Coulombian Model Research Articles

The design of negative stiffness spring for precision vibration isolation using axially magnetized permanent magnet rings

A negative stiffness element is always employed to generate high-static–low-dynamic stiffness characteristic of the vibration isolator, reduce the resonance frequency of the isolator, and improve the vibration isolation performance under low and ultra-low frequency excitation. In this paper, a new compact negative stiffness permanent magnetic spring (NSPMS) that is composed of two axial-magnetized permanent magnetic rings is proposed. An analytical expression of magnetic negative stiffness of the NSPMS is deduced by using the Coulombian model. After analyzing the effect of air-gap width, air-gap position, height difference between the inner ring and outer ring on the negative stiffness characteristic, a design procedure is proposed to realize the negative stiffness characteristic with a global minimum linear component and uniformity stiffness near the equilibrium position. Finally, an experimental prototype is developed to validate the effectiveness of the NSPMS. The experimental results show that combining a vibration isolator with the NSPMS in parallel can lower the natural frequency and improve the isolation performance of the isolator.

A hybrid (permanent magnet and foil) bearing set for complete passive levitation of high-speed rotors

This paper presents the design and development of a hybrid bearing set for complete passive levitation of a typical rotor. A hybrid bearing set consists of permanent magnet thrust bearing and radial discrete bump foil bearings. The permanent magnet thrust bearing is made up of three pairs of ring magnets arranged in rotation magnetized direction. The mathematical model to determine the force and stiffness in rotation magnetized direction configuration is presented using Coulombian model and vector approach. Bump foil bearings are designed and developed for rotor weight to provide the radial support to the rotor system. The proposed bearing set with rotor is analysed using finite element analysis for rotor dynamic characteristics. The experiments are conducted on the fabricated rotor-bearing configuration by rotating the rotor up to the speeds of 40,000 r/min. The system response is acquired using advanced rotor-dynamic data acquisition system. The experimental results show that the rotor is completely airborne and stable at the desired speed.

Generalized Three-Dimensional Mathematical Models for Force and Stiffness in Axially, Radially, and Perpendicularly Magnetized Passive Magnetic Bearings With “n” Number of Ring Pairs

This work deals with generalized three-dimensional (3D) mathematical model to estimate the force and stiffness in axially, radially, and perpendicularly polarized passive magnetic bearings with “n” number of permanent magnet (PM) ring pairs. Coulombian model and vector approach are used to derive generalized equations for force and stiffness. Bearing characteristics (in three possible standard configurations) of permanent magnet bearings (PMBs) are evaluated using matlab codes. Further, results of the model are validated with finite element analysis (FEA) results for five ring pairs. Developed matlab codes are further utilized to determine only the axial force and axial stiffness in three stacked PMB configurations by varying the number of rings. Finally, the correlation between the bearing characteristics (PMB with only one and multiple ring pairs) is proposed and discussed in detail. The proposed mathematical model might be useful for the selection of suitable configuration of PMB as well as its optimization for geometrical parameters for high-speed applications.

Magnetic Bearing Using Rotation Magnetized Direction Configuration

Passive magnetic bearing (PMB), made of high remanence rare earth permanent magnets, is brittle in nature; therefore, precautions must be taken to reduce the chances of vibration transmitting to the permanent magnets. In the present work, a rotation magnetized direction (RMD) structure made of aluminum ring and square shaped magnetic pieces has been proposed. A comparative study of load carrying capacities of sector magnets and square magnets has been presented. Three-dimensional (3D) Coulombian model was solved to estimate the load carrying capacity. Theoretical and experimental studies on the load carrying capacities of full ring magnet (more prone to cracking) and the proposed structure have been presented to prove the superiority of the proposed structure. In addition to load capacity, comparison between amplitudes of vibration at different frequencies, orbit plots, and time taken for breakage of the magnets at the resonance frequency has been presented.

3-D Analysis for the Torque of Permanent Magnet Coupler

This paper first presents a 3-D analysis model for calculating the torque transmitted between the permanent radial magnet couplers based on the Coulombian model. Then, a 3-D finite-element analysis model is constructed to analyze the spatial distributions of the magnetic field and end effect of the axial length of the magnet coupler. Besides, the parametric studies of the magnet coupler are taken using the two 3-D models, respectively, and the simulation results obtained are analyzed and compared. Finally, two kinds of real applied magnet couplers are tested to compare with the results with the 3-D analysis model. It is concluded that the 3-D analytical model of the torque of the radial magnet couplers is viable and could be used for the design of magnet couplers.

Theoretical and Experimental Study for Hybrid Journal Bearing

Journal bearings are well known for their lowest wear rate and high damping coefficients under fully developed hydrodynamic lubrication mechanism. However, metal-metal contact at start/stop and low speed condition, causing bearing wear and excessive power loss, are major disadvantages of hydrodynamic bearing. Therefore, a concept of hybrid (hydrodynamic-permanent magnetic) bearing for the radial applied load has been developed. In such bearings the static load is taken by magnetic bearing and dynamic load by hydrodynamic bearing. However, due to the unstable nature of passive magnetic bearings and larger axial thrust compared to radial thrust emphasis for the requirement of extra subassembly and proper alignment compared to conventional bearings test setup. In the present work, using 3D Coulombian model, the radial and axial forces of a passive magnetic bearing for the given dimension having different arc length of the top stator magnet have been analyzed. To reduce the unstable force, the required axial offset was estimated. After obtaining the required configuration from the theoretical results, the hybrid bearing was developed. Experiments were performed on the preliminary developed experimental setup and the challenges faced for development of setup and problems faced after performing the experiment have been reported. A detailed investigation was carried out to overcome the problems and various solutions have been provided.

ANALYSIS OF RADIAL MAGNETIZED PERMANENT MAGNET BEARING CHARACTERISTICS FOR FIVE DEGREES OF FREEDOM

This paper presents a simple mathematical model to determine the force, stifiness and moment parameters in Permanent Magnet (PM) bearings made of radial magnetized ring magnets using Coulombian model and vector approach for flve degrees of freedom. MATLAB codes are written to evaluate the bearing characteristics for three translational (x, y and z) and two angular (» and ∞) degrees of freedom of the rotor magnet. The results of the mathematical model are compared with the results of Finite Element Analysis (FEA) using ANSYS and experiments for a PM bearing with one ring pair, thereby the presented mathematical model is validated. Furthermore, the PM bearing with three ring pairs with alternate radial polarizations is analysed by extending the presented mathematical model and also using ANSYS. Finally, the 5£5 stifiness matrix consisting of principal and cross coupled values is presented for the elementary structure as well as for the stacked structure with three ring pairs.

ANALYSIS OF RADIAL MAGNETIZED PERMANENT MAGNET BEARING CHARACTERISTICS

The use of permanent magnets as bearings has gained attention of researchers nowadays. The characteristics of forces and moments have to be analysed thoroughly for the proper design of permanent magnet bearings. This paper presents a mathematical model of an axially magnetized permanent magnet bearing (ring magnets) using Coulombian model and a vector approach to estimate the force, moment and stifiness. A MATLAB code is developed for evaluating the parameters for flve degrees of freedom (three translational and two rotational) of the rotor. Furthermore, it is extended to analyse stacked ring magnets with alternate axial polarizations. The proposed model is validated with the available literature. Comparison of force and stifiness results of the presented model with the results of three dimensional (3D) flnite element analysis using ANSYS shows good agreement. Finally, the cross coupled stifiness values in addition to the principal stifiness values are presented for elementary structures and also for stacked structures with three ring permanent magnets.

ANALYSIS OF AXIALLY MAGNETIZED PERMANENT MAGNET BEARING CHARACTERISTICS

The use of permanent magnets as bearings has gained attention of researchers nowadays. The characteristics of forces and moments have to be analysed thoroughly for the proper design of permanent magnet bearings. This paper presents a mathematical model of an axially magnetized permanent magnet bearing (ring magnets) using Coulombian model and a vector approach to estimate the force, moment and stifiness. A MATLAB code is developed for evaluating the parameters for flve degrees of freedom (three translational and two rotational) of the rotor. Furthermore, it is extended to analyse stacked ring magnets with alternate axial polarizations. The proposed model is validated with the available literature. Comparison of force and stifiness results of the presented model with the results of three dimensional (3D) flnite element analysis using ANSYS shows good agreement. Finally, the cross coupled stifiness values in addition to the principal stifiness values are presented for elementary structures and also for stacked structures with three ring permanent magnets.

Cylindrical Magnets and Coils: Fields, Forces, and Inductances

This paper presents a synthesis of analytical calculations of magnetic parameters (field, force, torque, stiffness) in cylindrical magnets and coils. By using the equivalence between the amperian current model and the coulombian model of a magnet, we show that a thin coil or a cylindrical magnet axially magnetized have the same mathematical model. Consequently, we present first the analytical expressions of the magnetic field produced by either a thin coil or a ring permanent magnet whose polarization is axial, thus completing similar calculations already published in the scientific literature. Then, this paper deals with the analytical calculation of the force and the stiffness between thin coils or ring permanent magnets axially magnetized. Such configurations can also be modeled with the same mathematical approach. Finally, this paper presents an analytical model of the mutual inductance between two thin coils in air. Throughout this paper, we emphasize why the equivalence between the coulombian and the amperian current models is useful for studying thin coils or ring permanent magnets. All our analytical expressions are based on elliptic integrals but do not require further numerical treatments. These expressions can be implemented in Mathematica or Matlab and are available online. All our models have been compared to previous analytical and semianalytical models. In addition, these models have been compared to the finite-element method. The computational cost of our analytical model is very low, and we find a very good agreement between our analytical model and the other approaches presented in this paper.

Performances and design of ironless loudspeaker motor structures

This paper presents several kinds of structures of ironless loudspeaker motors. The proposed devices go from simple structures, in terms of manufacturing process, to structures in which the magnet rings can be stacked, with different magnetization directions or values. The structures are studied and compared according to their created induction level and uniformity across the coil displacement range, which are directly linked to the loudspeaker performances (force factor). The model used is the coulombian model of magnets to calculate the magnetic field created, therefore all the calculations are analytical. The paper shows that no structure is universal when it comes to the loudspeaker design: some are well adapted to micro displacements of the coil while other structures are adapted to large displacements.

Force and Stiffness of Passive Magnetic Bearings Using Permanent Magnets. Part 2: Radial Magnetization

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> This paper deals with the calculation of the force and the stiffness between two ring permanent magnets whose polarization is radial. Such a configuration corresponds to a passive magnetic bearing. The magnetic force exerted between ring permanent magnets is determined by using the Coulombian model. The expressions obtained are semianalytical and we show that it is not possible to find an exact analytical expression of the force between two ring permanent magnets. Then, thanks to these semianalytical calculations, the ring dimensions are optimized in order to have a great force or a great stiffness. Moreover, we show that the relative position of the rings for which the force is the strongest depends on the air gap dimension. This result is new because the curvature effect is taken into account in this paper. We can say that such semianalytical expressions are more precise than the numerical evaluation of the magnetic forces obtained with the finite-element method. Moreover, semianalytical expressions have a low computational cost whereas the finite-element method has a high one. Thereby, as shown in this paper, such calculations allow an easy optimization of quadripolar lenses or devices using permanent magnets. </para>

Force and Stiffness of Passive Magnetic Bearings Using Permanent Magnets. Part 1: Axial Magnetization

This paper deals with the calculation of the force and the stiffness between two ring permanent magnets whose polarization is axial. Such a configuration corresponds to a passive magnetic bearing. All the calculations are determined by using the Coulombian model. The paper also discusses the optimal ring dimensions that provide a large force or stiffness between the rings. Such properties are commonly sought in passive magnetic bearings. We propose a three-dimensional method for optimizing these parameters. An important result is established in this paper: the exact relative position of the rings for which the force is the strongest depends on the air gap dimension. As the expressions in this paper give this exact relative position, manufacturers can easily optimize their passive magnetic bearings. This result is new because the curvature effect is taken into account. Such semianalytical expressions are more precise than the numerical evaluation of the magnetic forces obtained with the finite-element method. In addition, semianalytical expressions have a low computational cost whereas the finite-element method is computation-intensive. Such calculations make it easy to optimize quadripolar lenses and other devices utilizing permanent magnets.

Magnetic Field Created by Tile Permanent Magnets

This paper presents an analytical calculation of the three components of the magnetic field created by tile permanent magnets whose magnetization is either radial or axial. The calculations are based on the Coulombian model of permanent magnets. The magnetic field is directly calculated, without the magnetic potential. Both axial and radial magnetization of the tiles are considered. The expressions obtained give the magnetic field in all the space. Such analytical expressions are very useful for the design and optimization of many industrial applications.

Permanent Magnet Couplings: Field and Torque Three-Dimensional Expressions Based on the Coulombian Model

This paper presents three-dimensional expressions for the optimization of permanent-magnet couplings. First, we give a fully analytical expression of the azimuthal field created by one arc-shaped permanent magnet radially polarized which takes into account its magnetic pole volume density. Such an expression has a very low computational cost and is exact for all points in space. Then, we propose two semianalytical expressions of the azimuthal force and the torque exerted between two arc-shaped permanent magnets. These expressions are valid for thick or thin arc-shaped permanent magnets. Furthermore, this approach allows us to realize easily parametric studies and optimizations. The analytical approach taken in this paper, based on the Coulombian model, is a good alternative compared to the finite element method generally used to study such configurations.

Torque in permanent magnet couplings: Comparison of uniform and radial magnetization

We present a three-dimensional study of the torque transmitted between tile permanent magnets uniformly magnetized. All this study is based on the Coulombian model. The torque is calculated semianalytically by considering all the surface densities that appear on the tiles. In addition, no simplifying assumptions are done in the expressions given in this paper. Consequently, the evaluation of the torque is very accurate and allows us to show the drawbacks of using tile permanent magnets uniformly magnetized instead of using tile permanent magnets radially magnetized. Such an approach is useful because it allows us to realize easily parametric studies.

ANALYTICAL EXPRESSION OF THE MAGNETIC FIELD CREATED BY TILE PERMANENT MAGNETS TANGENTIALLY MAGNETIZED AND RADIALS CURRENT IN MASSIVE DISKS

In this paper, we present new expressions for calculating the magnetic field produced by either tile permanent magnets tangentially magnetized or by radial currents in massive disks. These expressions are fully analytical, that is, we do not use any special functions for calculating them. In addition, they are three-dimensional and can be used for calculating the magnetic field for all regular points in space. The expressions commonly used for calculating the magnetic field produced by radial currents in massive disks are often based on elliptic integrals or semi-analytical forms. We propose in this paper an alternative analytical method that can also be used for tile permanent magnets. Indeed, by using the analogy between the coulombian model and the amperian current model, radial currents in massive disks can be represented by using the fictitious magnetic pole densities that are located on two faces of a tile permanent magnet tangentially magnetized. The two representations are equivalent and thus, the shape of magnetic field produced is the same for all points in space, with a smaller value in the case of it is produced by radial currents in massive disks. Such expressions can be used for realizing easily parametric studies.

INTRODUCING FICTITIOUS CURRENTS FOR CALCULATING ANALYTICALLY THE ELECTRIC FIELD IN CYLINDRICAL CAPACITORS

The aim of this paper is to show the interest of using equivalence models for calculating the electric fleld produced by cylindrical capacitors with dielectrics. To do so, we use an equivalent model, based on the dual Maxwell's Equations for calculating the two electric fleld components created inside the capacitor and outside it. This equivalent model uses flctitious currents generating a electric vector potential that allows us to determine the electric fleld components in all points in space. The electric fleld produced by charge distributions as capacitor with dielectrics is generally determined by using the coulombian model. Indeed, it is well known that the electric fleld derives from a scalar potential. By using the Maxwell's equations, this scalar potential is in fact linked to the existence of charge distributions that are generally located on the faces of the capacitors. However, this last model does not allow us to obtain reduced analytical expressions since it involves the calculation of charge volume density appearing in the dielectric material for arc- shaped cylindrical topologies. Consequently, it is interesting to look for another approach that gives analytical expressions with a lower computational cost. In this paper, we show that the use of flctitious currents instead of charges allow us to obtain 3D analytical reduced expressions with a lower computational cost. This analytical approach is compared to the coulombian model for showing the equivalence between the two approaches.

DISCUSSION ABOUT THE ANALYTICAL CALCULATION OF THE MAGNETIC FIELD CREATED BY PERMANENT MAGNETS

This paper presents an improvement of the calculation of the magnetic field components created by ring permanent magnets. The three-dimensional approach taken is based on the Coulombian Model. Moreover, the magnetic field components are calculated without using the vector potential or the scalar potential. It is noted that all the expressions given in this paper take into account the magnetic pole volume density for ring permanent magnets radially magnetized. We show that this volume density must be taken into account for calculating precisely the magnetic field components in the near-field or the far-field. Then, this paper presents the component switch theorem that can be used between infinite parallelepiped magnets whose cross-section is a square. This theorem implies that the magnetic field components created by an infinite parallelepiped magnet can be deducted from the ones created by the same parallelepiped magnet with a perpendicular magnetization. Then, we discuss the validity of this theorem for axisymmetric problems (ring permanent magnets). Indeed, axisymmetric problems dealing with ring permanent magnets are often treated with a 2D approach. The results presented in this paper clearly show that the two-dimensional studies dealing with the optimization of ring permanent magnet dimensions cannot be treated with the same precisions as 3D studies.

MAGNETIC FIELD PRODUCED BY A TILE PERMANENT MAGNET WHOSE POLARIZATION IS BOTH UNIFORM AND TANGENTIAL

This paper presents the exact 3D calculation of the magnetic field produced by a tile permanent magnet whose polarization is both tangential and uniform. Such a calculation is useful for optimizing magnetic couplings or for calculating the magnetic field produced by alternate magnet structures. For example, our 3D expressions can be used for calculating the magnetic field produced by a Halbach structure. All our expressions are determined by using the coulombian model. This exact analytical approach has always proved its accuracy and its usefulness. As a consequence, the tile permanent magnet considered is represented by using the fictitious magnetic pole densities that are located on the faces of the magnet. In addition, no simplifying assumptions are taken into account for calculating the three magnetic field components. Moreover, it is emphasized that the magnetic field expressions are fully three-dimensional. Consequently, the expressions obtained are valid inside and outside of the tile permanent magnet, whatever its dimensions. Such an approach allows us to realize easily parametric studies.