# Electromechanical Performance Analysis of the Hybrid Piezoelectric-Electromagnetic Energy Harvester under Rotary Magnetic Plucking Excitation

Authors: Hongyan Wang; Jiarui Hu; Gang Sun; Liying Zou
Journal: Shock and Vibration
 (2021)

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## Abstract

This paper presents an analysis of the hybrid piezoelectric-electromagnetic energy harvester (P-EMEH) driven by contactless rotary magnetic plucking. A lumped-parameter model of the hybrid P-EMEH is developed, and the model parameters are determined from the finite element analysis (FEA) method. A parametric study is conducted to investigate the effects of driving force parameters, load resistance, and electromechanical coupling strengths (EMCSs) on the maximal displacements and velocities, average power inputs and outputs, and energy efficiencies of the system for indicating the performance of the hybrid P-EMEH. The results show that the hybrid P-EMEH can obtain the improved power inputs by reducing the gyration radii of the rotary magnet and shortening the gaps between the two magnets. The structural vibrations can be strongly suppressed owing to the optimal piezoelectric power outputs, which can lead to the occurrence of valleys’ power of the electromagnetic element. At weak coupling, the hybrid P-EMEH can achieve higher power outputs than the single piezoelectric energy harvester (PEH) and the single electromagnetic energy harvester (EMEH). At strong coupling, the use of the PEH is more advantageous for energy harvesting due to wider power bandwidths at high dimensionless frequencies when compared with the hybrid P-EMEH. This work provides a fundamental understanding on the effect of load resistance and EMCSs on the dynamic and electrical characteristics of the magnetically plucked hybrid P-EMEH.


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## Body

## 1. Introduction

With the rapid development of the self-sustained sensor networks and wireless communication technology, scavenging energy from the ambient environment to generate sustainable electricity for the low-power electronic devices and the wireless sensor nodes has received extensive attention. Various types of energy sources, such as light, heat, radio frequency, and mechanical motions, can be extracted from the ambient environment [1]. Among these, mechanical vibrations have received the most attention in the past few years because of their abundance in the ambient environment [2]. The rotational vibration is often visible in the ambient environment, ranging from fluid flow [3] and machinery operation [4] to human motion [5]. The rotational vibration energy can be converted into electricity by various transduction mechanisms including piezoelectric [6], electromagnetic [7], electrostatic [8], and triboelectric [9]. Among these, the piezoelectric and electromagnetic energy harvesters have been paid much more attention from researchers due to their advantages of simple configuration and high output performance.The researchers have proposed various types of rotational energy harvesters for vibration energy harvesting from rotating environments, such as the torsional piezoelectric vibration energy harvesters attached to a rotary oil drilling shaft [10], the rotating piezoelectric vibration energy harvesters based on the gravitational and centrifugal forces’ actions [11–13], the mechanically plucked piezoelectric energy harvesters [14], the magnetically plucked piezoelectric energy harvesters [15, 16], and the rotational electromagnetic energy harvesters based on rotational electromagnetic generator designs [7, 17]. Among these, the frequency upconversion rotational energy harvesters through magnetic plucking have gained more attention in recent years as they have the advantages of the contactless excitation design and high capability of problem solving in the frequency mismatch between the excitation and the harvester. In terms of the influence of system parameters, the studies on magnetically plucked rotational energy harvesters have mainly focused on excitation frequencies [16, 18], magnet arrangements [16–21], and rectified power estimation [22], while less attention has been paid on the effect of the load resistance and the mechanical coupling strength on the performance of the system, which is significant to design and optimize the magnetically plucked rotational energy harvester.This work aims to conduct a comprehensive analysis of the effect of the load resistance and the EMCSs on the performance of the hybrid piezoelectric-electromagnetic energy harvester (P-EMEH) driven by the rotary magnetic plucking. The proposed harvester in this study consists of a clamped-clamped piezoelectric beam, three magnets, and a stationary coil. A lumped-parameter model of the hybrid P-EMEH is established to analyze the performance of the harvester. The mechanical parameters and the piezoelectric EMCS are determined from the coupled electromechanical FEA. The electromagnetic electromechanical coupling coefficient (EMCC) and the equivalent magnetization of the magnets for the magnetic driving force calculation are determined from the magnetic field finite element analysis. Based on the determined parameters, a parametric study is then performed to analyze the effects of the dimensionless load resistance and dimensionless EMCSs on the maximal displacements and velocities, average power inputs and outputs, and energy conversion efficiencies for indicating the performance of the hybrid P-EMEH. The power performance of the hybrid P-EMEH is also investigated by comparison with the single PEH and the single EMEH at weak and strong couplings.


## 2. Configuration of the Hybrid P-EMEH

Figures1(a) and 1(b) show the physical model of a hybrid P-EMEH and its simplified model, respectively. It can be seen from Figure 1(a) that the hybrid P-EMEH consists of a clamped-clamped metal beam with two piezoelectric plates located at two sides of a central magnetic mass on the beam and an external induction coil fixed to a holder. The central magnetic mass includes a lower cuboidal magnet (magnet 2), an upper cylindered magnet (magnet 3), and a middle magnet holder. Magnet 2 is used to match an external rotary cuboidal magnet (magnet 1) to generate the magnet driving force. Magnet 3 is used to match a fixed coil to generate the induction current. The hybrid P-EMEH can be simplified as a single-degree-of-freedom mass-spring-damper model with a piezoelectric transducer and an electromagnetic transducer, as shown in Figure 1(b). The parameters m, k, and c are the equivalent mass, the equivalent stiffness, and the viscous damping of the harvester, respectively. Cp is the capacitance of the two piezoelectric plates in parallel. Lc and Rc are the internal induction and the internal resistance of the coil, respectively. θp and θe are the EMCCs related with the piezoelectric and electromagnetic elements, respectively. R1 and R2 are the load resistance connected to the piezoelectric and electromagnetic elements, respectively. Fz and u are the driven force and the vibration displacement of the mass m, respectively.Figure 1
(a) Physical model and (b) simplified model of the hybrid P-EMEH.
(a)(b)Figures2(a) and 2(b) show the relative positions among the three magnets and a coil in the three-dimensional spatial direction. From Figure 2(a), it can be seen that magnets 1 and 2 have the sizes of 2A1 × 2B1 × 2C1 and 2A2 × 2B2 × 2C2, respectively. Magnet 3 has the radius of r3 and the height of h3. J1, J2, and J3 are the magnetizations of magnets 1, 2, and 3, respectively. The interaction of magnets 1 and 2 is repulsive due to the opposite direction of the magnetization. The interaction of magnets 2 and 3 is attractive due to the same direction of the magnetization. dx, dy, and dz represent the central distances between magnets 1 and 2 in three axes, respectively. z3 presents the central distance between magnets 2 and 3 in the Z-axis. It can be seen from Figure 2(b) that the fixed coil has the inner radius of rmin, the outer radius of rmax, and the height of hc. When magnet 3 has the vibration displacement of u, the distance from the center of magnet 3 to the upper surface of the fixed coil is zc = zc0 − u, where zc0 is the initial position of magnet 3 in the Zc-axis.Figure 2
Relative positions among magnets 1, 2, and 3 and the fixed coil. (a) Magnets 1, 2, and 3. (b) Magnet 3 and the coil.
(a)(b)


## 3. Theoretical Model

The governing differential equation of the hybrid P-EMEH with the magnetic force excitation can be written as(1)mu¨=Fz−cu˙−ku−θpVp−θeIe,where (·) = ∂/∂t, Vp is the voltage across the load resistor R1, and Ie is the current through the load resistor R2. In this research, the magnetic interaction force is induced by the interaction between magnets 1 and 2 in the z-direction, which can be written as [23](2)Fz=J1⋅J24πμ0∑i=01∑j=01∑k=01∑l=01∑p=01∑q=01−1i+j+k+l+p+qϕz,where μ0 is the vacuum permeability. The parameters i, j, k, l, p, and q are equal to 0 or 1 according to the specific corner. The parameter ϕz is a function of the magnet dimensions and their gaps in 3 axes, which is given by(3)ϕz=−uijwpqlnr−uij−vklwpqlnr−vkl+uijvkltan−1uijvklrwpq−rwpq,where the lengths uij, vkl, and wpq correspond to the distance between the cube corners and their projections on the axes. r is a function of the parameters of uij, vkl, and wpq. These parameters are expressed as [23](4)uij=dx+−1jA2−−1iA1,vkl=dy+−1lB2−−1kB1,wpq=dz+−1qC2−−1pC1,r=uij2+vkl2+wpq21/2.When the revolving disk with magnet 1 rotates with the motor shaft (see Figure1(a)), the gaps dx, dy, and dz between magnets 1 and 2 can be expressed as [16](5)dx=rm+dx0−rmcosωrt−φ0,dy=rmsinωrt−φ0+dy0,dz=dz0+u,where ωr is the rotational angular frequency of the motor shaft, rm is the gyration radius of magnet 1, dx0, dy0, and dz0 are the initial gaps between magnets 1 and 2 in the x-, y-, and z-axes, respectively, and ϕ0 is the initial angular position of magnet 1 relative to magnet 2. In this paper, we determine dx0 = dy0 = ϕ0 = 0.The electromagnetic EMCC,θe, can be obtained by calculating the gradient of the magnetic flux in the coil:(6)θe=−dΦdu,where Φ is the total magnetic flux in the coil, Φ=NcξΦa, where Φa is the magnetic flux per unit cross-sectional area of the coil. Nc is the number of turns in the coil. ξ is the fill factor of the coil, ξ=NcAw/Ac, where Aw is the conductive area of a single wire, Aw=πdw2/4, where dw is the conductive diameter of a single wire. Ac is the cross-sectional area of the coil, Ac = (rmax − rmin) hc.The magnetic field at the position of a point of the coil is given by [24](7)B⟶=−μ04π∇⟶σ⟶⋅s⟶s3=μ04πs53σ⟶⋅s⟶s⟶−r2σ⟶,where ∇⟶ is a vector gradient, σ⟶ is the magnetic dipole moment of magnet 3, s⟶ is a vector from the magnetic core to a point of the coil, and s is its magnitude. According to Figure 2(b), s⟶=re^r+z20−u−z1e^z, where r is the radius of the wire loop and e^r and e^z are the unit vector along the xc- and zc-axis, respectively. Substituting the vector s⟶=re^r+z20−u−z1e^z into equation (7) and simplifying this equation, the magnetic field B⟶ can be rewritten as(8)B⟶=μ0σ4π3rzc−ze^r−r2−2zc−z2e^zr2+zc−z25/2,where σ is the magnitude of the magnetic dipole moment of magnet 3. Dot multiplication of vectors B⟶ and e^z and then taking integration of the surface of a circle enclosed by a wire loop, the normal component of the magnetic flux through a wire loop with the radius r can be calculated as(9)Φz=∫02π∫0rB⟶e^zrdrdθ=μ0σ2r2r2+zc−z23/2.Integrating Φz given in equation (9) over the whole cross section of the coil and dividing by the area Ac, the average magnetic flux through the cross section of the coil can be written as(10)Φa=1Ac∫0hc∫rminrmaxΦzdrdz=1Ac∫0hc∫rminrmaxμ0σ2r2r2+zc−z23/2drdz.Sincezc = zc0 − u and zc is not relative to z, returning to equation (6), we can rewrite the electromagnetic EMCC as(11)θe=−NcξdΦadu=NcξdΦadzc−z.Substituting the expression of Φa in equation (10) into equation (11) and then taking the derivative with respect to zc − z, the electromagnetic EMCC, θe, can be written by [25](12)θe=Ncξμ0σ2Ac∑i,j=12−1i+jlnrci+zij−rcizij,where zij2=rci2+zj−u2, where rc1 = rmin, rc2 = rmax, z1 = zc0, and z2 = zc0 − hc.Considering the equivalent electrical model of the electromagnetic transducer [26], the circuit governing equation for electromagnetic transduction of the system can be written as(13)θeu˙=LcI˙e+IeRc+R2.Neglecting the inductance of the coil, which is commonly accepted for low frequencies [27, 28], the following equation can be derived from equations (1) and (13):(14)mu¨=Fz−c+θeRc+R2u˙−ku−θpVp.Considering the equivalent electrical model of the piezoelectric transducer [26], the circuit governing equation for piezoelectric transduction of the system can be written as(15)θpu˙=CpV˙p+VpR1.Defining the state-space vector(16)x1x2x3T=uu˙VpT,we can write the governing equations of the hybrid P-EMEH from equations (14) and (15) in the state-space form as(17)x˙1x˙2x˙3=x2Fzm−2ζ+ke21+r2ωnx2−ωn2x1−ωnkpx3Cpmωnkpx2mCp−ωnr1x3,where ωn=k/m, ζ=c/2km, kp2=θp2/kCp, ke2=θe2/mωnRc, r1=R1Cpωn, and r2=R2/Rc.The average electromagnetic coupling strengthk¯e2 in a rotational cycle is(18)k¯e2=ωr2π∫t0t0+2π/ωrke2utdt,where ωr is the rotational angular frequency of the driving magnet and t0 is the initial time for a rotational cycle.The average power input of the system in a rotational cycle is(19)P¯in=ωr2π∫t0t0+2π/ωrFzt⋅u˙tdt.The average power outputs on the resistorsR1 and R2 in a rotational cycle are(20)P¯R1=ωr2πR1∫t0t0+2π/ωrVp2tdt=ωrCpωn2πr1∫t0t0+2π/ωrVp2tdt,(21)P¯R2=ωrR22π∫t0t0+2π/ωIe2tdt=ke2mωnωr2π1+r2∫t0t0+2π/ωu˙2tdt.The total average power output of the hybrid energy harvesting system is(22)P¯total=P¯R1+P¯R2.The efficiency of the system is(23)η=P¯totalP¯in.When the coil circuit is open (r2 = 0 and θe = 0), equation (17) is reduced to(24)x˙1x˙2x˙3=x2Fzm−2ζωnx2−ωn2x1−ωnkpx3Cpmωnkpx2mCp−ωnr1x3,which is the governing equation of a single PEH in the state-space form.When the surface of the piezoelectric element is electrically shorted (r1 ⟶ ∞ and θp = 0), equation (17) is reduced to(25)x˙1x˙2=x2Fzm−2ζ+ke21+r2ωnx2−ωn2x1,which is the governing equation of a single EMEH in the state-space form.


## 4. System Parameter Determination

Based on the material and geometric parameters of the piezoelectric clamped-clamped beam listed in Table1 and the parameters of magnets and the coil listed in Table 2, in this section, the model parameters of the hybrid P-EMEH, including ωn, m, kp2, J1, J2, and σ, are determined based on the results from finite element physical field analysis.Table 1
Material and geometric parameters of the piezoelectric clamped-clamped beam.
ParameterValueClamped length of the metal beam (mm)85Thickness of the metal beam (mm)0.4Length of single piezoelectric plate,Lp (mm)15Thickness of single piezoelectric plate,hp (mm)0.2Width of the metal beam and piezoelectric plate,b (mm)12Length of the magnet holder (mm)12Width of the magnet holder (mm)12Central mass of the beam (including the magnet holder and magnets 2 and 3) (kg)0.005742Density of the metal beam (kg/m3)8920Density of the piezoelectric plate (kg/m3)7500Elastic modulus of the metal beam (GPa)113Poisson’s ratio of the metal beam0.35Vacuum permittivity of the piezoelectric plate,e0 (10−12 F/m)8.85Relative permittivity of the piezoelectric plate,ε11S/ε0, ε33S/ε01700, 1470Piezoelectric constants,d31, d33, and d15 (10−12 C/N)−274, 593, 741Piezoelectric compliance constants,ε11E, ε12E, ε13E, ε33E, ε44E, and ε66E (10−12 m2/N)16.5, −4.78, −8.45, 20.7, 43.5, 42.6Table 2
Parameters of magnets and the coil.
ParameterValueSize of magnet 1,A1 × B1 × C1 (mm3)4 × 4 × 1.5Size of magnet 2,A2 × B2 × C2 (mm3)4 × 4 × 1Radius of magnet 3,r3 (mm)5Height of magnet 3,h3 (mm)5Central distance between magnet 2 and magnet 3 in thez-axis, z3 (mm)8.5Residual flux density,Br (T)1.141Coercive force,Hc (A/m)−908775Permeability of vacuum,µ0 (N/A2)4π × 10−7Height of the coil,hc (mm)5Inner radius of the coil,rmin (mm)6Outer radius of the coil,rmax (mm)9Turns of the coil,Nc1000Conductive diameter of the wire,dw (mm)0.1Diameter of the wire (mm)0.123Internal resistance (Ω)90The theoretical parametersωn, m, and kp2 are determined from the coupled electromechanical FEA. Figure 3(a) shows the FE model of the clamped-clamped piezoelectric beam. When the circuit is shorted, the node voltage on the upper and lower surface of the two piezoelectric plates is coupled together and set to zero. When the circuit is open, the node voltage on the upper and lower surface of the two piezoelectric plates is coupled separately, and the voltage of the lower surface is set to zero. In FE modal analysis, the short-circuit and open-circuit resonance frequencies are obtained as fsc = 122.15 Hz and foc = 122.76 Hz, respectively. According to the equation kp2 = (ωoc2/ωsc2) − 1 [29, 30], the piezoelectric EMCS can be calculated as kp2 = 0.01, where ωsc and ωoc are the short-circuit and open-circuit resonant angular frequencies, respectively. The parameters ωsc = 2πfsc and ωoc = 2πfoc. In FE harmonic response analysis, based on the electrical short-circuit condition of the piezoelectric plate and the assumed damping ratio ζ = 0.02, a force of F = 1 N is applied to the center of the magnetic mass in the z-direction. Figure 3(b) shows the displacement response amplitudes, Uz, of the magnetic mass versus the dimensionless rotational frequency γ. The parameter γ = ω/ωn, where ω and ωn are the excitation angular frequency and the natural angular frequency, respectively. The parameter ωn = ωsc. It can be seen from Figure 3(b) that Uz = 5.697 mm at γ = 1. The equivalent stiffness of the beam can be calculated as k = 4388 N/m according to the equation of the magnification factor given in [31].(26)UzF/k=11−γ22+2ζγ2.Figure 3
FE model of the piezoelectric beam and displacement response of magnetic mass under 1 N applied force. (a) FE model. (b) Displacement versus dimensionless rotational frequencyγ.
(a)(b)The equivalent mass of the beam can be calculated asm = 0.0074 kg according to the equation m = k/ωn2. Since the two piezoelectric plates on the metal beam are electrically connected in parallel, the total capacitance of the piezoelectric element is Cp=2ε33Sε0Lpb/hp = 23.4 nF.The model parametersJ1 and J2 and σ are determined by the FE magnetic field analysis. Note that the magnetic parameters Br = 1.141 T and Hc = −908775 A/m given in Table 2 are provided for FE magnetic field analysis. The magnetic driving force obtained from FEA is used to determine the magnetizations J1 and J2 used in equation (2). The magnetic flux density obtained from FEA is used to determine the parameter σ used in equation (12). Figures 4(a) and 4(b) show the magnetic flux density distribution in the XOZ plane with the magnetic vector plots for the two-magnet and the three-magnet configurations, respectively. It can be seen from Figures 4(a) and 4(b) that, after magnet 3 is introduced to the original magnetic field induced by magnets 1 and 2, the magnetic flux density around magnets 1 and 2 is changed. It implied that magnets 1, 2, and 3 can interact with each other. In this paper, we consider the interaction force between magnets 2 and 3 as the internal force, and thus, equation (2) can be used to calculate the magnetic driving force in the field of the three-magnet configuration. Figures 4(c) and 4(d) show the theoretical and FE magnetic driving force Fz versus the gaps dz and dy, respectively. It can be seen from Figures 4(c) and 4(d) that the theoretical magnetic driving force with the parameters J1 = −J2 = 1.141 T and J1 = −J2 = 1.26 T, respectively, shows good agreements with FEA results for the two-magnet (magnets 1 and 2) and three-magnet (magnets 1, 2, and 3) arrangements. In Section 5, the parameter J1 = −J2 = 1.26 T will be used for the theoretical analysis of the dynamics and electrical characteristics of the hybrid P-EMEH.Figure 4
Magnetic flux density distribution in theXOZ plane for the (a) two-magnet configuration and (b) three-magnet configuration from FEA. (c) Magnetic driving force versus dz at dy = 0. (d) Magnetic driving force versus dy at dz = 0.
(a)(b)(c)(d)The magnetic flux Φz through an arbitrary wire loop in the normal direction can be obtained from FE magnetic field analysis and also be calculated from the theoretical model (see equation (9)). With a series of selected wire loops, we can determine the average magnetic flux density in the coil, Bave, as(27)Bave=1n1n2∑i=1n1∑j=1n21πri2Φzri,zj,where n1 is the number of wire loops in the xc-axis, n2 is the number of cross sections of the coil in the zc-axis, Φzri,zj is the magnetic flux through a given wire loop, where ri (i = 1, 2, 3, …, n1) is the radius of an arbitrary wire loop, and zj (j = 1, 2, 3, …, n2) is the location of the wire loop with respect to the upper surface of the coil along the zc-axis (see Figure 2(b)). In FEA, ri (i = 1, 2, 3, and 4) is given as 6 mm, 7 mm, 8 mm, and 9 mm, respectively. zj (j = 1, 2, 3, and 4) is given as 0 mm, 1 mm, 2 mm, 3 mm, 4 mm, and 5 mm, respectively. Thus, n1 = 4 and n2 = 6. In the theoretical model, we consider increments of 0.123 mm from r1 = rmin = 6 mm to rn1 = rmax = 9 mm in the positive xc-axis direction and from z1 = 0 to zn2 = hc = 5 mm in the positive z2-axis direction; thus, n1 = 26 and n2 = 41. Set z20 = 8.5 mm (corresponding to u = 0). Figure 5 shows the theoretical average magnetic flux densities as a function of the displacement u, which is matched to FEA results. It can be seen from Figure 5 that the theoretical average magnetic flux densities with the parameter σ = 0.3428 Am2 show good agreements with the results from FEA. The electromagnetic EMCS can be calculated by using the determined value of σ.Figure 5
Average magnetic flux densities in the coil versus vibration displacement of the harvester.


## 5. Parametric Study

In the parametric studies, the differential equations are solved using a Runge–Kutta method. The number of vibration cycles and the number of time steps in one period are given as 20 and 1000, respectively. The constant parameters are given asm = 0.0074 kg, wn = 2πfsc = 767.5 rad/s, ζ = 0.02, Cp = 23.4 nF, and Rc = 90 Ω. The rational radius of magnet 1 and the central distance between magnets 1 and 2 are originally set to be rm(0) = 30 mm and dz0(0) = 5 mm, respectively. The dimensionless parameters Ω = ωr/ωn, α = rm/rm(0), and β = dz0/dz0(0) are used in the parametric analysis. In Section 5.1, the effect of Ω, α, and β on the magnetic driving force and the average power inputs are analyzed. In Sections 5.2 and 5.3, the effect of the dimensionless load resistance and the EMCSs on the system performance is analyzed, respectively. In Section 5.4, the power outputs of the hybrid P-EMEH at strong and weak couplings are compared with those from the single PEH and the single EMEH.


### 5.1. Effect of Ω,α, and β on Magnetic Driving Force and Average Power Inputs

In this section, the electrical and the electromechanical parameters used for analysis are given asr1 = r2 = 1, kp2 = 0.01, and σ = 0.3428 Am2. Figures 6(a)–6(c) show the time responses of the magnetic driving force and the vibration velocity of the beam for three different Ω, respectively. It can be seen from Figures 6(a)–6(c) that when Ω = 1, the harvester works in the resonant mode with a natural vibration period of 8.19 ms, in which the rotational frequency of magnet 1 is synchronized with the beam vibration. When Ω = 1/4 and Ω = 1/8, it is observed that although given different excitation periods, 32.75 ms and 65.5 ms, the harvester vibrates at its natural period of 8.19 ms, which implies that the dominant vibration frequency of the harvester is equal to its natural frequency. As we all know, a maximal amplitude of the vibration occurs when a structure vibrates at its natural frequency. It is concluded that a multiple-peak frequency response is available in the nonresonant excitation case. In Figures 6(b) and 6(c), the gradual decay of vibration amplitude between the two excitation cycles indicates that the harvester works in a damped-free-vibration mode after each excitation. It should be noted that the velocity of the beam at off-resonance (see Figures 6(b) and 6(c)) is larger than that of the resonant case (Figure 6(a)). To illustrate the phenomenon, we substitute the parameter Vp in equation (15) into (14) and rearrange equation (14) as(28)u¨+2ωnζ+ζp+ζeu˙+ωnr2u=Fzm,where ωnr is the resonant frequency of the system, ωnr=ωn1+Ω2kp2r12/1+Ω2r12. ζp is the piezoelectric damping coefficient, ζp=kp2r1/21+Ω2r12. ζe is the electromagnetic damping coefficient, ζe=ke2/21+r2. Considering the average electromagnetic EMCS in a rotational period, the electromagnetic damping coefficients can be written as ζe=k¯e2/21+r2. It can be seen from equation (28) that, with a decrease in Ω, ζp increases and ωnr decreases. Higher ζp can lead to a faster decay of the velocity u˙ between two excitations. Lower ωnr means a lower stiffness of the system, which can lead to a higher initial velocity after each excitation. This is the reason why the velocity of the beam at off-resonance (see Figures 6(b) and 6(c)) is larger than that of the resonant case (Figure 6(a)). Figure 6(d) shows the frequency response of the average power inputs of the system in an excitation cycle. It can be seen from Figure 6(d) that there are multiple power peaks in a wide range of the frequency ratios. Define the power peaks from right to left, namely, the 1st peak frequency is the dimensionless resonance frequency Ω1; it can be seen from Figure 6(d) that the ith dimensionless peak frequency Ωi is almost equal to Ω1/i (i = 1, 2, 3, …, 9). The maximal average power inputs occur at the nonresonant frequency Ω4 and not at the resonant frequency Ω1. It shows that the power inputs from a nonresonant excitation can be more efficient than those from a resonant excitation. It is noted that the input power is affected by both the driving force Fz and the velocity u˙ (see equation (19)). Due to the coupling between Fz and u˙, the driving force Fz at each excitation is affected by the initial velocity of the beam (see Figures 6(a)–6(c)). It can be seen from Figures 6(a) and 6(b) that when Ω = 1 and 1/4, the effect of u˙ on Fz is not obvious. The difference is on that when Ω = 1/4, the direction of u˙ is the same with the direction of Fz, which leads to a higher average power input than that at Ω = 1 (see Figure 6(d)). It can be seen from Figures 6(b) and 6(c) that a lower initial velocity at Ω = 1/8 leads to a lower driving force peak when compared with that at Ω = 1/4. Thus, the average power input at Ω = 1/8 is lower than that at Ω = 1/4 (see Figure 6(d)). It means that the optimal power input can occur at a dimensionless frequency with a higher vibration velocity u˙ but with less effect on the input force Fz. In addition, higher average input power is achieved when u˙ and Fz have the same direction. In this analysis, the optimal average input power occurs at Ω4.Figure 6
Time response of input force and velocity and frequency responses of average power inputs. (a) Ω = 1. (b) Ω = 1/4. (c) Ω = 1/8. (d) Average power inputs for various Ω.α = β = 1, r1 = r2 = 1, kp2 = 0.01, and σ = 0.3428 Am2.
(a)(b)(c)(d)Figures7(a) and 7(b) show 3D plots of the average power inputs of the system with a series of Ω for various α and β, respectively. It can be seen from Figures 7(a) and 7(b) that, at an arbitrary peak frequency, the average power inputs increase with a decrease in α and β, which implies that a larger dynamic response can occur at smaller α or at smaller β. It is noted that the proper selections of α and β are necessary because the collision of magnet 2 with magnet 1 is easier to occur in a large displacement of the beam. Figure 8 shows the displacements and electromagnetic EMCS versus Ω at α = β = 1. It can be seen from Figure 8 that the range of the movement of the mass m is from −2 mm to 2 mm for the given parameter configuration. Since the absolute displacements of the mass m are less than the initial gaps between two magnets (|u|<2.5 mm), the collision between magnets 2 and 1 does not occur. In Sections 5.2–5.4, the constant parameters α = β = 1 will be used for the analysis of the effect of the load resistance and the EMCSs on the system performance, respectively.Figure 7
Average power inputs with a series of Ω,α, and β. (a) β = 1. (b) α = 1. r1 = r2 = 1, kp2 = 0.01, and σ = 0.3428 Am2.
(a)(b)Figure 8
Displacement response of the massm versus Ω at α = β = 1, r1 = r2 = 1, kp2 = 0.01, and σ = 0.3428 Am2.


### 5.2. Effect of Dimensionless Load Resistance on the System Performance

Let the electromechanical parameters be given askp2 = 0.01 and σ = 0.3428 Am2. A parametric sweep simulation is performed in this analysis. Figures 9(a)–9(f) show 3D plots of the maximal displacements, the maximal velocities, the average power inputs, the piezoelectric average power outputs, the electromagnetic average power outputs, and the power conversion efficiencies of the hybrid P-EMEH with a series of Ω and r1 at r2 = 1, respectively. It can be seen from Figures 9(a)–9(e) that, at arbitrary Ωi = Ω1/i (i = 1, 2, 3, …, 9), the displacements umax, the velocities u˙max, the power inputs P¯in, and the electromagnetic power outputs P¯R2 generate the valleys (see Figures 9(a)–9(c) and 9(e)). The piezoelectric power outputs P¯R1 and the power conversion efficiencies η generate the peaks (see Figure 9(d)). The occurrence of the peaks or the valleys is induced by the forward or backward actions of electromechanical energy conversion. When optimal r1 is given to the piezoelectric transducer, the piezoelectric transducer can obtain the maximal power outputs. Under the backward action of the piezoelectric high power outputs, the vibrations of the harvester are relatively strongly suppressed, which leads to an obvious decrease in P¯in and P¯R2. The valleys occur at optimal r1 (see Figures 9(a)–9(c) and 9(e)). In Figure 9(f), the highest energy efficiency of the system appears at Ω4 and not at Ω1, which shows that the nonresonant energy harvesting can achieve higher power outputs than the resonant energy harvesting.Figure 9
3D plots for dynamic and electrical outputs with a series of Ω andr1. (a) Maximal displacements. (b) Maximal velocities. (c) Average power inputs. (d) Piezoelectric average power outputs. (e) Electromagnetic average power outputs. (f) Power conversion efficiencies. r2 = 1, kp2 = 0.01, and σ = 0.3428 Am2.
(a)(b)(c)(d)(e)(f)Figures10(a)–10(f) show 3D plots of the maximal displacements, the maximal velocities, the average power inputs, the piezoelectric average power outputs, the electromagnetic average power outputs, and the efficiencies of the hybrid P-EMEH with a series of Ω and r2 at r1 = 1, respectively. It can be seen from Figures 10(a)–10(d) that, at arbitrary Ωi = Ω1/i (i = 1, 2, 3, …, 9), with an increase in r2, the displacements umax, the velocities u˙max, the power inputs P¯in, and the piezoelectric power outputs P¯R1 increase. The electromagnetic power outputs P¯R2 and the power conversion efficiencies η generate the peaks around r1 = 1. The variation tendency can also be explained from the forward and backward actions of electromechanical energy conversion. When optimal r2 is given to the electromagnetic element, the electromagnetic element can obtain the optimal power output (see Figure 10(e)). From the expression term of ζe given in equation (28), it can be seen that an increase in r2 can lead to the decrease in ζe. The total damping coefficient of the system decreases accordingly. Therefore, an increase in r2 can induce the increase in u and u˙ (see Figures 10(a) and 10(b)). According to equations (19), (15), and (20), an increase in u˙ and u can induce the increase in Pin and P¯R1, respectively. Thus, P¯in and P¯R1 increase with the increase of r2 (see Figures 10(c) and 10(d)). It can be seen from Figure 10(f) that the highest power conversion efficiency appears at Ω4.Figure 10
3D plots for dynamic and electrical outputs with a series of Ω andr2. (a) Maximal displacements. (b) Maximal velocities. (c) Average power inputs. (d) Piezoelectric average power outputs. (e) Electromagnetic average power outputs. (f) Power conversion efficiencies. r1 = 1, kp2 = 0.01, and σ = 0.3428 Am2.
(a)(b)(c)(d)(e)(f)


### 5.3. Effect of Electromechanical Coupling Strengths on the System Performance

Figures11(a)–11(f) show 3D plots of the maximal displacements, the maximal velocities, the average power inputs, the piezoelectric average power outputs, the electromagnetic average power outputs, and the efficiencies of the hybrid P-EMEH with a series of Ω and kp2 at r1 = r2 = 1 and σ = 0.3428 Am2, respectively. It can be seen from Figures 11(a)–11(e) that, at arbitrary Ωi = Ω1/i (i = 1, 2, 3, …, 9), the displacements umax, the velocities u˙max, the power inputs P¯in, and the electromagnetic power outputs P¯R2 decrease with an increase in kp2. The piezoelectric average power outputs P¯R1 at each Ωi (i = 1, 2, …, 9) can increase until the saturation. The low-frequency saturation power is significantly higher than the high-frequency saturation power. Hence, the use of the hybrid P-EMEH with higher kp2 is advantageous for scavenging rotational energy at low frequencies. In Figure 11(f), the power conversion efficiencies of the system have the same tendency with the piezoelectric power outputs given in Figure 11(d). It shows that the piezoelectric energy conversions are dominant in the hybrid energy harvesting system due to the use of higher kp2.Figure 11
3D plots for dynamic and electrical outputs with a series of Ω andkp2. (a) Maximal displacements. (b) Maximal velocities. (c) Average power inputs. (d) Piezoelectric average power outputs. (e) Electromagnetic average power outputs. (f) Power conversion efficiencies. r1 = r2 = 1 and σ = 0.3428 Am2.
(a)(b)(c)(d)(e)(f)To study the effect of the electromagnetic EMCS on the system performance, the average electromagnetic EMCSs in a rotational period are calculated. Figures12(a) and 12(b) show the instantaneous and average electromagnetic EMCSs versus Ω at σ = 0.3428 Am2 and σ = 2.5313 Am2, respectively. It can be seen from Figures 12(a) and 12(b) that, for an arbitrary given value of σ, the average electromagnetic EMCS can keep almost a constant value for various Ω. A higher value of σ can achieve higher k¯e2. Figures 13(a)–13(f) show 3D plots of the maximal displacements, the maximal velocities, the average power inputs, the piezoelectric average power outputs, the electromagnetic average power outputs, and the efficiencies of the hybrid P-EMEH with a series of Ω and k¯e2 at r1 = r2 = 1 and kp2 = 0.01, respectively. It can be seen from Figures 13(a)–13(e) that the maximal displacements umax, the maximal velocities u˙max, the average power inputs P¯in, and the piezoelectric average power outputs P¯R1 decrease with the increase in k¯e2. The average electromagnetic power outputs P¯R2 can increase until the saturation at each Ωi (i = 1, 2, …, 9). Because low-frequency saturation power is significantly higher than high-frequency saturation power for the electromagnetic energy harvesting, it is favorable for the hybrid P-EMEH with higher k¯e2 to harvest rotational energy at low frequencies. In Figure 13(f), the energy efficiencies of the system show the same tendency with the electromagnetic power outputs given in Figure 13(e). It shows that the electromagnetic energy conversions are dominant in the hybrid energy harvesting system due to the use of higher k¯e2.Figure 12
Instantaneous and average electromagnetic EMCSs versus Ω atr1 = r1 = 1 and kp2 = 0.01. (a) σ = 0.3428 Am2. (b) σ = 2.5313 Am2.
(a)(b)Figure 13
3D plots for dynamic and electrical outputs with a series of Ω andk¯e2. (a) Maximal displacements. (b) Maximal velocities. (c) Average power inputs. (d) Piezoelectric average power outputs. (e) Electromagnetic average power outputs. (f) Power conversion efficiencies. r1 = r1 = 1 and kp2 = 0.01.
(a)(b)(c)(d)(e)(f)


### 5.4. Power Comparison for the PEH, EMEH, and Hybrid P-EMEH at Weak and Strong Couplings

In this section, the power performance of the hybrid P-EMEH at weak and strong EMCS is compared with that from the single PEH and single EMEH. In order to distinguish strong coupling regions from weak ones, Shu and Lien [32] proposed using the parameter γ1=kp2/ζ as the indicator of the EMCS of the PEH and defined a criterion of strong coupling, γ1≥4ζ+1, for a pure resistor circuit. They also referred to that the EMCS is very weak if γ1≪1. Challa et al. [33] proposed using the parameter γ2=θe2/RcC as the indicator of the EMCS of the EMEH and defined the weak coupling (γ2≪1) and the strong coupling (γ2≫1). Letting ke2=θe2/mωnRc and ζ=C/2km, γ2 can be rewritten as γ2=ke2/2ζ. Li et al. [34] proposed using the parameter γ3=kp2+ke2/2ζ as the indicator of the EMCS of the hybrid P-EMEH and defined the weak coupling (kp2+ke2/ζ≪2) and the strong coupling (kp2+ke2/ζ≫2). Based on these criteria above and letting ke2 = k¯e2 and supposing ζ = 0.02 in this work, the weak coupling is confined in the regions of kp2≪0.02, k¯e2≪0.04, and kp2+k¯e2≪0.04, respectively, for the PEH, the EMEH, and the hybrid P-EMEH. The strong coupling is confined in the regions of kp2≥0.0816, k¯e2≫0.04, and kp2+k¯e2≫0.04, respectively, for the PEH, the EMEH, and the hybrid P-EMEH. Suppose kp2 = k¯e2 = 0.002 for the weak coupling and kp2 = k¯e2 = 0.2 for the strong coupling in this research.Figures14(a)–14(c) show the average power inputs, power outputs, and efficiencies for three different types of energy harvesters (PEH, EMEH, and hybrid P-EMEH) with optimal r1 and r2 at kp2 = k¯e2 = 0.002, respectively. In this study, optimal r1 and r2 are derived from the optimal average power outputs of the PEH and the EMEH, respectively, and they are used as the load resistance of the hybrid P-EMEH. It can be seen from Figures 14(a)–14(c) that there are close peak power inputs for the PEH, the EMEH, and the hybrid P-EMEH. However, the hybrid P-EMEH has higher peak power outputs and efficiencies than the PEH and the EMEH. It shows that, under a weak coupling condition, higher kp2 can convert more mechanical energy into electricity with less effect on the dynamics of the system. Therefore, it is advantageous to use the hybrid P-EMEH to harvest the vibration energy under a weak electromechanical coupling condition.Figure 14
Power comparisons for the PEH, EMEH, and hybrid P-EMEH with optimalr1 and r2. (a) Average power inputs. (b) Total average power outputs. (c) Power conversion efficiencies. kp2 = k¯e2 = 0.002.
(a)(b)(c)Figures15(a)–15(c) show the average power inputs, power outputs, and efficiencies for three different types of energy harvesters (PEH, EMEH, and hybrid P-EMEH) with optimal r1 and r2 at kp2 = k¯e2 = 0.2, respectively. It can be seen from Figures 15(a) and 15(b) that, for the PEH and the hybrid P-EMEH, there are obvious double peaks around each higher dimensionless frequency Ωi (i = 1, 2, 3, 4). One is located at Ωi (i = 1, 2, 3, and 4), and another one is located at Ωi1+kp2 (i = 1, 2, 3, and 4). The occurrence of the double peaks can be reasonably explained from equation (28). In equation (28), when r1 ⟶ 0 (electrical short circuit), the dimensionless resonant frequency is Ω1. When r1 ⟶ ∞ (electrical open circuit), the dimensionless resonant frequency is Ω11+kp2. According to [35], the piezoelectric optimal power outputs can occur at both short-circuit and open-circuit resonant frequencies. In this analysis, the double-peak power can be observed owing to higher kp2. Nonresonant energy harvesting has similar characteristics with resonant energy harvesting owing to the fact that the model frequency of the harvester is dominant in the vibration energy harvesting process, but with the decrease in Ωi and Ωi1+kp2 (i = 2, 3, …, 9), the two nonresonant excitation frequencies become closer and closer to each other until the frequency coincidence. Thus, there are single-peak power outputs at lower Ωi. The EMEH shows single-peak power characteristics owing to the fact that the electromagnetic EMCS is not relative to the resonance frequency of the harvester. It can be seen from Figure 15(b) that the PEH has wider power bandwidths than the hybrid P-EMEH at relatively high Ωi (i = 1, 2, 3, 4), but it has close power outputs with the hybrid P-EMEH at relatively low Ωi (i = 5, 6, …, 9). The reason to this phenomenon can be explained by the power conversion efficiencies of the harvesters given in Figure 15(c). It can be seen from Figure 15(c) that, at higher Ωi (i = 1, 2, 3, 4), the PEH and the hybrid P-EMEH have almost the same peak power efficiency, which means that power saturation occurs with given parameters kp2 = k¯e2 = 0.2. In this case, the same power outputs are theoretically possible for the hybrid EMEH and the PEH. However, owing to the single-peak power characteristics of electromagnetic energy harvesting and its energy dissipation in the coil, the power outputs from the hybrid P-EMEH become lower than those from the PEH at higher Ωi (i = 1, 2, 3, and 4) (see Figure 15(b)). For lower Ωi (i = 5, 6, …, 9), it can be seen from Figure 15(c) that the peak power efficiency of the PEH is lower than that of the hybrid P-EMEH, which means that the piezoelectric power saturation cannot not be achieved with given parameter kp2 = 0.2 at lower Ωi (i = 5, 6, …, 9). Theoretically, the power outputs of the PEH should be lower than those of the hybrid P-EMEH. However, due to relatively less power loss in the piezoelectric element when compared with the electromagnetic element, the PEH can still provide almost the same power outputs with the hybrid P-EMEH. The results show that it is more advantageous to use the single PEH to harvest the rotational vibration energy than the EMEH and hybrid P-EMEH under a strong electromechanical coupling condition. It can be seen from Figure 15(c) that the power conversion efficiencies of the three types of harvesters increase with a decrease in Ωi (i = 1, 2, …, 9). A higher power conversion efficiency means that higher kp2 or k¯e2 is required to achieve power saturation. It is the reason why the PEH with the same parameter kp2 = 0.2 can achieve power saturation at higher Ωi (i = 1, 2, 3, 4), but not at lower Ωi (i = 5, 6, …, 9). It should be noted that, for the single EMEH, it has less advantage for magnetically plucked vibration energy harvesting at low frequencies due to relatively lower power outputs than those at middle and high frequencies (see Figure 15(b)).Figure 15
Power comparisons for the PEH, EMEH, and hybrid P-EMEH with optimalr1 and r2. (a) Average power inputs. (b) Total average power outputs. (c) Power conversion efficiencies. kp2 = k¯e2 = 0.2.
(a)(b)(c)The significance of the analysis of the piezoelectric weak and strong coupling is that it provides a more comprehensive understanding on the power performance of the PEH in different coupling regimes. The results from weak coupling analysis show the advantages of higher power outputs of the hybrid P-EMEH when compared with the PEH and EMEH (see Figure14(b)). The results from strong coupling analysis show the advantages of wider power bandwidths of the PEH at high dimensionless frequencies when compared with the hybrid P-EMEH and EMEH (see Figure 15(b)).


## 5.1. Effect of Ω,α, and β on Magnetic Driving Force and Average Power Inputs

In this section, the electrical and the electromechanical parameters used for analysis are given asr1 = r2 = 1, kp2 = 0.01, and σ = 0.3428 Am2. Figures 6(a)–6(c) show the time responses of the magnetic driving force and the vibration velocity of the beam for three different Ω, respectively. It can be seen from Figures 6(a)–6(c) that when Ω = 1, the harvester works in the resonant mode with a natural vibration period of 8.19 ms, in which the rotational frequency of magnet 1 is synchronized with the beam vibration. When Ω = 1/4 and Ω = 1/8, it is observed that although given different excitation periods, 32.75 ms and 65.5 ms, the harvester vibrates at its natural period of 8.19 ms, which implies that the dominant vibration frequency of the harvester is equal to its natural frequency. As we all know, a maximal amplitude of the vibration occurs when a structure vibrates at its natural frequency. It is concluded that a multiple-peak frequency response is available in the nonresonant excitation case. In Figures 6(b) and 6(c), the gradual decay of vibration amplitude between the two excitation cycles indicates that the harvester works in a damped-free-vibration mode after each excitation. It should be noted that the velocity of the beam at off-resonance (see Figures 6(b) and 6(c)) is larger than that of the resonant case (Figure 6(a)). To illustrate the phenomenon, we substitute the parameter Vp in equation (15) into (14) and rearrange equation (14) as(28)u¨+2ωnζ+ζp+ζeu˙+ωnr2u=Fzm,where ωnr is the resonant frequency of the system, ωnr=ωn1+Ω2kp2r12/1+Ω2r12. ζp is the piezoelectric damping coefficient, ζp=kp2r1/21+Ω2r12. ζe is the electromagnetic damping coefficient, ζe=ke2/21+r2. Considering the average electromagnetic EMCS in a rotational period, the electromagnetic damping coefficients can be written as ζe=k¯e2/21+r2. It can be seen from equation (28) that, with a decrease in Ω, ζp increases and ωnr decreases. Higher ζp can lead to a faster decay of the velocity u˙ between two excitations. Lower ωnr means a lower stiffness of the system, which can lead to a higher initial velocity after each excitation. This is the reason why the velocity of the beam at off-resonance (see Figures 6(b) and 6(c)) is larger than that of the resonant case (Figure 6(a)). Figure 6(d) shows the frequency response of the average power inputs of the system in an excitation cycle. It can be seen from Figure 6(d) that there are multiple power peaks in a wide range of the frequency ratios. Define the power peaks from right to left, namely, the 1st peak frequency is the dimensionless resonance frequency Ω1; it can be seen from Figure 6(d) that the ith dimensionless peak frequency Ωi is almost equal to Ω1/i (i = 1, 2, 3, …, 9). The maximal average power inputs occur at the nonresonant frequency Ω4 and not at the resonant frequency Ω1. It shows that the power inputs from a nonresonant excitation can be more efficient than those from a resonant excitation. It is noted that the input power is affected by both the driving force Fz and the velocity u˙ (see equation (19)). Due to the coupling between Fz and u˙, the driving force Fz at each excitation is affected by the initial velocity of the beam (see Figures 6(a)–6(c)). It can be seen from Figures 6(a) and 6(b) that when Ω = 1 and 1/4, the effect of u˙ on Fz is not obvious. The difference is on that when Ω = 1/4, the direction of u˙ is the same with the direction of Fz, which leads to a higher average power input than that at Ω = 1 (see Figure 6(d)). It can be seen from Figures 6(b) and 6(c) that a lower initial velocity at Ω = 1/8 leads to a lower driving force peak when compared with that at Ω = 1/4. Thus, the average power input at Ω = 1/8 is lower than that at Ω = 1/4 (see Figure 6(d)). It means that the optimal power input can occur at a dimensionless frequency with a higher vibration velocity u˙ but with less effect on the input force Fz. In addition, higher average input power is achieved when u˙ and Fz have the same direction. In this analysis, the optimal average input power occurs at Ω4.Figure 6
Time response of input force and velocity and frequency responses of average power inputs. (a) Ω = 1. (b) Ω = 1/4. (c) Ω = 1/8. (d) Average power inputs for various Ω.α = β = 1, r1 = r2 = 1, kp2 = 0.01, and σ = 0.3428 Am2.
(a)(b)(c)(d)Figures7(a) and 7(b) show 3D plots of the average power inputs of the system with a series of Ω for various α and β, respectively. It can be seen from Figures 7(a) and 7(b) that, at an arbitrary peak frequency, the average power inputs increase with a decrease in α and β, which implies that a larger dynamic response can occur at smaller α or at smaller β. It is noted that the proper selections of α and β are necessary because the collision of magnet 2 with magnet 1 is easier to occur in a large displacement of the beam. Figure 8 shows the displacements and electromagnetic EMCS versus Ω at α = β = 1. It can be seen from Figure 8 that the range of the movement of the mass m is from −2 mm to 2 mm for the given parameter configuration. Since the absolute displacements of the mass m are less than the initial gaps between two magnets (|u|<2.5 mm), the collision between magnets 2 and 1 does not occur. In Sections 5.2–5.4, the constant parameters α = β = 1 will be used for the analysis of the effect of the load resistance and the EMCSs on the system performance, respectively.Figure 7
Average power inputs with a series of Ω,α, and β. (a) β = 1. (b) α = 1. r1 = r2 = 1, kp2 = 0.01, and σ = 0.3428 Am2.
(a)(b)Figure 8
Displacement response of the massm versus Ω at α = β = 1, r1 = r2 = 1, kp2 = 0.01, and σ = 0.3428 Am2.


## 5.2. Effect of Dimensionless Load Resistance on the System Performance

Let the electromechanical parameters be given askp2 = 0.01 and σ = 0.3428 Am2. A parametric sweep simulation is performed in this analysis. Figures 9(a)–9(f) show 3D plots of the maximal displacements, the maximal velocities, the average power inputs, the piezoelectric average power outputs, the electromagnetic average power outputs, and the power conversion efficiencies of the hybrid P-EMEH with a series of Ω and r1 at r2 = 1, respectively. It can be seen from Figures 9(a)–9(e) that, at arbitrary Ωi = Ω1/i (i = 1, 2, 3, …, 9), the displacements umax, the velocities u˙max, the power inputs P¯in, and the electromagnetic power outputs P¯R2 generate the valleys (see Figures 9(a)–9(c) and 9(e)). The piezoelectric power outputs P¯R1 and the power conversion efficiencies η generate the peaks (see Figure 9(d)). The occurrence of the peaks or the valleys is induced by the forward or backward actions of electromechanical energy conversion. When optimal r1 is given to the piezoelectric transducer, the piezoelectric transducer can obtain the maximal power outputs. Under the backward action of the piezoelectric high power outputs, the vibrations of the harvester are relatively strongly suppressed, which leads to an obvious decrease in P¯in and P¯R2. The valleys occur at optimal r1 (see Figures 9(a)–9(c) and 9(e)). In Figure 9(f), the highest energy efficiency of the system appears at Ω4 and not at Ω1, which shows that the nonresonant energy harvesting can achieve higher power outputs than the resonant energy harvesting.Figure 9
3D plots for dynamic and electrical outputs with a series of Ω andr1. (a) Maximal displacements. (b) Maximal velocities. (c) Average power inputs. (d) Piezoelectric average power outputs. (e) Electromagnetic average power outputs. (f) Power conversion efficiencies. r2 = 1, kp2 = 0.01, and σ = 0.3428 Am2.
(a)(b)(c)(d)(e)(f)Figures10(a)–10(f) show 3D plots of the maximal displacements, the maximal velocities, the average power inputs, the piezoelectric average power outputs, the electromagnetic average power outputs, and the efficiencies of the hybrid P-EMEH with a series of Ω and r2 at r1 = 1, respectively. It can be seen from Figures 10(a)–10(d) that, at arbitrary Ωi = Ω1/i (i = 1, 2, 3, …, 9), with an increase in r2, the displacements umax, the velocities u˙max, the power inputs P¯in, and the piezoelectric power outputs P¯R1 increase. The electromagnetic power outputs P¯R2 and the power conversion efficiencies η generate the peaks around r1 = 1. The variation tendency can also be explained from the forward and backward actions of electromechanical energy conversion. When optimal r2 is given to the electromagnetic element, the electromagnetic element can obtain the optimal power output (see Figure 10(e)). From the expression term of ζe given in equation (28), it can be seen that an increase in r2 can lead to the decrease in ζe. The total damping coefficient of the system decreases accordingly. Therefore, an increase in r2 can induce the increase in u and u˙ (see Figures 10(a) and 10(b)). According to equations (19), (15), and (20), an increase in u˙ and u can induce the increase in Pin and P¯R1, respectively. Thus, P¯in and P¯R1 increase with the increase of r2 (see Figures 10(c) and 10(d)). It can be seen from Figure 10(f) that the highest power conversion efficiency appears at Ω4.Figure 10
3D plots for dynamic and electrical outputs with a series of Ω andr2. (a) Maximal displacements. (b) Maximal velocities. (c) Average power inputs. (d) Piezoelectric average power outputs. (e) Electromagnetic average power outputs. (f) Power conversion efficiencies. r1 = 1, kp2 = 0.01, and σ = 0.3428 Am2.
(a)(b)(c)(d)(e)(f)


## 5.3. Effect of Electromechanical Coupling Strengths on the System Performance

Figures11(a)–11(f) show 3D plots of the maximal displacements, the maximal velocities, the average power inputs, the piezoelectric average power outputs, the electromagnetic average power outputs, and the efficiencies of the hybrid P-EMEH with a series of Ω and kp2 at r1 = r2 = 1 and σ = 0.3428 Am2, respectively. It can be seen from Figures 11(a)–11(e) that, at arbitrary Ωi = Ω1/i (i = 1, 2, 3, …, 9), the displacements umax, the velocities u˙max, the power inputs P¯in, and the electromagnetic power outputs P¯R2 decrease with an increase in kp2. The piezoelectric average power outputs P¯R1 at each Ωi (i = 1, 2, …, 9) can increase until the saturation. The low-frequency saturation power is significantly higher than the high-frequency saturation power. Hence, the use of the hybrid P-EMEH with higher kp2 is advantageous for scavenging rotational energy at low frequencies. In Figure 11(f), the power conversion efficiencies of the system have the same tendency with the piezoelectric power outputs given in Figure 11(d). It shows that the piezoelectric energy conversions are dominant in the hybrid energy harvesting system due to the use of higher kp2.Figure 11
3D plots for dynamic and electrical outputs with a series of Ω andkp2. (a) Maximal displacements. (b) Maximal velocities. (c) Average power inputs. (d) Piezoelectric average power outputs. (e) Electromagnetic average power outputs. (f) Power conversion efficiencies. r1 = r2 = 1 and σ = 0.3428 Am2.
(a)(b)(c)(d)(e)(f)To study the effect of the electromagnetic EMCS on the system performance, the average electromagnetic EMCSs in a rotational period are calculated. Figures12(a) and 12(b) show the instantaneous and average electromagnetic EMCSs versus Ω at σ = 0.3428 Am2 and σ = 2.5313 Am2, respectively. It can be seen from Figures 12(a) and 12(b) that, for an arbitrary given value of σ, the average electromagnetic EMCS can keep almost a constant value for various Ω. A higher value of σ can achieve higher k¯e2. Figures 13(a)–13(f) show 3D plots of the maximal displacements, the maximal velocities, the average power inputs, the piezoelectric average power outputs, the electromagnetic average power outputs, and the efficiencies of the hybrid P-EMEH with a series of Ω and k¯e2 at r1 = r2 = 1 and kp2 = 0.01, respectively. It can be seen from Figures 13(a)–13(e) that the maximal displacements umax, the maximal velocities u˙max, the average power inputs P¯in, and the piezoelectric average power outputs P¯R1 decrease with the increase in k¯e2. The average electromagnetic power outputs P¯R2 can increase until the saturation at each Ωi (i = 1, 2, …, 9). Because low-frequency saturation power is significantly higher than high-frequency saturation power for the electromagnetic energy harvesting, it is favorable for the hybrid P-EMEH with higher k¯e2 to harvest rotational energy at low frequencies. In Figure 13(f), the energy efficiencies of the system show the same tendency with the electromagnetic power outputs given in Figure 13(e). It shows that the electromagnetic energy conversions are dominant in the hybrid energy harvesting system due to the use of higher k¯e2.Figure 12
Instantaneous and average electromagnetic EMCSs versus Ω atr1 = r1 = 1 and kp2 = 0.01. (a) σ = 0.3428 Am2. (b) σ = 2.5313 Am2.
(a)(b)Figure 13
3D plots for dynamic and electrical outputs with a series of Ω andk¯e2. (a) Maximal displacements. (b) Maximal velocities. (c) Average power inputs. (d) Piezoelectric average power outputs. (e) Electromagnetic average power outputs. (f) Power conversion efficiencies. r1 = r1 = 1 and kp2 = 0.01.
(a)(b)(c)(d)(e)(f)


## 5.4. Power Comparison for the PEH, EMEH, and Hybrid P-EMEH at Weak and Strong Couplings

In this section, the power performance of the hybrid P-EMEH at weak and strong EMCS is compared with that from the single PEH and single EMEH. In order to distinguish strong coupling regions from weak ones, Shu and Lien [32] proposed using the parameter γ1=kp2/ζ as the indicator of the EMCS of the PEH and defined a criterion of strong coupling, γ1≥4ζ+1, for a pure resistor circuit. They also referred to that the EMCS is very weak if γ1≪1. Challa et al. [33] proposed using the parameter γ2=θe2/RcC as the indicator of the EMCS of the EMEH and defined the weak coupling (γ2≪1) and the strong coupling (γ2≫1). Letting ke2=θe2/mωnRc and ζ=C/2km, γ2 can be rewritten as γ2=ke2/2ζ. Li et al. [34] proposed using the parameter γ3=kp2+ke2/2ζ as the indicator of the EMCS of the hybrid P-EMEH and defined the weak coupling (kp2+ke2/ζ≪2) and the strong coupling (kp2+ke2/ζ≫2). Based on these criteria above and letting ke2 = k¯e2 and supposing ζ = 0.02 in this work, the weak coupling is confined in the regions of kp2≪0.02, k¯e2≪0.04, and kp2+k¯e2≪0.04, respectively, for the PEH, the EMEH, and the hybrid P-EMEH. The strong coupling is confined in the regions of kp2≥0.0816, k¯e2≫0.04, and kp2+k¯e2≫0.04, respectively, for the PEH, the EMEH, and the hybrid P-EMEH. Suppose kp2 = k¯e2 = 0.002 for the weak coupling and kp2 = k¯e2 = 0.2 for the strong coupling in this research.Figures14(a)–14(c) show the average power inputs, power outputs, and efficiencies for three different types of energy harvesters (PEH, EMEH, and hybrid P-EMEH) with optimal r1 and r2 at kp2 = k¯e2 = 0.002, respectively. In this study, optimal r1 and r2 are derived from the optimal average power outputs of the PEH and the EMEH, respectively, and they are used as the load resistance of the hybrid P-EMEH. It can be seen from Figures 14(a)–14(c) that there are close peak power inputs for the PEH, the EMEH, and the hybrid P-EMEH. However, the hybrid P-EMEH has higher peak power outputs and efficiencies than the PEH and the EMEH. It shows that, under a weak coupling condition, higher kp2 can convert more mechanical energy into electricity with less effect on the dynamics of the system. Therefore, it is advantageous to use the hybrid P-EMEH to harvest the vibration energy under a weak electromechanical coupling condition.Figure 14
Power comparisons for the PEH, EMEH, and hybrid P-EMEH with optimalr1 and r2. (a) Average power inputs. (b) Total average power outputs. (c) Power conversion efficiencies. kp2 = k¯e2 = 0.002.
(a)(b)(c)Figures15(a)–15(c) show the average power inputs, power outputs, and efficiencies for three different types of energy harvesters (PEH, EMEH, and hybrid P-EMEH) with optimal r1 and r2 at kp2 = k¯e2 = 0.2, respectively. It can be seen from Figures 15(a) and 15(b) that, for the PEH and the hybrid P-EMEH, there are obvious double peaks around each higher dimensionless frequency Ωi (i = 1, 2, 3, 4). One is located at Ωi (i = 1, 2, 3, and 4), and another one is located at Ωi1+kp2 (i = 1, 2, 3, and 4). The occurrence of the double peaks can be reasonably explained from equation (28). In equation (28), when r1 ⟶ 0 (electrical short circuit), the dimensionless resonant frequency is Ω1. When r1 ⟶ ∞ (electrical open circuit), the dimensionless resonant frequency is Ω11+kp2. According to [35], the piezoelectric optimal power outputs can occur at both short-circuit and open-circuit resonant frequencies. In this analysis, the double-peak power can be observed owing to higher kp2. Nonresonant energy harvesting has similar characteristics with resonant energy harvesting owing to the fact that the model frequency of the harvester is dominant in the vibration energy harvesting process, but with the decrease in Ωi and Ωi1+kp2 (i = 2, 3, …, 9), the two nonresonant excitation frequencies become closer and closer to each other until the frequency coincidence. Thus, there are single-peak power outputs at lower Ωi. The EMEH shows single-peak power characteristics owing to the fact that the electromagnetic EMCS is not relative to the resonance frequency of the harvester. It can be seen from Figure 15(b) that the PEH has wider power bandwidths than the hybrid P-EMEH at relatively high Ωi (i = 1, 2, 3, 4), but it has close power outputs with the hybrid P-EMEH at relatively low Ωi (i = 5, 6, …, 9). The reason to this phenomenon can be explained by the power conversion efficiencies of the harvesters given in Figure 15(c). It can be seen from Figure 15(c) that, at higher Ωi (i = 1, 2, 3, 4), the PEH and the hybrid P-EMEH have almost the same peak power efficiency, which means that power saturation occurs with given parameters kp2 = k¯e2 = 0.2. In this case, the same power outputs are theoretically possible for the hybrid EMEH and the PEH. However, owing to the single-peak power characteristics of electromagnetic energy harvesting and its energy dissipation in the coil, the power outputs from the hybrid P-EMEH become lower than those from the PEH at higher Ωi (i = 1, 2, 3, and 4) (see Figure 15(b)). For lower Ωi (i = 5, 6, …, 9), it can be seen from Figure 15(c) that the peak power efficiency of the PEH is lower than that of the hybrid P-EMEH, which means that the piezoelectric power saturation cannot not be achieved with given parameter kp2 = 0.2 at lower Ωi (i = 5, 6, …, 9). Theoretically, the power outputs of the PEH should be lower than those of the hybrid P-EMEH. However, due to relatively less power loss in the piezoelectric element when compared with the electromagnetic element, the PEH can still provide almost the same power outputs with the hybrid P-EMEH. The results show that it is more advantageous to use the single PEH to harvest the rotational vibration energy than the EMEH and hybrid P-EMEH under a strong electromechanical coupling condition. It can be seen from Figure 15(c) that the power conversion efficiencies of the three types of harvesters increase with a decrease in Ωi (i = 1, 2, …, 9). A higher power conversion efficiency means that higher kp2 or k¯e2 is required to achieve power saturation. It is the reason why the PEH with the same parameter kp2 = 0.2 can achieve power saturation at higher Ωi (i = 1, 2, 3, 4), but not at lower Ωi (i = 5, 6, …, 9). It should be noted that, for the single EMEH, it has less advantage for magnetically plucked vibration energy harvesting at low frequencies due to relatively lower power outputs than those at middle and high frequencies (see Figure 15(b)).Figure 15
Power comparisons for the PEH, EMEH, and hybrid P-EMEH with optimalr1 and r2. (a) Average power inputs. (b) Total average power outputs. (c) Power conversion efficiencies. kp2 = k¯e2 = 0.2.
(a)(b)(c)The significance of the analysis of the piezoelectric weak and strong coupling is that it provides a more comprehensive understanding on the power performance of the PEH in different coupling regimes. The results from weak coupling analysis show the advantages of higher power outputs of the hybrid P-EMEH when compared with the PEH and EMEH (see Figure14(b)). The results from strong coupling analysis show the advantages of wider power bandwidths of the PEH at high dimensionless frequencies when compared with the hybrid P-EMEH and EMEH (see Figure 15(b)).


## 6. Conclusions

In this work, we investigate the dynamics and electrical characteristics of a hybrid P-EMEH driven by rotary magnetic plucking. The lumped-parameter model of the hybrid P-EMEH is established, and the model parameters includingωn, k, m, kp2, J1, J2, and σ are determined based on the results from FEA. A parametric study is performed to investigate the effects of the dimensionless magnetic force parameters, excitation frequencies, load resistance, and EMCSs on the maximal displacements and velocities, average power inputs and outputs, and energy efficiencies of the system for indicating the performance of the hybrid P-EMEH. The optimal power outputs of the hybrid P-EMEH are analyzed and compared with those from the single PEH and the single EMEH under weak and strong coupling conditions. The results show that the hybrid P-EMEH has peak power inputs at a series of dimensionless frequencies Ωi = Ω1/i (i = 1, 2, 3, …, 9), where Ω1 is the resonance frequency of the harvester. At each Ωi, the hybrid P-EMEH can obtain the improved average power inputs by reducing the rotational radii of the driving magnet and shortening the gaps between the two magnets. The structural vibrations can be strongly suppressed owing to the optimal piezoelectric power outputs, which can lead to the occurrence of valleys’ power of the electromagnetic element. At weak coupling, the hybrid P-EMEH can achieve higher power outputs than the PEH and the EMEH. The use of the PEH strong coupling is more advantageous for energy harvesting due to wider operating bandwidths at high dimensionless frequencies when compared with those from the hybrid P-EMEH. The studies also show that the nonresonant rotational energy harvesting at some frequencies can be more efficient than the resonant rotational energy harvesting. The parametric study provides a fundamental understanding on the effect of load resistance and EMCSs on the performance of the hybrid P-EMEH driven by rotary magnetic plucking.



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