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

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.


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]. e rotational vibration is often visible in the ambient environment, ranging from fluid flow [3] and machinery operation [4] to human motion [5]. e 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. e 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][12][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][17][18][19][20][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.
is 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. e 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. e mechanical parameters and the piezoelectric EMCS are determined from the coupled electromechanical FEA. e 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. e 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.

Configuration of the Hybrid P-EMEH
Figures 1(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. e 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. e 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). e parameters m, k, and c are the equivalent mass, the equivalent stiffness, and the viscous damping of the harvester, respectively. C p is the capacitance of the two piezoelectric plates in parallel. L c and R c are the internal induction and the internal resistance of the coil, respectively. θ p and θ e are the EMCCs related with the piezoelectric and electromagnetic elements, respectively. R 1 and R 2 are the load resistance connected to the piezoelectric and electromagnetic elements, respectively. F z and u are the driven force and the vibration displacement of the mass m, respectively.
Figures 2(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 2A 1 × 2B 1 × 2C 1 and 2A 2 × 2B 2 × 2C 2 , respectively. Magnet 3 has the radius of r 3 and the height of h 3 . J 1 , J 2 , and J 3 are the magnetizations of magnets 1, 2, and 3, respectively. e interaction of magnets 1 and 2 is repulsive due to the opposite direction of the magnetization. e interaction of magnets 2 and 3 is attractive due to the same direction of the magnetization. d x , d y , and d z represent the central distances between magnets 1 and 2 in three axes, respectively. z 3 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 r min , the outer radius of r max , and the height of h c . 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 z c � z c0 − u, where z c0 is the initial position of magnet 3 in the Z c -axis.

Theoretical Model
e governing differential equation of the hybrid P-EMEH with the magnetic force excitation can be written as where (·) � z/zt, V p is the voltage across the load resistor R 1 , and I e is the current through the load resistor R 2 . 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] where μ 0 is the vacuum permeability. e parameters i, j, k, l, p, and q are equal to 0 or 1 according to the specific corner. e parameter ϕ z is a function of the magnet dimensions and their gaps in 3 axes, which is given by where the lengths u ij , v kl , and w pq correspond to the distance between the cube corners and their projections on the axes. r is a function of the parameters of u ij , v kl , and w pq . ese parameters are expressed as [23] 2 Shock and Vibration When the revolving disk with magnet 1 rotates with the motor shaft (see Figure 1(a)), the gaps d x , d y , and d z between magnets 1 and 2 can be expressed as [16] where ω r is the rotational angular frequency of the motor shaft, r m is the gyration radius of magnet 1, d x0 , d y0 , and d z0 are the initial gaps between magnets 1 and 2 in the x-, y-, and z-axes, respectively, and ϕ 0 is the initial angular position of magnet 1 relative to magnet 2. In this paper, we determine d x0 � d y0 � ϕ 0 � 0. e electromagnetic EMCC, θ e , can be obtained by calculating the gradient of the magnetic flux in the coil: where Φ is the total magnetic flux in the coil, Φ � N c ξΦ a , where Φ a is the magnetic flux per unit cross-sectional area of the coil. N c is the number of turns in the coil. ξ is the fill factor of the coil, ξ � N c A w /A c , where A w is the conductive area of a single wire, A w � πd 2 w /4, where d w is the conductive diameter of a single wire. A c is the cross-sectional area of the coil, A c � (r max − r min ) h c . e magnetic field at the position of a point of the coil is given by [24] 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   (z 20 − u − z 1 )e z into equation (7) and simplifying this equation, the magnetic field B → can be rewritten as 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 Integrating Φ z given in equation (9) over the whole cross section of the coil and dividing by the area A c , the average magnetic flux through the cross section of the coil can be written as Since z c � z c0 − u and z c is not relative to z, returning to equation (6), we can rewrite the electromagnetic EMCC as Substituting the expression of Φ a in equation (10) into equation (11) and then taking the derivative with respect to z c − z, the electromagnetic EMCC, θ e , can be written by [25] where Considering the equivalent electrical model of the electromagnetic transducer [26], the circuit governing equation for electromagnetic transduction of the system can be written as 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): Considering the equivalent electrical model of the piezoelectric transducer [26], the circuit governing equation for piezoelectric transduction of the system can be written as Defining the state-space vector we can write the governing equations of the hybrid P-EMEH from equations (14) and (15) in the state-space form as where where ω r is the rotational angular frequency of the driving magnet and t 0 is the initial time for a rotational cycle. e average power input of the system in a rotational cycle is e average power outputs on the resistors R 1 and R 2 in a rotational cycle are e total average power output of the hybrid energy harvesting system is e efficiency of the system is 4 Shock and Vibration When the coil circuit is open (r 2 � 0 and θ e � 0), equation (17) is reduced to which is the governing equation of a single PEH in the statespace form.
When the surface of the piezoelectric element is electrically shorted (r 1 ⟶ ∞ and θ p � 0), equation (17) which is the governing equation of a single EMEH in the state-space form.

System Parameter Determination
Based on the material and geometric parameters of the piezoelectric clamped-clamped beam listed in Table 1 and the parameters of magnets and the coil listed in Table 2, in this section, the model parameters of the hybrid P-EMEH, including ω n , m, k 2 p , J 1 , J 2 , and σ, are determined based on the results from finite element physical field analysis. e theoretical parameters ω n , m, and k 2 p 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 shortcircuit and open-circuit resonance frequencies are obtained as f sc � 122.15 Hz and f oc � 122.76 Hz, respectively. According to the equation k 2 p � (ω oc 2 /ω sc 2 ) − 1 [29,30], the piezoelectric EMCS can be calculated as k 2 p � 0.01, where ω sc and ω oc are the short-circuit and open-circuit resonant angular frequencies, respectively. e parameters ω sc � 2πf sc and ω oc � 2πf oc . 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, U z , of the magnetic mass versus the dimensionless rotational frequency c. e parameter c � ω/ω n , where ω and ω n are the Shock and Vibration 5 excitation angular frequency and the natural angular frequency, respectively. e parameter ω n � ω sc . It can be seen from Figure 3(b) that U z � 5.697 mm at c � 1. e equivalent stiffness of the beam can be calculated as k � 4388 N/m according to the equation of the magnification factor given in [31].
e equivalent mass of the beam can be calculated as m � 0.0074 kg according to the equation m � k/ω n 2 . Since the two piezoelectric plates on the metal beam are electrically connected in parallel, the total capacitance of the piezoelectric element is C p � 2ε S 33 ε 0 L p b/h p � 23.4 nF. e model parameters J 1 and J 2 and σ are determined by the FE magnetic field analysis. Note that the magnetic parameters B r � 1.141 T and H c � − 908775 A/m given in Table 2 are provided for FE magnetic field analysis. e magnetic driving force obtained from FEA is used to determine the magnetizations J 1 and J 2 used in equation (2). e 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 F z versus the gaps d z and d y , respectively. It can be seen from Figures 4(c) and 4(d) that the theoretical magnetic driving force with the parameters J 1 � − J 2 � 1.141 T and J 1 � − J 2 � 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 J 1 � − J 2 � 1.26 T will be used for the 5 Central distance between magnet 2 and magnet 3 in the z-axis, z 3  Shock and Vibration theoretical analysis of the dynamics and electrical characteristics of the hybrid P-EMEH. e magnetic flux Φ z 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, B ave , as where n 1 is the number of wire loops in the x c -axis, n 2 is the number of cross sections of the coil in the z c -axis, Φ z (r i , z j ) is the magnetic flux through a given wire loop, where r i (i � 1, 2, 3, . . ., n 1 ) is the radius of an arbitrary wire loop, and z j (j � 1, 2, 3, . . ., n 2 ) is the location of the wire loop with respect to the upper surface of the coil along the z c -axis (see Figure 2(b)). In FEA, r i (i � 1, 2, 3, and 4) is given as 6 mm, 7 mm, 8 mm, and 9 mm, respectively. z j (j � 1, 2, 3, and 4) is given as 0 mm, 1 mm, 2 mm, 3 mm, 4 mm, and 5 mm, respectively. us, n 1 � 4 and n 2 � 6. In the theoretical model, we consider increments of 0.123 mm from r 1 � r min � 6 mm to r n1 � r max � 9 mm in the positive x c -axis direction and from z 1 � 0 to z n2 � h c � 5 mm in the positive z 2 -axis direction; thus, n 1 � 26 and n 2 � 41. Set z 20 � 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 Am 2 show good agreements with the results from FEA. e electromagnetic EMCS can be calculated by using the determined value of σ.

Parametric Study
In the parametric studies, the differential equations are solved using a Runge-Kutta method. e number of vibration cycles and the number of time steps in one period are given as 20 and 1000, respectively. e constant parameters are given as m � 0.0074 kg, w n � 2πf sc � 767.5 rad/s, ζ � 0.02, C p � 23.4 nF, and R c � 90 Ω. e rational radius of magnet 1 and the central distance between magnets 1 and 2 are originally set to be r m(0) � 30 mm and d z0(0) � 5 mm, respectively. e dimensionless parameters Ω � ω r /ω n , α � r m / r m(0) , and β � d z0 /d z0(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.

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 as r 1 � r 2 � 1,  Shock and Vibration 7 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 V p in equation (15) into (14) and rearrange equation (14) as where ω nr is the resonant frequency of the system, . ζ e is the electromagnetic damping coefficient, ζ e � k 2 e /(2(1 + r 2 )). Considering the average electromagnetic EMCS in a rotational period, the electromagnetic damping coefficients can be written as ζ e � k 2 e /(2(1 + r 2 )). It can be seen from equation (28) that, with a decrease in Ω, ζ p increases and ω nr decreases. Higher ζ p can lead to a faster decay of the velocity _ u between two excitations. Lower ω nr means a lower stiffness of the system, which can lead to a higher initial velocity after each excitation. is 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(  8 Shock and Vibration 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 1 st peak frequency is the dimensionless resonance frequency Ω 1 ; it can be seen from Figure 6(d) that the ith dimensionless peak frequency Ω i is almost equal to Ω 1 /i (i � 1, 2, 3, . . ., 9). e maximal average power inputs occur at the nonresonant frequency Ω 4 and not at the resonant frequency Ω 1 . 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 F z and the velocity _ u (see equation (19)). Due to the coupling between F z and _ u, the driving force F z 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 F z is not obvious. e difference is on that when Ω � 1/4, the direction of _ u is the same with the direction of F z , 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. us, 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 F z . In addition, higher average input power is achieved when _ u and F z have the same direction. In this analysis, the optimal average input power occurs at Ω 4 . Figures 7(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.

Effect of Dimensionless Load Resistance on the System
Performance. Let the electromechanical parameters be given as k 2 p � 0.01 and σ � 0.3428 Am 2 . 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 r 1 at r 2 � 1, respectively. It can be seen from Figures 9(a)-9(e) that, at arbitrary 1, 2, 3, . . ., 9), the displacements u max , 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)). e piezoelectric power outputs P R1 and the power conversion efficiencies η generate the peaks (see Figure 9(d)). e occurrence of the peaks or the valleys is induced by the forward or backward actions of electromechanical energy conversion. When optimal r 1 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 . e valleys occur at optimal r 1 Shock and Vibration (see Figures 9(a)-9(c) and 9(e)). In Figure 9(f), the highest energy efficiency of the system appears at Ω 4 and not at Ω 1 , which shows that the nonresonant energy harvesting can achieve higher power outputs than the resonant energy harvesting. Figures 10(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 r 2 at r 1 � 1, respectively. It can be seen from Figures 10(a)-10(d) that, at arbitrary 1, 2, 3, . . ., 9), with an increase in r 2 , the displacements u max , the velocities _ u max , the power inputs P in , and the piezoelectric power outputs P R1 increase. e electromagnetic power outputs P R2 and the power conversion efficiencies η generate the peaks around r 1 � 1.
e variation tendency can also be explained from the forward and backward actions of electromechanical energy conversion. When optimal r 2 is given to the electromagnetic element, the electromagnetic element can obtain the optimal power output (see Figure 10(e)). From the expression term of ζ e given in equation (28), it can be seen that an increase in r 2 can lead to the decrease in ζ e . e total damping coefficient of the system decreases accordingly. erefore, an increase in r 2 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 P in and P R1 , respectively. us, P in and P R1 increase with the increase of r 2 (see Figures 10(c) and 10(d)). It can be seen from Figure 10(f) that the highest power conversion efficiency appears at Ω 4 .    Figures 11(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 k 2 p at r 1 � r 2 � 1 and σ � 0.3428 Am 2 , respectively. It can be seen from Figures 11(a)-11(e) that, at arbitrary Ω i � Ω 1 /i (i � 1, 2, 3, . . ., 9), the displacements u max , the velocities _ u max , the power inputs P in , and the electromagnetic power outputs P R2 decrease with an increase in k 2 p . e piezoelectric average power outputs P R1 at each Ω i (i � 1, 2, . . ., 9) can increase until the saturation. e low-frequency saturation power is significantly higher than the high-frequency saturation power. Hence, the use of the hybrid P-EMEH with higher k 2 p 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 k 2 p . To study the effect of the electromagnetic EMCS on the system performance, the average electromagnetic EMCSs in a rotational period are calculated. Figures 12(a) and 12(b) show the instantaneous and average electromagnetic EMCSs versus Ω at σ � 0.3428 Am 2 and σ � 2.5313 Am 2 , 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 2 e . 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 2 e at r 1 � r 2 � 1 and k 2 p � 0.01, respectively. It can be seen from Figures 13(a)-13(e) that the maximal displacements u max , 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 2 e . e average electromagnetic power outputs P R2 can increase until the saturation at each Ω i (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 2 e 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

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 c 1 � k 2 p /ζ as the indicator of the EMCS of the PEH and defined a criterion of strong coupling, c 1 ≥ 4(ζ + 1), for a pure resistor circuit. ey also referred to that the EMCS is very weak if c 1 ≪ 1. Challa et al. [33] proposed using the parameter c 2 � θ 2 e /(R c C) as the indicator of the EMCS of the EMEH and defined the weak coupling (c 2 ≪ 1) and the strong coupling (c 2 ≫ 1). Letting k 2 e � θ 2 e /(mω n R c ) and ζ � C/(2 ��� km √ ), c 2 can be rewritten as c 2 � k 2 e /(2ζ). Li et al. [34] proposed using the parameter c 3 � (k 2 p + k 2 e )/(2ζ) as the indicator of the EMCS of the hybrid P-EMEH and defined the weak coupling ((k 2 p + k 2 e )/ζ ≪ 2) and the strong coupling ((k 2 p + k 2 e )/ζ ≫ 2). Based on these criteria above and letting k 2 e � k  Figures 14(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 r 1 and r 2 at k 2 p � k 2 e � 0.002, respectively. In this study, optimal r 1 and r 2 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 k 2 p can convert more mechanical energy into electricity with less effect on the dynamics of the system. erefore, it is advantageous to use the hybrid P-EMEH to harvest the vibration energy under a weak electromechanical coupling condition. Figures 15(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 r 1 and r 2 at k 2 p � k 2 e � 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 Ω i (i � 1, 2, 3, 4). One is located at Ω i (i � 1, 2, 3, and 4), and another one is located 1, 2, 3, and 4). e occurrence of the double peaks can be reasonably explained from equation (28). In equation (28), when r 1 ⟶ 0 (electrical short circuit), the dimensionless resonant frequency is Ω 1 . When r 1 ⟶ ∞ (electrical open circuit), the dimensionless resonant frequency is Ω 1 ����� 1 + k 2 p . According to [35], the piezoelectric optimal power outputs can occur at both shortcircuit and open-circuit resonant frequencies. In this analysis, the double-peak power can be observed owing to higher k 2 p . 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 Ω i and Ω i ����� 1 + k 2 p (i � 2, 3, . . ., 9), the two nonresonant excitation frequencies become closer and    closer to each other until the frequency coincidence. us, there are single-peak power outputs at lower Ω i . e 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 Ω i (i � 1, 2, 3, 4), but it has close power outputs with the hybrid P-EMEH at relatively low Ω i (i � 5, 6, . . ., 9). e 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 Ω i (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 k 2 p � k 2 e � 0.2. In this case, the same power outputs are theoretically possible for the hybrid EMEH and the PEH. However, owing to the singlepeak 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 Ω i (i � 1, 2, 3, and 4) (see Figure 15(b)). For lower Ω i (i � 5, 6, . . ., 9), it can be seen    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 k 2 p � 0.2 at lower Ω i (i � 5, 6, . . ., 9). eoretically, 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. e 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 Ω i (i � 1, 2, . . ., 9). A higher power conversion efficiency means that higher k 2 p or k 2 e is required to achieve power saturation. It is the reason why the PEH with the same parameter k 2 p � 0.2 can achieve power saturation at higher Ω i (i � 1, 2, 3, 4), but not at lower Ω i (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)).   e 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. e 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 Figure 14(b)). e 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)).

Conclusions
In this work, we investigate the dynamics and electrical characteristics of a hybrid P-EMEH driven by rotary magnetic plucking.
e lumped-parameter model of the hybrid P-EMEH is established, and the model parameters including ω n , k, m, k p 2 , J 1 , J 2 , 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. e 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. e results show that the hybrid P-EMEH has peak power inputs at a series of dimensionless frequencies Ω i � Ω 1 /i (i � 1, 2, 3, . . ., 9), where Ω 1 is the resonance frequency of the harvester. At each Ω i , 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. e 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. e 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. e studies also show that the nonresonant rotational energy harvesting at some frequencies can be more efficient than the resonant rotational energy harvesting. e 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.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.