THEORY AND SIMULATION OF ELECTROMAGNETIC DAMPERS FOR EARTHQUAKE ENGINEERING APPLICATIONS by Siavash Haji Akbari Fini A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Civil Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) February 2016 © Siavash Haji Akbari Fini, 2016 ii Abstract The present study develops applications of electromagnetic devices in Civil Engineering. Three different types of electromagnetic system are investigated through mathematical and numerical models. Chapter 3 deals with Coil-Based Electromagnetic Damper (CBED). CBEDs can operate as passive, semi-active and active systems. They can also be considered as energy harvesting systems. However, results show that CBEDs cannot simultaneously perform as an energy harvesting and vibration control system. In order to assess the maximum capacity of CBEDs, an optimization is conducted. Results show that CBEDs can produce high damping density only when they are considered as a passive vibration control system. Chapter 4 deals with the development of a novel Eddy Current Damper (ECD). The eddy current damper uses permanent magnets arranged in a circular manner to create a strong magnetic field, where specially shaped conductive plates are placed between the permanent magnets to cut through the magnetic fields. Detailed analytical equations are derived and verified using the finite element analysis program Flux. The verified analytical models are used to optimize the damper design to reach the maximum damping capacity. The analytical simulation shows that the proposed eddy current damper can provide a high damping density up to 2,733 kN-s/m4. The Hybrid Electromagnetic Damper (HEMD) are developed and designed in Chapter 5. The idea is to couple the CBED and ECD with the aim of designing a semi-active, active and energy harvesting electromagnetic damper. The simulation results show that it is feasible to manufacture hybrid electromagnetic dampers for industrial applications. iii Preface The author of this thesis is responsible for the literature review, model development computational analysis, data processing, and result presentation. iv Table of contents Abstract .................................................................................................................................... ii Preface ................................................................................................................................... iii Table of contents .................................................................................................................... iv List of tables ........................................................................................................................... vii List of figures ........................................................................................................................ viii Acknowledgments ....................................................................................................................x Dedication ............................................................................................................................... xi 1. Introduction ..........................................................................................................................1 1.1 Introduction ...................................................................................................................1 1.2 Literature review ...........................................................................................................4 1.3 Scope of thesis ...............................................................................................................5 1.4 Overview of thesis .........................................................................................................6 2. Review of electromagnetic theory .......................................................................................7 2.1 Introduction ...................................................................................................................7 2.2 Electromagnetic theories ...............................................................................................7 2.2.1 Maxwell’s equations ..............................................................................................7 2.2.2 Lorentz’s law ..........................................................................................................9 2.2.3 Ohm’s law ..............................................................................................................9 2.2.4 Constitutive equation of permanent magnet ..........................................................9 2.3 Simplified electromagnetic damper .............................................................................11 2.4 Conclusion ...................................................................................................................13 3. Coil-based electromagnetic damper .................................................................................14 3.1 Introduction .................................................................................................................14 v 3.2 Mathematical model for coil-based electromagnetic damper .....................................16 3.2.1 Operation point of permanent magnets ................................................................16 3.2.2 Induced voltage ....................................................................................................19 3.2.3 Damping force ......................................................................................................21 3.2.4 Passive, semi-active and active CBED ................................................................22 3.3 Finite element analysis ................................................................................................25 3.4 Optimization procedure ...............................................................................................26 3.5 Conclusion ...................................................................................................................34 4. Development of eddy current damper ..............................................................................35 4.1 Introduction .................................................................................................................35 4.2 Analytical model of the proposed eddy current damper .............................................37 4.2.1 Magnetic flux density produced by an arc segment permanent magnet ..............37 4.2.2 Eddy current damping force calculation ..............................................................39 4.3 Validation of the analytical model using finite element analysis ................................46 4.4 Optimization ................................................................................................................48 4.5 Thermal analysis ..........................................................................................................50 4.6 Conclusion ...................................................................................................................56 5. Electromagnetic dampers with applications in structural engineering ........................57 5.1 Introduction .................................................................................................................57 5.2 Coil-based electromagnetic damper ............................................................................59 5.3 Eddy current damper ...................................................................................................61 5.4 Hybrid electromagnetic damper ..................................................................................62 5.5 Conclusion ...................................................................................................................65 6. Summary and conclusion ...................................................................................................66 6.1 Conclusion ...................................................................................................................66 vi 6.2 Suggestions for future research ...................................................................................68 Bibliography ...........................................................................................................................69 Appendices ..............................................................................................................................73 Appendix A: Analytical expression for magnetic potential ..................................................73 Appendix B: Details of finite element model .......................................................................74 vii List of tables Table 3-1. Material properties. .................................................................................................................................... 19 Table 3-2. Damper parameters for the finite element analysis model. ........................................................................ 25 Table 3-3. Optimum parameters for the CBED. .......................................................................................................... 30 Table 4-1. Device parameters. ..................................................................................................................................... 50 Table 4-2. Optimal design values. ............................................................................................................................... 50 Table 4-3. Flow and thermal conditions. ..................................................................................................................... 55 Table 5-1. Optimum design parameters for the CBED. ............................................................................................... 60 Table 5-2. Comparison between bench mark damper and CBED. .............................................................................. 61 Table 5-3. Optimum parameters for ECD. .................................................................................................................. 61 Table 5-4. Comparison between CBED and ECD. ...................................................................................................... 62 Table 5-5. HEMD parameters. ..................................................................................................................................... 64 Table 5-6. Comparison between developed dampers. ................................................................................................. 65 Table 5-7. A summary of the developed electromagnetic dampers. ............................................................................ 67 viii List of figures Figure 1-1. Different typs of electromagetic dampers: (a) CBED, (b) ECD. ................................................................ 3 Figure 2-1. Conductive wire and magnetic field: (a). 3D view, (b). Section view. ....................................................... 8 Figure 2-2. Electrical field induced in a conductive wire. ............................................................................................. 8 Figure 2-3. Lorentz’s law. ............................................................................................................................................. 9 Figure 2-4. The relationship between magnetic field and magnetic flux inside a permanent magnet. ........................ 11 Figure 2-5. Simplified electromagnetic damper. ......................................................................................................... 12 Figure 3-1. Coil-based electromagnetic damper: (a) Entire damper, (b) Stationary part, (c) Mover. .......................... 15 Figure 3-2. The two pole CBED: (a). 3D view, (b). Section view. .............................................................................. 17 Figure 3-3. Faraday’s law for the two pole CBED: (a). Two pole machine, (b). Magnetic flux. ................................ 20 Figure 3-4. Cross section of a coil. .............................................................................................................................. 21 Figure 3-5. CBED: (a). Passive mode, (b). Semi-active mode, (c). Active mode. ...................................................... 23 Figure 3-6. Force-displacement relationship for a typical passive CBED. .................................................................. 24 Figure 3-7. Mathematical and FEM results for the magnetic flux density inside a magnet......................................... 26 Figure 3-8. Mathematical and FEM results for the magnetic flux density inside a coil. ............................................. 26 Figure 3-9. The two pole system: (a). Two pole system with three coils, (b). Equivalent one coil system, (c). Active coil wire. ............................................................................................................................................................ 29 Figure 3-10. Optimization procedure........................................................................................................................... 31 Figure 3-11. Penalty value for genetic algorithm. ....................................................................................................... 31 Figure 3-12. Additional resistance effects on CBED. .................................................................................................. 33 Figure 3-13. A four phase CBED. ............................................................................................................................... 34 Figure 4-1. ECD configurations: (a). Conventional ECD (Ebrahimi et al., 2008), (b). Proposed ECD. ..................... 37 Figure 4-2. An arc segment of the permanent magnet. ................................................................................................ 38 Figure 4-3. Conductive plate subjected to one row of permanent magnets on both sides. .......................................... 39 Figure 4-4. Y- Component of magnetic flux at the mid-plane of a sample conductive plate. ...................................... 41 Figure 4-5. Illustration of the pole projection area when a conductive plate is subjected to a uniform magnetic flux. ........................................................................................................................................................................... 42 Figure 4-6. Eddy current density over the conductive plate with finite length. ........................................................... 44 Figure 4-7. A conductive plate subjected to two magnetic fields: (a). 3D view, (b). Mathematical model. ................ 45 Figure 4-8. Eddy currents in a conductive plate. ......................................................................................................... 46 Figure 4-9. Sample comparison of the magnetic flux in the azimuthal and radial directions calculated using analytical and FEM simulations. ........................................................................................................................ 47 Figure 4-10. Plots of velocity versus time and power loss versus time for the ECD under the dynamic load. ........... 48 Figure 4-11. Convergence of genetic algorithm. ......................................................................................................... 49 Figure 4-12. Forced convection internal Flow, (a). 3D view, (b). Cross section. ........................................................ 51 Figure 4-13. Equivalent hydraulic circular section for infinite channel. ..................................................................... 53 ix Figure 5-1. Hybrid electromagnetic damper. ............................................................................................................... 59 Figure 5-2. Four phase CBED. .................................................................................................................................... 60 x Acknowledgments I would like to express my deep gratitude to my survivors Professor Tony Yang and Professor Kang Li for their valuable support and enthusiastic encouragement. Their understanding and willingness to dedicate their time so generously has been a constant motivation for me to tackle all obstacles and achieve the standards of the academic research. It is my pleasure to thank the members of the Smart Structure Research Group at UBC and my friends and colleagues, who helped me with their comments and resources. My thanks also go to all the wonderful staff of Graduate Office of Civil Engineering Department of UBC. Last but not least, I would like to thank my parents who gave me their true love and support in every step of my way. xi Dedication To my beloved parents for their unconditional love and supports & In memorial of my best friend, Sina who I missed his smile forever 1 1. Introduction 1.1 Introduction In conventional seismic design, the sudden surge of the earthquake energy is dissipated through yielding of the structural elements. This result a significant damage to the structure which leads to hefty financial losses and prolong repair time. In recent years, considerable attentions have been paid to research and development of structural control systems that can improve the performance of structure under extreme earthquake loads. Vibration control systems can be divided into four groups: 1. Passive 2. Active 3. Semi-active 4. Hybrid Passive control systems are realized by adding energy dissipating devices and/or isolating the structures from earthquake ground motion. Passive systems do not require an external power source. Such systems are widely used because of their simplicity and reliability. However, they are effective only over a limited operating range dictated by the tuning condition of their parameters. An active control system is defined as a system which typically requires a large power source for operation. Control forces are developed based on feedback from sensors that measure the excitation and/or the response of the structure. By using the active vibration control, the dynamic characteristic of a system can be altered, hence a greater reduction in vibration levels can be achieved. However, an incorrectly designed active control system can lead to an increased level of vibration in the controlled system. 2 A semi-active control system requires a small external power source for operation. This system utilizes the motion of the structure to develop the control forces. The magnitude of forces can be adjusted by the external power source. Similar to active control systems, their mechanical properties are typically adjusted based on measured feedback from the structural. The three major classes of control systems described above can be combined to form hybrid control systems. Hybrid control systems consist of combined passive and active devices or combined passive and semi-active, and have been designed to take advantages of passive, active and semi-active control systems. Electromagnetic damper is known as one of the most recent developed vibration control system. Electromagnetic dampers can operate as passive, semi-active and active systems, or can function in combination with other devices as hybrid systems. In general, there are two types of electromagnetic dampers (see Figure 1-1), Coil-Based Electromagnetic Damper (CBED) and Eddy Current Dampers (ECD). In both systems, electrical current and the damping force are generated when relative motion happens between the permanent magnets and the conductive material. In CBED, the electrical current is diverted into the electrical coil and can be stored in a battery, while in the ECD the electrical current will appear in a swirling pattern in the conductive material. The swirling electrical current is commonly known as the eddy current, hence this type of damper is known as ECD. The following section presents a brief literature review of previous study. 3 Figure 1-1. Different typs of electromagetic dampers: (a) CBED, (b) ECD. 4 1.2 Literature review Applications of electromagnetic devises have been explored over the past decades. Coil-based electromagnetic systems can be employed as both micro scale generators and/or large scale dampers. Lu et al., (2005) developed a tubular electromagnetic micro-motor for robotic applications. Buren & Troster, (2007) designed a micro power generator to supply power for sensors. Ebrahimi et al., (2008) and Palomera-Arias et al., (2008) studied the application of CBEDs as a vibration control systems in vehicles and structures. Nakamura et al., (2014) have proposed a new type of CBED with rotating inertial mass to control the vibrations of structures. Wang & Hua, (2013) studied the application of the passive CBED for bridge stay cables through experimental tests. The application of CBEDs for energy harvesting has been recently investigated. Zhu et al., (2012) discussed the dynamic characteristic of CBED for both energy harvesting and damping force. In addition, Shen & Zhu, (2015) investigated the CBED for energy harvesting in a bridge stay cables. Zuo et al., (2010) also proposed a new type of electromagnetic energy harvester for vehicle suspension systems. Eddy current systems are employed for different applications. Lee & Park, (1999) applied the eddy current phenomena to propose a non-contact braking system. Bae et al., (2005) and Sodano et al., (2005) studied the application of an ECD in controlling the vibrations of small cantilever beam. Kienholz et al., (1996) used the eddy current systems as an isolation system for a space shuttle payload. Ebrahimi et al., (2008) proposed an ECD for the vehicle suspension systems. Wang et al., (2012) investigated the feasibility of using eddy current devices in tuned mass dampers (TMD). 5 Efforts have been made to improve the damping performance of ECDs. For instance, Zuo et al., (2011) and Ebrahimi et al., (2010) studied the influence of permanent magnets arrangements to increase the damping density of ECDs. Sodano et al., (2006) studied the influence of different type of magnetic flux cycle to improve the performance of ECDs. Hybrid electromagnetic damper (HEMD) is also investigated to take the advantages of both CBED and ECD. Gysen et al., (2010) utilized the coli spring and the electromanetic system to improve the vehicle suspension systems. Martins et al., (1999) studied the hybrid devices integrating the active and passive systems. 1.3 Scope of thesis In Chapter 3 the performance of CBEDs is analyzed by means of detailed mathematical and finite element method (FEM). A novel ECD with circumferential permanent magnets is proposed in Chapter 4. Detailed analytical model of the ECD is derived using electromagnetic theory under quasi-static condition. The dynamic response of ECD is also verified using FEM. In addition, a genetic algorithm (GA) is applied to optimize the CBED and ECD capacity. A HEMD is proposed in Chapter 5 to integrate the features of CBED and ECD. The contributions of the present study are summarized below: 1. Performance evaluation of CBEDs 2. Investigation on the optimum configuration and maximum capacity of CBEDs 3. Development of a novel ECD with high damping density 4. Quantifying the magnetic field produced by an arc segment permanent magnet 5. Investigation on the optimum configuration and maximum capacity of ECDs 6. Proposing the HEMD 6 1.4 Overview of thesis The material presented in the remainder of this thesis is organized into five chapters. Chapter 2 briefly reviews the electromagnetic theories. In Chapter 3, the CBEDs are studied in details through mathematical and numerical models. In addition, the maximum capacity and applications of CBEDs are discussed. In Chapter 4, a new type of ECD is developed. As will be shown, the proposed ECD can provide high damping force comparable to a viscous damper. In Chapter 5, the possible application of electromagnetic systems are evaluated and compared with the benchmark damper. The HEMD is also proposed in Chapter 5 to integrate feature of CBED and ECD. Finally, Chapter 6 summarizes the findings, and presents suggestions for future studies. 7 2. Review of electromagnetic theory 2.1 Introduction This chapter presents electromagnetic theories for the quasi-static magnetic field condition. The quasi-static condition describes a slowly varying magnetic field. This condition can be considered for the design of electromechanical devices which operate at low frequencies and relatively low velocities. For a better understanding of electromagnetic theories, the mathematical model for a simplified electromagnetic damper is also derived here. 2.2 Electromagnetic theories 2.2.1 Maxwell’s equations Equations (2-1) to (2-3) define the Maxwell’s equations under the quasi-static magnetic field condition (Furlani, 2001): . .C SH dl J ds Ampere’s law (2-1) . .C SBE dl dst Faraday’s law (2-2) . 0SB ds Continuity condition (2-3) Where H , J , E and B define the magnetic field strength, current density, electrical field and magnetic flux density, respectively. t is a partial derivation with respect to time. Ampere’s law explains the relationship between the magnetic field strength in space and the electrical current passing through a conductor. Figure 2-1 shows a conductive wire and the magnetic field produced by a wire. According to Ampere’s law, the amount of a magnetic field at an arbitrary line, for example contour C in Figure 2-1, is determined by integrating the current density over the surface which is defined by contour C (equation 2-1). 8 Figure 2-1. Conductive wire and magnetic field: (a). 3D view, (b). Section view. Faraday’s law determines the induced electrical field due to the time-varying magnetic field. When the amount of magnetic flux varies in a closed loop, for instance contour C in Figure 2-2, the electrical field will be generated in the conductor. The electrical field is defined by the integral over the contour C and the surface defined by C (equation 2-2). Figure 2-2. Electrical field induced in a conductive wire. 9 2.2.2 Lorentz’s law Lorentz’s law quantifies the amount of damping force. When a particle electrical charge, q, with the velocity v moves in the magnetic field, the particle will be subjected to the force. The force is equal to cross product of velocity and magnetic field as: ( )F q v B (2-4) Where F is known as the electromagnetic force. Figure 2-3 shows a conductive wire in a magnetic field. Integrating the cross product of current density and magnetic field over the wire volume, V, defines the electromagnetic force as: ( )VF J B dV Lorentz’s Law (2-5) Figure 2-3. Lorentz’s law. 2.2.3 Ohm’s law The Ohm’s law describes the relationship between the current density and electrical field in a conductive material as: J E (2-6) Where defines the electrical conductivity of material. 2.2.4 Constitutive equation of permanent magnet Following Haus et al., (2008), the constitutive equation for magnetic flux density inside a permanent magnet, Bm, is: 0 ( ) m mB H M (2-7) 10 Where 0 is the permeability of the free space, Hm is the magnetic field inside the permanent magnet and the M is the magnetization of the permanent magnet. Magnetization is a function of magnetic field, and whenever the magnetic field is removed, Hm=0, it will be equal to the initial magnetization as M0. Figure 2-4 shows the relationship between Bm and Hm. In other words, when permanent magnets are located in free space or in the domain with non-magnet material such as copper, the magnetic flux density inside the magnets is only proportional to M0 as: 0 0rB M (2-8) Where Br is the residual magnetic flux density defined in Figure 2-4. However, when permanent magnets operate in the presence of ferromagnetic material such as iron, the magnetic flux, Bm, cannot be defined by equation (2-7). In this condition, the magnets operate as a source of energy and magnetize the ferromagnetic material. As a consequence, the magnets are demagnetized and operate in the second quarter of Figure 2-4. It is more convenient to approximate the demagnetization curve using the straight line as: m r m mB B H (2-9) rmcBH (2-10) Where Br and Hc define the residual magnetic flux density and coercivity magnetic field as shown in Figure 2-4. The above-mentioned equation can be generalized to define the relationship between the magnetic field and magnetic flux density inside all materials using equations (2-11) and (2-12): B H (2-11) o r (2-12) 11 Where and r are the permeability and relative permeability of material. r for the non-magnet material is approximately equal to 1 and for the ferromagnetic material is much greater than unity. Figure 2-4. The relationship between magnetic field and magnetic flux inside a permanent magnet. 2.3 Simplified electromagnetic damper The electromagnetic theories are applied to derive a mathematical expression for a simplified electromagnetic damper. Figure 2-5 illustrates a simplified electromagnetic damper. The device consists of a conductive loop with the length and width of x and L, respectively. It is also assumed that the cross sectional area of the mover part is 1 m2. The system is subjected to a uniform magnetic flux density, B, as illustrated in Figure 2-5.The translation motion of the mover causes a variation in the magnetic flux inside the loop. Consequently, the current is induced in the conductors. According to Lorentz’s law, when the conductor carrying a current is located in a 12 magnetic field, it will be subjected to a force. The governing equations will be described in the following paragraphs. Figure 2-5. Simplified electromagnetic damper. The magnetic flux, TB passes the conductive loop is given by equation (2-13). .BT B dA (2-13) Where B defines the magnetic flux density over the loop as shown in Figure 2-5. Considering Figure 2-5, the total magnetic flux is equal to equation (2-14). BT BLx (2-14) According to Faraday’s law (equation 2-2), the variation of magnetic flux with respect to the time produces an electrical field as stated in equation (2-15). BTEt (2-15) 13 Substituting equation (2-14) into equation (2-15) the electrical force can be defined by equation (2-16). E BLx BLvt (2-16) Where v is the velocity. The current density, J, is defined using the Ohm’s law (equation 2-6) and equation (2-16) as: E BLvJR R (2-17) Where R (1/ ) represents the resistance of the conductive loop. The amount of damping force is quantified using Lorentz’s law (equation 2-5) as: ( ) VF J B dV (2-18) Where V is the mover volume. Substituting equation (2-17) into (2-18) the damping force for the simplified electromagnetic damper can be rewritten as: 2( )BLF vR (2-19) The damping coefficient, C, in this case clearly appears as: RBLC2)( (2-20) 2.4 Conclusion In this chapter, the fundamental electromagnetic theories are described under the quasi-static condition. In addition, a mathematical expression is derived for a simplified electromagnetic damper. In the following chapters the electromagnetic theories will be applied to investigate the CBED and ECD. 14 3. Coil-based electromagnetic damper 3.1 Introduction The loss of life and infrastructure damage from earthquakes around the world reveals the importance of understanding and controlling the structural responses. In the past decades, significant efforts have been devoted to developing new vibration control systems such as viscous and visco-elastic dampers (Samali & Kwok, 1995), buckling restrained brace systems (Yang et al., 2014) and MR dampers (Yang et al., 2002). In all common dampers, the vibration energy is dissipated by converting into heat. Coil-Based Electromagnetic Damper (CBED) is considered to be one of the most recent developed vibration control systems. CBEDs can be utilized as a damper or energy harvester system. The CBED converts the kinetic energy into electricity that can be stored in an external battery through an energy harvesting circuit attached to a damper. This chapter investigates the performance and optimum configuration of CBEDs. Figure 3-1 illustrates the machine configuration. In this configuration, the mover consists of axial permanent magnets while coils are stationary and located in the damper shell. The damping force is produced as a result of the relative movements of permanent magnets and coils. Furthermore, considering Faraday’s law (equation 2-2), the voltage is induced in the coils. This chapter is divided into two parts. The first part derives the mathematical model for EMDs. In general, three approaches are available to express the mathematical model: 1. Lumped equivalent magnetic circuit applied by Lu et al., (2005), Ebrahimi et al., (2008) and Palomera-Arias et al., (2008). 2. Numerical solution applied by Basak & Shirkoohi, (1990) and Zuo et al., (2010). 3. Analytical solution, separation of variables, which applied by Wang et al., (2004) and Tsai & Chiang, (2010). 15 The first method is utilized here to derive the mathematical model. The formulation is then verified by finite element analysis. In the second part, the genetic algorithm in Matlab toolbox (Mathworks, 2013) is used to investigate the maximum capacity of CBEDs using a verified mathematical model. Results show that the maximum achievable damping density for the CBED is 3,061 kN-s/m4. In addition, corresponding damper details for the optimum design configuration are provided. Figure 3-1. Coil-based electromagnetic damper: (a) Entire damper, (b) Stationary part, (c) Mover. 16 3.2 Mathematical model for coil-based electromagnetic damper 3.2.1 Operation point of permanent magnets The permanent magnets are used in the design of CBEDs. A magnetic circuit will be analyzed to calculate the damping force and induced voltage. Magnetic circuit is similar to electrical circuit, however in a magnetic circuit a permanent is used as a source of energy instead of a battery in an electrical circuit. To analyze a magnet circuit, the operation point of permanent magnets should be determined. The operation point defines the amount of energy produced by permanent magnets in a circuit. Figure 3-2 shows a two pole CBED with two active coils. An active coil is a coil which is subjected to the magnetic flux. The magnet circuit analysis is conducted for a two pole machine to determine the operation point of permanent magnets. It should be noted that the results can be utilized for a several pole device. Ampere’s law (equation 2-1) is applied over the magnetic flux closed loop as indicated in Figure 3-2(b). Since the active coils experience different magnetic fields both upward and downward, the direction of the currents inside the active coils run in opposite directions, which means the left side of Ampere’s law is zero. 17 Figure 3-2. The two pole CBED: (a). 3D view, (b). Section view. And the right side of Ampere’s law is: . 2 2 2m pole air coil shellC Magnet Pole Air Coil ShellH dl H dl H dl H dl H dl H dl (3-1) Where H and its subscript describes the magnetic field in various regions. It should be noted that the magnetic field is axial inside the magnet and shell and is radial inside the iron pole, air gap and coils. Using the consecutive law of material (equations 2-11 and 2-12) above equation can be modified as: 0 0 0 00. 2 2 2 ( )m a cm ar r rpole air coil shellm m m iFe cu FeC r rB B B BH dl H t dr dr dr t t (3-2) 18 Where B and its subscript describes the magnetic field in various regions. cu and Fe are the relative permeability of copper and iron. Table 3-1 shows the material properties used in this study. Also rm, ra, rc, tm and ti are the dimensions described in Figure 3-2(b). Using equation (2-9), Hm can be defined in terms of Bm: m rmmB BH (3-3) The continuity condition, equation (2-3), can be applied to determine the magnetic flux density in the different regions in terms of Bm: 02mm m pole pole pole mirBA B A B B r rt (3-4) 2 2m mm m air air air m air BA B A B B r r rrt (3-5) 2 2m mm m coil coil coil a cir BA B A B B r r rrt (3-6) 22 2 ( )m mm m shell shell shells cr BA B A B Br r (3-7) The integrals in equation (3-2) can be calculated as: 20 004mrpole mm pole mFe Fe iB rdr B k Bt (3-8) 20 0ln2amrair m am air mi mrB r rdr B k Bt r (3-9) 20 0ln2carcoil m cm coil mcu cu i arB r rdr B k Bt r (3-10) 22 20 0( )( )shell mm i m shell mFe s c FeB rt t B k Br r (3-11) 19 Substituting equations (3-8) to (3-11) in equation (3-2) the operation point of permanent magnets is derived as: (2 2 2 )r mmm m pole air coil shellB tBt k k k k (3-12) The derived Bm defines the amount of magnetic flux produces by a permanent magnet in a CBED. Table 3-1. Material properties. Parameters Description Values 0 Permeability of Free Space 74 10 cu Relative Permeability of Copper 1 Fe Relative Permeability of Iron 2000 rB Residual Magnetic Flux Density 1.2 cH Coercivity Magnetic Field 3900 10 Electrical Conductivity 75.65 10 3.2.2 Induced voltage Faraday’s law (equation 2-2) is applied to determine the induced voltage in the CBED shown in Figure 3-3(a). Equation (3-13) shows the left side of Faraday’s law for a single arbitrary wire at the (x, r) position in the active coil region shown in Figure 3-3(b). B . 2 2 coil coil coil a cdds B rx B rv r r rt dt (3-13) Where Bcoil is defined in terms of Bm using equation (3-6) and v is the velocity. The right side of Faraday’s law describes the induced voltage. The induced voltage for a single wire, Ewire, can be stated by substituting equation (3-6) in equation (3-13) as: 20 2 m mwireir BE vt (3-14) As a result, for the Np active coli and Na active wire turns in each coil the total induced voltage, EEM is: 2 m mEM p air BE N N vt (3-15) Figure 3-3. Faraday’s law for the two pole CBED: (a). Two pole machine, (b). Magnetic flux. The resistant of the coil should also be determined to define the current in the electromagnetic damper. The resistance of a wire, R, with a length and cross-sectional area of l and Awire is: wirelRA (3-16) Where is the electrical conductivity. Considering Figure 3-4 the average radius, rave, and length, lave, of a single wire used in a coil are: 2a caver rr (3-17) 2 ( )2a caver rl (3-18) 21 And the cross-sectional area of a wire is: 2wire wA r (3-19) Where rw is the radius of a wire. The average resistance can be calculated using equations (3-16) to (3-19) as: 2( )a cavewr rRr (3-20) Figure 3-4. Cross section of a coil. And the total resistance of the device is: 2( )a cEM w cwr rR N Nr (3-21) Where Nw and Nc define the total number of wire turns in each coil and the number of coils in a CBED, respectively. 3.2.3 Damping force Lorentz’s law (equation 2-5) is applied to quantify the force experienced by the mover part due to electrical currents inside the coils. Equation (2-5) can be rewritten for the force applied to a single wire, Fwire as: ( ) wire coillF JAB dl (3-22) 22 Where l and A are the length and cross-sectional area of a single wire, and Bcoil is the magnetic flux density inside the coil. JA also is equal to the current, i, inside the coil. Therefore, the force applied to a single wire is: 2 wire coil coillF iB dl riB (3-23) Bcoil can be explained in terms of Bm using equation (3-6) as: 2 2m mcoil a cir BB r r rrt (3-24) Substituting equation (3-24) into equation (3-23) the Fwire is: 2 m mwireir B iFt (3-25) And the total damping force, FEM, is: 2 m mEM p air BF N N it (3-26) 3.2.4 Passive, semi-active and active CBED The CBED can perform as a passive, semi-active and active vibration control system based on an electrical circuit attached to the damper. In this section, the different modes of damper operation are briefly investigated. The constitutive equations for the damper are repeated here as: 2 m mEM p a EMir BE N N v K vt (3-27) 2( )a cEM w cwr rR N Nr (3-28) 2m mEM p a EMir BF N N i K it (3-29) Where KEM is known as the damper constant. 23 Figure 3-5(a) shows a passive CBED. When the damper operates in a passive mode, the machine parameters cannot be adjusted during the vibration and the circuit equation follows simple Ohm’s law as: EM EME R i (3-30) It should be noted that the inductance of the coil, LEM, is neglected for the low frequencies vibration such as earthquake excitations (1-10 Hz) and it is not considered in the circuit equation. The damping force for the passive modes of vibration, FPEM, is derived by substituting equations (3-27) and (3-28) in equation (3-29) as: 2EMPEMEMKF vR (3-31) Figure 3-5. CBED: (a). Passive mode, (b). Semi-active mode, (c). Active mode. The CBED can be considered as a semi-active system by adding an external resistance as shown Figure 3-5(b). The circuit equation and damping force are derived using Kirchhoff voltage law as: EM EM addE R R i (3-32) And the semi-active damping force, FSEM, is derived using equation (3-27) to (3-28) as: 24 2EMSEMEM addKF vR R (3-33) Similar to semi-active mode, the CBED can operate as an active system by adding an additional source of energy, ESE, as illustrated in Figure 3-5(c). Like passive and semi-active modes, the damping force in the active mode, FAEM, is: EM EM SEE R i E (3-34) 2 EM SEEMAEMEM EMK EKF vR R (3-35) When the CBED performs as a passive device, its operation can be modeled as a linear damper with a constant damping coefficient (equation 3-31). Figure 3-6 illustrates the force-displacement relationship for a sample passive CBED under harmonic excitation. Figure 3-6. Force-displacement relationship for a typical passive CBED. 25 3.3 Finite element analysis The mathematical model for the CBED has been defined in the previous sections. The analytical model for Bm has been stated in equation (3-12). In the present section, a finite element analysis is conducted to validate the previous formulations. A 2D axisymmetric model of the mover is analyzed under the magneto-static condition using finite element software, Flux (Cedrat, 2013). Table 3-2 defines the finite element analysis parameters. Figure 3-7 compares the numerical and mathematical results for the magnetic flux density inside the permanent magnets for the various range of mover to magnet length ratios, RL. Also, when RL is equal to 2, the variation of magnetic flux density inside the coil, Bcoil,, is studied in Figure 3-8. As illustrated, the error between the two models is constant and equal to 2.77 percent. Table 3-2. Damper parameters for the finite element analysis model. Parameters Description Values rs Damper Shell Radius 121.1 mm rc Coil Radius 109.4 mm ra Air Gap Radius 101 mm rm Magnet Radius 100 mm tl Mover Length 80.5 mm RL Mover to Magnet Length Ratio [1.2-2.5] tm Magnet Length tl/RL ti Iron Pole Length (tl-tm)/2 tc Coil Length ti 26 Figure 3-7. Mathematical and FEM results for the magnetic flux density inside a magnet. Figure 3-8. Mathematical and FEM results for the magnetic flux density inside a coil. 3.4 Optimization procedure The derived analytical equations quantify the damping coefficient on the basis of mechanical and material properties. The verified formulations are used to investigate the maximum damping capacity of CBED. 27 Since the two pole system is repeated along the CBED (see Figure 3-1), only the two pole damper is considered for the optimization. The measure that is assessed is a damping density, defined as a variable determining the achievable damping force in a unite machine volume. The volume, VEM, and damping coefficient, CPEM, for the passive CBED can be stated using equation (3-36) to (3-37) as: 2222( )( ) m mp aiEMPEMa cEMw cwr BN NtKCr rRN Nr (3-36) 22pEM l sNV t r (3-37) The term NwNc defines the total number of coil wire turns in the device which is equal to NpNt/2. Nt defines the number of coil turns between two pole. Figure 3-9(a) and (b) shows the definition of Nc, Nw and Nt. Moreover, the term Na/ti is equal to Nt/tl, since the coil turns are uniformly distributed over the machine length (see Figure 3-9(b) and (c)). Therefore equation (3-36) can be rewritten as: 2222( )( )2 m mp tlEMPEMa cEMwr BN NtKCr rRr (3-38) Using equations (3-36) and (3-37), the damping density for the passive CBED, CVPEM, is: 4 2322( )4t m mlPEMPEMa cEMswN r BtCCVr rVrr (3-39) The Nt can be determined by equation (3-40) proposed by Zhu et al., (2012): 2 3ctwANA (3-40) 28 Where Ac and Aw are the cross-sectional areas of the coil and conductive wire. The genetic algorithm (GA) is Matlab toolbox (Mathworks, 2013) used to optimize the damping density for the range of parameters defined in Table 3-3 and Figure 3-9(a). To ensure accurate results, the population size, generation and tolerance function for GA optimization are considered as 150, 200 and 10E-11, respectively. It should be pointed out that iterations are needed to calculate the damping density in each step of GA. In other words, all the machine parameters can be determined based on the information provided in Table 3-3, except the damper shell radius, rs. Another constraint, magnetic flux saturation condition, is applied to find the minimum required damper shell radius. A magnetic flux saturation condition defines the maximum magnetic flux that can pass through the material in a magnetic circuit. Using equation (3-7), the iron shell radius can be determined as: 22 2 ( )m mm m shell shell shells cr BA B A B Br r (3-41) Whenever the Bshell is equal to the saturated magnetic flux, Bsat, the minimum damper shell will be reached. Figure 3-10 describes the optimization procedure. 29 Figure 3-9. The two pole system: (a). Two pole system with three coils, (b). Equivalent one coil system, (c). Active coil wire. 30 Table 3-3. Optimum parameters for the CBED. Parameters Description Values Optimum Value Variable tl Mover Length [10-500] mm 80.5 mm RL Mover to Magnet Length Ratio [1.2-2.5] 1.96 tm Magnet Length tl/RL 41 mm ti Iron Pole Length (tl-tm)/2 19.7 mm tc Coil Length tl 80.5 mm rm Magnet Radius [10-100] mm 100 mm hg Air Gap Thickness [1-5] mm 1 mm ra Air Gap Radius rm+hg 101 mm hw Coil Thickness [10-100] 8.4 mm rc Coil Radius ra+hw 109.4 rs Damper Shell Radius Saturation Condition 121.1 mm rw Wire Radius [0-40] AWG 21 AWG Target Function CVPEM Passive Damping Density 3,061 kN-s/m4 31 Figure 3-10. Optimization procedure. Figure 3-11. Penalty value for genetic algorithm. 32 Figure 3-11 and Table 3-3 illustrate the GA iteration and the optimum parameters. The maximum achievable damping density for the passive CBED is 3,061 kN-s/m4 which is in the order of regular oil dampers that are currently used in the structures (2,800-4,200 kN-s/m4 (Taylor, 2003). However, when the CBEDs are considered as semi-active or active systems, the additional resistance or power which is applied to the damper circuit alters the damping force. For instance, in the semi-active control mode the damping force, FSEM, is: 2EMSEMEM addKF vR R (3-42) Assuming the additional resistance as a portion of damper resistance, the damping force is stated as: add EMR R (3-43) 21EMSEMEMKF vR (3-44) Figure 3-12 shows the ratio of semi-active to passive mode damping force. It can be concluded that by adding the additional resistance, which is inevitable during a control algorithm, the damping force is rapidly decreased. A similar discussion is applicable when the CBEDs are employed as an energy harvester. Shen & Zhu, (2015) proposed the optimum additional resistance, ROpt,E to reach to the maximum harvesting capability as: 2, 1EMOpt E EMp EMKR RC R (3-45) Where Cp is a constant and defines the mechanical losses in CBEDs due to some imperfection such as friction. 33 In light of the above discussions, when the CBEDs operate either as semi-active or active dampers or as a harvesting energy system, the total resistance is no longer equal to REM. Therefore, the damping force significantly affected by additional resistance and will be decreased. Figure 3-12. Additional resistance effects on CBED. In the optimization procedure, equation (3-39), a single equivalent coil is used between two poles. However, in practice, when the coil length is equal to the magnetic flux cycles, the voltage will not be generated since both positive and negative voltage will be induced in the same coil segments. As discussed in Table 3-3 the maximum damping density occurs when the mover to magnet length ratio, RL, is approximately 2. It is therefore logical to consider coil length the same as magnet length. As a result, all the coils in the passive CBED perform as a four phase electrical machine. As Figure 3-13 shows, the first and third phases generate the minimum positive and negative voltage at the indicated position, and the second and fourth phases have zero voltage. It 34 is worth to mention that, although the voltage or power of each coil depends on the relative position of coil and mover, the total generated power is constant. Figure 3-13. A four phase CBED. 3.5 Conclusion In this chapter, the mathematical model for the CBED is derived and verified by means of finite element analysis. Different applications of CBED as passive, semi-active and active dampers are briefly studied. The verified formulation is optimized through the genetic algorithm (GA) to find the maximum damping density and its corresponding machine configuration. As a result, the four phase CBED is designed which can generate 3,061 kN-s/m4 damping density in its passive mode. As discussed, the capacity of passive CBED is within the range of conventional oil dampers. However, when they are implemented as semi-active and active systems or as an energy harvester, their damping force are no longer enough to be used in structural systems. In chapter 4, another type of passive electromagnetic damper, the Eddy Current Damper (ECD) will be examined. Chapter 4 will suggest ways of improving the performance of electromagnetic dampers for large scale applications such as structural systems. 35 4. Development of eddy current damper 4.1 Introduction In the previous chapter, the concept of Coil-Based Electromagnetic Damper (CBED) was discussed. An optimum device configuration was suggested in order to reach the maximum capacity of CBED. The present chapter explores the development of new types of a passive electromagnetic damper identified as the Eddy Current Damper (ECD). The eddy current phenomena appears due to the relative motion of magnets and conductors which induces an electrical field and therefore a current in the conductors. Considering Lorentz’s law (equation 2-5) the eddy currents produce the reverse force which is treated as a damping force. In other words, the ECD converts the kinetic energy into the heat through the eddy currents in the conductor without any contacts. Traditional ECD has two significant disadvantages: 1) the ECD usually have small damping density in comparison with other structural dampers (Warmerdam, 2000). 2) The conventional ECD usually requires iron poles and shells to guide the magnetic flux in a loop (see Figure 4-1(a)). The iron can be easily magnetized. During vibration, the iron shell in the conventional ECD is subjected to alternating magnetic fields, which cause the direction of the magnetism within the iron pole to change frequently, and eventually losses its ability to guide the magnetic flux. In this chapter, a novel ECD design with the circumferential permanent magnets, (see Figure 4-1(b)), is proposed. In the proposed ECD, the permanent magnets are placed in a circumferential pattern, which can produce high magnetic flux without the use of iron, which will produce a higher damping density. Simulations show that the proposed ECD design can produce a high damping density (2,733 kN-s/m4) in the order of viscous fluid dampers (2,400 kN-s/m4 to 4,200 kN-s/m4) (Taylor, 2003). 36 This chapter is divided into three sections. In the first section, the analytical model of the proposed ECD is derived using electromagnetic theory under quasi-static condition. The mathematical model is verified through the finite element method (FEM). In section two, the verified mathematical model is used for the optimization procedure using the genetic algorithm (GA). As previously stated, in ECDs the kinetic energy is converted into heat. A thermal analysis is therefore carried out in section three in order to investigate the permanent magnets temperature. 37 Figure 4-1. ECD configurations: (a). Conventional ECD (Ebrahimi et al., 2008), (b). Proposed ECD. 4.2 Analytical model of the proposed eddy current damper 4.2.1 Magnetic flux density produced by an arc segment permanent magnet Figure 4-2 illustrates an arc permanent magnet and the coordinate system used in this study. In this arc permanent magnet, the magnetization is formulated in a circumference manner as indicated 38 by the vector M . Equation (4-1) defines the magnetization, M , of the arc permanent magnet in the θ direction. M θ M (4-1) Where M is magnitude of the magnetization determined from the property of the permanent magnet. In this study, NdFeB N35 magnet with the magnetization of 9.5E5 Ampere/meter is used in ECD design. Figure 4-2. An arc segment of the permanent magnet. Equation (4-2) shows the magnetic potential function, , of the arc segment of the magnet as illustrated in Figure 4-2. 1 .M 1 M.n( , , )4 4 SVr z dV dSd d (4-2) Where is the Nabla operator; d is the relative distance between points m and p (see Figure 4-2); p is a point of interest; m is an arbitrary point in the magnet and n i is the unit normal vector to 39 the ith surface of the magnet. In Figure 4-2, 1n and 2n are the normal vectors to the north and south poles, respectively. Due to the special magnetization pattern shown in Figure 4-2, .M = 0 , 1M.n M , 2M.n M and M.n 0i for the other surfaces. Using the parameters defined in Figure 4-2, the magnetic potential for the arc magnet is presented in the Appendix A. The associate magnetic field, H , and flux density, B , can be derived using equations (4-3) and (4-4), respectively. H ( , , ) ( , , ) r z r z (4-3) 0B( , , ) H( , , )r z r z (4-4) Where 0 is the permeability of air (70 4 10 ). 4.2.2 Eddy current damping force calculation Figure 4-3 shows a conductive plate placed between two arc permanent magnets. In addition, Figure 4-3 illustrates the dimension and coordinate system used for the following derivations. Figure 4-3. Conductive plate subjected to one row of permanent magnets on both sides. 40 The current density, J , induced in the conductive plate due to the relative movement of conductive plates and permanent magnets is described in equations (4-5)–(4-7). J= E (4-5) c bE=(E +E ) (4-6) bE =v×B (4-7) Where is the electrical conductivity of the plate (in this study, the copper plate with electrical conductivity of 5.8E7/Ohm is used); cE is the electrostatic field generated by the coulomb charges, and bE is the electromotive field induced by the relative movement of the conductive plate at the relative velocity v . B is the magnetic flux density with rB , θB and zB components which can be defined by equation (4-4). Due to the anti-symmetric arrangement of the permanent magnets on both side of the conductive plate, the radial component of magnetic flux, rB , produced by each magnet cancels each other. On the other hand, the azimuthal component of magnetic flux, θB , adds up. Because the relative movement of the conductive plate and the permanent magnets is in the Z direction, the Z component of the magnetic flux density, zB , does not contribute to bE . To simplify the derivation, a new Cartesians coordinate system with the origin O at the center of the pole projection area (see Figure 4-3) is used. Because the air gap is small, the angle is approximately zero and the vector yB can be approximated using θB . Similarly, xB is almost zero. The relative velocity between permanent magnets and plates is also in Z direction, vz. Therefore, bE (equation 5-7) can be simplified to: b bxE = E = .z × .yz yv B (4-8) 41 Figure 4-4 shows a sample illustration of the amount of magnetic flux density, yB , at the mid-plane of a conductive plate (with the parameters as shown below). As shown in this figure, the proposed magnets configuration can produce a high magnetic flux density in the conductive plate. Figure 4-4. Y- Component of magnetic flux at the mid-plane of a sample conductive plate. Figure 4-5 shows the pole projection area and the reference axes on a conductive plate when the plate is subjected to a uniform magnetic flux density. Equation (4-9) shows the electrostatic field generated by the coulomb charges in the x-direction, cxE , (Heald, 1988). , arctan arctan arctan arctan2 z avecxv B z b z b z b z bE x zx a x a x a x a (4-9) Where Bave is the average magnetic flux density over the pole projection area defined using the dimension 2b and 2a as shown in Figure 4-5. 42 Figure 4-5. Illustration of the pole projection area when a conductive plate is subjected to a uniform magnetic flux. The current density in the x-direction, xJ , inside and outside of the pole projection can be described using equations (4-10) and (4-11), respectively. ( ) x cx bxJ E E Inside the pole projection area (4-10) x cxJ E Outside the pole projection area (4-11) Where bxE and cxE are stated by equations (4-8) and (4-9), respectively. The eddy current force, zF , can be calculated using equation (4-12). zF = .x × .y x yJ B dV (4-12) It should be note that the amount of eddy current force, zF , can also be expressed as equation (4-13), which is linearly proportional to the relative velocity, zv . z zF C v (4-13) Where 2 11 arctan arctan arctan arctan2 yz b z b z b z bC Bx a x a x a x a 43 Equation (4-13) is only valid if the conductive plate has infinite length. On the other hand, the conductive plate of the proposed ECD (shown in Figure 4-3) has a finite length and zero eddy current density at its edges. To consider the edges effect, the image method proposed by Lee & Park (2001) is used. In this method, the current density is first calculated using conductive plate with infinite length, then a mirror image of the current density with respect to the edge of the conductive plate is subtracted. Equation (4-14) shows the modified current density for the proposed ECD where the edge effects are included. ' 1 2 3 x x x xJ J J J (4-14) Where 1xJ is the eddy current for the conductive plate with infinite plate; 2xJ and 3xJ are the imaginary currents at the two edges of the conductive plate calculated using equations (4-15) and (4-16), respectively. 2 1 12 , x xJ J r x z (4-15) 3 1 22 , x xJ J r x z (4-16) Where r1 and r2 are the distances to the edges as illustrated in Figure 4-5. Figure 4-6 shows the eddy current density over a sample conductive plate. The results confirmed that the eddy currents for the finite plate are zero at the edges. 44 Figure 4-6. Eddy current density over the conductive plate with finite length. Substituting equation (4-14) into equation (4-12), the damping force for the proposed ECD can be calculated using equation (4-17). 1 2 3 b az p x x x yb aF t J J J B dydx (4-17) Where tp is the plate thickness as shown in Figure 4-3. It should be note that equation (4-17) does not account for the configuration with multiple rows of magnets. To account for multiple rows of magnets (see Figure 4-7), the net electrical field, 'cxE, for the infinite plate can be written as equation (4-18). ' 1 2, cx cx cxE x z E E (4-18) Where 1cxE and 2cxE present the electrostatic field produced by the magnets in the first and second row, respectively. 1cxE is stated using equation (9) and 2cxE can be defined using equation (4-19). 2 , arctan arctan arctan arctan2 avecxvB z b D z b D z b D z b DE x zx a x a x a x a (4-19) Where D is a distance between the center of pole projection areas as illustrated in Figure 4-7(b). 45 Figure 4-7. A conductive plate subjected to two magnetic fields: (a). 3D view, (b). Mathematical model. Substituting the net electrical field, 'cxE (equation 4-18), into cxE in equation (4-10), the damping force for the conductive plate subjected to two rows of arc magnetic fields can be described using equation (4-20). 21 2 31 b az p ix ix ix yi b aF t J J J B dzdx (4-20) Equation (4-21) shows the generalized damping force on the conductive plate subjected to the multiple rows of arc permanent magnets: 1 2 31 MagnetRows b az p ix ix ix yi b aF t J J J B dzdx (4-21) 46 4.3 Validation of the analytical model using finite element analysis The analytical model is verified using 3D finite element software, Flux (Cedrat, 2013). The permanent magnet is modelled using the Linear Magnetic material with tetrahedral triangle element with automatic mesh generator. The mesh is manually adjusted locally to consider the skin depth effect in the conductive plate. The skin depth effect refers to a phenomenon that the current density at the conductor surface is greater than that at the core (Hayt & Buck, 2001). Figure 4-8 shows a sample 3D finite element result when a conductive plate is subjected to a cyclic displacement loading. As illustrated, the proposed circumferential permanent magnet configuration can produce high current density on the conductive plate. Figure 4-8. Eddy currents in a conductive plate. 47 Figure 4-9(a) and (b) illustrate the comparison of the analytical model and finite element simulation for the magnetic flux density produced by an arc permanent magnet in the azimuthal and radial directions, respectively. The results show that the error between these two simulations is less than 2.5 percent. Figure 4-9. Sample comparison of the magnetic flux in the azimuthal and radial directions calculated using analytical and FEM simulations. Figure 4-10(a-c) show the comparison of the power loss between the analytical and finite element simulation at different loading conditions. The result shows that the analytical model over predicts the power loss compared to the finite element simulation. The discrepancy between these two modeling approaches is highest when the velocity is at its maximum level. Such discrepancy can be attributed by the time variant property of the eddy currents. According to Faraday’s law, because the eddy current is time variant, it will produce additional magnetic fields. It should be noted that this phenomena is not considered in the analytical solution which is based on the quasi-static condition. Therefore, the accuracy of analytical models may be affected by the influence of the time variant property of the eddy currents. 48 Figure 4-10. Plots of velocity versus time and power loss versus time for the ECD under the dynamic load. 4.4 Optimization Genetic algorithm toolbox as presented in Matlab (Mathworks, 2013) is used to optimize the ECD design to maximize the damping density. The air gap and thickness of conductive plates (as shown in Figure 4-3) are considered as fixed variables due to manufacturing limitations. Table 4-1 shows the range of parameters included in the optimization. In order to carry out an optimization procedure, it is necessary to constrain the device length by defining the number of magnet rows, Nmr. Therefore, a series of optimizations are conducted for the damper with different Nmr. Figure 4-11 shows a sample convergence of the genetic algorithm. In order to achieve accurate results, the population size, generation and tolerance functions for optimization are considered as 150, 200 and 10E-11, respectively. The result shows the genetic algorithm converged to a solution efficiently. Table 4-2 summarizes the optimal parameters for the range of parameters shown in 49 Table 1 for the different Nmr cases. As illustrated, as Nmr goes above 16, the influence of Nmr values has less influence on the optimal design of the proposed ECD. In addition, the optimum magnet length, tm (see Figure 4-3) converges to 23 mm. The proposed configuration is able to achieve a damping density of 2,733 kN-s/m4. It worth to mention that the damping density for fluid dampers is between 2,400 kN-s/m4 to 4,200 kN-s/m4 (Taylor, 2003). This shows the proposed ECD can have high potential for earthquake engineering applications. Figure 4-11. Convergence of genetic algorithm. 50 Table 4-1. Device parameters. Parameters Description Values Fixed gA Air Gap 1 mm pt Conductive Thickness 5 mm Variable θ (Np) Magnet Angle (Number of Conductive Plate ) [10o-90o] (36-4) 2R Outer Magnet Radius [10-150] mm 1 2S=R /R Ratio of Inner to Other Magnet Radius [0-0.95] L Device Length Defined by Nmr m mrt =L/N Thickness of Magnets in Z direction [0.5-100] mm mrN Number of Magnet Rows [2-100] Table 4-2. Optimal design values. mrN θ (Degree) 2R (mm) S mt (mm) ECCV (kN-s/m4) 2 30 144.2 0.11 20 2,537 4 32.7 138.4 0.1 21 2,615 8 36 133.6 0.097 21 2,675 12 36 130.2 0.091 23 2,704 16 36 125.5 0.088 23 2,718 25 36 125.5 0.088 23 2,733 50 36 125.5 0.088 23 2,733 100 36 125.5 0.088 23 2,733 4.5 Thermal analysis The eddy current damper converts the kinetic energy into the heat through the induced currents in conductive plates. Due to the high capacity of ECDs, high temperatures are likely to be reached during the operation. All permanent magnet materials are temperature sensitive and it is important 51 to take this into consideration. Heat transfer analysis is therefore required to predict the temperature of the device. Two approaches are available for this purpose: 1. Considering the air flow in different parts of the device by using the computational fluid mechanics applied by Zhao et al., (2015). 2. Simplified analytical models based on empirical equations applied by Negrea & Rosu, (2001). In the first method, the fluid and electromagnetic system are modeled and solved simultaneously, which is generally time-consuming. In this research the second approach is utilized. It is assumed that only the air passing through air gaps dissipates the heat in conductive plates, the heat transfer through permanent magnets is not taken into consideration. The device is treated as a steady state forced convection internal flow problem in heat transfer analysis (Incropera et al., 2006). Figure 4-12 describes the assumptions and the cross section of air channels. Figure 4-12. Forced convection internal Flow, (a). 3D view, (b). Cross section. 52 Newton’s cooling law states the relationships between the heat flux, surface and air temperature as: ( )s fq h T T (4-21) where q is thermal flux produced by eddy currents in conductive plates, h is a local convection heat transfer coefficient, Ts is the plate surface temperature and Tf is the ambient temperature which is considered as 293.15o K (25o Celsius). Whenever the h is defined, it is possible to estimate the plate surface temperature. The convection coefficient is defined using the dimensionless Nusselt number (Incropera et al., 2006). The Nusselt number is well defined for the circular cross section channel for both laminar and turbulence flow conditions. However, many circular channel results can be applied for noncircular channels using the equivalent hydraulic sections. The channel shown in Figure 4-12 is considered as an infinitive plate due to the large length to width ratio of channel cross section. Figure 4-13 shows the equivalent hydraulic section for the infinite channel. The Nusselt number for the equivalent hydraulic section is stated as: hairhDNuk (4-22) Where kair is the thermal conductivity of the air and is equal to 0.0269 W/m K and Dh is the equivalent hydraulic diameter. 53 Figure 4-13. Equivalent hydraulic circular section for infinite channel. To quantify the heat transfer coefficient, it is required to define the flow and thermal conditions. The Reynolds number defines the flow condition as: ReRe<2300 Laminar Flowa hV D (4-23) where Va, Dh and are defined as the air velocity, hydraulic diameter and kinetic viscosity (see Table 4-3). The thermal condition is considered as a constant surface heat flux condition, since the condition is steady state and the thermal flux due to the eddy currents in the conductive plate is constant. Due to the results in Table 4-3, the flow is treated as a fully developed laminar flow with a constant heat flux. According to the flow and thermal conditions, the Nusselt number for an 54 infinitive channel with one side isolated boundary condition is stated by Bejan, (2013) as 5.39. Therefore, considering equation (4-44) the heat convection coefficient is calculated as 72.5 W/m2 K. The heat flux on the surface of each plate for the maximum damping density, 2,733 kN-s/m4, at the average device velocity, 0.25 m/s, is 2.67E3 W/m2 (see Table 4-3). Using Newton’s cooling law the temperature of plate surfaces is calculated as 62o Celsius, which is within the operation range of permanent magnet, 80o Celsius (MMG, 2006). It should be noted that this temperature is an upper bound estimation, since the radiation and heat transfer through magnets and other parts of the device are neglected. 55 Table 4-3. Flow and thermal conditions. Parameters Descriptions Value Formulation Flow condition CW Channel Width 1 mm Ag R2 Magnets Outer Radius 68 mm R1 Magnets Inner Radius 6 mm CL Channel Length 62 mm R2-R1 Rc Channel Ratio 62 CL/CW Dh Hydraulic Diameter 2 mm 2CW tp Plate Thickness 5 mm Vp Plate Velocity 0.25 m/s Steady State Condition Va Air Velocity 0.625 m/s Vp.tp/(2CW)- Conservation of Mass Air Viscosity 1.75E-5 m2/s Re Reynolds Number Re<2300 VaDh/ Thermal condition CVEC Maximum Damping Density 2,733 kN-s/m4 qm Maximum Power Density 170k W/m3 CVEC.(Vp)2 A Device Cross Section Area 0.0195 m2 Q Total Power for unit Length 3315 W qm.A Np Number of Plates 10 Ns Number of Plate Surfaces 20 2.Np q Heat Flux for each Plate Surface 2.67E3 W/m2 Q/(Ns.CL) 56 4.6 Conclusion A novel ECD with circumferential permanent magnets is proposed in this study. The circumferential arrangement of permanent magnets and conductive plates can produce a high strength magnetic field without the use of iron, hence a higher damping force can be achieved. A theoretical model of the proposed ECD is constructed using electromagnetic theory under quasi-static condition. The magnetic flux and eddy current damping force are quantified analytically and validated using finite element simulation. Although the analytical model neglects the effects of the time-variant property of the eddy currents, the analytical model is still able to approximate the damping force of the proposed ECD accurately. The developed analytical model is used to optimize the ECD design using genetic algorithm. The result shows that the proposed ECD can produce a high damping density up to 2,733 kN-s/m4, which is feasible to be used for structural applications. Since the ECDs convert kinetic energy into heat, a thermal analysis for the optimum ECD is carried out in order to evaluate the permanent magnets temperature. It is shown that not only can the proposed device provide a high damping density but also can perfectly transfer the thermal energy to the air surrounding the device. 57 5. Electromagnetic dampers with applications in structural engineering 5.1 Introduction In structural engineering the mitigation of damage induced by sever dynamic loads such as earthquakes and strong winds is an eminent interest. Over the past decades, finding an effective system to protect structures has been one of the major challenges for civil engineers. Among the various techniques, structural control methods show a great potential for reducing the damage effects of seismic excitations. Therefore, several control devices and algorithm have been investigated recently. In general, control systems can be categorized into four groups according to their operations. Passive devices, which require no external power, never destabilize structures. However, they have low adaptability to changes in external loads. On the other hand, active control devices are adaptive to the varying applied loads, but their stability and large power consumption are still major problems. Semi-active devices are the third group of structural control systems, in which mechanical properties such as stiffness can be modified to improve their operations. Similar to active devices, semi-active systems require control algorithms to adapt their performance in response to vibration loads. However, they require only a slight amount of energy. Semi-active systems only dissipate the energy, and do not input forces to structures to affect their stability. In the last group of control systems, active or semi-active devices are integrated with passive devices as a hybrid system, thereby offering the adaptability of active and semi-active devices combined with the features of passive devices. Electromagnetic systems can operate as passive (Chapter 3 and 4), semi-active and active (Chapter 3) control devices. A newly developed passive eddy current dampers (Chapter 3) can 58 provide a high damping force, but their performances are not adjustable. To resolve the difficulties, a hybrid electromagnetic damper (HEMD) is proposed. The present chapter deals with applications of electromagnetic systems for structural engineering. CBED and ECD prototypes are designed here. Moreover, the pros and cons of each system are briefly discussed and compared with a bench mark conventional viscous damper. In order to comprise the unique features of active, semi-active and passive dampers, the HEMD is proposed. Figure 5-1 illustrates the proposed HEMD. This configuration consists of two parts: external and internal. The CBED is considered as an internal part, surrounded by the developed ECD as an external part. Therefore, the internal part operates as an active or semi-active device while the external part performs as a passive damper. 59 Figure 5-1. Hybrid electromagnetic damper. 5.2 Coil-based electromagnetic damper The CBED can be considered as passive, semi-active and active control devices. The four phase optimum passive CBED was discussed in Chapter 3. Table 5-1 and Figure 5-2 illustrate the damper parameters. 60 Table 5-1. Optimum design parameters for the CBED. Parameters Description Value tm Magnet Length 40 mm ti Iron Pole Length 20 mm tc Coil Length 40 mm rm Magnet Radius 100 mm ra Air Gap Radius 101 mm rc Coil Radius 109.5 rs Damper Shell Radius 121 mm rw Wire Radius 21 AWG CVPEM Passive Damping Density 3,061 kN-s/m4 Nm Number of required Magnets 81 C Provided Damping Coefficient 906 kN-s/m Figure 5-2. Four phase CBED. For the purposes of comparison, the capacity of a viscous fluid damper that has been manufactured by Taylor Device and installed in San Bernardino Country Medical Center is chosen as a bench mark damper (Taylor, 2003). Table 5-2 compares the capacity of CBED with the conventional viscous damper assuming a linear force-velocity relationship. As shown, the optimum passive CBED can operate similar to regular oil dampers. However, when the CBED is considered as semi-active, active or energy harvester systems, their damping density will be 61 decreased due to the additional resistance or power which is imposed to the device circuit (Chapter 3). Table 5-2. Comparison between bench mark damper and CBED. Damper Damping Coefficient (kN-s/m) Volume (m3) Damping Density (kN-s/m4) San Bernardino 906 0.33 2746 CBED 906 0.3 3061 5.3 Eddy current damper The novel ECD which is classified as a passive control device was developed in Chapter 4. As stated, the proposed ECD can provide a high damping density as 2,733 kN-s/m4. Table 5-3 review the optimum design parameters for the ECD. Table 5-3. Optimum parameters for ECD. Parameters Description Value Ag Air Gap 1 mm tp Conductor Thickness 5 mm θ (Np) Magnet Angle (Number of Conductive Plate ) 36o (10) R2 Outer Magnets Radius 125.5 mm R1 Inner Magnets Radius 11 mm Lp Conductor Plate Length 125.5 mm r0 Plates Intersection Radius 7 mm tm Thickness of Magnets in Z direction 23 mm Nmr Number of Magnet Rows 291 CVEC Damping Density 2,733 kN-s/m4 C Provided Damping Coefficient 906 kN-s/m 62 Table 5-4 compares the novel ECD with both the bench mark oil damper and CBED. It can be inferred that the novel ECD can produce a damping force in the range of regular oil dampers. However, the passive control systems are not adaptable to the load patterns. For this reason, a HEMD is proposed in the next section with the aim of integrating the innovative features of both electromagnetic systems. Table 5-4. Comparison between CBED and ECD. Damper Damping Coefficient (kN-s/m) Volume (m3) Damping Density (kN-s/m4) San Bernardino 906 0.33 2,746 CBED 906 0.3 3,061 ECD 906 0.34 2,733 5.4 Hybrid electromagnetic damper The HEMD is investigated in this section. Figure 5-1 illustrates the HEMDs configuration. CBED and ECD are treated as internal and external parts which operate as two parallel systems. The total damping force produced by HEMD, FHEMD, is the sum of the damping force produced by each part: HEMD GEMD ECDF F F (5-1) The parameters described in Table 5-1 for the optimum four phase CBED are considered as the internal part, while the external part parameters, ECD, will be defined based on mathematical model developed in chapter 4. Table 5-5 defines the hybrid electromagnetic damper parameters. 63 Figure 5-3. A sample HEMD, (a). 3D view, (b). Cross section. 64 Table 5-5. HEMD parameters. Parameters Description Value tm CBED Magnet Length 40 ti Iron Pole Length 20 tc Coil Length 40 rm CBED Magnet Radius 100 mm ra Air Gap Radius 101 mm rc Coil Radius 109.5 rs CBED Shell Radius and Inner ECD Magnet Radius 121 mm rw Coil Wire Radius 21 AWG Nm Number of Magnets in CBED 44 Ag ECD Air Gap 1 mm tp Conductor Plate Thickness 5 mm θ (Np) Magnet Angle (Number of Conductive Plate ) 36 (10) tecm ECD Magnet Thickness 23 mm Nmr Number of Magnet Rows in ECD 153 CVHEM Damping Density 2,049 kN-s/m4 C Provided Damping Coefficient 906 kN-s/m As Table 5-5 shows, the HEMD damping capacity is less than the ECD and its volume is 1.34 times greater than the bench mark damper. However, the HEMDs offer the controllable vibration systems which can be also utilized as a harvesting energy system. 65 Table 5-6. Comparison between developed dampers. Damper Damping Coefficient (kN-s/m) Volume (m3) Damping Density (kN-s/m4) San Bernardino 906 0.33 2,746 Passive CBED 906 0.3 3,061 ECD 906 0.34 2,733 HEMD Total 906 0.44 2,049 External Part 411 Internal Part 495 5.5 Conclusion The application of electromagnetic devices for structural systems has been investigated in the present study. CBED and ECD prototype are designed and compared with the viscous damper. The results show that it is feasible to employ the electromagnetic devices as a vibration control systems for the structural applications. However, in order to improve their performance, a HEMD consisting of CBED and ECD dampers is proposed and designed. The HEMD offers both a stable and adjustable damping force. The application of HEMDs can also be extended as the energy harvesting system. 66 6. Summary and conclusion 6.1 Conclusion The present research deals with the concept of electromagnetic systems and their applications in Civil Engineering. Three different electromagnetic devices, Coil-Based Electromagnetic Damper (CBED), Eddy Current Damper (ECD) and Hybrid Electromagnetic Damper (HEMD) were investigated through mathematical and numerical models. The fundamental electromagnetic theories, Lorentz and Faraday’s laws, are utilized to formulate a mathematical model. Numerical simulations are conducted using the finite element program, Flux. Results show that both modeling approaches are consistent with each other. Finally, the verified analytical models and Genetic Algorithm (GA) are employed in order to examine the maximum damping capacity and their corresponding geometrical parameters. After a brief introduction on vibration control systems and the concept of electromagnetic theories in chapter 1 and chapter 2, CBEDs are investigated in chapter 3. CBEDs can operate as passive, semi-active and active systems. They can also be considered as an energy harvesting system that convert kinetic energy into electricity using an additional circuit attached to the damper. Conducting an optimization led to the optimum design of four phase passive CBED. However, the operation of CBEDs relies on the additional resistance attached to the damper. Therefore, when they are employed as a semi-active, active or an energy harvester, their damping capacity is sharply reduced. An innovative ECD with high damping density is proposed in chapter 4. The arc segment permanent magnets with circumferential magnetization are used to produce a high strength magnetic field. A theoretical model of the proposed ECD is constructed using electromagnetic theory under quasi-static condition. The magnetic flux and eddy current damping force are 67 quantified analytically and validated using finite element simulation. Although the analytical model neglects the effects of the time-variant property of the eddy currents, the analytical model is still able to approximate the damping force of the proposed ECD accurately. The developed analytical model is used to optimize the ECD design using genetic algorithm. The result shows that the proposed ECD can produce a high damping density up to 2,733 kN-s/m4, which is feasible to be used for structural applications. With the design of CBEDs, the idea of adding the ECDs is presented and developed as a HEMD. The HEMDs consist of two parts, the CBED forms the internal part, while the ECD is located externally. The present study indicates that coupling EDCs and CBEDs can produce adequate damping force. Furthermore, these devices can be considered as an energy harvesting system. Table 5-7 summarizes the capacity of the systems developed in the present research. Table 5-7. A summary of the developed electromagnetic dampers. Damper Damping Coefficient (kN-s/m) Volume (m3) Damping Density (kN-s/m4) Passive CBED 906 0.3 3,061 ECD 906 0.34 2,733 HEMD Total 906 0.44 2,049 External Part 411 Internal Part 495 68 6.2 Suggestions for future research Based on the findings of the present study, the following research is recommended for the future: 1. The detailed analytical and numerical model were provided for the developed eddy current damper. It will be important to verify the theories through experimental investigations. 2. This study only compared the three types of electromagnetic system in the component level. To extend this study, the developed devices can be implemented in a real bench mark structure for the system level investigation. 3. In the developed eddy current damper, the magnetic field was produced using the arc segment permanent magnet. The coil system can be used instead of permanent magnets to generate the magnetic field. Therefore, by controlling the current inside the coils the active eddy current damper can be presented. 4. Future research should examine the variations of eddy currents in the conductive sheet through the analytical solution. 5. An empirical method was employed for the heat transfer analysis. It may possible to model and solve the entire system using computational fluid mechanics. 6. Development of the controller system for the final hybrid damper designs is another important subject for future research. Controllers can be designed for both energy harvesting and damping applications. 7. Development of an industrial version of the hybrid electromagnetic damper is also strongly recommended as a topic for subsequent research. 69 Bibliography Bae, J. S., Kwak, M. K. & Inman, D. J., 2005. Vibration Suppression of a Cantilever Beam using Eddy Current Damper. Journal of Sound and Vibration, Volume 284, pp. 805-824. Basak, A. & Shirkoohi, G. H., 1990. Computation of Magnetic Field in D. C. 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Smart Materials and Structures, Volume 10. 73 Appendices Appendix A: Analytical expression for magnetic potential 2 2 2( )1 1 1( , , ) ( 1) , ,4 i j km i j ki j kMr z R z (A-1) 22 222 2, , 2 cos cos arctan2 cos cos i k j k i i j i jki i j i jr z z r R rR R rz zr R rR R r (A-2) 22 22 22 2 2 222 22 22 cos coscos arctan2 cos cos 2 coslog cos 2 coscos log 2 i i j i ji j ki i j i j i i j kk i j i i j ki j k ir R rR R rR r z zr R rR R r r R rR z zz z R r r R rR z zR r z z r R rR 2cos i j kz z 74 Appendix B: Details of finite element model ################################################################################ GENERAL DATA ################################################################################ Project name : | C:\Windows\Temp\3DConfiguration.FLU Project dir size : | 1543.41 Mbytes Software version : | Flux3D (11.2) Application : | MagneticTransient3D Last modification : | 26/11/2015 00:41:47 Date of the report : | 26/11/2015 00:45:12 ------------------------------------------------------------------------------------- ========== checkGeometry ========== The geometry has been modified during the solving process to take into account the mechanical sets The verification of confused points is not allowed. No superimposed points No abnormal lines ========== checkMesh ========== Volume elements : List of poor quality elements : 26 364 415 676 826 1030 1303 1307 1595 1599 1687 1688 1690 1781 1802 1832 1834 1838 1901 1902 1904 1905 1906 1907 1908 1910 2013 2020 2022 2023 2024 2064 2073 2074 2075 2124 2125 2126 2127 2128 2129 2131 2132 2139 2212 2233 2241 2242 2251 2252 2253 2254 2255 2257 2259 2260 2261 2262 2311 2318 2323 2334 2361 2374 2378 2379 2380 2381 2382 2383 2384 2389 2391 2395 2467 2471 2517 2518 2522 2530 2532 2534 2565 2613 2656 2658 2682 2689 2694 2695 2704 2710 2711 2787 2797 2805 2825 2846 2856 2875 75 2877 2907 2950 2952 2968 3004 3008 3016 3018 3021 3022 3037 3043 3045 3047 3052 3055 3080 3094 3107 3190 3208 3209 3284 3287 3288 3300 3360 3418 3419 3420 3423 3426 3434 3472 3487 3488 3491 3495 3500 3501 3502 3505 3514 3542 3547 3548 3557 3608 3611 3613 3645 3662 3691 3729 3736 3738 3746 3748 3749 3750 3751 3796 3845 3854 3950 4006 4015 4019 4091 4092 4093 4102 4103 4105 4107 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58798 58811 58825 58849 58910 58911 58934 58938 59008 59038 59064 59233 59239 59256 59262 59263 59268 59274 59282 59283 59284 59289 59291 59299 59303 59304 59305 59307 59308 59309 59310 59354 59360 59361 59526 59568 59584 59649 59652 59837 59900 59950 59971 59996 60013 60219 60221 60224 60246 60247 60250 60257 60260 60264 60267 60268 60271 60280 60283 60286 60287 60297 60300 60304 60323 60324 60325 60335 60350 60387 60394 60396 60397 60409 60426 60713 60722 80 60811 60812 61205 61649 61752 61825 61929 62000 62076 62238 62302 62306 62309 62313 62414 62714 62981 62998 63144 63178 63211 63215 63557 63560 63561 63565 63609 63624 63644 63712 64021 64267 64278 64398 64472 64597 65127 65194 65206 65214 65218 65503 65834 66021 66058 66131 66132 66134 66135 66136 66162 66165 66169 66172 66177 66180 66187 66232 66247 66303 66305 66312 66322 66323 66326 66330 66361 66438 66646 66681 66691 66717 66718 66719 66720 66725 66728 66729 66734 66736 66737 66738 66752 66755 66756 66758 66759 66761 66810 66816 66932 67022 67053 67258 67328 67371 67428 67606 67607 67617 67619 67627 67635 67637 67640 67672 67680 67690 67716 67734 67776 67778 67780 67786 67795 67807 67820 67834 67850 67857 67870 67872 68147 68188 68256 68260 68267 68321 68384 68651 68747 68769 68968 68970 69133 69237 69257 69614 69685 69689 69692 69717 69729 69998 70123 70192 70480 70505 70822 70938 70941 70958 71081 71209 71287 71293 71318 71319 71467 71483 71560 71784 71815 71976 72445 72469 72585 72945 73069 73094 73106 73110 73185 73201 73239 73241 73334 73418 73420 73466 73566 73610 73636 73704 73705 73707 73714 73723 73734 73781 73785 73789 73791 73802 73806 73878 73893 73895 73913 73914 73922 73948 73980 73982 74009 74010 74013 74015 74221 74261 74338 74339 74381 74388 74391 74399 74401 74403 74433 74440 74444 74447 74452 74463 74478 74479 74480 74495 74516 74518 74562 74570 74706 74905 75271 75375 75443 75449 75469 75527 75550 75554 75556 75557 75560 75561 75571 75572 75854 75858 75883 75972 75973 75988 75989 75990 75991 76004 76007 76015 76032 76033 76097 76100 76163 76297 76515 76525 76603 76770 81 77216 77241 77272 77318 77319 77328 77332 77336 77337 77390 77473 77648 77762 77766 77873 77890 77914 78552 78614 78636 78638 78768 78935 78948 78957 78995 79021 79023 79059 79165 79218 79277 79320 79392 79401 79772 79776 79942 79972 80282 80293 80303 80317 80467 80476 80514 80626 80796 81158 81192 81247 81273 81341 81366 81394 81399 81416 81488 81492 81500 81503 81609 81787 81811 81819 81820 81823 81829 81860 81863 81864 81866 81885 81887 81964 82099 82100 82145 82263 82295 82317 82351 82360 82372 82392 82418 82421 82491 82551 82591 82706 82764 82821 82971 82992 83036 83041 83058 83256 83306 83363 83380 83386 83388 83389 83392 83398 83401 83419 83571 83617 83776 83830 83832 83833 83834 83851 83853 83897 83989 84003 84005 84016 84018 84019 84034 84043 84045 84413 84424 84446 84910 84920 84927 84930 84934 84936 84951 84958 84986 84987 84996 85019 85025 85026 85032 85033 85034 85062 85064 85136 85161 85163 85167 85171 85506 85513 85533 85569 85613 85621 85787 86082 86093 86139 86187 86314 86533 86552 86677 86911 87152 87241 87273 87335 87559 87618 87652 87865 87887 87976 88166 88168 88248 88259 88349 88798 88869 89004 89056 89058 89196 89275 89338 89587 89654 89673 89676 89679 89694 89713 89717 89722 89725 89731 89753 89775 89777 89808 89817 89822 89827 89829 89830 89837 89922 89923 90191 90238 90304 90313 90315 90340 90342 90428 90586 90588 90589 90590 90597 90598 90600 90605 90609 90637 90663 90797 90949 91155 91183 91187 91189 91206 91210 91220 91226 91228 91237 91257 91267 91274 91281 91482 91602 91662 91680 91836 91837 91839 91843 91846 91847 91859 91864 91873 91892 91899 91902 91947 91969 82 91988 92112 92153 92260 92591 92649 92662 92666 92679 92698 92800 92812 92814 92863 92868 92899 93023 93029 93030 93038 93072 93272 93320 93340 93573 93590 93618 93960 94027 94511 94817 94827 94840 94845 95243 95381 95384 95451 95454 95464 95467 95469 95633 96312 96316 96336 96361 96406 96716 96720 96966 97081 97092 97096 97101 97103 97107 97111 97115 97136 97172 97196 97287 97301 97322 97325 97330 97331 97339 97342 97345 97352 97353 97402 97488 97490 97635 97647 97686 97691 97692 97725 97726 97727 97733 97749 97777 97921 97922 97923 97930 97931 97935 97946 97947 97948 97951 97956 97957 98015 98030 98221 98254 98416 98422 98534 98535 98542 98633 98811 98895 99045 99056 99060 99073 99116 99119 99121 99126 99127 99129 99130 99132 99134 99176 99247 99254 99280 99282 99295 99296 99304 99305 99306 99362 99493 99593 100161 100166 100191 100198 100366 100648 100686 100696 100721 100731 100736 100779 100786 100810 101176 101252 101259 101299 101321 101499 101734 101753 101766 101770 101787 101791 101803 101848 101955 102037 102065 102115 102159 102317 102356 102666 102669 102929 103039 103146 103190 103203 103342 103343 103344 103353 103428 103498 103630 103872 103927 103942 103948 103956 103971 103994 104021 104043 104047 104066 104067 104148 104226 104265 104432 104436 104438 104514 104516 104535 104581 104667 104702 104704 104705 104739 104749 104771 104830 104876 105001 105216 105242 105243 105248 105304 105325 105445 105482 105668 105738 105746 105748 105763 105769 105772 105777 105841 105845 105849 105853 105860 105861 105863 105870 105876 105882 105884 105946 105961 106127 106131 106133 106143 106212 106227 106238 106248 106330 106405 106688 106875 107191 107356 107366 107386 107762 107975 108001 108003 108196 108222 108268 83 108274 108319 108333 108933 109164 109295 109303 109418 109450 109475 109507 109508 109697 109728 109748 109772 109832 110012 110084 110125 110250 110390 110417 110432 110460 110463 110512 110542 110603 110753 110762 110787 110797 111000 111011 111054 111102 111267 111275 111416 111464 111723 111762 111816 111835 111838 111842 111851 111864 111876 111889 112020 112276 112277 112296 112332 112503 112574 112586 112600 112851 112906 113012 113090 113108 113346 113396 113624 113719 113781 113954 114046 114050 114233 114241 114304 114357 114495 114503 114521 114526 114533 114536 114544 114557 114765 114897 114937 115134 115172 115187 115188 115242 115398 115545 115570 115594 115656 116004 116236 116388 116581 116618 116658 116704 116721 116867 117151 117155 117214 117255 117282 117288 117524 117615 117621 117705 117869 118091 118095 118110 118467 118469 118509 118578 118693 118820 118857 118868 118983 119001 119107 119164 119193 119226 119249 119294 119330 119442 119572 119586 119668 119728 119857 120329 120496 120598 120600 120986 120992 121019 121085 121198 121245 121261 121443 121571 121628 121641 121666 121799 121896 122258 122507 122630 122833 122851 122897 122923 123108 123135 123152 123164 123170 123212 123415 123453 123727 123950 124070 124260 124327 124400 124407 124423 124443 124629 Volume elements : Number of elements not evaluated : 0 % Number of excellent quality elements : 13.02 % Number of good quality elements : 63.99 % Number of average quality elements : 21.02 % Number of poor quality elements : 1.96 % Number of nodes : 244586 Number of line elements : 4518 Number of surface elements : 79851 Number of volume elements : 124713 Mesh order : 2nd order Verifying of linked faces ... 89 face(s) respect strongly (ELEMENTS) linked mesh generator. 84 List of linked faces of which NODES do not verify linked mesh generator: 27 31 39 43 111 114 121 124 List of linked faces without source face (geometry): 119 129 167 ========== checkPhysic ========== Begin of physical check ... Multiplying coefficient for the flux in the coils and for coupling with electrical circuits : Automatically calculated value = 1 The mechanical set PLATES does not have an internal characteristic (inertia, frictions). The internal friction losses torque will not be computed End of physical check. Multiplying coefficient for the flux in the coils and for coupling with electrical circuits : Automatically calculated value = 1 The mechanical set PLATES does not have an internal characteristic (inertia, frictions). The internal friction losses torque will not be computed ========== checkCircuit ========== ========== Loaded macros ========== EXTRUDEFACEREGIONS AUTOMATICREPORT ################################################################################ GEOMETRICAL PROPERTIES ################################################################################ CoordSys Num | Name | Comment | Type | Parent | Origin | Rotations (deg) -------------------------------------------------------------------------------------------------------------------------- 1 | XYZ1 | Standard Cartesian system of coordinates | Cartesian | GlobalUnits | ['0', '0', '0'] | ['0', '0', '0'] 2 | Z_ON_OY | Cartesian system with z axis on -OY | Cartesian | GlobalUnits | ['0', '0', '0'] | ['90', '0', '0'] 85 3 | Z_ON_OX | Cartesian system with z axis on OX. | Cartesian | GlobalUnits | ['0', '0', '0'] | ['90', '90', '0'] 4 | XY1 | Cartesian system of coordinates. | Cartesian | GlobalUnits | ['0', '0', '0'] | ['0', '0', '0'] -------------------------------------------------------------------------------------------------------------------------- InfiniteBoxCylinderZ size = ['150','180','400','450'] -------------------------------------------------------------------------------- DomainType3D lengthUnit = LengthUnit['MILLIMETER'] angleUnit = AngleUnit['DEGREE'] INFINITE_EXP = 2.0 -------------------------------------------------------------------------------- TransfTranslationVector name = 'MAGNET' coordSys = CoordSys['XYZ1'] vector = ['0','0','50'] TransfTranslationVector name = 'PLATE1' coordSys = CoordSys['XYZ1'] vector = ['0','0','50'] TransfTranslationVector name = 'PLATE2' coordSys = CoordSys['XYZ1'] vector = ['0','0','-50'] PointCoordinates Num | CoordSys | uvw | Nature | Mesh ------------------------------------------------------------------------------------------------------------------ 1 | XY1 | ['0', '0', '0'] | STANDARD | AIDED_MESHPOINT001 2 | XY1 | ['75', '0', '0'] | STANDARD | AIDED_MESHPOINT001 3 | XY1 | ['0', '2.375', '0'] | STANDARD | AIDED_MESHPOINT001 4 | XY1 | ['0', '-2.375', '0'] | STANDARD | AIDED_MESHPOINT001 5 | XY1 | ['75', '-2.375', '0'] | STANDARD | AIDED_MESHPOINT001 6 | XY1 | ['75', '2.375', '0'] | STANDARD | AIDED_MESHPOINT001 7 | XY1 | ['0', '3.175', '0'] | STANDARD | AIDED_MESHPOINT001 86 8 | XY1 | ['53.0330085889911', '56.2080085889911', '0'] | STANDARD | AIDED_MESHPOINT001 9 | XY1 | ['75', '3.175', '0'] | STANDARD | AIDED_MESHPOINT001 10 | XY1 | ['0', '-3.175', '0.704991620637E-15'] | STANDARD | AIDED_MESHPOINT001 11 | XY1 | ['75', '-3.175', '0.704991620637E-15'] | STANDARD | AIDED_MESHPOINT001 12 | XY1 | ['53.0330085889911', '-56.2080085889911', '0.124806850608E-13'] | STANDARD | AIDED_MESHPOINT001 ------------------------------------------------------------------------------------------------------------------ PointPropagated Num | Pt orig | Transf | Nature | Mesh ----------------------------------------------------- 13 | 10 | MAGNET | STANDARD | AIDED_MESHPOINT 14 | 11 | MAGNET | STANDARD | AIDED_MESHPOINT 15 | 12 | MAGNET | STANDARD | AIDED_MESHPOINT 16 | 7 | MAGNET | STANDARD | AIDED_MESHPOINT 17 | 8 | MAGNET | STANDARD | AIDED_MESHPOINT 18 | 9 | MAGNET | STANDARD | AIDED_MESHPOINT 19 | 1 | PLATE1 | STANDARD | AIDED_MESHPOINT 20 | 3 | PLATE1 | STANDARD | AIDED_MESHPOINT 21 | 6 | PLATE1 | STANDARD | AIDED_MESHPOINT 22 | 2 | PLATE1 | STANDARD | AIDED_MESHPOINT 23 | 5 | PLATE1 | STANDARD | AIDED_MESHPOINT 24 | 4 | PLATE1 | STANDARD | AIDED_MESHPOINT 25 | 1 | PLATE2 | STANDARD | AIDED_MESHPOINT 26 | 3 | PLATE2 | STANDARD | AIDED_MESHPOINT 27 | 6 | PLATE2 | STANDARD | AIDED_MESHPOINT 28 | 2 | PLATE2 | STANDARD | AIDED_MESHPOINT 29 | 5 | PLATE2 | STANDARD | AIDED_MESHPOINT 30 | 4 | PLATE2 | STANDARD | AIDED_MESHPOINT 31 | 16 | MAGNET | STANDARD | AIDED_MESHPOINT 32 | 17 | MAGNET | STANDARD | AIDED_MESHPOINT 33 | 18 | MAGNET | STANDARD | AIDED_MESHPOINT 34 | 31 | MAGNET | STANDARD | AIDED_MESHPOINT 35 | 32 | MAGNET | STANDARD | AIDED_MESHPOINT 36 | 33 | MAGNET | STANDARD | AIDED_MESHPOINT 37 | 34 | MAGNET | STANDARD | AIDED_MESHPOINT 38 | 35 | MAGNET | STANDARD | AIDED_MESHPOINT 39 | 36 | MAGNET | STANDARD | AIDED_MESHPOINT 40 | 13 | MAGNET | STANDARD | AIDED_MESHPOINT 41 | 14 | MAGNET | STANDARD | AIDED_MESHPOINT 42 | 15 | MAGNET | STANDARD | AIDED_MESHPOINT 43 | 40 | MAGNET | STANDARD | AIDED_MESHPOINT 44 | 41 | MAGNET | STANDARD | AIDED_MESHPOINT 45 | 42 | MAGNET | STANDARD | AIDED_MESHPOINT 46 | 43 | MAGNET | STANDARD | AIDED_MESHPOINT 47 | 44 | MAGNET | STANDARD | AIDED_MESHPOINT 48 | 45 | MAGNET | STANDARD | AIDED_MESHPOINT 49 | 19 | MAGNET | STANDARD | AIDED_MESHPOINT 50 | 20 | MAGNET | STANDARD | AIDED_MESHPOINT 87 51 | 21 | MAGNET | STANDARD | AIDED_MESHPOINT 52 | 22 | MAGNET | STANDARD | AIDED_MESHPOINT 53 | 23 | MAGNET | STANDARD | AIDED_MESHPOINT 54 | 24 | MAGNET | STANDARD | AIDED_MESHPOINT 55 | 49 | MAGNET | STANDARD | AIDED_MESHPOINT 56 | 50 | MAGNET | STANDARD | AIDED_MESHPOINT 57 | 51 | MAGNET | STANDARD | AIDED_MESHPOINT 58 | 52 | MAGNET | STANDARD | AIDED_MESHPOINT 59 | 53 | MAGNET | STANDARD | AIDED_MESHPOINT 60 | 54 | MAGNET | STANDARD | AIDED_MESHPOINT 61 | 55 | MAGNET | STANDARD | AIDED_MESHPOINT 62 | 56 | MAGNET | STANDARD | AIDED_MESHPOINT 63 | 57 | MAGNET | STANDARD | AIDED_MESHPOINT 64 | 58 | MAGNET | STANDARD | AIDED_MESHPOINT 65 | 59 | MAGNET | STANDARD | AIDED_MESHPOINT 66 | 60 | MAGNET | STANDARD | AIDED_MESHPOINT 67 | 61 | MAGNET | STANDARD | AIDED_MESHPOINT 68 | 62 | MAGNET | STANDARD | AIDED_MESHPOINT 69 | 63 | MAGNET | STANDARD | AIDED_MESHPOINT 70 | 64 | MAGNET | STANDARD | AIDED_MESHPOINT 71 | 65 | MAGNET | STANDARD | AIDED_MESHPOINT 72 | 66 | MAGNET | STANDARD | AIDED_MESHPOINT ----------------------------------------------------- LineSegment Num | def points | Nature ----------------------------- 1 | 1,3 | STANDARD 2 | 1,4 | STANDARD 3 | 4,5 | STANDARD 4 | 3,6 | STANDARD 5 | 5,2 | STANDARD 6 | 2,6 | STANDARD 7 | 7,8 | STANDARD 9 | 7,9 | STANDARD 10 | 10,11 | STANDARD 12 | 10,12 | STANDARD 138 | 73,81 | STANDARD 140 | 75,83 | STANDARD 142 | 77,85 | STANDARD 144 | 79,87 | STANDARD 145 | 89,73 | STANDARD 146 | 91,75 | STANDARD 148 | 93,77 | STANDARD 150 | 95,79 | STANDARD 153 | 97,81 | STANDARD 154 | 99,83 | STANDARD 156 | 89,97 | STANDARD 157 | 101,85 | STANDARD 159 | 91,99 | STANDARD 160 | 103,87 | STANDARD 162 | 93,101 | STANDARD 164 | 95,103 | STANDARD ----------------------------- 88 LineArc3PTS Num | def points | Nature ------------------------------ 133 | 73,74,75 | STANDARD 134 | 75,76,77 | STANDARD 135 | 77,78,79 | STANDARD 136 | 79,80,73 | STANDARD 137 | 81,82,83 | STANDARD 139 | 83,84,85 | STANDARD 141 | 85,86,87 | STANDARD 143 | 87,88,81 | STANDARD 147 | 89,90,91 | STANDARD 149 | 91,92,93 | STANDARD 151 | 93,94,95 | STANDARD 152 | 95,96,89 | STANDARD 155 | 97,98,99 | STANDARD 158 | 99,100,101 | STANDARD 161 | 101,102,103 | STANDARD 163 | 103,104,97 | STANDARD ------------------------------ LinePropagated Num | Transf | Appli nb | Line orig | Nature ------------------------------------------------ 15 | MAGNET | 1 | 10 | STANDARD 17 | MAGNET | 1 | 11 | STANDARD 18 | MAGNET | 1 | 12 | STANDARD 21 | MAGNET | 1 | 7 | STANDARD 23 | MAGNET | 1 | 8 | STANDARD 24 | MAGNET | 1 | 9 | STANDARD 27 | PLATE1 | 1 | 1 | STANDARD 29 | PLATE1 | 1 | 4 | STANDARD 31 | PLATE1 | 1 | 6 | STANDARD 33 | PLATE1 | 1 | 5 | STANDARD 35 | PLATE1 | 1 | 3 | STANDARD 36 | PLATE1 | 1 | 2 | STANDARD 39 | PLATE2 | 1 | 1 | STANDARD 41 | PLATE2 | 1 | 4 | STANDARD 43 | PLATE2 | 1 | 6 | STANDARD 45 | PLATE2 | 1 | 5 | STANDARD 47 | PLATE2 | 1 | 3 | STANDARD 48 | PLATE2 | 1 | 2 | STANDARD 49 | MAGNET | 1 | 19 | STANDARD 50 | MAGNET | 1 | 20 | STANDARD 51 | MAGNET | 1 | 21 | STANDARD 52 | MAGNET | 1 | 22 | STANDARD 53 | MAGNET | 1 | 23 | STANDARD 54 | MAGNET | 1 | 24 | STANDARD 55 | MAGNET | 1 | 49 | STANDARD 56 | MAGNET | 1 | 50 | STANDARD 57 | MAGNET | 1 | 51 | STANDARD 58 | MAGNET | 1 | 52 | STANDARD 59 | MAGNET | 1 | 53 | STANDARD 89 60 | MAGNET | 1 | 54 | STANDARD 61 | MAGNET | 1 | 55 | STANDARD 62 | MAGNET | 1 | 56 | STANDARD 63 | MAGNET | 1 | 57 | STANDARD 64 | MAGNET | 1 | 58 | STANDARD 65 | MAGNET | 1 | 59 | STANDARD 66 | MAGNET | 1 | 60 | STANDARD 67 | MAGNET | 1 | 13 | STANDARD 68 | MAGNET | 1 | 14 | STANDARD 69 | MAGNET | 1 | 15 | STANDARD 70 | MAGNET | 1 | 16 | STANDARD 71 | MAGNET | 1 | 17 | STANDARD 72 | MAGNET | 1 | 18 | STANDARD 73 | MAGNET | 1 | 67 | STANDARD 74 | MAGNET | 1 | 68 | STANDARD 75 | MAGNET | 1 | 69 | STANDARD 76 | MAGNET | 1 | 70 | STANDARD 77 | MAGNET | 1 | 71 | STANDARD 78 | MAGNET | 1 | 72 | STANDARD 79 | MAGNET | 1 | 73 | STANDARD 80 | MAGNET | 1 | 74 | STANDARD 81 | MAGNET | 1 | 75 | STANDARD 82 | MAGNET | 1 | 76 | STANDARD 83 | MAGNET | 1 | 77 | STANDARD 84 | MAGNET | 1 | 78 | STANDARD 85 | MAGNET | 1 | 25 | STANDARD 86 | MAGNET | 1 | 26 | STANDARD 87 | MAGNET | 1 | 27 | STANDARD 88 | MAGNET | 1 | 28 | STANDARD 89 | MAGNET | 1 | 29 | STANDARD 90 | MAGNET | 1 | 30 | STANDARD 91 | MAGNET | 1 | 31 | STANDARD 92 | MAGNET | 1 | 32 | STANDARD 93 | MAGNET | 1 | 33 | STANDARD 94 | MAGNET | 1 | 34 | STANDARD 95 | MAGNET | 1 | 35 | STANDARD 96 | MAGNET | 1 | 36 | STANDARD 97 | MAGNET | 1 | 85 | STANDARD 98 | MAGNET | 1 | 86 | STANDARD 99 | MAGNET | 1 | 87 | STANDARD 100 | MAGNET | 1 | 88 | STANDARD 101 | MAGNET | 1 | 89 | STANDARD 102 | MAGNET | 1 | 90 | STANDARD 103 | MAGNET | 1 | 91 | STANDARD 104 | MAGNET | 1 | 92 | STANDARD 105 | MAGNET | 1 | 93 | STANDARD 106 | MAGNET | 1 | 94 | STANDARD 107 | MAGNET | 1 | 95 | STANDARD 108 | MAGNET | 1 | 96 | STANDARD 109 | MAGNET | 1 | 97 | STANDARD 110 | MAGNET | 1 | 98 | STANDARD 111 | MAGNET | 1 | 99 | STANDARD 112 | MAGNET | 1 | 100 | STANDARD 113 | MAGNET | 1 | 101 | STANDARD 90 114 | MAGNET | 1 | 102 | STANDARD 115 | MAGNET | 1 | 103 | STANDARD 116 | MAGNET | 1 | 104 | STANDARD 117 | MAGNET | 1 | 105 | STANDARD 118 | MAGNET | 1 | 106 | STANDARD 119 | MAGNET | 1 | 107 | STANDARD 120 | MAGNET | 1 | 108 | STANDARD 121 | MAGNET | 1 | 109 | STANDARD 122 | MAGNET | 1 | 110 | STANDARD 123 | MAGNET | 1 | 111 | STANDARD 124 | MAGNET | 1 | 112 | STANDARD 125 | MAGNET | 1 | 113 | STANDARD 126 | MAGNET | 1 | 114 | STANDARD 127 | MAGNET | 1 | 115 | STANDARD 128 | MAGNET | 1 | 116 | STANDARD 129 | MAGNET | 1 | 117 | STANDARD 130 | MAGNET | 1 | 118 | STANDARD 131 | MAGNET | 1 | 119 | STANDARD 132 | MAGNET | 1 | 120 | STANDARD ------------------------------------------------ LineExtruded Num | Transf | option | point | Nature ------------------------------------------ 13 | MAGNET | 2 | 10 | STANDARD 14 | MAGNET | 2 | 11 | STANDARD 16 | MAGNET | 2 | 12 | STANDARD 19 | MAGNET | 2 | 7 | STANDARD 20 | MAGNET | 2 | 8 | STANDARD 22 | MAGNET | 2 | 9 | STANDARD 25 | PLATE1 | 2 | 1 | STANDARD 26 | PLATE1 | 2 | 3 | STANDARD 28 | PLATE1 | 2 | 6 | STANDARD 30 | PLATE1 | 2 | 2 | STANDARD 32 | PLATE1 | 2 | 5 | STANDARD 34 | PLATE1 | 2 | 4 | STANDARD 37 | PLATE2 | 2 | 1 | STANDARD 38 | PLATE2 | 2 | 3 | STANDARD 40 | PLATE2 | 2 | 6 | STANDARD 42 | PLATE2 | 2 | 2 | STANDARD 44 | PLATE2 | 2 | 5 | STANDARD 46 | PLATE2 | 2 | 4 | STANDARD ------------------------------------------ 91 ################################################################################ MESH PROPERTIES ################################################################################ MeshPoint Num | Name | Comment | Length unit | Value | Color ------------------------------------------------------------------------------------------------- 1 | SMALL | Small mesh size | MILLIMETER | 1.1875 | Yellow 2 | MEDIUM | Medium mesh size | MILLIMETER | 2.98433409657833 | Turquoise 3 | LARGE | Large mesh size | MILLIMETER | 7.50000000000001 | Red 4 | AIDED_MESHPOINT | MeshPoint of aided mesh | MILLIMETER | Dynamic() | Cyan 5 | AIDED_MESHPOINT001 | MeshPoint of aided mesh | MILLIMETER | Dynamic() | Cyan ------------------------------------------------------------------------------------------------- MeshLineRelativeDeviation Num | Name | Comment | Color | Deviation value ---------------------------------------------------------------------------- 1 | AIDED_MESHLINE | Mesh line by deviation | Cyan | 0.5 2 | AIDED_MESHLINE001 | Mesh line by deviation | Cyan | 0.5 ---------------------------------------------------------------------------- MeshGeneratorAutomaticRelativeDeviation name = 'AIDED_MESHGENERATOR : Automatic mesh by deviation' relativeDeviation = 0.5 MeshGeneratorAutomaticInactiveDeviation name = 'AUTOMATIC : Automatic mesh: triangles, tetraedra elements' -------------------------------------------------------------------------------- MeshGeneratorNoMesh name = 'NO_MESH : No mesh on the faces or volumes' -------------------------------------------------------------------------------- MeshGeneratorMapped name = 'MAPPED : Mapping mesh: rectangles, hexaedra elements' -------------------------------------------------------------------------------- MeshGeneratorExtrusive 92 name = 'MeshGeneratorExtrusive_MAGNET' transf = Transf['MAGNET'] MeshGeneratorExtrusive name = 'MeshGeneratorExtrusive_PLATE1' transf = Transf['PLATE1'] MeshGeneratorExtrusive name = 'MeshGeneratorExtrusive_PLATE2' transf = Transf['PLATE2'] -------------------------------------------------------------------------------- MeshGeneratorLinked name = 'MeshGeneratorLinked_MAGNET' transf = Transf['MAGNET'] ################################################################################ PHYSICAL PROPERTIES ################################################################################ ApplicationPreprocessor3D ApplicationMagneticTransient3D formulationModel = MagneticTransient3DFormulationModelAutomatic approximation = VectorPotentialApproximationEdge scalarVariableOrder = ScalarVariableAutomaticOrder vectorNodalVariableOrder = VectorNodalVariableAutomaticOrder coilCoefficient = CoilCoefficientAutomatic transientInitialization = TransientInitializationStaticComputation -------------------------------------------------------------------------------- DomainType3D lengthUnit = LengthUnit['MILLIMETER'] angleUnit = AngleUnit['DEGREE'] INFINITE_EXP = 2.0 -------------------------------------------------------------------------------- Material name = 'MAGNET' propertyBH = PropertyBhMagnetOneDirection br = '1.2' mur = '1.06' Material name = 'PLATE' propertyBH = PropertyBhLinear mur = '1' propertyJE = PropertyJeLinear 93 rho = '1.724E-8' -------------------------------------------------------------------------------- MechanicalSetFixed name = 'MAGNETS' MechanicalSetCompressibleRemeshing name = 'AIR' MechanicalSetTranslation1Axis name = 'PLATES' kinematics = LinearImposedPosition Position = '0.02*Sin(2*Pi*TIME)' initialVelocity = '0' translationAxis = TranslationZAxis coordSys = CoordSys['XYZ1'] RegionVolume name = 'MAGNETS1' magneticTransient3D = MagneticTransient3DVolumeMagnetic material = Material['MAGNET'] formulation = Formulation['MT3SCA'] color = Color['Turquoise'] visibility = Visibility['INVISIBLE'] mechanicalSet = MechanicalSet['MAGNETS'] RegionVolume name = 'PLATES' magneticTransient3D = MagneticTransient3DVolumeSolidConductor material = Material['PLATE'] formulation = Formulation['MT3TWOM'] color = Color['Turquoise'] visibility = Visibility['VISIBLE'] mechanicalSet = MechanicalSet['PLATES'] RegionVolume name = 'MAGNETS2' magneticTransient3D = MagneticTransient3DVolumeMagnetic material = Material['MAGNET'] formulation = Formulation['MT3SCA'] color = Color['Turquoise'] visibility = Visibility['INVISIBLE'] mechanicalSet = MechanicalSet['MAGNETS'] RegionVolume name = 'AIR' magneticTransient3D = MagneticTransient3DVolumeVacuum formulation = Formulation['MT3SCA'] color = Color['Turquoise'] visibility = Visibility['INVISIBLE'] mechanicalSet = MechanicalSet['AIR'] RegionVolume 94 Num | Name | Color | Type | Mechanical set --------------------------------------------------------------------------------------- 1 | MAGNETS1 | Turquoise | MagneticTransient3DVolumeMagnetic | MAGNETS 2 | PLATES | Turquoise | MagneticTransient3DVolumeSolidConductor | PLATES 3 | MAGNETS2 | Turquoise | MagneticTransient3DVolumeMagnetic | MAGNETS 4 | AIR | Turquoise | MagneticTransient3DVolumeVacuum | AIR --------------------------------------------------------------------------------------- -------------------------------------------------------------------------------- ################################################################################ SPATIAL PROPERTIES ################################################################################ VariationParameterPilot Num | Name | Comment | Ref. value ----------------------------------------------------------------------------- 1 | TIME | Time (s) | 0.0 2 | LINPOS_PLATES | Linear pos. (m) of mechanical set PLATES | 0.0 ----------------------------------------------------------------------------- ################################################################################ SOLVING PROPERTIES ################################################################################ Scenario name = 'SCENARIO_1' pilots = [MultiValues parameter = VariationParameter['TIME'] intervals = [IntervalStepValue minValue = 0.0 maxValue = 1.1 stepValue = 0.05]] adaptive = InactivatedAdaptive -------------------------------------------------------------------------------- SolvingOptions parametersLinearSystemSolvers = ParametersLinearSystemSolvers 95 linearSystemSolverChoice = LinearSystemSolverAutomatic parametersIccg = ParametersIccgAutomatic parametersGmres = ParametersGmresAutomatic parametersBicgstab = ParametersBicgstabAutomatic parametersSuperlu = ParametersSuperluAutomatic parametersMumps = ParametersMumpsAutomatic parametersPardiso = ParametersPardisoAutomatic parametersAutomaticScaling = ParametersAutomaticScalingAutomatic userSubroutineLinearSolverParameters = ParametersLinearSolverByUserSubroutine preLinearSolverChoice = PreLinearSolverAutomatic newtonRaphsonParameters = ParametersNewtonRaphson precision = 1.0E-4 maximumIterationNumber = 100 relaxationFactorComputationMethod = ParametersNewtonRaphsonRelaxationFactorComputationMethodAutomatic electromagneticThermalCouplingParameters = ParametersElectromagneticThermalCoupling updatingPrecision = 0.01 minAndMaxUpdatingNumbers = [1,5] updatingsNonConvergenceBehavior = UpdatingsNonConvergenceContinue initialTemperature = 293.15 t0ComputationMethod = T0ComputationMethodAutomatic transientVariableSetToZeroType = TransientVariablesResettingAutomatic automaticSaving = autoFrequencySaving rescalingFactors = RescalingFactors i1RescalingFactor = 1.0 i2RescalingFactor = 1.0 u1RescalingFactor = 0.001 v1eitRescalingFactor = 1.0 v2eitRescalingFactor = 1.0 t1wRescalingFactor = 1.0 t2wRescalingFactor = 1.0 edgeElementGaugeMethod = EdgeElementGaugeMethodAutomatic MatMethodConstruction = BuildMatrixMethodAutomaticDetermination symFormulations = autoFormulations theta = 1.0 hj1ndSplittingMethod = Hj1ndComputationAutomaticSplittingDetermination nonlinearZsComputationMethod = NonlinearZsComputationMethod1DFEModel heElementNumber = 0 -------------------------------------------------------------------------------- AdaptiveSolverOptions energyErrorThreshold = MediumThresholdAdaptiveSolver iterationsMaxNumber = 4
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Theory and simulation of electromagnetic dampers for earthquake engineering applications Haji Akbari Fini, Siavash 2016
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Title | Theory and simulation of electromagnetic dampers for earthquake engineering applications |
Creator |
Haji Akbari Fini, Siavash |
Publisher | University of British Columbia |
Date Issued | 2016 |
Description | The present study develops applications of electromagnetic devices in Civil Engineering. Three different types of electromagnetic system are investigated through mathematical and numerical models. Chapter 3 deals with Coil-Based Electromagnetic Damper (CBED). CBEDs can operate as passive, semi-active and active systems. They can also be considered as energy harvesting systems. However, results show that CBEDs cannot simultaneously perform as an energy harvesting and vibration control system. In order to assess the maximum capacity of CBEDs, an optimization is conducted. Results show that CBEDs can produce high damping density only when they are considered as a passive vibration control system. Chapter 4 deals with the development of a novel Eddy Current Damper (ECD). The eddy current damper uses permanent magnets arranged in a circular manner to create a strong magnetic field, where specially shaped conductive plates are placed between the permanent magnets to cut through the magnetic fields. Detailed analytical equations are derived and verified using the finite element analysis program Flux. The verified analytical models are used to optimize the damper design to reach the maximum damping capacity. The analytical simulation shows that the proposed eddy current damper can provide a high damping density up to 2,733 kN-s/m⁴. The Hybrid Electromagnetic Damper (HEMD) are developed and designed in Chapter 5. The idea is to couple the CBED and ECD with the aim of designing a semi-active, active and energy harvesting electromagnetic damper. The simulation results show that it is feasible to manufacture hybrid electromagnetic dampers for industrial applications. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2016-03-01 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivs 2.5 Canada |
DOI | 10.14288/1.0225869 |
URI | http://hdl.handle.net/2429/57068 |
Degree |
Master of Applied Science - MASc |
Program |
Civil Engineering |
Affiliation |
Applied Science, Faculty of Civil Engineering, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2016-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/2.5/ca/ |
AggregatedSourceRepository | DSpace |
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