Magnet forces

This is crucial because a significant amount of energy is wasted due to friction in the main roller bearing. In order to overcome this, the concept of levitation has gained popularity. Levitation is achieved by employing the repelling forces between two opposing poles of a permanent magnet (PM), significantly reducing friction between the turbine stator and rotor. As a result, the overall energy production of the turbine increases. The use of passive permanent magnet bearings has several disadvantages, such as limited load-bearing capacity and rigidity. To address these limitations, we conducted numerical studies on four different configurations in order to enhance load-bearing capacity and stiffness. The results showed that the radial configuration outperformed the axial-type configuration in terms of stiffness and load-bearing capabilities in all four arrangements. Furthermore, it was revealed that radial passive magnetic bearings with adequate air gaps are not only more efficient but also less expensive than employing iron cores at the rear and between ring magnets for small-scale wind turbines.

We present exact three-dimensional semi-analytical expressions of the force exerted between two coaxial thick coils with rectangular cross-sections. Then, we present a semi-analytical formulation of their mutual inductance. For this purpose, we have to calculate six and seven integrations for calculating the force and the mutual inductance respectively. After mathematical manipulations, we can obtain semi-analytical formulations based on only two integrations. It is to be noted that such integrals can be evaluated numerically as they are smooth and derivable. Then, we compare our results with the filament and the finite element methods. All the results are in excellent agreement.

The magnetic field analysis based on the wavelet transform is performed. The Halbach array magnetic field analysis has been studied using many methods such as magnetic scalar potential, magnetic vector potential, Fourier analysis and Finite Element Methods. But these analyses cannot identify a transient oscillation at the beginning stage of levitation. The wavelet

New accurate approximation is proposed using integral expressions for evaluating the magnetic force between cylindrical permanent magnet arrays. The magnetic field distribution is calculated analytically by using Coulombian model. In this paper, every cylindrical magnet is divided into elementary cuboidal magnets. The accuracy can be controlled by regulating the value of elementary cuboidal permanent magnets "N ". The approximation can also be used to calculate the force interaction in the cylindrical linear single-axis-actuator. We confirm the validity of magnetic force calculation by comparing it with other methods and measurements. The calculation results are in very good agreement with measured values, which indicates the feasibility of our approximation.

We developed a simple method to calculate the axial force between concentric thin walled solenoids. To achieve this, the force between them was mapped as a function of their geometrical relations based on separation-to-diameter ratios. This resulted in an equation and a set of data. We used them together to calculate axial forces between two coaxial thin walled solenoids. With this method, direct evaluation of elliptical integrals was circumvented, and the forces were obtained with a simple expression. The results were validated against solutions obtained with an existing semi-analytical method and force measurements between high coercivity permanent magnets.

New accurate approximation is proposed using integral expressions for evaluating the magnetic force between cylindrical permanent magnet arrays. The magnetic field distribution is calculated analytically by using Coulombian model. In this paper, every cylindrical magnet is divided into elementary cuboidal magnets. The accuracy can be controlled by regulating the value of elementary cuboidal permanent magnets "N ". The approximation can also be used to calculate the force interaction in the cylindrical linear single-axis-actuator. We confirm the validity of magnetic force calculation by comparing it with other methods and measurements. The calculation results are in very good agreement with measured values, which indicates the feasibility of our approximation.

Analytical calculation of interaction forces between two cylindrical magnets is a difficult problem. Solutions can be obtained by elliptic integrals or by numerical computation. Thanks to the development of the analytical calculation for parallelepiped magnets (interaction energy, force and torque), the calculation of the force component between two cylinders can be obtained in analytical form by decomposition in a finite number of elements. This type of calculation allows the calculation of many systems that uses the forces of interaction between cylinder magnets. For example it can be used to directly calculate the force exerted in permanent magnet bearings in non-centred position.

The energy in permanent magnet is not a trivial problem because it exist two types of energy: the field energy and the demagnetizing energy. For parallelepiped shape, the magnet energy has been calculated by fully analytical expressions in 3D. The result has been obtained after four successive integrations of Arctangent and Logarithm functions. The analytically calculated energy corresponds to the demagnetization energy in the magnet. The analytical results have been compared with different terms of energy obtained by computation.

Usely, in analytical calculation of magnetic and mechanical quantities of Halbach systems, the authors use the Fourier series approximation because the exact calculations are more difficult. In this work the interaction forces between linear Halbach arrays are analytically calculated thanks to our recent development 3D exact calculation of forces between two cuboïdal magnets with parallel and perpendicular magnetization. We essentially describe the way to separately calculate the forces between two magnets, between one magnet and a Halbach array and between two Halbach systems

Up to now, the analytical calculation has been made only when the magnets own parallel magnetization directions. We have succeeded in two new results of first importance for the analytical calculation: the torque between two magnets, and the force components and torque when the magnetization directions are perpendicular. The last result allows the analytical calculation of the interactions when the magnetizations are in all the directions. The 3D analytical expressions are difficult to obtain, but the torque and force expressions are very simple to use. As example the analytical expression can be included in optimization software allowing to directly obtaining the shape optimization by a fast way.

Most of the systems working by magnet interactions can be calculated by superposition of the interactions between parallelepiped elementary magnets. Each elementary magnet is submitted to a force and a torque. By 3-D fully analytical calculation, up to now, only the force components problem has been solved. It was published for the first time in 1984. Until now, the torque components have never been analytically calculated because of the angular derivation. We have solved it, so that now all the interactions (energy, three forces components, and three torque components) can be expressed by analytical formulations. The only hypothesis is that the magnetizations and are supposed to be rigid and uniform in each magnet. The analytical calculation is made by replacing magnetizations by distributions of magnetic charges on the magnet poles. The torque is calculated for rotational movement of the second magnet around its center. The three components of the torque are written with functions based on logarithm and arctangent. The results have been verified and validated by comparison with finite-element calculation. Analytical calculation owns many advantages in comparison with other calculation methods. The 3-D analytical expressions are difficult to obtain, but the expressions of energy, force, and torque are very simple to use. For example, the analytical expression can be included in optimization software allowing to directly obtaining the shape optimization by a fast way.

The work is devoted to a geometrical configuration of permanent magnets on the basis of opposing geometrically linear assemblies (e.g. Halbach arrays) for the generation of strong magnetic fields, which have been theoretically modeled and experimentally verified. The implementation of these opposing assemblies using NdFeB magnets of a total weight of 3.75 kg provided a value of magnetic flux density in the middle of an air gap of a width of 20 mm that was higher by 56% in comparison with the simplest possible design. When the air-gap width was 3 mm, the flux density reached a value of 2.16 T, which represents an increase of more than 100%. Simultaneously, however, unlike in the simplest possible parallel configuration, opposing Halbach assemblies have shown, in the middle of an air gap, a significant decrease of the magnetic flux density values when passing from the middle of the assemblies in the direction parallel to the x-axis.

Even though the self-inductance calculation of the disk coil (pancake) has been given by many authors (Spielrein, Grover, Dwight, Kalantarov, Kajikawa, Babic, Akyel, Yu, Conway, Luo) it is also the challenge in this time to obtain the simpler form of presented formulas. The expressions for calculating the self-inductance of the pancake are given over the convergent series, complete elliptic integrals or generalized hypergeometric functions. Definitely the closed form doesn’t exist except in the special cases. The propose of this paper is to give relatively simple and fast method comfortable for the potential users such as physicist and engineers. With the Mathematica code all proposed different formulas give the same results for all range of the parameters describing the disk coil (pancake). All expressions are obtained over the complete elliptic integral of the first kind and two terms given by the simple integrals which converge on the whole interval of integration. These integral...

In this paper we calculate the mutual inductance and the magnetic force between the thick Bitter coil of rectangular cross section with the inverse radial current and the thin wall superconducting solenoid with the constant azimuthal current. The semi-analytical and the analytical expressions of these magnetic quantities are obtained over complete elliptic integrals of the first and second kind as well as Heuman's Lambda function. There is a simple integral which has to be solved numerically by some of numerical integrations. The results of this method are compared by those obtained by the modified filament method for the presented configuration. All results are in an excellent agreement.

Even though the self-inductance calculation of the disk coil (pancake) has been given by many authors (Spielrein, Grover, Dwight, Kalantarov, Kajikawa, Babic, Akyel, Yu, Conway, Luo) it is also the challenge in this time to obtain the simpler form of presented formulas. The expressions for calculating the self-inductance of the pancake are given over the convergent series, complete elliptic integrals or generalized hypergeometric functions. Definitely the closed form doesn’t exist except in the special cases. The propose of this paper is to give relatively simple and fast method comfortable for the potential users such as physicist and engineers. With the Mathematica code all proposed different formulas give the same results for all range of the parameters describing the disk coil (pancake). All expressions are obtained over the complete elliptic integral of the first kind and two terms given by the simple integrals which converge on the whole interval of integration. These integral...

New accurate approximation is proposed using integral expressions for evaluating the magnetic force between cylindrical permanent magnet arrays. The magnetic field distribution is calculated analytically by using Coulombian model. In this paper, every cylindrical magnet is divided into elementary cuboidal magnets. The accuracy can be controlled by regulating the value of elementary cuboidal permanent magnets "N ". The approximation can also be used to calculate the force interaction in the cylindrical linear single-axis-actuator. We confirm the validity of magnetic force calculation by comparing it with other methods and measurements. The calculation results are in very good agreement with measured values, which indicates the feasibility of our approximation.

Applications of permanent magnets bearings have gained a new interest thanks to the development of rare earth materials, characterised by residual magnetic induction greater than 1 T. The present paper proposes a new geometry for permanent magnets bearings with V-shaped elements, both for a plane slide and for cylindrical bearings. The aim of this geometry is to give new possibilities to the application of these bearing systems, by reducing its inherent instability. A design method, involving Finite Elements and Magnetic Field Integral Equations analyses, is also described for defining the most suitable V-opening angle and the two magnetisation directions. These parameters can be varied in order to reduce the unstable force in the coupling, and to reach the desired force and stiffness in the stable direction. The design is founded on the evaluation of four "geometric" vectors, that depend on the geometry of the elements. Some results are reported for a reference geometry for both the slide and the cylindrical bearings.

Up to now, the analytical calculation has been made only when the magnets own parallel magnetization directions. We have succeeded in two new results of first importance for the analytical calculation: the torque between two magnets, and the force components and torque when the magnetization directions are perpendicular. The last result allows the analytical calculation of the interactions when the magnetizations are in all the directions. The 3D analytical expressions are difficult to obtain, but the torque and force expressions are very simple to use. As example the analytical expression can be included in optimization software allowing to directly obtaining the shape optimization by a fast way.

New accurate approximation is proposed using integral expressions for evaluating the magnetic force between cylindrical permanent magnet arrays. The magnetic field distribution is calculated analytically by using Coulombian model. In this paper, every cylindrical magnet is divided into elementary cuboidal magnets. The accuracy can be controlled by regulating the value of elementary cuboidal permanent magnets "N ". The approximation can also be used to calculate the force interaction in the cylindrical linear single-axis-actuator. We confirm the validity of magnetic force calculation by comparing it with other methods and measurements. The calculation results are in very good agreement with measured values, which indicates the feasibility of our approximation.

The energy in permanent magnet is not a trivial problem because it exist two types of energy: the field energy and the demagnetizing energy. For parallelepiped shape, the magnet energy has been calculated by fully analytical expressions in 3D. The result has been obtained after four successive integrations of Arctangent and Logarithm functions. The analytically calculated energy corresponds to the demagnetization energy in the magnet. The analytical results have been compared with different terms of energy obtained by computation.

Up to now, the analytical calculation has been made only when the magnets own parallel magnetization directions. We have succeeded in two new results of first importance for the analytical calculation: the torque between two magnets, and the force components and torque when the magnetization directions are perpendicular. The last result allows the analytical calculation of the interactions when the magnetizations are in all the directions. The 3D analytical expressions are difficult to obtain, but the torque and force expressions are very simple to use. As example the analytical expression can be included in optimization software allowing to directly obtaining the shape optimization by a fast way.

Most of the systems working by magnet interactions can be calculated by superposition of the interactions between parallelepiped elementary magnets. Each elementary magnet is submitted to a force and a torque. By 3-D fully analytical calculation, up to now, only the force components problem has been solved. It was published for the first time in 1984. Until now, the torque components have never been analytically calculated because of the angular derivation. We have solved it, so that now all the interactions (energy, three forces components, and three torque components) can be expressed by analytical formulations. The only hypothesis is that the magnetizations and are supposed to be rigid and uniform in each magnet. The analytical calculation is made by replacing magnetizations by distributions of magnetic charges on the magnet poles. The torque is calculated for rotational movement of the second magnet around its center. The three components of the torque are written with functions based on logarithm and arctangent. The results have been verified and validated by comparison with finite-element calculation. Analytical calculation owns many advantages in comparison with other calculation methods. The 3-D analytical expressions are difficult to obtain, but the expressions of energy, force, and torque are very simple to use. For example, the analytical expression can be included in optimization software allowing to directly obtaining the shape optimization by a fast way.

Usely, in analytical calculation of magnetic and mechanical quantities of Halbach systems, the authors use the Fourier series approximation because the exact calculations are more difficult. In this work the interaction forces between linear Halbach arrays are analytically calculated thanks to our recent development 3D exact calculation of forces between two cuboïdal magnets with parallel and perpendicular magnetization. We essentially describe the way to separately calculate the forces between two magnets, between one magnet and a Halbach array and between two Halbach systems

The paper presents an analytical method for the determination of the magnetic force produced by a miniactuator with permanent magnets. The results are compared with those obtained by performing a numerical field analysis with COMSOL Multiphysics, showing a very good agreement. The study reveals that the actuator has two equilibrium points, one of which is stable and the other one unstable.

Permanent Magnets (PMs) are ever more used in high-performance applications such as electrical machines, linear servo actuators, contactless magnetic bearings, etc. This paper focuses on PM modeling for a contactless lithographic vibration isolation device with an integrated PM-based gravity compensator. This device needs to position a high mass (tons) with micrometer accuracy, with permanent magnets providing zero-power gravity compensation, and active devices ensuring stability and accurate positioning. To minimize the energy consumption of the active devices due to modeling errors it is essential that the PM modeling accuracy is maximized. Further, low calculation time is required for quick and efficient topology optimization. The 3D Finite Element Method (FEM), although accurate, consumes significant amounts of time, especially for models with small stroke compared to overall device dimensions. The 3D analytical charge model, on the other hand, provides field expressions which are exact for ironless structures, i.e. they are not approximations and, in contrary to FEM, remain very time inexpensive. Furthermore , these analytical field solutions exhibit significantly lower noise (150dB) compared to FEM solutions, and consequently provide an increased accuracy, especially at large magnetic field gradients (e.g. the magnet edges). Analytical charge modeling has been commonly used in literature to analytically describe the 3D magnetic fields induced by PMs, e.g. in [1,2] to analyze the field of and the interaction between cuboidal permanent magnets. It has been illustrated in [3] that the PM topology of the gravity compensator is of significant influence on the interaction force. More specifically, topologies which focus the magnetic flux to a small volume, such as the Halbach topology, provide the highest force density. In [3], solely cuboidal PMs were used to investigate the achievable force density for bidirectional arrays and Halbach arrays. Differently shaped PMs can contribute to increased flux focusing properties, hence, increase force density. This paper specifically presents novel analytical field expressions of the pyramidal frustum, or 3D trapezoidal PM (Fig. 1a), for the gravity compensator. It is defined such, that the bottom and top surface are rectangular, parallel, and share the vertical symmetry axis. By reducing the horizontal dimensions of the top surface towards zero, a pyramid is easily obtained. Fourteen rectangular and triangular shaped surfaces are distinguished on the contours of the PM. A charge is applied to each surface, and the respective magnetic fields are obtained analytically as described in [1] and [4], respectively. A summation of these fields results in the exact analytical expressions for the magnetic field exhibited by the 3D trapezoidal PM. The derivation of all surfaces, rotations, field expressions and the summations which result in the total field are elaborated on in the paper. The paper discusses anomalies in the mathematical field description, and methods to overcome those, which results in a smooth field solution. Comparison with 3D FEM solutions (Fig. 2b) demonstrates the high accuracy of the analytical solution (Fig. 2a) for the pyramidal frustum. Besides the pyramidal frustum, additional PM shapes, such as the hexagonal or octagonal prism (Fig. 1b), may contribute to an increased force density. The analytical model of such PMs is obtained by superposition of cuboidal PMs and triangular PMs, described in [1] and [4], respectively. In the final paper, a comparison with FEM shows the high accuracy for such PM shapes. Using the analytical field expressions obtained for exotic-shaped PMs, novel array topologies are investigated. Due to space considerations, this paper presents force density for a selection of possible topologies to be incorporated in the gravity compensator. [1] J.P. Yonnet and G. Akoun, " 3D analytical calculation of the forces exerted between two cuboidal magnets, " (a) pyramidal frustum and (b) top view of hexagonal and octagonal prism showing the cuboidal and triangular shapes (a): analytically obtained absolute value of the magnetic flux density, parallel to one of the side surfaces (projected in the figure). (b): difference with results obtained by FEM.

The interaction forces exerted between permanent magnets are used in many magneto-mechanical devices (magnetic bearings, couplings, etc ... ). By analytical calculation, 2D problems can be solved easily, when simple shaped magnets are used. Usually, the 3D problems are numerically computed, by using a finite element method for example.

Accurate vibration isolation and magnetic levitation are becoming ever more important in the high-precision industry. Nowadays, magnetic bearings based on permanent magnets are increasingly considered for vibration isolation. This article proposes a novel extension to the existing analytical equations for the interaction force between permanent magnets, suitable for the frequently occurring situations where their respective side surfaces are aligned. Further, the analytical stiffness matrix necessary for the design and analysis of such a magnetic bearing is proposed. The analytical force and stiffness expressions are derived for the cases where the conventional analytical expressions are difficult to solve and they are subsequently compared with results obtained using the Maxwell Stress method. With these adapted expressions it is possible to optimize planar bearing structures without having to correct for non-solving configurations.

Permanent Magnets (PMs) are ever more used in high-performance applications such as electrical machines, linear servo actuators, contactless magnetic bearings, etc. This paper focuses on PM modeling for a contactless lithographic vibration isolation device with an integrated PM-based gravity compensator. This device needs to position a high mass (tons) with micrometer accuracy, with permanent magnets providing zero-power gravity compensation, and active devices ensuring stability and accurate positioning. To minimize the energy consumption of the active devices due to modeling errors it is essential that the PM modeling accuracy is maximized. Further, low calculation time is required for quick and efficient topology optimization. The 3D Finite Element Method (FEM), although accurate, consumes significant amounts of time, especially for models with small stroke compared to overall device dimensions. The 3D analytical charge model, on the other hand, provides field expressions which are exact for ironless structures, i.e. they are not approximations and, in contrary to FEM, remain very time inexpensive. Furthermore , these analytical field solutions exhibit significantly lower noise (150dB) compared to FEM solutions, and consequently provide an increased accuracy, especially at large magnetic field gradients (e.g. the magnet edges). Analytical charge modeling has been commonly used in literature to analytically describe the 3D magnetic fields induced by PMs, e.g. in [1,2] to analyze the field of and the interaction between cuboidal permanent magnets. It has been illustrated in [3] that the PM topology of the gravity compensator is of significant influence on the interaction force. More specifically, topologies which focus the magnetic flux to a small volume, such as the Halbach topology, provide the highest force density. In [3], solely cuboidal PMs were used to investigate the achievable force density for bidirectional arrays and Halbach arrays. Differently shaped PMs can contribute to increased flux focusing properties, hence, increase force density. This paper specifically presents novel analytical field expressions of the pyramidal frustum, or 3D trapezoidal PM (Fig. 1a), for the gravity compensator. It is defined such, that the bottom and top surface are rectangular, parallel, and share the vertical symmetry axis. By reducing the horizontal dimensions of the top surface towards zero, a pyramid is easily obtained. Fourteen rectangular and triangular shaped surfaces are distinguished on the contours of the PM. A charge is applied to each surface, and the respective magnetic fields are obtained analytically as described in [1] and [4], respectively. A summation of these fields results in the exact analytical expressions for the magnetic field exhibited by the 3D trapezoidal PM. The derivation of all surfaces, rotations, field expressions and the summations which result in the total field are elaborated on in the paper. The paper discusses anomalies in the mathematical field description, and methods to overcome those, which results in a smooth field solution. Comparison with 3D FEM solutions (Fig. 2b) demonstrates the high accuracy of the analytical solution (Fig. 2a) for the pyramidal frustum. Besides the pyramidal frustum, additional PM shapes, such as the hexagonal or octagonal prism (Fig. 1b), may contribute to an increased force density. The analytical model of such PMs is obtained by superposition of cuboidal PMs and triangular PMs, described in [1] and [4], respectively. In the final paper, a comparison with FEM shows the high accuracy for such PM shapes. Using the analytical field expressions obtained for exotic-shaped PMs, novel array topologies are investigated. Due to space considerations, this paper presents force density for a selection of possible topologies to be incorporated in the gravity compensator. [1] J.P. Yonnet and G. Akoun, " 3D analytical calculation of the forces exerted between two cuboidal magnets, " (a) pyramidal frustum and (b) top view of hexagonal and octagonal prism showing the cuboidal and triangular shapes (a): analytically obtained absolute value of the magnetic flux density, parallel to one of the side surfaces (projected in the figure). (b): difference with results obtained by FEM.

– This paper presents novel analytical expressions which describe the 3D magnetic field of arbitrarily magnetized triangular-shaped charged surfaces. These versatile expressions are suitable to model triangular-shaped permanent magnets and can be expanded to any polyhedral shape. Many applications are very suitable to incorporate this modeling technique. Examples are given for the subclass of slotless synchronous linear actuators with a skewed permanent magnet structure and planar magnetic bearings. It is shown that these analytical equations are very suitable to obtain a variety of performance data for the 3D magnetic analysis of slotless linear machines, as well as planar machines and bearings.

Accurate vibration isolation and magnetic levitation are becoming ever more important in the high-precision industry. Nowadays, magnetic bearings based on permanent magnets are increasingly considered for vibration isolation. This article proposes a novel extension to the existing analytical equations for the interaction force between permanent magnets, suitable for the frequently occurring situations where their respective side surfaces are aligned. Further, the analytical stiffness matrix necessary for the design and analysis of such a magnetic bearing is proposed. The analytical force and stiffness expressions are derived for the cases where the conventional analytical expressions are difficult to solve and they are subsequently compared with results obtained using the Maxwell Stress method. With these adapted expressions it is possible to optimize planar bearing structures without having to correct for non-solving configurations.

– This paper presents novel analytical expressions which describe the 3D magnetic field of arbitrarily magnetized triangular-shaped charged surfaces. These versatile expressions are suitable to model triangular-shaped permanent magnets and can be expanded to any polyhedral shape. Many applications are very suitable to incorporate this modeling technique. Examples are given for the subclass of slotless synchronous linear actuators with a skewed permanent magnet structure and planar magnetic bearings. It is shown that these analytical equations are very suitable to obtain a variety of performance data for the 3D magnetic analysis of slotless linear machines, as well as planar machines and bearings.

In this paper a non-contact magnetic spring design is presented that uses inclined magnets to produce an adjustable relationship between load force and dynamic stiffness. With appropriate choice of parameters, the spring may either operate with a range of constant natural frequency against variable load forces, or a positive stiffness in one horizontal direction may be achieved in addition to having a positive vertical stiffness. Dynamic simulations are presented to assess the non-linear stability of a planar three degree of freedom version of the system; cross-coupling between horizontal and rotation motion is shown to compromise passive stability, in which case passive constraints or active control must be used to avoid instability. The design is scalable in that using larger magnets increases the load bearing capacity and decreases the natural frequency of the system.

A recently-published equation for calculating the force between coaxial cylindrical magnets is presented in simplified form. The revised equation is now very compact: it is defined in terms of fewer parameters and contains fewer terms than the original equation. The new equation is purely real, unlike the original which contained imaginary components. As a result of the simplifications, the new equation is demonstrably faster to evaluate than the original, improving its utility for parametric optimisation. A reference implementation is provided for Matlab and Mathematica.

Multipole magnet arrays have the potential to achieve greater forces than homogeneous magnets for linear spring applications. This paper investigates the effects of varying key parameters of linear multipole magnet arrays in relation to their force bearing potential. Equal sized arrays in repulsive configurations are vertically displaced to each other; only vertical forces are compared. The force versus displacement characteristic is dependent on the aspect ratio of the arrays, the wavelength of magnetisation, and the total number of magnets used in the array. Some general design guidelines for optimising the repulsive forces are established based on the results.

Recently, we wrote that to maximise the force between two magnets of fixed volume, the magnet dimensions should be chosen to be as thin as possible in the direction of magnetisation. This was incorrect, and we would like to clarify this point.

Vibration isolation systems incorporating linear mechanical springs exhibit the undesirable characteristic of changing resonance frequency with changing payload mass. Previous research at the University of Adelaide and elsewhere has demonstrated the theoretical feasibility of vibration isolation devices utilising magnetic springs as a means of achieving a constant resonance frequency across a range of payload masses due to their nonlinear force-displacement relationship. A conceptual prototype design is presented for a levitating magnet vibration isolation device which aims to achieve a load-independent resonance frequency across a range of payload masses via the use of inclined magnetic springs. Quasi-static and dynamic system models which informed the design process are presented, as well as a finiteelement model aimed at validating the assumptions of the quasi-static system model for a select set of system states. Challenges related to the conflicting design requirements of stability, low transmissibility and load-independent resonance frequency are addressed, and an experimental framework for testing the real prototype is outlined.

The paper presents an analytical method for the determination of the magnetic force produced by a miniactuator with permanent magnets. The results are compared with those obtained by performing a numerical field analysis with COMSOL Multiphysics, showing a very good agreement. The study reveals that the actuator has two equilibrium points, one of which is stable and the other one unstable.

Applications of permanent magnets bearings have gained a new interest thanks to the development of rare earth materials, characterised by residual magnetic induction greater than 1 T. The present paper proposes a new geometry for permanent magnets bearings with V-shaped elements, both for a plane slide and for cylindrical bearings. The aim of this geometry is to give new possibilities to the application of these bearing systems, by reducing its inherent instability. A design method, involving Finite Elements and Magnetic Field Integral Equations analyses, is also described for defining the most suitable V-opening angle and the two magnetisation directions. These parameters can be varied in order to reduce the unstable force in the coupling, and to reach the desired force and stiffness in the stable direction. The design is founded on the evaluation of four "geometric" vectors, that depend on the geometry of the elements. Some results are reported for a reference geometry for both the slide and the cylindrical bearings.

We present exact three-dimensional semi-analytical expressions of the force exerted between two coaxial thick coils with rectangular cross-sections. Then, we present a semi-analytical formulation of their mutual inductance. For this purpose, we have to calculate six and seven integrations for calculating the force and the mutual inductance respectively. After mathematical manipulations, we can obtain semi-analytical formulations based on only two integrations. It is to be noted that such integrals can be evaluated numerically as they are smooth and derivable. Then, we compare our results with the filament and the finite element methods. All the results are in excellent agreement.

This paper presents a novel method to obtain fully analytical expressions of the magnetic field created by a pyramidal-frustum shaped permanent magnet. Conventional analytical tools only provide expressions for cuboidal permanent magnets and this paper extends these tools to more complex shapes. A novel analytical model for the magnetic field induced by triangular charged surfaces is combined with the field expressions for rectangular charged surfaces. This provides a fully analytical expression for the magnetic field of a pyramidalfrustum shaped permanent magnet. Field and force obtained using this model are validated with Finite Element Modeling.

Up to now, the analytical calculation has been made only when the magnets own parallel magnetization directions. We have succeeded in two new results of first importance for the analytical calculation: the torque between two magnets, and the force components and torque when the magnetization directions are perpendicular. The last result allows the analytical calculation of the interactions when the magnetizations are in all the directions.

Most of the systems working by magnet interactions can be calculated by superposition of the interactions between parallelepiped elementary magnets. Each elementary magnet is submitted to a force and a torque. By 3-D fully analytical calculation, up to now, only the force components problem has been solved. It was published for the first time in 1984. Until now, the torque components have never been analytically calculated because of the angular derivation. We have solved it, so that now all the interactions (energy, three forces components, and three torque components) can be expressed by analytical formulations. The only hypothesis is that the magnetizations J and J ' are supposed to be rigid and uniform in each magnet. The analytical calculation is made by replacing magnetizations by distributions of magnetic charges on the magnet poles. The torque is calculated for rotational movement of the second magnet around its center. The three components of the torque are written with functions based on logarithm and arc-tangent. The results have been verified and validated by comparison with finite-element calculation. Analytical calculation owns many advantages in comparison with other calculation methods. The 3-D analytical expressions are difficult to obtain, but the expressions of energy, force, and torque are very simple to use. For example, the analytical expression can be included in optimization software allowing to directly obtaining the shape optimization by a fast way.

In this paper, we present analytic-numerical expressions for the calculation of the mutual inductance of two axisymetric circular coils with rectangular cross section in air. This original and new method may seem complicated but it is explicit, accurate, and fast, even though all expressions are obtained by the complete elliptic integrals of the first and second kind, Heuman's lambda function, and three terms that must be solved numerically. We confirm the validity of this approach by comparing it with other approaches (filament method and previously published data). We also compare the accuracy and the computational cost of this approach and that of the filament method. All results obtained by the various approaches are in excellent agreement.

Up to now, the analytical calculation has been made only when the magnets own parallel magnetization directions. We have succeeded in two new results of first importance for the analytical calculation: the torque between two magnets, and the force components and torque when the magnetization directions are perpendicular. The last result allows the analytical calculation of the interactions when the magnetizations are in all the directions. The 3D analytical expressions are difficult to obtain, but the torque and force expressions are very simple to use. As example the analytical expression can be included in optimization software allowing to directly obtaining the shape optimization by a fast way.

Usely, in analytical calculation of magnetic and mechanical quantities of Halbach systems, the authors use the Fourier series approximation because the exact calculations are more difficult. In this work the interaction forces between linear Halbach arrays are analytically calculated thanks to our recent development 3D exact calculation of forces between two cuboïdal magnets with parallel and perpendicular magnetization. We essentially describe the way to separately calculate the forces between two magnets, between one magnet and a Halbach array and between two Halbach systems

This paper presents a semi-analytical calculation of the interaction between magnetic bodies with ring shapes. It leads to fast and accurate evaluation of forces and torques but also to the symbolic expression of their gradients. Our method can be used to compute partial derivatives to address optimization needs. It has been implemented into an available software environment in order to be automatically available for end users. Efficient optimization can be done to size magnetic bearing systems.