Theoretical modeling and experimental verification of a broadband microvibrational energy harvesting system

To scavenge energy from imperceptible vibrations, this paper investigates the broadband response and output performance of a microvibrational piezoelectric energy harvesting system with mechanical stopper. The energy harvesting system comprises a cantilever beam made of piezoelectric material, which is affixed with a coil at its unbound end and a mechanical stopper. The coil is placed in a magnetic field to provide an ultra‐low level excitation. The electromechanical model is derived according to force integration method (FIM) and Hertz's contact theory, and numerical simulations are undertaken to evaluate the influence of the excitation level, and the gap on the performance. For the linear counterpart without stopper, experimental results indicate the system generates a peak power of 24.12 μW with matched resistance under excitation with a level of 0.003 N and a frequency of 200.3 Hz. When a polydimethylsiloxane (PDMS) stopper is introduced, the vibration of the piezoelectric beam exhibits an obvious nonlinearity with an amplitude of micron scale. Increasing the excitation level and decreasing the gap could efficiently broaden the response bandwidth. Experimental results demonstrate that a copper stopper with larger elastic modulus results in a wider response frequency range, and the half‐power bandwidth could reach 37.1 Hz under excitation with a level of 0.003 N.


| INTRODUCTION
In recent decades, the extensive application of wireless sensors and portable electronics has garnered significant research interest, and the proliferation of these electronic devices has captured a significant portion of the market's expansion. 1Although a tremendous advance has been made in lowering the power consumption of these electronics, the devices continue to rely on conventional batteries that necessitate frequent replacement or recharging. 2 Energy harvesting technology 3 is capable of transforming ambient energy into usable electrical power, 4 making it a potential alternative to batteries.This technology has gained considerable attention as a promising battery-free solution. 5To date, the scavenging of mechanical and vibrational kinetic energy has been achieved through the implementation of three prevailing energy conversion mechanisms, specifically, the piezoelectric, 6 electromagnetic, 7 and electrostatic 8 effects.Piezoelectric energy harvesters have gained considerable global attention as a result of their high energy density and ease of miniaturization in fabrication. 9Piezoelectric harvesters utilizing the linear resonance mechanism have been extensively studied in the past. 10However, the linear harvesters only perform well near the resonance frequency and little change in the frequency of vibration will greatly degrade the energy harvesting performance. 11he issue at hand has prompted a great deal of interest in the study of nonlinear vibrational harvesters, which can function over a broad spectrum of frequencies.Both theoretical analysis and empirical validation of such devices have received considerable attention. 12In particular, multistable oscillators 13 with different types of potential energy functions have been extensively investigated.As an example, Stanton et al. 14 introduced a monostable piezoelectric harvester with bidirectional hysteresis by incorporating a permanent magnet end mass, which interacted with oppositely poled stationary magnets.Neiss et al. 15 presented the analytical formulations for determining the jump-up and jump-down points, maximum power output under optimal resistive load, and 3 dB-bandwidth of a monostable piezoelectric harvester.Due to the large amplitude oscillation between two potential wells, bistable energy harvesters resulting from nonlinear and mechanical mechanisms have been widely investigated for broadband operation. 16Erturk et al. 17,18 presented a bistable piezomagnetoelastic device that aims to enhance piezoelectric power generation substantially.This mechanism exhibits superior performance over its linear counterpart, as evidenced by large-amplitude periodic oscillations observed across a wide frequency range.By coupling two rotatable magnets with the end magnet of a beam, Zhou et al. 19,20 proposed a novel bistable piezoelectric harvester and indicated the design could cover the broad low-frequency range of 4-22 Hz.Daqaq 21 investigated the response of bistable harvesters to white Gaussian excitations and noted that the superiority of bistable configurations over linear harvesters is dependent on the proper design of the potential function, which should be based on the known noise intensity.Exploring internal resonance, Fan delves into a U-shaped vibration-based energy harvester endowed with both broadband and bidirectional capabilities. 22Additionally, a wideband two-element piezoelectric energy harvester exhibiting both bistability and parametric resonance characteristics is presented. 23Furthermore, Fan introduced an internal resonance piezoelectric energy harvester featuring threedimensionally coupled bending and torsional modes. 24ore recently, Pan et al. 25 proposed a new concept by integrating bi-stability and swinging balls to harvest wind energy and experimental results indicated that the design achieved a significant improvement in electric outputs due to its ability to realize snap-through motions over a wide range of wind speed.To enhance the efficiency of energy harvesting from vibrations characterized by low excitation levels and a broad frequency spectrum, a variety of nonlinear configurations exhibiting tristable, [26][27][28] quad-stable, 29,30 and penta-stable 31,32 characteristics have been proposed and investigated theoretically and experimentally.
In addition to the magnetic coupling method mentioned above, introducing mechanical stoppers to the energy harvesting system to achieve broadband response has attracted significant attention. 33By positioning a stopper to a low-frequency cantilever nanogenerator, Song et al. 34 found that the bandwidth could be broadened by at least 200%.In their research, Zhou et al. 35,36 conducted a comparison of the efficacy of a piezoelectric energy harvesting device utilizing four different types of stoppers.Their objective was to determine the ideal impact configurations that would result in optimal performance.Their findings demonstrated that there were specific regions within the parameter space where the energy harvester could achieve an optimal balance between the bandwidth of the harvested energy and the average power generated.Instead of placing stoppers on the base, Hu et al. 37 proposed an impact engaged two degrees of freedom (DOF) piezoelectric harvester by positioning the stoppers onto the body with a primary DOF and theoretical investigation showed that the bandwidth and the opencircuit voltage could be enhanced by tuning the stopper distance.By combining magnetic coupling mechanism and mechanical stopper, the performance of piezoelectric harvesters can be enhanced more obviously.For instance, the researchers Fan et al. 38,39 presented a design for a monostable piezoelectric energy harvester that incorporates symmetric magnetic attraction to a cantilever beam and a pair of stoppers to restrict the maximum deflection of the beam.They demonstrated that this new design can produce a broader operating bandwidth and higher output voltage compared to the linear energy harvesting approach.The researchers, Wang et al. 40 introduced a novel and compact piezoelectric energy harvesting device featuring a hybrid nonlinear mechanism.This device was designed for use in condition monitoring systems for freight trains, with a specific focus on ultralow-frequency and broadband applications.Theoretical and experimental investigations demonstrated that this harvester could effectively operate within the frequency range of 1-11 Hz, thereby enabling it to power typical commercial wireless Bluetooth sensors.
Notably previous studies have been carried out under large excitations provided by the shaker to ensure that the amplitude of the response is relatively large, and the softening and hardening characteristics of the piezoelectric energy harvesters are more obvious. 41However, there are still many vibration sources with extremely weak amplitudes in real life. 42nder excitation of these imperceptible vibrations, little attention has been paid to the broadband response of the piezoelectric energy harvesting system, especially when the response amplitude is in the micron scale.Therefore, this paper investigates the broadband response and energy harvesting performance of a microvibrational piezoelectric energy harvesting system with mechanical stopper under excitation of imperceptible vibration.The proposed energy harvesting system is composed of a piezoelectric cantilever beam and a coil attached at the free end of the beam which is placed in a magnetic field to provide an ultra-low level excitation.Electromechanical model of the energy harvesting system is derived, following which numerical analysis is provided.Finally, experiments are undertaken, and the influence of the excitation level, the gap between the beam and the stopper, and the materials of the mechanical stopper are considered.
The subsequent sections of this paper are structured as follows: Section 2 provides a comprehensive explanation of the design, operational concept, and modeling of the microvibrational piezoelectric energy harvesting system.In Section 3, numerical investigations are conducted, and Section 4 outlines the experimental validation process.Lastly, the conclusions drawn from this study are presented in Section 5.

| Description and operating principle
Figure 1A illustrates the schematic diagram of the microvibrational energy harvesting system with a mechanical stopper for performance enhancement.It is composed of a piezoelectric cantilever beam, a coil attached at the free end, a mechanical stopper, and a pair of external electromagnets.The end coil is placed between the external electromagnets, and the electromagnets are powered by a DC source to provide a relatively uniform external magnetic field.On the contrary, an AC signal is sent to the coil, and an alternating magnetic force provided by the electromagnet will be exerted on the piezoelectric beam to drive it to oscillate.During the oscillation, the beam will impact with the mechanical stopper, thus nonlinearity is introduced to the system, and the output performance of the piezoelectric cantilever beam will be enhanced.When the cantilever beam is at rest, the gap between the beam and the mechanical stopper is denoted as d, and it is set to be adjustable as needed.Furthermore, the AC signal sent to the coil can also be adjusted in practical conditions to control the magnetic force exerted on the piezoelectric cantilever beam.
As noted, the electromagnets are applied to provide magnetic field in this paper, and the coil with current is utilized to produce alternating magnetic force.In practical application, permanent magnets can be used instead of electromagnets.As an alternative, the end coil can also be replaced by a permanent magnet, and then placed in an alternating magnetic field.Of course, there may be other combinations of coils, magnets, and electromagnets, and their ultimate goals are to provide alternating magnetic force exerted on the piezoelectric cantilever beam.

| Mathematical modeling
For the microvibrational energy harvesting system illustrated Figure 1, it can be simplified as a theoretical impact vibration model, which contains a cantilever beam and a stopper at the end of the beam.The piezoelectric beam in Figure 1 has a layered structure which is detailed shown in Figure 2A, which composes of a substrate and two piezoelectric layers.To model the transverse vibro-impact of the beam, a finite element model of a three-layer beam structure is proposed as shown in Figure 2B.In this particular model, the beam undergoes a process of discretization whereby it is divided into N distinct elements.The model employs a plane beam element that features two nodes.Each of the nodes is characterized by 2-DOF, namely a translational DOF and a rotational DOF.The force integration method (FIM) in the finite element framework can be used in the dynamic modeling.
The displacement vector δ e of each element is calculated as follows: where Y and θ are the translational and rotational displacement, respectively.A concentrated mass matrix M [ ] e of the beam element is written as follows.
where ρ and A represent the density, the cross-section area, respectively.It should also be noted that L p is the length of the piezoceramic layer and L b represents the length of the substrate layer.f 1 (l) is expressed as: but in practice, the actual vibrating beam length is l due to the clamping of the fixed end of the beam.Where N is shape function with ξ = x l .The stiffness matrix is expressed as 43 : where B is geometric function matrix of the element that is in the following form: D is the elastic matrix, which is E in the beam element problem in this paper.Where where y ˆis the distance from a point in the plane to the neutral axis of a global cross-section, I zA 1 , I ZA 2 , and I ZA 3 are the moments of inertia of A 1 , A 2 , and A 3 with respect to the neutral axis, respectively.

( )
, where represents the Young's modulus of the three layers, respectively.For the linear system without the stopper, the dynamic model after finite element discretization is expressed as Equation (10). where and C [ ] are the mass matrix, stiffness matrix, and Rayleigh damping matrix.δ { } is the displacement vector of the node, and P t ( ) is the vector for the excitation force vector.The assembling process of mass and stiffness matrices are shown in Figure 3.
The coil at the end of the beam is viewed as a concentrated mass applied to the corresponding location of the stiffness matrix of the beam.The excitation force is exerted at the end of the beam and it is in the y direction.The external excitation force vector can be calculated as follows.
The force is assumed to be exerted on the (2N − 3)th node.Furthermore, F e is the excitation force provided by the electromagnets.By applying the method of modal superposition, the first n modes are discretized and the displacement vector δ { } of the beam is expressed in Equation ( 12): F I G U R E 3 Assembling process of mass and stiffness matrices.
where η i is the generalized displacements, φ { } is the mode shape of the beam.By substituting Equation ( 12) into Equation (10), equation of motion about the generalized displacements can be obtained using the orthogonality of mode shapes.
where ξ i is the damping ratio, ω i is the ith natural frequency.In Equation ( 13), the damping term is given directly according to the set damping ratio ξ i .By solving Equation ( 13), the generalized displacements can be calculated.Subsequently, the displacement response of any node on the linear beam can be obtained by substituting the generalized displacements into Equation (13).For the piezoelectric cantilever beam, the electromechanical coupling characteristics should be considered.Therefore, according the Dai et al. 44 and Erturk and Inman, 45 the electromechanical model of the system with a mechanical stopper can be written as: where C P is the equivalent capacitance, θ i is the equivalent electromechanical coupling efficient, and R is the load resistance.C P and θ i can be identified by experiments.Furthermore, Q t ( ) is the impact force generated due to the impact of the beam by the stopper, which can be expressed as: where λ is the equivalent spring stiffness obtained from the Hertz contact model.It is assumed that the stopper is located at the Nth node on the beam, and δ (1) N represents the first freedom value of the Nth displacement vector, namely the lateral displacement of the point corresponding to the position of the stopper on the beam.The system parameters are as shown in the following table (Tables 1).

| NUMERICAL INVESTIGATION
Based on the mathematical model, numerical simulations are carried out, and the influence of the excitation level, and the gap between the beam and stopper on the energy harvesting performance is evaluated.The fourthorder Runge-Kutta algorithm is adopted to obtain the numerical results.During the simulation, the node displacement on the beam corresponding to the position of the stopper is monitored.Dichotomy method is applied to reduce the simulation step when the piezoelectric beam impacts with the stopper and disengages from the contact, to ensure the difference between the numerical result and the gap is less than the preset threshold.
Under forward-sweep frequency excitation with levels of 0.0012, 0.0018, 0.0022, 0.0026, and 0.0030 N, the open-circuit voltage response of the linear piezoelectric beam without stopper is illustrated in Figure 4A.At any excitation level, the output voltage first increases and then decreases, and obtains a maximum value at the frequency of 198.5 Hz.The peak voltages are respectively 1.78, 2.67, 3.26, 3.86, and 4.45 V for excitation levels of 0.0012, 0.0018, 0.0022, 0.0026, and 0.0030 N. In general, the output performance of the linear piezoelectric beam without stopper could be enhanced by increasing the excitation level.Under reverse-sweep frequency excitation, the open-circuit voltage response of the linear piezoelectric beam at various excitation levels are depicted in Figure 4B and there is almost no changes in the peak voltages compared with the results under forward-sweep frequency excitation.However, the frequency, at which the peak voltages are obtained, is 197.5 Hz and it is a little smaller compared with that under forward-sweep frequency excitation.
When the mechanical stopper is introduced to the system, the voltage response of the system under excitation level of 0.0030 N is applied to evaluate the influence of the gap between the beam and the stopper.For the gap equaling 0 μm, the response of the system under forward-sweep frequency excitation, shown in Figure 4C, indicates that the output voltage is relatively low and achieves a peak value at the frequency of 216.7 Hz.Compared with the results of the linear counterpart, there is an obvious shift in the frequency at which the peak voltage is obtained.On the contrary, the voltage response of the system with the gap equaling 140 μm is basically consistent with the output of the linear counterpart.On this occasion, the displacement response amplitude of the end of the cantilever beam is smaller than the gap between the beam and the stopper, and the response of the system is linear.
As the gap between the beam and stopper decreases to 120 μm, the voltage response under forward-sweep frequency excitation is slightly different from that of linear counterpart, and the system exhibits weak nonlinearity.Meanwhile, the displacement response of the end of the cantilever beam is close to the gap between the beam and stopper.Therefore, it can be reached that obvious nonlinearities will be introduced into the system by further decreasing the gap between the beam and stopper.For the gap equaling 100, 80, 60, 40, and 20 μm, the voltage response under forward-sweep frequency excitation, illustrated in Figure 4C, reveals that the decreasing of the gap could broaden the response frequency range, and the corresponding half-power bandwidth are 6.2, 7.9, 10.5, 15.From the numerical simulations, it is observed that the displacement response of the piezoelectric beam is in the micron scale, and the half-power bandwidth of the system can be broadened by decreasing the gap between the beam and stopper.Due to the decreasing of the gap, the response amplitude of the cantilever beam is limited, and the peak voltage exhibits a decreasing trend.To validate the numerical results calculated from the electromechanical model, experiments of the microvibrational piezoelectric energy harvesting system with mechanical stopper are carried out under different conditions, and the experiment setup is illustrated in Figure 5.In the experiment, a DC power supply is applied to provide current for the electromagnets, and an AC signal generated by a function generator (Agilent 33120 A) is sent to the coil to realize alternating magnetic force excitation.The amplitude of the magnetic force is determined by applying the Wheatstone bridge, and the output voltage of the piezoelectric beam is acquired by a data acquisition card (NI-USB-6259) and then processed by a computer.The piezoelectric beam used in the experiments is made of PZT with the dimension of 32 mm × 7.2 mm × 0.83 mm, and the coils applied has an external diameter of 8 mm, an inner diameter of 4 mm, and a thickness of 3 mm.Furthermore, the piezoelectric beam is fixed by a 3D printing fixture, and a precision sliding table with a positioning accuracy of 20 μm is fixed to the bottom of the 3D printing fixture to adjust the distance between the piezoelectric beam and the stopper.

| Calibration of magnetic force
Wheatstone bridge has a high sensitivity in resistance measurement, and the unbalanced bridge is commonly utilized to measure the variation in resistance.Wheatstone bridge can be categorized into single-bridge, halfbridge, and full-bridge, among which the full-bridge has the highest sensitivity.Therefore, the full-bridge circuit combined with resistive strain gauges is applied in this paper to calibrate the magnetic force exerted on the piezoelectric cantilever beam.To further improve the sensitivity of measurement, the piezoelectric cantilever beam is replaced by a plastic sheet with a small elastic modulus for calibration.The benefit of this is to make the cantilever beam deformable more easily under ultra-low electromagnetic excitation, thus causing the change in the resistance of the resistance strain gauge.
In the experiment, four resistive strain gauges (R 1 , R 2 , R 3 , and R 4 ) with the same size are applied as bridge arms and attached to the upper and lower surfaces at the root of a cantilever beam, as shown in Figure 6A.A source meter (Keithley 2400) provides the operating voltage to the Wheatstone bridge circuit, and the nano-voltmeter (Keith-ley2182A) measures the voltage between the parallel arms.Figure 6B shows the circuit diagram of the Wheatstone bridge where R 1 , R 2 , R 3 , and R 4 are the resistances of the four resistive strain gauges and they have the same value of R. The resistances of the strain gauges depend on the deformation of the cantilever beam, and the voltage between the parallel arms can be expressed as following.
where U is the voltage provided by the source meter.When a mass with known weight is placed at the end of the cantilever beam, the beam will bend and cause the change in the strain gauges.Then, the voltage between the parallel arms is calculated as Equation (18).
where R Δ i represents the change in the resistance of each strain gauge.For the reason that the deformation of the cantilever beam is small, the resulting change in resistance is also very tiny.Therefore, Equation ( 18) can be simplified as the following.
In small elastic deformation, the relative change in the resistance of the strain gauge is positively correlated with strain, as in Equation (20).
where K is the sensitivity coefficient of the resistance strain gauge, and ε is the strain at the root of the cantilever beam.In this condition, Equation ( 20) can be further simplified as following.
Accordingly, we can linearly convert the strain signal into voltage signal through the Wheatstone bridge.For the cantilever beam, the strain under the action of the mass block can be expressed as: where m is the mass of the block, g is the acceleration due to gravity, L is the distance between the strain gauge and the mass block, E is elastic modulus of cantilever beam, b is the width of the cantilever beam, and h is the thickness.According to Equations ( 21) and ( 22), the mass m is also linear with the voltage variation.Therefore, the relationship between the mass and the voltage can be obtained, as illustrated in Figure 7A.
When the coil at the free end of the cantilever beam is placed in the magnetic field, the deformation is stimulated by the electromagnetic force.In the experiment, the magnetic field at the position of the coil is measured with a Gauss meter (F.W.Ell 5180, F.W.ELL, USA).By changing the output current of DC power supply, the magnetic force exerted on the beam through the electromagnet can be adjusted.Then, the relationship between the magnetic field at the position of the coil and the electromagnetic force can be obtained according to the relationship between the voltage variation and the mass, as depicted in Figure 7B.Therefore, the electromagnetic force exerted on the cantilever beam can be calibrated, and it can be adjusted in the following experiments.For excitation levels of 0.0012, 0.0018, 0.0022, 0.0026, and 0.003 N, the open-circuit voltage response of the linear piezoelectric cantilever beam under forward sweep frequency excitations are illustrated in Figure 8A.At a certain excitation level, the response voltage first increases and then decreases with an increase in the excitation frequency, and achieves the maximum value near the resonance frequency.And the obtained maximum voltage for excitation levels of 0.0012, 0.0018, 0.0022, 0.0026, and 0.003 N are respectively, 1.43, 2.10, 2.62, 3.22, and 3.97 V at the frequencies of 202.8, 202.6, 202.1, 201.7, and 201 Hz.Obviously, the increase in the excitation level could enhance the output performance of the linear piezoelectric beam.To be noted, the frequency, corresponding to the maximum voltage, decreases with an increase in the excitation level.The reason for this phenomenon may be that the direction of the electromagnetic excitation in the experiment was not completely perpendicular to the piezoelectric beam, and there was a small component in the axis of the beam.Thus, the axial preload will decrease the resonant frequency.Under reverse sweep frequency excitation, the voltage response of the linear piezoelectric beam under various excitation levels is shown in Figure 8B.When the excitation frequencies are respectively 201. 6, 201.3, 200.9, 200.6, and 200.4 Hz, the piezoelectric beam under excitation levels of 0.0012, 0.0018, 0.0022, 0.0026, and 0.003 N achieve the maximum voltages and they are 1.46, 2.12, 2.64, 3.29, and 4.01 V, respectively.It can be observed that the output performance of the piezoelectric beam under the reverse sweep frequency excitation is consistent with that under the forward sweep frequency excitation, and there is only a little difference in the increasing trend near the resonance frequency.
Under excitation with a level of 0.003 N and a frequency of 200.3 Hz, the voltage response of the linear piezoelectric beam and corresponding frequency spectrum are, respectively, shown in Figure 9A,B.The achieved maximum voltage is 3.75 V, and the energy in the frequency spectrum mainly distributes at the frequency of 200.3 Hz, which is consistent with the excitation frequency.Obviously, the response of the piezoelectric beam exhibits a perfect periodic vibrational characteristic under electromagnetic excitation with ultra-low levels.
Additionally, different load resistances are connected to the circuit to investigate its influence on the energy harvesting performance of the linear piezoelectric beam.In the experiment, a parametric sweep was conducted for different resistance values ranging from 100 Ω to 900 KΩ.The measurements were performed under excitation with a frequency of 200.3 Hz and an amplitude of 0.003 N. Figure 10A,B, respectively, illustrate the variation of peak output voltage and maximum output power with the change of load resistance.With the increase of load resistance, the voltage gradually increases to an asymptotic value.While, the output power increases to a maximum value and then starts to decrease to an asymptotic value with an increase in the load resistance.The phenomenon indicates that the optimum load resistance of the piezoelectric beam is 50 kΩ and the obtained maximum output power is 24.12 μW.
Furthermore, the influence of the end mass and its position on the output performance of the linear piezoelectric beam was investigated.In the experiment, the end mass was provided by mass blocks with the same volume and masses of 0.1, 0.3, and 0.7 g, respectively.When the mass blocks were pasted at a position 20 mm away from the fixed end of the piezoelectric beam, the open-circuit voltage response of the system under forward sweep frequency | 2545 excitation with a level of 0.003 N is illustrated in Figure 11A.For the end mass of 0.1 g, the maximum voltage is 3.36 V and obtained at the frequency of 198 Hz.As the end mass increases to 0.3 and 0.7 g, the maximum voltage are, respectively, 3.4 V at the frequency of 193 Hz, and 3.0 V at the frequency of 181.9 Hz.Under reverse sweep frequency excitation, the voltage response of the system with different end masses is shown in Figure 11B.When the end mass are 0.1, 0.3 and 0.7 g, the obtained peak voltages are, respectively, 3.38, 3.42, and 3.0 V at the frequencies of 196.9, 192.1, and 181.1 Hz.In summary, the increase of the end mass is an effective method to reduce the resonant frequency of the system, which can be applied to adapt to different vibration frequencies.
To investigate the influence of the position of end mass on the energy harvesting performance, the end mass of 0.7 g was applied, and it was pasted at the position 10, 15, and 20 mm away from the fixed end of the piezoelectric beam.Under forward-sweep frequency excitation with a level of 0.003 N, the voltage response of the linear piezoelectric beam is exhibited in Figure 11C.With the end mass at the position 10, 15, and 20 mm away from the fixed end, the system achieved the peak voltages of 3.45, 3.32, and 2.97 V, respectively, at the resonance frequency of 199.7, 193.7, and 181.9 Hz.Under reverse-sweep frequency excitation, the maximum voltages are respectively obtained at 198.5, 192.8, and 181.2 Hz with the values of 3.44, 3.3, and 2.99 V, as shown in Figure 11D.From the results, it can be observed that the closer the end mass is to the free end of the piezoelectric cantilever, the lower is the resonance frequency of the system.By reasonably adjusting the end mass and its position on the linear piezoelectric beam, the system can be utilized to generate considerable electrical energy under excitation with different frequency.

| Response of nonlinear system with a mechanical stopper
In the experiment, the hemispherical mechanical stopper is made from PDMS, and had the diameter of 5 mm and elastic modulus of 1 MPa.When the gap between the beam and the stopper is 40 μm, the response of the nonlinear system is investigated under various excitation levels, and Figure 12A illustrates the open-circuit voltage response under forward sweep frequency excitation.When the excitation level is 0.0012 N, the response amplitude of the beam is smaller than 40 μm, and the response exhibits the same characteristics as the linear system, with a maximum output voltage of 1.32 V generated.For excitation levels of 0.0018, 0.0022, 0.0026, and 0.003 N, the maximum output voltages of the system are, respectively, 1.64, 1.73, 1.78, and 1.79 V, which are smaller than that of the linear piezoelectric beam.The reason for this phenomenon may be that the introduction of the stopper limits the response amplitude of the beam.Although, the response frequency range of the system is broadened, and the half-power bandwidth at excitation levels of 0.0018, 0.0022, 0.0026, and 0.003 N are, respectively, 4.3, 5.9, 7.7, and 9.9 Hz.In comparison, the halfpower bandwidth of the linear counterpart under excitation with various levels is about 3.0 Hz.Therefore, it can be conclude that the introduction of the mechanical stopper can widen the response frequency range of the system, and this phenomenon is even more pronounced with the increase of excitation level.Under reverse-sweep frequency excitation with different levels, the voltage response of the system is exhibited in Figure 12B, and the variation trend is almost consistent with the response under forward-sweep frequency excitation.To be noted, the jump phenomenon from the high-energy orbit to low-energy orbit under forward-sweep frequency excitation is more pronounced that the jump phenomenon from the low-energy orbit to high-energy orbit under reverse-sweep frequency excitation.
Under excitation with a level of 0.003 N, the influence of the gap between the beam and the stopper on the performance of the system is investigated.Under forward sweep frequency excitation, the open-circuit voltage response of the system with different gaps between the beam and the stopper is illustrated in Figure 13A.When the gap between the beam and the stopper is 140 μm, the voltage response is basically consistent with the response of the linear system, and the voltage reaches the maximum value (3.84 V) at the frequency of 200.9 Hz.In this case, the gap between the beam and the stopper is larger than the response amplitude of the beam, and there is no impact between the PDMS stopper and the beam.As the gap between the beam and the stopper decreases to 120 μm, there is a weak impact between the beam and the stopper, and the maximum voltage is achieved at the frequency of 201.1 Hz.As the gap further decreases to 100 μm, the voltage response of the piezoelectric beam exhibits obvious nonlinear characteristics.Before the beam impacts with the PDMS stopper, the voltage response exhibits linear characteristics and is similar to the results of linear counterpart.The impact between the beam and the stopper occurs at the frequency of 199.3 Hz, and the peak voltage shows an increasing trend as the frequency continues to increase, with a maximum voltage of 3.14 V achieved at the frequency of 201.7 Hz.After the frequency of 203.5 Hz, the impact phenomenon no longer occurs, and the response again exhibits linear characteristics.Furthermore, the halfpower bandwidth on this occasion is 5.8 Hz.As the gap decreases to 80, 60, 40, and 20 μm, the response amplitude of the beam and voltage is limited, and the maximum voltage are respectively 2.70, 2.10, 1.51, and 1.17 response range can be broadened with a decrease of the gap between the beam and the stopper, with a smaller maximum voltage generated.
In addition, the material of the mechanical stopper on the response voltage and half-power bandwidth of the system is studied in the experiment.A new mechanical stopper made from copper (elasticity modulus: about 100 GPa) is applied and it has the same dimension with the PDMS stopper utilized before.Herein, the experiments are carried out under sweep frequency excitation with a level of 0.003 N, and the gap between the beam and the stopper with the value of 40, 60, 80, and 100 μm is emphasized.When the gap between the beam and the stopper is 40 μm, the open-circuit voltage response of the system under forward and reverse sweep frequency excitation is illustrated in Figure 14A.Before the impact between the beam and the stopper happens, the voltage response is consistent with the results of linear counterpart.For forward-sweep frequency excitation, the impact occurs at the frequency of 192.2 Hz and terminates at the frequency of 229.2 Hz.A maximum voltage of 1.56 V is generated, and the half-power bandwidth is about 37.1 Hz.While, the frequency range where the impact occurs is from 192.2 to 202.2 Hz under reverse sweep frequency excitation.Regarding the output performance, the obtained maximum voltage is 1.42 V and the half-power bandwidth is 12.4 Hz.Compared to the results utilizing the PDMS stopper, it is observed that the copper stopper results in a much wider half-power bandwidth under forward-sweep frequency excitation, and this phenomenon may be contributed to the larger elastic modulus of the copper stopper.
With an increase in the gap between the beam and the stopper, the impact between them begins to occur at higher frequencies and ends up at lower frequencies under forward sweep frequency excitation.This phenomenon contributes to that the half-power bandwidth becomes narrower and they are respectively 31.9, 24, and 11.5 Hz when the gap is 60, 80, and 100 μm, as shown in Figure 14B-D.To be noted, the increase in the gap leads to an increase in the response amplitude and the maximum voltages under forward sweep frequency excitation are, respectively, 2.05, 2.64, and 3.0 V for gaps of 60, 80, and 100 μm.Under reverse sweep frequency excitation, the half-power bandwidth also shows a narrowing trend with an increase in the gap between the beam and the stopper, with the values of 8.4, 6.5, and 5.2 Hz for gaps of 60, 80, and 100 μm.And the corresponding maximum voltage are, respectively, 2.03, 2.45, and 2.95 V.In general, the increase in the elastic modulus of the mechanical stopper could broaden the response frequency range of the system, and the overall performance depends significantly on the gaps between the beam and the stopper.| 2549

| Comparison of energy harvesting devices
To compare performances of other energy harvesting devices and the energy harvesting device based on impact vibration proposed in this paper, a comparative table is given above.The device in this paper has advantages in frequency bandwidth and power density.Subsequently, higher output power density can be achieved by optimizing the volume and shape of the fixture Tables 2.

| CONCLUSION
In this work, a microvibrational piezoelectric energy harvesting system with mechanical stopper for performance enhancement has been designed, modeled, and investigated numerically and experimentally.The confirmation is composed of a piezoelectric cantilever beam, a mechanical stopper, and a coil attached at the free end of the beam which is placed in a magnetic field to provide an ultra-low level excitation.A theoretical model of the system has been established, and numerical simulations indicates that the increase in excitation level has a positive effect on the output and the nonlinearity resulted from the introduction of the mechanical stopper could broaden the half-power bandwidth.
In the experiments, the magnetic force in the excitation is calibrated through the Wheatstone bridge.Experimental results of the linear piezoelectric beam are consistent with the numerical simulation, and it is demonstrate that the vibration of the beam is in the micron scale.Under excitation with a level of 0.003 N and a frequency of 200.3 Hz, a peak output power of 24.12 μW is achieved with a matched load resistance of 50 KΩ.Furthermore, experiments verify that the frequency range for large-amplitude oscillation can be controlled by adjusting the end mass and its position on the beam.
When a PDMS stopper with an elastic modulus of about 1 MPa is utilized as the stopper, the vibration of the piezoelectric beam exhibits strong nonlinearity.Experimental results under different excitation conditions validate that the response bandwidth of the system can be broadened by increasing the excitation level and decreasing the gap between the beam and stopper.To be noted, the decrease in the gap limits the response amplitude of the beam and leads to a smaller peak voltage.To further investigate the material of the mechanical stopper on energy harvesting performance, a copper stopper is utilized to replace the PDMS stopper and it has a larger elastic modulus of 100 GPa.Experimental results under forward sweep frequency excitation with a level of 0.003 N reveal that the larger elastic modulus results in a wider response frequency range, and the half-power bandwidth could reach 37.1 Hz when the gap between the beam and the stopper is 40 μm.In general, the results provided in this paper can be referenced for the design of precision sensors, elastic modulus identification, surface flatness testing, and so on.

F I G U R E 1
Schematic diagram of the microvibrational energy harvesting system.
3, and 20.7 Hz, respectively.And the peak voltages are 3.0, 2.55, 2.04, 1.48, and 1.0 V, respectively, at the excitation frequencies of 200.2, 201.1, 202.4,205, and 208.2 Hz for the gap equaling 100, 80, 60, 40, and 20 μm.It can be observed that, due to the decreasing of the gap, the peak voltages are achieved at higher frequencies and show an obvious decreasing trend.Under reverse-sweep frequency excitation, the open-circuit voltage response of the system for various gaps are illustrated in Figure 4D, and the changing trend of voltage is almost same as that under forwardsweep frequency excitation.For the gap equaling 100, 80, 60, 40, and 20 μm, the half-power bandwidth are 6.4,8, 10.4,14.4, and 19.8 Hz, respectively.

F I G U R E 4
Numerical results of the linear and nonlinear system under sweep frequency excitation: (A) linear system, forward-sweep; (B) linear system, reverse-sweep; (C) nonlinear system, forward-sweep; and (D) nonlinear system, reverse-sweep.VALIDATION 4.1 | Experimental setup

F I G U R E 6
Magnetic force calibration: (A) schematic diagram of measurement principle; (B) circuit diagram for the Wheatstone bridge.

F I G U R E 7
Magnetic force calibration: (A) relationship between mass and voltage variation; (B) relationship between magnetic field and electromagnetic force.F I G U R E 8 Voltage response of the linear system under forward (A) and reverse (B) sweep frequency excitation with various levels.

F I G U R E 9
Voltage response and frequency spectrum at the excitation of 200.3 Hz and 0.003 N. F I G U R E 10 Influence of external load resistance on the peak voltage and maximum power at the excitation of 200.3 Hz and 0.003 N. WEI ET AL.

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Influence of end mass on the performance: (A) forward and (B) reverse sweep frequency excitation.Influence of end mass's position on the performance: (C) forward and (D) reverse sweep frequency excitation.F I G U R E 12 Influence of excitation level on the performance with a gap of 40 μm: (A) forward and (B) reverse sweep frequency excitation.
V at the frequencies of 202, 202.4,203.4,and 205.4 Hz.However, the half-power bandwidth increases with a decrease in the gap, and it is, respectively, 7.2, 10, 14.4, and 18.2 Hz for the gaps of 80, 60, 40, and 20 μm.Under reverse sweep frequency excitation, the voltage response of the system is illustrated in Figure 13B.It is seen that the response under reverse-sweep frequency excitation is similar to that under forward-sweep frequency excitation, while the jump phenomenon is not obvious.For gaps of 100, 80, 60, 40, and 20 μm, the maximum voltage are respectively achieved at the frequency of 201.2, 202, 202.3, 204.1, and 206.2 Hz, with the values of 3.2, 2.65, 2.14, 1.44, and 0.91 V. To the noted, the half-power bandwidth are respectively, 6.3, 7.9, 10.3, 15.8, and 25.1 Hz.From the experimental results, it is concluded that the frequency F I G U R E 13 Influence of the gap between the beam and stopper on the performance at an excitation level of 0.003 N: (A) forward and (B) reverse sweep frequency excitation.F I G U R E 14 Influence of the material of stopper of the stopper: (A) 40 μm; (B) 60 μm; (C) 80 μm; (D) 100 μm.
Physical properties of the structures.
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