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A Study of Nonlinear Piezoelectric Energy Harvester with Variable Damping using Thin Film MR Fluid
13th IFAC Workshop on Intelligent Manufacturing Systems August 12-14, 2019. Oshawa, Canada 13th IFAC IFAC Workshop on Intelligent Intelligent Manufacturing Systems Systems 13th Workshop on Manufacturing Available online at www.sciencedirect.com August 12-14, 2019. Oshawa, Canada 13th IFAC Workshop on Intelligent Manufacturing Systems August 12-14, 2019. Oshawa, Canada 13th IFAC Workshop on Intelligent Manufacturing Systems August 12-14, 2019. Oshawa, Canada August 12-14, 2019. Oshawa, Canada
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IFAC PapersOnLine 52-10 (2019) 394–399 A Study of Nonlinear Piezoelectric Energy Harvester with Variable Damping A Study of Nonlinear Piezoelectric Harvester Variable Damping A Study of Nonlinear Piezoelectric Energy Harvester with Variable Damping using ThinEnergy Film MR Fluid with A Study of Nonlinear Piezoelectric Energy Harvester with Variable Damping A Study of Nonlinear Piezoelectric Energy Harvester with Variable Damping using Thin Film MR Fluid using Thin Film MR Fluid Thin MR H. Azangbebil *,using S. S. Djokoto*, M. Agelin-Chaab*, using Thin Film Film MR Fluid FluidEgidijus Dragašius **
H. H. Azangbebil Azangbebil *, *, S. S. S. S. Djokoto*, Djokoto*, M. M. Agelin-Chaab*, Agelin-Chaab*, Egidijus Egidijus Dragašius Dragašius ** ** *, S. S. Djokoto*, M. Agelin-Chaab*, H. Azangbebil Egidijus Dragašius ** Science University of * Dept. of Automotive, Mechanical and *, Manufacturing Engineering, Faculty of Egidijus Engineering and Applied H. Azangbebil S. S. Djokoto*, M. Agelin-Chaab*, Dragašius ** * Dept. of of Automotive, Automotive, Mechanical and Manufacturing Manufacturing Engineering, Faculty of Engineering Engineering andL1H Applied Science University of of Ontario Institute of Technology, 2000 Simcoe Street, North, Oshawa, Canada ON 7K4 (Science e-mail:University * Dept. Mechanical and Engineering, Faculty of and Applied * Dept. of Automotive, Mechanical and Manufacturing Engineering, Faculty of Engineering and Applied Science University of Ontario Institute of Technology, 2000 Simcoe Street, North, Oshawa, Canada ON L1H 7K4 ( e-mail: [email protected]; [email protected]; [email protected] ). Ontario Institute of Technology, 2000 Simcoe Street, North, Oshawa, Canada ON 7K4 (Science e-mail:University of * Dept. of Automotive, Mechanical and Manufacturing Engineering, Faculty of Engineering andL1H Applied Ontario Institute of Technology, 2000 Simcoe Street, North, Oshawa, Canada ON L1H56, 7K4 ( ). e-mail: [email protected]; [email protected]; [email protected] ** Dept. of Production Engineering, Kaunas University of Technology, Studentu 51424, [email protected]; [email protected]; [email protected] Ontario Institute of Technology, 2000 Simcoe Street, North, Oshawa, Canada ON L1H 7K4 ( ). e-mail: [email protected]; ). **[email protected]; Dept. of of Production Production Engineering, Kaunas University of [email protected] Technology, Studentu Studentu 56, 56, 51424, 51424, Kaunas, Lithuania (e-mail:University [email protected]) ** Dept. Engineering, Kaunas of Technology, [email protected]; [email protected]; [email protected] ). ** Dept. of Production Engineering, of Technology, Studentu 56, 51424, Kaunas, LithuaniaKaunas (e-mail:University [email protected]) Kaunas, Lithuania (e-mail: [email protected]) ** Dept. of Production Engineering, Kaunas University of Technology, Studentu 56, 51424, Kaunas, Lithuania (e-mail: [email protected]) Kaunas, Lithuania (e-mail: [email protected]) Abstract: This paper presents a numerical study of the application of magnetorheological (MR) fluid in a Abstract: paper presents numerical study the of (MR) in piezoelectric energy based on of a cantilever architecture. The study of variable damping Abstract: This This paperharvesting presents aa system numerical study of the application application of magnetorheological magnetorheological (MR) fluid fluid in aa Abstract: This paper presents a system numerical study of the application ofthis magnetorheological (MR) fluid in a piezoelectric energy harvesting based on a cantilever architecture. The study of variable damping using a thin film magnetorheological (MR) fluid is presented in paper. MR fluids change their piezoelectric energy based on of a cantilever architecture. The study of variable damping Abstract: This paperharvesting presents a system numerical study the application of magnetorheological (MR) fluid in a piezoelectric energy harvesting system based on a cantilever architecture. The study of variable damping using thinproperties film magnetorheological magnetorheological (MR)of fluid is presented presented inThis thischange paper. MR fluids change their rheological upon thesystem application aa magnetic field. in rheology makes them using aa thin film (MR) fluid is in this paper. MR fluids change their piezoelectric energy harvesting based on cantilever architecture. The study of variable damping using a dampers thin filmwith magnetorheological (MR)offluid isThe presented in results thischange paper. MR change their rheological the aa magnetic field. in rheology makes variable aupon controllable damping force. simulated showed thatfluids the damping ratio rheological properties upon the application application magnetic field. inThis This in rheology makes them them using a thinproperties film magnetorheological (MR)offluid is presented thischange paper. MR fluids change their rheological properties upon the application of a magnetic field. This change in rheology makes them variable dampers with a controllable damping force. The simulated results showed that the damping ratio decreasesdampers from 0.02 to aupon 0.0014 upon the application ofThe a magnetic of 0.1T. Therheology decrease in damping variable with controllable damping simulated results showed that the damping ratio rheological properties the application offorce. a magnetic field. field This change in makes them variable with a0.0014 controllable damping force. The simulated results showed that the damping ratio decreases from 0.02 to the application of aa magnetic field of 0.1T. decrease in damping is showndampers to increase the powerupon output due to the increase in the quality ofThe the harvester. the decreases from 0.02 to upon the application ofThe magnetic field offactor 0.1T. The decrease in Also, damping variable dampers with a0.0014 controllable damping force. simulated results showed that the damping ratio decreases from 0.02 to 0.0014 upon thedue application of a magnetic field offactor 0.1T.ofThe decrease in Also, damping is shown to increase the power output to the increase in the quality the harvester. the application of the fluid has been shown to enhance frequency tuning. is shown to increase the0.0014 powerupon output to the increase in the quality thedecrease harvester. the decreases from 0.02 to thedue application of a magnetic field offactor 0.1T.ofThe in Also, damping is shown to of increase thehas power output due to the increase in the quality factor of the harvester. Also, the application the fluid been shown to enhance frequency tuning. application of the fluid has been shown to enhance frequency tuning. is shown to increase the power output due to the increase in the quality factor of the harvester. Also, the Keywords: Magnetorheological fluid, squeeze mode yield stress, piezoelectric energy harvester, variable © 2019, IFAC Federation Control) Hosting application of (International the fluid has been shownoftoAutomatic enhance frequency tuning.by Elsevier Ltd. All rights reserved. Keywords: Magnetorheological fluid, squeeze mode yield stress, piezoelectric energy harvester, variable application of the fluid has been shown to enhance frequency tuning. damping, frequency enhancement Keywords: Magnetorheological fluid, squeeze mode yield stress, piezoelectric energy harvester, variable Magnetorheological Keywords: fluid, squeeze mode yield stress, piezoelectric energy harvester, variable damping, enhancement damping, frequency frequency enhancement Keywords: Magnetorheological fluid, squeeze mode yield stress, piezoelectric energy harvester, variable damping, frequency enhancement damping, frequency enhancement There are basically three different ways of converting 1. INTRODUCTION There basically different ways converting mechanical into electrical These include 1. INTRODUCTION There are are vibrations basically three three different energy. ways of of converting 1. INTRODUCTION Energy harvesting from the environment has been around for There are vibrations basically three different ways of converting mechanical into electrical energy. These include 1. INTRODUCTION electromagnetic, electrostatic and piezoelectric (Liu et al., mechanical into electrical These include There are vibrations basically three different energy. ways of converting 1. INTRODUCTION Energy the around for centuries, dating from back the days has of been windmills mechanical vibrations into electrical energy. These include Energy harvesting harvesting from thetoenvironment environment has been aroundand for electromagnetic, electrostatic and piezoelectric (Liu et al., 2008). Electromagnetic transduction mechanisms work based electromagnetic, electrostatic and piezoelectric (Liu et al., vibrations into electrical energy. These include Energy harvesting from thetoenvironment been around for mechanical centuries, dating back the of windmills and waterwheels (Paradiso 2005). The has technology is even electromagnetic, electrostatic piezoelectric (Liu et al., centuries, dating backettheal., toenvironment the days days of windmills and transduction mechanisms work based Energy harvesting from has been around for 2008). on the Electromagnetic concept developed by and Faraday (Cheng and Arnold, 2008). Electromagnetic transduction mechanisms work based electromagnetic, electrostatic and piezoelectric (Liu et al., centuries, dating tomore theprominence days of windmills and 2008). Electromagnetic transduction mechanisms work based waterwheels (Paradiso et The technology is even recently being givenback much due to the waterwheels (Paradiso et al., al., 2005). The of technology is rapid even the concept developed by Faraday (Cheng and Arnold, centuries, dating back to 2005). the days windmills and on 2010). Electromagnetic transducers convert the relative on the concept developed by Faraday (Cheng and Arnold, 2008). Electromagnetic transduction mechanisms work based waterwheels (Paradiso et al., 2005). technology is rapid even recently given much more prominence due to reduction in the power consumption ofThe electronic and 2010). on the concept developed by Faraday (Cheng and Arnold, recently being being given much more prominence due devices to the the Electromagnetic transducers convert relative waterwheels (Paradiso et al., 2005). The technology is rapid even motion between a magnetic field and coil due to the the external 2010). Electromagnetic transducers convert the relative on the concept developed by Faraday (Cheng and Arnold, recently being given much more prominence due to the Also, rapid reduction in the power consumption of electronic devices and the development of autonomous self-powered devices. 2010). Electromagnetic transducers convert the relative reductionbeing in thegiven power consumption of electronic and motion between a magnetic field and coil due to the external recently much more prominence due devices to the rapid excitation into electrical power. Electrostatic transduction motion between a magnetic field and coil due to the external 2010). Electromagnetic transducers convert the relative reduction in the power consumption of electronic devices and the development of autonomous self-powered devices. Also, inherent problems with batteries needing constant motion between a magnetic field and coil due to the external the development of autonomous self-powered devices. into electrical power. Electrostatic transduction reduction in the power consumption of electronic devicesAlso, and excitation mechanisms employ the concept of a coil variable capacitance to excitation into electrical power. Electrostatic transduction motion between a magnetic field and due to the external the development of autonomous self-powered devices. Also, the inherent problems with batteries needing constant replacement given impetus to harnessing excitation into electrical power.ofmotion Electrostatic transduction inherent have problems with more batteries needing constant employ the concept a variable capacitance to the development ofeven autonomous self-powered devices. Also, mechanisms generate charges from the relative between two plates mechanismsinto employ the concept a variable capacitance to excitation electrical power.ofElectrostatic transduction the inherent problems withtomore batteries needing constant replacement have even to energy from the environment powerimpetus inaccessible wireless generate mechanisms employ the concept ofmotion a variable capacitance to replacement have even given given impetus to harnessing harnessing charges from the relative between two plates the inherent problems with more batteries needing constant (Boisseau et al., 2012). generate charges from the relative motion between two plates mechanisms employ the concept of a variable capacitance to replacement have even given more impetus to harnessing energy from the environment to power inaccessible wireless sensor nodes. The growing interest in energy harvesting from generate charges from the relative motion between two plates energy from the powerimpetus inaccessible wireless (Boisseau et al., 2012). replacement haveenvironment even giventomore to harnessing (Boisseau et al., 2012). generate charges from the relative motion between two plates energy from the environment to power inaccessible wireless Piezoelectric mechanisms use the concept of spontaneous sensor The growing in harvesting from both thenodes. academic fraternityinterest and industry players has (Boisseau et al., 2012). sensor nodes. Theenvironment growing interest in energy energy harvesting from energy from the to various power inaccessible wireless (Boisseau et al., 2012). mechanisms use concept of sensor nodes. The growing in energy harvesting from polarization certain materials mechanical stress is both the academic fraternity and various industry players has led to the development ofinterest commercial products such as Piezoelectric Piezoelectric by mechanisms use the thewhen concept of spontaneous spontaneous both thenodes. academic fraternity and various industry players has sensor The growing interest in energy harvesting from Piezoelectric mechanisms use thewhen concept of spontaneous polarization by certain materials mechanical stress is both thethe academic fraternity and various industry players has applied to them to convert a mechanical force into electric led to development of commercial products such as wearable devices, biomedical implants, etc. (Teresa et al., polarization bymechanisms certain materials mechanical stress is use thewhen concept of spontaneous led to development of and commercial products such has as Piezoelectric both thethe academic fraternity various industry players polarization by These certain materials whenpiezoelectric mechanical stress is to them to convert a mechanical force into electric led to thedevices, development of commercial products such as applied force (voltage). materials, called materials, wearable biomedical implants, etc. (Teresa et al., 2005) applied to them to convert a mechanical force into stress electric by certain materials when mechanical is wearable biomedical implants, etc. (Teresasuch et al., led to thedevices, development of commercial products as polarization applied to them to convert a mechanical force into electric wearable devices, biomedical implants, etc. (Teresa et al., force (voltage). These materials, materials, convert the mechanical strainaasmechanical acalled resultpiezoelectric of the pressure applied 2005) force (voltage). These materials, called piezoelectric materials, applied to them to convert force into electric 2005) wearable devices, biomedical implants, etc. (Teresa et al., Energy harvesting is the conversion of ambient environmental convert force (voltage). These materials, called piezoelectric materials, 2005) the mechanical strain as result of applied on them into electricity. Conversely, piezoelectric materials convert the mechanical strain as aacalled resultpiezoelectric of the the pressure pressure applied force (voltage). These materials, materials, 2005) Energy harvesting is conversion of environmental energy into electrical energy. Wireless sensors are designed to on convert the mechanical strain as a result of the pressure applied Energy harvesting is the the conversion of ambient ambient environmental them into electricity. Conversely, piezoelectric materials can also convert a voltage applied across their terminals into on them into electricity. Conversely, piezoelectric materials convert the mechanical strain as a result of the pressure applied Energy harvesting isand the conversion ambient environmental energy electrical energy. sensors to operate remotely are Wireless mostlyof bydesigned batteries. on them intoforce electricity. Conversely, piezoelectric materials energy into into electrical energy. Wireless sensors are are designed to can also convert a voltage applied across their terminals into Energy harvesting is the conversion ofpowered ambient environmental mechanical (this is known as indirect piezoelectric canthem also convert a voltageConversely, applied across their terminals into into electricity. piezoelectric materials energy into electrical energy. sensors designed to on operate remotely and are mostly powered by batteries. Batteries, however, need to beWireless recharged everyare now and then. can also(Garimella convert a etvoltage applied across their terminals into operateinto remotely and are mostly powered bydesigned batteries. force (this is known as indirect piezoelectric energy electrical energy. Wireless sensors are to mechanical effect) al., 2015). mechanical force (this is known as indirect piezoelectric can also convert a voltage applied across their terminals into operate remotely and are mostly powered by batteries. Batteries, however, need to be recharged every now and then. They also require periodic replacements. These make the use mechanical force (this is known as indirect piezoelectric Batteries,remotely however, and need are to bemostly recharged every now then. effect) (Garimella et al., 2015). operate powered by and batteries. effect) (Garimella et al., 2015). mechanical force (this is known as indirect piezoelectric Batteries, however, need to be recharged every now and then. Among(Garimella the three,et al., converting They also require periodic These make the use of batteries to power devices every quitenow problematic, 2015). mechanical vibrations into They also however, require periodic replacements. These make thethen. use effect) Batteries, need remote toreplacements. be recharged and effect) (Garimella et al., 2015). mechanical the three, converting vibrations into They also require periodic replacements. These make the use Among electrical energy using piezoelectric transducers is the most of batteries to power remote devices quite problematic, especially in environments where human beings have limited Among the three, converting vibrations into of batteries to power remote devices These quite make problematic, They also require periodic replacements. the use Among the three, converting mechanical mechanical vibrations into electrical energy using piezoelectric transducers is the most of batteries to power remote devices quite problematic, preferred method because of their higher power densities, especially in environments where human beings have limited access (Ju et.al., 2018). electrical energy using piezoelectric transducers is the most Among the three, converting mechanical vibrations into especially in environments wheredevices human beings have limited electrical energy using piezoelectric transducers is the most of batteries to power remote quite problematic, preferred method because of their higher power densities, especially in environments where human beings have limited higher feasibility for practical applications and also they can access (Ju et.al., 2018). preferred method because of their higher power densities, electrical energy using piezoelectric transducers is the most access (Ju et.al., 2018). especially inbe environments where human beings have limited preferred method because of their higher power densities, Energy canet.al., harvested feasibility for practical applications and also they can access (Ju 2018). from a variety of sources including higher easily be produced at both macroscale and microscale using higher feasibility practical alsodensities, they can preferred method for because of applications their higher and power access (Ju et.al., 2018). solar/light Energy be of including thermal, electromagnetic higher feasibility practical applications and also theyusing can Energy can canmechanical, be harvested harvested from from aa variety varietyand of sources sources including easily be produced at both macroscale and microscale thick and thin filmfor deposition techniques (Erturk, 2009). easily be produced at both macroscale and microscale higher feasibility for practical applications and also theyusing can Energy can be harvested from a variety of sources including thermal, mechanical, solar/light and electromagnetic radiations. Mechanical energy harvesting involves converting easily be produced at both macroscale and microscale using thermal,canmechanical, solar/light electromagnetic and thin film deposition techniques (Erturk, 2009). Energy be harvested from a varietyand of sources including thick thick and thin film deposition techniques (Erturk, 2009). easily be produced at both macroscale and microscale using thermal, mechanical, solar/light and electromagnetic Energy harvesting using piezoelectric materials is a very radiations. energy harvesting converting the motionMechanical or displacement of a vibrating structure or thick and thin film deposition techniques (Erturk, 2009). radiations. Mechanical energy harvesting involves converting thermal, mechanical, solar/light andinvolves electromagnetic thick andharvesting thin film deposition techniques (Erturk, Energy using piezoelectric materials is radiations. Mechanical energy harvesting involves converting technology. However, vibration energy2009). harvesting, the motion or displacement of a vibrating structure or vibrations caused by air or water flow into electrical energy. Energy harvesting using piezoelectric materials is aa very very the motion or displacement of a vibrating structure or promising radiations. Mechanical energy harvesting involves converting Energy harvesting using piezoelectric materials is a very promising technology. However, vibration energy harvesting, the motion or displacement of a vibrating structure or in general, produces maximum power only when the frequency vibrations caused by air or water flow into electrical energy. Energy harvesting dueairtoor vibration is most abundant form promising technology. However, vibration energy harvesting, harvesting using piezoelectric materials is a very vibrations caused by water into electrical energy. the motion or displacement of flow a the vibrating structure or Energy promising technology. However, vibration energy harvesting, in produces maximum power only the frequency vibrations caused by air or water into electrical energy. the source matches the frequency of thewhen harvester. This in Energy harvesting due vibration is most abundant form of energy in the environment (Tangflow al., 2010). in general, general, produces maximum power only when theharvesting, frequency promising technology. However, vibration energy Energy harvesting dueairto toor vibration isetthe the most abundant form of vibrations caused by water flow into electrical energy. in general, produces maximum power only when the frequency the source matches the frequency of the harvester. This in Energy harvesting due to vibration isetthe most abundant form of effect limits the amount of power generated since the natural of energy in the environment (Tang al., 2010). of general, the source matches the frequency thewhen harvester. This in produces maximum power of only the frequency of energy in the environment (Tangisetthe al.,most 2010). Energy harvesting due to vibration abundant form in of the limits sourcethe matches theoffrequency of the harvester. This in of energy in the environment (Tang et al., 2010). effect amount power generated since the natural effect limits the amount of power generated since the natural of the source matches the frequency of the harvester. This in of energy in the environment (Tang et al., 2010). effect limits the amount of power generated since the natural Copyright@ 2019 IFAC 394Hosting effect limits the amount power generated since the natural 2405-8963 © 2019, IFAC (International Federation of Automatic Control) by Elsevier Ltd. All of rights reserved. Copyright@ 2019 394 Peer review under responsibility of International Federation of Automatic Copyright@ 2019 IFAC IFAC 394Control. 10.1016/j.ifacol.2019.10.063 Copyright@ 2019 IFAC 394 Copyright@ 2019 IFAC 394
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frequency of vibrations is dynamic, requiring constant controlling of the frequency of the harvester to resonate with that of the source of excitation for maximum power production. Currently, two different techniques have been proposed by Zhu et al., 2010 to solve this limitation. The first is to tune the resonant frequency of a single generator periodically so that it matches the excitation frequency at all times. This, however, can be used to tune the harvester to respond to only one frequency at a time making it problematic since practical systems with frequency tuning require actuators or controllers to do the tuning. The power that will be expended by the actuators can defeat the purpose of the energy harvesting in environments where there are large variations in the excitation frequency. The second solution is to design the generator with a wider bandwidth. Current techniques in bandwidth widening involve complex mathematical models to describe their behavior making their implementation complex (Zhu et al., 2010).
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are non-magnetic, organic or aqueous liquid and silicone or mineral oils. Upon the application of the magnetic field, the magnetizable particles acquire a dipole moment which aligns with the external magnetic field and forms. The formation of these chains induces reversible yield stress in the fluid (Choi et al., 2012). The yield stress increases the stiffness of the system and enables frequency tuning and damping effect. This phenomenon is illustrated in Fig. 1 and Fig. 2.
Fig. 1 Non-activated magnetic field (Djokoto et al., 2019)
Mechanical frequency tuning is based on the well-known resonant frequency of a spring-mass system which relates the spring constant to the mass of the harvester (Zhu et al., 2010). The stiffness of the system depends on the dimensions of the spring-mass system. Thus, by altering any of the mechanical properties of the energy harvester, its frequency can be tuned.
Fig. 2 Direction of activated magnetic field magnetic (Djokoto et al., 2019)
In this article, an impact-based frequency enhancement with variable damping using a vibrating bimorph piezoelectric energy generator is proposed. In particular, a novel type of impacting object, which is an activated MR fluid is investigated. The MR fluid changes from a fluid state to a semi-solid state when activated due to its rheological properties. The goal is to design an energy generator with the capabilities of producing 2.5mW using a varying magnetic field applied to the MR fluid.
MR fluid-based devices are considered excellent dampers because of their long-range controllable damping force, fast adjustable response, and low energy consumption (Zhu et al., 2012). MR dampers have been used in many engineering applications such as landing gears for airplanes to absorb the impact energy and to minimize the impact force transmitted to the airplane due to drop impact at landing (Kim et al., 2018), high-speed train suspension system to improve the ride comfort of passengers (Ramasastry et al., 2016), seismic vibration control of engineering structures (Koo, 2003), gun recoil application (Singh et al., 2014), washing machines (Ulasyar et al., 2018), etc..
The model presented in this work is based on the squeeze mode of the MR fluid. This model has been analytically simulated and experimentally tested by Kaluvan et al., in the design of a resonant-based measurement system using a piezolaminated cantilever beam coupled to an electromagnetic coil. The shift in resonant frequency due to the change in viscosity of the MR fluid was measured and shown to be related to the yield stress of the fluid. This work, however, focusses on the impact of the damping force due to the MR fluid when used as an impacting object in piezoelectric energy harvesting.
Until recently, where there has been much emphasis on designing autonomous microsystems which harvest energy from their natural environment, the vibrational energy absorbed by dampers has always been dissipated in the environment because of the absence of viable harvesting mechanisms.
1.1 Viscous damping MR fluid MR fluids and electrorheological (ER) fluids are part of a special group of fluids termed smart fluids whose viscosity can change significantly by the application of a suitable stimulus (Stanway, 2004). MR fluids have the special property that their viscosity or resistance to flow can be changed from liquids to semi-solids by the application of a magnetic field. The major distinguishing feature between MR and ER fluids is the type of stimulus applied to them. MR fluids have attracted much more attention from the research community because they do not require high voltage to produce high damping force as in the case of ER fluids.
2. THEORY/METHODOLOGY The proposed energy harvester is shown in Fig.3. The system consists of a vibrating piezoelectric bimorph cantilever with an attached mass, two permanent magnets one serving as the end mass and the other attached to the bottom of the ferromagnetic fluid holder. The magnets are mounted on opposite sides of the beam with the MRF acting as the impact object attached to one of the magnets. The piezoelectric cantilever beam used consists of a shim layer sandwiched in between two piezoceramic layers from Johnson Matthey Piezo Products GmbH as shown in Fig. 4.
MR fluids are made up of magnetizable particles (such as pure iron, carbonyl iron, or cobalt power) and carrier fluids which 395
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𝑓𝑓(𝑡𝑡) = 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹(𝜔𝜔𝜔𝜔)
(2)
Where 𝐹𝐹 is the amplitude of the external force. The rheological properties of the MR fluid due to the intensity of the magnetic field determines the effective stiffness of the beam at the point of impact. At the point of impact with the MRF, the stiffness of the vibrating cantilever beam is increased by 𝐾𝐾2 . From this stiffness effect, the vibration of the cantilever is transformed from linear oscillations to nonlinear impact oscillations as a result of the displacement constraints given by the MR fluid stopper. The additional stiffness introduced by the MR fluid is nonlinear due to the nonlinear relationship between the tip displacement and the stiffness induced by the fluid. Fig. 3 Schematic diagram of the proposed variable-damping MR fluid energy harvester
In the study of the energy harvester using the MR fluid as an impact object, the most important parameter of the fluid to consider is the magnetic field induced yield stress. In this study, the MR fluid, 140-CG, was used (Lord Products, 2014). The properties of the fluid are governed by two empirical equations (Carlson, 2008). Equation (3) gives the magnitude of the induced yield stress, 𝜏𝜏(𝐻𝐻), as a function of the applied magnetic field intensity 𝐻𝐻 , whiles equation (4) gives the relationship between the magnetic flux density, 𝐵𝐵 and the applied field intensity in the fluid. 𝜏𝜏(𝐻𝐻) = C × 271700 × Φ1.5239 × tanh(6.33 × 10−6 × 𝐻𝐻) (3) (4) 𝐵𝐵 = 1.91 × Φ1.133 (1 − 𝑒𝑒 −10.97𝜇𝜇𝑜𝑜 𝐻𝐻 ) + 𝜇𝜇𝑜𝑜 𝐻𝐻
Fig. 4 Bimorph (Piezo-Bending Actuators 427.00085.11Z)
2.1 System Modelling
Where C is a coefficient dependent on the carrier fluid of the MRF and Φ is the particle volume fraction of the fluid.
The piezoelectric energy harvester is modeled as a single degree of freedom (SDOF) spring-mass-damper model to illustrate the behavior of the MR fluid energy harvester. The SDOF system is shown in Fig. 5.
The stiffness due to the fluid and squeeze mode damping coefficient depend on the force generated upon impact and the yield stress. The force generated due to the magnetic field exerted on the fluid is given by (Brigley et al., 2008) and (Kaluvan et al., 2014): 4𝜋𝜋𝑟𝑟 3 𝜏𝜏
𝐹𝐹𝑀𝑀𝑀𝑀𝑀𝑀 =
(5)
3(𝑑𝑑1 +𝑍𝑍)
Where r is the radius of the magnet, 𝑑𝑑1 is the initial gap between the beam and the fluid and z is the displacement of the beam. The magnitude of the stiffness due to the MR fluid in squeeze mode is calculated from equation (6) as shown below (Kaluvan et al., 2014): 𝐾𝐾2 =
Fig. 5 SDOF model of the energy harvesting system 𝑚𝑚𝑒𝑒𝑒𝑒𝑒𝑒 𝑍𝑍̈ + (𝐶𝐶1 + 𝐶𝐶2 )𝑍𝑍̇+(𝐾𝐾1 + 𝐾𝐾2 )𝑍𝑍 = 𝑓𝑓 (1) Where, 𝑚𝑚𝑒𝑒𝑒𝑒𝑒𝑒 , is the effective mass (which is equal to the mass of the cantilever beam and the tip mass), 𝐶𝐶1 is the damping coefficient due to the cantilever beam, 𝐶𝐶2 is the additional damping due to the MRF. 𝐾𝐾1 is the stiffness of the beam and 𝐾𝐾2 is the nonlinear stiffness introduced by the MRF. The external force is denoted as 𝑓𝑓. Z is the tip displacement of the cantilever beam at a distance 𝑑𝑑1 from the fixed end as shown in Fig 5.
𝜕𝜕𝐹𝐹𝑀𝑀𝑀𝑀𝑀𝑀 𝜕𝜕ℎ
=
4𝜋𝜋𝑟𝑟 3 𝜏𝜏
(6)
3(𝑑𝑑1 +𝑍𝑍)2
The stiffness of the cantilever beam alone is given by (Roundy et al., 2003): 𝐾𝐾1 =
3𝐸𝐸𝐸𝐸 𝑙𝑙𝑏𝑏 3
(7)
Where 𝐸𝐸 is Young’s modulus of the piezoelectric beam, 𝐼𝐼 is the moment of inertia and 𝑙𝑙𝑏𝑏 is the free vibrating length of the beam. The moment of inertia is given by (Beer et al., 2006): 𝐼𝐼 =
𝑤𝑤
12
𝐸𝐸𝑠𝑠ℎ
(
𝐸𝐸𝑝𝑝
𝑡𝑡𝑠𝑠ℎ 3 + 2𝑡𝑡𝑝𝑝 3 ) +
𝑤𝑤𝑤𝑤𝑝𝑝 2
(𝑡𝑡𝑠𝑠ℎ + 𝑡𝑡𝑝𝑝 )2
(8)
Where w is the width of the beam, 𝐸𝐸𝑠𝑠ℎ is the modulus of elasticity of the shim layer, 𝐸𝐸𝑝𝑝 is Young’s modulus of the
Under a harmonic excitation, the force due to the excitation is given by:
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piezoelectric material, 𝑡𝑡𝑠𝑠ℎ is the thickness of the shim layer, and 𝑡𝑡𝑝𝑝 is the thickness of piezoelectric material.
The cantilever is excited at 2.5 𝑚𝑚/𝑠𝑠 2 . The effective damping ratio is determined from the log decrement plot as 𝛿𝛿⁄2𝜋𝜋. Where 1 𝑍𝑍 𝛿𝛿 is given as log ( 𝑛𝑛 ) and n the number of cycles.
𝜔𝜔𝑛𝑛 = √
3. RESULTS AND DISCUSSION
(9)
𝑚𝑚𝑒𝑒𝑒𝑒𝑒𝑒
The squeeze mode yield stress of the energy harvester was determined analytically from the empirical formulas of the MR fluid under the application of different values of magnetic field density (0.03T, 0.06T, and 0.10T). The magnetic field intensity was calculated using Equation (4) for the different values of field density. The magnetic field intensity values determined were then used to compute the yield stress of the fluid in Equation (3). The properties and dimensions of the energy harvester are shown in Table 2 in the appendix. The properties of the fluid and the piezoelectric bimorph are as defined by the manufacturers.
Where 𝐾𝐾𝑒𝑒𝑒𝑒𝑒𝑒 is the effective stiffness of the generator given as the sum of the 𝐾𝐾1 and 𝐾𝐾2 and 𝑚𝑚𝑒𝑒𝑒𝑒𝑒𝑒 is the effective mass of the energy generator. Mechanical vibration energy is harvested through damping. The damping ratio of the system defines the mechanical dissipation due to the fluid squeeze force. The damping effect as a result of the squeeze force of the fluid is given as (Kaluvan et al., 2014): 𝐶𝐶2 =
3 𝜋𝜋𝜋𝜋𝑟𝑟 4
The effect of the thin film MR fluid on the nonlinear energy harvester is studied by varying the magnetic field applied to the fluid. Mechanical energy due to vibration is merely harvested through damping.
(10)
2 (ℎ+𝑍𝑍)3
Where 𝜂𝜂 is the viscosity of the fluid. The damping coefficient of the piezoelectric cantilever beam, on the other hand, is given as (Ali et. al., 2012): 𝐶𝐶1 = 2 × 𝜁𝜁 × 𝑚𝑚𝑒𝑒𝑒𝑒𝑒𝑒 × 𝜔𝜔𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏
It is observed from Fig. 6 that, as the magnetic field is increased, the additional damping effect due to the fluid causes the bandwidth of the harvester to decrease, however, the power output increases. Bandwidth, as used in this paper, is the 3-dB bandwidth, and it is defined as the frequency range over which the spectrum of the power response is greater than or equal to half of the maximum power harvested (Cammarano et al., 2014). The reduction in bandwidth is due to the fact that the effective damping ratio decreases and as a result the quality factor increases. This, in turn, results in bandwidth reduction as can be seen from the equation below:
(11)
Where 𝜁𝜁 is the damping ratio of the free vibration beam. When the external excitation is harmonic, the net electrical power generated can be calculated using (Williams et al., 1996): 𝑃𝑃 =
𝜔𝜔 3 𝑛𝑛 2 𝜔𝜔 2 𝜔𝜔 2 [1−( ) ] +[2𝜁𝜁 ] 𝜔𝜔𝑛𝑛 𝜔𝜔𝑛𝑛
𝑚𝑚𝑚𝑚𝑌𝑌 2 (𝜔𝜔 ) 𝜔𝜔3
(12)
𝑄𝑄 =
From equation (14), it is clear that maximum power is obtained when the excitation frequency is equal to the frequency of the energy harvester. Therefore, 𝑃𝑃𝑚𝑚𝑚𝑚𝑚𝑚 =
𝑚𝑚𝑌𝑌 2 𝜔𝜔3
1 2ζ
=
𝜔𝜔 𝐵𝐵𝐵𝐵
(16)
Where Q is the quality factor and BW is the bandwidth.
(13)
4𝜁𝜁
𝑍𝑍𝑛𝑛+1
𝑛𝑛
The resonant frequency of the energy harvester is, therefore, given as: 𝐾𝐾𝑒𝑒𝑒𝑒𝑒𝑒
397
The results of the damping ratio as the fluid is applied and by varying the magnetic field are shown in Table 1.
To determine the displacement amplitude when the cantilever is subjected to harmonic excitation, equation (3) is numerically simulated using MATLAB ode45 by defining the following state model: 𝑍𝑍̇ = 𝐴𝐴𝐴𝐴 + 𝐵𝐵𝐵𝐵 ,
𝑌𝑌 = 𝐶𝐶𝐶𝐶
(14)
A represents the system matrix, B is the input matrix, C is the output matrix, X is the state vector, u is the input excitation and Y is the system output. The state space form of equation (3) is: 0 𝐴𝐴 = [−(𝐾𝐾1+𝐾𝐾2) 𝑚𝑚𝑒𝑒𝑒𝑒𝑒𝑒
𝐵𝐵 = [ 𝑢𝑢 = 𝑓𝑓
0
1
𝑚𝑚𝑒𝑒𝑒𝑒𝑒𝑒
1
−(𝐶𝐶1 +𝐶𝐶2 )]
Fig. 6 Output power Vs Frequency of the energy harvester at different magnetic field densities.
𝑚𝑚𝑒𝑒𝑒𝑒𝑒𝑒
], 𝐶𝐶 = [0
𝑍𝑍 1], 𝑋𝑋 = [𝑍𝑍1 ] 2
Table 1: Summary of results Magnetic field density (T) 0 0.03 0.06 0.1 397
Damping ratio
Frequency (Hz)
0.02 0.0143 0.0044 0.0014
128 147 164 186
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3.1 Displacement Response of the Energy Harvester
4. CONCLUSIONS
The displacement responses of the energy harvesting with/without the application of the fluid are as shown below. As expected the amplitude of vibration of the system decreases as the magnetic field is increased and this can be attributed to the extra damping effect added to the system due to the application of the fluid.
The response of a vibration energy harvester due to the application of a thin film MR fluid has been analyzed in this paper. Using the thin film MR fluid in frequency tuning is also shown. The main intent of this paper was to investigate the impact of the variable damping due to the MRF on the frequency response and also the power output of the nonlinear energy harvester. It has been demonstrated that the power bandwidth of the energy harvester decreases as a result of the MR fluid induced damping. However, the power output increases due to the increase in the quality factor of the energy harvester. Using the fluid in frequency tuning of vibration energy harvesting has been demonstrated to be beneficial especially when fine-tuning is required to closely match the frequency of the source. The results of the simulation show a decrease in damping ratios from 0.02 without fluid to 0.0014 upon the application of the fluid with a magnetic field density of 0.1T.
Fig. 7 Displacement response without fluid
Though the initial goal of attaining a power output of 2.5mW could not be achieved, the impact of the additional damping force added to the system has been shown. Using the fluid to aid frequency tuning in energy harvesting is also highlighted. REFERENCES Ali W. G. and Ibrahim S. W. (2012) "Power Analysis for Piezoelectric Energy Harvester" Energy Power Eng., vol. 04, no. 06, pp. 496–505. Beards C. F. and Arnold E. (2013) Engineering vibration analysis with application to control systems, vol. 33, no.09. Beer P. F., Johnson E. R. and Dewolf T. J. (2006) Mechanics of Materials, 4th ed. New York: McGraw Hill. Blevins R. D. (1979) Formulas-for-Natural-Frequency-andMode-Shape, Van Nostrand Reinhold, New York Brigley M, Choi Y-T, and Wereley N. M. (2008) "Experimental and Theoretical Development of Multiple Fluid Mode Magnetorheological Isolators". J Guid Control Dyn ;31:449–59. doi:10.2514/1.32969. Boisseau S, Despesse G, and Ahmed B. (2012) "Electrostatic Conversion for Vibration Energy Harvesting". SmallScale Energy Harvest:1–39. doi:10.5772/51360. Cammarano A, Neild, S. A Burrow S. G, and Inman D. J (2014) "The bandwidth of optimized nonlinear vibrationbased energy harvesters" Smart Mater. Struct. 23 055019 (9pp) Carlson J. D. (2008) Magnetorheological fluids, in Smart Materials. New York: Taylor & Francis. Cheng S, and Arnold DP. (2010) "A study of a multi-pole magnetic generator for low-frequency vibrational energy harvesting". J Micromechanics & Microengineering; 20. doi:10.1088/0960-1317/20/2/025015. Choi S.-B. and Han Y.-M. (2012) Magnetorheological Fluid Technology: Applications in Vehicle Systems. CRS Press Taylor & Francis Group "Data Sheet Piezoceramic Trimorph Bending Actuator, Part No. 427.0085.11Z, Johnson Matthey Piezoproducts Bahnhofstraße 43, D- 96254 Redwitz," 2011 Djokoto S. S., Agelin-Chaab M., Jūrėnas V. and Dragašius E. (2019) "Experimental Investigation of Squeezed MRF
Fig. 8 Displacement response with a magnetic field density of 0.03 T
Fig. 9 Displacement response with a magnetic field density of 0.06 T
Fig. 10 Displacement response with a magnetic field density of 0.10 T. 398
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Film Stopper and Its Effect on Vibrating Bimorph for Frequency Tuning of an Energy Generator" IEEE SoutheastCon 2019 Proceedings Erturk A. (2009) "Electromechanical modeling of piezoelectric energy harvesters", Heal. San Fr., Garimella R. C, Sastry V. R, Mohiuddin M. S. and Piezo-Gen (2015) "An Approach to Generate Electricity from Vibrations". Procedia Earth Planet Sci; 11:445–56. doi:10.1016/j.proeps.2015.06.044. Ju S, and Ji C. (2018) "Impact-based piezoelectric vibration energy harvester". Appl Energy; 214:139–51. doi: 10.1016/j.apenergy.2018.01.076. Kaluvan S., Shah K., and Choi S. B., (2014) "A new resonant based measurement method for squeeze mode yield stress of magnetorheological fluids", Smart Mater. Struct., vol. 23, no. 11. Kim B., Han C., Choi S.-B., and Kang B., (2018) "Design and Analysis of a Magnetorheological Damper for Airplane Landing Gear", Proc. SPIE Active and Passive Smart Structures and Integrated Systems XII no., p. 55 Koo J., (2003) "Using Magneto-Rheological Dampers in Semiactive Tuned Vibration Absorbers to Control Structural Vibrations by Using Magneto-Rheological Dampers in Semiactive Tuned Vibration Absorbers to Control Structural Vibrations", Ph.D. Thesis, Virginia Polytechnic Institute, and State University Liu J. Q. Fang H-B, Zheng-Yi X, Mao X-H, Shen X-C, Chen D, Liao H. and Cai B-C (2008) "A MEMS-based piezoelectric power generator array for vibration energy harvesting", Microelectronics J., vol. 39, no. 5, pp. 802– 806. L.Corporation, MRF-140CG, Magneto-Rheological Fluids, http://www.lord.com/products and solutions/magnetorheological-(MR)/product.xml/1646. (2014) Paradiso J. A. and Starner T., (2005) "Energy scavenging for mobile and wireless electronics", IEEE Pervasive Comput., vol. 4, no. 1, pp. 18–27. Ramasastry D. V. A., Ramana K. V., Rao N. M., Kumar S. S., and Priyanka T. G. L., (2016) "Analysis of Train Suspension System Using MR dampers", IOP Conference Series: Materials Science and Engineering, vol. 149, no. 1. Stanway R., (2004) “Smart fluids: current and future developments,” Mater. Sci. Technol., vol. 20, no. 8, pp. 931–939. Singh H. J. and Wereley N. M., (2014) "Optimal control of gun recoil in direct fire using magnetorheological absorbers", Smart Mater. Struct., vol. 23, no. 5. Tang L., Yang Y., and Soh C. K. (2010) "Toward broadband vibration-based energy harvesting", J. Intell. Mater. Syst. Struct., vol. 21, no. 18, pp. 1867–1897. Teresa M, Guido F, Mastronardi V, Desmaele D, Epifani G, and Algieri L, (2017) "Piezoelectric MEMS vibrational energy harvesters: Advances and outlook". Microelectron Eng 2017;183–184:23–36. doi: 10.1016/j.mee.2017.10.005. Ulasyar A. and Lazoglu I., (2018) "Design and analysis of a new magnetorheological damper for washing machine", J. Mech. Sci. Technol., vol. 32, no. 4, pp. 1549–1561.
399
Williams C. B. and Yates R. B. (1996) "Analysis of a microelectric generator for Microsystems", Science (80-.)., vol. 52, pp. 8–11. Zhu D., Tudor M. J., and Beeby S. P., (2010) "Strategies for increasing the operating frequency range of vibration energy harvesters: A review", Meas. Sci. Technol., vol. 21, no. 2. Zhu X., Jing X., and Cheng L., (2012) "Magnetorheological fluid dampers: A review on structure design and analysis", Journal of Intelligent Material Systems and Structures, vol. 23, no. 8. pp. 839–873. APPENDIX A. Table 2: System properties and dimensions Symbol l lb w t tp tsh ρp ρsh 𝜀𝜀33 Esh
Φ µ0 𝜂𝜂 r h
C
mt
399
Description Total length of beam Free (vibrating) length Beam width Beam thickness Thickness of piezoelectric layer Shim layer thickness Piezoelectric layer density Shim layer density Piezoelectric dielectric constant Shim layer modulus of elasticity Particle volume fraction Permeability of free space Viscosity of MRF Radius of magnet Initial gap between MRF and Beam Coefficient dependent on the carrier fluid of MRF Tip mass
Value 49.95
Unit mm
29.5
mm
7.2 ± 0.05 0.78 ± 0.03 0.25 ± 0.05
mm mm mm
0.28 ± 0.05
mm
8000
Kg/m3
1800 30975e-12
Kg/m3 F/m
120e9
N/m2
0.4 1.25e-6
H/m
0.3 3.25 0.5
Pa·s mm mm
1
0.003
kg
ScienceDirect
IFAC PapersOnLine 52-10 (2019) 394–399 A Study of Nonlinear Piezoelectric Energy Harvester with Variable Damping A Study of Nonlinear Piezoelectric Harvester Variable Damping A Study of Nonlinear Piezoelectric Energy Harvester with Variable Damping using ThinEnergy Film MR Fluid with A Study of Nonlinear Piezoelectric Energy Harvester with Variable Damping A Study of Nonlinear Piezoelectric Energy Harvester with Variable Damping using Thin Film MR Fluid using Thin Film MR Fluid Thin MR H. Azangbebil *,using S. S. Djokoto*, M. Agelin-Chaab*, using Thin Film Film MR Fluid FluidEgidijus Dragašius **
H. H. Azangbebil Azangbebil *, *, S. S. S. S. Djokoto*, Djokoto*, M. M. Agelin-Chaab*, Agelin-Chaab*, Egidijus Egidijus Dragašius Dragašius ** ** *, S. S. Djokoto*, M. Agelin-Chaab*, H. Azangbebil Egidijus Dragašius ** Science University of * Dept. of Automotive, Mechanical and *, Manufacturing Engineering, Faculty of Egidijus Engineering and Applied H. Azangbebil S. S. Djokoto*, M. Agelin-Chaab*, Dragašius ** * Dept. of of Automotive, Automotive, Mechanical and Manufacturing Manufacturing Engineering, Faculty of Engineering Engineering andL1H Applied Science University of of Ontario Institute of Technology, 2000 Simcoe Street, North, Oshawa, Canada ON 7K4 (Science e-mail:University * Dept. Mechanical and Engineering, Faculty of and Applied * Dept. of Automotive, Mechanical and Manufacturing Engineering, Faculty of Engineering and Applied Science University of Ontario Institute of Technology, 2000 Simcoe Street, North, Oshawa, Canada ON L1H 7K4 ( e-mail: [email protected]; [email protected]; [email protected] ). Ontario Institute of Technology, 2000 Simcoe Street, North, Oshawa, Canada ON 7K4 (Science e-mail:University of * Dept. of Automotive, Mechanical and Manufacturing Engineering, Faculty of Engineering andL1H Applied Ontario Institute of Technology, 2000 Simcoe Street, North, Oshawa, Canada ON L1H56, 7K4 ( ). e-mail: [email protected]; [email protected]; [email protected] ** Dept. of Production Engineering, Kaunas University of Technology, Studentu 51424, [email protected]; [email protected]; [email protected] Ontario Institute of Technology, 2000 Simcoe Street, North, Oshawa, Canada ON L1H 7K4 ( ). e-mail: [email protected]; ). **[email protected]; Dept. of of Production Production Engineering, Kaunas University of [email protected] Technology, Studentu Studentu 56, 56, 51424, 51424, Kaunas, Lithuania (e-mail:University [email protected]) ** Dept. Engineering, Kaunas of Technology, [email protected]; [email protected]; [email protected] ). ** Dept. of Production Engineering, of Technology, Studentu 56, 51424, Kaunas, LithuaniaKaunas (e-mail:University [email protected]) Kaunas, Lithuania (e-mail: [email protected]) ** Dept. of Production Engineering, Kaunas University of Technology, Studentu 56, 51424, Kaunas, Lithuania (e-mail: [email protected]) Kaunas, Lithuania (e-mail: [email protected]) Abstract: This paper presents a numerical study of the application of magnetorheological (MR) fluid in a Abstract: paper presents numerical study the of (MR) in piezoelectric energy based on of a cantilever architecture. The study of variable damping Abstract: This This paperharvesting presents aa system numerical study of the application application of magnetorheological magnetorheological (MR) fluid fluid in aa Abstract: This paper presents a system numerical study of the application ofthis magnetorheological (MR) fluid in a piezoelectric energy harvesting based on a cantilever architecture. The study of variable damping using a thin film magnetorheological (MR) fluid is presented in paper. MR fluids change their piezoelectric energy based on of a cantilever architecture. The study of variable damping Abstract: This paperharvesting presents a system numerical study the application of magnetorheological (MR) fluid in a piezoelectric energy harvesting system based on a cantilever architecture. The study of variable damping using thinproperties film magnetorheological magnetorheological (MR)of fluid is presented presented inThis thischange paper. MR fluids change their rheological upon thesystem application aa magnetic field. in rheology makes them using aa thin film (MR) fluid is in this paper. MR fluids change their piezoelectric energy harvesting based on cantilever architecture. The study of variable damping using a dampers thin filmwith magnetorheological (MR)offluid isThe presented in results thischange paper. MR change their rheological the aa magnetic field. in rheology makes variable aupon controllable damping force. simulated showed thatfluids the damping ratio rheological properties upon the application application magnetic field. inThis This in rheology makes them them using a thinproperties film magnetorheological (MR)offluid is presented thischange paper. MR fluids change their rheological properties upon the application of a magnetic field. This change in rheology makes them variable dampers with a controllable damping force. The simulated results showed that the damping ratio decreasesdampers from 0.02 to aupon 0.0014 upon the application ofThe a magnetic of 0.1T. Therheology decrease in damping variable with controllable damping simulated results showed that the damping ratio rheological properties the application offorce. a magnetic field. field This change in makes them variable with a0.0014 controllable damping force. The simulated results showed that the damping ratio decreases from 0.02 to the application of aa magnetic field of 0.1T. decrease in damping is showndampers to increase the powerupon output due to the increase in the quality ofThe the harvester. the decreases from 0.02 to upon the application ofThe magnetic field offactor 0.1T. The decrease in Also, damping variable dampers with a0.0014 controllable damping force. simulated results showed that the damping ratio decreases from 0.02 to 0.0014 upon thedue application of a magnetic field offactor 0.1T.ofThe decrease in Also, damping is shown to increase the power output to the increase in the quality the harvester. the application of the fluid has been shown to enhance frequency tuning. is shown to increase the0.0014 powerupon output to the increase in the quality thedecrease harvester. the decreases from 0.02 to thedue application of a magnetic field offactor 0.1T.ofThe in Also, damping is shown to of increase thehas power output due to the increase in the quality factor of the harvester. Also, the application the fluid been shown to enhance frequency tuning. application of the fluid has been shown to enhance frequency tuning. is shown to increase the power output due to the increase in the quality factor of the harvester. Also, the Keywords: Magnetorheological fluid, squeeze mode yield stress, piezoelectric energy harvester, variable © 2019, IFAC Federation Control) Hosting application of (International the fluid has been shownoftoAutomatic enhance frequency tuning.by Elsevier Ltd. All rights reserved. Keywords: Magnetorheological fluid, squeeze mode yield stress, piezoelectric energy harvester, variable application of the fluid has been shown to enhance frequency tuning. damping, frequency enhancement Keywords: Magnetorheological fluid, squeeze mode yield stress, piezoelectric energy harvester, variable Magnetorheological Keywords: fluid, squeeze mode yield stress, piezoelectric energy harvester, variable damping, enhancement damping, frequency frequency enhancement Keywords: Magnetorheological fluid, squeeze mode yield stress, piezoelectric energy harvester, variable damping, frequency enhancement damping, frequency enhancement There are basically three different ways of converting 1. INTRODUCTION There basically different ways converting mechanical into electrical These include 1. INTRODUCTION There are are vibrations basically three three different energy. ways of of converting 1. INTRODUCTION Energy harvesting from the environment has been around for There are vibrations basically three different ways of converting mechanical into electrical energy. These include 1. INTRODUCTION electromagnetic, electrostatic and piezoelectric (Liu et al., mechanical into electrical These include There are vibrations basically three different energy. ways of converting 1. INTRODUCTION Energy the around for centuries, dating from back the days has of been windmills mechanical vibrations into electrical energy. These include Energy harvesting harvesting from thetoenvironment environment has been aroundand for electromagnetic, electrostatic and piezoelectric (Liu et al., 2008). Electromagnetic transduction mechanisms work based electromagnetic, electrostatic and piezoelectric (Liu et al., vibrations into electrical energy. These include Energy harvesting from thetoenvironment been around for mechanical centuries, dating back the of windmills and waterwheels (Paradiso 2005). The has technology is even electromagnetic, electrostatic piezoelectric (Liu et al., centuries, dating backettheal., toenvironment the days days of windmills and transduction mechanisms work based Energy harvesting from has been around for 2008). on the Electromagnetic concept developed by and Faraday (Cheng and Arnold, 2008). Electromagnetic transduction mechanisms work based electromagnetic, electrostatic and piezoelectric (Liu et al., centuries, dating tomore theprominence days of windmills and 2008). Electromagnetic transduction mechanisms work based waterwheels (Paradiso et The technology is even recently being givenback much due to the waterwheels (Paradiso et al., al., 2005). The of technology is rapid even the concept developed by Faraday (Cheng and Arnold, centuries, dating back to 2005). the days windmills and on 2010). Electromagnetic transducers convert the relative on the concept developed by Faraday (Cheng and Arnold, 2008). Electromagnetic transduction mechanisms work based waterwheels (Paradiso et al., 2005). technology is rapid even recently given much more prominence due to reduction in the power consumption ofThe electronic and 2010). on the concept developed by Faraday (Cheng and Arnold, recently being being given much more prominence due devices to the the Electromagnetic transducers convert relative waterwheels (Paradiso et al., 2005). The technology is rapid even motion between a magnetic field and coil due to the the external 2010). Electromagnetic transducers convert the relative on the concept developed by Faraday (Cheng and Arnold, recently being given much more prominence due to the Also, rapid reduction in the power consumption of electronic devices and the development of autonomous self-powered devices. 2010). Electromagnetic transducers convert the relative reductionbeing in thegiven power consumption of electronic and motion between a magnetic field and coil due to the external recently much more prominence due devices to the rapid excitation into electrical power. Electrostatic transduction motion between a magnetic field and coil due to the external 2010). Electromagnetic transducers convert the relative reduction in the power consumption of electronic devices and the development of autonomous self-powered devices. Also, inherent problems with batteries needing constant motion between a magnetic field and coil due to the external the development of autonomous self-powered devices. into electrical power. Electrostatic transduction reduction in the power consumption of electronic devicesAlso, and excitation mechanisms employ the concept of a coil variable capacitance to excitation into electrical power. Electrostatic transduction motion between a magnetic field and due to the external the development of autonomous self-powered devices. Also, the inherent problems with batteries needing constant replacement given impetus to harnessing excitation into electrical power.ofmotion Electrostatic transduction inherent have problems with more batteries needing constant employ the concept a variable capacitance to the development ofeven autonomous self-powered devices. Also, mechanisms generate charges from the relative between two plates mechanismsinto employ the concept a variable capacitance to excitation electrical power.ofElectrostatic transduction the inherent problems withtomore batteries needing constant replacement have even to energy from the environment powerimpetus inaccessible wireless generate mechanisms employ the concept ofmotion a variable capacitance to replacement have even given given impetus to harnessing harnessing charges from the relative between two plates the inherent problems with more batteries needing constant (Boisseau et al., 2012). generate charges from the relative motion between two plates mechanisms employ the concept of a variable capacitance to replacement have even given more impetus to harnessing energy from the environment to power inaccessible wireless sensor nodes. The growing interest in energy harvesting from generate charges from the relative motion between two plates energy from the powerimpetus inaccessible wireless (Boisseau et al., 2012). replacement haveenvironment even giventomore to harnessing (Boisseau et al., 2012). generate charges from the relative motion between two plates energy from the environment to power inaccessible wireless Piezoelectric mechanisms use the concept of spontaneous sensor The growing in harvesting from both thenodes. academic fraternityinterest and industry players has (Boisseau et al., 2012). sensor nodes. Theenvironment growing interest in energy energy harvesting from energy from the to various power inaccessible wireless (Boisseau et al., 2012). mechanisms use concept of sensor nodes. The growing in energy harvesting from polarization certain materials mechanical stress is both the academic fraternity and various industry players has led to the development ofinterest commercial products such as Piezoelectric Piezoelectric by mechanisms use the thewhen concept of spontaneous spontaneous both thenodes. academic fraternity and various industry players has sensor The growing interest in energy harvesting from Piezoelectric mechanisms use thewhen concept of spontaneous polarization by certain materials mechanical stress is both thethe academic fraternity and various industry players has applied to them to convert a mechanical force into electric led to development of commercial products such as wearable devices, biomedical implants, etc. (Teresa et al., polarization bymechanisms certain materials mechanical stress is use thewhen concept of spontaneous led to development of and commercial products such has as Piezoelectric both thethe academic fraternity various industry players polarization by These certain materials whenpiezoelectric mechanical stress is to them to convert a mechanical force into electric led to thedevices, development of commercial products such as applied force (voltage). materials, called materials, wearable biomedical implants, etc. (Teresa et al., 2005) applied to them to convert a mechanical force into stress electric by certain materials when mechanical is wearable biomedical implants, etc. (Teresasuch et al., led to thedevices, development of commercial products as polarization applied to them to convert a mechanical force into electric wearable devices, biomedical implants, etc. (Teresa et al., force (voltage). These materials, materials, convert the mechanical strainaasmechanical acalled resultpiezoelectric of the pressure applied 2005) force (voltage). These materials, called piezoelectric materials, applied to them to convert force into electric 2005) wearable devices, biomedical implants, etc. (Teresa et al., Energy harvesting is the conversion of ambient environmental convert force (voltage). These materials, called piezoelectric materials, 2005) the mechanical strain as result of applied on them into electricity. Conversely, piezoelectric materials convert the mechanical strain as aacalled resultpiezoelectric of the the pressure pressure applied force (voltage). These materials, materials, 2005) Energy harvesting is conversion of environmental energy into electrical energy. Wireless sensors are designed to on convert the mechanical strain as a result of the pressure applied Energy harvesting is the the conversion of ambient ambient environmental them into electricity. Conversely, piezoelectric materials can also convert a voltage applied across their terminals into on them into electricity. Conversely, piezoelectric materials convert the mechanical strain as a result of the pressure applied Energy harvesting isand the conversion ambient environmental energy electrical energy. sensors to operate remotely are Wireless mostlyof bydesigned batteries. on them intoforce electricity. Conversely, piezoelectric materials energy into into electrical energy. Wireless sensors are are designed to can also convert a voltage applied across their terminals into Energy harvesting is the conversion ofpowered ambient environmental mechanical (this is known as indirect piezoelectric canthem also convert a voltageConversely, applied across their terminals into into electricity. piezoelectric materials energy into electrical energy. sensors designed to on operate remotely and are mostly powered by batteries. Batteries, however, need to beWireless recharged everyare now and then. can also(Garimella convert a etvoltage applied across their terminals into operateinto remotely and are mostly powered bydesigned batteries. force (this is known as indirect piezoelectric energy electrical energy. Wireless sensors are to mechanical effect) al., 2015). mechanical force (this is known as indirect piezoelectric can also convert a voltage applied across their terminals into operate remotely and are mostly powered by batteries. Batteries, however, need to be recharged every now and then. They also require periodic replacements. These make the use mechanical force (this is known as indirect piezoelectric Batteries,remotely however, and need are to bemostly recharged every now then. effect) (Garimella et al., 2015). operate powered by and batteries. effect) (Garimella et al., 2015). mechanical force (this is known as indirect piezoelectric Batteries, however, need to be recharged every now and then. Among(Garimella the three,et al., converting They also require periodic These make the use of batteries to power devices every quitenow problematic, 2015). mechanical vibrations into They also however, require periodic replacements. These make thethen. use effect) Batteries, need remote toreplacements. be recharged and effect) (Garimella et al., 2015). mechanical the three, converting vibrations into They also require periodic replacements. These make the use Among electrical energy using piezoelectric transducers is the most of batteries to power remote devices quite problematic, especially in environments where human beings have limited Among the three, converting vibrations into of batteries to power remote devices These quite make problematic, They also require periodic replacements. the use Among the three, converting mechanical mechanical vibrations into electrical energy using piezoelectric transducers is the most of batteries to power remote devices quite problematic, preferred method because of their higher power densities, especially in environments where human beings have limited access (Ju et.al., 2018). electrical energy using piezoelectric transducers is the most Among the three, converting mechanical vibrations into especially in environments wheredevices human beings have limited electrical energy using piezoelectric transducers is the most of batteries to power remote quite problematic, preferred method because of their higher power densities, especially in environments where human beings have limited higher feasibility for practical applications and also they can access (Ju et.al., 2018). preferred method because of their higher power densities, electrical energy using piezoelectric transducers is the most access (Ju et.al., 2018). especially inbe environments where human beings have limited preferred method because of their higher power densities, Energy canet.al., harvested feasibility for practical applications and also they can access (Ju 2018). from a variety of sources including higher easily be produced at both macroscale and microscale using higher feasibility practical alsodensities, they can preferred method for because of applications their higher and power access (Ju et.al., 2018). solar/light Energy be of including thermal, electromagnetic higher feasibility practical applications and also theyusing can Energy can canmechanical, be harvested harvested from from aa variety varietyand of sources sources including easily be produced at both macroscale and microscale thick and thin filmfor deposition techniques (Erturk, 2009). easily be produced at both macroscale and microscale higher feasibility for practical applications and also theyusing can Energy can be harvested from a variety of sources including thermal, mechanical, solar/light and electromagnetic radiations. Mechanical energy harvesting involves converting easily be produced at both macroscale and microscale using thermal,canmechanical, solar/light electromagnetic and thin film deposition techniques (Erturk, 2009). Energy be harvested from a varietyand of sources including thick thick and thin film deposition techniques (Erturk, 2009). easily be produced at both macroscale and microscale using thermal, mechanical, solar/light and electromagnetic Energy harvesting using piezoelectric materials is a very radiations. energy harvesting converting the motionMechanical or displacement of a vibrating structure or thick and thin film deposition techniques (Erturk, 2009). radiations. Mechanical energy harvesting involves converting thermal, mechanical, solar/light andinvolves electromagnetic thick andharvesting thin film deposition techniques (Erturk, Energy using piezoelectric materials is radiations. Mechanical energy harvesting involves converting technology. However, vibration energy2009). harvesting, the motion or displacement of a vibrating structure or vibrations caused by air or water flow into electrical energy. Energy harvesting using piezoelectric materials is aa very very the motion or displacement of a vibrating structure or promising radiations. Mechanical energy harvesting involves converting Energy harvesting using piezoelectric materials is a very promising technology. However, vibration energy harvesting, the motion or displacement of a vibrating structure or in general, produces maximum power only when the frequency vibrations caused by air or water flow into electrical energy. Energy harvesting dueairtoor vibration is most abundant form promising technology. However, vibration energy harvesting, harvesting using piezoelectric materials is a very vibrations caused by water into electrical energy. the motion or displacement of flow a the vibrating structure or Energy promising technology. However, vibration energy harvesting, in produces maximum power only the frequency vibrations caused by air or water into electrical energy. the source matches the frequency of thewhen harvester. This in Energy harvesting due vibration is most abundant form of energy in the environment (Tangflow al., 2010). in general, general, produces maximum power only when theharvesting, frequency promising technology. However, vibration energy Energy harvesting dueairto toor vibration isetthe the most abundant form of vibrations caused by water flow into electrical energy. in general, produces maximum power only when the frequency the source matches the frequency of the harvester. This in Energy harvesting due to vibration isetthe most abundant form of effect limits the amount of power generated since the natural of energy in the environment (Tang al., 2010). of general, the source matches the frequency thewhen harvester. This in produces maximum power of only the frequency of energy in the environment (Tangisetthe al.,most 2010). Energy harvesting due to vibration abundant form in of the limits sourcethe matches theoffrequency of the harvester. This in of energy in the environment (Tang et al., 2010). effect amount power generated since the natural effect limits the amount of power generated since the natural of the source matches the frequency of the harvester. This in of energy in the environment (Tang et al., 2010). effect limits the amount of power generated since the natural Copyright@ 2019 IFAC 394Hosting effect limits the amount power generated since the natural 2405-8963 © 2019, IFAC (International Federation of Automatic Control) by Elsevier Ltd. All of rights reserved. Copyright@ 2019 394 Peer review under responsibility of International Federation of Automatic Copyright@ 2019 IFAC IFAC 394Control. 10.1016/j.ifacol.2019.10.063 Copyright@ 2019 IFAC 394 Copyright@ 2019 IFAC 394
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frequency of vibrations is dynamic, requiring constant controlling of the frequency of the harvester to resonate with that of the source of excitation for maximum power production. Currently, two different techniques have been proposed by Zhu et al., 2010 to solve this limitation. The first is to tune the resonant frequency of a single generator periodically so that it matches the excitation frequency at all times. This, however, can be used to tune the harvester to respond to only one frequency at a time making it problematic since practical systems with frequency tuning require actuators or controllers to do the tuning. The power that will be expended by the actuators can defeat the purpose of the energy harvesting in environments where there are large variations in the excitation frequency. The second solution is to design the generator with a wider bandwidth. Current techniques in bandwidth widening involve complex mathematical models to describe their behavior making their implementation complex (Zhu et al., 2010).
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are non-magnetic, organic or aqueous liquid and silicone or mineral oils. Upon the application of the magnetic field, the magnetizable particles acquire a dipole moment which aligns with the external magnetic field and forms. The formation of these chains induces reversible yield stress in the fluid (Choi et al., 2012). The yield stress increases the stiffness of the system and enables frequency tuning and damping effect. This phenomenon is illustrated in Fig. 1 and Fig. 2.
Fig. 1 Non-activated magnetic field (Djokoto et al., 2019)
Mechanical frequency tuning is based on the well-known resonant frequency of a spring-mass system which relates the spring constant to the mass of the harvester (Zhu et al., 2010). The stiffness of the system depends on the dimensions of the spring-mass system. Thus, by altering any of the mechanical properties of the energy harvester, its frequency can be tuned.
Fig. 2 Direction of activated magnetic field magnetic (Djokoto et al., 2019)
In this article, an impact-based frequency enhancement with variable damping using a vibrating bimorph piezoelectric energy generator is proposed. In particular, a novel type of impacting object, which is an activated MR fluid is investigated. The MR fluid changes from a fluid state to a semi-solid state when activated due to its rheological properties. The goal is to design an energy generator with the capabilities of producing 2.5mW using a varying magnetic field applied to the MR fluid.
MR fluid-based devices are considered excellent dampers because of their long-range controllable damping force, fast adjustable response, and low energy consumption (Zhu et al., 2012). MR dampers have been used in many engineering applications such as landing gears for airplanes to absorb the impact energy and to minimize the impact force transmitted to the airplane due to drop impact at landing (Kim et al., 2018), high-speed train suspension system to improve the ride comfort of passengers (Ramasastry et al., 2016), seismic vibration control of engineering structures (Koo, 2003), gun recoil application (Singh et al., 2014), washing machines (Ulasyar et al., 2018), etc..
The model presented in this work is based on the squeeze mode of the MR fluid. This model has been analytically simulated and experimentally tested by Kaluvan et al., in the design of a resonant-based measurement system using a piezolaminated cantilever beam coupled to an electromagnetic coil. The shift in resonant frequency due to the change in viscosity of the MR fluid was measured and shown to be related to the yield stress of the fluid. This work, however, focusses on the impact of the damping force due to the MR fluid when used as an impacting object in piezoelectric energy harvesting.
Until recently, where there has been much emphasis on designing autonomous microsystems which harvest energy from their natural environment, the vibrational energy absorbed by dampers has always been dissipated in the environment because of the absence of viable harvesting mechanisms.
1.1 Viscous damping MR fluid MR fluids and electrorheological (ER) fluids are part of a special group of fluids termed smart fluids whose viscosity can change significantly by the application of a suitable stimulus (Stanway, 2004). MR fluids have the special property that their viscosity or resistance to flow can be changed from liquids to semi-solids by the application of a magnetic field. The major distinguishing feature between MR and ER fluids is the type of stimulus applied to them. MR fluids have attracted much more attention from the research community because they do not require high voltage to produce high damping force as in the case of ER fluids.
2. THEORY/METHODOLOGY The proposed energy harvester is shown in Fig.3. The system consists of a vibrating piezoelectric bimorph cantilever with an attached mass, two permanent magnets one serving as the end mass and the other attached to the bottom of the ferromagnetic fluid holder. The magnets are mounted on opposite sides of the beam with the MRF acting as the impact object attached to one of the magnets. The piezoelectric cantilever beam used consists of a shim layer sandwiched in between two piezoceramic layers from Johnson Matthey Piezo Products GmbH as shown in Fig. 4.
MR fluids are made up of magnetizable particles (such as pure iron, carbonyl iron, or cobalt power) and carrier fluids which 395
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𝑓𝑓(𝑡𝑡) = 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹(𝜔𝜔𝜔𝜔)
(2)
Where 𝐹𝐹 is the amplitude of the external force. The rheological properties of the MR fluid due to the intensity of the magnetic field determines the effective stiffness of the beam at the point of impact. At the point of impact with the MRF, the stiffness of the vibrating cantilever beam is increased by 𝐾𝐾2 . From this stiffness effect, the vibration of the cantilever is transformed from linear oscillations to nonlinear impact oscillations as a result of the displacement constraints given by the MR fluid stopper. The additional stiffness introduced by the MR fluid is nonlinear due to the nonlinear relationship between the tip displacement and the stiffness induced by the fluid. Fig. 3 Schematic diagram of the proposed variable-damping MR fluid energy harvester
In the study of the energy harvester using the MR fluid as an impact object, the most important parameter of the fluid to consider is the magnetic field induced yield stress. In this study, the MR fluid, 140-CG, was used (Lord Products, 2014). The properties of the fluid are governed by two empirical equations (Carlson, 2008). Equation (3) gives the magnitude of the induced yield stress, 𝜏𝜏(𝐻𝐻), as a function of the applied magnetic field intensity 𝐻𝐻 , whiles equation (4) gives the relationship between the magnetic flux density, 𝐵𝐵 and the applied field intensity in the fluid. 𝜏𝜏(𝐻𝐻) = C × 271700 × Φ1.5239 × tanh(6.33 × 10−6 × 𝐻𝐻) (3) (4) 𝐵𝐵 = 1.91 × Φ1.133 (1 − 𝑒𝑒 −10.97𝜇𝜇𝑜𝑜 𝐻𝐻 ) + 𝜇𝜇𝑜𝑜 𝐻𝐻
Fig. 4 Bimorph (Piezo-Bending Actuators 427.00085.11Z)
2.1 System Modelling
Where C is a coefficient dependent on the carrier fluid of the MRF and Φ is the particle volume fraction of the fluid.
The piezoelectric energy harvester is modeled as a single degree of freedom (SDOF) spring-mass-damper model to illustrate the behavior of the MR fluid energy harvester. The SDOF system is shown in Fig. 5.
The stiffness due to the fluid and squeeze mode damping coefficient depend on the force generated upon impact and the yield stress. The force generated due to the magnetic field exerted on the fluid is given by (Brigley et al., 2008) and (Kaluvan et al., 2014): 4𝜋𝜋𝑟𝑟 3 𝜏𝜏
𝐹𝐹𝑀𝑀𝑀𝑀𝑀𝑀 =
(5)
3(𝑑𝑑1 +𝑍𝑍)
Where r is the radius of the magnet, 𝑑𝑑1 is the initial gap between the beam and the fluid and z is the displacement of the beam. The magnitude of the stiffness due to the MR fluid in squeeze mode is calculated from equation (6) as shown below (Kaluvan et al., 2014): 𝐾𝐾2 =
Fig. 5 SDOF model of the energy harvesting system 𝑚𝑚𝑒𝑒𝑒𝑒𝑒𝑒 𝑍𝑍̈ + (𝐶𝐶1 + 𝐶𝐶2 )𝑍𝑍̇+(𝐾𝐾1 + 𝐾𝐾2 )𝑍𝑍 = 𝑓𝑓 (1) Where, 𝑚𝑚𝑒𝑒𝑒𝑒𝑒𝑒 , is the effective mass (which is equal to the mass of the cantilever beam and the tip mass), 𝐶𝐶1 is the damping coefficient due to the cantilever beam, 𝐶𝐶2 is the additional damping due to the MRF. 𝐾𝐾1 is the stiffness of the beam and 𝐾𝐾2 is the nonlinear stiffness introduced by the MRF. The external force is denoted as 𝑓𝑓. Z is the tip displacement of the cantilever beam at a distance 𝑑𝑑1 from the fixed end as shown in Fig 5.
𝜕𝜕𝐹𝐹𝑀𝑀𝑀𝑀𝑀𝑀 𝜕𝜕ℎ
=
4𝜋𝜋𝑟𝑟 3 𝜏𝜏
(6)
3(𝑑𝑑1 +𝑍𝑍)2
The stiffness of the cantilever beam alone is given by (Roundy et al., 2003): 𝐾𝐾1 =
3𝐸𝐸𝐸𝐸 𝑙𝑙𝑏𝑏 3
(7)
Where 𝐸𝐸 is Young’s modulus of the piezoelectric beam, 𝐼𝐼 is the moment of inertia and 𝑙𝑙𝑏𝑏 is the free vibrating length of the beam. The moment of inertia is given by (Beer et al., 2006): 𝐼𝐼 =
𝑤𝑤
12
𝐸𝐸𝑠𝑠ℎ
(
𝐸𝐸𝑝𝑝
𝑡𝑡𝑠𝑠ℎ 3 + 2𝑡𝑡𝑝𝑝 3 ) +
𝑤𝑤𝑤𝑤𝑝𝑝 2
(𝑡𝑡𝑠𝑠ℎ + 𝑡𝑡𝑝𝑝 )2
(8)
Where w is the width of the beam, 𝐸𝐸𝑠𝑠ℎ is the modulus of elasticity of the shim layer, 𝐸𝐸𝑝𝑝 is Young’s modulus of the
Under a harmonic excitation, the force due to the excitation is given by:
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piezoelectric material, 𝑡𝑡𝑠𝑠ℎ is the thickness of the shim layer, and 𝑡𝑡𝑝𝑝 is the thickness of piezoelectric material.
The cantilever is excited at 2.5 𝑚𝑚/𝑠𝑠 2 . The effective damping ratio is determined from the log decrement plot as 𝛿𝛿⁄2𝜋𝜋. Where 1 𝑍𝑍 𝛿𝛿 is given as log ( 𝑛𝑛 ) and n the number of cycles.
𝜔𝜔𝑛𝑛 = √
3. RESULTS AND DISCUSSION
(9)
𝑚𝑚𝑒𝑒𝑒𝑒𝑒𝑒
The squeeze mode yield stress of the energy harvester was determined analytically from the empirical formulas of the MR fluid under the application of different values of magnetic field density (0.03T, 0.06T, and 0.10T). The magnetic field intensity was calculated using Equation (4) for the different values of field density. The magnetic field intensity values determined were then used to compute the yield stress of the fluid in Equation (3). The properties and dimensions of the energy harvester are shown in Table 2 in the appendix. The properties of the fluid and the piezoelectric bimorph are as defined by the manufacturers.
Where 𝐾𝐾𝑒𝑒𝑒𝑒𝑒𝑒 is the effective stiffness of the generator given as the sum of the 𝐾𝐾1 and 𝐾𝐾2 and 𝑚𝑚𝑒𝑒𝑒𝑒𝑒𝑒 is the effective mass of the energy generator. Mechanical vibration energy is harvested through damping. The damping ratio of the system defines the mechanical dissipation due to the fluid squeeze force. The damping effect as a result of the squeeze force of the fluid is given as (Kaluvan et al., 2014): 𝐶𝐶2 =
3 𝜋𝜋𝜋𝜋𝑟𝑟 4
The effect of the thin film MR fluid on the nonlinear energy harvester is studied by varying the magnetic field applied to the fluid. Mechanical energy due to vibration is merely harvested through damping.
(10)
2 (ℎ+𝑍𝑍)3
Where 𝜂𝜂 is the viscosity of the fluid. The damping coefficient of the piezoelectric cantilever beam, on the other hand, is given as (Ali et. al., 2012): 𝐶𝐶1 = 2 × 𝜁𝜁 × 𝑚𝑚𝑒𝑒𝑒𝑒𝑒𝑒 × 𝜔𝜔𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏
It is observed from Fig. 6 that, as the magnetic field is increased, the additional damping effect due to the fluid causes the bandwidth of the harvester to decrease, however, the power output increases. Bandwidth, as used in this paper, is the 3-dB bandwidth, and it is defined as the frequency range over which the spectrum of the power response is greater than or equal to half of the maximum power harvested (Cammarano et al., 2014). The reduction in bandwidth is due to the fact that the effective damping ratio decreases and as a result the quality factor increases. This, in turn, results in bandwidth reduction as can be seen from the equation below:
(11)
Where 𝜁𝜁 is the damping ratio of the free vibration beam. When the external excitation is harmonic, the net electrical power generated can be calculated using (Williams et al., 1996): 𝑃𝑃 =
𝜔𝜔 3 𝑛𝑛 2 𝜔𝜔 2 𝜔𝜔 2 [1−( ) ] +[2𝜁𝜁 ] 𝜔𝜔𝑛𝑛 𝜔𝜔𝑛𝑛
𝑚𝑚𝑚𝑚𝑌𝑌 2 (𝜔𝜔 ) 𝜔𝜔3
(12)
𝑄𝑄 =
From equation (14), it is clear that maximum power is obtained when the excitation frequency is equal to the frequency of the energy harvester. Therefore, 𝑃𝑃𝑚𝑚𝑚𝑚𝑚𝑚 =
𝑚𝑚𝑌𝑌 2 𝜔𝜔3
1 2ζ
=
𝜔𝜔 𝐵𝐵𝐵𝐵
(16)
Where Q is the quality factor and BW is the bandwidth.
(13)
4𝜁𝜁
𝑍𝑍𝑛𝑛+1
𝑛𝑛
The resonant frequency of the energy harvester is, therefore, given as: 𝐾𝐾𝑒𝑒𝑒𝑒𝑒𝑒
397
The results of the damping ratio as the fluid is applied and by varying the magnetic field are shown in Table 1.
To determine the displacement amplitude when the cantilever is subjected to harmonic excitation, equation (3) is numerically simulated using MATLAB ode45 by defining the following state model: 𝑍𝑍̇ = 𝐴𝐴𝐴𝐴 + 𝐵𝐵𝐵𝐵 ,
𝑌𝑌 = 𝐶𝐶𝐶𝐶
(14)
A represents the system matrix, B is the input matrix, C is the output matrix, X is the state vector, u is the input excitation and Y is the system output. The state space form of equation (3) is: 0 𝐴𝐴 = [−(𝐾𝐾1+𝐾𝐾2) 𝑚𝑚𝑒𝑒𝑒𝑒𝑒𝑒
𝐵𝐵 = [ 𝑢𝑢 = 𝑓𝑓
0
1
𝑚𝑚𝑒𝑒𝑒𝑒𝑒𝑒
1
−(𝐶𝐶1 +𝐶𝐶2 )]
Fig. 6 Output power Vs Frequency of the energy harvester at different magnetic field densities.
𝑚𝑚𝑒𝑒𝑒𝑒𝑒𝑒
], 𝐶𝐶 = [0
𝑍𝑍 1], 𝑋𝑋 = [𝑍𝑍1 ] 2
Table 1: Summary of results Magnetic field density (T) 0 0.03 0.06 0.1 397
Damping ratio
Frequency (Hz)
0.02 0.0143 0.0044 0.0014
128 147 164 186
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3.1 Displacement Response of the Energy Harvester
4. CONCLUSIONS
The displacement responses of the energy harvesting with/without the application of the fluid are as shown below. As expected the amplitude of vibration of the system decreases as the magnetic field is increased and this can be attributed to the extra damping effect added to the system due to the application of the fluid.
The response of a vibration energy harvester due to the application of a thin film MR fluid has been analyzed in this paper. Using the thin film MR fluid in frequency tuning is also shown. The main intent of this paper was to investigate the impact of the variable damping due to the MRF on the frequency response and also the power output of the nonlinear energy harvester. It has been demonstrated that the power bandwidth of the energy harvester decreases as a result of the MR fluid induced damping. However, the power output increases due to the increase in the quality factor of the energy harvester. Using the fluid in frequency tuning of vibration energy harvesting has been demonstrated to be beneficial especially when fine-tuning is required to closely match the frequency of the source. The results of the simulation show a decrease in damping ratios from 0.02 without fluid to 0.0014 upon the application of the fluid with a magnetic field density of 0.1T.
Fig. 7 Displacement response without fluid
Though the initial goal of attaining a power output of 2.5mW could not be achieved, the impact of the additional damping force added to the system has been shown. Using the fluid to aid frequency tuning in energy harvesting is also highlighted. REFERENCES Ali W. G. and Ibrahim S. W. (2012) "Power Analysis for Piezoelectric Energy Harvester" Energy Power Eng., vol. 04, no. 06, pp. 496–505. Beards C. F. and Arnold E. (2013) Engineering vibration analysis with application to control systems, vol. 33, no.09. Beer P. F., Johnson E. R. and Dewolf T. J. (2006) Mechanics of Materials, 4th ed. New York: McGraw Hill. Blevins R. D. (1979) Formulas-for-Natural-Frequency-andMode-Shape, Van Nostrand Reinhold, New York Brigley M, Choi Y-T, and Wereley N. M. (2008) "Experimental and Theoretical Development of Multiple Fluid Mode Magnetorheological Isolators". J Guid Control Dyn ;31:449–59. doi:10.2514/1.32969. Boisseau S, Despesse G, and Ahmed B. (2012) "Electrostatic Conversion for Vibration Energy Harvesting". SmallScale Energy Harvest:1–39. doi:10.5772/51360. Cammarano A, Neild, S. A Burrow S. G, and Inman D. J (2014) "The bandwidth of optimized nonlinear vibrationbased energy harvesters" Smart Mater. Struct. 23 055019 (9pp) Carlson J. D. (2008) Magnetorheological fluids, in Smart Materials. New York: Taylor & Francis. Cheng S, and Arnold DP. (2010) "A study of a multi-pole magnetic generator for low-frequency vibrational energy harvesting". J Micromechanics & Microengineering; 20. doi:10.1088/0960-1317/20/2/025015. Choi S.-B. and Han Y.-M. (2012) Magnetorheological Fluid Technology: Applications in Vehicle Systems. CRS Press Taylor & Francis Group "Data Sheet Piezoceramic Trimorph Bending Actuator, Part No. 427.0085.11Z, Johnson Matthey Piezoproducts Bahnhofstraße 43, D- 96254 Redwitz," 2011 Djokoto S. S., Agelin-Chaab M., Jūrėnas V. and Dragašius E. (2019) "Experimental Investigation of Squeezed MRF
Fig. 8 Displacement response with a magnetic field density of 0.03 T
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399
Williams C. B. and Yates R. B. (1996) "Analysis of a microelectric generator for Microsystems", Science (80-.)., vol. 52, pp. 8–11. Zhu D., Tudor M. J., and Beeby S. P., (2010) "Strategies for increasing the operating frequency range of vibration energy harvesters: A review", Meas. Sci. Technol., vol. 21, no. 2. Zhu X., Jing X., and Cheng L., (2012) "Magnetorheological fluid dampers: A review on structure design and analysis", Journal of Intelligent Material Systems and Structures, vol. 23, no. 8. pp. 839–873. APPENDIX A. Table 2: System properties and dimensions Symbol l lb w t tp tsh ρp ρsh 𝜀𝜀33 Esh
Φ µ0 𝜂𝜂 r h
C
mt
399
Description Total length of beam Free (vibrating) length Beam width Beam thickness Thickness of piezoelectric layer Shim layer thickness Piezoelectric layer density Shim layer density Piezoelectric dielectric constant Shim layer modulus of elasticity Particle volume fraction Permeability of free space Viscosity of MRF Radius of magnet Initial gap between MRF and Beam Coefficient dependent on the carrier fluid of MRF Tip mass
Value 49.95
Unit mm
29.5
mm
7.2 ± 0.05 0.78 ± 0.03 0.25 ± 0.05
mm mm mm
0.28 ± 0.05
mm
8000
Kg/m3
1800 30975e-12
Kg/m3 F/m
120e9
N/m2
0.4 1.25e-6
H/m
0.3 3.25 0.5
Pa·s mm mm
1
0.003
kg