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In this paper, a theoretical mathematical model in conjunction with an electrical generation model is examined. Using a simulated algorithm, the optimal design of a two-mass energy harvester that finds the maximal electrical power will be assessed. Before the optimal design is performed, the influence of the electrical power with respect to design param- eters such as the magnet’s height, the diameter, the stiffness of the lower springs, the stiffness of the upper springs, the revolution of the lower coil, the revolution of the upper coil, the diameter of the coil’s wire, and the electrical resistance of the loading will be analyzed. Results reveal that the design parameters play essential roles in maximizing electrical power. The two mode shapes of the two-mass energy harvester also occur at the targeted forcing frequencies. The electrical power is optimally extracted at the two primary forcing frequencies, i.e. 12 and 30 Hz. Moreover, it is obvious that the induced electrical power of the two-mass energy harvester will be superior to that of the one-mass energy harvester. Keywords Two-mass, vibration, harvester Introduction Because portable devices are unusually popular, durable electrical power for these devices is a concern. However, there is the problem of electrical exhaust during prolonged use. So as to provide sufficient and continuous electrical power, interest in portable energy harvesters with a spring–mass system has surfaced. In 1997, Williams et al., doing research with small generators, invented a wireless sensing device that was triggered by vibrational energy using a sensor that was fixed onto a bridge to detect structural status. Now, there are many energy resources including mechanical energy, thermal energy, and potential energy. Conventional work in the vibration control is to eliminate the vibrational energy using various active vibration control methods and vibra- 2–5 tion absorbers. As a view point of energy saving, it is an energy loss for the vibrational equipment system; therefore, along with these, there are numerous energy extraction methods such as the piezoelectric, electromag- 6,7 netic, magnetostrictive, and electrostatic transducers. The electrical application of the above electrical genera- tion devices that do not need continuous amounts of electricity is still sufficient even though the electrical power created from the vibration-based generator is small. However, because of progress in the semiconductor field, MEMS (Micro-Electro-Mechanical Systems) along with various integrated circuit (IC) devices with low electrical Department of Mechanical and Automation Engineering, Chung Chou University of Science and Technology, Yuanlin City, Taiwan Department of Mechanical Engineering, Tatung University, Taipei, Taiwan Corresponding author: Min-Chie Chiu, Department of Mechanical and Automation Engineering, Chung Chou University of Science and Technology, Yuanlin City, Changhua County 51003, Taiwan. Email: minchie.chiu@msa.hinet.net Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (http://www. creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/open-access-at-sage). Chiu et al. 91 power was developed. An example of the MEMS would be the application of vibration-based ICs used for the diagnosis of equipment using a vibrational signal via a remote monitoring system. In order to extract more energy from environmental vibrational energy, many energy harvesters have been proposed. Also, various piezoelectric energy harvesters with low electrical power have been created. However, they are fragile and therefore risky when used at lower frequencies to extract energy of higher vibrational ampli- tude. So, to extract energy from a lower frequency vibrational source, an electromagnetic energy harvester was presented. In previous studies, a portable one-mass vibration-based energy harvester used in extracting vibrational energy from equipment has been developed. Sapinski developed a generator for a linear magnetorheological damper using permanent magnets and coil with foil winding. However, electrical power with one-tone resonation 11 12 has been limited and is insufficient. Later, Chiu et al. assessed three kinds of portable one-mass pendulum-arm energy harvesters using experimental tests. But, because only a resonating vibration will be induced for a one-mass vibrating system, the energy extracted from the one-mass vibrational system was limited to vibration with one forcing frequency. Therefore, in order to extract more vibrational energy from equipment having multitone forcing vibration, the development of a multimass energy harvester used to extract the vibrational energy at the multitone resonating frequencies is essential. There are many related works found in literatures. In this paper, in order to expand the electrical power of various frequencies (12 and 30 Hz), a portable vibration-based two-mass energy harvester is proposed. To maximize the extracted energy, two resonant frequen- cies of the two-mass vibrational system will be similarly tuned as two external vibrating frequencies emitted from the vibrational source. Eight kinds of design parameters—the magnet’s height (H ), the diameter (D ), the m m stiffness of the lower springs (k ), the stiffness of the upper springs (k ), the revolution of the lower coil (N ), 1 2 1 the revolution of the upper coil (N ), the diameter of the coil’s wire (d ), and the electrical resistance of the loading 2 w (R )—used in tuning the system’s natural frequencies are adopted. The optimization of a two-mass electro- load magnetic energy harvester is performed by using the simulated annealing (SA) method. Consequently, a compar- ison of the outputted electrical power between the one-mass energy harvester and the two-mass energy harvester will also be assessed and discussed. Mathematical models Motion of the two-mass vibration system The two-dimensional vibration-based electromagnetic energy harvester is depicted in Figures 1 and 2. In case of a ix t ix t 1 2 base excitation is inspired simultaneously by y ðtÞ¼ Y e and y ðtÞ¼ Y e , the related displacement f f f f 1 1 2 2 responses with respect to magnet 1 and magnet 2 are assumed as ix t ix t 1 2 x ðtÞ¼ X ðe Þþ X ðe Þ (1a) 1 1f 1f 1 2 ix t ix t 1 2 x ðtÞ¼ X ðe Þþ X ðe Þ (1b) 2 2f 2f 1 2 The motion equation for a two-mass harvester using differential operation is m x ¼ðx yÞk ðx _ y_Þc þðx x Þk þðx _ x _ Þc 1 1 1 1 1 2 1 2 2 1 2 (2) m x ¼ðx x Þk ðx _ x _ Þc 2 2 1 2 2 1 2 Plugging equation (1) into equation (2) and performing Laplace transform yields s m ½X þ X ¼ X Y k X Y k X Y sc X Y sc > 1 1f 1f 1f f 1 1f f 1 1f f 1 1f f 1 1 2 1 1 2 2 1 1 2 2 þ X X k þ X X k þ X X sc þ X X sc 2f 1f 2 2f 1f 2 2f 1f 2 2f 1f 2 1 1 2 2 1 1 2 2 (3) > 2 > s m ½X þ X ¼ X X k X X 2 2f 2f 2f 1f 2 2f 1f > 1 2 1 1 2 2 X X sc X X sc 2f 1f 2 2f 1f 2 1 1 2 2 92 Journal of Low Frequency Noise, Vibration and Active Control 37(1) Figure 1. Two-dimensional base excitation with two supporting springs and two coils. (a) A mechanism of two-mass energy harvester and (b) experimental equipment for a vibration-based electromagnetic energy harvester. The matrix form of equation (3) is "#"# "# m s þðc þ c Þs þ k þ k ðk þ c sÞ X þ X k þ c s 1f 1f 1 1 1 1 2 1 2 2 2 1 2 ¼ ðY þ Y Þ (4) f f 1 2 ðk þ c sÞ m s þ k þ c s X þ X 0 2f 2f 2 2 2 2 2 1 2 Or Chiu et al. 93 Figure 2. Kinetic diagram for a two-mass base-excitation energy harvester. Using the Cramer’s rule in solving X and X yields 1f ;f 2f ;f 1 2 1 2 ðk þ c sÞðm s þ c s þ k Þ 1 1 2 2 2 X ¼ Y 1f ;f f ;f 1 2 1 2 2 2 ðk þ c sÞ ðm s þ c s þ k Þ½ m s þðc þ c Þs þ k þ k (5a) 2 2 2 2 2 1 1 2 1 2 ¼ A þ jB 1f ;f 1f ;f 1 2 1 2 ðk þ c sÞðc s þ k Þ 1 1 2 2 X ¼Y 2f ;f f ;f 1 2 1 2 2 2 ðk þ c sÞ ðm s þ c s þ k Þ½ m s þðc þ c Þs þ k þ k 2 2 2 2 2 1 1 2 1 2 (5b) ¼ A þ jB 2f ;f 2f ;f 1 2 1 2 The resulting solution of x ðtÞ yields 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X X s t jðx tþa Þ i 2 2 i i x ðtÞ¼ X e ¼ A þ B e (6a) 1f ;f 1f 1 2 i 1f 1f i i i¼1 i¼1 where a ¼ tan ðB =A Þ (6b) i 1f 1f i i Similarly, the resulting solution of x ðtÞ is 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X X s t jðx tþb Þ i 2 2 i x ðtÞ¼ X e ¼ A þ B e (7a) 2f ;f 2f 1 2 i 2f 2f i i i¼1 i¼1 where b ¼ tan ðB =A Þ (7b) 2f 2f i i i The relative displacements of the lower magnet z ðtÞ and the upper magnet z ðtÞ are 1 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 X X jðx tþaÞ x t 2 2 i i z ðtÞ¼ x ðtÞ y ðtÞ¼ A þ B e Y ðe Þ (8a) 1f ;f 1f f f 1 2 i i 1 1 i i¼1 i¼1 94 Journal of Low Frequency Noise, Vibration and Active Control 37(1) 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X X 2 2 jðx tþbÞ x t i i z ðtÞ¼ x ðtÞ y ðtÞ¼ A þ B e Y ðe Þ (8b) 2f ;f 2f f f 1 2 i i i 2 2 i¼1 i¼1 In case of an excitation yðtÞ¼ Y sinx t þ Y sinx t, the relative displacement of the lower magnet and the f 1 f 2 1 2 upper magnet is () qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jðx tþaÞ x t 2 2 i i z ðtÞ¼ Imfz ðtÞg ¼ Im A þ B e Y ðe Þ (9a) 1 1f ;f f 1 2 1 1 i i¼1 () qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jðx tþbÞ x t 2 2 i i z ðtÞ¼ Imfz ðtÞg ¼ Im A þ B e Y ðe Þ (9b) 2 2f ;f f 1 2 2 2 i i¼1 To simplify the simulation, the diameter and resistance of the coil’s wires around the upper magnet and the lower magnet are similarly designed. Therefore, for a single-layer coil, the relative damping coefficients (c and c ) 1 2 are expressed as 2 2 K ðN Bd Þ 1 1 c ¼ ¼ (10a) 4N D R þ R 1 c i1 load q þ R load 2 2 K ðN Bd Þ 2 2 c ¼ ¼ 4N D (10b) R þ R 2 c i2 load q þ R c load where l l c1 c2 K ¼ N Bl ; K ¼ N Bl ; R ¼ q ; and R ¼ q (10c) 1 1 c1 2 2 c2 i1 i2 c c A A c1 c2 For n-layer coil, the related damping coefficient yields ðnN Bd Þ c ¼ (11a) R þ R i1 load ðnN Bd Þ c ¼ (11b) R þ R i2 load where 1 D 8N þðn 1Þd 1 c R ¼ q (11c) i1 c n¼1 L D 2 c 8N þðn 1Þd 2 c R ¼ q (11d) i2 n¼1 The magnetic flux for a permanent magnet For a cylindrical permanent magnet, it is assumed that the inner magnetization is uniform. The coupled currents method is then applied for calculating the magnetic field of the permanent magnet. The effective current dis- tribution is the same as an ideal solenoid. Since the magnetic field is along the z-axis only, integration of the Chiu et al. 95 magnetic field along the z-axis is obtained and shown in equation (12a) using the Biot–Savart law. The distribution flux density through the center of cylindrical axis is expressed in equation (13a) Z Z m Z 2p l I sinh sinh 1 2 dB ¼ Rdhdz ub br (12a) z / 2 2 2 m 4pr r r 1 1 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 D H D m m m where R ¼ ; r ¼ z þ þ ; 2 2 2 (12b) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 H D m m r ¼ z þ 2 2 2 3 D D m m z þ z 6 7 2 2 B ðzÞ¼ B qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C (13a) z r4 5 2 2 H H m m 2 2 4 z þ þ D 4 z þ D m m 2 2 where H H m m C ¼ 1 < z < 2 2 (13b) > H H m m : C ¼ 0 > z; z > 2 2 The electromagnetic energy output According to Wang’s analysis and experimental work, a best allocation for the coil can be obtained. Results reveal that the magnetic voltage will increase if the layer of the coil increases. In order to simplify the analysis, only a one-layer solenoid/winding coil is considered. In addition, the magnet will be allocated at the center of the coil. According to Faraday’s law, the individual induced voltage with respect to each circular coil yields d/ dB dz e ¼ ¼A (14) dt dz dt For a single-layer coil, the cross area can be expressed as A ¼ p (15) For an n-layer coil, the related cross area of each layer yields A ¼ p þðÞ a 1 d ;fg a 2 N : a > 0 (16) a c In the case of a single-layer coil, the relative displacement d between a moving magnet and each turn of a fixed coil with N coil revolutions can be expressed as equation (17) N þ 1 d ¼ Z d n (17) n c where n is the index number of each revolution from the top to the bottom. 96 Journal of Low Frequency Noise, Vibration and Active Control 37(1) Subsequently, the induced voltage of each turn is 2 3 2 2 6 7 D D dz m m 6 7 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e ¼A B (18) n c r 4 5 hi hi 2 2 dt 2 2 H H m 2 m 2 4 d þ þ D 4 d þ D n n 2 m 2 m Moreover, the total induced voltage yields 2 3 6 7 6 7 N N 2 2 X X 6 7 D D dz m m 6 7 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V ¼ e ¼ A B total n c r (19) 6 7 "# "# u u 2 2 dt 6 2 2 7 n¼1 n¼1 u u H H 4 5 m m t t 2 2 4 d þ þ D 4 d þ D n n m m 2 2 Likewise, in the case of an n-layer coil, M is the number of coil layers and N is the number of revolutions of each coil layer. The total induced voltage yields 2 3 6 7 6 7 M N M N 2 2 XX XX 6 7 D D dz m m 6 7 V ¼ e ¼ A B vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (20) total n a r 6"# "# u u 2 dt 6 2 2 7 a¼1 n¼1 a¼1 n¼1 u u H H 4 m m 5 t t 2 2 4 d þ þ D 4 d þ D n n m m 2 2 According to Faraday’s law, the individual induced voltage with respect to each circular coil yields d/ pD dB dz e ¼ ¼ (21a) dt 4 dz dt where 2 3 2 2 6 7 dB ðzÞ D D m m 6 7 ¼ B rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (21b) 4 5 hi hi dz 3 3 2 2 H H m 2 m 2 4 z þ þ D 4 z þ D 2 m 2 m Consequently, the total voltage induced by a moving magnet will be obtained by summing up all the coil’s induced electrical voltages N 1 V ðD ; H ; k ; k ; N ; N ; d ; R Þ¼ e (22) total m m 1 2 1 2 c load n n¼0 Objective function The induced system’s electrical voltage will be maximized if the system’s two natural frequencies (x and x ) are d1 d2 the same as the two external inputted frequencies (x and x ). Here, x is smaller than x . In order to reach ex1 ex2 ex1 ex2 the targeted natural frequencies, an objective function of the summation of the frequency’s squared deviation is established 2 2 (23) OBJðD ; H ; k ; k ; N ; N ; d ; R Þ¼ ðx x Þ þðx x Þ m m 1 2 1 2 c load ex1 d1 ex2 d2 Chiu et al. 97 Table 1. Given data of the vibrating system and the two-mass vibration-based electromag- netic energy harvester. Item Properties Unit Upper mass m ¼ 16:02 g Lower mass m ¼ 14:91 g Permanent magnet Diameter D ¼ 12 mm Height H ¼ 16 mm Coil Number of layers L ¼ 4 layers Turns of each layers N ¼ 34 turns Height of each layer h ¼ 73 mm Supporting spring k ¼ 88.2528 N/m Loading resistor R ¼ 200 X load Excited amplitude Y ¼ 500 mm The optimal design data where the system’s two natural frequencies are very close to the two targeted fre- quencies (x and x ) will be obtained by adjusting eight parameters (D , H , k , k , N , N , d , R ) during ex1 ex2 m m 1 2 1 2 c load the optimization process. Model check To verify the accuracy of the mathematical model and experimental data, the experimental work of a two-mass vibration-based electromagnetic energy harvester in conjunction with a vibrating source shown in Figure 1 has been established. As indicated in Figure 1, two NdFeB-made cylindrical permanent magnets serve as a vibrator. The vibrator that has a 12 mm diameter and is 16 mm in height is compressed by two springs (Stainless A313) at both ends. The selected spring is about 71 mm in l (free length of the spring), 0.5 mm in d (diameter of the s s spring’s wire), and 11.57 mm in D . Before the theoretical formula is calculated, some physical conditions will be preset and experimentally measured. The physical conditions are given and shown in Table 1. As indicated in Figure 1, to obtain the spring’s stiffness and damping ratio, experimental work using the logarithmic decrement method in conjunction with a piezoelectric pressure sensor is carried out. The damping coefficient can be obtained by using an experimental measurement without the coil. The damping ratio relationships are shown as below pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi c ¼ 21 km ¼ c þ c ¼ 21 km þ c (24a) sys m e e sys m pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi c ¼ 21 km 21 km ¼ 2 kmð1 1 Þ (24b) sys m sys m where damping ratio ð1 Þ for a mechanical motion is 0.01184. The logarithmic, d, can be calculated using the logarithmic decrement method 1 V d ¼ ln (25) n V where V is the first peak amplitudes and V is the nth period peak amplitude. Therefore, the related damping o n ratio is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ¼ (26) sys 2p 1 þ Using the above data in the theoretical calculation, the profile of the theoretically induced electrical voltage and the experimental voltage with respect to frequency is obtained and shown in Figure 3. The lower spring stiffness will decrease and the upper spring stiffness will increase due to the magnetic effect on the springs. In addition, because the center axis of the helical spring, the center axis of cylindrical magnet, and the center axis of the vibration source may not be set up on the same line, some vibrational energy might be dissipated in the uniaxial 98 Journal of Low Frequency Noise, Vibration and Active Control 37(1) Figure 3. Performance of a two-mass vibration-based electromagnetic energy harvester when comparing the theoretical simulation to the experimental work. (a) The response of the lower magnet and (b) the response of the upper magnet. Table 2. The ranges of the design parameters. Optimization parameter defined Optimized range Unit D 0.01–0.03 m H 0.01–0.03 m H H m m N b c 0:2b c 2 d d c c H H m m N b c 0:2b c 2 d d c c k 0:5x 2x N/m 1 ex1 ex2 k 0:5x 2x N/m 2 ex1 ex2 d 0.0001–0.0001 m R 100–10,000 X load direction. Therefore, the practical energy detected in the experimental work will be smaller than that of the theoretical prediction. As depicted in Figure 3, the performance curve of the theoretical and experimental data is in agreement. Moreover, the maximum electromagnetic voltage is tuned to the desired frequencies of 7 and 18.4 Hz. Therefore, the mathematical model is acceptable. Consequently, the model linked by the SA numerical method is used for optimizing the unit-induced electrical voltage of a two-mass vibration-based electromagnetic energy harvester in the following section. Case study There are two kinds of base vibrating sources (excited frequencies of 12 and 30 Hz) with the same displacement amplitude of 0.1 mm occurring in the equipment. To optimally extract these vibrations, a two-mass electromag- netic energy harvester is adopted. To simplify the simulation, the dimensions of two magnets and the diameter of two coal wires are similarly designed. Eight kinds of design parameters—the magnet’s height (H ), the diameter (D ), the stiffness of the lower m m springs (k ), the stiffness of the upper springs (k ), the revolution of the lower coil (N ), the revolution of the upper 1 2 1 coil (N ), the diameter of the coil’s wire (d ), and the electrical resistance of the loading (R )—used in tuning the 2 w load system’s natural frequencies are adopted. The optimization of a two-mass electromagnetic energy harvester is performed using the SA method described in the following sections. The related ranges of the design parameters are shown in Table 2. Moreover, the data for the vibrating system and the vibration-based electromagnetic energy harvester are shown in Table 3. Chiu et al. 99 Table 3. The related physical property of the magnet and the coil. Neodymium permanent magnet Symbol Description 3 3 q Neodymium magnet density 7:4 10 kg=m B Residual magnetic flux density 1.2 T Wounding coil Figure 4. Flow diagram of a SA optimization. Optimization process The OBJ function is linked to the SA method. The concept behind SA was first introduced by Metropolis et al. and later developed by Kirkpatrick et al. The optimization flow diagram using the SA method is shown in Figure 4. 100 Journal of Low Frequency Noise, Vibration and Active Control 37(1) For the SA optimization process, a new random solution (X ) will be chosen from the neighborhood of the current solution (X). If the change in the objective function is negative (i.e. DF 0), a new solution will be acknowledged as the new current solution with the transition property pb(X ) of 1; if it is not negative (i.e. DF > 0), 0 0 then a new transition property (pb(X )) varying from 0 to 1 will be calculated using the Boltzmann factor (pb(X ) =exp(DF/CT)) as shown in equation (27) 1 ;DF 0 pbðX’Þ¼ (27a) DF exp ;DF > 0 CT DF ¼ OBJðX Þ OBJðX Þ (27b) where C and T are the Boltzmann constant and the current temperature. Simultaneously, a new random prob- ability of rand(0, 1) will be given to compare with the above new transition property (pb(X )). If the transition property (pb(X )) is greater than the random number of rand(0, 1), the new uphill solution, which results in a higher energy condition, will then be accepted; if not, it will be rejected. Here, the uphill solution with an inferior solution will then have a better possibility of escaping from the local optimum. The algorithm reiterates the perturbation of the current solution and the measurement of change in the objective function. Each successful swap of the new current solution will point to the decay of the current temperature as T ¼ kk T (28) new old where kk is the cooling rate. The process is reiterated until the predetermined number (iter ) of the outer loop is max reached. Table 4. Optimal OBJ for the vibration-based electromagnetic energy harvester at various kk and iter (at targeted tones of 12 and max 30 Hz with an amplitude of 0.01 m). SA control parameter Design parameter Result kk iter D H d k k N N n n R OBJ max m m c 1 2 1 2 1 2 load m m m N/m N/m X 100 0.9 0.023 0.0233 0.0004 672.749 1066.768 38 18 35 25 8028 54.964 100 0.91 0.0294 0.0105 0.0007 873.416 673.695 5 23 3 2 3458 96.435 100 0.92 0.0135 0.0261 0.0009 1150.708 1099.709 45 23 26 16 9707 10.664 100 0.93 0.0229 0.0167 0.0006 685.782 691.118 41 24 23 7 4235 11.186 100 0.94 0.0269 0.0153 0.0009 709.65 943.682 8 4 47 11 5817 49.227 100 0.95 0.0244 0.0151 0.0008 728.773 730.911 17 28 39 49 791 69.609 100 0.96 0.0277 0.0128 0.0003 843.845 745.468 23 35 17 23 2070 10.017 100 0.97 0.0136 0.0231 0.0005 1114.461 843.545 44 57 33 27 1452 41.089 100 0.98 0.0241 0.0165 0.0004 1066.093 616.93 47 59 44 12 6749 33.668 100 0.99 0.022 0.0247 0.0009 1054.002 897.155 28 13 30 3 5909 12.185 1000 0.96 0.023 0.0233 0.0004 672.749 1066.768 38 18 35 2 8028 7.885 1000 0.96 0.0294 0.0105 0.0007 873.416 673.695 5 23 3 2 3458 1.659 2000 0.96 0.0135 0.0261 0.0009 1150.708 1099.709 45 23 26 16 9707 0.098 3000 0.96 0.0258 0.0155 0.0002 635.718 879.902 97 54 29 8 8038 0.638 4000 0.96 0.0269 0.0153 0.0009 709.65 943.682 8 4 47 11 5817 0.154 5000 0.96 0.0244 0.0151 0.0008 728.773 730.911 17 28 39 49 791 0.399 6000 0.96 0.0277 0.0128 0.0003 843.845 745.468 23 35 17 23 2070 0.767 7000 0.96 0.0136 0.0231 0.0005 1114.461 843.545 44 57 33 27 1452 0.458 8000 0.96 0.0241 0.0165 0.0004 1066.093 616.93 47 59 44 12 6749 0.069 9000 0.96 0.022 0.0247 0.0009 1054.002 897.155 28 13 30 3 5909 0.029 OBJ: objective function; SA: simulated annealing. Chiu et al. 101 Results and discussion For a base-vibrating system excited by two frequencies (f ¼ 12 Hz, f ¼ 30 Hz), the optimal results by varying d1 d2 the SA’s control parameters (kk, iter ) are shown in Table 4. As indicated in Table 4, the optimal design data max can be obtained when the SA parameters at (kk, iter ) ¼ (0.96, 10,000) are applied. As indicated in Table 4, the max optimal design data occurring at the 20th set are (D ¼ 0:022, H ¼ 0:0247, d ¼ 0:0009, k ¼ 1054:002, m m c 1 k ¼ 897:155, N ¼ 28, N ¼ 13, n ¼ 30, n ¼ 3, R ¼ 5909). The related system’s natural frequencies of f 2 1 2 1 2 load ex1 Figure 5. Comparison of the frequency response between the target tones (f and f ) and the designed tones (f and f ). d1 d2 ex1 ex2 Figure 6. The displacements of two magnets with respect to time at the external excitation frequency of 12 Hz. Figure 7. The displacements of two magnets with respect to time at the external excitation frequency of 30 Hz. 102 Journal of Low Frequency Noise, Vibration and Active Control 37(1) and f are 12.048 and 30 Hz, respectively. The designed tones (f and f ) for two magnets at x and x are ex2 ex1 ex2 1 2 plotted in Figure 5. The response of the displacements of two magnets with respect to time at the external excitation frequency of 12 Hz is shown in Figure 6. Additionally, the response of the displacements of two magnets with respect to time at the external excitation frequency of 30 Hz is shown in Figure 7. Inputting the Figure 8. The induced voltage of two coils with respect to time at the external excitation frequency of 12 Hz during one period of time. Figure 9. The induced voltage of two coils with respect to time at the external excitation frequency of 30 Hz during one period of time. Figure 10. The induced voltage with respect to time of two-mass vibration system excited by two frequencies of 12 and 30 Hz. Chiu et al. 103 Table 5. Optimal OBJ for the one-mass vibration-based electromagnetic energy harvester (at targeted tones of 12 and 30 Hz with an amplitude of 0.01 m). m (kg) d (m) k (N/m) c (N m/s) N (turns) L (layers) 1 c 1 1 1 1 0.0134 0.0002 76.319 0.10073 18 11 OBJ: objective function. Figure 11. The induced voltage with respect to time of one-mass vibration system excited by two frequencies of 12 and 30 Hz. Figure 12. The induced voltage with respect to various design parameters (D , H , k , k , d , N , N , R ). (a) The induced voltage m m 1 1 c 1 2 load with respect to D while other parameters are fixed, (b) the induced voltage with respect to H while other parameters are fixed, (c) m m the induced voltage with respect to k while other parameters are fixed, (d) the induced voltage with respect to k while other 1 2 parameters are fixed, (e) the induced voltage with respect to d while other parameters are fixed, (f) the induced voltage with respect to N while other parameters are fixed, (g) the induced voltage with respect to N while other parameters are fixed, and (h) the 1 2 induced voltage with respect to R while other parameters are fixed. load 104 Journal of Low Frequency Noise, Vibration and Active Control 37(1) design data into the electrical power’s calculation, the resulting electrical power of the magnets with respect to time is plotted in Figures 8 and 9. Moreover, considering two external excitation frequencies of 12 and 30 Hz simultaneously acting on the two- mass energy harvester, the electrical power of a two-magnet harvester with respect to time is shown in Figure 10. As indicated in Table 4, the optimal design data can be obtained when the iteration number (iter ) increases max and the SA’s cooling rate (kk) is 0.96. In addition, Figure 5 indicates that the system’s optimal natural frequencies (f and f ) are very close to the external forcing frequencies (f =12 Hz, f =30 Hz). Obviously, the system’s ex1 ex2 d1 d2 motion will resonate at the external forcing frequencies (f , f ) of 12 and 30 Hz. The induced resonant motion d1 d2 will result in a large reciprocating motion between the magnets and the related coils. With this, the induced electrical power will be maximized. As indicated in Figure 6, the amplitude of the upper magnet (x ) is larger than that of the lower magnet (x ) when the external forcing frequency is at 12 Hz. The phase angle difference between x and x is 0 . Moreover, as indicated Figure 7, the amplitude of the lower magnet (x ) is larger than that of the 1 2 1 upper magnet (x ) when the external forcing frequency is at 30 Hz. The phase angle difference between x and x is 2 1 2 180 . Obviously, the phase angle difference of 0 shown in Figure 6 represents the first mode shape that occurs during the reciprocating motion of the two-mass vibrational system. Additionally, the phase angle difference of 180 shown in Figure 7 represents the second mode shape that occurs during the reciprocating motion of the two-mass vibrational system. For a two-mass vibrational system, the physical phenomena of two mode shapes during the resonant motion are thus verified. As can be seen in Figures 8 and 9, the immediate electrical power for two magnets varies periodically when two kinds of external base excitation forcing frequencies (12 and 30 Hz) are added individually. Considering two external excitation frequencies of 12 and 30 Hz simultaneously acting on the two-mass energy harvester, the electrical power of two magnets with respect to time is shown in Figure 10. To appreciate the electrical efficiency of one-mass and two-mass harvesters in extracting the vibrational energy from a piece of vibrating equipment with two excitation frequencies of 12 and 30 Hz, the electrical power of a one- magnet harvester with respect to time is optimally assessed and presented in Table 5 and Figure 11. As indicated in Figure 12. Continued. Chiu et al. 105 Figures 10 and 11, it is obvious that the electrical efficiency of the two-mass harvester is superior to that of the one- mass harvester. To realize the influence of parameters (D , H , k , k , d , N , N , R ) with respect to the induced voltage has m m 1 2 c 1 2 load been assessed and shown in Figure 12. As indicated in Figure 12(a) and (b), there are two peaks of voltage occurred at two specified values of D (the magnet’s diameter) and H (the magnet’s height) due to a two-mode m m resonating system. And, as illustrated in Figure 12(c) and (d), a peak of voltage occurred at a specified k and k . 1 2 Also, result in Figure 12(e) shows that the induced electrical voltage (V) will increase if d (coil’s wire diameter) decreases. As illustrated in Figure 12(f) and (g), the peak of voltage will occur at a compromised number of coil turns (N and N ). It is because that the retarding force to the magnetic motion will increase when number of coil 1 2 turns (N and N ). Therefore, to reach maximal electrical voltage, the coil turns must be compromised. 1 2 Consequently, as indicated in Figure 12(h), the electrical voltage will remain the same when R (the electrical load resistance) is beyond a specified value. Conclusion It has been shown that SA can be used in the optimization of a two-mass vibration-based electromagnetic energy harvester. The SA parameters of the kk (cooling rate) and the iter (maximum iteration number) are essential max during the SA optimization. The higher iter will result in a better solution. max Concerning the geometric allocation, eight parameters (H : the magnet’s height; D : the magnet’s diameter; m m k : the stiffness of the lower springs; k : the stiffness of the upper springs; N : the revolution of the lower coil; N : 1 2 1 2 the revolution of the upper coil; d : the diameter of the coil’s wire; R : the electrical resistance of the loading) w load which are tightly related to the induced electrical voltage are adopted for maximization of electrical power. To increase the electrical power extracted from vibrating equipment with two primary forcing frequencies (f =12 d1 Hz, f =30 Hz), a two-mass energy harvester induced to resonate via two natural frequencies (f and f ) is built d2 ex1 ex2 and optimized using an SA optimizer. The resulting motions of two magnets at two individual forcing frequencies are obtained and shown in Figures 6 and 7. The figures indicate that the two mode shapes of the two-mass energy harvester occur at the targeted forcing frequencies (f =12 Hz, f =30 Hz). The frequency response in Figure 5 d1 d2 also indicates that the two-mass energy harvester is optimally tuned at the two primary forcing frequencies of the vibrating equipment. Considering two external excitation frequencies of 12 and 30 Hz simultaneously acting on the two-mass energy harvester, the electrical power of two magnets with respect to time is shown in Figure 10. Consequently, as indicated in Figures 10 and 11, the electrical efficiency of the two-mass harvesters is superior to that of the one-mass harvester. Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors acknowledge the financial support of the National Science Council (NSC 100-2221-E-036-019, ROC). References 1. Williams CB, Pavic A, Crouch RS, et al. Feasibility study of vibration-electric generator for bridge vibration sensor, Proc. SPIE Vol.3243, Proceeding of the 16th International Modal Analysis Conference, 1998, p1111. 2. Snamina J and Orkisz P. Active control strategy for reduction of vibrations in mast exposed to ground motions. Low Freq Noise Vib Active Control 2013; 32: 157–166. 3. Ladipo I and Muthalif A. 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Appendix Notation A cross section area of coil (m ) A cross section area of magnet (m ) B average magnetic flux density (T) B residual flux density at z-axis (T) B flux density at z-axis (T) c electromagnetic damping coefficient of the coil (N s/m) c mechanical damping coefficient (N s/m) c damping coefficient of vibration system (N s/m) sys d coil’s wire diameter (m) D coil diameter (m) D diameter of permanent magnet (m) H height of magnet (m) iter maximal iteration in the SA method max k spring constant (N/m ) k spring constant of vibration system (N/m) sys kk cooling rate in the SA method l length of a coil’s wire (m) m ; m upper ward and down ward mass of permanent magnet (kg) 1 2 N ; N number of coil’s turns 1 2 R electrical resistance of the coil’s wire (X) iner R electrical resistance of the loading for the electromagnetic energy harvester (X) load R electrical resistance of wire resistance and loading resistance (R þ R )(X) total load iner Y input amplitude for the exciting base (m) Z relative displacement of a magnet with respect to the excited base (m) e induced voltage (V) q electrical resistivity coefficient of coil ðX mÞ 1 damping ratio of a vibration system sys 1 damping ratio for a mechanical motion 1 electromagnetic damping ratio / magnetic flux (Wb)
"Journal of Low Frequency Noise, Vibration and Active Control" – SAGE
Published: Feb 19, 2018
Keywords: Two-mass; vibration; harvester
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