Acoustics
, Volume 1 (2) – Apr 18, 2019

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acoustics Article Study on the Output Performance of a Nonlinear Hybrid Piezoelectric-Electromagnetic Harvester under Harmonic Excitation 1 , 2 1 2 , 3 Haipeng Liu , Shiqiao Gao , Junru Wu * and Ping Li State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing 100081, China; HaipengLiu@uvm.edu (H.L.); ShiqiaoGao@UVB.CN (S.G.) Department of Physics, The University of Vermont, Burlington, VT 05405, USA Beijing Research Institute of Mechanical Equipment, Beijing 100854, China; PingLi@UMB.CN * Correspondence: jwu@uvm.edu Received: 25 February 2019; Accepted: 15 April 2019; Published: 18 April 2019 Abstract: The nonlinear energy harvester has become a hot topic due to its broad bandwidth and lower resonant frequency. Based on the preliminary test and analyses in our previous work, further analyses and tests on the inﬂuence of parameters, including the nonlinear magnetic force of the hybrid energy harvesting structure on its output performance under harmonic excitation, are performed in this paper, which will provide powerful support for structural optimization. For designing a nonlinear piezoelectric-electromagnetic hybrid energy harvester, the state equation of electromechanical coupling, the harmonic response and average output power, voltage, and current of a nonlinear hybrid energy harvester under harmonic excitation are derived by the harmonic balance method. The eects of the excitation acceleration and the external load on the output performance of the nonlinear hybrid energy harvester are veriﬁed through experimental tests. The results showed that the output power of the nonlinear hybrid energy harvester increases with the increase in the acceleration of harmonic excitation, and the increase is aected by external load. When the piezoelectric-electromagnetic hybrid harvester operates at the optimal load and the resonant frequency, the average output power reaches its maximum value and the increase of the load of the piezoelectric unit makes the resonant frequency of the energy harvesting system increase. Compared with linear harvesting structures, the nonlinear hybrid harvester has better ﬂexibility of environmental adaptability and is more suitable for harvesting energy in low-frequency environments. Keywords: nonlinearity; hybrid energy harvester; piezoelectric; electromagnetic; harmonic excitation 1. Introduction Vibration energy harvesters can convert vibration energy into electrical energy by piezoelectric, electromagnetic, and electrostatic mechanisms. This process has been improved signiﬁcantly with the use of low-power electronic devices [1–6]. Broader bandwidth and lower resonant frequency are two main challenges when the harvester is used in practice [7–10]. Therefore, researchers have proposed many methods to solve these problems. Many researchers have introduced nonlinearity into energy harvesting to widen frequency bandwidth and improve performance, for example, adding stoppers to the structure, adopting active tuning structure, utilizing material nonlinearity or introducing nonlinear force. Of these, the nonlinear energy harvesting technique is one of the most feasible solutions [11–14]. Mann [15] and Foisal et al. [16] designed a nonlinear magnetic spring electromagnetic energy harvester. By analyzing the size of the magnets and the distance between the magnets, the frequency bandwidth of the harvester can be optimized. Luigi and Maryam [17] proposed a vibration energy harvesting model based on nonlinear vibration-damping. The output power and harvesting frequency Acoustics 2019, 1, 382–392; doi:10.3390/acoustics1020021 www.mdpi.com/journal/acoustics Acoustics 2019, 2, x FOR PEER REVIEW 2 of 12 bandwidth of the harvester can be optimized. Luigi and Maryam [17] proposed a vibration energy Acoustics 2019, 1 383 harvesting model based on nonlinear vibration-damping. The output power and harvesting frequency bandwidth of this model are better than those of linear energy harvester. Waleed [18] designed a piezoelectric harvester with adjustable resonant frequency by changing the force between bandwidth of this model are better than those of linear energy harvester. Waleed [18] designed a the magnets. The experiment results showed that the resonant frequency of the harvester can be piezoelectric harvester with adjustable resonant frequency by changing the force between the magnets. adjusted up by 70%. Marcin [19] designed a MEMS piezoelectric energy harvester with adjustable The experiment results showed that the resonant frequency of the harvester can be adjusted up by 70%. stiffness, with which the harvesting frequency bandwidth can be increased by 36%. In addition, Marcin [19] designed a MEMS piezoelectric energy harvester with adjustable stiness, with which the Challa [5] designed a nonlinear piezoelectric harvester with adjustable resonant frequency by harvesting frequency bandwidth can be increased by 36%. In addition, Challa [5] designed a nonlinear changing the system’s effective stiffness. The experimental tests demonstrated that the natural piezoelectric harvester with adjustable resonant frequency by changing the system’s eective stiness. frequency of the linear energy harvester can be adjusted by ± 20%. The experimental tests demonstrated that the natural frequency of the linear energy harvester can be In order to improve the energy conversion efficiency and broaden the application bandwidth of adjusted by20%. the vibration energy harvester, a hybrid energy harvesting technique of combining piezoelectric (PE) In order to improve the energy conversion eciency and broaden the application bandwidth and electromagnetic (EM) mechanisms was proposed, which benefits from the advantages of the two of the vibration energy harvester, a hybrid energy harvesting technique of combining piezoelectric techniques simultaneously [20–23]. Karami and Inman [24] established a unified approximation (PE) and electromagnetic (EM) mechanisms was proposed, which beneﬁts from the advantages of the method for linear, softly nonlinear, and bi-stable nonlinear energy harvesters. The method quantifies two techniques simultaneously [20–23]. Karami and Inman [24] established a uniﬁed approximation equivalent changes in the damping and excitation frequency of the mechanical subsystem that method for linear, softly nonlinear, and bi-stable nonlinear energy harvesters. The method quantiﬁes resembles the backward coupling from energy harvesting. Hamdi and Belhaq [25] introduced time equivalent changes in the damping and excitation frequency of the mechanical subsystem that resembles delay in the mechanical component of the delayed hybrid energy harvester, consisting of a nonlinear the backward coupling from energy harvesting. Hamdi and Belhaq [25] introduced time delay in the mono-stable harvester device. The results show that the powers output performance of the harvester mechanical component of the delayed hybrid energy harvester, consisting of a nonlinear mono-stable is significantly improved at certain ranges of time delay and delay amplitude. In this paper, a harvester device. The results show that the powers output performance of the harvester is signiﬁcantly nonlinear PE-EM hybrid energy harvester with adjustable stiffness is proposed and its governing improved at certain ranges of time delay and delay amplitude. In this paper, a nonlinear PE-EM hybrid equations are established. By means of the harmonic balance method, expressions of the vibration energy harvester with adjustable stiness is proposed and its governing equations are established. amplitude, output voltage, current, and average power of hybrid harvester under the harmonic By means of the harmonic balance method, expressions of the vibration amplitude, output voltage, excitation are derived. The effects of excitation acceleration, load resistance to vibration amplitude, current, and average power of hybrid harvester under the harmonic excitation are derived. The eects output power, and the resonant frequency of nonlinear hybrid energy harvesters are investigated of excitation acceleration, load resistance to vibration amplitude, output power, and the resonant experimentally and the results are compared with the output performances of linear energy frequency of nonlinear hybrid energy harvesters are investigated experimentally and the results are harvesters. compared with the output performances of linear energy harvesters. 2. Nonlinear Hybrid Energy Harvester Structure 2. Nonlinear Hybrid Energy Harvester Structure An illustration of the nonlinear hybrid PE-EM energy harvester is shown in Figure 1. The central An illustration of the nonlinear hybrid PE-EM energy harvester is shown in Figure 1. The central movable magnet is supported by two identical clamped hybrid beams on both sides, and two coils movable magnet is supported by two identical clamped hybrid beams on both sides, and two coils are are placed above and below the movable magnet, respectively. Besides this, two magnets are placed above and below the movable magnet, respectively. Besides this, two magnets are mounted mounted inside the two coils, and forces between the movable magnets and fixed magnets are an inside the two coils, and forces between the movable magnets and ﬁxed magnets are an attractive force, attractive force, so the forces decrease the natural frequency of harvester by reducing the effective so the forces decrease the natural frequency of harvester by reducing the eective spring constant. spring constant. In addition, piezoelectric layers (PZT) mounted on the top of the movable magnet In addition, piezoelectric layers (PZT) mounted on the top of the movable magnet are polarized are polarized in the beam thickness direction (vertical direction, shown in Figure 1), and based on in the beam thickness direction (vertical direction, shown in Figure 1), and based on piezoelectric piezoelectric effect and electromagnetic induction law, PZT layers and coils will output electrical eect and electromagnetic induction law, PZT layers and coils will output electrical energy under energy under external excitation. external excitation. Figure 1. Illustration of a nonlinear energy harvester structure. Figure 1. Illustration of a nonlinear energy harvester structure. Cylindrical magnets are used in the design to apply the desired magnetic force, and the magnetic Cylindrical magnets are used in the design to apply the desired magnetic force, and the magnetic force between any two cylindrical magnets is given by Equation (1) [26], force between any two cylindrical magnets is given by Equation (1) [26], 3 M V M V 0 0 0 1 1 F = , (1) 2d Acoustics 2019, 1 384 where = 4 10 H/m is magnetic permeability, V and V are volumes of two cylindrical magnets, 0 1 0 M and M are magnetization of the two magnets, respectively, and d is the static distance between the 1 0 magnets. For the designed nonlinear hybrid energy harvester shown above, the total force generated between the top ﬁxed magnet and the movable magnet, and that between the bottom ﬁxed magnet and the movable magnet is given by Equation (2), M V M V t t b b F (z) = F F = 3 M V [ ], (2) m m1 m2 0 m m 4 4 2(d + z) 2(d z) where d is the equilibrium distance between the top (or bottom) magnet and the movable magnet; z is the vertical displacement of the movable magnet under the external excitation; M and V are m m the magnetization and volume of the central magnet, respectively; M and V are the magnetization t t and volume of the top magnet respectively; and M and V are the magnetization and volume of the b b bottom magnet respectively. To avoid the irreversible plastic deformation of the beams, the nonlinear force should be less than the elastic restoring force of the beams, which means F (z) < jk zj, (3) m b where k is the stiness of the beam. For the designed nonlinear hybrid energy harvester, the top and bottom magnets are the same, and their equilibrium distances from the central magnet are equal. By the method of Taylor expansion [27], when neglecting high-order terms and z < d, Equation (2) can be expressed as: F (z) = k z + k z , (4) m 1 3 where: M V b b k = 12 M V , (5) 1 0 m m k = k . (6) 3 1 According to Figure 2, and combining the former study for linear hybrid energy harvesters [28], the governing equations of a nonlinear hybrid PE and EM energy harvester can be illustrated as: .. . .. ( ) ( ) ( ) ( ( )) ( ) ( ) ( ) m z t + c z t + k z t + F z t + g I t + V t = m y t , (7) e m m e em p e V (t) . p . ( ) ( ) + C V t z t = 0, (8) p p L I (t) + (R + R )I (t) g z(t) = 0, (9) c em c m em e .. where y(t) is the excitation acceleration; m is the equivalent mass of the vibration system; c and k are e m the damping coecient and linear stiness of the vibration system, respectively; R and R are the p m load resistance of the PE and EM energy harvesting unit, respectively; C is the equivalent capacitance of the PE layer; V is the output voltage of the PE energy harvesting unit; I is the output current of p em the EM energy harvesting unit; R and L refers to the resistance and inductance of coils; and and c c g are the PE and EM transfer factors, respectively. These parameters are dependent on the material constants and the design of the energy harvester, which can be derived by standard model analysis. Acoustics 2019, 2, x FOR PEER REVIEW 4 of 12 parameters are dependent on the material constants and the design of the energy harvester, which Acoustics 2019, 1 385 can be derived by standard model analysis. Figure 2. Experimental setup and a hybrid structure. Figure 2. Experimental setup and a hybrid structure. 3. Vibration Responses and Output of Nonlinear Energy Harvester under Harmonic Excitation 3. Vibration Responses and Output of Nonlinear Energy Harvester under Harmonic Excitation Substituting Equation (4) into Equation (7), Equation (7) becomes the typical Dung equation, Substituting Equation (4) into Equation (7), Equation (7) becomes the typical Duffing equation, if the external excitation is harmonic excitation, and the inductance eect of the coil is ignored if the external excitation is harmonic excitation, and the inductance effect of the coil is ignored in the in the low frequency vibration. When the excitation acceleration input to the energy harvester is .. low frequency vibration. When the 2 K excitation acceleration i mnput to the e energy ha 1 rvester is () = y(t) = Y cos(!t + ), let ! = , K = k + k , 2! = , = , = . According to 1 n m m R +R R C e e c m p p c g 2 K 1 m e the harmonic balance method [29,30], considering the weak nonlinear condition, and assuming that − ( + ) , let ω = ， K = k + k ， 2ζω = ， β = ， μ = . According to n 1 n m m R + R R C the response of the energy harvesting e system is mainly ethe single harmonic c m frequency p p , the vibration response of the energy harvester, the output voltage of the piezoelectric energy harvesting unit, and the harmonic balance method [29,30], considering the weak nonlinear condition, and assuming that the electromagnetic energy harvesting unit are shown in Equations (10)–(12), respectively: the response of the energy harvesting system is mainly the single harmonic frequency, the vibration response of the energy harvester, the output voltage of the piezoelectric energy harvesting unit, and z(t) = A cos!t, (10) the electromagnetic energy harvesting unit are shown in Equations (10)–(12), respectively: z(t) = A cosωt , (10) V (t) = A cos(!t + ), (11) p 1 2 1 I (t) = A cos(!t + ). (12) em 3 2 V (t) = A cos(ωt + φ ) , (11) p 2 1 Substituting Equations (10)–(12) into Equation (7), according to I (t) = A cos(ωt + φ ) . (12) em 3 2 3 1 cos !t = cos!t + cos 3!t, (13) 4 4 Substituting Equations (10)–(12) into Equation (7), according to and ignoring high-order harmonic terms, the following equation can be obtained: 3 3 1 cos ωt = cosωt + cos 3ωt , (13) 4 4 2 2 A (! ! ) cos!t 2A ! ! sin!t+ 1 n 1 and ignoring high-order harmonic t 3k e Arms, the following equation can be obtained: 1 2 . (14) cos!t + cos(!t + ' )+ 4m m e e A g 3 e A() ωω−−cosωtA2 ζωωsin ωt+ cos(!t + ' ) = Y cos(!t + ) 11nn 3kA A θ 31 2 By making the coecients ofco the sωω twott++ sides of cothe s( equationϕ )+equal, we obtain: 4mm (14) ee 3k e 2 Ag 2 3 3 e A ! ! + A + A cos' + A cos' = Y cos, (15) 1 2 1 3 2 n cos(ωϕtY += ) cos(ωt+φ ) 4m m m e e e 2A ! ! + A sin + A sin = Y sin', (16) n 2 3 2 By making the coefficients 1 of the two sides of the eq 1 uation equal, we obtain: m m e e and the six-order nonlinear expression of vibration amplitude z as: ( − ) + + cos + cos = cos , M (15) ( ) 2 2 2 ! 3k g 2 2 2 2 2 2 3 ! e z = A = Y ! ! + z + + 2! ! + + ! . (17) n 2 2 2 2 4m m m m M 1 M e e C (! + ) e C (! + ) e p p Acoustics 2019, 1 386 In the actual calculation, the vibration amplitude z can be calculated using Equation (17), according to the designed characteristic parameters. The output voltage of the piezoelectric energy harvesting unit is: V = z . (18) p M C ! + ( ) R C p p Similarly, the output current of the electromagnetic energy harvesting unit is: g ! I = z . (19) em M R + R c m The output powers of the PE and EM energy harvesting units are: 2 2 1 p ! P = = z , (20) 2 R 1 2 2 2R C [! + ( ) ] R C p p 2 2 1 R g ! 2 2 P = R I = z . (21) em m em 2(R + R ) c m Therefore, the total output power of the designed nonlinear PE-EM hybrid energy harvester is: P = P + P p em 8 9 > > > > 2 2 2 2 . (22) < g ! = ! e 2 = + z > > 2 2 > > 1 M : 2R (R +R ) ; 2R C [! +( ) ] m c m p p R C p p 4. Experimental Tests In order to test and study the output characteristics of the nonlinear PE-EM hybrid energy harvester, the designed energy harvester structure and the experimental test system are shown in Figure 2. 4.1. Inﬂuence of Magnetic Force on the Output of Nonlinear Hybrid Harvesting Structure The magnetic force in the harvesting energy structure can change the vibration state of the cantilever beam, and can improve the environmental adaptability of the harvesting energy structure. Figure 3 shows that under the action of magnet attractive force, as the distance between the ﬁxed magnet and movable magnet, d, increases, the peak frequency of the nonlinear energy harvester incr Acoustics eases 2019 and , 2the , x FO total R PEER R output EVIEW power increases. The dots are the experimental mean values of thr 6 of ee 12 tests, the solid lines are the best ﬁts of the data. 2.5mm 3mm 4mm 0 20 40 60 80 100 Frequency/Hz Figure 3. Output of harvester under the attractive force. Figure 3. Output of harvester under the attractive force. When the magnetic force is a repulsive force, the relationship between the total output power of nonlinear energy harvester and the peak frequency at different distance is as shown in Figure 4. As the distance between the magnets increases, the repulsive force decreases, the peak frequency of the harvesting structure decreases, and the 3db bandwidth decreases. However, when the distance between magnets increases to a certain extent, the change of the output power is no longer obvious. 2mm 3mm 4mm 40 60 80 100 120 140 160 180 Frequency/Hz Figure 4. Output of harvester under the repulsive force. From the above analyses, the influence of the distance between the magnets on the bandwidth and output power should be considered when designing a nonlinear hybrid energy harvester. Under the same environmental conditions, selecting the magnetic attractive force is more conducive to reducing the peak frequency of the harvesting structure, increasing the bandwidth, and improving the environmental adaptability of the harvesting structure. 4.2. Effects of Excitation Acceleration on Energy Harvesting Characteristics When the external acceleration from the shaker is 0.3, 0.45, and 0.6 g, the total output power of the nonlinear hybrid energy harvester changes with the excitation frequency, as shown in Figure 5, where g is the gravitational acceleration. P/mW P/mW Acoustics 2019, 2, x FOR PEER REVIEW 6 of 12 2.5mm 3mm 4mm 0 20 40 60 80 100 Frequency/Hz Figure 3. Output of harvester under the attractive force. Acoustics 2019, 1 387 When the magnetic force is a repulsive force, the relationship between the total output power of When the magnetic force is a repulsive force, the relationship between the total output power nonlinear energy harvester and the peak frequency at different distance is as shown in Figure 4. As of nonlinear energy harvester and the peak frequency at dierent distance is as shown in Figure 4. the distance between the magnets increases, the repulsive force decreases, the peak frequency of the As the distance between the magnets increases, the repulsive force decreases, the peak frequency of harvesting structure decreases, and the 3db bandwidth decreases. However, when the distance the harvesting structure decreases, and the 3db bandwidth decreases. However, when the distance between magnets increases to a certain extent, the change of the output power is no longer obvious. between magnets increases to a certain extent, the change of the output power is no longer obvious. 2mm 3mm 4mm 40 60 80 100 120 140 160 180 Frequency/Hz Figure 4. Output of harvester under the repulsive force. Figure 4. Output of harvester under the repulsive force. From the above analyses, the inﬂuence of the distance between the magnets on the bandwidth From the above analyses, the influence of the distance between the magnets on the bandwidth and output power should be considered when designing a nonlinear hybrid energy harvester. Under and output power should be considered when designing a nonlinear hybrid energy harvester. Under the same environmental conditions, selecting the magnetic attractive force is more conducive to the same environmental conditions, selecting the magnetic attractive force is more conducive to reducing the peak frequency of the harvesting structure, increasing the bandwidth, and improving the reducing the peak frequency of the harvesting structure, increasing the bandwidth, and improving environmental adaptability of the harvesting structure. the environmental adaptability of the harvesting structure. 4.2. Eects of Excitation Acceleration on Energy Harvesting Characteristics 4.2. Effects of Excitation Acceleration on Energy Harvesting Characteristics When the external acceleration from the shaker is 0.3, 0.45, and 0.6 g, the total output power of the When the external acceleration from the shaker is 0.3, 0.45, and 0.6 g, the total output power of nonlinear hybrid energy harvester changes with the excitation frequency, as shown in Figure 5, where the nonlinear hybrid energy harvester changes with the excitation frequency, as shown in Figure 5, Acoustics 2019, 2, x FOR PEER REVIEW 7 of 12 g is the gravitational acceleration. where g is the gravitational acceleration. Figure 5. Output power of a nonlinear hybrid energy harvester under dierent excitation accelerations. Figure 5. Output power of a nonlinear hybrid energy harvester under different excitation accelerations. According to the test results shown in Figure 3, both the output power and the bandwidth of the energy harvester increase with the increase in the excitation acceleration. When the external According to the test results shown in Figure 3, both the output power and the bandwidth of the acceleration is 0.3, 0.45, and 0.6 g, the maximum output power of the energy harvester is 0.52, 1.76, and energy harvester increase with the increase in the excitation acceleration. When the external 3.54 mW, respectively. At the same time, the resonant frequency of the energy harvester decreases with acceleration is 0.3, 0.45, and 0.6 g, the maximum output power of the energy harvester is 0.52, 1.76, the increase in the excitation acceleration. As shown in in Figure 3, when the excitation acceleration and 3.54 mW, respectively. At the same time, the resonant frequency of the energy harvester decreases with the increase in the excitation acceleration. As shown in in Figure 3, when the excitation acceleration increases from 0.3 to 0.6 g, the resonant frequency decreases from 113.2 to 110.5 Hz. This tells us the stiffness of this system is softening as amplitude increases. Since the peak frequency of the nonlinear hybrid energy harvester changes with the excitation acceleration, the optimal load of the energy harvester also changes. At the resonant frequency, when the acceleration is 0.2 and 0.45 g, the change of the output peak voltage and power of the piezoelectric energy harvesting unit with the piezoelectric load is as shown in Figure 6, and the electromagnetic energy harvesting unit is the optimal load at this time. The data shown in Figure 6 are averaged over several tests. 0.45g 0.2g 0 100 200 300 400 500 600 Resistance /Kohm Figure 6. Piezoelectric energy harvesting unit output varies with piezoelectric load. According to the results shown in Figure 6, the higher the acceleration, the higher the output voltage of the piezoelectric unit. When the excitation accelerations are 0.2 and 0.45 g, the maximum output powers of the piezoelectric unit are 0.085 and 0.5 mW and the corresponding optimal loads Voltage /V P/mW P/mW Acoustics 2019, 2, x FOR PEER REVIEW 7 of 12 Figure 5. Output power of a nonlinear hybrid energy harvester under different excitation accelerations. According to the test results shown in Figure 3, both the output power and the bandwidth of the energy harvester increase with the increase in the excitation acceleration. When the external Acoustics 2019, 1 388 acceleration is 0.3, 0.45, and 0.6 g, the maximum output power of the energy harvester is 0.52, 1.76, and 3.54 mW, respectively. At the same time, the resonant frequency of the energy harvester decreases with the increase in the excitation acceleration. As shown in in Figure 3, when the excitation increases from 0.3 to 0.6 g, the resonant frequency decreases from 113.2 to 110.5 Hz. This tells us the acceleration increases from 0.3 to 0.6 g, the resonant frequency decreases from 113.2 to 110.5 Hz. This stiness of this system is softening as amplitude increases. tells us the stiffness of this system is softening as amplitude increases. Since the peak frequency of the nonlinear hybrid energy harvester changes with the excitation Since the peak frequency of the nonlinear hybrid energy harvester changes with the excitation acceleration, accelerat the ion, t optimal he optimal load loof ad o the f the ener ener gy gy h harvester arvester also ch also changes. anges. At the re At the sonant fre resonant quenc frequency y, when , when the acceleration is 0.2 and 0.45 g, the change of the output peak voltage and power of the piezoelectric the acceleration is 0.2 and 0.45 g, the change of the output peak voltage and power of the piezoelectric energy harvesting unit with the piezoelectric load is as shown in Figure 6, and the electromagnetic energy harvesting unit with the piezoelectric load is as shown in Figure 6, and the electromagnetic energy harvesting unit is the optimal load at this time. The data shown in Figure 6 are averaged over energy harvesting unit is the optimal load at this time. The data shown in Figure 6 are averaged over several tests. several tests. 0.45g 0.2g 0 100 200 300 400 500 600 Resistance /Kohm Figure 6. Piezoelectric energy harvesting unit output varies with piezoelectric load. Figure 6. Piezoelectric energy harvesting unit output varies with piezoelectric load. Acoustics Accor 2019ding , 2, x FO toR PE theER R results EVIEW shown in Figure 6, the higher the acceleration, the higher the output 8 of 12 According to the results shown in Figure 6, the higher the acceleration, the higher the output voltage of the piezoelectric unit. When the excitation accelerations are 0.2 and 0.45 g, the maximum voltage of the piezoelectric unit. When the excitation accelerations are 0.2 and 0.45 g, the maximum are 140 and 190 kΩ, respectively. Therefore, the peak load of the piezoelectric unit increases as the output out powers put poof wers of t the piezoelectric he piezoelectric unit unit ar are e 0.085 0.085 an and d 0. 0.5 5 mmW W and and the corresp the corr onding op esponding tima optimal l loads loads acceleration increases. are 140 and 190 kW, respectively. Therefore, the peak load of the piezoelectric unit increases as the At the resonant frequency point of the system, when the excitation accelerations are 0.2 and 0.45 acceleration increases. g, the changes in the output voltage and power of the electromagnetic energy harvesting unit with At the resonant frequency point of the system, when the excitation accelerations are 0.2 and 0.45 g, the electromagnetic load are as shown in Figure 7, wherein the piezoelectric energy harvesting unit the changes in the output voltage and power of the electromagnetic energy harvesting unit with the is connected with the optimal load. When the excitation accelerations are 0.2 and 0.45 g, the maximum electromagnetic load are as shown in Figure 7, wherein the piezoelectric energy harvesting unit is output powers of the electromagnetic unit are 0.14 and 1.19 mW and the corresponding optimal loads connected with the optimal load. When the excitation accelerations are 0.2 and 0.45 g, the maximum are 17.5 and 21 Ω, respectively, that is, the optimal load of the electromagnetic unit also increases output powers of the electromagnetic unit are 0.14 and 1.19 mW and the corresponding optimal loads with the increasing of excitation acceleration. The data shown in Figure 7 are averaged over several are 17.5 and 21 W, respectively, that is, the optimal load of the electromagnetic unit also increases with tests. the increasing of excitation acceleration. The data shown in Figure 7 are averaged over several tests. (a) (b) Figure Figure 7. 7. Electr Electr omagnetic omagnetic unit output vari unit output varies es with with ele electr ctromagnetic loa omagnetic load. d.( ( aa )) Volt Voltage, age, ( (b b) Po ) Power wer. . 4.3. Effect of Load on Energy Harvesting Characteristics When the excitation acceleration is 0.45 g and the loads of the piezoelectric unit are 50 and 170 kΩ, the output power of the PE energy harvester unit is as shown in Figure 8. In addition, when the loads of the electromagnetic unit are 5 and 15 Ω, the output power of the EM energy harvester unit is as shown in Figure 9. Figure 8. Effect of piezoelectric load on the PE energy harvesting unit. Voltage /V Acoustics 2019, 2, x FOR PEER REVIEW 8 of 12 are 140 and 190 kΩ, respectively. Therefore, the peak load of the piezoelectric unit increases as the acceleration increases. At the resonant frequency point of the system, when the excitation accelerations are 0.2 and 0.45 g, the changes in the output voltage and power of the electromagnetic energy harvesting unit with the electromagnetic load are as shown in Figure 7, wherein the piezoelectric energy harvesting unit is connected with the optimal load. When the excitation accelerations are 0.2 and 0.45 g, the maximum output powers of the electromagnetic unit are 0.14 and 1.19 mW and the corresponding optimal loads are 17.5 and 21 Ω, respectively, that is, the optimal load of the electromagnetic unit also increases with the increasing of excitation acceleration. The data shown in Figure 7 are averaged over several tests. (a) (b) Acoustics 2019, 1 389 Figure 7. Electromagnetic unit output varies with electromagnetic load. (a) Voltage, (b) Power. 4.3. Effect of Load on Energy Harvesting Characteristics 4.3. Eect of Load on Energy Harvesting Characteristics When the excitation acceleration is 0.45 g and the loads of the piezoelectric unit are 50 and 170 When the excitation acceleration is 0.45 g and the loads of the piezoelectric unit are 50 and 170 kW, kΩ, the output power of the PE energy harvester unit is as shown in Figure 8. In addition, when the the output power of the PE energy harvester unit is as shown in Figure 8. In addition, when the loads loads of the electromagnetic unit are 5 and 15 Ω, the output power of the EM energy harvester unit of the electromagnetic unit are 5 and 15 W, the output power of the EM energy harvester unit is as is as shown in Figure 9. shown in Figure 9. Acoustics 2019, 2, x FOR PEER REVIEW 9 of 12 Figure 8. Eect of piezoelectric load on the PE energy harvesting unit. Figure 8. Effect of piezoelectric load on the PE energy harvesting unit. Figure 9. Eect of electromagnetic load on the EM energy capture harvesting unit. Figure 9. Effect of electromagnetic load on the EM energy capture harvesting unit. According to the test results shown in Figures 8 and 9, the loads of the piezoelectric and According to the test results shown in Figures 8 and 9, the loads of the piezoelectric and electromagnetic unit can aect the output power of the energy harvester, and they also aect the energy electromagnetic unit can affect the output power of the energy harvester, and they also affect the harvesting frequency bandwidth. When the electromagnetic load remains unchanged, the energy energy harvesting frequency bandwidth. When the electromagnetic load remains unchanged, the harvesting frequency bandwidth of the piezoelectric unit at the load of 170 KW is larger than that at energy harvesting frequency bandwidth of the piezoelectric unit at the load of 170 KΩ is larger than 50 KW; at the same time, when the piezoelectric load remains unchanged, the electromagnetic energy that at 50 KΩ; at the same time, when the piezoelectric load remains unchanged, the electromagnetic harvesting frequency bandwidth at the load of 15 W is larger than that at 5 W. In addition, the resonant energy harvesting frequency bandwidth at the load of 15 Ω is larger than that at 5 Ω. In addition, the frequency of the energy harvester is increased by 0.5 Hz, but the electromagnetic load hardly aects resonant frequency of the energy harvester is increased by 0.5 Hz, but the electromagnetic load hardly the resonant frequency of the energy harvester. affects the resonant frequency of the energy harvester. In addition, the eects of piezoelectric and electromagnetic loads on the output power of the In addition, the effects of piezoelectric and electromagnetic loads on the output power of the electromagnetic unit and the piezoelectric unit are shown in Figures 10 and 11. The data shown in electromagnetic unit and the piezoelectric unit are shown in Figures 10 and 11. The data shown in Figures 10 and 11 are averaged over several tests. The output power of the piezoelectric unit increases Figures 10 and 11 are averaged over several tests. The output power of the piezoelectric unit increases with the increase in the electromagnetic load, and the output power of the electromagnetic unit with the increase in the electromagnetic load, and the output power of the electromagnetic unit decreases ﬁrst and then increases with the increase in the piezoelectric load, and up to the minimum decreases first and then increases with the increase in the piezoelectric load, and up to the minimum value at the optimal load of the piezoelectric unit. The reason for this is that the coupling eect of value at the optimal load of the piezoelectric unit. The reason for this is that the coupling effect of the the electromagnetic unit to the energy harvesting system is mainly represented by the damping eect. electromagnetic unit to the energy harvesting system is mainly represented by the damping effect. Therefore, the vibration amplitude of the energy harvesting system is increased, and the output power Therefore, the vibration amplitude of the energy harvesting system is increased, and the output power of the piezoelectric unit is increased. The output power of the piezoelectric unit is up to the maximum value when the load is the optimal load. At this time, the stiffness feedback to the energy harvesting system is maximized. Figure 10. Effect of electromagnetic load on maximum piezoelectric power. Acoustics 2019, 2, x FOR PEER REVIEW 9 of 12 Figure 9. Effect of electromagnetic load on the EM energy capture harvesting unit. According to the test results shown in Figures 8 and 9, the loads of the piezoelectric and electromagnetic unit can affect the output power of the energy harvester, and they also affect the energy harvesting frequency bandwidth. When the electromagnetic load remains unchanged, the energy harvesting frequency bandwidth of the piezoelectric unit at the load of 170 KΩ is larger than that at 50 KΩ; at the same time, when the piezoelectric load remains unchanged, the electromagnetic energy harvesting frequency bandwidth at the load of 15 Ω is larger than that at 5 Ω. In addition, the resonant frequency of the energy harvester is increased by 0.5 Hz, but the electromagnetic load hardly affects the resonant frequency of the energy harvester. In addition, the effects of piezoelectric and electromagnetic loads on the output power of the electromagnetic unit and the piezoelectric unit are shown in Figures 10 and 11. The data shown in Figures 10 and 11 are averaged over several tests. The output power of the piezoelectric unit increases with the increase in the electromagnetic load, and the output power of the electromagnetic unit decreases first and then increases with the increase in the piezoelectric load, and up to the minimum value at the optimal load of the piezoelectric unit. The reason for this is that the coupling effect of the Acoustics 2019, 1 390 electromagnetic unit to the energy harvesting system is mainly represented by the damping effect. Therefore, the vibration amplitude of the energy harvesting system is increased, and the output power of t of the piezoelectric he piezoe unit lectric is incr unit eased. is incre The ased. The ou output power tput power of of the piezoelectric the piezunit oeleis ctri up c uni to the t is up maximum to the ma value ximum when val the ue when the load is the loptimal oad is the op load.ti At ma this l loa time, d. Atthe this sti ti me, the sti ness feedback ffness to feedba the ener ck to the energy gy harvesting harvestin system isg maximized. system is maximized. Acoustics 2019, 2, x FOR PEER REVIEW 10 of 12 Figure 10. Eect of electromagnetic load on maximum piezoelectric power. Figure 10. Effect of electromagnetic load on maximum piezoelectric power. Figure 11. Eect of piezoelectric load on maximum electromagnetic power. Figure 11. Effect of piezoelectric load on maximum electromagnetic power. When the excitation acceleration is 0.2 g and the force between the magnets is the attraction force, When the excitation acceleration is 0.2 g and the force between the magnets is the attraction through the experimental tests, the optimal loads of the PE and EM units of nonlinear energy harvester force, through the experimental tests, the optimal loads of the PE and EM units of nonlinear energy are 152.5 kW and 17.3 W. The optimal loads of the PE and EM units of the corresponding linear energy harvester are 152.5 kΩ and 17.3 Ω. The optimal loads of the PE and EM units of the corresponding harvester (without ﬁxed magnets and the nonlinear magnetic forces is free) are 133 kW and 15.5 W, linear energy harvester (without fixed magnets and the nonlinear magnetic forces is free) are 133 kΩ respectively. At the same time, the output powers at the optimal load of the nonlinear hybrid energy and 15.5 Ω, respectively. At the same time, the output powers at the optimal load of the nonlinear harvester and the corresponding linear energy harvester are shown in Figure 12. hybrid energy harvester and the corresponding linear energy harvester are shown in Figure 12. According to the results shown in Figure 10, the output power of the nonlinear energy harvester is larger than that of the corresponding linear energy harvester. Under the same excitation acceleration, the output powers are 0.44 and 0.4 mW, respectively. Due to the nonlinear magnetic force, the resonant frequency of the nonlinear energy harvester decreased, and the resonant frequency of the linear energy harvester and the nonlinear energy harvester are measured to be 119 Hz and 113.5 Hz, respectively. Therefore, the nonlinear energy harvester can increase the output power and reduce the resonant frequency, and it is more suitable for low-frequency vibration energy harvesting. Figure 12. Comparison between output of linear and nonlinear hybrid energy harvester. According to the results shown in Figure 10, the output power of the nonlinear energy harvester is larger than that of the corresponding linear energy harvester. Under the same excitation acceleration, the output powers are 0.44 and 0.4 mW, respectively. Due to the nonlinear magnetic force, the resonant frequency of the nonlinear energy harvester decreased, and the resonant frequency of the linear energy harvester and the nonlinear energy harvester are measured to be 119 Hz and 113.5 Hz, respectively. Therefore, the nonlinear energy harvester can increase the output power and reduce the resonant frequency, and it is more suitable for low-frequency vibration energy harvesting. 5. Conclusions According to the electromechanical coupling equation of the PE-EM hybrid harvester, the expressions of the vibration response, output voltage, current, and power of the nonlinear hybrid energy harvester under harmonic excitation were derived. The output characteristics of the nonlinear Acoustics 2019, 2, x FOR PEER REVIEW 10 of 12 Figure 11. Effect of piezoelectric load on maximum electromagnetic power. When the excitation acceleration is 0.2 g and the force between the magnets is the attraction force, through the experimental tests, the optimal loads of the PE and EM units of nonlinear energy harvester are 152.5 kΩ and 17.3 Ω. The optimal loads of the PE and EM units of the corresponding linear energy harvester (without fixed magnets and the nonlinear magnetic forces is free) are 133 kΩ and 15.5 Ω, respectively. At the same time, the output powers at the optimal load of the nonlinear Acoustics 2019, 1 391 hybrid energy harvester and the corresponding linear energy harvester are shown in Figure 12. Figure 12. Comparison between output of linear and nonlinear hybrid energy harvester. Figure 12. Comparison between output of linear and nonlinear hybrid energy harvester. 5. Conclusions According to the results shown in Figure 10, the output power of the nonlinear energy harvester is larger than that of the corresponding linear energy harvester. Under the same excitation According to the electromechanical coupling equation of the PE-EM hybrid harvester, acceleration, the output powers are 0.44 and 0.4 mW, respectively. Due to the nonlinear magnetic the expressions of the vibration response, output voltage, current, and power of the nonlinear force, the resonant frequency of the nonlinear energy harvester decreased, and the resonant hybrid energy harvester under harmonic excitation were derived. The output characteristics of the frequency of the linear energy harvester and the nonlinear energy harvester are measured to be 119 nonlinear hybrid harvester under harmonic excitation were also studied by the experimental testing. Hz and 113.5 Hz, respectively. Therefore, the nonlinear energy harvester can increase the output From the results of theoretical analysis and experimental tests, it was observed that: (1) The average power and reduce the resonant frequency, and it is more suitable for low-frequency vibration energy power, output voltage, and output current of a nonlinear hybrid energy harvester increases with the harvesting. increase in excitation acceleration under harmonic excitation, and the output power increases linearly with the acceleration, and is also aected by the load. (2) When the PE-EM hybrid energy harvester 5. Conclusions operates at the optimal load and the peak frequency, the average output power can reach the maximum According to the electromechanical coupling equation of the PE-EM hybrid harvester, the value, and the increase in the load of the piezoelectric unit makes the peak frequency of the energy expressions of the vibration response, output voltage, current, and power of the nonlinear hybrid harvesting system increase. (3) Compared with linear structures, the designed nonlinear PE-EM hybrid energy harvester under harmonic excitation were derived. The output characteristics of the nonlinear energy harvesting has better environmental adaptability and is more suitable for harvesting energy in low-frequency environments. Author Contributions: H.L., S.G., P.L. did experiments and data analyses, J.W. helped editing and revised the manuscript. H.L. and J.W. discussed the concept of the article. Funding: This research received no external funding. 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Nonlinear Vibration; Tianjin Science and Technology Press: Tianjin, China, 1983. © 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

Acoustics – Multidisciplinary Digital Publishing Institute

**Published: ** Apr 18, 2019

**Keywords: **nonlinearity; hybrid energy harvester; piezoelectric; electromagnetic; harmonic excitation

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