High-performance low-frequency bistable vibration energy harvesting plate with tip mass blocks

Energy 180 (2019) 737e750

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High-performance low-frequency bistable vibration energy harvesting plate with tip mass blocks Yi Li a, Shengxi Zhou a, *, Zhichun Yang a, Tong Guo a, Xutao Mei b a b

School of Aeronautics, Northwestern Polytechnical University, Xi’an, 710072, China Institute of Industrial Science, The University of Tokyo, Tokyo, 153-8505, Japan

a r t i c l e i n f o

a b s t r a c t

Article history: Received 30 November 2018 Received in revised form 26 April 2019 Accepted 1 May 2019 Available online 8 May 2019

Bistable vibration energy harvesting plate owns large-amplitude output voltage and high output power because of its large asymmetric deformation. In order to enhance low-frequency energy harvesting, this paper designs a high-performance low-frequency bistable vibration energy harvesting plate with tip mass blocks. The bistable substrate is made from carbon fiber/epoxy pre-preg with an asymmetric stacking sequence and heat treatment. The dynamic response and the snap-through behavior of the presented energy harvesting plate are analyzed by the Finite Element Method (FEM). The effect of uncertainties induced by errors in manufacturing process on the open circuit voltage is studied by interval analysis. Basing on these information from the finite element analysis, the complete experiment is conducted, in which the similar dynamic response and snap-through behavior are observed. And more importantly, the broadband characteristic is experimentally obtained. Meanwhile, the experimental results show that, the presented energy harvesting plate can produce the output power higher than 1 mW, which is enough to power some wireless sensors embedded in smart infrastructures used for monitoring structural health. © 2019 Elsevier Ltd. All rights reserved.

Keywords: Piezoelectric energy harvesting Nonlinear Bistable plate Finite element analysis Low-frequency

1. Introduction Piezoelectric vibration energy harvesting becomes a research hotspot because of its great application potential for powering wireless sensors and other small embedded microelectromechanical devices [1e3]. Based on different structural designs, a series of vibration energy harvesters were performed and verified to be efficient to work in harmonic base excitations [4,5], harmonic random excitations [6,7], rotating excitations [8,9], vortex-induced excitations [10e12], galloping [13,14], human motion excitations [15,16], heartbeat excitations [17], etc. Their theoretical models have been proposed to describe the dynamic characteristics and predict the output power [2]. In order to improve broadband energy harvesting performance, nonlinear energy harvesters have been presented based on external magnetic force [18e20], internal resonance mechanism [21e23], geometric nonlinearity [24,25], electric circuit [26,27], etc. As one of the most efficient nonlinear energy harvesters, the bistable energy harvesting plate (BEHP) works based on morphing

* Corresponding author. E-mail address: [email protected] (S. Zhou). https://doi.org/10.1016/j.energy.2019.05.002 0360-5442/© 2019 Elsevier Ltd. All rights reserved.

or adaptive structure concepts [28e30]. Traditionally, the substrate layer of the BEHP is made of metal material based on a special thermal treatment. Piezoelectrical materials are usually attached to the center part of the substrate layer, where the stress is large. When the BEHP is forced to continuously vibrate across two stable states, the attached piezoelectrical materials will generate large electrical energy because of the large shape change [30,31]. Thus, the BEHP with nonlinear characteristics has a high level of the output power over a wide excitation frequency range. More importantly, such harvester has a great application potential in complex environment, such as inside aircrafts. Arrieta et al. [31] designed a BEHP and made experimental investigations of its snap-through behavior, and they demonstrated that actuation can be induced by vibrating the plates at their natural frequencies. Thus, the broadband energy harvesting performance was realized. Meanwhile, Betts et al. [32] experimentally investigated the surface profiles of a series of arbitrary layup asymmetric laminates. The results shown that the existing modelling methods are suitable for accurately predicting roomtemperature shapes. Based on this research, Betts et al. [33] designed a BEHP combining the bistable composites and piezoelectric materials for broadband vibration energy harvesting. They

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Nomenclature BEHP BEHPT

Bistable Energy Harvesting Plate Bistable Energy Harvesting Plate with Tip Mass Blocks FEM Finite Element Method MFC Macro Fiber Composite PZT Piezoelectric Ceramic Transducer NLGEOM NL stands for nonlinear, GEOM stands for geometry PIA Parameterized Interval Analysis MEMS Micro-Electro-Mechanical Systems

demonstrated that the BEHP could be designed to occupy smaller physical space than the corresponding magnetic cantilever based energy harvesting systems. Betts et al. [34] and Syta et al. [35] experimentally investigated the nonlinear dynamic characteristics of the BEHP based on time-domain outputs, frequency spectrum, bifurcation diagrams, phase portraits, and 0e1 test. Kuder et al. [36] presented the optimal positioning of the bistable structures while assessing the energy required for morphing under aerodynamic loading. The concurrent design and the optimization framework can be further used to develop bistable energy harvesters and actuators by taking the maximum advantage of the available stiffness variability of the bistable laminates. Lee et al. [37] experimentally characterized the vibration energy harvesting performance of the multifunctional bistable laminate to enable morphing. They observed snap-through phenomenon in intermittencies and subharmonic, chaotic, and limit cycle oscillations, which may benefit high-efficiency vibration energy harvesting. Although the excellent energy harvesting performance of the BEHP was numerically and experimentally verified, there is relatively less research work of the BEHP compared with that of cantilever beam based energy harvesters. In order to improve energy harvesting performance in lowfrequency vibration environments, this paper designs a highperformance low-frequency bistable vibration energy harvesting plate with tip mass blocks (BEHPT) and its energy harvesting performance is verified theoretically and experimentally. In Section 2, the design concept and a prototype of the BEHPT are provided.

Meanwhile, the processing mechanism to fabricate bistable composite plates is introduced. In Section 3, the main theoretical equations for building the dynamic model of the BEHPT via FEM are formulated. Afterwards, numerical simulations are performed to predict the vibration energy harvesting performance of the BEHPT. In Section 4, experimental setup and tests are presented to verify the BEHPT design. Finally, a summary of the presented studies is given in the concluding section. 2. Design and processing mechanism 2.1. Design concept and prototype The structure diagram of the BEHPT is shown in Fig. 1. The bistable substrate layer is made of the composite materials with reasonable residual thermal stress, which will be explained later. Two piezoelectric patches are attached on the central part of the substrate layer, and they are subjected for parallel connection in this study. Different from existing BEHPs, four pairs of identical tip mass blocks are installed at the four corners of the BEHPT to lower its working frequency range. As the side view shown in Fig. 1 (a), the central position of the BEHPT will be connected with a vibration exciter to get external excitations. As shown in Fig. 1 (b), 3D view is provided to clearly show the BEHPT. In addition, Fig 2 shows the prototype of the fabricated BEHPT which is used in experimental tests 2.2. Processing mechanism of bistable composite plate It is well known that, bistable composite plate is made based on asymmetrical carbon fiber reinforced plastic laminates [38]. For the composite laminate with an asymmetrical stacking sequence, the thermal stresses produced during the curing process can cause an out of plane curvature. This deformation can be maintained without any external load, thus, it is recognized as one stable equilibrium position of composite plate. Then external forces can make it snap through to another stable equilibrium position. The stacking sequence and the ply thickness will be adjusted to make the composite laminate have more than one stable equilibrium position [39,40]. These stable equilibrium positions can be held without an applied load. One stable equilibrium position can be changed into another stable equilibrium position as long as the critical snap-through load is applied. The required snap-through

Fig. 1. The structural diagram of the BEHPT: (a) Side view; (b) 3D view.

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3. Numerical simulation and uncertain analysis In this paper, the FEM [38,40,42] is employed to predict the stable equilibrium positions of the BEHPT and simulate the snapthrough phenomenon between two stable states. As shown in Fig. 1, the dynamic model of the BEHPT under harmonic base excitations can be built basing on the FEM. Firstly, the piezoelectric constitutive equations [1] should be used to describe the relationship between the electrical and mechanical properties of the piezoelectric patches, as follows:




T D






cE e

¼

e εS


 S E

(1)

where S is strain. T is stress. E is the intensity of electric field. D is electric displacement. e is the piezoelectric coefficient. cE is the elastic stiffness constant under a constant electric field. εS is the dielectric constant under the constant strain. The finite element model governing the mechanical response of the BEHPT can be expressed by:

Fig. 2. Prototype of the fabricated BEHPT.

load can be calculated by using the semi-analytical approach [39] or finite element method (FEM) [38,41] latter. Carbon fiber/epoxy pre-preg is used to stack a rectangle laminate (150
149 mm2) asymmetrically (00 =900 ). This square laminate of two layers is heated up to about 125  C in a vacuum autoclave, and then it is cooled down to the room temperature (about 20  C). Consequently, the square laminate is deformed into a shell with a curvature, and it has two stable equilibrium positions in experiments. When a concentrated force is applied on the central points of straight sides, another stable equilibrium position can be attained by snap-through buckling. The above process is simulated by the FEM, the S4R element in ABAQUS software is used to model the composite laminate. The mechanical properties of the material are listed in Table 1. The boundary condition is depicted in Fig. 3. For the processing mechanism simulation, in the first step, the temperature field is imposed on all the shell elements, and the temperature is decreased from 125  C to 20  C. Due to an asymmetrical stacking sequence of the composite laminate, the thermal stresses produced during the cooling process consequently make the square plate converge into a stable cylinder shape at the end of this step, as the whole process shown in Fig. 3. In the second step, the room temperature (20  C) is held and a small concentrated force (3 N) is applied on the central line (points E and F) of the upper and lower straight sides. At the end of the second step, the composite plate converges into the stable equilibrium position B, as shown in Fig. 3. When the concentrated force is acted on the points M and N, respectively, the bistable composite plate can snap-through back to the stable equilibrium position A.

€ þ C u_ þ Ku ¼ FðtÞ Mu

(2)

where M is the mass matrix. C is damping matrix. K is stiffness € are time, matrix and FðtÞ is equivalent force vector. t, u, u_ and u displacement, velocity and acceleration vectors, respectively. In ABAQUS software, the implicit method is available for time integration of the general equation of motion include the operator defined by Hibbitt [43] and Hilber et al. [44] and the backward Euler operator. The Hilber-Hughes-Taylor operator definition is completed based on the Newmark formulae for displacement and velocity integration: nþ1 K nþ1 ¼ Rnþ1 eff u eff

utþDt ¼ ut þ Dt u_ t þ

(3) 

 1 € t þ bDt 2 u € tþDt  b Dt 2 u 2

(4)

€ t þ gDt u € tþDt u_ tþDt ¼ u_ t þ ð1  gÞDt u

(5)

where


K nþ1 eff ¼

1 g C þK Mþ bDt bDt 2

nþ1 þ ðM þ C gDtÞ Rnþ1 eff ¼ F

 (6)


   1 1 n 2 €n _n þ  u þ D t u b D t u 2 bDt 2

n

€   C½u_ n þ ð1  gÞDt u (7) The parameter a is used to control the damping. Meanwhile, the following inequations should be met:

Table 1 Mechanical properties of the substrate layer (bistable composite plate). Value

Tensile modulus E1 Tensile modulus E2 Poisson’s ratio n12 Shear modulus G12 Shear modulus G13 Shear modulus G23 Thermal expansion coefficient a1

144 GPa 9.6 GPa 0.303 4.2 GPa 4.2 GPa 3.366 GPa

Thermal expansion coefficient a2 Thickness tply Density r

1 4

(8)

1 2

(9)

b ¼ ð1  aÞ2

Property

 1
107 = C 105 = C

1:9
0.14 mm

1600 Kg =m3

g¼  a 1  a0 3

(10)

If a ¼ 0, there is no damping and the operator is the trapezoidal rule. For a ¼  13, a significant damping level is obtained.

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Fig. 3. Curing process and transformation between two stable equilibrium positions.

Based on above equations and the FEM, the numerical analyses will be conducted by using the FEM. The modeling and simulation will be performed via ABAQUS software. The S4R element is used to model the BEHPT. The mechanical properties of the substrate layer and piezoelectric materials are listed in Tables 1 and 2, respectively. The C3D8RE element with an extra degree of freedom for electrical potential is used to simulate the Macro Fiber Composite (MFC). According to the IEEE Standard [45], the constitutive relations of the piezoelectric energy harvesting device are typically defined by equation (1), and the material properties of the MFC are given in Table 2. Two MFCs are bonded perfectly on the central area of the bistable composite plate. The bonded surfaces are modeled by the Tie constraint in ABAQUS software. The lumped elements located in the four corners of the bistable composite plate are used to model the metal blocks. The mass of each lumped element is 0.004 Kg which is the same in the consequential experiments. The finite element model is shown in Fig. 4. Two step analysis in ABAQUS are conducted, in the first step an uniform temperature (125  C) field is applied to the composite plate, and is decreased to 20  C, the central node of composite plate is fixed. The NLGEOM (nonlinear

Table 2 Material properties of the MFC. Property

Value

Tensile modulus E1 Tensile modulus E2 Tensile modulus E3 Poisson’s ratio n12 Shear modulus G12 Piezoelectric constant d33

30.3 GPa 15.9 GPa 15.9 GPa 0.303 5.5 GPa

Piezoelectric constant d31 Dielectric constant ε Density r

4:0
1010 C=N  1:7


1010

xI ¼ ½x; x

(11)

C=N

1:31
108 F=m 5440 Kg =m3

geometry) is activated to account for geometric nonlinearity and obtain the accurate deformation, and the corresponding deformed mesh (the stable equilibrium position A) is exported, and then imported into a new model, tie the C3D8RE elements (are used to simulate the MFC), four lumped elements are connected on each corner of the composite plate separately, consequently the finite element model for the second step analysis is built. In the second step, the freedom degrees of the central nodes are fixed except for the Z direction, and the excitation is applied along the Z direction. The residual stress field from the first step is propagated into the composite shell elements. The electric potential, displacement responses are calculated by the implicit dynamic analysis in ABAQUS. Due to errors in manufacturing process, the mechanic properties of composite and mass of metal block are uncertain variable, and these uncertainties can alter the electric potential responses of the BEHPT, thus we should quantify the effects of uncertainties before carrying out the experiment. According to the researches [46], the mechanic properties of composite are random variable, and their bounds generally deviate from the design value by 10e15%. In this paper, we use interval number to express these uncertainties, listed in Table 3 (these mechanic properties with the strong influence on the bending deformation of composite plate are considered), and use the improved interval extension based on the 1st order Taylor series and parameter [47e49] to predict the bounds of voltage responses of the BEHPT. And this method will be introduced in detail below. The uncertainties in energy harvesting system can be described by the interval variable.

where x and x define the lower and upper bounds of interval, respectively. And its central value and deviation:

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Fig. 4. Finite element model of the BEHPT at the stable equilibrium position B.

Variables

Value

Lower bound

Upper bound

the central values) with respect to the interval variable. We substitute Eqs. (13) and (14) into Eq. (16) to build a continuous function with the 1st order independent variables:

Tensile modulus E1 Tensile modulus E2 Thickness tply Mass of metal block

144 GPa 9.6 GPa 0.14 mm 0:004Kg

129.6 GPa 8.64 GPa 0.123 mm 0.0036 Kg

158.4 GPa 10.56 GPa 0.157 mm 0.0044 Kg

n   X   vVðXc Þ

V X I z V aj ¼ VðXc Þ þ Dxi sinaj vx i i¼1

Table 3 Uncertain variables in the BEHPT.

(17)

Eq. (17) can be rewritten:

xc ¼

xþx 2

Dx ¼

(12)

xx 2

(13)

According to the Parameterized Interval Analysis (PIA) [48,49], an interval xI can be represented by its central value and deviation, i.e. Eq. (11) can be rewritten as:

xI ¼ xc þ Dxsina

(14)

h p pi a2  ;

(15)

2 2

The 1st order Tay lor expansion of voltage VðX I Þ around the central value of uncertainties Xc is: n    X vVðXc Þ  I xi  xic V X I z VðXc Þ þ vxi i¼1

vV ðXc Þ vxi

n  X  vVðXc Þ

Dxi sinaj G aj ¼ vx i i¼1

(18)

(19)

where j is defined according to the number of categories of uncertainties, the same type of uncertain variables should be expressed by a single a. In this paper, a1 is used to express the uncertain mechanic properties of composite plate, a2 is used to account for the uncertain mass of metal block. And then the bounds of voltage or displacement VðX I Þ can be obtained by maximizing and minimizing this continuous function Gðaj Þ, as shown in Eqs. (20) and (21).

    V X I ¼ VðXc Þ þ minimum G aj

(20)

   V X I ¼ VðXc Þ þ maximum G aj

(21)

(16)

where. X I ¼ xI1 ; xI2 ; …; xIn ; Xc ¼ x1c ; x2c ; …; xnc : In above equation, VðXc Þ is the voltage response at the central values of intervals.

    V X I z V aj ¼ VðXc Þ þ G aj

is the 1st order sensitivity of response (at

Meanwhile, the 1st order sensitivities can be calculated by the finite difference method.

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Fig. 5. Measurement locations of composite plate after curing process.

Table 4 Displacement of composite plate after curing process and uncertain analysis. Displacement H

Value /mm

e

e

Measurement

17.55 Central value /mm 15.81

e Lower bound /mm 14.01

e Upper bound /mm 17.61

e Simulation

V ’ ðxÞ ¼

Vðx þ hÞ  Vðx  hÞ 2h

(22)

where h is called the finite-difference interval. Before carrying out dynamic response analysis, the finite element model of bistable composite plate should be verified at first. We calculate the displacement of the corner point relative to the central point according to the deformation field from the first step analysis, and the bounds are also predicted by the improved interval extension (only considering the uncertain material properties of composite in Table 3), and compare them with the measured values from the composite test piece cured in autoclave, as shown in Fig. 5. The detail values are listed in Table 4, the measure value is dropped within the interval of displacement from the static analysis and interval extension. It is demonstrated that

the initial stiffness induced by residual thermal stress and the geometrical shape of stable equilibrium position A can be simulated reasonably by the finite element model in this paper. Basing on this finite element model, the electric potential, displacement responses calculated in the second step analysis are reliable, and the results from the second step analysis can be used to guide the experiment, at same time the uncertain analysis should be conducted to investigate the effects of uncertain parameters listed in Table 3 on the open circuit voltage of the BEHPT. Secondly, the open circuit voltage responses of the BEHPT with uncertainties listed in Table 3 are predicted by the implicit dynamic finite element analysis and the improved interval extension, while the frequencies of external excitation (2.6 g) are 8 Hz, 12 Hz, 14 Hz and 16 Hz. The bounds of the minimum and maximum of the stable open circuit voltage responses are given at the cases (8 Hz and 12 Hz), as shown in Figs. 6 and 7. Interestingly, we found out that tiny change (0.1%) of the modules E1 can make the BEHPT occur snap through deformation and go into the vibration state between two stable equilibrium positions from the vibration state of oscillating in one stable equilibrium position, when frequency of the excitation is 14 Hz, as shown in Fig. 8. And the slight change (0.2%) of the modules E1 can make the shapes of voltage response curves alter dramatically, when the BEHPT snapping through and back between two stable equilibriums positons (frequency of the excitation is 16 Hz), as shown in Fig. 9. However, when the BEHPT

Fig. 6. Numerical open circuit output voltage and its bounds of the maximum and minimum of stable response (frequency of excitation is 8 Hz).

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Fig. 7. Numerical open circuit output voltage and its bounds of the maximum and minimum of stable response (frequency of excitation is 12 Hz).

vibrates in one stable equilibrium position, the shape of voltage response curve has slightly change (frequency of the excitation is 8 Hz), as shown in Fig. 10. After nonlinear static analysis (the first step), implicit dynamic analysis (the second step) and uncertain analysis, the finite element model of the BEHPT is reasonable and can be used to obtain the information about excitation and response to guide the experiment. Consequently, the dynamic behavior and electric characteristic of the BEHPT at the excitations with different frequencies are investigated basing on the finite element model. It is well known that nonlinear systems have multi-solution ranges, and the highenergy oscillations benefit for vibration energy harvesting [50,51]. In simulations, the excitation level is set as 2.6 g (g is gravitational acceleration). Displacement and velocity of the location where the lumped mass elements are located (node S) along the z direction are calculated by the implicit dynamic analysis. It should note that

the BEHPT is subjected to open-circuit conditions in simulations. The numerical output voltage and phase trajectory are plotted in Figs. 11e14. When the excitation frequency is 8 Hz or 12 Hz, the BEHPT vibrates around one stable equilibrium position, and the output voltage is no more than 20 V, as shown in Fig. 11. The corresponding phase trajectories shown in the second plot of both Fig. 11 (a) and (b) demonstrate that the final stable vibration orbit is relatively small. When the excitation frequency is increased to 14 Hz, the BEHPT makes snap-through oscillations between the two stable equilibrium positions at the beginning (0~1s), and the peak output voltage reaches 73 V as shown in Fig. 12 (a). At last, the BEHPT converges to the vibration around the stable equilibrium position A, and the steady-state output voltage amplitude is 12.2 V. However, when the excitation frequency is further increased to 16 Hz, the BEHPT continuously make snap-through oscillations, and the output voltage amplitude is as large as 72.2 V, as show in Fig. 12

Fig. 8. Numerical open circuit output voltages, when modules E1 is changed by 0.1% (frequency of excitation is 14 Hz).

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Fig. 9. Numerical open circuit output voltages, when modules E1 is changed by 0.2% (frequency of excitation is 16 Hz).

Fig. 10. Numerical open circuit output voltages, when modules E1 is changed by 0.2% (frequency of excitation is 8 Hz).

Fig. 11. Numerical output voltage and phase trajectory of the BEHPT at: (a) 8 Hz; (b) 12 Hz.

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Fig. 12. Numerical output voltage and phase trajectory of the BEHPT at: (a) 14 Hz; (b) 16 Hz.

(b). Meanwhile, Fig 13 shows the transformation between two stable equilibrium positions and the output voltage for the excitation frequency being increased from the 14 Hz to the 16 Hz. Fig. 14 compares the output voltage and the phase trajectory from the BEHPT and the traditional BEHP at the excitation frequency of 18 Hz. For the BEHPT, the output voltage amplitude is larger than 65 V, while the output voltage amplitude of the traditional BEHP without tip mass blocks is smaller than 2 V. The phase trajectory of the former is also much larger than that of the latter. This demonstrates the BEHPT has better energy harvesting performance under low-frequency excitations.

hysteresis phenomenon of the BEHPT becomes more obvious, as the results shown in Fig. 17. The total effective frequency bandwidth is increased to 8.2 Hz, and the maximum output power is increased to 1.96 mW. When the excitation level is further increased to 1.8 g, the nonlinear hysteresis phenomenon of the BEHPT becomes much more obvious than above other two cases, as the results shown in Fig. 18. The total effective frequency bandwidth is increased to 9.3 Hz, and the maximum output power is larger than 2 mW. From 10.7 to 20 Hz, the output power of the BEHPT under up-sweep excitation is larger than 0.1 mW at the excitation level of 1.8 g. Meanwhile, there is an additional effective frequency range around

4. Experimental verification Fig. 15 shows the overall view of experimental setup. The BEHPT is directly connected with a vibration exciter. The data collection system is employed to get experimental data. The substrate layer is made of the carbon fiber/epoxy pre-preg with the size of 150
149
0:28 mm3. Two prepackaged piezoe-lectric transducers (M-2814-P2) are used to generate electrical energy because of stress-strain effect. Eight annular NdFeB magnets (the weight of each magnet is 0.002 Kg, thus, each tip mass block is 0.004 Kg) are used as the edge mass blocks for improving the energy harvesting performance from low-frequency excitations. The optimal load resistance is about 262 kU and this value is used in experiments. Linearly increasing frequency sweep excitation (up-sweep) and decreasing frequency sweep excitation (down-sweep) experiments are performed over the excitation frequency range of 5e20 Hz for qualitatively exploring the energy harvesting performance of the BEHPT. The base acceleration amplitude values of 1 g, 1.5 g and 1.8 g are selected as the three excitation levels for swept-frequency experiments, and the corresponding results are shown in Figs. 16e18.  2 As shown in Fig. 16, the maximum output power (P ¼ pVffiffiffi ) 2R

of the BEHPT at the excitation level of 1 g is 1.7 mW. In the frequency range of 11.5e15.1 Hz, the output power of the BEHPT under both up-sweep and down-sweep excitations is larger than 0.1 mW, which is large enough to power some small wireless sensors and micro-electro-mechanical systems. In this frequency range, different with the down-sweep excitation, the output voltage of the BEHPT under up-sweep excitation has an additional effective frequency range of 9.0e9.5 Hz. This is a common nonlinear hysteresis phenomenon that the nonlinear system will vibrate in different orbits in the multi-solution range under different initial conditions. When the excitation level is increased to 1.5 g, the nonlinear

Fig. 13. Output voltage for the excitation frequency being increased from the 14 Hz to the 16 Hz.

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Fig. 14. Comparison of output voltage and phase trajectory: (a) The BEHPT; (b) the traditional BEHP without tip mass blocks.

Fig. 15. Experimental setup: (a) Overview of main experimental devices; (b) data collection system.

8.7 Hz. Above experimental results demonstrate that the presented BEHPT has broadband characteristics under low-frequency excitations. It is well known that the dynamic response of nonlinear systems is more complex than that of linear systems [40,42]. Fast Fourier Transform (FFT) can be used to analyze the characteristics of nonlinear responses. Meanwhile, nonlinear energy harvesters need a larger excitation level for achieving the high-energy vibration

branch under the constant frequency excitations. Therefore, 2.6 g is selected as the excitation level for the following experiments under the constant frequency excitations, as the experimental results shown in Figs. 19e22. At the 8 Hz excitation, the time-domain output voltage is obviously asymmetric because of the asymmetric oscillation of the BEHPT, as shown in Fig. 19 (a). In the spectrum of the output voltage, the first (8 Hz) and second (16 Hz) harmonics are found in the output voltage, as shown in Fig. 19 (b).

Fig. 16. Experimental results of the BEHPT under up-sweep and down-seep at 1 g: (a) Output voltage; (b) output power.

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Fig. 17. Experimental results of the BEHPT under up-sweep and down-seep at 1.5 g: (a) Output voltage; (b) output power.

Fig. 18. Experimental results of the BEHPT under up-sweep and down-seep at 1.8 g: (a) Output voltage; (b) output power.

An interesting phenomenon is that there are a series of different frequency components of 92 Hz, 100 Hz, 108 Hz, and 116 Hz. The frequencies of these harmonics are not the integer multiples of the excitation frequency, while the frequency interval is the excitation

frequency. The amplitude of the 108 Hz harmonic is highest in the spectrum. When the excitation frequency is increased to 12 Hz, the output voltage becomes almost symmetric, as shown in Fig. 20. The

Fig. 19. Experimental results under the constant excitation frequency of 8 Hz: (a) Output voltage; (b) spectrum.

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Fig. 20. Experimental results under the constant excitation frequency of 12 Hz: (a) Output voltage; (b) spectrum.

amplitude of the first harmonic of the output voltage is highest in the spectrum. Meanwhile, the harmonics (88 Hz, 100 Hz, 112 Hz, 124 Hz) with the frequency interval of 16 Hz are still found in the spectrum. As shown in Fig. 21, the response of the BEHPT at the 14 Hz excitation is similar with that from the 12 Hz excitation. At the 16 Hz excitation, the output voltage becomes smaller but still symmetric, as shown in Fig. 22. The amplitude of the first harmonic is highest in the spectrum. The harmonics (84 Hz, 100 Hz, 116 Hz, 132 Hz) with the frequency interval of 16 Hz are still in the spectrum. Therefore, the harmonic with the frequency of 100 Hz and a series of side frequency harmonics with the frequency interval of 16 Hz (the excitation frequency) are found in the output voltage of the BEHPT. Overall, the high-performance energy harvesting and broadband characteristics of the presented BEHPT are numerically and experimentally verified. The nonlinear response characteristics are analyzed based on spectrum analysis. Fig. 21. Experimental results under the constant excitation frequency of 14 Hz: (a) Output voltage; (b) spectrum.

Fig. 22. Experimental results under the constant excitation frequency of 16 Hz: (a) Output voltage; (b) spectrum.

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5. Conclusions This paper originally presents a high-performance bistable vibration energy harvesting plate with tip mass blocks for enhanced low-frequency energy harvesting. Carbon fiber/epoxy pre-preg is used to stack a rectangle laminate which is used as the substrate layer based on asymmetric stacking sequence and heat treatment. Through the implicit method, the backward Euler operator and the Newmark formulae, the dynamic response and the snap-through behavior of the presented energy harvesting plate are analyzed via the finite element method. Uncertain analysis is conducted to investigate the effect of uncertain parameters in the BEHPT on the open circuit voltage and dynamic behavior, especially it’s found out that a slight change of mechanic properties can active the BEHPT to snap through and back between two stable equilibrium positions, when the frequency and amplitude of external excitation is close to the ones that can induce the snap-through behavior. And when the BEHPT vibrate between two stable equilibrium positions, the dynamic response is more sensitive to the mechanic properties than the case of vibrating in one stable equilibrium position. In experiments, the effective working frequency range of 10.7e20 Hz is observed, which verifies the broadband characteristic of the presented energy harvesting plate. More importantly, the presented energy harvesting plate can produce an output power larger than 1 mW, which can be used to power some wireless sensors embedded in smart infrastructure used for monitoring structural health. The future work will focus on investigating the complete nonlinear dynamic mechanism and the optimization design for maximizing the output power. Acknowledgements This project has been supported by the National Natural Science Foundation of China (Grant Nos. 51606171 and 11802237), and the Fundamental Research Funds for the Central Universities (Grant No. G2018KY0306). References [1] Erturk A, Inman DJ. Piezoelectric energy harvesting. UK: John Wiley & Sons; 2011. [2] Yang Z, Zhou S, Zu J, Inman DJ. High-performance piezoelectric energy harvesters and their applications. Joule 2018;2:642e97. [3] Madinei H, Haddad Khodaparast H, Friswell MI, Adhikari S. Minimising the effects of manufacturing uncertainties in MEMS Energy harvesters. Energy 2018;149:990e9. [4] Zhou S, Cao J, Inman DJ, Lin J, Liu S, Wang Z. Broadband tristable energy harvester: modeling and experiment verification. Appl Energy 2014;133: 33e9. [5] Fan K, Liu S, Liu H, Zhu Y, Wang W, Zhang D. Scavenging energy from ultralow frequency mechanical excitations through a bi-directional hybrid energy harvester. Appl Energy 2018;216:8e20. [6] Litak G, Friswell MI, Adhikari S. Magnetopiezoelastic energy harvesting driven by random excitations. Appl Phys Lett 2010;96(21):214103. [7] Zhou Z, Qin W, Zhu P. Harvesting acoustic energy by coherence resonance of a bi-stable piezoelectric harvester. Energy 2017;126:527e34. [8] Fu H, Yeatman EM. Rotational energy harvesting using bi-stability and frequency up-conversion for low-power sensing applications: Theoretical modelling and experimental validation. Mech Syst Signal Pr; DOI: https://doi. org/10.1016/j.ymssp.2018.04.043. [9] Zhang Y, Zheng R, Shimono K, Kaizuka T, Nakano K. Effectiveness testing of a piezoelectric energy harvester for an automobile wheel using stochastic resonance. Sensors 2016;16(10):1727. [10] Zhang L, Dai H, Abdelkefi A, Wang L. Experimental investigation of aerodynamic energy harvester with different interference cylinder cross-sections. Energy 2019;167:970e81. [11] Zhang B, Song B, Mao Z, Tian W, Li B. Numerical investigation on VIV energy harvesting of bluff bodies with different cross sections in tandem arrangement. Energy 2017;133:723e36. [12] Shan X, Song R, Liu B, Xie T. Novel energy harvesting: a macro fiber composite piezoelectric energy harvester in the water vortex. Ceram Int 2015;41:

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