3.1 Experimental measurement
The mechanical vibration experiment was carried out to verify the validity and accuracy of simulation result, and the schematic diagram is given in Fig. 8(a) for the measurement setup.25 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. 8(b) and Fig. 8(c). 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.
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 ,22 wherethe 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 excitation,17, 19 and the δf is the 3dB bandwidth of PEHDCB.33
3.2 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 results,22 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. 9(a) and Fig. 9(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. 9(a), 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 frequency.34 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 reports.15, 35 The energy harvesting efficiency could be generally evaluated by the half-power bandwidth value,17, 19 but the proportion of half-power bandwidth should be more suitable to evaluate the harvesting efficiency for PEH at the multiple-frequency excitation.23 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.
3.3 Effect of width splitting method on the energy harvesting performances
In order to verify the effect of width splitting method on the energy harvesting performances,17 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 Figs. 10(a) and 10(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 29.8V /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 reports.17, 19 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.
3.4 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. Generally, 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 monostable nonlinear energy harvester36, 37 and 5–10 % for the piezoelectric energy harvester for harnessing energy from flow-induced vibration,38 and the Up−max error is approximately at the range of 10–26 % for the rotational mechanical plucking energy harvester.39 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.
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 result.40 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.