Study of the effects of magneto-mechanical coupling in the performance of electromagnetic vibration energy harvesters

MASTER’S FINAL THESIS Study of the effects of magnetomechanical coupling in the performance of electromagnetic vibration energy harvesters Author: Arnau Arasanz Armengol Director / Co-director: Robert Arcos Villamarin / Victor Ordoñez Izquierdo Degree: Master’s degree in research in Mechanical Engineering (MUREM) Examination session: Spring, 2023 Document: Report i Abstract Electromagnetic vibration energy harvesters convert mechanical energy, in the form of vibrations, into electricity. They are, therefore, devices with huge potential but also with major drawbacks regarding adaptability, efficiency and return of investment. Their behaviour is complex to characterize, especially when there is a significant magnetic interaction between the device and external ferromagnetic or magnetic components, which could result in strong non-linear behaviours that might affect the performance of the device. The aim of this project is to understand better the influence of these magnetic forces on the mechanical system response by developing a simulation tool which can predict the behaviour of a harvester under different scenarios of interest. The proposed model is a one-degree-of-freedom spring-damper-mass system with applied magnetic forces. Its system response is computed for a sinusoidal excitation input and through two different methods. The former being a time domain integration method and the latter known as harmonic balance method, which is performed in the frequency domain. A weak coupling between magnetic and mechanical phenomena is assumed by performing the electromagnetic simulation independently and later inputting the results in the equation of motion of the system. Several experimental tests representing the different case scenarios are carried out in order to validate the simulation tool. Both experimentally and numerically, when magnetic forces are being applied, the harvester is seen to experience a significant shift in its resonant frequency, i.e., a change in its stiffness. This shift results in a softening effect if the oscillation magnet is subjected to attraction forces and in a hardening effect if, on the contrary, it is subjected to repulsion forces. Els recol·lectors electromagnètics d’energia de vibracions converteixen energia mecànica, en forma de vibracions, a electricitat. Són per tant, dispositius amb un gran potencial però també amb grans desavantatges pel que fa a adaptabilitat, eficiència i cost. El seu comportament és complex de caracteritzar, especialment quan hi ha una interacció magnètica significativa entre el propi dispositiu i components magnètics o electromagnètics externs, fet que podria derivar en un comportament no-lineal i afectar considerablement al seu rendiment. L’objectiu principal d’aquest projecte és el d’entendre millor la influencia d’aquestes forces magnètiques al rendiment mecànic d’aquest tipus de dispositius desenvolupant una eina de simulació que permeti predir el comportament d’un recol·lector sota diferents escenaris d’interès. El model que es proposa és el d’un sistema massa-esmorteïdor-molla d’un grau de llibertat amb forces magnètiques aplicades. La resposta del sistema és calculada per a una excitació d’entrada sinusoidal i a través de dos mètodes diferents. El primer és un mètode d’integració temporal mentre que el segon, conegut com a harmonic balance method, es calcula en l’espai freqüencial. Per unir la part magnètica amb la mecànica s’empra un acoblament dèbil; calculant en primer lloc les forces magnètiques que rep el dispositiu per posteriorment introduir-les en l’equació de moviment del sistema. Diferents tests experimentals representatius dels diversos escenaris a estudiar es duen a terme per tal de validar l’eina de simulació. Tant experimentalment com numèricament, en aquells casos en que hi ha forces magnètiques aplicades, s’observa un canvi substancial en la freqüència de ressonància del sistema, és a dir, un canvi en la seva rigidesa. Aquest canvi implica una reducció en la rigidesa del sistema quan l’imant del recol·lector està subjecte a forces d’atracció i un augment quan les forces són de repulsió. Study of the effects of magneto-mechanical coupling in the performance of electromagnetic vibration energy harvesters ii Acknowledgements Firstly, I would like to express my gratitude to both Dr. Robert Arcos and Dr. Victor Ordoñez for their assistance throughout the completion of this thesis. Their technical and personal contribution to the project has been essential to achieve the presented results and their implication and willingness to help at any time needed has always been remarkable. It has been a pleasure working with them on such an exciting project and getting to learn that much in such a short period of time. Last but not least, I want to extend my gratitude to my friends and family. Especially to my mother and sister, who have always supported me during adverse moments and encouraged me when taking new challenges in life. iii Table of contents ABSTRACT .................................................................................................................................................... I ACKNOWLEDGEMENTS ............................................................................................................................... II TABLE OF CONTENTS .................................................................................................................................. III LIST OF TABLES ............................................................................................................................................ V LIST OF FIGURES .......................................................................................................................................... V LIST OF ABBREVIATIONS............................................................................................................................. VI LIST OF SYMBOLS ...................................................................................................................................... VII 1. INTRODUCTION ....................................................................................................................................... 1 1.1 OBJECT ............................................................................................................................................... 1 1.2 SCOPE................................................................................................................................................. 1 1.3 REQUIREMENTS .................................................................................................................................... 2 1.4 JUSTIFICATION ...................................................................................................................................... 3 2 BACKGROUND AND STATE-OF-THE-ART .............................................................................................. 4 2.1 VIBRATION ENERGY HARVESTING ............................................................................................................. 4 2.1.1 Transduction mechanisms ........................................................................................................... 4 2.1.2 Applications ................................................................................................................................. 5 2.1.3 Highlights and challenges ............................................................................................................ 6 2.2 ELECTROMAGNETIC VIBRATION ENERGY HARVESTING ................................................................................... 6 2.2.1 Working principle ......................................................................................................................... 6 2.2.2 Architecture ................................................................................................................................. 7 2.2.3 Magnet configuration .................................................................................................................. 7 2.2.4 Case study .................................................................................................................................... 9 2.3 JUSTIFICATION OF THE RESEARCH ............................................................................................................ 10 3 METHODOLOGY ................................................................................................................................ 11 3.1 MODELLING ....................................................................................................................................... 12 3.1.1 Mechanical model ..................................................................................................................... 12 3.1.2 Magnetic model ......................................................................................................................... 16 3.1.3 Coupling ..................................................................................................................................... 17 3.2 IMPLEMENTATION ............................................................................................................................... 21 3.2.1 Software .................................................................................................................................... 21 3.2.2 Algorithm and architecture........................................................................................................ 22 3.3 EXPERIMENTAL ................................................................................................................................... 28 3.3.1 Setup .......................................................................................................................................... 28 3.3.2 Prototype redesign..................................................................................................................... 30 3.3.3 Methodology ............................................................................................................................. 33 4 RESULTS ............................................................................................................................................ 37 4.1 SIMULATION RESULTS ........................................................................................................................... 37 4.1.1 Magnetic forces ......................................................................................................................... 37 4.1.2 Coupling ..................................................................................................................................... 38 4.1.3 Discussion .................................................................................................................................. 39 4.2 EXPERIMENTAL RESULTS ........................................................................................................................ 42 4.2.1 Model A ..................................................................................................................................... 42 4.2.2 Model B ...................................................................................................................................... 42 4.2.3 Model C ...................................................................................................................................... 42 4.2.4 Discussion .................................................................................................................................. 43 4.3 VALIDATION ....................................................................................................................................... 44 Study of the effects of magneto-mechanical coupling in the performance of electromagnetic vibration energy harvesters iv 5 BUDGET SUMMARY .......................................................................................................................... 46 6 ANALYSIS AND ASSESSMENT OF ENVIRONMENTAL AND SOCIAL IMPLICATIONS ............................... 47 7 CONCLUSIONS .................................................................................................................................. 48 7.1 DISCUSSION ....................................................................................................................................... 48 7.2 FURTHER WORK .................................................................................................................................. 49 8 REFERENCES ..................................................................................................................................... 50 v List of tables TABLE 1. COMPARISON BETWEEN PVEH, EVEH, AND EMVEH TRANSDUCTION METHODS [1] [5] [6]. .................................. 5 TABLE 2. STUDY CASE MODELS. ............................................................................................................................... 11 TABLE 3. MECHANICAL MODEL PARAMETERS. ............................................................................................................ 12 TABLE 4. MAGNETIC MODEL PARAMETERS. ................................................................................................................ 16 TABLE 5. AVAILABLE SIMULATIONS. .......................................................................................................................... 21 TABLE 6. EVHAST SIMULATION TOOL OPTIONS. ......................................................................................................... 23 TABLE 7. CONVERSION OF INPUT AMPLITUDES TO DISPLACEMENTS. ................................................................................ 25 TABLE 8. POST PROCESSING OPTIONS. ...................................................................................................................... 28 TABLE 9. BASE RE-DESIGN FEATURES. ....................................................................................................................... 30 TABLE 10. HARVESTER RE-DESIGN FEATURES. ............................................................................................................. 31 TABLE 11. BILL OF MATERIALS OF THE PROTOTYPE. ..................................................................................................... 32 TABLE 12 COMPARISON BETWEEN 3M-HA-P AND STUDY CASE PROTOTYPE PHYSICAL PROPERTIES. ....................................... 33 TABLE 13. EXPERIMENTAL TESTS. ............................................................................................................................. 33 TABLE 14. MODELS M, B & C SIMULATION OPTIONS. ................................................................................................. 38 TABLE 15. MODELS M, B & C SIMULATION PARAMETERS. ............................................................................................ 38 TABLE 16. MODEL M SIMULATION RESULTS. .............................................................................................................. 39 TABLE 17. MODEL B SIMULATION RESULTS. ............................................................................................................... 39 TABLE 18. MODEL C SIMULATION RESULTS. ............................................................................................................... 39 TABLE 19. MODEL A EXPERIMENTAL RESULTS. ............................................................................................................ 42 TABLE 20. MODEL B EXPERIMENTAL RESULTS. ............................................................................................................ 42 TABLE 21. MODEL C EXPERIMENTAL RESULTS. ............................................................................................................ 42 TABLE 22. SIMULATION VS EXPERIMENTAL RESULTS FOR MODELS M/A, B & C. ................................................................ 45 TABLE 23. BUDGET SUMMARY. ............................................................................................................................... 46 List of figures FIGURE 2.1. MAGNET IN-LINE COIL ARCHITECTURES. ..................................................................................................... 7 FIGURE 2.2. MAGNET ACROSS COIL ARCHITECTURES. ..................................................................................................... 7 FIGURE 2.3. LINEAR (A) AND CIRCULAR (B) HALBACH MAGNET ARRAY. .............................................................................. 8 FIGURE 2.4. 3M-HA-P PROTOTYPE CROSS-SECTION (SOURCE: [25]) ............................................................................... 9 FIGURE 2.5. 3M-HA-P HALBACH ARRAY .................................................................................................................... 9 FIGURE 3.1. MECHANICAL MODEL. .......................................................................................................................... 13 FIGURE 3.2. FREE BODY DIAGRAM OF THE MECHANICAL MODEL..................................................................................... 14 FIGURE 3.3. MAGNETIC MODEL. ............................................................................................................................. 16 FIGURE 3.4. FREE BODY DIAGRAM OF THE WEAKLY-COUPLED MODEL. ............................................................................. 18 FIGURE 3.5. MODEL B (A) AND C (B) STABILITY POINTS. ............................................................................................... 18 FIGURE 3.6. TRANSMISSIBILITY UNSTABLE BRANCH DUE TO STIFFNESS SOFTENING. ............................................................ 19 FIGURE 3.7. TEMPORAL RESPONSE OF MODEL B AT THE RESONANT FREQUENCY FOR STATIC INITIAL CONDITIONS. .................... 20 FIGURE 3.8. EVHAST SIMULATION TOOL ARCHITECTURE. ............................................................................................ 22 FIGURE 3.9. EVHAST_MF SCRIPT ALGORITHM. ........................................................................................................ 23 FIGURE 3.10. EMVEH SCRIPT ALGORITHM. .............................................................................................................. 24 FIGURE 3.11. FEMM MAGNETIC MODEL (A) AND MAGNETIC FORCES SIMULATION (B). ........................................................ 24 FIGURE 3.12. EVHAST_S1.M SCRIPT ALGORITHM. .................................................................................................... 25 FIGURE 3.13. EVHAST_S2 SCRIPT ALGORITHM. ........................................................................................................ 25 FIGURE 3.14. EVHAST_S2_ODE SCRIPT ALGORITHM. ............................................................................................... 26 FIGURE 3.15. EVHAST_S3 SCRIPT ALGORITHM. ........................................................................................................ 26 FIGURE 3.16. EXPERIMENTAL SETUP FLOWCHART. ...................................................................................................... 28 FIGURE 3.17. ACCELEROMETERS LAYOUT. ................................................................................................................. 29 FIGURE 3.18. ASSEMBLING SEQUENCE (A) AND ASSEMBLY CROSS-SECTION (B) ................................................................. 31 FIGURE 3.19. TRANSMISSIBILITY OBTAINED WITH SINE (A) AND SWEEP SINE (B) EXCITATIONS. .............................................. 34 FIGURE 3.20. TAS,TBS,TCS TESTS WORKFLOW. ......................................................................................................... 36 FIGURE 3.21. TASSX,TBSSX,TCSSX TESTS WORKFLOW. ............................................................................................. 36 FIGURE 4.1. MODEL B MAGNETIC FORCES RAW RESULTS. ............................................................................................. 37 Study of the effects of magneto-mechanical coupling in the performance of electromagnetic vibration energy harvesters 4 2 Background and state-of-the-art Energy harvesting (EH) is a process in which environmental energy is scavenged and converted into electricity. This technology differs from regular renewable energy generation in that it mostly focuses on low-powered devices (in the range of mW) such as sensors and other monitoring instruments. The technology is implementable to various energy sources, e.g., thermal, solar and kinetic amongst others, covering a wide range of applications. EH devices are of special interest in self-powered wireless devices, being an alternative to batteries and consequently reducing the environmental impact on their disposal. Moreover, they help reducing maintenance or replacement operations, which are usually required in battery-powered devices. Biggest drawback of this technology is the huge dependency on the energy source and the need to adapt, or even to specifically design, the device to the environment constraints and conditions. The present document focuses on kinetic EH, specifically in devices that scavenge mechanical energy in the form of vibrations. 2.1 Vibration Energy Harvesting The interest in Vibration Energy Harvesting (VEH) devices is rapidly growing, one of the main reasons being the increasing need for autonomous sensors or other ultra-low-power devices to be installed in remote, limited access or hazardous locations. Thus, this technology is in direct relation with Internet of Things (IoT) and Wireless Sensors Network (WSN) applications and its growth inside the industry during the last two decades. Scavenging environmental or residual mechanical energy usually implies working with low amplitude excitations. For these devices to generate sufficient power, the input energy needs to be amplified. In VEH technology this is done through mechanical resonance. Apart from the transducer system, which is the responsible for converting the kinetic energy into electricity, VEH devices can also feature a power management unit and an energy storage device. Several technologies can be used for converting the kinetic energy intro electricity, mainly by electromagnetic, piezoelectric, and electrostatic transductions mechanisms. 2.1.1 Transduction mechanisms Piezoelectric Energy Harvesters (PVEHs): Piezoelectric Harvesters [2] use the piezoelectricity phenomena that occurs in certain materials, such as crystals and ceramics, which produces a voltage due to a structural imbalance caused by an external mechanical stress or deformation. Since in this case the vibration occurs on the material, it needs to be taken into account that its stiffness will be considerably high and therefore the usual resonant frequencies will be around 1 kHz. This limits their application range to mostly industrial environments. Moreover, the Nonlinear stiffness of the materials plays a huge role in the response of these devices to external excitations and therefore needs to be considered when modelling them. To maximize the energy produced, PVEHs usually utilize compliant mechanisms, which allow great stress/strain while having low mass, are low cost, and easy to integrate in MEMS devices. Electrostatic Energy Harvesters (EVEHs): In EVEHs devices [3], the conversion into electrical energy occurs as the working area of a capacitator varies as a response of an external mechanical excitation. According to their working principle, EVEH devices can either be electret-free or electret-based. The first type requires additional energy to convert the mechanical into electric energy.The main advantages of these devices are its suitability at small scales and its reduced cost. 5 Electromagnetic Vibration Energy Harvesters (EMVEHs): EMVEH devices [4] use the principle of Faraday’s law of induction to convert the kinetic energy of a magnet moving relatively to a coil into induced current. Further information can be found from Section 2.2 onwards. Table 1. Comparison between PVEH, EVEH, and EMVEH transduction methods [1] [5] [6]. PVEH EVEH EMVEH Output + High voltage High voltage High current - Low current Low current Low current Scalability + Suitable for small-micro scales (MEMS integration) Suitable for small-micro scales (MEMS integration) - Suitable for mediumlarge scales Cost + Unexpensive - Expensive (smart material and complying mechanisms) Potentially expensive (magnets) Architecture + Simple Simple, versatile, robust - Complex Lifetime + Long lifetime and robustness - Aging Aging Frequency range >1 kHz >1 Hz >5 Hz Powerless Yes No (electret-free) Yes 2.1.2 Applications Vibrations are found in many scenarios, usually being residual energy from industrial activity, traffic, or even natural sources. The implementation of VEH devices and the scavenging of vibrations, therefore, is of great interest in a wide range of applications where that energy would otherwise be lost. According to the range of frequencies where the device is designed to work at, the suitable applications might differ. Most devices currently being commercialized are aiming for industrial applications, where the resonant frequencies commonly range from 25 Hz upwards. The current research in this area is mainly focusing on characterization and optimization in order to be able to scavenge with maximum efficiency [7] [8]. On the other hand, while not as widely implemented yet, ultra-low frequency devices are of increasing interest due to their distinct applications: from scavenging natural sources such as ocean waves, tall buildings swaying, to traffic induced vibrations [9]. These devices need to be considerably more compact to achieve lower resonant frequencies (ranging from 1 Hz to 10 Hz), and therefore they must be designed according to the mechanisms and scaling laws applicable [10] to such scales. Wearable vibration harvesters [11] [12] [13] which are currently a strong line of research due to the need of autonomously monitor patients health, are also considered ultra-low VEH devices since they must have resonant frequencies compatible with human motion (ranging from 4.5 Hz to 6 Hz). Study of the effects of magneto-mechanical coupling in the performance of electromagnetic vibration energy harvesters 6 2.1.3 Highlights and challenges The current pros and cons of this technology limit the feasibility to implement these devices in real-world, industrial and user-level applications. One of the strongest points of the technology is its performance under circumstances where other EH or renewable energy devices fall short, such as for indoor or mobile applications. This is due to the fact that they have low reliance on unpredictable environmental sources such as sunlight or wind, and due to their architectural simplicity and compactness. Moreover, the three different transduction methods previously introduced can cover a wide range of applications and problematics. On the other hand, one of the biggest drawbacks (applicable especially to electromagnetic and piezoelectric devices) is that the peak of output power is extremely narrow, having its maximum efficiency near the resonance frequency of the system and rather negligible power output across the rest of the spectrum due to their transduction mechanism, which relies on mechanical resonance principle. This drawback implies that, currently, these devices must be designed ad hoc for a specific application, having little to no adaptability to background, random or varying source vibrations. One solution to this problematic is increasing the bandwidth of the system response, i.e., the range where the system generates high power. These devices are referred to as wide bandwidth vibration energy harvesters and can be developed, for instance, by adding a movable mass which alters the effective mass of the system during operation [14], or by implementing a multi-band resonator, resulting in two spikes of maximum efficiency [15]. Another solution and current line of research is to study and implement tunability capabilities in EMVEH devices. This is a high value feature that would potentially increase the return of the investment since it would increase the adaptability of the devices. Such tunability can be achieved, for instance, by modifying the position of the spring [16], by hybridization of transduction methods [17], by employing liquid metal [18] or by simply changing the mechanical properties of the system [19]. 2.2 Electromagnetic Vibration Energy Harvesting In this project the focus will be on VEH devices that uses electromagnetism as a transduction mechanism mainly due to the advantages previously discussed. 2.2.1 Working principle As presented early, EMVEH devices use the induced current, generated on a coil when a varying magnetic field is applied, as a transduction method between mechanical and electrical energy. The voltage produced directly depends on the number of loops of the coil and on the rate of change of magnetic flux [4]: ɛ=−𝑁𝑐∆Փ ∆𝑡 (2.1) Where ɛ is the induced voltage, Nc the number of loops of the coil and Փ the magnetic flux. Due to the fact that their working principle relies on resonance, these systems are considered of second order and have a narrow spike of maximum vibration amplitude and hence, maximum power for excitations near the resonant frequency. This theoretical maximum power achievable in a mass-spring can be found through the next expression [4]: 𝑃𝑚𝑎𝑥=𝑚𝑌2𝜔3 4𝜉 (2.2) Where m is the oscillating mass, Y the vibration amplitude, ω the vibration frequency and ξ the damping coefficient. 7 2.2.2 Architecture The rate of change of the magnetic field on the coil is commonly achieved by having a relative movement between the conductor material and the magnet. Depending on the moving components and layout of the system the transducer architecture can be categorized into [20]: - Magnet in-line coil: The centre axis of magnet and coil is parallel to the oscillation direction. This architecture is especially convenient with cylindrical components as shown in Figure 2.1. Figure 2.1. Magnet in-line coil architectures. - Magnet across coil: The centre axis of the magnet and coil is orthogonal to the oscillation direction. This architecture is mainly applicable to rectangular magnet configurations as observed in Figure 2.2. Figure 2.2. Magnet across coil architectures. The most suitable architecture will depend on the application and its dimensional or environmental constraints. 2.2.3 Magnet configuration When designing an EMVEH device the magnet configuration and layout is of special importance since it can define how well the system will perform. Geometrical parameters, material, magnetization direction and number of magnets are just the basic considerations to take into account. Regarding geometry, whether the magnet has a cylindrical or a planar shape, the resulting magnetic field will obviously differ [20]. Generally speaking, it is not clear if any geometry proves to be especially advantageous to the other since it will mainly depend on the specific application and layout of the system and it will usually be the architecture of the system what ends up defining it. Study of the effects of magneto-mechanical coupling in the performance of electromagnetic vibration energy harvesters 8 The magnetization direction i.e., the direction of the pole in the magnet (from north to south) is crucial to direct the magnetic field towards regions of interest such as the coil and hence, to maximise the generated power. For cylindrical permanent magnets the most common magnetization directions are axial and radial. While for rectangular are through-length and through thickness. When scaling it to multipole magnets other combinations can be found with interesting properties. Lastly, by arranging several permanent magnets in a specific layout, the magnetic field can be intensified on one side and attenuated (or even suppressed) on the other. These are known as Halbach arrays. 2.2.3.1 Halbach array Halbach arrays consists in an indefinite series of magnets where the magnetization direction is arranged in a rotating pattern. This array can be achieved by means of a circular or a linear layout, as illustrated in Figure 2.3: Figure 2.3. Linear (a) and circular (b) Halbach magnet array. The dark arrows from Figure 2.3 point the direction of the magnetization for each of the permanent magnets while the clearer arrows portray the resulting magnetic flux. As it can be observed, the magnetic field is intensified in those regions where the magnets with a magnetization direction parallel (or tangent) to the array match the direction of the magnetic flux and, on the contrary, it is attenuated where the directions are opposing. Amongst the multiple interesting applications of this concept (such as on DC motors, particle accelerators or refrigerator magnets), it is of special interest on EMVEH devices, where the output power, and consequently the efficiency of the device, is maximised. Several harvesters have been developed exploiting the benefits of Halbach arrays and further discussion about its benefits and drawbacks can be found in [21] [22] [23] [24]. (a) (b) 9 2.2.4 Case study This project is a continuation of the work developed in [1], [25], [21] and [7]. The present case of study is based on the device studied in [1] identified as 3M-HA-P (Figure 2.4). However, the results obtained during the present thesis will not be comparable to those of the 3M-HA-P device since some physical modifications are made to the current prototype that will slightly affect the system response. A more in-depth explanation of the implemented features is carried out in Section 3.3.2. Focusing on the transducer subsystem, 3M-HA-P was conceived as a simple, and therefore affordable, high-power and compact EMVEH device to be employed in the context of wireless sensor networks for civil or industrial applications. The device was indeed tested in real environment applications, such in a railway metro tunnel and in a water distribution system [7]. The main characteristics of this device, which are also applicable to the present study case, are: Transduction mechanism: The transduction mechanism is electromagnetic, and its mechanical setup can be conceived as a basic spring-mass system. The mass consists in a case encompassing the magnet array with two helical springs anchored at each end. Architecture: Referring to the definitions of Section 2.2.2, the device follows a magnet inline coil architecture i.e., the centre axis of both the coil and the magnet, which in this case are concentric, are parallel to the oscillation direction. In this case the fixed component is the coil and the moving part the magnet, which is encircling it. Magnet configuration: The magnet in this prototype is a linear Halbach array composed of three permanent magnets. Upper and lower magnets have axial magnetization direction while the middle one has radial polarization as seen in Figure 2.5. This configuration concentrates the magnetic field on the inner region of the device, where the coil is located, and attenuates it at the exterior. Figure 2.4. 3M-HA-P Prototype cross-section (Source: [25]) Figure 2.5. 3M-HA-P Halbach array The original springs and magnet array will be cannibalized for the new version of the prototype, with the main focus of the re-design being on the mechanical interfaces and on achieving an accurate relative positioning between components. Since the present thesis studies the mechanical response of the system, the coil, which was part of the 3M-HA-P prototype will be removed. Study of the effects of magneto-mechanical coupling in the performance of electromagnetic vibration energy harvesters 10 2.3 Justification of the research No documentation exploring the effects of externally induced magnetic interactions on the performance of EMVEH devices has been found. Predicting these behaviours would imply having more control over the performance of such devices on field. Moreover, if a shift in frequencies is indeed observed in both the numerical simulations and the experimental tests, it could open a line of research into exploring tunability features or increasing the bandwidth of the system response. Obtaining numerical results comparable to the ones observed experimentally would also suggest that these effects can be studied employing a weak coupling method and thus, that these systems can be easily characterized with a low computational demanding tool. While aiming to contribute to the same line of research as the work developed in [1], this thesis will cover aspects of the modelling that were left out of the scope in the cited works; with the idea of shedding light on some of the magneto-mechanical coupling phenomena that has significant impact on their performance. Besides complementing the theoretical and modelling part of the research, during this project a re-design of the 3M-HA-P prototype’s external mechanical interfaces will be made; allowing to test various cases scenarios as well as increasing the captured data, and consequently the accuracy of the measures. by installing more accelerometers. The ultimate goal of the present thesis is not to provide a hard-coded tool but rather a scalable and improvable one; allowing for future refinement updates to include some of the less-critical aspects that, due to limitations of time, will be left out of the scope in the present project. It is, hopefully, a steppingstone towards a better understanding of EMVEH devices. 11 3 Methodology The methodology that is followed to fulfil the objectives presented in Section 1.1 can be split into three distinct phases: I) Modelling: An analytical model is obtained from the physical problem (re-design of prototype 3M-HA-P). Some simplifications, assumptions and hypotheses are introduced to reduce complexity. II) Implementation: The analytical model is shaped into a numerical simulation tool which complies with the requirements established in Section 1.3. To develop such tool, both commercial and open-source software are employed. III) Experimentation: The prototype is developed and tested to compare its experimental response with the simulation tool results and therefore, validate both the modelling and implementation phases. The different scenarios that are studied throughout this thesis are labelled as models and are introduced in Table 2: Table 2. Study case models. Model M (MM) Model A (MA) Model B (MB) Model C (MC) Sim. Exp. Sim. Exp. Sim. Exp. Sim. Exp. X X X X X X Mass Non-magnetic Magnet Magnet Magnet Base Non-magnetic Non-magnetic Magnet (attraction force) Magnet (repulsion force) - Model M: In Model M, the oscillating magnets on the harvester are replaced by steel equivalent mass. Having no magnetic forces applied and hence, no internal or external magnetic interactions, Model M serves as a reference on how the magnetism affects the mechanical response of the system. This model is left out of the scope during the experimental tests since it would imply a significant physical modification (by changing the spring or the oscillating mass to a nonmagnetic material) that would potentially affect the dynamic performance of the device and therefore provide results non-comparable to the others. It is therefore conceived as a theoretical model towards the comparison of the simulated and experimental results. - Model A: This model is representative of a scenario where the harvester is placed on a base with no ferromagnetic or magnetic properties. The difference between models A and M is essentially that, in the former, the internal magnetic interaction between the mass and spring is active (due to the magnets of the harvester) while in the latter it is not. This magnetic interaction needs to be taken into account only when the oscillating mass is a magnet, and the spring is made of a ferromagnetic material. For this reason, this model will just be studied experimentally. In the simulation, since the system is axisymmetrically represented, the spring cannot be modelled and hence, such interaction can be computed. - Model B: Model B portraits a likely scenario where the harvester is placed on a ferromagnetic surface, for instance, on top of a steel machine chassis, and hence, the oscillating magnet experiences magnetic attraction forces. This is of special interest since here the magnetic effects are amplified and therefore, it is where a bigger change on the performance is expected. Study of the effects of magneto-mechanical coupling in the performance of electromagnetic vibration energy harvesters 12 For academic purposes, and in order to maximise the effects of this interaction, instead of considering a simple metallic base, this model is set up with a radial magnet with axial magnetization on its base. If working with other magnet configurations other than a Halbach array (Section 2.2.3), similar results could potentially be obtained by just placing a ferromagnetic base since the magnetic field would not be as much axially attenuated. The result in this case is a considerably higher force applied to the harvester’s oscillating magnet and also the inception of another configuration which is labelled as Model C. - Model C: Model C does not represent a realistic case but rather an interesting one, aiming to explore potential tunability capabilities in EMVEH devices using external magnetism. In this case the magnetization direction of the radial magnet placed on the base is opposing the one from the nearer Halbach magnet, inducing a repulsive force at the harvester’s magnet. 3.1 Modelling In this section, both the theoretical background applicable and the modelling followed for the underlying mechanical and magnetic systems are presented. The general assumptions applicable to all levels of the modelling are: i) Rigid body: The oscillating magnet does not experience deformation. ii) Purely vertical displacement: The oscillating magnet does not experience rotational displacements in any axis. iii) Linear spring behaviour: The behaviour of the spring is assumed to be linear. Real springs show slight nonlinearities due to internal material behaviours, which for the scope on this project will be negligible. iv) Viscous damping model: The mechanical damping will follow a viscous model. These hypotheses will allow considering the electromagnetic model to be axisymmetric and the mechanical model as a 1DOF system, resulting in a simple system convenient for study the targeted magneto-mechanical couplings. 3.1.1 Mechanical model Ignoring any magnetic interaction, and following the general hypothesis presented above, the harvester introduced in Section 2.2.4 can be modelled as a rather simple 1 DOF springdamper-mass system with base excitation. The resulting mechanical model is represented in Figure 3.1, with the parameters and its corresponding values applicable to the case study being presented in Table 3: Table 3. Mechanical model parameters. Description Model value Units 𝐿 Total case inner length 0.051 m 𝑙0 Spring natural length 0.011 m 𝑙𝑚 Magnet height 0.023 m 𝑒𝑜 Outer case thickness 0.0025 m 𝑒𝑚 Magnet case thickness 0.004 m 𝑚 Magnet mass 0.07375 kg 𝑘 Spring total stiffness 10645.27 N/m 𝑐 Spring total damping 0.140 N/ms 13 Figure 3.1. Mechanical model. 3.1.1.1 Damping In a generic mechanical system, the total viscous damping is a sum of the mechanical damping cm the Coulomb damping cc and the damping induced by an external fluid cf. 𝑐=𝑐𝑚+𝑐𝑐+𝑐𝑓 (3.1) The mechanical damping is intrinsic of the spring due to its material absorbing energy over each oscillation. This value will be found experimentally exciting the harvester at low input amplitudes and without external ferromagnetic components, to minimise any distortion in the system response. Coulomb damping, which is induced due to friction, will be neglected since this specific harvester is conceived to work vertically. It is therefore assumed that the mass oscillates concentrically to the outer shell without contacting it. To ensure this, when performing the experimental tests, it will be especially important to check the device’s parallelism to the floor. Lastly, any mass moving through a viscous fluid or gas such air suffers from a viscous damping induced by the resistance to move through the fluid. The damping induced by air is usually just relevant at micro or nanoscales and therefore it will also be neglected during the modelling of this macro-scale device. 3.1.1.2 External excitation The external excitation is the harmonic movement of the base (or floor) of the harvester, which is defined by the following equation: 𝑦(𝑡)=𝐴 cos (𝜔𝑒𝑡+𝜑) (3.2) Where 𝐴 is the amplitude of the excitation, 𝜔𝑒 the excitation frequency, 𝑡 the time and 𝜑 the phase. This model might not fully represent the input that a device would receive in a real environment since no random or residual vibrations (involving multiple frequencies) are taken into account, but in most cases such vibrations are negligible to the performance of the harvester since they are far from the resonance of the system. Study of the effects of magneto-mechanical coupling in the performance of electromagnetic vibration energy harvesters 20 3.1.3.4 Solving methods To solve Eq. (3.18) two different methods are proposed: - Brute Force method: The O.D.E from Eq.(3.18) is solved using a Range-Kutta nonlinear time integrator. The system response is computed for a downward or upward sweep of forcing frequencies. This method has some interesting benefits, such as the ability to output the response of the magnet over time, as shown in Figure 3.7, including both unsteady and steady phases, but also some major drawbacks regarding computational cost and the inability to capture unstable branches in the response. Figure 3.7. Temporal response of Model B at the resonant frequency for static initial conditions. - Harmonic Balance method [28]: The Harmonic Balance method is based on the concept that any periodic solution of an ODE can be represented by a combination of sinusoids (Fourier series). This limits its applicability to only periodic cases and not to transient or random problems. Contrary to the previous method, this one is solved in the frequency domain. The equation of movement of the system, therefore, needs to be converted by employing the Fourier transform (Section 3.1.1.4). HBM solves the dynamic equilibrium of forces for an ansatz (assumed solution to the system’s equation) and obtains a residue. This process is iterated until this residue, which can also be represented in a Fourier series, is close to 0 up to the same truncation order of the proposed solution. Considering the present study case, HBM proves to be superior to the BFM in the following areas: - Numerical stability: Time integration metho<ds can induce to instabilities or to numerical damping due to approximation errors during the computation. This might be palliated by refining tolerance errors, but this will have a huge impact on the numerical cost. Since on the Harmonic Balance Method the obtained solution is always sinusoidal and hence, periodic, no growth or decrease in energy will happen. - Computation of unstable branches: Unlike all of the non-linear time integration methods, harmonic balance can compute unstable branches of the system response. Even isolated branches can be accessed, either with a frequency sweep of by inducing an impact or perturbation. - Computational efficiency: This method is considerably more computational effective than time integration mainly since it avoids computational effort on transient terms, and it usually require a Fourier series low truncation order. Despite these clear advantages both methods are implemented in the numerical tool in order to compare their results and computational efficiency. Transient response Steady response Transient response Steady response 21 3.1.3.5 Natural and resonant frequencies As discussed in Section 3.1.3.3, magnetic forces induce a softening or hardening effect on the stiffness on the system. The difference in natural and resonant frequencies comes from this new equivalent stiffness 𝑘𝑤(𝑥) , which thorough this document will be referred as weighted stiffness. 𝑘𝑤(𝑥)=𝑘−𝑑𝑓𝑚(𝑥) 𝑑𝑥 (3.19) Since this weighted stiffness is dependent on 𝑥 , the values of natural and resonant frequencies will also be dependent on the oscillation amplitude: 𝜔𝑛,𝑤(𝑥)=√𝑘𝑤(𝑥) 𝑚 (3.20) 𝜔𝑟,𝑤(𝑥)=𝜔𝑛,𝑤(𝑥)√1−2𝜉2 (3.21) The frequencies that can be found algebraically and hence, computed beforehand, are for considerable low input/output oscillation amplitudes, where it can be assumed that the slope of the magnetic forces curve is constant since the difference in 𝑥 is negligible. In that case no unstable branch appear in the system response of the system. 3.2 Implementation This section presents the different tools used to develop the simulation script as well as a discussion of its architecture and the main characteristics of each layer. The simulation tool is named EVHAST, being an acronym of “Electromagnetic Vibration Harvester Simulation Tool”. Throughout this section, the basic algorithms for each of the script that compose EVHAST are discussed with the aim of complementing the notes and information already present on the scripts to ease its comprehension. The three simulations that the script is able to perform are presented in Table 5: Table 5. Available simulations. Simulation Description Coupling Magnetic Forces Models (Table 2) S1 Frequency Domain Method (FDM) No No M S2 Brute Force Method (BFM) Weak Available M/B/C S3 Harmonic Balance Method (HBM) Weak Available M/B/C 3.2.1 Software The main script is developed in Matlab R2021b [29]. The capabilities that are exploited are: - User interface: Input parameters and system options. - Data processing: Processing of the magnetic forces data. - Main computations: Brute force and Harmonic balance methods. - Post-processing: Plots and numerical results. The HBM computation employs an adapted version of NLvib V1.3 [30] Matlab tool, developed by Dr. Malte Krack and Dr. Johann Gross from the University of Stuttgart. The scripts are modified from the Duffing oscillator that is included when downloading the software as a 1-DOF example. Study of the effects of magneto-mechanical coupling in the performance of electromagnetic vibration energy harvesters 22 - Numerical integration: unconditionally stable Newmark integrator. - Numerical solution and path continuation: predictor-corrector method with Newton-type solver (MATLAB’s fsolve) and analytical gradients. - Analysis type: frequency response analysis. Lastly, to compute the magnetic forces, the open-source software that is employed is femm [31], which can be run inside Matlab’s environment, making it much easier to integrate with the rest of the script. This software adds the following capabilities: - 2D modelling. - Static magnetic simulation. - Magnetic forces and output voltage computation. 3.2.2 Algorithm and architecture The simulation tool is divided in several sub-scripts according to their functionality. The aim is to simplify the main script in order to make it more intuitive and ready-to-use for any external user. The algorithm and architectures governing the simulation tool are represented in Figure 3.8: Figure 3.8. EVHAST simulation tool architecture. 23 3.2.2.1 EVHAST_main.m (User interface) This script is the first layer of the simulation tool. It is therefore conceived to be the user interface, where the main parameters and options of the system can be modified. The main tunable options of the script are stored in a struct and are presented in Table 6: Table 6. EVHAST Simulation tool options. EVHASTopts Variable User input Definition HEinput ‘disp’ The base harmonic excitation values are inputted in displacement [m]. ‘vel’ The base harmonic excitation values are inputted in velocity [m/s]. ‘acc’ The base harmonic excitation values are inputted in acceleration [g]. HEval ‘amp’ HEinput values are inputted in amplitude (total) value. ‘rms’ HEinput values are inputted in RMS value. MF ‘yes’ Electromagnetism is computed. (Model B) ‘no’ Electromagnetism is not computed (Model M/A). MFdata ‘no’ No magnetic data is available and hence, a simulation needs to be done. ‘<filename>’ Magnetic data is available and stored under in ‘<filename>’ file. s1 ‘yes’ Temporal response in frequency domain simulation is performed. ‘no’ Temporal response in frequency domain simulation is not performed. s2 ‘yes’ Brute force method is performed. ‘no’ Brute force method is not performed. s3 ‘yes’ Harmonic Balance method is performed. ‘no’ Harmonic Balance method is not performed. 3.2.2.2 EVHAST_MF (MF processing) This second layer script is the responsible of processing the magnetic forces inputted from the external magnetic simulation tool (femm in this case). If the user specifies in the EVHASTopts struct that there is existing data to import, the output parameters kdw, x0, {fit} are directly be computed. If not, a subscript is previously run executing the software femm with the geometrical parameters defined in EVHAST_main. Figure 3.9. EVHAST_MF script algorithm. The discrete MF data needs to be converted to a continuous function, either by interpolating or by computing a curve fit, to be able to obtain a value for any x within the applicable range. In this case, a curve fitting is done from a 5th order rational model which is the one that represents better the tendency of the results curve. 𝑓𝑚=𝑝1𝑥5+𝑝2𝑥4+𝑝3𝑥3+𝑝4𝑥2+𝑝5𝑥+𝑝6 𝑥5+𝑞1𝑥4+𝑞2𝑥3+𝑞3𝑥2+𝑞4𝑥+𝑞5 (3.22) Study of the effects of magneto-mechanical coupling in the performance of electromagnetic vibration energy harvesters 24 Computing Eq.(3.22) at each iteration in the solver is seen to be much more computationally efficient than doing so with the curve fit function, hence, {fit} will be an array containing p and q parameters. To ensure the quality of the regression model obtained, a for loop is implemented checking that the Coefficient of Determination (R-squared) has a value greater than 0,995. x0, as seen in Section 3.1.3.2, is the height, referenced from the harvester base, at which the static forces are cancelled and therefore the oscillating mass is at equilibrium. This parameter is found by solving the non-linear Eq. (3.17) with Matlab’s fsolve tool. If EVHASTopts.MF is set to ‘no’ the value of xfloor is instead computed following Eq. (3.5). kdw is the weighted stiffness (see Section 3.1.3.5). As discussed earlier, the system experiences a softening effect due to the mere presence of magnetic forces. kdw is found by computing the slope of the curve fit, i.e., the x derivative of Eq.(3.22) at x0. If no magnetic forces are being considered (Model M) kdw takes the same value as the mechanical system’s stiffness k. 3.2.2.3 EMVEH (MF simulation) EMVEH is an adaptation of a script done by Dr. Victor Ordoñez [1] and it performs the magnetic simulation in case no data is already available. All the 2D design, materials and other settings are configured in Matlab‘s environment and executed in femm. This static simulation computes the magnetic force at the magnet at different points between the minimum (plate) and maximum height Xmax established. A general representation of its algorithm is shown in Figure 3.10: Figure 3.10. EMVEH script algorithm. First, the model presented in 0 is transferred into femm. All the geometrical parameters are being inputted from the main script. Next, each component is identified as a block in order for the software to recognise them as an entity. Following the block labelling, the materials are applied using femm existing library. Lastly, the simulation is run (Figure 3.11): Figure 3.11. femm magnetic model (a) and magnetic forces simulation (b). (a) (b) 25 3.2.2.4 EVHAST_s1 (FDM) The first simulation computes the transmissibility of the system in the frequency domain, that is, after applying the Fourier Transform to the equation of movement (Section 3.1.1.4) and algebraically solving Eq.(3.9). No magnetic forces are taken into account and therefore the provided results correspond to the Model M case. This script follows the algorithm presented in Figure 3.12. Figure 3.12. EVHAST_s1.m script algorithm. The output variables are: - {T1}: Transmissibility of the mechanical system. - {T1w}: Transmissibility of the mechanical system assuming low input amplitudes and taking into account the weighted stiffness kdw. 3.2.2.5 EVHAST_s2 (BFM) EVHAST_s2 (Figure 3.13) prepares the data to enter in the time integration script and postprocesses the resulting temporal response (in the form of {x} and {t} vectors). Figure 3.13. EVHAST_s2 script algorithm. The script starts by converting to displacement the amplitude that has been inputted by the user. This is done by checking the parameters HEinput and HEval contained in the VIEHopts struct. Depending on the {HEarray} input that the user has chosen, this computation varies as shown in Table 7: Table 7. Conversion of input amplitudes to displacements. EVHASTopts.HEinput EVHASTopts.HEval ‘disp’ [m] ‘vel’ [m/s] ‘acc’ [m/s] ‘amp’ [m] ‘rms’[m] 𝑦=𝑦 𝑦=𝑦󰇗𝜔 𝑦=𝑦󰇘·𝑔 𝜔2 𝑦=𝑦 𝑦=√2·𝑦 The output of the EVHAST_s2_ODE script is the computed temporal response of the system: {x} being the instant value of the oscillating mass position and {t} the corresponding time vector. To only process the stabilized or steady part of the temporal response (since the first instants will depend on the initial conditions inputted), the last 5% of the vector {x} are filtered in the variable {x_ste}. The root mean square (RMS) {x_rms} is done from the previously computed steady response and finally the transmissibility {T2} is computed. Study of the effects of magneto-mechanical coupling in the performance of electromagnetic vibration energy harvesters 26 3.2.2.6 EVHAST_s2_BFM This script solves Eq.(3.16) in the Brute Force Method using the ode45 function from Matlab’s library. ode45 uses Runge-Kutta integration method with variable time-step. The general algorithm is presented: Figure 3.14. EVHAST_s2_ODE script algorithm. To work with this function, the ODE needs to be of first order. Hence, Eq.(3.16) order needs to be reduced by employing the state variable method: {𝑥󰇗}=[𝐴]{𝑥}+{𝐵} (3.23) First, the state vector containing state variables is defined (which is a column vector the size of the order of the O.D.E.): {𝑥}={𝑥1 𝑥2}={𝑥 𝑑𝑥 𝑑𝑡} (3.24) Next, the time derivative of the previously found state vector is defined: {𝑥󰇗}={𝑥1 𝑥2󰇗󰇗}={𝑑𝑥 𝑑𝑡 𝑑2𝑥 𝑑𝑡} (3.25) x2’, which is the second order time derivative of the equation of motion can be obtained isolating simply isolating the term on Eq.(3.16), resulting in the following expression: 𝑥󰇗2=𝑥󰇘=(𝑘𝑦+𝑐𝑦󰇗−𝑐𝑥󰇗−𝑘𝑥−𝐿+𝑙𝑚 2−𝑚𝑔+𝑓𝑚) 𝑚 (3.26) Where {fm} is computed following Eq.(3.22). 3.2.2.7 EVHAST_s3 (HBM) EVHAST_s3 prepares the necessary parameters to later access the Harmonic Balance method solver. It is an adapted version of a NLvib’s [28] duffing oscillator script. It follows the same general algorithm as EVHAST_s2 script, as seen in Figure 3.15, where the input values are first converted to displacements. Figure 3.15. EVHAST_s3 script algorithm. 27 An initial guess is first computed by neglecting magnetic forces. Since this method works in the frequency domain, the initial guess is computed following Eq.(3.9). 𝑋𝑖=−𝜔2𝑚+𝑖𝜔𝑐+𝑘 (𝑘+𝑖𝑤𝑐)𝑌 (3.27) With this value, the main solver, composed by the scripts solve_and_continue and EVHAST_s3_HB is accessed. The output is the oscillation of the mass at its steady state {a}, from which the transmissibility {T3} can be obtained. 3.2.2.8 EVHAST_s3_HBM This script computes the residual value of the dynamic force equilibrium of the system in the frequency domain for a sinusoidal ansatz i.e., the difference between internal and external forces: 𝑟(𝑋,𝑤)=𝑓𝑖𝑛−𝑓𝑒𝑥 (3.28) The internal forces are composed of a linear and a non-linear term. 𝑟(𝑋,𝑤)=(𝑓𝑙+𝑓𝑛𝑙)−𝑓𝑒𝑥 (3.29) In this case of study, the linear term is defined by the stiffness matrix of the system: 𝑓𝑙= (𝑘+𝑖𝜔𝑐−𝜔2𝑚) 𝑋 (3.30) The only non-linear term acting on the system is the dynamic part of the magnetic forces: 𝑓𝑛𝑙=𝑓𝑚(𝑥−𝑥0−𝑦) (3.31) And the external forces are those of the harmonic excitation, resulting in the equation of motion seen in previous sections. 𝑓𝑒𝑥=(𝑘+𝑖𝜔𝑐)𝑌 (3.32) If after computing the residual the tolerance is not met, the solve_and_continue is accessed again, computing the next {x} iteration for the x DOF. Once an acceptable residual is reached, the same procedure is followed for the next frequency value. 3.2.2.9 EVHAST_post (post-processing) EVHAST_post has the role of doing the post-processing of the simulations i.e., the plots and the computation of relevant parameters that will be used to analyse the results. The output parameters that are available in the present version of the software are the resonant frequencies, transmissibility peaks, maximum displacements, maximum speeds, and maximum accelerations for each of the simulations. This allows a direct comparison between methods as well as identifying beforehand if the inputted excitation amplitudes are within the working ranges of the device. These are outputted in the EVHASTresults variable. A struct similar to the general tool options, and presented in Table 8, is implemented in order to ease user customization: Study of the effects of magneto-mechanical coupling in the performance of electromagnetic vibration energy harvesters 28 Table 8. Post processing options. POSTopts Variable User input Definition scale ‘lin’ Defines the Y-axis scale as linear. ‘log’ Defines the Y-axis scale as logarithmic. S1 ‘yes’ Plots s1 results, if available. ‘no’ Does not plot s1 results. S2 ‘yes’ Plots s2 results, if available. ‘no’ Does not plot s2 results. S3 ‘yes’ Plots s2 results, if available. ‘no’ Does not plot s2 results. compareS1 ‘yes’ Compares S2 and/or S3 with S1 ‘no’ - compareS2S3 ‘yes’ Compares S2 with S3, if both were computed. ‘no’ - compareEXP ‘yes’ Compares the results with experimental data. ‘no’ - 3.3 Experimental Experimental tests are of paramount importance to validate the modelling and implementation phases presented in Sections 3.1 and 3.2. The desirable outcome would be for the developed tool to represent as close as possible the real behaviour of the device, even while taking into account the several hypotheses and assumptions taken during the modelling procedure (Section 3.1). If a mismatch between both results is observed, an analysis should be done to try to pinpoint the cause of it, potentially invalidating the numerical model. 3.3.1 Setup A flowchart of the setup used during the experimentation phase is presented in Figure 3.16: Figure 3.16. Experimental setup flowchart. 29 3.3.1.1 Software The software used is LMS TestXpress by Siemens, which acts as the interface between the user and the different input (accelerometers and voltage) and output (modal exciter) channels. The software features that are exploited during the tests are: - Channel setup: setting up channel sensitivity and measure units. - Visualisation: Real-time visualization of the temporal signal. - Real-time signal processing: RMS, averaging and filtering of the signals. - Data recording: Recording and exporting the data into .mat file. - Modal exciter controller: Setting up excitation profiles (sine or sweep sine). 3.3.1.2 Data acquisition device The data acquisition device is the physical interface between the sensors and exciter and the software. It should have at least 9 input channels, to cover the 8 accelerometers and a possible voltage input (if applicable) and 1 output channel, to control the modal exciter.The used device (SCADAS Mobile acquisition device by Siemens) can operate with 1 output and 24 input channels, covering the requirements specified above. 3.3.1.3 Power amplifier The amplifier needed to drive the modal shaker is the Type 2720 by Brüel & Kjær. The main characteristic is that it can work with low-force shakers since it has an adjustable RMS output-current limit, which makes it suitable for this setup. Also, it has a usable frequency range from 40 Hz to 15 kHz, which vastly covers the frequencies of interest of the harvester studied in this report. 3.3.1.4 Modal exciter The modal exciter is the device responsible for inputting a harmonic excitation to the harvester. The device will be fixed to its stem using a base as an interface (Section 3.3.2.1). The Type 4825 by Brüel & Kjær can work with up to 5 kHz. Moreover, its internal mechanism is designed in order to minimize forces drop-off when testing resonant frequencies. 3.3.1.5 Accelerometers A total of 8 accelerometers are employed: four on the base, to capture the modal exciter real input, and the rest on the oscillating magnet, to obtain the output vibration of the system. These accelerometers are distributed according Figure 3.17: Figure 3.17. Accelerometers layout. Study of the effects of magneto-mechanical coupling in the performance of electromagnetic vibration energy harvesters 36 there will always likely be some residual rotational component simply due to the fact that only 4 discrete points are being analysed. ii) In addition to the averaging, a high filter is applied to the resulting signal so that the sinusoids are centred. That is to ensure that he mean value is as close as possible to 0. If this wasn’t done, the RMS, which depends on the value of the amplitude, could carry some slight error. iii) Finally, the RMS of this processed channel is outputted using a visualization gadget. 3.3.3.4 Measurements The firsts test to be performed should be the ones employing sine profiles input (TAS, TBS, TCS) since its output data will be needed to perform the rest of them. [TAS / TBS / TCS]: For sine input tests, the RMS should be adjusted before any measure is recorded, following the procedure presented in Figure 3.20, to match the amplitude of the test being performed. This is due to the fact that changing the excitation frequency will modify the real acceleration input that the harvester is receiving. Figure 3.20. TAS,TBS,TCS tests workflow. [TASSX / TBSSX / TCSSX]: The RMS adjustment on the sweep sine tests needs to be also done before recording the results. In this case a full cycle must be first run to match the highest output value to the resonant values obtained during the first tests. Figure 3.21. TASSX,TBSSX,TCSSX tests workflow. 3.3.3.5 Post-processing [TAS / TBS / TCS]: Using the RMS and other visualization gadgets, the transmissibility for the sine excitation cases can be obtained in situ doing a simple algebraic computation between input and output values, but for sweep sines it must be post-processed. [TASSX / TBSSX / TCSSX]: For the sweep sine tests the data must be first post-processed. This is done by employing Matlab’s tfestimate internal function. This function approximates the transfer function of the system through its input and output signals. 37 4 Results In this section both the simulation and experimental results are presented as well as a validation of the simulation tool with the gathered data. 4.1 Simulation results 4.1.1 Magnetic forces The magnetic simulations are run according to the models and parameters discussed in 3.1.2. Both are performed with 250 x points ranging from 0 mm to 25 mm (referenced from the base magnet top surface). 4.1.1.1 Model B Figure 4.1. Model B magnetic forces raw results. 4.1.1.2 Model C Figure 4.2. Model C magnetic forces raw results. Study of the effects of magneto-mechanical coupling in the performance of electromagnetic vibration energy harvesters 38 4.1.2 Coupling The simulations cases reduce to three, each one involving different system options as specified in Table 14: Table 14. Models M, B & C simulation options. Model M Model B Model C Variable User input HEinput ‘acc’ ‘acc’ ‘acc’ HEval ‘rms’ ‘rms’ ‘rms’ MF ‘no’ ‘yes’ ‘yes’ MFdata - ‘ModelB’ ‘ModelC’ s1 ‘yes’ ‘yes’ ‘yes’ s2 ‘yes’ ‘yes’ ‘yes’ s3 ‘yes’ ‘yes’ ‘yes’ All simulations and models are run at an input excitation value of 0.005 g since it is observed that higher amplitudes, within the working range of the harvester, provide similar numerical results. A significant difference would only be observed if inputting excitations outside the current prototype capabilities or if spring non-linearities were implemented on the simulation tool. Other parameters, that depend on the model that is being tested, are presented in Table 15: Table 15. Models M, B & C simulation parameters. Model M Model B Model C Variable Description User input wmin Minimum frequency value [Hz]. 58 57 59 wmax Maximum frequency value [Hz]. 63 62 64 wnum Number of frequency points to be computed. 150 150 - tend End time of the simulation (BFM). - 15 - H Harmonic order (HBM). - - 7 N Number of samples per period (HBM). - - 107 The results are presented for all three models and simulations available in the following section. For models B and C, the obtained results are compared to those of S1 even though this simulation does not compute magnetic forces. Thus, when analysing the graphs, It must be considered that S1 always portrays Model M results and consequently, it is expected that its response won’t match with the ones obtained from BFM and BHM for models B & C. This is done to graphically observe the effects of magnetic forces on the system response. 39 Figure 4.3. Model A S1, S2 & S3 transmissibilities. Table 16. Model M simulation results. Sim. 𝜔𝑟 [Hz] 𝑇𝑚𝑎𝑥 𝑡𝑠𝑖𝑚 [s] S1 (FDM) 60.48 199.7 <0.01 S2 (BFM) 60.48 201.8 142.6 S3 (HBM) 60.47 200.1 2.35 4.1.2.1 Model B Figure 4.4. Model B S1, S2 & S3 transmissibilities. Table 17. Model B simulation results. Sim. 𝜔𝑟 [Hz] 𝑇𝑚𝑎𝑥 𝑡𝑠𝑖𝑚 [s] S1 (FDM) 60.48 199.7 <0.01 S2 (BFM) 59.75 200.2 153.95 S3 (HBM) 59.75 198.7 5.60 4.1.2.2 Model C Figure 4.5. Model C S1, S2 & S3 transmissibilities. Table 18. Model C simulation results. Sim. 𝜔𝑟 [Hz] 𝑇𝑚𝑎𝑥 𝑡𝑠𝑖𝑚 [s] S1 (FDM) 60.48 199.7 <0.01 S2 (BFM) 61.18 204.5 158.66 S3 (HBM) 61.18 202.5 6.07 º 4.1.3 Discussion Model M - EVHAST Model B - EVHAST Model C - EVHAST Study of the effects of magneto-mechanical coupling in the performance of electromagnetic vibration energy harvesters 40 4.1.3.1 Magnetic forces results As it can be observed, both in Figure 4.1 and Figure 4.2, especially for measure points further from the base, there is considerable numerical noise as a result of the magnetic forces computational method. As discussed in Section 3.2.2.2, this is fully palliated when doing the curve fit of the discrete data, avoiding any potential inaccuracies when later computing the system response. The curve fit for Models B and C is shown in Figure 4.6: Figure 4.6. Model B (a) and C (b) magnetic forces curve fit results (logarithmic scale). Both curves are observed to be practically symmetrical between each other, with the maximum force being of around 180 N (of attraction in Model B and of repulsion in Model C) at an x of 0 mm. The computational effort is considerably high for a 250 measure points simulation (950 seconds). Besides not being optimal, this does not have other remarkable implications since it is not expected for the electromagnetic model to suffer from persistent architectural iterations unlike the mechanical model. Despite this, considering that the device’s oscillation stroke does not come closer than 11.5 mm to the plate and hence, that it works at a height where the magnetic force can be approximated as linear, the points can be further reduced without affecting the accuracy of the results. Figure 4.7 is obtained by making the MF mesh coarser and analysing the resultant resonant frequency for Model B. As it can be seen, the tool provides accurate results up to, approximately, 75 measure points. Figure 4.7. Simulation accuracy vs MF computational cost. If the device were to work at a closer range, or a stronger magnet was used, a computational efficient solution could be locally refining the mesh at the exponential-growth region and making it coarser at the rest of the curve. (a) (b) 41 4.1.3.2 Coupling results As it can be observed in Figure 4.3, Figure 4.4 and Figure 4.5, in all three models the transmissibility obtained with BFM and HBM methods does match quite accurately. This is partly due to the fact that the device is being simulated under low excitation amplitudes, where the response of the system can be approximated as linear. The most important phenomena that can be inferred from the presented results is the stiffness softening (for model B) and hardening (for model C) as described in Section 3.1.3.3. Model B suffers from a shift of -0.68 Hz and Model C of +0.7 Hz from the reference model (with no magnetic forces applied). This change in the stiffness appears to be almost symmetrical, which is coherent with the MF curves being also symmetrical between models. Figure 4.8. Transmissibility comparison between Models M, B & C for Harmonic Balance Method. A direct consequence of this are unstable branches, which appear for higher amplitudes as the stiffness of the system is no longer constant. Unfortunately, such amplitudes are not physically achievable with the current prototype and hence, this behaviour will not be able to be validated experimentally. The following figures are obtained for an acceleration of 0.5 g at the base (± 10 mm of oscillation amplitude on the magnets at the resonant frequency): Figure 4.9. BFM vs HBM for 0.5 g input acceleration, models B (a) and C (b). Figure 5.9 portraits the superiority of the Harmonic Balance Method against the Brute Force Method, which fully captures the unstable branch of the transmissibility while also being considerably more computationally efficient. Similar to the MF case, the BFM computational time could be reduced by refining the mesh at the peak of transmissibility and making it coarser elsewhere, but even in that case, its computational method would by far exceed the one from HBM, which for the presented simulations is, approximately, 25 times higher. (a) Model B - EVHAST (b) Model C - EVHAST Study of the effects of magneto-mechanical coupling in the performance of electromagnetic vibration energy harvesters 42 4.2 Experimental results The results for both the harmonic and sweep sines tests are presented in this section, followed by a brief discussion regarding observations that are not seen on the simulations. 4.2.1 Model A Figure 4.10. Model A experimental transmissibilities. Table 19. Model A experimental results. Test 𝜔𝑟 [Hz] 𝑇𝑚𝑎𝑥 TASS 60.48 187.7 TASS1 60.44 160.1 TASS2 60.25 271.2 4.2.2 Model B Figure 4.11. Model B experimental transmissibilities. Table 20. Model B experimental results. Test 𝜔𝑟 [Hz] 𝑇𝑚𝑎𝑥 TBSS 59.35 222.9 TBSS1 59.32 196.7 TBSS2 59.09 263.8 4.2.3 Model C Figure 4.12. Model C experimental transmissibilities. Table 21. Model C experimental results. Test 𝜔𝑟 [Hz] 𝑇𝑚𝑎𝑥 TCSS 61.24 189.8 TCSS1 61.2 209.7 TCSS2 60.98 296.1 43 4.2.4 Discussion The first observation in Figure 4.10, Figure 4.11 and Figure 4.12 is the inconsistency in the transmissibility peaks across models and tests. This is thought to be mainly because of the fact that the magnet on the prototype is not vertically guided. In addition, the springs ends are observed to have a bad parallelism, causing the magnet to be slightly tilted at its static position. The combination of these two aspects leads to a slight rotational movement of both the mass and the plate, especially when excited at frequencies near the resonant one. This rotational movement is indeed observed during the tests due to a discrepancy in both the base and harvester accelerometers outputs, as seen in Figure 4.13 (a) and (b). Figure 4.13. Normalized base (a) and magnet (b) accelerations for Model C (TCS test). In Figure 4.13 (b) a difference of a 10% from the nominal acceleration value is observed for accelerometers 2 and 3. Due to the fact that they are placed at 180º from each other, it pinpoints a rotational movement on their perpendicular axis i.e., the axis defined by accelerometers 4 and 5, which provide almost identical results. Figure 4.13 (a), where some accelerometers are seen to have an output signal of up to 5 times the nominal acceleration, depicts a loss of energy induced by the harvester, which, especially near the resonant frequency is transmitting part of the input energy to the base. All of this does not comply with the hypothesis of purely vertical displacement introduced in Section 3.1 and consequently entails that transmissibility peak values obtained from the experimental tests are non-conclusive. Another unexpected phenomenon is the stiffness shift of around -0.2 Hz that can be observed in all three models when exciting the harvester at higher amplitudes (TASS2, TBSS2, TCSS2). This is highly likely due to a non-linear behaviour of the springs, since this change appears to be uniform across all the measure range. If this was otherwise a consequence of the magnetic force interaction (due to the mass oscillating nearer the base magnet) it is believed that this change of stiffness would be much more concentrated at the peak as seen in Figure 4.9 (a) and (b) since the MF curve resembles an exponential growth. The implication of this observation is that, if aiming to fully predict the behaviour of this harvester, the spring should be fully characterised first. By doing this, not only the nonlinearity would be identified, but also it would be ensured whether the viscous model hypothesis introduced previously properly represents the behaviour of the spring. On a positive note, sine tests (TAS, TBS, TCS) and low amplitude sweep sines (TASS1, TBSS1, TCSS1) as expected, are observed to match accurately, especially regarding their resonant frequencies. The maximum deviation observed is of 0.04 Hz for both Models A and C. Moreover, it can be observed that the stiffness softening and hardening effect also appears on the experimental prototype. In this case, for the low amplitude tests, Model B suffers from a -1.12 Hz shift while Model C from a +0.76 Hz one. (a) (b) Study of the effects of magneto-mechanical coupling in the performance of electromagnetic vibration energy harvesters 44 This difference in the resonant frequency can be clearly seen in Figure 4.14, which portrays the comparison between the transmissibility obtained for the three models: Figure 4.14. Low (a) and high (b) amplitude sweep sine tests results for Models A, B and C. As observed, the change in the rigidity is significant, especially considering that the oscillating magnet is a Halbach array, which greatly attenuates the magnetic flux over the base magnet’s field of action. 4.3 Validation A graphical and numerical comparison between the simulation and experimental results is presented in Figure 4.15 and Table 22. Due to some of the unexpected behaviour and issues observed during the experimental tests for higher amplitudes (discussed in Section 4.2.1) the validation is done for the experimental data corresponding to low input amplitudes and with the profiles obtained from sweep sine tests. Figure 4.15. Simulation vs experimental transmissibilities for Models M/A, B & C. (a) (b) 45 Table 22. Simulation vs experimental results for Models M/A, B & C. Model Results 𝜔𝑟 [𝐻𝑧] Shift [Hz] Diff. [Hz] M/A Experimental [TASS1] 60.44 0 0.03 Simulation [HBM] 60.47 0 B Experimental [TBSS1] 59.32 -1.12 0.4 Simulation [HBM] 59.75 -0.72 C Experimental [TCSS1] 61.2 +0.76 0.05 Simulation [HBM] 61.18 +0.71 Comparing the numerical and experimental results (Figure 4.15) it can be seen how the system response is accurately predicted for Models M/A and C, with a maximum error of 0.05 Hz in the resonant frequency of the former. On the other hand, the model clearly falls short on predicting Model B response (by 0.4 Hz) even though it can anticipate its softening tendency. There are several hypotheses that could, separately or combined, explain such error in the simulation: i) Springs magnetic interaction: It is thought that some portion of such error is due to the magnetic interaction between the springs, which are not included in the model, and the magnets on the prototype. Of special interest would be studying the lower spring, which is the one nearer the base magnet and could be the responsible for the observed asymmetry in the frequency shift observed between models B and C. ii) Springs length discrepancy: A difference in length between upper and lower springs, which would draw the magnet closer to the base at its equilibrium position and consequently decrease resonant frequencies, could be responsible for part of the observed difference. iii) Springs behaviour: Slight differences in the stiffness, damping, or nonlinearities between the upper and lower spring could also induce the observed asymmetry in the observed system response. iv) Magnet rotation: The magnet’s rotation discussed previously could increase the stiffness softening effect since it implies that, at least partially, the magnet would momentarily oscillate nearer the base. v) Coupling: There could be some dynamic coupling phenomena that is not being considered due to the assumptions taken when implementing a weak coupling. Study of the effects of magneto-mechanical coupling in the performance of electromagnetic vibration energy harvesters 52 [31] D. C. Meeker, Finite Element Method Magnetics, Version 4.2, https://www.femm.info, Accessed: January 2023.