HARVESTING MECHANICAL VIBRATIONS ENERGY USING NONLINEAR ELECTROMAGNETIC MINIGENERATORS A SURVEY OF CONCEPTS AND PROBLEMS

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1 POZNAN UNIVE RSITY OF TE CHNOLOGY ACADE MIC JOURNALS No 90 Electrical Engineering 2017 DOI /j Marcin KULIK* Mariusz JAGIEŁA* HARVESTING MECHANICAL VIBRATIONS ENERGY USING NONLINEAR ELECTROMAGNETIC MINIGENERATORS A SURVEY OF CONCEPTS AND PROBLEMS The state of knowledge in the field of conversion of energy of mechanical vibrations into electrical energy using nonlinear electromagnetic generators is presented. The principle of operation of the considered converters is based on the Faraday law. The electromotive force is induced by the relative movement of the coil or permanent magnets under impact of externally applied vibrations. In order to diminish the disadvantages of conventional generators, namely the narrow frequency bandwidth, in recent years the nonlinear systems were introduced that exhibit the nonlinear resonance phenomenon. Broadening the frequency bandwidth, in which the power generated by the system is relatively high, is realized via introduction of nonlinear force into system kinematics. Designing such systems becomes a big challenge. Based on thorough survey of recent publications as well as on own expertise in the field, the work compares a few concepts of nonlinear electromechanical minigenerators in term of their functional characteristics and design problems. Sample calculations of frequency characteristics using time- and frequency-domain models are presented. KEYWORDS: energy harvesting, nonlinear resonance, bistability, electromagnetic induction 1. INTRODUCTION Harvesting energy from sources available in the environment, has recently attracted considerable attention [1, 2]. This fact is related with continuously increasing number of wireless systems requiring complete autonomy of power supply as well as with the need to increase the share of renewable systems in generation of electric power. One of the most widespread are the systems harvesting energy from mechanical vibrations [1] as the sources of vibrations are present in almost every environment. The most commonly considered type of vibration energy harvesters are the inertial ones which use a few kinds of physical phenomena for conversion of vibrations into electricity, namely: * Opole University of Technology.

2 348 Marcin Kulik, Mariusz Jagieła electromagnetic, piezoelectric, magnetoelectric and electrostatic [1, 2]. This work focuses on the electromagnetic generators which are characterised by the highest power density per unit volume and relatively high currents and the output voltages of nearly 1 V [2]. The main disadvantage of such systems relates with the need to design the mechanical system of the harvester so as its structural resonance frequency matches frequency of the external vibrations. A configuration of the kinematics of such the system is presented in Fig. 1. The vibrations are applied to the moving base which moves with respect to the reference point along y axis. The relative motion z( of mass m is limited by stiffness k and damping d of the spring element that connects vibrating mover with permanent magnets with the base. As the permanent magnets move with respect to the coil fixed to the base the back emf is induced in the winding. The equations of electromechanical balance for the considered systems are in form [2] 2 d z( dz( m d kz( F( (1a) 2 dt dt di( t ) Lc ( RL RC )i( t ) (t ) (1b) dt where, F( is the total applied force, L c and R c the coil inductance and resistance, respectively, R L the load resistance and i( the electric current through the coil winding. The theoretical point of maximum generated electrical power is reached when the frequency of the excitation force matches the structural resonance frequency given by k f 1 res 2 m (2) Fig. 1. Kinematics of linear electromagnetic vibration energy harvester

3 Harvesting mechanical vibrations energy using nonlinear Figure 2 displays typical frequency characteristics of the linear harmonic oscillator for two magnitudes of the excitation frequency. Theoretically in this type of harvesters the resonance frequency depends only on the mechanical structure of the system that determines the stiffness k. In practice, beside the externally applied excitation force F ext ( the total force F( contains the electromagnetic force Fe(i) produced by the action of the magnetic flux on current. The latter affects the stiffness of the system. As a consequence, even though the kinematics does not contain any nonlinear components, the system characteristics expose weak nonlinear effects [8]. Fig. 2. Frequency characteristics of linear harmonic oscillator for two magnitudes of the external force The majority of sources of mechanical vibrations have the non stationary frequency rich spectra [1, 2, 3]. The only ways to tune the harvester with respect to the signal is to change the number of turns in the coil and the load resistance, but both are the same extremely impractical, if not impossible methods [3]. The latter was the motivation to focus attention on nonlinear systems, where the frequency bandwidth is wider than in the linear systems. 2. NONLINEAR ELECTROMECHANICAL MINIGENERATORS 2.1. Concepts and structures The nonlinear electromechanical minigenerators excited with permanent magnets develop the internal force F mag that makes impact on the overall stiffness of the system. This force is brought by action of the magnets on the ferromagnetic core [4] or by action of the magnets on another magnets [5]. This effect can also be accomplished by utilisation of pure mechanical effects such as those produced by the guided beams [6]. In this work we focus the attention of systems where the nonlinearity is brought by the electromagnetic force. Figure 3 displays the kinematics of the nonlinear harvester, where the spring constant is augmented by the nonlinear

4 350 Marcin Kulik, Mariusz Jagieła stiffness k mag (z) dependent on the displacement. In such case the equations of electromechanical balance are in form 2 d z( dz( m d [ k k ( z)] z( F( 2 mag (3a) dt dt di( t ) dz( t ) ( RL RC )i(t ) (3b) i dt z dt where is the flux linkage of the coil. Equation (3b) is a more general case which takes account for systems where the nonlinearity is brought by action of the magnets on the ferromagnetic core [4]. The relationships are nonlinear not only due to k mag (z) = F mag /z, but also to (i,z). If the second case is considered, namely when the nonlinearity is a result of force between magnets without the presence of ferromagnetic core, the flux linkage is decoupled from current, then const Lc which greatly simplifies the designing process. When i neglecting the impact of electromagnetic force due to current coupled with, damping d, and assuming that z( is a single harmonic function of time, equation (3a) has the following nonlinear eigenvalue ~ 1 k kmag ( Z ) f r ( Z ) (4) 2 m where f ~ r is the time averaged frequency, and Z the magnitude of displacement. The resonance frequency that the system would exhibit neglecting the nonlinear effects is that given by (2) and is called the natural frequency f n. From (4) it is clear that the frequency has to be found iterating between solution of (3) and equation (4). In many cases the force can be approximated by the polynomial [7] 3 F s ( z) k1z k2z (5) making possible to apply the existing theory of the nonlinear oscillator based on the Duffing equation [7]. Fig. 3. Kinematics of nonlinear vibration energy harvester

5 Harvesting mechanical vibrations energy using nonlinear Usually the magnetic system that is responsible for relationship (5) can be reconfigured such that coefficients a and b can be positive or negative. With respect to system kinematics the nonlinear stiffness causes the hardening (k mag (z) > 0) or the softening action (k mag (z) < 0). Figure 4a depicts the theoretical frequency characteristics for both cases. The complexity of the frequency characteristics is beyond the dependence of the resonance point on the magnitude of the external force. The force (5) is a non monotonic function of displacement which causes complex behavior of the frequency characteristics. When the force reaches the knee point, the characteristic experiences a jump o value. The jump appears at different points when increasing or decreasing frequency. Another consequence of this phenomenon is thus the hysteresis of the characteristics as explained by arrows in Fig. 4. a) b) Fig. 4. Frequency characteristics of nonlinear oscillators: a) for hardening and softening magnetic stiffness, b) for softening magnetic stiffness and the two different magnitudes of external forces In the beginning of the XXI century some configurations of nonlinear harvesters were presented. One of the proposed systems had a single permanent magnet connected to a stretching membrane that brought nonlinear stiffness into system kinematics [8]. Work [9] presents the results of mathematical modeling of a nonlinear electromechanical generator, composed of the two permanent magnets connected to the base and the two moving in a tube with a coil wound on the surface. The model used a single equation of motion where the results of the 2d finite element computations of the magnetic force were used. The impact of the load resistance was investigated. The maximum determined power was equal to 6 mw. Works [10] and [11] present a low frequency harvester. The construction uses a moving coil and stationary magnets. The nonlinearity is brought by nonlinear spring made of a special polymer. Works [6] and [12] present another solution, in which the nonlinearity is brought by the guided beam. There also examples of hybrid generation systems such as that in [7, 13] where the nonlinear beam made of a piezoelectric material increases the amount of the generated power. Another group are

6 352 Marcin Kulik, Mariusz Jagieła systems combining the nonlinear stiffness of mechanical and magnetic origins simultaneously [14]. Widening the frequency bandwidth is due to the system bistability caused by the magnetic forces and mechanical stretching. Works [5, 15] presents such the system designed for the operating frequency equal to 35 Hz that has the frequency bandwidth wider by some 5 Hz with respect to linear system. Configuration of such the harvester is depicted in Fig. 5. Fig. 5. Nonlinear vibration energy harvester with bistability caused by magnetic force [5] A generator with stationary coil and the ferromagnetic core was presented in [16, 17]. By means of coupled analysis using partial finite element computations and the dynamic equation of motion it was demonstrated that the frequency bandwidth can be as much as 30 Hz wide. It is also shown that at higher frequencies the system exhibits chaotic motion. A modified version of such the generator is presented in [18]. Another example of the system with nonlinearity brought by the magnetic forces is presented in [19]. The considered systems consists of a singly supported cantilever beam with permanent magnet attached to the end. The coil is attached to the base. The generator uses additional "magnet on magnet" system to bring the nonlinearity. The vibrating beam is covered with the piezoelectric material to increase the efficiency of harvesting. The system of this type constructed by the authors [4, 20, 21] is shown in Fig. 6. Fig. 6. Electromagnetic minigenerator with cored coil [20]

7 Harvesting mechanical vibrations energy using nonlinear Design problems Designing the harvesters with nonlinear resonance phenomenon should be started from providing appropriate variation of the potential energy which depends on both mechanical and magnetic stiffness [22]. If bistable systems are considered, then it should be noticed that an appropriate balance between the mechanical and magnetic energy should be provided [7, 23]. To ensure transition between the local stability points (see Fig. 7), an appropriate design of the magnetic system should be carried out [24, 25, 26]. a) b) Fig. 7. Potential energy vs displacement for: a) monostable system, b) bistable system The characteristic in Fig. 7a has a single point of local stability where the moving part of the harvester stops at static conditions and starts its operation from. For the generators with the ferromagnetic cores shown in Fig. 6 the potential energy has two stable minima and one unstable point. By appropriate design of the magnetic system the potential energy can be reverted giving one stable and two unstable points [21]. This task can be accomplished using the finite element analysis of the magnetic field distribution [4, 20]. Beside the local stability one should take account of the global stability so as the system generates electric power that is proportional to power of the excitation force and does not hang in local equilibrium points. This task, first of all, requires application of structural analysis [21] in order to determine the equivalent mechanical stiffness. Then, based on equations (3) the analysis of global stability should be carried out using the computational methods for nonlinear systems such as the Lyapunov methods [16, 17]. This regards especially systems with bistability which expose tendency to oscillate around or even hang in the stable equilibrium points. The chaos phenomenon defined by sensivity of system to initial conditions and lack of stable trajectory, can be detected using the method of Lyapunov exponents [16]. The presence of chaos

8 354 Marcin Kulik, Mariusz Jagieła is generally a determinant of instability regarding the long term prediction of performance for nonlinear systems. For the short time predictions, the systems exposing chaotic behaviour can, however generate the electric power which is proportional to the power of the excitation force [17]. In order to determine the correct conditions of operation every system should be considered individually taking account of not only the system itself but also a type of signal representing the vibrations. 3. CALCULATION OF FREQUENCY CHARACTERISTICS The frequency characteristics of the system response to the excitation force are the main descriptors of performance for the energy harvesters. In determination of these characteristics both the time and frequency domain models are used. The former are based on solution of the ode systems defined by equations (3) using the methods of numerical integration. The frequency domain models are formulated using the Galerkin harmonic balance approach or the complex envelope function approach [27]. The time domain analysis provides higher accuracy of predictions, although involves severe numerical costs. The frequency domain models are far more computationally efficient, but provide lower accuracy of predictions especially for the systems exposing significant chaos and bistability. Often they, however provide the accuracy that is sufficient for basic design purposes. The advantage of the frequency domain models is the possibility to illustrate the jump and hysteresis phenomena. To compare the performance of both models the analysis was carried out for the system illustrated in Fig. 6 having the variation of the magnetic stiffness as shown in Fig. 8. The results were obtained using the 2d magnetostatic analysis. a) b) Magnetic stiffness kmag [N\m] Magnetic potential energy [J] Displacement [m] Displacement [m] Fig. 8. Characteristics of magnetic system: a) magnetic stiffness vs. displacement, b) potential energy vs. displacement. The characteristics are even functions of displacement

9 Harvesting mechanical vibrations energy using nonlinear As seen in figure the systems is magnetically unstable at 0 point and exposes two stable equilibria at 5 mm. The remaining parameters are (see equations (3)): m = kg, d = 10 5 Ns/m. The system is analysed at open circuit conditions, thus equation (3b) is not considered. Figures 9 10 present the frequency characteristics calculated using both approaches for k = 825 N/m and k = 425 N/m, respectively. The time domain analysis was carried out using the continuous sweep of the frequency from 0 to 50 Hz. The computations were carried out using Matlab ode23s function. The equations of the frequency domain model were derived using the complex envelope method and solved using the fixed point technique. The shaded area in Figs is enclosed between the upper and lower envelope of the obtained waveforms. a) b) Fig. 9. Comparison of characteristics determined using time and frequency domain models for system with small chaos: a) frequency charatceristics, b) spectrogram of displacement for the time domain analysis a) b) Fig. 10. Comparison of characteristics determined using time and frequency domain models for system with significant chaos: a) frequency charatceristics, b) spectrogram of displacement for the time domain analysis

10 356 Marcin Kulik, Mariusz Jagieła The information on the harmonic spectrum of the waveforms is presented in spectrograms in Figs. 9b 10b. As it can be seen the system with smaller k exposes chaos at small frequencies. After jump of displacement both systems are no longer proportional. For the system with higher k the predicted frequency of a jump diverges by some 2.5 Hz, though for higher magnitudes these are qualitatively different. The differences become clear from Fig. 9b as after jump the change of frequency cannot be handled by the single harmonic frequency domain model. For the chaotic system the fixed point iterations are divergent and the frequency domain model does not provide any solution (see Fig. 10a). 4. CONCLUSIONS In this work we presented the most important problems involving application of nonlinear electromechanical minigenerators as the systems harvesting electric power from mechanical vibrations. The attention was focused on the systems using ferromagnetic core elements as means to bring nonlinearity. The basic computational methods used for predictions of the performances of the considered systems were compared showing their advantages and drawbacks. Currently, using the presented methods the authors work on designing an improved system which is to be presented shortly. REFERENCES [1] Mitcheson P. D., Yeatman E. M., Rao G. K., Holmes A. S., Green T. C., Energy Harvesting From Human and Machine Motion for Wireless Electronic Devices, Proc. IEEE, vol. 96, no. 9, pp , Sep [2] Beeby S. P., Tudor M. J., White N. M., Energy harvesting vibration sources for microsystems applications, Meas. Sci. Technol., vol. 17, no. 12, pp. R175 R195, Dec [3] Beeby S. P., Torah R. N., Tudor M. J., Glynne Jones P., O Donnell T., Saha C. R., Roy S., A micro electromagnetic generator for vibration energy harvesting, J. Micromechanics Microengineering, vol. 17, no. 7, pp , Jul [4] Jagieƚa M., Kulik M., Considerations on frequency characteristics of an electromechanical vibration energy harvesting converter with nonlinear parametric resonance, Int. J. Appl. Electromagn. Mech., vol. 53, no. 1, pp , Jan [5] Podder P., Amann A., Roy S., Combined Effect of Bistability and Mechanical Impact on the Performance of a Nonlinear Electromagnetic Vibration Energy Harvester, IEEE/ASME Trans. Mechatronics, vol. 21, no. 2, pp , Apr [6] Boisseau S., Despesse G., Seddik B. A., Nonlinear H Shaped Springs to Improve Efficiency of Vibration Energy Harvesters, J. Appl. Mech. Asme, vol. 80, no. 6, p , 2013.

11 Harvesting mechanical vibrations energy using nonlinear [7] Elvin N., Erturk A., Advances in Energy Harvesting Methods, ISBN New York, NY: Springer New York, [8] Williams C. B., Shearwood C., Harradine M. A., Mellor P. H., Birch T. S., Yates R. B., Development of an electromagnetic micro generator, IEE Proc. Circuits, Devices Syst., vol. 148, no. 6, p. 337, [9] Dallago E., Marchesi M., Venchi G., Analytical Model of a Vibrating Electromagnetic Harvester Considering Nonlinear Effects, IEEE Trans. Power Electron., vol. 25, no. 8, pp , Aug [10] Sardini E., Serpelloni M., An efficient electromagnetic power harvesting device for low frequency applications, Sensors Actuators A Phys., vol. 172, no. 2, pp , Dec [11] Mallick D., Amann A., Roy S., Analysis of Nonlinear Spring Arm for Improved Performance of Vibrational Energy Harvesting Devices, J. Phys. Conf. Ser., vol. 476, no. 1, p , Dec [12] Cottone F., Basset P., Vocca H., Gammaitoni L., Electromagnetic Buckled Beam Oscillator for Enhanced Vibration Energy Harvesting, in 2012 IEEE International Conference on Green Computing and Communications, 2012, pp [13] Ab Rahman M. F., Kok S. L., Ali N. M., Hamzah R. A., Aziz K. A. A., Hybrid vibration energy harvester based on piezoelectric and electromagnetic transduction mechanism, in 2013 IEEE Conference on Clean Energy and Technology (CEAT), 2013, pp [14] Podder P., Mallick D., Roy S., Bandwidth widening in nonlinear electromagnetic vibrational generator by combined effect of bistability and stretching, J. Phys. Conf. Ser., vol. 557, no. 1, p , Nov [15] Podder P., Amann A., Roy S., A bistable electromagnetic micro power generator using FR4 based folded arm cantilever, Sensors Actuators A Phys., vol. 227, pp , May [16] Sato T., Igarashi H., A New Wideband Electromagnetic Vibration Energy Harvester with Chaotic Oscillation, J. Phys. Conf. Ser., vol. 476, no. 1, p , Dec [17] Sato T., Igarashi H., A chaotic vibration energy harvester using magnetic material, Smart Mater. Struct., vol. 24, no. 2, p , Feb [18] Sugisawa T., Igarashi H., Properties of chaotic vibration energy harvester: comparison of numerical results with experiments, Int. J. Numer. Model. Electron. Networks, Devices Fields, no. April, pp. 1 6, [19] Xu Z. L., Shan X. B., Song R. J., Xie T., Electromechanical modeling and experimental verification of nonlinear hybrid vibration energy harvester, in 2014 Joint IEEE International Symposium on the Applications of Ferroelectric, International Workshop on Acoustic Transduction Materials and Devices & Workshop on Piezoresponse Force Microscopy, 2014, pp [20] Kulik M., Jagieła M., Coupled dynamic FE analysis of permanent magnet mechanical vibration energy harvesting converter, Pozn. Univ. Technol. Acad. Journals. Electr. Eng., vol. 85, pp , 2016.

12 358 Marcin Kulik, Mariusz Jagieła [21] Jagieła M., Kulik M., Cogging force and frequency bandwidth of a vibration energy harvester with nonlinear electromechanical resonance, Przegląd Elektrotechniczny, vol. 1, no. 1, pp , Jan [22] Gammaitoni L., Vocca H., Neri I., Travasso F., Orfei F., Vibration Energy Harvesting: Linear and Nonlinear Oscillator Approaches, in Sustainable Energy Harvesting Technologies Past, Present and Future, InTech, 2011, pp [23] Harne R. L., Wang K. W., A review of the recent research on vibration energy harvesting via bistable systems, Smart Mater. Struct., vol. 22, no. 2, p , Feb [24] Lan C., Qin W., Enhancing ability of harvesting energy from random vibration by decreasing the potential barrier of bistable harvester, Mech. Syst. Signal Process., vol. 85, pp , Feb [25] Kumar A., Balpande S. S., Anjankar S. C., Electromagnetic Energy Harvester for Low Frequency Vibrations Using MEMS, Procedia Comput. Sci., vol. 79, pp , [26] Zhou Z., Qin W., Zhu P., Improve efficiency of harvesting random energy by snap through in a quad stable harvester, Sensors Actuators A Phys., vol. 243, pp , Jun [27] Peng Z.K., Lang Z.Q., Billings S.A., Tomlinson G.R., Comparisons between harmonic balance and nonlinear output frequency response function in nonlinear system analysis, J. of Sound and Vibration, vol. 311, pp , (Received: , revised: )