Introduction

In recent years, harvesting ambient energy using piezoelectric effect is considered to be an important energy conversion technology applicable to future industries along with photovoltaics1, thermoelectrics2, radio frequencies3, and triboelectric technology4. Vibration-based piezoelectric energy harvester is a potential candidate to replace existing power sources such as the batteries which have a limited energy storage capacity and lifetime for some applications5. Based on Taguchi orthogonal method, the variation in the excitation frequency, the thickness of piezoelectric layers and the acceleration of the bimorph piezoelectric energy harvester (PEH) were optimized by maximum output peak voltage (Up-max) and signal-to-noise ratio as the optimization objectives6, and the ultimate goal is that one could obtain the Up-max with lesser trials7. Most of PEHs with the different beam shapes, such as the rectangles8,9,10,11 and triangles12,13 have been focused on the energy harvesting characteristics from bending vibration, and the main problem is the significant dropping of output power when the excitation frequency slightly away from the natural frequency of the harvesters. The durability of PEH is importance to sustainable power wireless sensors, and some researchers have focused on the fatigue analysis for PEH under different excitation cycles14 and levels of base excitation15.

PEH can effectively harvest electric power from torsional vibration by attaching a simple cantilever beam structure on the top surface of a rotating shaft generate16. A unimorph cantilever beam undergoing bending-torsion vibrations has been proposed by asymmetry increasing under a transverse harmonic base excitation to narrow the resonance frequency difference (Δf0) between first and second modes, therefore it allows the harvesting of electrical power from multiple-frequency excitation17. The novel asymmetric vortex-induced piezoelectric harvester for capturing wind energy at low wind speed is of lower natural frequency and smaller electromechanical coefficient than those of the conventional vortex-induced piezoelectric energy harvester18. The effect of width-splitting on the harvested power was theoretically and experimentally investigated for a fixed dimension piezoelectric materials of PEH undergoing bending vibration, and the width-splitting method could be used to increase the harvested voltage19 and power20 over a wider range of excitation frequencies. The single piezoelectric beam with the similar total width was folded equally and then split a given dimension of piezoelectric materials with the predefined dimensions into the array of smaller-width beams, and there is a substantial increase in harvested power21. There are many reports on the evaluation methods of half-power bandwidth and quality factor for PEHs undergoing bending vibration19,21, however relatively few researches are focus on the evaluation method for PEH undergoing coupled bending-torsion vibrations. The best performance occurs at a single or very narrow frequency range for most resonating harvesters, and then a critical issue is how to adjust the resonance frequency (f0) flexibility and maximize harvesting power because of environment vibrations composed of some broadband random or multiple-frequency excitation22,23. In a word, the width-splitting method could be used to improve energy harvesting performances of PEH undergoing bending vibration, however one could not know whether it is applicable to harvest more ambient energy from multiple-frequency excitation by using PEH undergoing coupled bending-torsion vibrations.

In this paper, PEH with single-cantilever-beam (SCB) is equally split into PEH with double-cantilever-beam (DCB) in order to harvest ambient energy from multiple-frequency excitation, which is verified comparing with the experimental and simulation results of the Up-max and Δf0. After the geometrical dimensions including the split gap d, primary beam L and substrate thickness ts were determined by using Up-max and Δf0 combination in multi-factors analysis based on orthogonal test method, we had fabricated PEHSCB and PEHDCB undergoing coupled bending-torsion vibrations. The mechanical vibration experiments were performed by shaker to measure Up vs frequency (Up-f) curves of PEHs undergoing coupled bending-torsion vibrations, and the half-power bandwidth proportion and materials mechanical quality factors are used to evaluate the harvesting efficiency from the multiple-frequency excitation. We expect that the research could significantly offer some useful guidelines to design high-performance PEH, which could harvest more ambient energy under multiple-frequency excitation.

Design and simulation

Construction of PEHDCB split from PEHSCB

The single/double T-shaped brass shims and commercial piezoelectric ceramic PZT bulks were provided by Wuxi Hui Feng Electronics Co. Ltd, and the material properties parameters24,25,26 are listed in Table 1. PZT element was formed of PZT wafer, top and bottom electrodes, before PEHSCB was designed by bonding the elements on the primary beam via conductive adhesive, as shown in Fig. 1a. PEHDCB is constructed by splitting equally the primary beam of PEHSCB, as shown in Fig. 1b, and the split gap d determined by the width-splitting method19,21 is listed in Table 2.

Table 1 The materials properties.
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Figure 1
figure 1

Schematics of PEH (a) SCB and (b) DCB undergoing coupled bending-torsion vibrations.

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Table 2 The geometrical dimensions.
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There are two variable mass blocks M2 and M1 placed at tip of the crossbeam, and M2/M1 is defined as the mass ratio m* and when the mass ratio changes, the mass sum of mass blocks remains unchanged at both of the beam ends. The geometrical dimensions of PEHSCB include the length L, width b, thickness ts, piezoelectric ceramic thickness tp for the primary beam and the length L1, L2, width a, mass ratio m* for the crossbeam, and all of them are listed in Table 2. The polarization direction of piezoelectric ceramic PZT-53 bulk is marked by the red arrows along Z-axis. Learning from the series connection structure27, two PZT elements split from the primary beam of PEHSCB are poled oppositely in the perpendicular to length direction to form the series connection shown in Fig. 1b. In Cartesian coordinate system (X, Y, Z), one ends of the single and double beams are fixed, and the crossbeams are at free end.

Simulation

Modeling on PEHSCB and PEHDCB

The Up and f0 of PEHs are simulated by using the commercial analysis package ANSYS. The 8-node hexahedral coupled-field element SOLID5 and the 8-node linear structural element SOLID45 are used for piezoelectric materials and non-piezoelectric materials, and piezoelectric circuit element CIRCU94 is used for load resistor. The displacement degrees of freedom are constrained to be zero for fixed end of the primary beam of PEH. The electrode connections are made by using the “couple” commands28. As for PEHDCB, the potentials of top surface of substrate layer are coupled with bottom surface of piezoelectric layer, and they are individually coupled for top surfaces of two piezoelectric layers. The potential is constrained to be zero for one top surface of piezoelectric layers, and a load resistor is connected between the top surfaces of two piezoelectric layers, as shown in Fig. 1. The output voltage across load resistor is solved for the potential of top surface of piezoelectric layer, when T-shaped beam is loaded with the inertial acceleration 9.8 m/s2.

In order to obtain a convergent solution by using mesh refinement29, the modal stress diagrams at the first mode of PEHDCB are performed by considering the four different mesh densities 0.7, 0.6, 0.5 and 0.4 mm, and they are respectively given as in Fig. 2a,b,c,d. Obviously, the difference of the maximum stress is less than 6% for the last two mesh densities, as shown in Fig. 2c,d, therefore it could be regarded as a convergent solution when mesh density 0.5 mm.

Figure 2
figure 2

The modal stress diagrams of convergent judge at the mesh densities for PEHDCB (a) 0.7, (b) 0.6, (c) 0.5 and (d) 0.4 mm.

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The modal stress analysis

In order to satisfy the yield strength of piezoelectric materials and brass30, the modal analyses on PEHs with asymmetric mass are performed under the mesh density 0.5 mm, and the modal stress diagrams are presented in Fig. 3a,b for PEHSCB and Fig. 3c,d for PEHDCB. Obviously, the maximum stresses of piezoelectric layer 37.3 MPa and brass layer 163 MPa occur at the first mode for PEHDCB as shown in Fig. 3c, and they are smaller than the yield strengths of 60 MPa and 200 MPa, indicating yield strength reliability.

Figure 3
figure 3

The modal stress diagrams of yield strength (a,b) for PEHSCB and (c,d) for PEHDCB.

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Optimization on asymmetry PEHs with the different mass ratios

To investigate the effect of m* on the harvesting performance of PEHSCB, the Up-f curves were simulated under the different mass ratios m* = 19, 9, 4 and 1.5 when b = 12 mm, L1 = L2 = 22 mm, and they are described as Fig. 4a. With the asymmetry increase of m* from 1.5 to 19, the Up-max1 and f01 at the first mode decrease from 33.6 to 29.0 V and 88 to 72 Hz. On the contrary, the Up-max2 and f02 at the second mode increase from 5.1–11.6 V and 118 to 177 Hz. As for m* > 1, all of the f01 and f02 are far lower than the nature frequency 200 Hz and the Δf0 is narrowed. Therefore it allows the harvesting of electrical power from the multiple-frequency excitation, and they are agreement with the previous results17. Drawing the Up-max and f0 at the certain m* from Fig. 4a, the Up-max vs m* and f0 vs m* curves are described as Fig. 4b for the first mode and Fig. 4c for the second mode. Both of the Up-max1 and f01 decrease with the asymmetry increase of m*, as shown in Fig. 4b, however the f02 increases and the Up-max2 increases to reach a peak value at m* = 9 and then decreases by contrary, as shown in Fig. 4c. The optimization mass ratio m* is determined as 9 by considering the trade-off between Up-max and f0, in order that PEHSCB could achieve not only the higher Up-max but also both the lower f01 and f02. According to the previous analysis method31, we analogously determine the mass ratios m* = 9 to investigate harvest ambient energy of PEHDCB undergoing coupled bending-torsion vibrations from a multiple-frequency excitation as follow.

Figure 4
figure 4

The Up-f curves (a), Up-max and f0 vs m* curves at the first (b) and second (c) modes of PEHSCB.

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Effect of single-factor on PEHDCB

To understand the effect of single geometrical parameter on energy harvesting performance of PEHDCB, the Up-f curves were simulated under the different split gaps d, such as 2, 3, 4, 5 and 6 mm, and they are described as Fig. 5a when m* = 9, b = 12 mm, L1 = L2 = 22 mm. With the d increase from 2 to 6 mm, the Up-max1, f01 and f02 increase from 31.8 to 37.3 V, 75 to 80 Hz and167 to 177 Hz however the Up-max2 decreases from 14.6 to 13.3 V. The dependencies of Up-max and f0 on m* are analogously discussed as shown in Fig. 4b,c, and the Up-max and f0 vs d curves are described as Fig. 5b for the first mode and Fig. 5c for the second mode. With the increase of d, both of the Up-max1 and f01 increase monotonously, however the Up-max2 decreases and the f02 increases by contrary. By adjusting the split gap d, PEHDCB can harvest energy more evenly at the two first modes, and it is in an agreement with the reported result32.

Figure 5
figure 5

The Up-f curves (a), Up-max and f0 vs m* curves at the first (b) and second (c) modes.

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The Up-f curves were simulated under the different primary beam lengths L, such as 26, 27, 28, 29 and 30 mm, L1 = L2 = 22 mm, and they are described as Fig. 6a when m* = 9, b = 12 mm. The d dependencies of Up-max and f0 are analogously discussed, as shown in Fig. 5b,c, and the Up-max and f0 vs L curves are described as Fig. 6b for the first mode and Fig. 6c for the second mode. With the increase of L, the Up-max1 monotonously increases but the Up-max2 slightly decreases at the second mode, however both of the f01 and f02 decrease. It indicates that the primary beam length can be adjusted by considering the trade-off between the Up-max and f0, in order that PEHDCB could harvest energy more evenly under the multiple-frequency excitation.

Figure 6
figure 6

The Up-f curves (a), the Up-max and f0 vs L curves at the first (b) and second (c) modes.

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The Up-f curves were simulated under the different substrate thicknesses ts, such as 0.4, 0.5, 0.6, 0.7 and 0.8 mm, and they are described as Fig. 7a, when m* = 9, b = 12 mm, L1 = L2 = 22 mm. Analogously, the Up-max and f0 vs ts curves are described as Fig. 7b for the first mode and Fig. 7c for the second mode. With the increase of ts, the Up-max1 increases to reach the peak at ts = 0.6 mm however the Up-max2 decreases monotonously. Both the f01 and f02 increase. It could guide to choose a suitable ts to achieve high Up-max and low f0 of PEHDCB under the multiple-frequency excitation.

Figure 7
figure 7

The Up-f curves (a), the Up-max and f0 vs ts curves at the first (b) and second (c) modes of PEHDCB.

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Multi-factor analysis via orthogonal test method

Generally, the effect of single-factor on performance of PEHDCB could not be taken into account the cross factor effect33, therefore we carry out the multi-factors analysis to optimize energy harvesting performance of PEHDCB by using orthogonal test method. The orthogonal factor/level list and the L25 (56) corresponding orthogonal list are shown in Tables 3 and 4.

Table 3 The orthogonal factor and level list.
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Table 4 L25 (56) orthogonal list.
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There are the three geometrical dimensions d, ts and L corresponding to 5 levels in Table 3. Distinguished from the traditional orthogonal test method8, the Up-max1, Up-max2 and Δf0 were chosen as the optimization objectives to analyze the influences of the three factors on the performance, and the last three factors are kept empty corresponding to the L25 (56) lists in Table 434. As for the optimal combination of Case 21 in Table 4, the Up-max1 and Up-max2 are 37.3 V and 13.3 V, and the Δf0 is 97.0 Hz. Correspondingly, the d, L and ts are optimally determined as 6, 30, and 0.4 mm, in order that PEHDCB achieves not only high Up-max but also narrow Δf0 for easily harvesting electrical power from the multiple-frequency excitation17.

Experiment

Experimental measurement

The mechanical vibration experiment was carried out to verify the validity and accuracy of simulation result, and the schematic diagram and power measurement circuit are given in Fig. 8a,b for the measurement setup27. In order to investigate the effect of width splitting method on the energy harvesting performances, PEHDCB was constructed by splitting equally the primary beam, and both of PEHSCB and PEHDCB were fabricated according to the optimal combination Case 21 in Table 4, as shown in Fig. 8c,d. After clamped by the fixtures, PEHSCB and PEHDCB were fixed on the shaker (JZk-5, Sinocera Piezotronics, Inc., Jiangsu, China), and the acceleration was obtained through the accelerometer (CA-DR-005, Sinocera Piezotronics, Inc., Jiangsu, China) placed on the top of the shaker. The sinusoidal vibration from the function generator (Tektronix AFG 3021B, Tektronix) was amplified by the power amplifier (YE5871A, Sinocera Piezotronics, Inc., Jiangsu, China), and the shaker was applied at 2.1 A and 1 V to generate the mechanical energy.

Figure 8
figure 8

Schematic diagram (a). Power measurement circuit (b). The vibration component prototypes of PEHSCB (c) and PEHDCB (d).

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All the output peak voltages of PEHSCB and PEHDCB were measured at the certain resistance of 100 kΩ by Digital Oscilloscope (Tektronix TDS 1002, Tektronix), and the output power was calculated by using the formula \(P = U_{P}^{2} /2R_{L}\) 24, where \(R_{L}\) the load resistance. Generally, the proportion of half-power bandwidth and the quality factor (Q = f0/δf) are used to evaluate the collection efficiencies of PEHDCB under a single-frequency excitation and multi-frequency excitation19,21, and the δf is the 3 dB bandwidth of PEHDCB35.

Test and verify on energy harvesting efficiency of PEHSCBs

The energy harvesting efficiency of PEHSCB undergoing coupled bending- torsion vibrations is better than that of PEHSCB undergoing bending vibration in the simulation results24, therefore we hope to verify the previous results by comparing with the Δf0 and proportion of half-power bandwidth in the mechanical vibration experiment as follow. The output peak voltages were measured under b = 12 mm, L1 = L2 = 22 mm, RL = 100 kΩ, and the Up-f and P-f curves are given in Fig. 9a,b for PEHSCBs undergoing coupled bending-torsion vibrations with m* = 9 and bending vibration with m* = 1. Here, the simulation results are represented by the solid lines, and the experimental results are described by the dots for the latter and the triangles for the former. From Fig. 9a, the f01 of PEHSCB with m* = 9 is lower than that of PEHSCB with m* = 1, which indicates that PEH undergoing coupled bending-torsion vibrations is more suitable for harvesting the electrical power from the ambient vibration composed of low frequency36. On the other hand, the Δf0 is 90.0 Hz for PEHSCB undergoing coupled bending-torsion vibrations, however it is far much narrower than that of PEHSCB undergoing bending vibration. Obviously, the coupled bending-torsion vibrations could decrease the Δf0 to allow the harvesting of electrical power from the multiple-frequency excitation, and the results are similar with the experimental observations in the previous reports17,37. The energy harvesting efficiency could be generally evaluated by the half-power bandwidth value19,21, but the proportion of half-power bandwidth should be more suitable to evaluate the harvesting efficiency for PEH at the multiple-frequency excitation25. Here, the proportions of half-power bandwidth are 5.2% at the first mode and 3.8% and at the second mode for PEHSCB with m* = 9, and the proportion of half-power bandwidth is 5.1% at the first mode for PEHSCB with m* = 1. Obviously, the proportion of half-power bandwidth of PEHSCB undergoing coupled bending-torsion vibrations is superior to that of PEHSCB undergoing bending vibration at the first mode, meanwhile the former could far more easy to harvest the electrical power from the ambient vibration because of the resonance lack for the latter at the second mode. The energy harvesting efficiency of PEHSCB undergoing coupled bending-torsion vibrations must be much better than that of PEHSCB undergoing bending vibration, since the environmental vibrations are generally composed of the multiple-frequency.

Figure 9
figure 9

The Up-f (a) and P-f (b) curves of PEHSCBs undergoing coupled bending-torsion vibrations and bending vibration.

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Effect of width splitting method on the energy harvesting performances

In order to verify the effect of width splitting method on the energy harvesting performances19,21, the Up-max and Pp-max of PEHDCB are compared with those of PEHSCB, meanwhile the quality factors related with Δf0 are compared because of damping change induced width splitting method. The Up-max were measured for PEHDCB with d = 6 mm, b = 12 mm, L1 = L2 = 22 mm, m* = 9, RL = 100 kΩ, and the Up-f and P-f curves are given in Fig. 10a,b for both of PEHSCB and PEHDCB. The simulation/experimental Up-max1 and Up-max2 results are 37.3 V/35.1 V and 13.3 V/12.1 V for PEHDCB, and they are 29.8 V /28.4 V and 11.6 V/ 10.7 V for PEHSCB. The Pp-max1 and Pp-max2 are 6.97 mW/ 6.16 mW and 0.89 mW/ 0.73 mW for PEHDCB, and they are 4.43 mW/ 4.03 mW and 0.65 mW/ 0.57 mW for PEHSCB. For the same undergoing coupled bending-torsion vibrations, the Up-max and Pp-max of PEHDCB are obviously larger than those of PEHSCB, and they are agreement with the simulation reports19,21. As for the first and second modes, the simulation/experiment Q1 and Q2 results are 32.13/25.08 and 35.4/30.17 for PEHDCB, and they are 29.72/24.33 and 33.15/25.2 for PEHSCB. Obviously, the Q1 and Q2 of PEHDCB undergoing coupled bending-torsion vibrations are larger than those of PEHSCB undergoing bending vibration, and it indicates that the former could easily harvest the energy from the ambient vibration. In a word, all of the energy harvesting performances of PEHDCB undergoing coupled bending-torsion vibrations have been enhanced by the width-splitting method. Our result shows that bending torsional coupled vibration PEH can harvest more power than bending vibration PEH, which is consistent with the result reported38.

Figure 10
figure 10

Effect of width splitting method on the Up-f (a) and P-f (b) curves of PEHs undergoing coupled bending-torsion vibrations.

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Energy harvesting performances of asymmetry PEHDCBs

The Up-f curves of the PEHDCBs were measured under the different mass ratios m*, such as 1.5, 4, 9 and 19, and they are given in Fig. 11, when b = 12 mm, L1 = L2 = 22 mm, d = 6 mm, RL = 100 kΩ. Here, the simulation and experimental results are represented by the lines and dots, in order that the validity could be confirmed by analyzing the relative deviations of Up-max and f0 at the first and second modes. Obviously, the maximum relative deviations of Up-max and f0 are 6.8% and 2.6% for the first mode, and they are 14.2% and 2.5% for the second mode. There are generally the incomplete clamping of the experimental fixture and the unavoidable assembly errors in the manufacture39, therefore the simulation results could be not exactly equal to the resonance frequency experiment results for PEHs with the asymmetric mass and width-splitting beam in Figs. 8, 9 and 10. The clamped end of beam is not completely rigid in mechanical vibration experimental, however the fixed boundary conditions in simulation make the beam slightly stiffer. The incomplete clamping and inevitable assembly frequency errors are approximately at the ranges of 2–7% for the nonstable nonlinear energy harvester40,41 and 5–10% for the piezoelectric energy harvester for harnessing energy from flow-induced vibration39, and the Up-max error is approximately at the range of 10–26% for the rotational mechanical plucking energy harvester42. The Up-max and f0 deviations are 14.2% and 2.6% for the PEHDCBs under the different mass ratios, as shown in Fig. 11, therefore the results are acceptable for the mechanical vibration experiment.

Figure 11
figure 11

The Up-f curves under the different mass ratios m* for the PEHDCBs.

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Drawn from Fig. 11, the Up-max and m* are served as the vertical and horizontal ordinates to understand the effect of asymmetric mass on energy harvesting performance, and the Up-max1 and Up-max2 vs m* curves are described as Fig. 12. With the asymmetric increase of m* from 1.5 to 9, the simulation/experimental Up-max1 values decrease from 41.0 V/38.2 V to 36.5 V/34.8 V, and the simulation/experimental Up-max2 values increase to the reach peak values of 13.3 V/12.8 V and 12.1 V/11.1 V. Obviously, the energy harvesting efficiency is slightly decreased under the multiple-frequency excitation when the mass ratio m* is larger than 9. Considering the trade-off of Up-max1 and Up-max2, PEHDCB with m* = 9 can harvest energy more evenly, and it is similar with the reported result43. In a word, there is a useful strategy to enhance the Up-max1 and Up-max2 by adjusting the asymmetric mass ratio under the multiple-frequency excitation. There is an impact of fatigue life on the energy harvest from environmental vibration with a multiple-frequency excitation, and the fatigue life could be chosen as the indicative parameters at the multi-factors analysis the further work including experiment and simulation.

Figure 12
figure 12

The Up-max vs m* curves at the first and second modes for PEHDCB.

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Conclusion

In summary, the optimization geometrical parameters of PEHDCB are determined by considering the Up-max and Δf0 in the multi-factor analysis based on the orthogonal test, and the Δf0 narrowed of PEHDCB undergoing coupled bending-torsion vibrations results in harvesting the more electrical power from the ambient vibration composed of low frequency. The quality factors Q1 and Q2 of PEHDCB undergoing coupled bending-torsion vibrations are larger than those of PEHSCB under the multi-frequency excitation, and the Up-max1 and Pp-max1 at the first mode are increased 25.2% and 57.3% for PEHDCB. The width-splitting method could be successfully used to improve the energy harvesting performances, and the measurement reliability is acceptable considering the incomplete clamping, damping and inevitable assembly effects. Considering the trade-off of Up-max1 and Up-max2, the asymmetry PEHDCB with the mass ratio determined as 9 can harvest energy more evenly, and the results are effective to predict the energy harvesting performances for PEHDCB undergoing coupled bending-torsion vibrations under the multi-frequency excitation. As for the purpose to explore the effect of width splitting method on the energy harvesting performances in this work, we should consider the parameters related with damping change such as the width splitting size d into electromechanical coupling dynamic equation, and one could obtain a closed form solution and comparing to give better insights on how the harvested power varies by varying the parameters in future research.