses of the tri-stable PEH; in Section 5, the experimental validations are conducted. Finally, key conclusions obtained by simulations and experiments are summarized.

2. Model of the Tri-stable PEH by considering the GNL and GE Fig. 1 shows the schematic sketch of a magnet-coupled tri-stable PEH considered in this study [21]. The tri-stable PEH mainly consists of a piezoelectric beam, three magnets (denoted as A, B and C) and a base. The piezoelectric beam, which is tightly clamped at the base, is composed of a metal substrate, two piezoelectric (PZT) patches, and magnet A (which is called tip magnet). The two PZT patches with identical sizes and opposite polarization in thickness direction are completely bonded to the upper and lower surfaces of the metal substrate root and connected in series to a low power electronic device with a resistance R. Magnet A is attached at the tip of the piezoelectric beam. Magnets B and C (which are called external magnet) are fixed at the base. The horizontal distance (or separation distance) between Magnet A and Magnets B, C is d. The vertical distance between Magnet B and Magnet C is dg. GM is the gravity of the tip magnet and Dw is the static deformation of the piezoelectric beam caused by the gravity of the tip magnet; and €z0 ðtÞ ¼ AcosðxtÞ is the base acceleration with amplitude A and frequency x.

G. Wang et al. / Mechanical Systems and Signal Processing 138 (2020) 106571

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Fig. 1. Schematic of the tri-stable energy harvester.

2.1. Strain generated in the piezoelectric beam by considering the GNL We have noted that most of the previous models of the PEH are based on the linear Euler-Bernoulli beam theory that the linear inextensible strain generated in the beam is proportional to the second spatial derivative of the bending coordinate [21–24]. In contrast to the aforementioned works, we take into account the geometric nonlinearities (GNL) of the piezoelectric beam, as shown in Fig. 2 [25]. In this figure, @x is the length of a differential element (PQ ) of the piezoelectric beam before deformation, the coordinate of P is ½uðx; tÞ; 0, and Q is ½uðx; tÞ þ @x; 0. While @s and # are the length and rotating angle of the differential element (P 0 Q 0 ) after deformation, the coordinate of P 0 is ½uðx; tÞ; wðx; tÞ and Q 0 is ½uðx; tÞ þ @x þ @uðx; tÞ; wðx; tÞ þ @wðx; tÞ, where uðx; tÞ and wðx; tÞ are the axial displacement and the transverse displacement of the piezoelectric beam at horizontal position x , and duðx; tÞ and dwðx; tÞ are the extensions in the axial direction and the transverse direction, respectively. Then we can calculate the length @s of the differential element after deformation.

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @s ¼ ð@x þ @uÞ2 þ dw2 ¼

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  2 @u @w þ 1þ @x @x @x

ð1Þ

The axial strain of the differential element can be obtained

@s  @x ¼ e¼ @x

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  2 @u @w 1 1þ þ @x @x

ð2Þ

As the piezoelectric beam is inextensional, that is to say e ¼ 0, then we can obtain

u0 ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  w02  1

ð3Þ

@ @ where the prime notations ðÞ0 and ðÞ00 are the shorthand for @x and @x 2 respectively. And the rotating angle of the differential element can also be obtained 2

tan# ¼

w0 w0 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ u0 1  w02

ð4Þ

Fig. 2. Deformation of a differential element of the piezoelectric beam.

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Expanding Eq. (4) with Taylor series, we can then obtain

  w0 1 # ¼ arctan pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ w0 þ w03 þ Oðw05 Þ 02 6 1w

ð5Þ

As the relationship between the curvature q and rotating angle h is

1 2

q ¼ #0 ¼ w00 þ w02 w00

ð6Þ

Hence, the strain caused by the bending deformation is

  1 e1 ¼ zq ¼ z w00 þ w02 w00 2

ð7Þ

Therefore, the total strain of the differential element can be obtained

  1 S ¼ e þ e1 ¼ zq ¼ z w00 þ w02 w00 2

ð8Þ

The second term of Eq. (8) represents the strain generated by the geometric nonlinearity of the cantilever beam. 2.2. Nonlinear dynamic model of the tri-stable PEH by considering GNL and GE For modeling, we assume that: (1) the thickness of the piezoelectric beam is small compared to its length so that the shear deformation and rotary inertia can be neglected; (2) the piezoelectric beam is inextensible; and (3) the beam’s first mode of vibration is particularly dominant and has the greatest strain profile nearest the clamp location. The electromechanical behavior of the PZT patches is governed by the following constitutive equations [26].

T 1 ¼ C 11 S1  e31 E3

ð9Þ

D3 ¼ e31 S1 þ e33 E3

where T 1 and S1 represents mechanical stress and strain, E3 and D3 denote the electric field and displacement, e31 is the piezoelectric constant, C 11 is the elastic stiffness tested in a zero electric field, and e33 denotes the permittivity constant of the piezoelectric material at zero strain. Focusing on the configuration of the tri-stable PEH shown in Fig. 1, we derive the dynamic model according to Hamilton Principle [27] that the variational indicator should be zero all the time through Lagrange functions, as shown in the following expression.

Z

t2



 dðT  U s  U m  U g þ W p Þ þ dW dt ¼ 0

ð10Þ

t1

where d is variational symbol, T is the system kinetic energy, U i ði ¼ s; m; gÞ is the potential energy generated by the elastic deformation of the piezoelectric beam, the magnetic field and the gravity of the tip magnet, respectively, W p is potential energy of the electric field generated in the PZT patches, and dW is the system variational work. The kinetic energy of the tri-stable PEH is the sum of the kinetic energies of the beam and the tip magnet, which is expressed as

1 T¼ 2

Z Vs

qs w_ 2 ðx; tÞdV s þ 2 

1 2

Z

!2 1 1 @ 2 wðx; tÞ 2 2 _ _ qp w ðx; tÞdV p þ mt w ðL; tÞ þ It jx ¼ L 2 2 @x@t Vp

ð11Þ

where V i and qi ði ¼ s; pÞ represent the volume and density of the substrate and the PZT patches, respectively; mt and It are the mass and the rotary inertia of the tip magnet, and L represents the substrate length. As the two PZT patches connecting in series and the continuous electrode pairs covering the top and the bottom faces are perfectly conductive, so that a single electric potential difference can be defined between them. Therefore, the instantaneous electric fields induced in the two PZT patches are assumed to be uniform throughout their lengths. Let VðtÞ be the harvested voltage across the electrodes, so the electrical field in the PZT patches can be defined as

E3 ¼ w0 VðtÞ ¼ 

1 VðtÞ 2hp

ð12Þ

where w0 ¼ 1=2hp is the potential distributed function in the PZT patches. Then the elastic potential energy of the tri-stable PEH can be calculated by following expression.

Us ¼

1 2

Z

T s Ss dV s þ 2  Vs

1 2

Z

T 1 S1 dV p Vp

Substituting Eqs. (8), (9) and (12) into Eq. (13), we can obtain

ð13Þ

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G. Wang et al. / Mechanical Systems and Signal Processing 138 (2020) 106571

Us ¼

 2 1 C s zw00 ðx; tÞ  zw00 ðx; tÞw02 ðx; tÞ dV s 2 Vs   Z  1 1 þ zC 11 w00 ðx; tÞ  C 11 zw00 ðx; tÞw02 ðx; tÞ þ e31 w0 VðtÞ zw00 ðx; tÞ  zw00 ðx; tÞw02 ðx; tÞ dV p 2 2 Vp 1 2

Z

ð14Þ

where C s and C 11 are the elastic modulus of the substrate and the PZT patches, respectively. The potential energy generated by the electric field of the PZT patches is expressed as

Wp ¼

  Z 2 Z 1X 1 E3;i D3;i dV p;i ¼ w0 VðtÞ ze31 w00 ðx; tÞ  ze31 w00 ðx; tÞw02 ðx; tÞ  e33 w0 VðtÞ dV p 2 i¼1 V p;i 2 Vp

ð15Þ

As shown in Fig. 1, the gravitation potential energy of the tip magnet can be expressed as

U g ¼ GM wðx; tÞjx¼L ¼ Mt gwðx; tÞjx¼L

ð16Þ

where g is the gravity acceleration. The magnetic repulsing force exerted on the tip magnet can be calculated with our proposed magnetic model as shown in Fig. 3. The details about the magnetic model can be referred to our previous work [21]. The magnetic force can be expressed as

8 >

9 >

= l MA SA MB SB < a1 a1 a3 a3 Fm ¼ 0 h þ  þ i h i h i 3=2 3=2 3=2 3=2 > 2 > 4p 2 2 : ðb þ dÞ2 þ a2 ; ðd þ a2 Þ ð2b þ dÞ þ a2 ðb þ dÞ þ a2 1

1

2

2

3

3

8 9 > > = l0 MA SA MC SC < a2 a2 a4 a4   þ þ h i h i h i 3=2 3=2 3=2 3=2 > 2 > 4p 2 2 : ðb þ dÞ2 þ a2 ; ðd þ a24 Þ ð2b þ dÞ þ a2 ðb þ dÞ þ a2

ð17Þ

4

with a1 ¼ wðL; tÞ  0:5dg , a2 ¼ wðL; tÞ þ 0:5dg , a3 ¼ wðL; tÞ þ bh  0:5dg , a4 ¼ wðL; tÞ þ bh þ 0:5dg .where l0 is the magnetic permeability constant; Mi and Si ði ¼ A; B; and; CÞ are the magnetization intensity and the surface area of magnet i respectively;b is the length of the magnet; and h ¼ w0 ðL; tÞ is the rotating angle of the tip magnet. Therefore, the potential energy generated by the magnetic field can be obtained by

Z Um ¼

ð18Þ

F m dx

Finally, the system variational work dW can be calculated by the following expression.

Z dW ¼ 0

L

ðmðxÞ€z0 ðtÞdwðx; tÞÞdx þ ðmt €z0 ðtÞÞdwðx; tÞjx¼L 

Z

VðtÞ dVðtÞdt R

ð19Þ

where mðxÞ is the mass per unit length of the piezoelectric beam;€z0 ðtÞ is the base excitation. dwðx; tÞ and dVðtÞ are the variational displacement applied to the base and the variational voltage applied to external resistance, respectively. The first two items in Eq. (19) are inertial forces that they can be introduced in the kinetic energy, and the last term is the electrical power dissipated in the external resistance.

Fig. 3. The improved magnetic force calculating method [21].

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G. Wang et al. / Mechanical Systems and Signal Processing 138 (2020) 106571

2.3. Reduced-order model To develop a reduced-order model of the tri-stable PEH, we use the Galerkin procedure. Based on the expansion theorem, wðx; tÞ can be represented by an absolutely and uniformly convergent series of the eigenfunctions as

wðx; tÞ ¼

n X

/i ðxÞr i ðtÞ

ð20Þ

i¼1

where /i ðxÞ and r i ðtÞ are the mode function that satisfies the boundary conditions of the given beam and the modal coordinates, respectively. As the beam’s first mode of vibration particularly dominant when the beam is bending with a low frequency, so the transverse displacement wðx; tÞ can be simplified as

wðx; tÞ ¼ /1 ðxÞrðtÞ  with /1 ðxÞ ¼

ð21Þ

/11 ðxÞ; 0  x < Lp ; and /12 ðxÞ;Lp  x  L

/11 ðxÞ ¼ C 1 cosðb1 xÞ þ C 2 coshðb1 xÞ þ C 3 sinðb1 xÞ þ C 4 sinhðb1 xÞ /12 ðxÞ ¼ D1 cosðb2 xÞ þ D2 coshðb2 xÞ þ D3 sinðb2 xÞ þ D4 sinhðb2 xÞ where /11 ðxÞ and /12 ðxÞ are the modes of the beam with and without the PZT patches, respectively. C i and Di ði ¼ 1; 2; 3; 4Þ are the constant coefficients determined by the boundary conditions of the given beam, b1 and b2 are the eigenvalues of the characteristics equation, Lp is the PZT patches length. Substituting Eq. (21) into Eqs. (11)–(19) and using the extended Hamilton principle Eq. (10), we can obtain the reducedorder model of the Tri-stable PEH by considering the GNL and GE.

8 €0 < M€r þ C r_ þ Kr þ n1 r5 þ n2 r 3  hV  n3 Vr2 þ n4 Vr þ F m þ M t g/12 ðLÞ ¼ Bf w V : hr_  n4 Vr r_ þ 3n5 r 2 r_ þ C p V_ þ ¼ 0 R

where

Z M ¼ ðqs As þ 2qp Ap Þ  K ¼ C s Is þ 2C 11 Ip

Lp

0

Z

Lp

0

bp bs hs ; Ip ¼ Is ¼ 12 3

h¼

h hs

e31 bp ðhs þ hp Þ 2

Z

Lp

 n2 ¼ 2C s Is þ 4C 11 Ip

Lp

1 e31 bp 2

Lp

Z

L

Lp

hs 3 i 2

; Cp ¼

0

Z /11 ðxÞdx þ qs As

Z 0

Z

Lp

0

n4 ¼



L

/212 ðxÞdx þ mt /212 ðLÞ þ It /02 12 ðLÞ

/002 12 ðxÞdx

e33 bp Lp 2hp

/0011 ðxÞdx

0

3 3 C s Is þ C 11 Ip 4 2

3bp e31 n3 ¼ 8hp

3

3

Z



/002 11 ðxÞdx þ C s I s

þ hp

2

Bf ¼ ðqs As þ 2qp Ap Þ

n1 ¼

Z /211 ðxÞdx þ qs As

Lp

L

/12 ðxÞdx þ mt /12 ðLÞ Lp

3 04 /002 11 ðxÞ/11 ðxÞdx þ C s I s 4

02 /002 11 ðxÞ/11 ðxÞdx þ 2C s Is

Z

Z

L

Lp L

Lp

02 /002 12 ðxÞ/12 ðxÞdx

" 2  2 # Z Lp hs hs /0011 ðxÞ/02 þ hp  11 ðxÞdx 2 2 0 Z 0

Lp

/02 11 ðxÞdx

04 /002 12 ðxÞ/12 ðxÞdx

ð22Þ

G. Wang et al. / Mechanical Systems and Signal Processing 138 (2020) 106571

n5 ¼

 1 bp e31 hs þ hp 8

Z

Lp

0

7

/0011 ðxÞ/02 11 ðxÞdx

hi and bi ði ¼ s; pÞ are the height and width of the substrate and the PZT patches, respectively; and C is the system damping coefficient. 3. Static characteristics of the tri-stable PEH The static characteristics of the tri-stable PEH mainly include the bifurcations of the equilibrium solutions and the potential wells, which can help us to deeply understand the form mechanism of multi-stability state of the tri-stable PEH. 3.1. System’s equilibrium solution bifurcation The bifurcation mechanism of the equilibrium solution of the tri-stable PEH depending on the geometric parameters of d and dg has been studied by Kim and Seok [16] and Wang et al. [21]. However, the GE of the tip magnet is not considered. In this subsection, the bifurcation behaviors of the system’s equilibrium solutions are further investigated by considering the GE of the tip magnet. The equilibrium solutions can be obtained by solving the following expression.

Kq þ F m þ n1 q5 þ n2 q3 þ Mt g/12 ðLÞ ¼ 0

ð23Þ

In Eq. (23), the third and the fourth terms present the nonlinear force caused by the geometric nonlinearity (GNL) of the piezoelectric beam, and the last term denotes the gravitation effect (GE) of the tip magnet. It shows that the equilibrium solutions greatly depend on the GNL and the GE except for the horizontal distance (d) and the vertical distance (dg). Therefore, the tri-stable PEH under consideration may exhibit different bifurcation behaviors with respect to the previous works. The simulating parameters are listed in Table 1. Fig. 4 shows the bifurcation diagrams plotted in ðd; wÞ space for the equilibrium solution of the tri-stable PEH by considering the GE. Here, the vertical distance dg is 25 mm and Mt = [0, 3, 4.1, 7, 9.1, 10] g, respectively. The solid lines stand for stable equilibriums and the dotted lines for unstable ones. When the tip magnet mass is zero, i.e., Mt = 0 [Fig. 4(a)], the bifurcation of the considered tri-stable PEH are similar to those of neglecting the GE [16,21], two saddle-node bifurcations at point SN and a pitchfork bifurcation at point BP1 of the equilibrium solutions are observed. In the range of d > dSN, only a trivial but stable zero equilibrium solution exists. Therefore, the tri-stable PEH in this range becomes a mono-stable system. Within the range of dSN > d > dBP1, there are two non-trivial but stable solution branches, two unstable solution branches and a trivial but stable zero equilibrium solution. The two stable and unstable solution branches are symmetrical from each other about the zero equilibrium solution, and the distance between these two stable or unstable equilibrium solutions gradually increases as the horizontal distance (d) decreases, and then tends to steadily decrease after it reaches a certain critical value. In this range, the tri-stable PEH becomes a tri-stable system. In the range of d < dBP1, a pitchfork bifurcation at point BP1 is appeared, from which the stable but trivial zero equilibrium solution starts bifurcate into two unstable non-trivial solutions and a stable but trivial zero equilibrium solution. Consequently, the tri-stable PEH is also a tri-stable system in this region. We can see from Fig. 4(a) that the most remarkable feature of the tri-stable PEH with Mt = 0 is that the bifurcations are symmetric with respect to the zero equilibrium position. When the tip magnet mass increases to 3 g [Fig. 4(b)], the symmetries of the bifurcation behaviors shown in Fig. 4(a) are broken, and asymmetrical phenomena in bifurcation behaviors are observed. Due to gravitation effect of the tip magnet, the

Table 1 Model parameters used for numerical calculations. Parameter

Symbol

Value

Length of the substrate Width of the substrate Height of the substrate Material density of the substrate Elastic modulus of the substrate Length of the piezoelectric layer Width of the piezoelectric layer Height of the piezoelectric layer Material density of piezoelectric layer

L bs hs

qp

70 mm 10 mm 0.15 mm 7800 kg/m3 212 GPa 10 mm 10 mm 0.5 mm 7450 kg/m3

Elastic modulus of piezoelectric layer Coupling coefficient of piezoelectric layer Permittivity constant of piezoelectric layer Modal damping ratio of the beam Mass of tip magnet Magnetization of magnets (A, B or C) Volume of magnet (A, B or C) Length of magnet (A,B or C)

Ep e31 e33 n Mt MA , MB , MC V A, V B, VC b

21.45 GPa 4.0773 1700 0.01 4.1 g 1.20 T 550 mm3 2.75 mm

qs Es Lp bp hp

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G. Wang et al. / Mechanical Systems and Signal Processing 138 (2020) 106571

Fig. 4. Bifurcation behaviors of the tri-stable PEH with different horizontal distance (d) by considering the GE when Mt = (a) 0, (b) 3 g, (c) 4.1 g, (d) 7 g, (e) 9.1 g, and (f) 10 g.

trivial but stable equilibrium solution is not zero, but deviates downward from zero to a certain value (Dw), as well as the two lower branches of the non-trivial stable and unstable equilibrium solutions move downward. Besides, the upper saddlenode bifurcation shifts left from point SN to point SN1; and the lower one shifts right from point SN to point SN2. Therefore, a bi-stable system with a trivial but stable equilibrium solution, an un-trivial but stable equilibrium solution and an unstable equilibrium solution is formed in the range of dSN1 < d < dSN2. Due to the magnetic repulsion force less than the gravity of the tip magnet in this region, the bi-stable system only exhibits a local interwell motion between the trivial but stable equilibrium solution and the un-trivial but stable equilibrium solution. In the range of d > dSN2, the tri-stable PEH becomes a monostable system with a trivial but stable non-zero equilibrium solution. However, in the range of d < dSN1, two non-trivial but stable solution branches, two unstable solution branches and a trivial but stable non-zero equilibrium solution are observed, and the tri-stable PEH becomes a tri-stable system which oscillates in a global interwell motion due to the magnet repulsion force is larger than the gravity of the tip magnet as the horizontal distance decreases to a certain value. It can also be found from Fig. 4(b) that the distance between the trivial but stable equilibrium solution and zero gradually decreases with the increase of the horizontal distance (d), and then tends to increase when it reaches a critical value. This is mainly attributed to the magnetic repulsion effect between the tip and external magnets. As the horizontal distance (d) deceases, the magnetic repulsion effect strengthens and plays a role in confining the tip magnet, leading to the decrease of the distance between the

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trivial but stable equilibrium solution and zero. However, when the horizontal distance decreases to a certain critical value, the magnetic revulsion effect plays a role in promoting the tip magnet, leading to the increase of the distance between the trivial but stable equilibrium solution and zero. In addition, a new saddle-node bifurcation appears at point SN3, which is caused by the degeneration of the pitchfork bifurcation at point BP1. Further increase the tip magnet mass to 4.1 g and 7 g [Fig. 4(c) and (d)], respectively, the asymmetry of the bifurcations become more obvious. The distance between the trivial but stable equilibrium solution and zero continuously increases. The upper saddle-node bifurcation point SN1 continuously moves left and the lower saddle node bifurcation point SN2 continuously moves right, leading to a large margin between these two points. This means that the space of the tri-stable PEH exhibiting a local bi-stable behavior becomes wider. When the tip magnet mass increases to 9.1 g [Fig. 4(e)], the local bi-stable range becomes wider, the trivial but stable equilibrium solution moves downward continuously and the lower saddle-node bifurcation solution [at point SN2] moves right continuously, and they happen to collide with each other at point BP2, leading to wider range for a bi-stable motion of the tri-stable PEH. With a further increase in Mt [Fig. 4(f)], the two bifurcation branches become to separate and then gradually moves away. Consequently, the tri-stable PEH becomes a mono-stable (when d > dSN2), bi-table (dSN1 < d < dSN2), or tristable (d < d SN1) system. 3.2. Potential wells analysis for the tri-stable PEH The influence mechanism of the GE on the potential wells of the tri-stable PEH can be investigated using the potential energy functions of the system. Figs. 5 and 6 show the potential wells of the system in bi- and tri-stable state respectively with five different tip magnet masses, i.e., Mt = 0, 5, 10, 15, and 20 g. It can be found that the tri-stable PEH has asymmetrical potential wells when the tip magnet mass is considered, the asymmetry of the potential wells becomes more and more significant with the increase of the tip magnet mass. This is consistent with the results obtained from the bifurcation analyses. In addition, it is also found that the difference between asymmetrical and symmetrical [Mt = 0] tri-stable PEH is that the vibration center of the former is asymmetrical, while the latter is symmetrical. Forwarding the asymmetrical potential wells shown in Figs. 5 and 6, each of them has a solely highest potential barrier and a lowest potential barrier, and the highest potential barriers increase but the lowest potential barriers decrease with the increase of the tip magnet mass. Besides, the width between the highest and the lowest potential barriers in tri-stable state is larger than that in bi-stable state. Both the highest and the lowest potential barriers in tri-stable state [Fig. 6] are lower than those of in bi-stable state [Fig. 5]. For example, the highest and lowest potential barriers in tri-stable state are 6.06 mJ and 0.7 mJ respectively when the tip magnet mass is 20 g; while in bi-stable state, they are 9.5 mJ and 1.7 mJ respectively. This means that the tri-stable PEH can easily overcome the potential barrier to oscillate in a tri-stable interwell motion than in a bi-stable interwell motion with lower external excitation [19]. As a comparison, the potential wells of a symmetrical tri-stable PEH [Mt = 0 g] are also shown in Figs. 5 and 6. In this case, the tri-stable PEH has two [in bi-stable state] and three [in tri-stable state] same highest potential barriers which may lead to a stronger impediment to larger-amplitude interwell oscillations. While the tri-stable PEH with asymmetrical potential wells can easily move across the lowest potential well and then go over the highest potential well to produce a larger-amplitude global interwell motion due to the positive work of the gravitation effects of the tip magnet on the system. Therefore, the tristable PEH may exhibit a tri-stable motion with large amplitude under a lower external excitation. In contrast, it needs large external ex citation to go over the highest barrier and move to the lowest well due to the negative work of the gravitation effects of the tip magnet on the system. In this case, the tri-stable PEH may exhibit a mono-stable motion with small

Fig. 5. Potential energy in bi-stable state with different tip magnet masses.

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Fig. 6. Potential energy in tri-stable state with different tip magnet masses.

amplitude when the external excitation is small, or a bi-stable local interwell motion or a tri-stable global interwell motion with large amplitude when the external excitation is large enough to overcome the highest barrier. 4. Nonlinear dynamic characteristics of the tri-stable PEH _ Let x1 ¼ qðtÞ, x2 ¼ qðtÞ, and x3 ¼ VðtÞ, then Eq. (22) can be rewritten as following form.

3 x2 x_ 1 6 Bf 7 n1 n2 h n3 n4 F m Mt x21 x1  2nx1 x2  x51  x31 þ x3 þ x21 x3  x1 x3   g/12 ðLÞ þ €z0 7 6_ 7 6 M M M M M M M M 7 4 x2 5 ¼ 6 7 6 5 4 n 3n h 1 4 5 2 x_ 3 x1 x2 x3  x1 x2  x2  x3 Cp RC p Cp Cp 2

3

2

ð24Þ

pffiffiffiffiffiffiffiffiffiffiffi where x1 ¼ K=M is the resonant frequency, n ¼ C=2x1 is the damping coefficient. The asymmetric feature by considering the GNL and GE is beneficial to the nonlinear dynamic characteristics of the tristable PEH and can be analyzed with Eq. (24) by numerical Runge-Kutta method. 4.1. Influences of the GE on the dynamic characteristics In this subsection, several studies are performed to reveal the influence mechanisms of the GE of the tip magnet on the dynamic characteristics of the tri-stable PEH. We firstly investigated the effects of the GE on the snap-though behaviors of the tri-stable PEH as shown in Fig. 7. In this figure, the threshold values of acceleration amplitude for the tri-stable PEH jumping from an intrawell motion to an interwell motion are showed for three different tip magnet masses, i.e., Mt = [0, 4.1 g, and 7 g]. It can be seen that the threshold value of acceleration amplitude for the tri-stable PEH oscillating from a

Fig. 7. The snap-though with the acceleration amplitude for different Mt.

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intrawell motion to an interwell motion is 9.5 m/s2 when the tip magnet mass is Mt = 0 g, while that is 6.4 m/s2 and 3 m/s2 when the tip magnet masses are 4.1 and 7 g, respectively. The threshold value by considering the GE [i.e., Mt = 4.1 g and 7 g] is smaller than that of neglecting the GE [Mt = 0 g]; this indicates that the tri-stable PEH by considering the GE can more easily snap through and go into an interwell motion to achieve large-amplitude outputs under lower acceleration amplitude. Particularly, when the tip magnet mass is 7 g, two jump-through are observed. When the acceleration amplitude A is 3 m/s2, the tri-stable PEH firstly jumps from a mono-stable motion to a local bi-stable interwell motion and the output voltage sharply increases from 0 V to 1.27 V. When the acceleration amplitude increases to 8.6 m/s2, the second jump-through occurs, leading to a global tri-stable interwell oscillation with large amplitude and the output voltage further increases from 1.27 V to 2.4 V. This indicates that the asymmetrical wells caused by the GE of the tip magnet can enhance the dynamic characteristics at a lower excitation level. The benefits of the GE of the tip magnet mass on the nonlinear dynamic characteristics can also been verified by the frequency response of the tri-stable PEH, as shown in Fig. 8. In the figure, the sweeping frequency ranges from 0 to 15 Hz. At the case of A = 7 m/s2 and Mt = 1 g [Fig. 8(a)], the tri-stable PEH exhibits an intrawell motion with small tip displacement, output voltage and output power within most of the interest frequency domain. Only in a narrow frequency range of 45 Hz, it oscillates in an interwell motion with a maximum tip displacement, output voltage and output power of 15 mm, 1.5 V and 0.003 mW, respectively. However, the tri-stable PEH oscillates in interwell motions with large tip displacement, output voltage and output power within a wider frequency range of 0.665.6 Hz at the same excitation strength when the tip magnet mass is increased to 4.1 g [Fig. 8(b)]. The maximum tip displacement, output voltage and output power reach 35 mm, 4 V and 0.016 mW, respectively. This indicates that increasing the tip magnet mass can enhance the energy harvesting ability and broad the frequency bandwidth of the tri-stable PEH. Further increase A to 10 m/s2, the tri-stable PEH exhibits a global interwell motion with large amplitude over the frequency range of 3.4–5.8 Hz when the tip magnet is 1 g [Fig. 8(c)], the maximum tip displacement; the output voltage and associated electric power reach 42 mm, 6 V, and 0.036 mW respectively. However, the tri-stable PEH oscillates in a global interwell motion within wider frequency range of 0.156.8 Hz when the tip magnet mass is 4.1 g [Fig. 8(d)]; the tip displacement, output voltage and generated electric power reach 43 mm, 6.2 V, and 0.037 mW. The results demonstrate that the increasing the tip magnet mass is beneficial to enhance the energy harvesting ability over a wide frequency range. Moreover, the results also indicate that increasing the excitation level can not only improve the energy harvesting ability but also broad the frequency bandwidth of the tri-stable PEH. The advantages of the GE on the dynamic characteristics can also be further investigated by the history responses of the tri-stable PEH at a given the base excitation. Fig. 9 shows the history responses of the tip displacement, harvested voltage and phase portrait of the tri-stable PEH with two different tip magnet masses of M t ¼ 1 g (Left column) and M t ¼ 4:1 g (Right column). Here, the system parameters are d ¼ 13 mm, dg ¼ 17 mm and R = 1 MX, and the base excitation is a harmonic signal with amplitude A = 7 m/s2 and frequency f = 6 Hz. As a comparison, we can find that increasing the tip magnet mass has a significant superiority in the dynamic characteristics under the same system parameters and exciting condition. When the tip magnet mass Mt is 4.1 g, the tri-stable PEH oscillates in a global interwell motion with tip displacement 22 mm, tip velocity 800 mm/s and output voltage 2.2 V respectively. While it oscillates in an intrawell motion with small tip displacement and output voltage at the tip magnet mass Mt = 1 g. From the tip displacements (first row in Fig. 9) we can find that the oscillating center of the tri-stable PEH is not zero, but has a deviation. The deviation of oscillating center from zero becomes bigger with the increase of the tip magnet mass. Besides, we also can see that the phase portraits of the tri-stable PEH are asymmetrical with the increase of the tip magnet mass which is consistent with the analysis results shown in Fig. 4. 4.2. Influences of GNL on the dynamic characteristics As mentioned above, the geometric nonlinearity (GNL) is mainly generated by the large deformation of the cantilever beam when the tri-stable PEH oscillates in large-amplitude interwell motions. In this subsection, several cases of phase portrait are depicted to reveal the influence mechanism of the GNL on the dynamic characteristics of the tri-stable PEH. For the sake of comparison, it also presents the phase portraits of the tri-stable PEH obtained by the previous works that the dynamic model is derived with the linear Euler-Bernoulli beam theory, as shown below [13–22].

3 2 x2 3 _x1 6 n1 n2 h F m Bf 7 x21 x1  2nx1 x2  x51  x31 þ x3  þ €z0 7 6_ 7 6 M M M M M 7 4 x2 5 ¼ 6 7 6 5 4 h 1 x_ 3  x2  x3 Cp RC p 2

ð25Þ

where the system parameters and the state variables are same as those of Eq. (24). As focusing on the effects of the GNL on the dynamic characteristics of the tri-stable PEH in this subsection, the tip magnet mass is set to be a small value (i.e., Mt = 0.1 g) that its effect on the tri-stable PEH characteristics is very small. Fig. 10 shows the phase portraits of the tri-stable PEH for three different acceleration amplitudes, i.e., A = 3 m/s2, 10 m/s2 and 15 m/s2. It can be seen that the phase portrait obtained by Eq. (24) is consistent with that obtained by Eq. (25) when the acceleration amplitude is A = 3 m/s2 [Fig. 10(a)]. For this case, the excitation strength is low and the tri-stable PEH oscillates in an intrawell motion with small amplitude, leading to a very small geometric deformation in the piezoelectric beam.

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Fig. 8. (First row) Tip displacement, (Second row) output voltage and (Third row) the associated electric power of the tri-stable PEH with different exciting amplitudes and tip magnet masses, (a) Mt = 1 g and A = 7 m/s2, (b) Mt = 4.1 g and A = 7 m/s2, (c) Mt = 1 g and A = 10 m/s2 , and (d) Mt = 4.1 g and A = 10 m/s2.

Therefore, the effect of the GN on the dynamic characteristics of the tri-stable PEH is very small and can be ignored. However, when the excitation strength increases to A = 10 m/s2 [Fig. 10(b)], the tri-stable PEH goes into an interwell motion with large amplitude, a large deformation with geometric nonlinearity is generated in the piezoelectric beam that degrades the oscillating amplitude of the tip magnet, leading to an obvious error between the two phase portraits. For example, the tip displacements obtained by previous model and our model are 28 mm and 24.5 mm respectively, the error is 3.5 mm. Not only that, the error gradually increases with the increase of the excitation strength, as shown in Fig. 10 (c). The tip displacement error between previous model and our model reaches 5 mm when the acceleration amplitude is A = 15 m/s2. Due to the high excitation strength, the oscillating amplitude of the tri-stable PEH is large, leading to the large geometric deformation and nonlinearity in the piezoelectric beam. Thus, the effect of the GNL on the dynamic characteristics is further enhanced. The numerical results indicate that the derived dynamic model with considering the GNL brings higher accuracy and can better reflect the practical dynamic characteristics, especially when the tri-stable PEH oscillates in an interwell motion with large amplitude. 5. Experimental investigation To validate the theoretical results obtained by the numerical calculation, corresponding experiments are performed. The experimental platform is shown in Fig. 11. The tri-stable PEH consists of a piezoelectric cantilever beam, three magnets and a base. The piezoelectric beam is clamped at one end and free to oscillate at the other. On the upper and lower surfaces of the clamped end of the piezoelectric beam, two PZT patches are completely boned, and a magnet is attached at its tip end. The tri-stable PEH is mounted on the base that is excited by a shaker (JZK-5). Two external magnets are fixed at the base and separated from the tip magnet with a horizontal distance (d), and are aligned parallel to each other with a vertical distance

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Fig. 9. Displacement response (first row), voltage response (second row) and phase portrait (third row) of the Tri-stable PEH with system parameters of mm, dg ¼ 17 mm when the excitation amplitude is 7 m/s2 and frequency is 6 Hz. (a) Mt ¼ 1 g (Left column) and (b) Mt ¼ 4:1 g (Right column).

(dg). The signal generator (AFG3102C, Tektronix) is used to produce a harmonic signal with adjustable frequency and amplitude to excite the shaker. An accelerometer (CA-1) fixed at the base is used to test the vibration amplitude of the shaker. A laser vibrometer (LK-G3000, KEYENCE) is used to measure the vibration displacement of the tri-stable PEH, and the harvested voltage is collected with an oscilloscope. In order to validate the mode shapes and the associated eigenvalues of the piezoelectric beam, the frequency response of the tip displacement per unit acceleration of the tri-stable PEH without the external magnets is firstly performed, as show in

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Fig. 10. Phase portraits of the tri-stable PEH when (a) A = 3 m/s2, (b) A = 10 m/s2, and (c) A = 15 m/s2.

Laser vibrometer

External magnets Accelerometer Piezoelectric beam Tip magnet Base

Shaker

Fig. 11. Experimental platform of the tri-stable PEH.

Fig. 12. The resonant frequencies and the associated eigenvalues obtained by experiment and calculating results with Eq. (21) are listed in Table. 2. It can be seen that the results obtained by experiment are good agreeable with the calculating results, which testifies that the mode shapes adopted in the model is correct. The experimental validation for the magnetic force and the potential energy function of the tri-stable PEH is then performed. In this test, the horizontal distance between the tip magnet and the external magnet is d = 17.8 mm, and the vertical distance between the two external magnets is dg = 25 mm. The mass of the tip magnet is Mt = 4.1 g. The magnetic force

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Fig. 12. Frequency responses of the cantilever beam obtained by experiment and calculation.

Table 2 Resonant frequencies and associated eigenvalues.

Experimental results Calculating results

First order resonant frequency x1 (rad/s)

Eigenvalue b1

Eigenvalue b2

43.96 43.53

8.82 8.77

13.99 13.93

Fig. 13. (a) Magnetic restoring force and (b) Potential energy function obtained by experiment and calculation.

Fig. 14. Experimental displacement frequency–response results for a base excitation of (a) 7 m/s2 and (b) 10 m/s2.

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Fig. 15. Experimental results of tip displacement, output voltage and phase portraits of the tri-stable PEH oscillating in bi-stable motion (a) and tri-stable motion (b).

exerted on the tip magnet is measured by a digital force dynamometer (HF-10). For a comparison, the nonlinear magnetic force obtained by theoretical calculation is also plotted, as shown in Fig. 13(a). It indicates that the calculating results are consistent with the experimental results. The elastic potential energy of the tri-stable PEH and the magnetic potential energy generated between the tip magnet and the two external magnets can be obtained by calculating Eqs. (14) and (18), respectively. The experimental potential energy can be obtained by integrating the magnetic force obtained from the experiment within the range of given displacement, as shown in Fig. 13(b). The results illustrate that the magnetic potential energy obtained by experiment are well consistent with the calculating result during the range of 20 mm to 20 mm; the results

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also show that the potential well of the tri-stable PEH are asymmetric, which is consistent with the theoretical result. In addition, the elastic potential energy obtained by experiment is good agreement with the calculating result in the whole testing range; the error of the elastic potential energy between the experiment and calculation is small, even when the tri-stable PEH deforms with large amplitude. This indicates that derived model by considering the GNL and the GE has high accuracy. To validate the effective bandwidth of the tri-stable PEH, the frequency response of the tip displacement of the tri-stable PEH are performed experimentally. In the experiment, two different base excitations are considered, i.e., 7 m/s2 and 10 m/s2, the tip magnet mass is 4.1 g, and the sweeping frequency ranges from 0 to 8 Hz with a step of 0.15 Hz. The experimental and calculating results are shown in Fig. 14. It shows that the experimental results are consistent with the calculating results in most of the frequency ranges. When the base acceleration is 7 m/s2 as shown in Fig. 14(a), the tri-stable PEH exhibits largeamplitude interwell oscillation within the range of 0.7 Hz  5.55 Hz, and the calculating frequency range is from 0.66 Hz to 5.6 Hz. Further increase the base acceleration to 10 m/s2 as shown in Fig. 14(b), the tri-stable PEH oscillates in an interwell motion with larger output displacement and wider frequency bandwidth. The frequency range obtained by experiment is from 0.15 Hz to 7.7 Hz, and the calculating result ranges from 0 to 6.8 Hz. Compared to the calculating results shown in Fig. 8, it indicates that the GE can benefit the effective bandwidth and energy harvesting enhancement of the tri-stable PEH. Finally, the dynamic responses in time history of the tri-stable PEH with different system parameters and exciting conditions are tested as shown in Fig. 15. Fig. 15(a) shows the experimental results of the tip displacement, harvested voltage and phase portrait of the tri-stable PEH excited by a base acceleration with amplitude of 15 m/s2 and frequency of 6 Hz. It shows that the tri-stable PEH exhibits a bi-stable motion with asymmetric tip displacement and inclining phase portrait. Adjusting the amplitude and frequency of the base acceleration to 6 m/s2 and 9 Hz, the experimental tip displacement, output voltage and phase portrait are shown in Fig. 15(b). It indicates that the tri-stable PEH exhibits a tri-stable motion; the tip displacement, the output voltage and the phase portrait are also asymmetric, and the center of the tip displacement is not at zero, but a deviation from zero as shown in Fig. 15(b). This is in qualitative agreement with the theoretical results shown in Fig. 9(b). Besides, we note that the tri-stable PEH oscillates in a bi-stable motion with large base excitation amplitude of 15 m/s2, which is 2.5 times of that when the tri-stable PEH oscillates in a tri-stable motion. This further proves that the tri-stable PEH with considering the GE has lower base excitation to oscillate in tri-stable motion.

6. Conclusions In this paper, we derived a global nonlinear distributed-parameter model of the tri-stable PEH, which includes the geometric nonlinearity (GNL) of the cantilever beam and the gravitation effect (GE) of the tip magnet. The effects of the GNL and GE on the bifurcations, the potential energy function and the dynamic responses of the tri-stable PEH were theoretically and experimentally analyzed. Unlike the tri-stable PEH with symmetric potential wells in the previous works, the tri-stable PEH by considering the GNL and GE has asymmetric potential wells that can enhance the output performances and broaden the effective frequency bandwidth with lower base excitations. The asymmetrical potential well has a solely highest potential barrier and a lowest potential barrier, and the highest potential barriers increase but the lowest potential barriers decrease with the increase of the tip magnet mass. The width between the highest and the lowest potential barriers in tri-stable state is larger than that in bi-stable state. Both the highest and the lowest potential barriers in tri-stable state are lower than those of in bi-stable state. This makes the tri-stable PEH more easily overcome the potential barrier to oscillate in a tri-stable interwell motion than in bi-stable interwell motion with lower base excitation. Due to the GNL and GE, the bifurcation behaviors of the tri-stable PEH are asymmetrical with respect to the zero equilibrium position. With the tip magnet mass increase, the asymmetry of the bifurcation behaviors becomes more obvious. The distance between the trivial but stable equilibrium solution and zero continuously increases. The upper saddle-node bifurcation point SN1 continuously moves left and the lower saddle node bifurcation point SN2 continuously moves right, leading to a large margin of bi-stable behavior. At last, the model derived by considering the GNL and GE has higher accuracy, especially when the tri-stable PEH oscillates in a large-amplitude interwell motion. The conclusions can be referred to design tri-stable energy harvesters with nonlinear technologies for the purpose of enhancing energy harvesting under various low excitation conditions.

Author contribution Guangqing Wang derived the dynamic model, designed experiments and wrote the manuscript; Zexiang Zhao carried out experiments; Wei-Hsin Liao checked the manuscript; Jiangping Tan analyzed experimental results; Yang Ju analyzed the numerical results with simulation method; Yin Li plotted the figures of the manuscript. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Acknowledgements The authors acknowledge the financial support of the National Natural Science Foundation of China (Grant No. 51777192), Zhejiang Provincial Natural Science Foundation of China (Nos. LY20E070001, LGG19F010005), and the Research Grants Council of the Hong Kong Special Administrative Region, China (CUHK14205917). References [1] N. Tran, M.H. Ghayesh, M. Arjomandi, Ambient vibration energy harvesters: A review on nonlinear techniques for performance enhancement, Int. J. Eng. Sci. 127 (2018) 162–185. [2] H.X. Zou, L.C. Zhao, Q.H. Gao, L. Zuo, F.R. Liu, T. Tan, K.X. Wei, W.M. Zhang, Mechanical modulations for enhancing energy harvesting: Principles, methods and applications, Appl. Energy 255 (2019) 113871. [3] C. Wei, X. Jing, A review comprehensive on vibration energy harvesting: Modeling and realization, Renew. Sustain. Energy Rev. 74 (2017) 1–18. [4] S. Roundy, P.K. Wright, A piezoelectric vibration based generator for wireless electronics, Smart Mater. Struct. 13 (5) (2004) 1131. [5] L. Kuang, M. Zhu, Characterizations of a knee-joint energy harvester powering a wireless communication sensing node, Smart Mater. Struct. 25 (5) (2016) 055013. [6] A.A. Babayo, Mh. Anisi, I. Ali, A review on energy management schemes in energy harvesting wireless networks, Renew. Sustain. Energy Rev. 76 (2017) 1176–1184. [7] S.P. Beeby, R.N. Torah, M.J. Tudor, P. Glynne-Jones, T. Odonnell, C.R. Saha, S. Roy, A micro electromagnetic generator for vibration energy harvesting, J. Micromech. Microeng. 17 (7) (2007) 1257–1265. [8] G. Wang, W. Liao, B. Yang, X. Wang, W. Xu, X. Li, Dynamic and energetic characteristics of a bistable piezoelectric vibration energy harvester with an elastic magnifier, Mech. Syst. Sig. Process. 105 (2018) 427–446. [9] G.Q. Wang, W.H. Liao, A bistable piezoelectric oscillator with an elastic magnifier for energy harvesting enhancement, J. Intell. Mater. Syst. Struct. 28 (3) (2017) 392–407. [10] B.L. Ooi, J.M. Gilbert, Design of wideband vibration-based electromagnetic generator by means of dual-resonator, Sens. Actuat. A Phys. 213 (2014) 9– 18. [11] P. Firoozy, S.E. Khadem, S.M. Pourkiaee, Broadband energy harvesting using nonlinear vibrations of a magnetopiezoelastic cantilever beam, Int. J. Eng. Sci. 111 (2017) 113–133. [12] P. Kim, J. Seok, A multi-stable energy harvester: dynamic modeling and bifurcation analysis, J. Sound Vib. 333 (21) (2014) 5525–5547. [13] J. Jung, P. Kim, J. Lee, J. Seok, Nonlinear dynamic and energetic characteristics of piezoelectric energy harvester with two rotatable external magnets, Int. J. Mech. Sci. 92 (2015) 206–222. [14] A.H. Hosseinloo, K. Turitsyn, Non-resonant energy harvesting via an adaptive bistable potential, Smart Mater. Struct. 25 (2015) 15010–15018. [15] S. Zhou, J. Cao, D.J. Inman, J. Lin, S. Liu, Z. Wang, Broadband tristable energy harvester: modeling and experiment verification, Appl. Energy 133 (2014) 33–39. [16] P. Kim, J. Seok, Dynamic and energetic characteristics of a tri-stable magnetopiezoelectric energy harvester, Mech. Mach. Theory 94 (2015) 41–63. [17] P. Zhu, X. Ren, W. Qin, Y. Yang, Theoretical and experimental studies on the characteristics of a tri-stable piezoelectric harvester, Archm Applm Mech. 87 (2017) 1541–1554. [18] M. Panyam, M.F. Daqaq, Characterizing the effective bandwidth of tri-stable energy harvesters, J. Sound Vib. 386 (2017) 336–358. [19] S. Zhou, L. Zuo, Nonlinear dynamic analysis of asymmetric tristable energy harvesters for enhanced energy harvesting, Commun. Nonlinear Sci. Numer. Simul. 61 (2018) 271–284. [20] P. Xu, Y. Jin, Stochastic resonance in an asymmetric tristable system driven by correlate noises, Appl. Mathemat. Model. 77 (2020) 408–425. [21] G. Wang, W. Liao, Z. Zhao, J. Tan, S. Cui, H. Wu, W. Wang, Nonlinear magnetic force and dynamic characteristics of a tri-stable piezoelectric energy harvester, Nonlinear Dyn. 97 (2019) 2371–2397. [22] W. Deng, Y. Wang, Systematic parameter study of a nonlinear electromagnetic energy harvester with matched magnetic orientation: Numerical simulation and experimental investigation, Mech. Syst. Sig. Process. 85 (2017) 591–600. [23] W.J. Su, J. Zu, Y. Zhu, Design and development of a broadband magnet-induced dual-cantilever piezoelectric energy harvester, J. Intell. Mater. Syst. Struct. 25 (4) (2014) 430–432. [24] S.C. Stanton, C.C. McGehee, B.P. Mann, Nonlinear dynamics for broadband energy harvesting: investigation of bistable piezoelectric inertial generator, Phys. D 239 (2010) 640–653. [25] A. Abdelkefi, A.H. Nayfeh, M.R. Hajj, Global nonlinear distributed-parameter model of parametrically excited piezoelectric energy harvesters, Nonlinear Dyn. 67 (2012) 1147–1160. [26] ANSI/IEEE, ANSI/IEEE Std 176-1987: IEEE Stand on Piezoelectricity Institute for Electrical and Electronics Engineers, 1987. [27] Erturk A, Inman DJ. Piezoelectric Energy Harvesting. 2011 (Wiley: Chichester, UK).