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A frequency and bandwidth tunable piezoelectric vibration energy harvester using multiple nonlinear techniques
Applied Energy 190 (2017) 368–375
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Applied Energy journal homepage: www.elsevier.com/locate/apenergy
A frequency and bandwidth tunable piezoelectric vibration energy harvester using multiple nonlinear techniques Xiang Wang a,b, Changsong Chen b, Na Wang b, Haisheng San b,⇑, Yuxi Yu c, Einar Halvorsen d, Xuyuan Chen d a
Department of Mechanical and Electrical Engineering, Xiamen University, Xiamen 361005, China Pen-Tung Sah Institute of Micro-Nano Science and Technology, Xiamen University, Xiamen 361005, Fujian, China c College of Materials, Xiamen University, Xiamen 361005, Fujian, China d Department of Micro- and Nanosystem Technology, University College of Southeast Norway, Borre 3184, Norway b
h i g h l i g h t s An ultra-wide bandwidth vibration energy harvester was built with tunable frequency and bandwidth. Multiple nonlinear tuning mechanisms were realized by tuning the gap between spring-plate and stopper. The theory models and numerical simulations were presented with experimental verification.
a r t i c l e
i n f o
Article history: Received 12 September 2016 Received in revised form 22 December 2016 Accepted 31 December 2016
Keywords: Nonlinear vibration energy harvester Piezoelectric Tunable frequency and bandwidth
a b s t r a c t This article presents a compact piezoelectric vibration energy harvester (VEH) using multiple nonlinear techniques to tuning the resonant frequency and broadening the bandwidth. The device was designed as a parallel-plate structure consisting of a suspended spring-plate with proof mass and piezoelectric transducers, a tunable stopper-plate, and supporting frames. By mechanical adjusting the vertical gap between the spring-plate and the stopper-plate (GBSS), the VEH can realize tuning of the resonant frequency and the bandwidth by multiple nonlinear effects. Experimentally, the piezoelectric VEH was assembled as a metal prototype that can be operated in three kinds of work states corresponding to the configurations of large-GBSS, small-GBSS, and over-GBSS. The sweeping-frequency measurement results show that the work frequency, bandwidth, and output-voltage of VEH depend on tuning of GBSS and excitation levels, indicating that the multiple nonlinear effects, such as Duffing-spring effect, impact effect, preload effect, and air elastic effect, have significant influence on the dynamic behaviors of VEH. The comparisons of numerical simulations with the experimental results were used to verify the validity of mathematical modeling on VEH with multiple nonlinear tuning techniques. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction The rapid growth of wireless sensor networks and mobile electronic platforms have led to the developments of self-powered devices using energy harvesting techniques. Energy harvesting is a method to generate electrical power from external energy sources such as solar, thermal, wind, vibration, radio frequency (RF) energy, human body heat, and human movements, and so on. Among these energies, vibration-based energy harvesting has received significant attention in recent years. As a typical example of wasted energy that may be harvested, vibrations can be available easily in our surroundings, such as different types of commer⇑ Corresponding author. E-mail address: [email protected] (H. San). http://dx.doi.org/10.1016/j.apenergy.2016.12.168 0306-2619/Ó 2017 Elsevier Ltd. All rights reserved.
cial and industrial machines, vehicles, buildings, various structures (bridges, railways), home appliances, and motion of biological systems [1]. The idea of vibration-to-electricity conversion was proposed by Williams and Yates in 1996 [2]. From then on, vibration energy harvesting has been an attractive technique for charging or powering low-power micro-electronics [3]. Typical vibration energy harvesters (VEHs) may be viewed as a simple oscillator (mass-spring system), excited by an external vibration source, with one electromechanical transducer used for vibration-to-electricity conversion [4]. The principal transduction mechanisms in VEHs are piezoelectric, electromagnetic, and electrostatic transductions [5]. While each of techniques mentioned above can provide a useful amount of energy, the piezoelectric technique has gained most attention due to their abilities to directly convert applied strain energy into usable electric energy
X. Wang et al. / Applied Energy 190 (2017) 368–375
and the ease to be integrated into a system [6]. From the power output viewpoint, actually the piezoelectric materials were not popular as an energy harvester due to its small power output. However, modern electronic devices are getting tiny and consume less power than before. Therefore the most recent developments of piezoelectric applications are related to energy harvesting [7]. Regardless of the transduction mechanisms, a primary issue in VEHs is that the maximum energy output usually occurs in its fundamental resonance frequency. The power output will be decreased significantly when the resonant frequency of VEHs do not matches the ambient vibration, this will seriously restrains the applications of VEHs [8]. There are two possible methods to optimize the output power under random vibration source: tuning the resonant frequency and broadening the bandwidth [9]. In principle, frequency tuning can be realized by controlling the mass and the spring constant of oscillator [7]. Some possible methods are: adding a tip mass [10], moving the center of gravity [11], changing the dimension [12] or shape of cantilever [13], or altering the stiffness [14] or damping [15] of the harvester. As it is impractical to change the dimensions or shape of cantilever, most of methods are to change the mass, damping or the stiffness by manual or autonomous tuning [7]. According to the practicable methods in application, our literature investigation on the frequency tuning focus on four kinds of tuning strategies: resonant frequency tuning, multimodal energy harvesting, nonlinear energy harvesting, and electrical damping tuning [16]. Resonant frequency tuning can be based on active or passive actuators for tuning frequency [17]. The active tuning actuators provide a continuous force proportional to displacement, acceleration, or velocity to alter effective stiffness, effective mass, or effective damping accordingly in a mass-spring system [17]. In contrast, the passive tuning actuators exert force only during altering the resonance frequency and no force occurs while maintaining the new resonance frequency [18–20]. Regardless of active and passive actuators, they need external energy (that is often larger than the harvesting energy) to generate the applied forces [17]. The multimodal energy harvesting is a potential effective-bandwidthincreasing approach. In practice, multimodal systems can be realized by exploiting multiple bending modes of a continuous beam or by an array of cantilevers [21,22]. The multimodal VEHs can achieve an energy harvesting by a multi-frequency resonance with an add advantage of requiring no tuning and thus the external energy. However, the effective bandwidth of multimodal systems are usually discrete, which may only be helpful when the vibration source has a rather wide frequency spectrum [16]. Furthermore, in order to avoid the mode-shape-dependent voltage cancelation in a continuous beam or the voltage cancelation caused by the phase difference between cantilevers in array configurations, multimodal VEHs usually requires complex interface circuit [16]. Nonlinear energy harvesting has shown a tremendous potential for the design of broadband VEHs in recent years. Nonlinear configurations for broadband energy harvesting focus on the generation of nonlinear force by a Duffing-type oscillator or a piecewise-linear impact oscillator. Duffing-type oscillators include monostable [23], bistable [24], and tristable [25] nonlinear configurations with nonlinear stiffness typically introduced by using magnets. Such a mechanism is applied to perturb and drive the system from a low-energy vibration branch into a high-energy orbit, resulting in a much higher output power. Piecewise-linear impact is another type of nonlinear vibration which can be introduced by adding a mechanical stopper to make a piecewise linear stiffness in vibration system [26]. It was found that the tuning configuration using mechanical stopper generates jump phenomenon and results in the increase of bandwidth during frequency sweep. However, the bandwidth is increased at the expense of lowering output power in comparison with the configurations without mechanical stopper
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[27]. The frequency tuning using electrical damping was investigated previously by Muriuki [15]. It is found that the stiffness and natural frequency of the beam can be adjusted when a capacitor is added in parallel to the piezoelectric element. Electrical damping can be varied by adjusting the electrical load. As the resistive load reduces the efficiency of power transfer while the inductive load is difficult to be varied, the varying of a capacitive load is preferable [7]. It is noted that above-mentioned broadband harvesting techniques are only preferable in specific conditions. Anyway the most suitable method for frequency tuning depends on the intended application. For example, the car-tire vibration spectrum spreads from 300 Hz to 700 Hz [28] and the railway tunnels vibration spectrum is distributed from 200 Hz to 600 Hz and 1200 Hz to 2000 Hz [29]. In these vibration environments, the wireless sensor nodes (WSN) powered by VEHs with a single adjusting configuration cannot be sufficient to tune over such a wide frequency range. Therefore the VEHs using multiple nonlinear tuning mechanisms in a device are considered to be a suitable solution due to its large tunability over work frequency range. According to the requirements of frequency matching over a wide frequency range, VEHs with multiple nonlinear tuning mechanisms are capable of tuning the resonant frequency with the accompaniment of broadening the bandwidth. Up to now there are few reports on the VEHs using multiple nonlinear tuning mechanisms. In this work, we present a compact piezoelectric VEH using multiple nonlinear tuning mechanisms to tune the resonant frequency and broaden the bandwidth synchronously. The device was designed and fabricated as a metal prototype with compact parallel-plate structure consisting of supporting frames, a suspended spring-plate with proof mass, a tunable stopper-plate. The piezoelectric sheet was integrated on one of the beams of spring-plate. This device can be assembled and adjusted according to the requirements of flexible study options. The multiple nonlinear effects, including the Duffing-spring effect, impact effect, preload effect, and air elastic effect can be achieved in this device by mechanical tuning the stopper-plate. Furthermore, this device has high reliability and repairability due to the high-reliable and interchangeable metal parts and the amplitude-limiting stopper, making this device a strong shock-resistibility for practical engineering applications.
2. Design and modeling of the VEH The targeted piezoelectric VEH is designed as a parallel-plate spring-mass system as schematically shown in Fig. 1(a), which consists of a suspension spring-plate with a proof mass, a rigid stopper-plate above the spring-plate, three structural frames, and connecting bolts. The spring-plate is designed as a hooked-crossshaped suspension configuration with four identical doubly clamped L-type beams (see Fig. 1(a)) by hollow-cutting a 0.2 mm thick and 45 mm long square bronze plate, such design brings about a compact structure as well as a nonlinear Duffing-spring effect. A 0.4 mm thick, 3 mm wide, and 24 mm long piezoelectric ceramic sheet is attached to the top surface of the L-shaped suspension beam. The prototype can be assembled by connecting all parts with four bolts, and the vertical gap between the springplate and the stopper-plate (GBSS) can be changed by adjusting the number of spacers in the bottom center of the stopper-plate. By varying the GBSS, the VEH can be operated in three kinds of work states, namely large-GBSS, small-GBSS, and over-GBSS, as shown in Fig. 1(b–d). The electromechanical model corresponding to the case of small-GBSS is schematically shown in Fig. 1(e). In this model, the proof mass M oscillates with the relative displacement z(t); Km,
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Fig. 1. (a) Schematic configuration of a tunable piezoelectric VEH. The work states of the VEH with (b) large-GBSS, (c) small-GBSS, and (d) over-GBSS. (e) Electromechanical model corresponding to the case of small-GBSS.
and bm represent the stiffness and damping constant of oscillator, respectively. The stopper-plate is shown as a parallel springdamping system with stiffness constant Ks and damping constant bs [30]. The attached PZT sheet generates a piezoelectric voltage V(t), which drives the external load R. From Fig. 1(e), the lumped-parameter equations of the electromechanical model can be described as:
(
mzðtÞ þ bm zðtÞ þ F r þ HVðtÞ ¼ mzb ðtÞ HzðtÞ ¼ 0 C pe VðtÞ þ VðtÞ R
;
ð1Þ
where zb(t) is the displacement of vibration station, H is the piezoelectric coupling coefficients and Cpe is the effective capacitance between the piezoelectric electrodes. Fr is the total restoring force which can be expressed as Fr = Fk + Fs. Fk is the elastic restoring force arised from the spring-plate and Fs is the impact force arised from the stopper-plate. Based on the spring-damping system, Fs can be expressed as:
FS ¼
K S ðz z0 Þ bS z_ if z P z0 0
if z 6 z0
;
ð2Þ
where z0 is the initial value of GBSS according to the origin of coordinate in the top surface of mass. By means of the finite-element method (FEM), we can calculate the dependence of elastic restoring force of spring-plate on displacement of proof mass. Fig. 2(a) shows the COMSOL Multiphysics FEM model of the spring-plate with specific dimension parameters. H1 is the predeflection of the spring-plate induced by the vertical preload. We simulated the dependence of the restoring force F on displacement z when VEH is in the case of large-GBSS, small-GBSS, and overGBSS, respectively. In the case of large-GBSS, z0 is set sufficiently large so that the proof mass never hit the stopper-plate during free vibration. As shown in Fig. 2(b), the spring-plate with H1 = 0 mm (without pre-deflection) displays a linear mechanical behavior with a constant stiffness kl = 611 N/m from 0.1 mm to 0.1 mm (see the inset of Fig. 2(b)). Beyond this small range of displacement, the spring-plate exhibits a nonlinear mechanical behavior with hardening characteristic in the positive and negative directions. The FEM simulation results are fitted using an odd polynomial function, the elastic restoring force Fk can be expressed as:
F r ¼ 607:2709z þ 3:8127 108 z3
ð3Þ
When the vertical gap z0 is further adjusted to a small value, namely, the case of small-GBSS, the spring-plate would hit the stopper-plate within its displacement over the applied frequency range. As the effective mass of the stopper-plate is negligible in comparison with the proof mass, the stopper-plate vibrates with the spring-mass system together during the impact, in which the effective restoring force of system is substituted by impact force Fs. As shown in Fig. 2(c), the restoring force abruptly varies from Fk to Fs at the impact point z = 0.025 mm (see the inset of Fig. 2 (c)), thereafter the nonlinear vibrations are introduced by impact force Fs. Fitting the FEM curve by an odd polynomial function, the total restoring force of spring-plate with small-GBSS can be expressed as:
(
Fr ¼
2193:31z þ 3:8127 108 z3
if z 6 0:025 103
3
50000 ðz 0:025 10 Þ if z P 0:025 103
ð4Þ
It should be noted in Eq. (4) that the linear function in the case of z P 0.025 mm is obtained by fitting the experimental data based on the impact model given by Eq. (2). If the GBSS is adjusted to a negative value by increasing the number of spacers in the stopper-plate, a tensile stress induced by the pre-deflection will be exerted on the spring-plate when all parts were assembled together. This means that the spring-plate is in the case of over-GBSS. As shown in Fig. 2(d), with an initial pre-deflection of H1 = 0.005 mm in the spring-plate, the simulated elastic restoring force Fk exhibits a nonlinear dynamic behavior with bistable characteristic. If the spring-plate is located an unstable position corresponding to a negative stiffness from 0.03 mm to 0.03 mm (see Fig. 2(d)), it will spontaneously move into the stable equilibrium position towards positive or negative direction. On account of the restriction of stopper-plate, the spring-plate would hit the stopper at pre-deflected position (z = 0.005 mm) when the spring-plate move towards the positive stable equilibrium position. As a result, the spring-plate was brought into a piecewise linear stiffness region. The total restoring force of spring-plate with over-GBSS can be expressed as:
(
Fr ¼
4008:23z þ 1:7638 1012 z3 50000 ðz þ 0:005 103 Þ
if z 6 0:005 103 if z P 0:005 103 ð5Þ
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Fig. 2. FEM model of the spring-plate (a) and the simulated dependence of the restoring force F on displacement z when VEH is in the case of (b) large-GBSS, (c) small-GBSS, and (d) over-GBSS, respectively.
By observing the coefficient of the first term of polynomial function in Eq. (5), it is found that the stiffness of spring-plate with over-GBSS is twice as large as that with small-GBSS. This implies that the principal resonance frequency of VEH with over-GBSS pffiffiffi should be 2 times as large as that with small-GBSS. Substituting Eqs. (3), (4) and (5) into Eq. (1), respectively, and solving these differential equations using MATLAB/Simulink models, we can simulate the acceleration-dependent of dynamic output characteristics of VEHs.
output port of piezoelectric transducer, and the output voltage (VR) across the resistor was measured through a high-impedance buffer amplifier. The output voltages of VEH prototype and accelerometer were read by an data acquisition board (NI-USB-6211 DAQ) and LabView program of National Instruments. The first step in the measuring sequences is to experimentally determine the optimal load-resistance for maximum power output. A tunable resistor was connected between the electrodes of the piezoelectric transducer. The value Rl of the resistor and voltage VR across the load were measured. The instantaneous power delivered by the energy harvester can be obtained according to
3. Experimental results and discussion
P ¼ V 2R =Rl . Though the optimal load resistance can be found by identifying the value at maximum output power, the optimal load resistance changes depending on the work state of device and its excitation frequency. Considering the multiple work states of VEH in practical application, it is reasonable to evaluate the device properties using a fixed optimal load for all work states. We stepped the load resistance and measured the output voltage for
Experimentally, the VEH prototype (see Fig. 3(a)) was mounted onto the vibration station (TIRA vibration exciter) as shown in Fig. 3(b). The shaker was driven using a power amplifier with output of DC-biased sinusoidal wave. An accelerometer (Piezotronics Inc. model 352A56A) was attached to VEH prototype for measuring the acceleration of variation. A load-resistor was connected to the
Fig. 3. (a) Photographs of the spring-plate of the VEH prototype and (b) experimental setup.
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the VEH with large-GBSS in its resonant frequency. Fig. 4 shows dependences of the output voltage and power on load-resistance. The optimal load-resistance was approximately 45 kX. Next, the instantaneously generated voltage over the load-resistor were measured by a chirp excitation with a linear sweep of 25 Hz/s from 0 to 20 s. Fig. 5 shows the time domain output waveforms of VEH prototypes in three kinds of work states at acceleration of 1.5 g. The maximum peak-to-peak voltages are approximately 2.449 V, 2.278 V, and 0.733 V for the cases of large-GBSS, small-GBSS, and over-GBSS at an acceleration of 1.5 g, respectively. It can be seen that the waveforms of VEH with small-GBSS is more efficient in energy conversion than those in the cases of large-GBSS and over-GBSS by comprehensive considering the magnitude and bandwidth of output-voltage at resonance region. In order to obtain insight into the dynamic behavior of VEH, the output waveforms corresponding to the VEH with large-GBSS, small-GBSS, and over-GBSS at the I, II, and III positions in the resonance regions shown in Fig. 5 are enlarged and shown in Fig. 6, respectively. For the case of large-GBSS as shown in Fig. 6(a), a symmetrical output waveform is generated according to the amplitude and shape in the negative and positive directions, whereas the output waveforms are asymmetric for the other two cases as shown in Fig. 6(b) and (c) because of the various degrees of single-side impact. Additionally, it is found from Fig. 6(b) that a weak highfrequency signal is modulated to the output waveform. It is considered that the high-frequency vibration of the rigid stopper-plate reacts back to the spring-plate, resulting in the vibration coupling of low- and high-frequency. By the coupling waveform, the vibration frequency of the stopper-plate can be deduced to be approximately 1800 Hz. However, similar behavior is not found in the VEH with over-GBSS. It is considered that the stiffness of spring-plate is greatly increased due to the preload effect, enabling the springplate to quickly bounces off the stopper-plate at the impact moment and thus resulting in the failure of the mechanical coupling between two collision objects. In the waveform shown in Fig. 6(c), a small peak between two large peaks appears to support our opinion. The frequency response characteristics of VEH can be obtained by a time-to-frequency conversion. Fig. 7 shows the frequency response of the root-mean-square (RMS) output-voltage of VEH under different acceleration levels. As shown in Fig. 7(a), the resonance frequency and 3 dB bandwidth of the VEH with large-GBSS are approximately 102 Hz and 13 Hz at 0.1 g acceleration, respectively. As the acceleration varies from 0.1 g to 1.5 g, the resonant peak is slightly up-shifted to 112 Hz and the bandwidth is broadened to 24 Hz for the frequency up-sweeps. It can be understood that with a low acceleration level of 0.1 g, the deflection of the spring-plate of VEH is located in the linear region from 0.1 mm
Fig. 5. Measured output waveforms of the VEH prototype with large-GBSS, smallGBSS, and over-GBSS at the sweep rate of 25 Hz/s and an acceleration of 1.5 g.
Fig. 6. Enlarged output waveforms corresponding to the VEH with large-GBSS (green color), small-GBSS (cyan color) and over-GBSS (orange color) at the I, II, and III positions in the resonance regions shown in Fig. 5 (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.).
Fig. 4. Output voltage and output power versus load resistance.
to 0.1 mm as shown in Fig. 2(b), resulting in the occurrence of linear free vibration. With the increase of acceleration level, the spring-plate is located in the range of large deflection (z < 0.1 mm or z > 0.1 mm), resulting in the up-shift of resonant frequency and jump phenomena as shown in Fig. 7(a). This demonstrates the hardening characteristics of the nonlinear spring [31]. Further measurements were performed in the VEH with smallGBSS. In Fig. 7(b), it is seen that a standard narrow-band free vibration occur at acceleration of 0.1 g, indicating that the vibration displacement of proof mass is too small for spring-plate to impact the stopper, and thus it is taken for granted that the dynamic behavior of the VEH with small-GBSS is same as the VEH with large-GBSS.
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Fig. 7. Experimental frequency responses of the RMS output voltage of the prototype with (a) large-GBSS, (b) small-GBSS, and (c) over-GBSS for different acceleration levels of a = 0.1 g, 0.5 g, 1.0 g, and 1.5 g.
However, the measured resonant frequency of the VEH with smallGBSS at 0.1 g is 173 Hz instead of the free vibration frequency of 102 Hz. This phenomenon could be explained by the air elastic effect [32]. For a pair of parallel plates as shown in Fig. 1(c), the resistive force to the spring-plate moving normally against the stationary stopper-plate is caused by the air damping pressure between the two plates. On account of the extremely small GBSS, the gas film is compressed but fails to escape when the plate oscillates with a very high frequency, or moves with a high speed. In this case, the gas film works like a bellows to exert an elastic damping force in spring-plate, affecting the dynamic behavior of the vibration system in air. The equations calculating the resonant frequency of system in air xres and the additional coefficient of elastic damping force ke are given by:
8 < x2res ¼ ðk0 þ ke Þ=m ¼ x20 þ ke =m 2 2 3 : ke ¼ 65 l Pxz5A
;
ð6Þ
0 0
where x is the radial frequency of the oscillation, xres is a function of x because ke is a function of x, l is the coefficient of viscosity of the air fluid (l = 1.81 105 Pa s at room temperature of 20 °C), P0 is air pressure (P0 = 105 Pa), and A is the area of square plate. z0 is 25 lm referring to actual assembly structure. By the Eq. (6), the ke is calculated to be 1511.3 N/m. According to ke, the principal resonance frequency is calculated to be 172.753 Hz which basically
373
agree with the experimental result of 175.65 Hz as shown in Fig. 7(b). The small error of 1.65% for the prediction model is mainly attributed to the measurement error in actual z0 value. With so small an air–gap, the spring-plate is driven to hit the stopper when the excited acceleration is increased to a suitable value, for example, a = 0.5 g. As shown in Fig. 7(b), the output voltage diverges from its normal behavior with a linear and moderate increase in the voltage amplitude during impact. The resonance bandwidth is extended over a wide range and is increased in both sides of the resonance region with the increase of acceleration. The 3 dB bandwidth is measured to be approximately 55 Hz at the acceleration of 1.5 g. Similar to the Duffing-type hardening configuration, jump-down phenomena can be observed during the frequency upsweep, which is attributed to piecewise-linear stiffness induced by the stopper impact. As the number of spacers adhered to stopper-plate was increased enough to contact the spring-plate and exert a pressure on it, the mass-spring system is brought into the case of over-GBSS. Fig. 7(c) shows the frequency response of the RMS output-voltage of VEH with over-GBSS. The tensile stress induced by the initial pre-deflection of H1 = 0.005 mm increases the spring-plate stiffness, enabling the resonance frequency of oscillator to be upshifted to about 340 Hz. With the increase in excited level, the jump-up and jump-down phenomena occur and the bandwidth prefers to broaden toward the low-frequency direction. It can be seen from Fig. 7(c) that the response bandwidths reach 70 Hz and 100 Hz for acceleration a = 1.0 g and a = 1.5 g, respectively, which are almost twice as wide as the bandwidth of the VEH with large-GBSS or small-GBSS. As depicted in the theory part, a preloaded spring-plate brings about a Duffing-type bistable nonlinear characteristic in mass-spring system. With a low excitation level, for example, a = 0.1 g, the spring-plate will oscillate with low amplitude and narrow bandwidth in monostable position below the stopper. With a high excitation level, for example, a = 1.0 g, the expected bistable oscillation is restrained by the stopperplate, resulting in a wide-band monostable impact-vibration. On account of the increase of spring-plate stiffness and the amplitude limit by stopper, it can be found that the amplitude of the output voltage is greatly decreased at same excitation level in comparison with other two cases. Of course, the total output power of device can be further increased when all four beams are integrated with piezoelectric transducers. In order to verify the validity of mathematical modeling on VEH with multiple nonlinear tuning mechanisms, the numerical simulations to the dynamic behaviors of VEH were performed using SIMULINKÒ model based on Eqs. (1) and (2). The restoring forces Fr used in simulation are these functions as shown in Eqs. (3)– (5). According to different GBSS used in the VEH, the simulated results of the VEH at the excitation level of a = 1.0 g are shown in Fig. 8, respectively. It can be seen that from Fig. 8 that the calculated curves generally agree well with the experimental curves, demonstrating the validity of our VEH modeling. The slight mismatches in amplitude over special frequency range are attributed to giving no consideration to the higher order terms of the elastic restoring force and the additional unmodeled dissipation [33]. To further understand the dynamic behaviors of VEH, we investigated the dependence of the RMS output voltage of VEH on the acceleration level. As shown in Fig. 9, with a 173 Hz of excitation signal and acceleration sweep from 0 to 5.0 g, the RMS outputvoltage of VEH with small-GBSS quickly increases and reaches a critical saturation point at a = 0.38 g, thereafter the saturation output slowly increases until the sweep stops. In contrast, the VEH with over-GBSS driven by a 340 Hz of excitation signal exhibits a slow increase in the RMS output voltage and reach a same saturation-output-level as the case of small-GBSS at a = 2.1 g. The critical saturation points imply that the dynamic stiffness of
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few practical applications. It is suggested to decrease the overall stiffness of spring-plate in order to shift the critical saturation point to a low acceleration value, realizing the applications of VEH with over-GBSS under the condition of low frequency and low acceleration level. 4. Conclusions
Fig. 8. Simulation and experimental frequency responses of the RMS output voltage of the prototype with (a) large-GBSS, (b) small-GBSS, and (c) over-GBSS for acceleration level of a = 1.0 g.
In summary, we present a compact piezoelectric vibration energy harvester (VEH) using multiple nonlinear techniques to tuning the resonant frequency and broadening the bandwidth. The device was designed as a parallel-plate structure consisting of a suspended spring-plate with proof mass and piezoelectric transducers, a tunable stopper-plate, and supporting frames. By mechanical adjusting the vertical gap between the spring-plate and the stopper-plate (GBSS), the VEH can realize tuning of the resonant frequency and the bandwidth by multiple nonlinear effects. On account of the high-reliable and interchangeable metal parts and the amplitude-limiting stopper, the VEH has high reliability and repairability capable of applying in a strong shock environments. If the device is out of work due to the damages of the structure or piezoelectric sheet, it can be repaired by replacing the parts with new one. Therefore, the device has a good economic viability. Experimentally, the piezoelectric VEH was assembled as a metal prototype that can be operated in three kinds of work states corresponding to the configurations of large-GBSS, small-GBSS, and over-GBSS. By frequency and acceleration sweeping measurements, it is found that the work frequency, 3 dB bandwidth, and output voltage of the VEH depend on tuning of GBSS and excitation levels. The mathematic modeling and numerical simulation based on FEM and MATLAB/SIMULINKÒ explain the different dynamic behaviors of VEH operated in different work states, indicating that the multiple nonlinear effects, such as Duffing-spring effect, impact effect, preload effect, and air elastic effect, have significant influence on characteristics of vibration energy harvesting. It is suggested to decrease the overall stiffness of spring-plate to realize the practical applications of VEHs under the condition of low frequency and low acceleration level. Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant Nos. 61574117, 61274120, and 51675452). References
Fig. 9. Comparison of the RMS output voltage of the VEH with small GBSS at 173 Hz and over GBSS at 340 Hz at an acceleration sweep from 0 to 5.0 g.
mass-spring system change from Duffing-nonlinear stiffness to piecewise-linear stiffness. While the VEH assembled using metal parts has strong shock-resistibility for high shock environments, the acceleration larger than 2.0 g is too large to be found in the ambient vibration, thus the case of over-GBSS could meet very
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Applied Energy journal homepage: www.elsevier.com/locate/apenergy
A frequency and bandwidth tunable piezoelectric vibration energy harvester using multiple nonlinear techniques Xiang Wang a,b, Changsong Chen b, Na Wang b, Haisheng San b,⇑, Yuxi Yu c, Einar Halvorsen d, Xuyuan Chen d a
Department of Mechanical and Electrical Engineering, Xiamen University, Xiamen 361005, China Pen-Tung Sah Institute of Micro-Nano Science and Technology, Xiamen University, Xiamen 361005, Fujian, China c College of Materials, Xiamen University, Xiamen 361005, Fujian, China d Department of Micro- and Nanosystem Technology, University College of Southeast Norway, Borre 3184, Norway b
h i g h l i g h t s An ultra-wide bandwidth vibration energy harvester was built with tunable frequency and bandwidth. Multiple nonlinear tuning mechanisms were realized by tuning the gap between spring-plate and stopper. The theory models and numerical simulations were presented with experimental verification.
a r t i c l e
i n f o
Article history: Received 12 September 2016 Received in revised form 22 December 2016 Accepted 31 December 2016
Keywords: Nonlinear vibration energy harvester Piezoelectric Tunable frequency and bandwidth
a b s t r a c t This article presents a compact piezoelectric vibration energy harvester (VEH) using multiple nonlinear techniques to tuning the resonant frequency and broadening the bandwidth. The device was designed as a parallel-plate structure consisting of a suspended spring-plate with proof mass and piezoelectric transducers, a tunable stopper-plate, and supporting frames. By mechanical adjusting the vertical gap between the spring-plate and the stopper-plate (GBSS), the VEH can realize tuning of the resonant frequency and the bandwidth by multiple nonlinear effects. Experimentally, the piezoelectric VEH was assembled as a metal prototype that can be operated in three kinds of work states corresponding to the configurations of large-GBSS, small-GBSS, and over-GBSS. The sweeping-frequency measurement results show that the work frequency, bandwidth, and output-voltage of VEH depend on tuning of GBSS and excitation levels, indicating that the multiple nonlinear effects, such as Duffing-spring effect, impact effect, preload effect, and air elastic effect, have significant influence on the dynamic behaviors of VEH. The comparisons of numerical simulations with the experimental results were used to verify the validity of mathematical modeling on VEH with multiple nonlinear tuning techniques. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction The rapid growth of wireless sensor networks and mobile electronic platforms have led to the developments of self-powered devices using energy harvesting techniques. Energy harvesting is a method to generate electrical power from external energy sources such as solar, thermal, wind, vibration, radio frequency (RF) energy, human body heat, and human movements, and so on. Among these energies, vibration-based energy harvesting has received significant attention in recent years. As a typical example of wasted energy that may be harvested, vibrations can be available easily in our surroundings, such as different types of commer⇑ Corresponding author. E-mail address: [email protected] (H. San). http://dx.doi.org/10.1016/j.apenergy.2016.12.168 0306-2619/Ó 2017 Elsevier Ltd. All rights reserved.
cial and industrial machines, vehicles, buildings, various structures (bridges, railways), home appliances, and motion of biological systems [1]. The idea of vibration-to-electricity conversion was proposed by Williams and Yates in 1996 [2]. From then on, vibration energy harvesting has been an attractive technique for charging or powering low-power micro-electronics [3]. Typical vibration energy harvesters (VEHs) may be viewed as a simple oscillator (mass-spring system), excited by an external vibration source, with one electromechanical transducer used for vibration-to-electricity conversion [4]. The principal transduction mechanisms in VEHs are piezoelectric, electromagnetic, and electrostatic transductions [5]. While each of techniques mentioned above can provide a useful amount of energy, the piezoelectric technique has gained most attention due to their abilities to directly convert applied strain energy into usable electric energy
X. Wang et al. / Applied Energy 190 (2017) 368–375
and the ease to be integrated into a system [6]. From the power output viewpoint, actually the piezoelectric materials were not popular as an energy harvester due to its small power output. However, modern electronic devices are getting tiny and consume less power than before. Therefore the most recent developments of piezoelectric applications are related to energy harvesting [7]. Regardless of the transduction mechanisms, a primary issue in VEHs is that the maximum energy output usually occurs in its fundamental resonance frequency. The power output will be decreased significantly when the resonant frequency of VEHs do not matches the ambient vibration, this will seriously restrains the applications of VEHs [8]. There are two possible methods to optimize the output power under random vibration source: tuning the resonant frequency and broadening the bandwidth [9]. In principle, frequency tuning can be realized by controlling the mass and the spring constant of oscillator [7]. Some possible methods are: adding a tip mass [10], moving the center of gravity [11], changing the dimension [12] or shape of cantilever [13], or altering the stiffness [14] or damping [15] of the harvester. As it is impractical to change the dimensions or shape of cantilever, most of methods are to change the mass, damping or the stiffness by manual or autonomous tuning [7]. According to the practicable methods in application, our literature investigation on the frequency tuning focus on four kinds of tuning strategies: resonant frequency tuning, multimodal energy harvesting, nonlinear energy harvesting, and electrical damping tuning [16]. Resonant frequency tuning can be based on active or passive actuators for tuning frequency [17]. The active tuning actuators provide a continuous force proportional to displacement, acceleration, or velocity to alter effective stiffness, effective mass, or effective damping accordingly in a mass-spring system [17]. In contrast, the passive tuning actuators exert force only during altering the resonance frequency and no force occurs while maintaining the new resonance frequency [18–20]. Regardless of active and passive actuators, they need external energy (that is often larger than the harvesting energy) to generate the applied forces [17]. The multimodal energy harvesting is a potential effective-bandwidthincreasing approach. In practice, multimodal systems can be realized by exploiting multiple bending modes of a continuous beam or by an array of cantilevers [21,22]. The multimodal VEHs can achieve an energy harvesting by a multi-frequency resonance with an add advantage of requiring no tuning and thus the external energy. However, the effective bandwidth of multimodal systems are usually discrete, which may only be helpful when the vibration source has a rather wide frequency spectrum [16]. Furthermore, in order to avoid the mode-shape-dependent voltage cancelation in a continuous beam or the voltage cancelation caused by the phase difference between cantilevers in array configurations, multimodal VEHs usually requires complex interface circuit [16]. Nonlinear energy harvesting has shown a tremendous potential for the design of broadband VEHs in recent years. Nonlinear configurations for broadband energy harvesting focus on the generation of nonlinear force by a Duffing-type oscillator or a piecewise-linear impact oscillator. Duffing-type oscillators include monostable [23], bistable [24], and tristable [25] nonlinear configurations with nonlinear stiffness typically introduced by using magnets. Such a mechanism is applied to perturb and drive the system from a low-energy vibration branch into a high-energy orbit, resulting in a much higher output power. Piecewise-linear impact is another type of nonlinear vibration which can be introduced by adding a mechanical stopper to make a piecewise linear stiffness in vibration system [26]. It was found that the tuning configuration using mechanical stopper generates jump phenomenon and results in the increase of bandwidth during frequency sweep. However, the bandwidth is increased at the expense of lowering output power in comparison with the configurations without mechanical stopper
369
[27]. The frequency tuning using electrical damping was investigated previously by Muriuki [15]. It is found that the stiffness and natural frequency of the beam can be adjusted when a capacitor is added in parallel to the piezoelectric element. Electrical damping can be varied by adjusting the electrical load. As the resistive load reduces the efficiency of power transfer while the inductive load is difficult to be varied, the varying of a capacitive load is preferable [7]. It is noted that above-mentioned broadband harvesting techniques are only preferable in specific conditions. Anyway the most suitable method for frequency tuning depends on the intended application. For example, the car-tire vibration spectrum spreads from 300 Hz to 700 Hz [28] and the railway tunnels vibration spectrum is distributed from 200 Hz to 600 Hz and 1200 Hz to 2000 Hz [29]. In these vibration environments, the wireless sensor nodes (WSN) powered by VEHs with a single adjusting configuration cannot be sufficient to tune over such a wide frequency range. Therefore the VEHs using multiple nonlinear tuning mechanisms in a device are considered to be a suitable solution due to its large tunability over work frequency range. According to the requirements of frequency matching over a wide frequency range, VEHs with multiple nonlinear tuning mechanisms are capable of tuning the resonant frequency with the accompaniment of broadening the bandwidth. Up to now there are few reports on the VEHs using multiple nonlinear tuning mechanisms. In this work, we present a compact piezoelectric VEH using multiple nonlinear tuning mechanisms to tune the resonant frequency and broaden the bandwidth synchronously. The device was designed and fabricated as a metal prototype with compact parallel-plate structure consisting of supporting frames, a suspended spring-plate with proof mass, a tunable stopper-plate. The piezoelectric sheet was integrated on one of the beams of spring-plate. This device can be assembled and adjusted according to the requirements of flexible study options. The multiple nonlinear effects, including the Duffing-spring effect, impact effect, preload effect, and air elastic effect can be achieved in this device by mechanical tuning the stopper-plate. Furthermore, this device has high reliability and repairability due to the high-reliable and interchangeable metal parts and the amplitude-limiting stopper, making this device a strong shock-resistibility for practical engineering applications.
2. Design and modeling of the VEH The targeted piezoelectric VEH is designed as a parallel-plate spring-mass system as schematically shown in Fig. 1(a), which consists of a suspension spring-plate with a proof mass, a rigid stopper-plate above the spring-plate, three structural frames, and connecting bolts. The spring-plate is designed as a hooked-crossshaped suspension configuration with four identical doubly clamped L-type beams (see Fig. 1(a)) by hollow-cutting a 0.2 mm thick and 45 mm long square bronze plate, such design brings about a compact structure as well as a nonlinear Duffing-spring effect. A 0.4 mm thick, 3 mm wide, and 24 mm long piezoelectric ceramic sheet is attached to the top surface of the L-shaped suspension beam. The prototype can be assembled by connecting all parts with four bolts, and the vertical gap between the springplate and the stopper-plate (GBSS) can be changed by adjusting the number of spacers in the bottom center of the stopper-plate. By varying the GBSS, the VEH can be operated in three kinds of work states, namely large-GBSS, small-GBSS, and over-GBSS, as shown in Fig. 1(b–d). The electromechanical model corresponding to the case of small-GBSS is schematically shown in Fig. 1(e). In this model, the proof mass M oscillates with the relative displacement z(t); Km,
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Fig. 1. (a) Schematic configuration of a tunable piezoelectric VEH. The work states of the VEH with (b) large-GBSS, (c) small-GBSS, and (d) over-GBSS. (e) Electromechanical model corresponding to the case of small-GBSS.
and bm represent the stiffness and damping constant of oscillator, respectively. The stopper-plate is shown as a parallel springdamping system with stiffness constant Ks and damping constant bs [30]. The attached PZT sheet generates a piezoelectric voltage V(t), which drives the external load R. From Fig. 1(e), the lumped-parameter equations of the electromechanical model can be described as:
(
mzðtÞ þ bm zðtÞ þ F r þ HVðtÞ ¼ mzb ðtÞ HzðtÞ ¼ 0 C pe VðtÞ þ VðtÞ R
;
ð1Þ
where zb(t) is the displacement of vibration station, H is the piezoelectric coupling coefficients and Cpe is the effective capacitance between the piezoelectric electrodes. Fr is the total restoring force which can be expressed as Fr = Fk + Fs. Fk is the elastic restoring force arised from the spring-plate and Fs is the impact force arised from the stopper-plate. Based on the spring-damping system, Fs can be expressed as:
FS ¼
K S ðz z0 Þ bS z_ if z P z0 0
if z 6 z0
;
ð2Þ
where z0 is the initial value of GBSS according to the origin of coordinate in the top surface of mass. By means of the finite-element method (FEM), we can calculate the dependence of elastic restoring force of spring-plate on displacement of proof mass. Fig. 2(a) shows the COMSOL Multiphysics FEM model of the spring-plate with specific dimension parameters. H1 is the predeflection of the spring-plate induced by the vertical preload. We simulated the dependence of the restoring force F on displacement z when VEH is in the case of large-GBSS, small-GBSS, and overGBSS, respectively. In the case of large-GBSS, z0 is set sufficiently large so that the proof mass never hit the stopper-plate during free vibration. As shown in Fig. 2(b), the spring-plate with H1 = 0 mm (without pre-deflection) displays a linear mechanical behavior with a constant stiffness kl = 611 N/m from 0.1 mm to 0.1 mm (see the inset of Fig. 2(b)). Beyond this small range of displacement, the spring-plate exhibits a nonlinear mechanical behavior with hardening characteristic in the positive and negative directions. The FEM simulation results are fitted using an odd polynomial function, the elastic restoring force Fk can be expressed as:
F r ¼ 607:2709z þ 3:8127 108 z3
ð3Þ
When the vertical gap z0 is further adjusted to a small value, namely, the case of small-GBSS, the spring-plate would hit the stopper-plate within its displacement over the applied frequency range. As the effective mass of the stopper-plate is negligible in comparison with the proof mass, the stopper-plate vibrates with the spring-mass system together during the impact, in which the effective restoring force of system is substituted by impact force Fs. As shown in Fig. 2(c), the restoring force abruptly varies from Fk to Fs at the impact point z = 0.025 mm (see the inset of Fig. 2 (c)), thereafter the nonlinear vibrations are introduced by impact force Fs. Fitting the FEM curve by an odd polynomial function, the total restoring force of spring-plate with small-GBSS can be expressed as:
(
Fr ¼
2193:31z þ 3:8127 108 z3
if z 6 0:025 103
3
50000 ðz 0:025 10 Þ if z P 0:025 103
ð4Þ
It should be noted in Eq. (4) that the linear function in the case of z P 0.025 mm is obtained by fitting the experimental data based on the impact model given by Eq. (2). If the GBSS is adjusted to a negative value by increasing the number of spacers in the stopper-plate, a tensile stress induced by the pre-deflection will be exerted on the spring-plate when all parts were assembled together. This means that the spring-plate is in the case of over-GBSS. As shown in Fig. 2(d), with an initial pre-deflection of H1 = 0.005 mm in the spring-plate, the simulated elastic restoring force Fk exhibits a nonlinear dynamic behavior with bistable characteristic. If the spring-plate is located an unstable position corresponding to a negative stiffness from 0.03 mm to 0.03 mm (see Fig. 2(d)), it will spontaneously move into the stable equilibrium position towards positive or negative direction. On account of the restriction of stopper-plate, the spring-plate would hit the stopper at pre-deflected position (z = 0.005 mm) when the spring-plate move towards the positive stable equilibrium position. As a result, the spring-plate was brought into a piecewise linear stiffness region. The total restoring force of spring-plate with over-GBSS can be expressed as:
(
Fr ¼
4008:23z þ 1:7638 1012 z3 50000 ðz þ 0:005 103 Þ
if z 6 0:005 103 if z P 0:005 103 ð5Þ
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Fig. 2. FEM model of the spring-plate (a) and the simulated dependence of the restoring force F on displacement z when VEH is in the case of (b) large-GBSS, (c) small-GBSS, and (d) over-GBSS, respectively.
By observing the coefficient of the first term of polynomial function in Eq. (5), it is found that the stiffness of spring-plate with over-GBSS is twice as large as that with small-GBSS. This implies that the principal resonance frequency of VEH with over-GBSS pffiffiffi should be 2 times as large as that with small-GBSS. Substituting Eqs. (3), (4) and (5) into Eq. (1), respectively, and solving these differential equations using MATLAB/Simulink models, we can simulate the acceleration-dependent of dynamic output characteristics of VEHs.
output port of piezoelectric transducer, and the output voltage (VR) across the resistor was measured through a high-impedance buffer amplifier. The output voltages of VEH prototype and accelerometer were read by an data acquisition board (NI-USB-6211 DAQ) and LabView program of National Instruments. The first step in the measuring sequences is to experimentally determine the optimal load-resistance for maximum power output. A tunable resistor was connected between the electrodes of the piezoelectric transducer. The value Rl of the resistor and voltage VR across the load were measured. The instantaneous power delivered by the energy harvester can be obtained according to
3. Experimental results and discussion
P ¼ V 2R =Rl . Though the optimal load resistance can be found by identifying the value at maximum output power, the optimal load resistance changes depending on the work state of device and its excitation frequency. Considering the multiple work states of VEH in practical application, it is reasonable to evaluate the device properties using a fixed optimal load for all work states. We stepped the load resistance and measured the output voltage for
Experimentally, the VEH prototype (see Fig. 3(a)) was mounted onto the vibration station (TIRA vibration exciter) as shown in Fig. 3(b). The shaker was driven using a power amplifier with output of DC-biased sinusoidal wave. An accelerometer (Piezotronics Inc. model 352A56A) was attached to VEH prototype for measuring the acceleration of variation. A load-resistor was connected to the
Fig. 3. (a) Photographs of the spring-plate of the VEH prototype and (b) experimental setup.
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the VEH with large-GBSS in its resonant frequency. Fig. 4 shows dependences of the output voltage and power on load-resistance. The optimal load-resistance was approximately 45 kX. Next, the instantaneously generated voltage over the load-resistor were measured by a chirp excitation with a linear sweep of 25 Hz/s from 0 to 20 s. Fig. 5 shows the time domain output waveforms of VEH prototypes in three kinds of work states at acceleration of 1.5 g. The maximum peak-to-peak voltages are approximately 2.449 V, 2.278 V, and 0.733 V for the cases of large-GBSS, small-GBSS, and over-GBSS at an acceleration of 1.5 g, respectively. It can be seen that the waveforms of VEH with small-GBSS is more efficient in energy conversion than those in the cases of large-GBSS and over-GBSS by comprehensive considering the magnitude and bandwidth of output-voltage at resonance region. In order to obtain insight into the dynamic behavior of VEH, the output waveforms corresponding to the VEH with large-GBSS, small-GBSS, and over-GBSS at the I, II, and III positions in the resonance regions shown in Fig. 5 are enlarged and shown in Fig. 6, respectively. For the case of large-GBSS as shown in Fig. 6(a), a symmetrical output waveform is generated according to the amplitude and shape in the negative and positive directions, whereas the output waveforms are asymmetric for the other two cases as shown in Fig. 6(b) and (c) because of the various degrees of single-side impact. Additionally, it is found from Fig. 6(b) that a weak highfrequency signal is modulated to the output waveform. It is considered that the high-frequency vibration of the rigid stopper-plate reacts back to the spring-plate, resulting in the vibration coupling of low- and high-frequency. By the coupling waveform, the vibration frequency of the stopper-plate can be deduced to be approximately 1800 Hz. However, similar behavior is not found in the VEH with over-GBSS. It is considered that the stiffness of spring-plate is greatly increased due to the preload effect, enabling the springplate to quickly bounces off the stopper-plate at the impact moment and thus resulting in the failure of the mechanical coupling between two collision objects. In the waveform shown in Fig. 6(c), a small peak between two large peaks appears to support our opinion. The frequency response characteristics of VEH can be obtained by a time-to-frequency conversion. Fig. 7 shows the frequency response of the root-mean-square (RMS) output-voltage of VEH under different acceleration levels. As shown in Fig. 7(a), the resonance frequency and 3 dB bandwidth of the VEH with large-GBSS are approximately 102 Hz and 13 Hz at 0.1 g acceleration, respectively. As the acceleration varies from 0.1 g to 1.5 g, the resonant peak is slightly up-shifted to 112 Hz and the bandwidth is broadened to 24 Hz for the frequency up-sweeps. It can be understood that with a low acceleration level of 0.1 g, the deflection of the spring-plate of VEH is located in the linear region from 0.1 mm
Fig. 5. Measured output waveforms of the VEH prototype with large-GBSS, smallGBSS, and over-GBSS at the sweep rate of 25 Hz/s and an acceleration of 1.5 g.
Fig. 6. Enlarged output waveforms corresponding to the VEH with large-GBSS (green color), small-GBSS (cyan color) and over-GBSS (orange color) at the I, II, and III positions in the resonance regions shown in Fig. 5 (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.).
Fig. 4. Output voltage and output power versus load resistance.
to 0.1 mm as shown in Fig. 2(b), resulting in the occurrence of linear free vibration. With the increase of acceleration level, the spring-plate is located in the range of large deflection (z < 0.1 mm or z > 0.1 mm), resulting in the up-shift of resonant frequency and jump phenomena as shown in Fig. 7(a). This demonstrates the hardening characteristics of the nonlinear spring [31]. Further measurements were performed in the VEH with smallGBSS. In Fig. 7(b), it is seen that a standard narrow-band free vibration occur at acceleration of 0.1 g, indicating that the vibration displacement of proof mass is too small for spring-plate to impact the stopper, and thus it is taken for granted that the dynamic behavior of the VEH with small-GBSS is same as the VEH with large-GBSS.
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Fig. 7. Experimental frequency responses of the RMS output voltage of the prototype with (a) large-GBSS, (b) small-GBSS, and (c) over-GBSS for different acceleration levels of a = 0.1 g, 0.5 g, 1.0 g, and 1.5 g.
However, the measured resonant frequency of the VEH with smallGBSS at 0.1 g is 173 Hz instead of the free vibration frequency of 102 Hz. This phenomenon could be explained by the air elastic effect [32]. For a pair of parallel plates as shown in Fig. 1(c), the resistive force to the spring-plate moving normally against the stationary stopper-plate is caused by the air damping pressure between the two plates. On account of the extremely small GBSS, the gas film is compressed but fails to escape when the plate oscillates with a very high frequency, or moves with a high speed. In this case, the gas film works like a bellows to exert an elastic damping force in spring-plate, affecting the dynamic behavior of the vibration system in air. The equations calculating the resonant frequency of system in air xres and the additional coefficient of elastic damping force ke are given by:
8 < x2res ¼ ðk0 þ ke Þ=m ¼ x20 þ ke =m 2 2 3 : ke ¼ 65 l Pxz5A
;
ð6Þ
0 0
where x is the radial frequency of the oscillation, xres is a function of x because ke is a function of x, l is the coefficient of viscosity of the air fluid (l = 1.81 105 Pa s at room temperature of 20 °C), P0 is air pressure (P0 = 105 Pa), and A is the area of square plate. z0 is 25 lm referring to actual assembly structure. By the Eq. (6), the ke is calculated to be 1511.3 N/m. According to ke, the principal resonance frequency is calculated to be 172.753 Hz which basically
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agree with the experimental result of 175.65 Hz as shown in Fig. 7(b). The small error of 1.65% for the prediction model is mainly attributed to the measurement error in actual z0 value. With so small an air–gap, the spring-plate is driven to hit the stopper when the excited acceleration is increased to a suitable value, for example, a = 0.5 g. As shown in Fig. 7(b), the output voltage diverges from its normal behavior with a linear and moderate increase in the voltage amplitude during impact. The resonance bandwidth is extended over a wide range and is increased in both sides of the resonance region with the increase of acceleration. The 3 dB bandwidth is measured to be approximately 55 Hz at the acceleration of 1.5 g. Similar to the Duffing-type hardening configuration, jump-down phenomena can be observed during the frequency upsweep, which is attributed to piecewise-linear stiffness induced by the stopper impact. As the number of spacers adhered to stopper-plate was increased enough to contact the spring-plate and exert a pressure on it, the mass-spring system is brought into the case of over-GBSS. Fig. 7(c) shows the frequency response of the RMS output-voltage of VEH with over-GBSS. The tensile stress induced by the initial pre-deflection of H1 = 0.005 mm increases the spring-plate stiffness, enabling the resonance frequency of oscillator to be upshifted to about 340 Hz. With the increase in excited level, the jump-up and jump-down phenomena occur and the bandwidth prefers to broaden toward the low-frequency direction. It can be seen from Fig. 7(c) that the response bandwidths reach 70 Hz and 100 Hz for acceleration a = 1.0 g and a = 1.5 g, respectively, which are almost twice as wide as the bandwidth of the VEH with large-GBSS or small-GBSS. As depicted in the theory part, a preloaded spring-plate brings about a Duffing-type bistable nonlinear characteristic in mass-spring system. With a low excitation level, for example, a = 0.1 g, the spring-plate will oscillate with low amplitude and narrow bandwidth in monostable position below the stopper. With a high excitation level, for example, a = 1.0 g, the expected bistable oscillation is restrained by the stopperplate, resulting in a wide-band monostable impact-vibration. On account of the increase of spring-plate stiffness and the amplitude limit by stopper, it can be found that the amplitude of the output voltage is greatly decreased at same excitation level in comparison with other two cases. Of course, the total output power of device can be further increased when all four beams are integrated with piezoelectric transducers. In order to verify the validity of mathematical modeling on VEH with multiple nonlinear tuning mechanisms, the numerical simulations to the dynamic behaviors of VEH were performed using SIMULINKÒ model based on Eqs. (1) and (2). The restoring forces Fr used in simulation are these functions as shown in Eqs. (3)– (5). According to different GBSS used in the VEH, the simulated results of the VEH at the excitation level of a = 1.0 g are shown in Fig. 8, respectively. It can be seen that from Fig. 8 that the calculated curves generally agree well with the experimental curves, demonstrating the validity of our VEH modeling. The slight mismatches in amplitude over special frequency range are attributed to giving no consideration to the higher order terms of the elastic restoring force and the additional unmodeled dissipation [33]. To further understand the dynamic behaviors of VEH, we investigated the dependence of the RMS output voltage of VEH on the acceleration level. As shown in Fig. 9, with a 173 Hz of excitation signal and acceleration sweep from 0 to 5.0 g, the RMS outputvoltage of VEH with small-GBSS quickly increases and reaches a critical saturation point at a = 0.38 g, thereafter the saturation output slowly increases until the sweep stops. In contrast, the VEH with over-GBSS driven by a 340 Hz of excitation signal exhibits a slow increase in the RMS output voltage and reach a same saturation-output-level as the case of small-GBSS at a = 2.1 g. The critical saturation points imply that the dynamic stiffness of
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few practical applications. It is suggested to decrease the overall stiffness of spring-plate in order to shift the critical saturation point to a low acceleration value, realizing the applications of VEH with over-GBSS under the condition of low frequency and low acceleration level. 4. Conclusions
Fig. 8. Simulation and experimental frequency responses of the RMS output voltage of the prototype with (a) large-GBSS, (b) small-GBSS, and (c) over-GBSS for acceleration level of a = 1.0 g.
In summary, we present a compact piezoelectric vibration energy harvester (VEH) using multiple nonlinear techniques to tuning the resonant frequency and broadening the bandwidth. The device was designed as a parallel-plate structure consisting of a suspended spring-plate with proof mass and piezoelectric transducers, a tunable stopper-plate, and supporting frames. By mechanical adjusting the vertical gap between the spring-plate and the stopper-plate (GBSS), the VEH can realize tuning of the resonant frequency and the bandwidth by multiple nonlinear effects. On account of the high-reliable and interchangeable metal parts and the amplitude-limiting stopper, the VEH has high reliability and repairability capable of applying in a strong shock environments. If the device is out of work due to the damages of the structure or piezoelectric sheet, it can be repaired by replacing the parts with new one. Therefore, the device has a good economic viability. Experimentally, the piezoelectric VEH was assembled as a metal prototype that can be operated in three kinds of work states corresponding to the configurations of large-GBSS, small-GBSS, and over-GBSS. By frequency and acceleration sweeping measurements, it is found that the work frequency, 3 dB bandwidth, and output voltage of the VEH depend on tuning of GBSS and excitation levels. The mathematic modeling and numerical simulation based on FEM and MATLAB/SIMULINKÒ explain the different dynamic behaviors of VEH operated in different work states, indicating that the multiple nonlinear effects, such as Duffing-spring effect, impact effect, preload effect, and air elastic effect, have significant influence on characteristics of vibration energy harvesting. It is suggested to decrease the overall stiffness of spring-plate to realize the practical applications of VEHs under the condition of low frequency and low acceleration level. Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant Nos. 61574117, 61274120, and 51675452). References
Fig. 9. Comparison of the RMS output voltage of the VEH with small GBSS at 173 Hz and over GBSS at 340 Hz at an acceleration sweep from 0 to 5.0 g.
mass-spring system change from Duffing-nonlinear stiffness to piecewise-linear stiffness. While the VEH assembled using metal parts has strong shock-resistibility for high shock environments, the acceleration larger than 2.0 g is too large to be found in the ambient vibration, thus the case of over-GBSS could meet very
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