12108 FACTA UNIVERSITATIS Series: Mechanical Engineering Vol. 23, No 4, 2025, pp. 667 - 686 https://doi.org/10.22190/FUME231102003S © 2025 by University of Niš, Serbia | Creative Commons License: CC BY-NC-ND Original scientific paper MULTI-OBJECTIVE OPTIMIZATION AND PERFORMANCE ANALYSIS OF BIMORPH MAGNETO-ELECTRO-ELASTIC ENERGY HARVESTERS Meisam Moory Shirbani, Sayed Ehsan Alavi Faculty of Engineering, Shohadaye Hoveizeh Campus of Technology, Shahid Chamran University of Ahvaz, Dashte Azadegan, Iran ORCID iDs: Meisam Moory Shirbani https://orcid.org/0000-0001-5121-4138 Sayed Ehsan Alavi https://orcid.org/0000-0002-5669-7498 Abstract. This study thoroughly investigates the multi-objective optimization of a magneto-electro-elastic (MEE) harvester in bimorph configurations and by the new method of Harris Hawk’s optimization (HHO). The harvesters are configured in both series and parallel connections and under harmonic excitation to explore the effects of various parameters on the performance of the harvesting system. The primary objective is to maximize the total harvested power. Optimization involves various parameters, including dimensions, relative displacement changes, voltage, and current values. The Pareto fronts from the HHO method reveal optimal points in different configurations and scenarios. Notably, the optimal points are selected based on the criterion of maximum total power. The results reveal distinct optimal points for each objective function, demonstrating trade-offs between performance metrics. These findings provide valuable insights into the design and operation of efficient energy harvesters in MEE systems. The parallel configuration outperforms the series connection in terms of the current generation. Moreover, the evaluation of the overall performance of the energy harvesters in terms of total harvested power indicated that both series and parallel connections could lead to promising outcomes. However, the series connection exhibited a more dominant effect on maximizing the total harvested power, proving its relevance in pursuing the highest possible power output. Key words: Multi-objective optimization, MEE harvester, Harris Hawk’s optimization (HHO), Series and parallel connections Received: November 02, 2023 / Accepted January 14, 2024 Corresponding author: Meisam Moory Shirbani Faculty of Engineering, Shohadaye Hoveizeh Campus of Technology, Shahid Chamran University of Ahvaz, Dashte Azadegan, B.P 78986-64418, Iran E-mail: m.mooryshirbani@scu.ac.ir https://orcid.org/0000-0001-5121-4138 https://orcid.org/0000-0002-5669-7498 668 M.M. SHIRBANI, S.E. ALAVI 1. INTRODUCTION Energy harvesting involves converting environmental sources such as wind, heat, and vibrations into electrical energy. Mechanical vibrations are present in almost every industrial device as a reliable energy source. Energy harvesting from ambient vibrations has gained substantial popularity due to its accessibility in various places and times [1-3]. This field involves multiple aspects of engineering disciplines, including mechanics, materials science, and electronics. Smart materials that can couple at least two fields, such as piezoelectric materials [4-6], MEE materials [7,8], and others [9,10], are employed in this energy conversion process. The potential applications of this technology were extensive, with one of its most efficient domains being wireless sensor networks. Piezoelectric energy harvesters have been used, often as cantilever beams and usually made of two layers [11-13]. Among notable initial works, the design and fabrication of a generator by Roundy et al. [14] in 2003 is noteworthy. This generator can produce 375 μW of power per 1 cm3 with a density of 5.8 mW/gcm3, sufficient output power for many low- power microelectronic devices. Among the most significant works conducted in the analytical and numerical modeling of energy harvesting from piezoelectric materials, the contributions of Erturk and Inman are noteworthy [15,16]. They delved into the analytical modeling of piezoelectric energy harvesters by considering electrical and mechanical parameters, subsequently comparing and validating their results with experimental data. Utilizing a simple cantilever beam structure and operating in a frequency range of 450 Hz, they successfully achieved a power density of 6.8 mW/gcm3. Researchers are employing various techniques, such as changing the type of piezoelectric material, altering electrode patterns, and layering energy harvesters to enhance the harvested power [17,18]. The MEE composites exhibit sensitivity to a magnetic field in addition to their sensitivity to an electric field [19-21]. With these dual sensitivities, these materials can interconvert mechanical, electrical, and magnetic energies, positioning them as superior and more versatile than piezoelectric materials. In 2018, Shirbani et al. [22] proposed using MEE composites in energy harvesters to enhance the electrical power density. They provided analytical modeling of the coupled mechanical, electrical, and magnetic dynamic responses of the suggested energy harvesters. An important outcome of their work was the positive impact of the new structure and circuit design used for the external coils around the composite layers, resulting in a 43% increase in the total harvested power. They also explored the influence of temperature differences between the different layers of MEE energy harvesters [23]. The optimization technique holds significant importance in mechanics, as it strives to discover the most optimal solutions to intricate problems while weighing various constraints and objectives [24-26]. Mechanics employ optimization techniques to design and analyze structures, systems, and processes that can efficiently withstand forces, stresses, and dynamic conditions. Optimization algorithms help engineers find the best configuration or parameters that lead to better performance, lower costs, and better safety. This paper aims to provide a comprehensive overview of the bimorph MEE energy harvester research, mainly focusing on applying multi-objective optimization using the HHO method. The optimization process involves the simultaneous consideration of various objective functions, enabling the identification of optimal solutions that balance competing design criteria. Multi-objective optimization offers a holistic approach to achieving the Multi-Objective Optimization and Performance Analysis of Bimorph... 669 highest levels of efficiency and performance, catering to different operational scenarios and requirements. The HHO is a relatively recent optimization algorithm inspired by the hunting behavior of Harris’s hawks [27], a type of raptor bird. The algorithm was proposed to solve complex optimization problems by simulating the social hunting behaviors of these birds. HHO aims to balance exploration and exploitation, leveraging the collaborative behavior of hawks during hunting. The HHO algorithm was introduced in a research paper by Heidari et al. [27] in 2019. The algorithm was designed to address optimization problems by imitating the cooperative hunting strategies of Harris’s hawks. The HHO algorithm has gained attention due to its unique optimization approach inspired by nature. However, like any optimization algorithm, its effectiveness depends on the problem and its characteristics. Researchers continue to explore its potential and refine its techniques for various applications [28,29]. 2. ENERGY HARVESTER EQUATIONS AND OPTIMIZATION APPROACH In the present work, energy harvesters are analyzed as cantilever beams under the assumptions of the Euler-Bernoulli beam theory, coupled with layers possessing active MEE properties. In this context, energy harvesting refers to the extraction of vibrational energy derived from the vibration of the base of the beam. With transverse vibration, the beam undergoes strain. As a result of electrical and magnetic polarization, an electric and magnetic potential difference is established along the polar axis of the beam, enabling the generation of electrical power. The beam features embedded electrically conductive electrodes (for utilizing the electric field generated and extracting electrical energy) with minimal thickness, spanning the entire beam length. Additionally, external coils (for utilizing the induced magnetic field and extracting magnetic energy) are present. Other assumptions include the linear behavior of magnetic, electric, and elastomer properties concerning strain, electric, and magnetic fields. Figures 1a and 1b illustrate the analyzed configurations. These configurations feature symmetric arrangements with layer connections in series (Fig. 1a) and parallel (Fig. 1b). In Fig. 1, iE, RE, VM, iM, and RM represent the voltage, current generated, and external electrical resistance across the electrodes and the voltage, induced current and external electrical resistance across the external coils. Furthermore, wbT denotes the transverse base excitation. The fundamental equations that characterize mechanical, electrical, and magnetic behaviors of materials with active MEE properties are expressed as follows [22]: i ik k ik k ik kC e E f H = + + (1a) k ik i kj j kj jD e h E g H= + + (1b) k ik i kj j kj jB f g E H = + + (1c) In these equations, σi, Di, and Bi represent the stress tensor, electric displacement, and magnetic flux density. Also, Cij, hij, and μij denote the tensor of elastic stiffness, and electric and magnetic permeability coefficients, respectively. The distinguishing feature of this class of materials from piezoelectric materials lies in the simultaneous influence of the strain fields εk, electric field Ek, and magnetic field Hk on stress, electric displacement, and 670 M.M. SHIRBANI, S.E. ALAVI magnetic flux density, characterized by the piezoelectric coefficients eij, piezomagnetic coefficients fij, and magnetoelectric coefficients gij. The strain component along the beam ε1 according to the relative displacement wrel is expressed as: 2 1 ,2 ( , )rel rel xx w x t z zw x   = − = −  (2) Fig. 1 Symmetric configurations of the MEE energy harvester; (a) Series connection, (b) Parallel connection of harvester layers 2.1. Equations Governing Mechanical Motion Coupling for Bimorph MEE Energy Harvesters In this section, the Euler-Bernoulli cantilever beam with the configurations depicted in Fig. 1 is considered the energy harvester. By applying a base excitation in the transverse direction wbT, the overall displacement change of each point along the beam w(x,t) is as follows: ( , ) ( ) ( , )bT relw x t w t w x t= + (3) The governing differential equation for the forced vibrations of the Euler-Bernoulli cantilever beam under base excitation is expressed by Eq. (4) [7]: 2 22 2 2 2 ( , ) ( , ) ( , )( , ) [ ]rel rel bT m w x t w x t w x tM x t c m m tx t t    + + = −    (4) The bending moment M(x,t) is the area moment of inertia, cm is the damping coefficient, m is the mass per unit length L, and t is the time. The structural equations for expressing the coupling relationship in the MEE layers (denoted by the superscript M) in the axial direction using Eqs. (5a) to (5c), are presented as follows: 1 11 1 31 3 31 3 M M M M M M MC e E f H = − − (5a) Multi-Objective Optimization and Performance Analysis of Bimorph... 671 3 31 1 33 3 33 3 M M M M M M MD e h E g H= + + (5b) 3 31 1 33 3 33 3 M M M M M M MB f g E H = + + (5c) Also, the stress-strain constitutive equation for the sublayer with homogeneous properties (marked with superscript h) is expressed as follows: 1 11 1 h h hC = (6) It should be noted that based on the polarities indicated in Fig. 1, for the series connection (denoted by the superscript s), the electric field E3 in both upper and lower layers is the same. For the parallel connection (denoted by the superscript p), the electric fields E3 in both the upper and lower layers are equal in magnitude but opposite in direction [7]. 3 ( ) ,0.5 0.5 , 0.5 0.5 2 s s E h M h M h h M V E t h z h h h h z h h = −   + − −   − (7a) 3 3 ( ) ,0.5 0.5 2 ( ) , 0.5 0.5 2 p p E h M h M p p E M h h M V E t h z h h h V E t h h z h h  = −   +   = + − −   −  (7b) Furthermore, the magnetic field H3 generated in the MEE layers is calculated using the induced voltage across the terminal of the external wire coils VM and Ampere’s law, along with the number of wire coil turns N, as given by Eq. (8) [7]: 3 ( ) ( ) ( ) M M M M M Ni t NV t H t h R h = = (8) The bending moments for each connection are calculated and expressed using the structural equations, electric and magnetic fields, and Euler-Bernoulli assumptions: 2 2 ( , ) ( , ) ( , ) ( , ) ( , ) ( )[ ( ) ( )] ( )[ ( ) ( )] elec mag elas rel eq EM E MM M M x t M x t M x t w x t M x t EI V t H x H x L V t H x H x L x  = + − − + − −    (9) where Melas, Melec, and Mmag are the components of the bending moments that comprise elastic, electric, and magnetic components. The electric-mechanical and magnetic- mechanical coupling coefficients ΘEM and ΘMM are calculated for each connection. 312 ( )p s M EM EM M hbe h h =  = + (10a) 31 ( ) M p s MM MM M h M Nf b h h R  =  = + (10b) The equivalent flexural rigidity of the composite section EIeq and mass per unit length are calculated for the bimorph configuration. 672 M.M. SHIRBANI, S.E. ALAVI , 3 3 311 11( ) ( 0.5 ) (0.5 ) 12 3 h M s p eq h M h h C b C b EI h h h h = + + −  (11a) , (2 )s p M M h hm b h h = + (11b) The densities of the MEE and homogeneous layers are represented by ρM and ρh. Furthermore, b, hM, and hh represent the beam’s width, the MEE layer’s thickness, and the homogeneous layer’s thickness, respectively. The derived motion equations can be solved using traditional modal analysis methods. By using variable separation techniques, the displacement variation of each point on the beam with respect to its base can be expressed through Eq. 12 [15]: 0 ( , ) ( ) ( )rel r r r w x t W x T t  = = (12) The time response of the beam and the shape of the r-th mode one-dimensional vibration of the cantilever beam are determined by Tr(t) and Wr(x). The coupled mechanical equations of motion in modal coordinates are derived after substituting Eq. (12) into Eq. (9) and then applying the orthogonality conditions of the mode shapes. 2 2 2 ( ) ( ) 2 ( ) ( ) ( ) ( )r r r r r r EMr E MMr M r d T t dT t T t V t V t F t dtdt     + + + + = (13) The modal electric-mechanical and magnetic-mechanical coupling coefficients αEMr, αMMr are defined here and are calculated for each connection as follows: 31 ( ) 2 ( )p s M r EMr EMr M h x L dW x be h h dx   =   = = +     (14a) 31 ( ) ( ) M p s r MMr MMr M h M x L f N dW x b h h R dx   =   = = − +     (14b) The natural frequency of the r-th mode is calculated using Eq. (15) [7]: 2 2 4 eq r r EI ml  = (15) The damping ratio ζr and the mechanical force function Fr(t) in modal coordinates are also defined as follows: 2 m r r c m   = (16a) 2 2 0 ( ) ( ) ( ) L bT r r d w t F t m W x dx dt = −  (16b) Multi-Objective Optimization and Performance Analysis of Bimorph... 673 2.2. Governing Electrical Coupling Equations of Electrodes in Bimorph MEE Energy Harvesters The differential equation governing the electrical circuit involving the two electrodes around the MEE layers is extracted in this section. The two electrode terminals and the external wire coils are connected to external electrical resistances RE and RM to use the generated voltages VE and VM. The electric charge created at the two electrode terminals qE(t) can be determined through Gauss’s law, which stipulates that the electric charge equals the integral of the electric displacement D3 M over the electrode surfaces Ae. Furthermore, the electric current iE(t) equals the time derivative of the electric charge produced [7]. 3 ( ) ( ) ( , ) e ME E e A dq t d i t D x t dA dt dt = =  (17) Consequently, the following equation is obtained for each connection using Eqs. (5b) and (17) and applying Ohm’s law (V = RI). 1 ( ) ( ) ( ) ( )E M RE r M C M E MEr ME rE dV t dT t dV t C V t dt R dt dt      = + = + (18) This equation defines CM, υMEr, and χME as the internal capacitance, modal mechanical- electrical, and magnetic-electrical coupling coefficients of the MEE layers, respectively. In addition to these parameters, the constants γCM and γRE for each connection are obtained as follows: 33 M M M h bL C h = (19a) , 31 ( ) 0.5 ( )s p M r MEr M h x L dW x e h h b dx  = = − + (19b) , 31 ( ) 0.5 ( )s p M r MEr M h x L dW x e h h b dx  = = − + (19c) 2 1 M M p s C C = = (19d) 2 2 E E p s R R = = (19e) 2.3. Governing Electrical Coupling Equations of External Coils in Bimorph MEE Energy Harvesters During vibrational excitation, the relative motion between the MEE layers and the external wire coils causes a change in the magnetic flux passing through the coils, resulting in an electric voltage. For the present energy harvester beam, the induced voltage in the wire coils results from changes in the magnetic flux density B3 M. Consequently, VM can be expressed as follows [7]: 674 M.M. SHIRBANI, S.E. ALAVI 3( ) ( , ) i M M A d V t B x t dA dt =  (20) By substituting Eqs. (12) and (20) into Eq. (5c), the differential equation for the electrical coupling in the circuit of the two wire coils is obtained: 1 ( ) ( ) ( ) ( )M r E c M M MMr EM r dV t dT t dV t L R V t dt dt dt    = + = + (21) In these equations, Lc, υMMr, and χEM represent the self-inductance coefficients of the wire coils, the modal mechanical-magnetic and electric-magnetic coupling coefficients, respectively. These coefficients are derived for each connection as follows: 2 33 M c M bL L N h = (22a) , 31 ( ) 0.5 ( )s p M r MEr M h x L dW x e h h b dx  = = − + (22b) , 33 M s p ME M M g N bL R h  = (22c) 2.4. Frequency Response of Bimorph MEE Energy Harvesters under Harmonic Base Excitation The base excitation of the beam in the transverse direction at the excitation frequency ω is given as wbT(t) = WAT ejωt, where WAT represents the amplitude of the base transverse displacement. In this case, the expressions for the modal coordinates response of the beam, the voltages VE(t) and VM(t) are assumed as follows: ( ) , ( ) , ( )j t j t j t r Ar E AE M AMT t T e V t V e V t V e  = = = (23) where TAr, VAE, and VAM represent the amplitudes of the modal coordinate response of the beam, the voltage across the two electrodes, and the voltage across the external coils, respectively. Also, the modal mechanical force function is rewritten as follows: ( )2 0 ( ) ( ) L j t j t r Ar AT rF t F e m W W x dx e = = −  (24) By substituting Eq. (23) into Eqs. (13), (18), and (21), the following expressions for VE(t), VM(t), and wrel(x,t) are obtained: 2 2 1 2 2 2 2 1 1 ( ) (2 ) ( ) 1 ( ) ( ) (2 ) ( ) (2 ) M E j tMEr Ar r r r r E MEr EMr MEr MMr C M ME ME R E r rr r r r r r j F e j V t j j j C n R j j                          =   = =  − +  =        + + + −       − + − +           (25) Multi-Objective Optimization and Performance Analysis of Bimorph... 675 2 2 1 2 2 2 2 1 1 ( ) (2 ) ( ) ( ) ( ) (2 ) ( ) (2 ) j tMMr Ar r r r r M MMr MMr MMr EMr M c EM EM r rr r r r r r j F e j V t j j R j L n j j                        =   = =  − +  =        + + + −       − + − +           (26) 2( ) ( ) ( ) ( , ) ( ) c M EM E rel r MM r R j L V t V t w x t W x j    + − = (27) where the parameters nME and nEM represent the ratios of VM(t) to VE(t) and vice versa, respectively, have been obtained as follows: 31 31 31 31 1 ( ) ( ) 1 ( ) ( ) M E M M M C M EM R EM ME M M EM E M ME M c f NR j C j e RV n n V j f NR e R j L           + −     = = =  − +  (28) Finally, by having the expressions for VE(t) and VM(t), the analytical relationships for the harvested electrical power PE(t), the harvested magnetic power PM(t), and the total harvested magneto-electric power PME(t) by the MEE energy harvesting beam can be obtained. These equations provide a quantitative understanding of the energy harvesting performance of the MEE beam. They can be used to optimize their design and operation for various applications in wireless sensor networks and biomedical implants. 2( ) ( ) 2 E E E V t P t R = (29) 2( ) ( ) 2 M M M V t P t R = (30) ( ) ( ) ( )ME E MP t P t P t= + (31) 2.5. Multi-objective Optimization of Bimorph MEE Harvesters by New Method of HHO Algorithm Optimization techniques are mathematical tools used to find the best solution for a given problem within a set of constraints. These techniques are widely used in various domains, such as engineering, economics, biology, and artificial intelligence. They help solve complex problems by finding the optimal solution that maximizes or minimizes a specific objective function. Optimization techniques have become increasingly popular in industries and academia due to their ability to provide efficient and effective solutions to problems. One such innovative approach is the HHO algorithm, a population-based method inspired by the hunting behavior of Harris’s hawks. It can be used for various optimization problems by utilizing appropriate formulas. The HHO simulates the behaviors of Harris’s hawks in hunting scenarios to optimize solutions. The algorithm involves exploration and exploitation stages based on the prey’s energy levels. Different strategies are modeled to mimic the prey’s escape patterns and the Hawk’s pursuit behaviors [27]. The principles of the HHO algorithm are summarized and illustrated in Fig. 2. 676 M.M. SHIRBANI, S.E. ALAVI Fig. 2 Flowchart of the HHO algorithm for the determination of the best design variables In the following, the bimorph MEE harvesters have been optimized using the novel algorithm of HHO and multi-objective optimizations. Five different multi-objective optimization cases, as introduced in the chart displayed in Fig. 3, are specified on the plots to determine the best performance of the energy harvesting device. Fig. 3 Five different multi-objective cases and corresponding functions optimized by the HHO method Multi-Objective Optimization and Performance Analysis of Bimorph... 677 Also, the geometric and electrical design variables and their limits in terms of optimization are listed as follows: 6 100 10 & 1 & 10 10 h M E M l mm b mm h h mm R R N       (32) 3. RESULTS AND DISCUSSION This section presents the numerical results of the optimal connections for the series and parallel configurations of the energy harvesting beams. Table 1 provides the values of the properties of the harvesting layers. It should be noted that a composite material consisting of BaTiO3-CoFe2O4 with a volumetric percentage of 50% is selected as the active layer, while aluminum is chosen as the homogenous layer. BaTiO3-CoFe2O4 is a composite material with potential applications in energy harvesting due to its piezoelectric and magnetic properties. Table 1 Properties of the MEE energy harvester layers 11 hC (Nm-2) 75×109 g33 (NsV-1C-1) 2600×10-12 11 MC (Nm-2) 225×109 μ33 (Ns2C-2) 90×10-6 e31 (Cm-2) -3.5 ρh (kgm-3) 2707 f31 (NA-1) 350 ρM (kgm-3) 5550 h33 (C2N-1m-2) 6.3×10-9 ζ 0.01 During the continuation of the analysis, the results of multi-objective optimization of the MEE energy harvester in series and parallel connections using the HHO algorithm were extracted and presented in Tables 2 and 3, which will help analyze the performance of the energy harvesting device and make informed decisions to optimize its efficiency. Also, as an example, in Figs. 4a and 4b, the multi-objective optimization results and the Pareto front for the first optimization case are shown. In these figures, points A, B, and C are optimal in this research phase. The power of PM is maximized for the series connection at point A, while the power of PE is minimal. Neither of the two harvested powers is maximized at point B, but their values are in the desired optimal state. At point C, the power of PE is maximized, while the power of PM is almost minimized. Point C was selected as the optimal point based on the maximum total power PME criteria. In Fig. 4b, point A was selected for parallel connection as the result of optimization due to the maximum of both electrical powers. Other optimized variables have been obtained with another similar method and are listed in Tables 2 and 3. 678 M.M. SHIRBANI, S.E. ALAVI Fig. 4 Multi-objective optimization results and the Pareto front for the first case of optimization case; (a) Series connection, (b) Parallel connection Table 2 Optimal multi-objective optimization variables for series connection b (mm) L (mm) hh (mm) hM (mm) N RM (Ω) RE (Ω) Series layers 5.61 90.50 0.857 0.143 6 0.008733 107.992 Case1 8.80 92.72 0.878 0.122 10 0.0156 95.568 Case2 8.92 90.01 0.872 0.128 9 0.0212 126.744 Case3 8.63 89.12 0.888 0.112 7 876.35 31681.40 Case4 9.91 98.81 0.857 0.143 4 51.48 34750.10 Case5 Table 3 Optimal multi-objective optimization variables for parallel connection After obtaining the values of the optimized variables with the help of Eq. (25), the frequency response of the voltage VE generated by each energy harvesting beam connection is plotted against the dimensionless excitation frequency η, considering the vicinity of their respective first natural frequencies. The results are depicted in Fig. 5, where Fig. 5a corresponds to the series connection, and Fig. 5b corresponds to the parallel connection. The parameters of the electrical circuits of the two-terminal electrodes and external coils, as well as the geometric values of the harvesting beams, are selected based on the values obtained from Tables 1 and 2 to plot these results. These figures show that the maximum voltage VE is generated near the first natural frequency of the beam harvester. Considering Fig. 5a, the maximum value of VE for the series connection in the second optimization case is measured as 0.808 Vs2/m. This value is notably higher than the minimum VE generated in the third case, indicating that the second optimization case can b (mm) L (mm) hh (mm) hM (mm) N RM (Ω) RE (Ω) Parallel layers 8.61 89.74 0.898 0.102 5 18.17 15940.02 Case1 9.53 91.11 0.857 0.143 7 0.0156 95.568 Case2 10 95.04 0.896 0.104 8 0.0212 126.744 Case3 10 92.01 0.900 0.100 6 651.52 10011.50 Case4 6.91 96.41 0.900 0.100 8 10.13 10020.31 Case5 Multi-Objective Optimization and Performance Analysis of Bimorph... 679 create a significantly higher VE. Similarly, Fig. 5b demonstrates that the maximum optimal value of VE for the parallel connection in the second optimization case is measured as 0.5771 Vs2/m. This configuration generates up to 44.67% higher voltage VE compared to the minimum optimization case, which is the third one. Furthermore, it is observed that the maximum voltage occurs for excitation ratios less than η = 1. The optimal excitation ratios for the series and parallel connections are measured as 0.99904 and 0.99944, respectively. This phenomenon implies that the electrical circuits connected to the energy harvesting beams increase the damping ratio of the beams. This excitation ratio is defined as the damping ratio of the harvesting beams in the presence of electrical coupling, denoted as ηdam. Utilizing the relation ηdam = (1-ζElec 2)0.5, where ζElec represents the electrical damping ratio of the harvesting beams, the values of ζElec are determined as 0.044 and 0.033 for the series and parallel connections, respectively. The series connection, characterized by a lower electrical damping ratio, exhibits a higher maximum voltage VE, indicating the advantageous utilization of coupled electrical circuits for this configuration. Fig. 5 Frequency response of the generated voltage VE over the electrodes in five cases of optimization; (a) Series connection, (b) Parallel connection of bimorph MEE layers By using Eq. (26), the frequency response of the voltage VM is plotted for both the series (Fig. 6a) and the parallel connections (Fig. 6b). The maximum values of VM are calculated as 0.025 Vs2/m and 0.0212 Vs2/m for the series and parallel connections, respectively. These values represent a 23% difference in the better performance of the series connection. Both of these maximum values occur in the third case. Furthermore, the maximum values of VM are generated at the excitation ratios of 0.99986 and 0.99957 for the series and parallel connections, respectively. By utilizing the relation ηdam = (1-ζMag 2)0.5 and the excitation ratios obtained from the results of Fig. 6, the magnetic damping ratio of the harvesting beams, ζMag, can be determined. The values of ζMag for the series and parallel connections are measured as 0.017 and 0.029, respectively. The series connection, characterized by a higher magnetic damping ratio, exhibits a lower maximum voltage VM. In the following, the frequency responses of currents iE and iM are obtained using Ohm’s law and Eqs. (25) and (26). Then, the frequency response of iE is plotted for both series and parallel connections in Figs. 7a and 7b, respectively. The optimal values of iE are obtained 680 M.M. SHIRBANI, S.E. ALAVI for the series and parallel connections in the fifth and third optimization cases, respectively, as 22.321 μAs2/m and 37.328 μAs2/m. These values signify a 71.72% better parallel connection performance than the series connection. As a result, the parallel connection is essential for designing a new harvesting device with the highest possible amount of current, iE. Fig. 6 Frequency response of the generated voltage VM over the external coil in five cases of optimization; (a) Series connection, (b) Parallel connection of bimorph MEE layers Fig. 7 Frequency response of the generated current iE over the electrodes in five cases of optimization; (a) Series connection, (b) Parallel connection of bimorph MEE layers According to Fig. 8a, the maximum value of iM for the series connection in the fifth optimization case is 200.903 μAs2/m. This value indicates an increase of about 21.55 times compared to the first case. Also, Fig. 8b illustrates that the maximum amount of iM in the parallel connection occurs in the first optimization case, measuring 480.71 μAs2/m. This value is 20.83 times higher than the minimum value in the fourth case. Also, the values Multi-Objective Optimization and Performance Analysis of Bimorph... 681 obtained for the maximum values of iM show that the parallel connection produces 1.39 times more electric current iM . Fig. 8 Frequency response of the generated current iM over the external coil in five cases of optimization; (a) Series connection, (b) Parallel connection of bimorph MEE layers The comparison of different connection’s performance examines how the arrangement of MEE layers, geometric dimensions, and electrical circuit parameters affect the harvested power by the electrodes PE and external coils PM. These results aid in selecting a more suitable and improved structure for these harvesters. The frequency response of PE is plotted for both series and parallel connections in Figs. 9a and 9b. The optimal amount of PE for the series connection occurs when the fifth optimization case is implemented. It has been observed that the maximum value of PE will be attained in the third case for the parallel connection. The power values achieved are 8.66 μWs4/m2 and 7.453 μWs4/m2, respectively, indicating that the series connection produces around 16.19% more power PE than the parallel connection. As a result, the series connection is preferred for achieving maximum generated power PE. The results related to the frequency response of the harvested power by the external coils PM are extracted in the next step. The displayed results in Fig. 10a for the series connection still emphasize the significance of the fifth optimization case, which exhibits a remarkably more significant difference than other cases. This difference is so substantial that it indicates an 8.43 times difference with the fourth optimization case. The results for the parallel connection in Fig. 10b depict the superior performance of the first optimization case and a 77.07% difference with the second optimization case. The comparison between both connections demonstrates that the parallel connection performs better in generating 102.31% more harvested power PM. 682 M.M. SHIRBANI, S.E. ALAVI Fig. 9 Frequency response of the harvested power PE by the electrodes in five cases of optimization; (a) Series connection, (b) Parallel connection of bimorph MEE layers Fig.10 Frequency response of the harvested power PM by the external coil in five cases of optimization; ) a) Series connection, (b) Parallel connection of bimorph MEE layers As mentioned, the ultimate goal in the design of harvesters is to achieve the maximum total harvested power. To this end, the total MEE harvested power PME results for both connections are depicted in Figs. 11a and 11b, respectively. The maximum values of PME have been measured for the series and parallel connections as 9.701 μWs4/m2 and 7.833 μWs4/m2, respectively. The highest value of PME in the series connection, achieved in the fifth optimization case, is 23.85% higher than the value obtained in the first case for the parallel connection. These results show the importance of series connection to achieve the highest total harvested power PME, which is the most crucial goal of a harvester. Multi-Objective Optimization and Performance Analysis of Bimorph... 683 Fig. 11 Frequency response of the total MEE harvested power PME in five cases of optimization; (a) Series connection, (b) Parallel connection of bimorph MEE layers In the final step, using Eq. (27), the tip relative displacement of the beam wrel(l) is calculated for both connections. The frequency responses of wrel(l) are then depicted in Figs. 12a and 12b, respectively. One of the reasons for examining this parameter is its fundamental role in the design of the harvester’s volume. In other words, besides the dimensions of the beam, a specific range should be left empty for the beam’s deflection so that the existing vibrations in the environment can be fully utilized. The results in Figs. 12a and 12b reveal that both connections exhibit similar maximum relative displacement changes in the fifth case. The maximum wrel(l) values for the series and parallel connections are 132.617 μms2/m and 131.457 μms2/m, respectively. The choice of different objective functions in different optimization cases has led to 47.95% and 37.55% greater deflection wrel(l) in the series and parallel connections, respectively. Fig. 12 Frequency response of the relative tip displacement wrel(l) in five cases of optimization; (a) Series connection, (b) Parallel connection of bimorph MEE layers 684 M.M. SHIRBANI, S.E. ALAVI For a better comparison, the maximum values extracted from the plotted graphs in Tables 4 and 5 are provided for both connections, respectively. In the fifth optimization case, where the maximum power PME occurs, the contribution of PE to the total power PME is 89.33% for the series connection. In this case, the optimal maximum value of the current iM is 9.01 times the generated current, iE. In the first optimization case of the parallel connection, the contribution of electrical power PE to the maximum total harvested power PME is 73.16%, indicating a larger share of the circuit connected to the external coils in power generation compared to the series case. The optimal maximum value of iM is 17.93 times the amount of the iE, which is a significant amount for producing electrical current in a harvesting system. Table 4 Optimal values of the objective functions in the series connection Case5 Case4 Case3 Case2 Case1 Series layers 9.701 5 2.9258 6.922 3.902 PME (µWs4/m2) 8.666 4.88 2.619 6.605 3.866 PE (µWs4/m2) 1.038 0.123 0.031 0.0312 0.003561 PM (µWs4/m2) 22.321 17.56 19.967 16.354 10.874 iE (µAs2/m) 200.903 16.812 30.277 27.99 8.906 iM (µAs2/m) 0.7759 0.5561 0.2622 0.808 0.7104 VE (Vs2/m) 0.0103 0.0146 0.025 0.0223 0.007978 VM (Vs2/m) 132.617 93.969 89.639 108.862 91.859 Wrel (µms2/m) Table 5 Optimal values of the objective functions in the parallel connection 4. CONCLUSIONS This study comprehensively investigates the multi-objective optimization of a MEE energy harvester operating in both series and parallel connections. The main goal is to maximize the total harvested power by considering various objective functions, such as harvested power, voltage, and current. The study uses the HHO algorithm and looks at points on the Pareto fronts for each connection and objective function. The best candidate in each iteration is considered the target or a solution close to the optimum in the HHO algorithm. The algorithm randomly places hawks in different places based on specific identification strategies. The resulting equations describing the beam’s mechanical displacement response, the harvested powers, voltages, and currents were formulated and presented according to the specified parameters. The frequency response of the bimorph configuration of the MEE energy harvesting system under harmonic excitation was Case5 Case4 Case3 Case2 Case1 Parallel layers 7.833 6.578 7.716 1.258 2.455 PME (µWs4/m2) 5.731 5.392 7.453 1.099 1.436 PE (µWs4/m2) 2.1 1.186 0.2606 0.157 1.017 PM (µWs4/m2) 26.814 18.684 37.328 4.688 5.365 iE (µAs2/m) 480.71 151.318 24.489 22.024 451.033 iM (µAs2/m) 0.4273 0.5771 0.3989 0.4688 0.5367 VE (Vs2/m) 0.00873 0.0156 0.0212 0.0143 0.00431 VM (Vs2/m) 107.992 95.568 126.744 104.142 131.457 Wrel (µms2/m) Multi-Objective Optimization and Performance Analysis of Bimorph... 685 investigated after finding and using the optimal points. These findings provide valuable insights into the design and operation of efficient MEE energy harvesters. Specifically, the investigation emphasizes the importance of selecting the optimal point based on the criterion of maximum total power. 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