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(a) Two permanent magnets with parallel magnetization vectors. (b) Two permanent magnets with orthogonal magnetization vectors.
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This paper concerns the analysis of the interaction force among the permanent magnets (PM) in a Halbach array as used in a moving-magnet planar actuator. The goal is to analyze the magnitude of these forces and to what extent the mechanical construction of an aluminium plate, on which the permanent magnets are mounted, is deformed by the interactio...
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Context 1
... a relative permeability , which will result in a slight error due to a difference compared to the actual value of . Using these analytical models, the interaction force can be calculated with zero distance between two permanent magnets. Consider the two permanent magnets as shown in Fig. 3 with two possible magnetization variations as shown in Fig. 4. For the two magnets shown in Fig. 4(a), the force between the magnets is given by [3] (1) ...
Context 2
... result in a slight error due to a difference compared to the actual value of . Using these analytical models, the interaction force can be calculated with zero distance between two permanent magnets. Consider the two permanent magnets as shown in Fig. 3 with two possible magnetization variations as shown in Fig. 4. For the two magnets shown in Fig. 4(a), the force between the magnets is given by [3] (1) ...
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Citations
... On the other hand, in [23], a Halbach array was developed, consisting of two smaller arrays differing in a span of a horizontally magnetized magnet. In [24,25], non-symmetrical Halbach arrays with wide vertically magnetized PM magnets are described. The authors mainly consider forces with a vertical component. ...
In high-speed magnetic railways, it is necessary to create the forces that lift the train. This effect is achieved by using active (EMS) or passive (EDS) magnetic systems. In a passive system, suspension systems with permanent magnets arranged in a Halbach array can be used. In this paper, an original Halbach array with various alternately arranged horizontally and vertically magnetized magnets is proposed. Correctly selected geometry allows us to obtain higher values of levitation forces and lower braking forces in relation to a system with identical horizontally and vertically magnetized elements. The effect of such a shape of the magnetic arrangement is the reduction of instantaneous power consumption while traveling due to the occurrence of lower braking forces. In order to perform a comparative analysis of the various geometries of the Halbach array, a simulation model was developed in the ANSYS Maxwell program. The performed calculations made it possible to determine the optimal dimensions of horizontally and vertically magnetized elements. The results of calculations of instantaneous power savings for various cruising speeds are also included.
... The interaction force's analytical models are derived from the interaction energy between two cuboidal PMs under parallel magnetization directions. Rovers et al. use the analytical models presented in [17] to calculate static forces between cuboidal PMs in a Halbach array [18]. According to the principle of virtual work and the interaction energy, Allag et al. have contributed to the interaction forces and torques' analytical expressions between two cuboidal PMs [19][20][21], verified by the finite element simulation. ...
Cuboidal permanent magnets (PMs) are commonly used as elementary magnets for most magnetic systems that work by superposition of magnetic forces. This paper proposed a new method that combines the analytical model with the numerical calculation method to calculate the magnetic forces between two cuboidal PMs with parallel and perpendicular magnetization directions. The assumption was that the magnets were ideal with constant and homogeneous magnetizations. The method was also valid for other magnetization directions and any reference points in space, only needed to rotate the coordinate system and give corresponding distance expressions. An important result was that this method had the advantages of high accuracy and low computational cost, enabling rapid studies of the magnetic force characteristics related to magnet positions, geometric dimensions, and magnetic properties, which were easy to calculate by mathematical software, requiring very little running time. Meanwhile, the calculation results have been validated by comparison with the finite element simulation and experimental measurement. The effects of relative positions, dimensions, and magnetic properties on magnetic force characteristics between the two cuboidal PMs have been studied and discussed. Moreover, the improved magnetization distribution on the PM’s surface was made taking the PM’s relative permeability into account, and corresponding results have been given. These results indicated that the proposed method could help for a fast design and optimization of many PM devices that rely on magnetic forces, such as magnetic bearings, magnetic suspensions, magnetic actuators, etc.
... See [20] for another reference on the self-forces and deformation in a (2D) Halbach array. ...
In the work presented here, the suitability of an unusual energy storage medium is investigated. The energy storage system is based on the forceful compression of two magnetic Halbach arrays. The mass and volume energy density is obtained and compared to existing common energy storage systems. The charge and discharge times and depths are also discussed. In addition, limits and considerations, which are needed for practical implementation, e.g., risk of demagnetization, internal mechanical stresses, heating of the magnetic structure and financial efficiency are investigated.
... The deformation of a linear motor due to thermal effects was discussed in [8], the deformation of a planar actuator due to thermal effects was discussed in [9]. The deformation due to the static force among the permanent magnets in the magnet array of the actuator shown in Fig. 1 is discussed in [10]. This paper presents the deformation of the magnet plate due to the forces acting on the magnets in the magnet array due to the commutated coil set. ...
This paper presents a method to calculate the deformation of the magnet plate of a commutated magnetically levitated planar actuator using a linked electromagnetic–mechanical model. The force and torque distribution on the moving magnet array is obtained from an electromagnetic model based on the surface charge method and the Lorentz force and torque integral. The mechanical model is a state-space model derived from FEM. This mechanical model uses the force and torque distribution to determine the deformation of the magnet plate during movement due to the commutated coil set.
... To be able to determine the force and torque distribution on the magnet array, expressions describing the forces and torques acting on each permanent magnet are needed. The static forces among the permanent magnets in a Halbach array were determined in [20]. In addition, analytic and semianalytic expressions describing the force between a permanent magnet and a coil were presented in [21] and [22]. ...
This paper concerns the analysis of the dynamic forces and torques acting on the magnets in a Halbach permanent magnet array of a magnetically levitated moving-magnet planar actuator. A new analysis tool is presented which predicts the dynamic force and torque distribution on the magnet array. This design tool uses lookup table data, which are generated by numerically solving the Lorentz force and torque integral, to describe the force and torque between each magnet and coil in the topology. It offers a fast and accurate solution for the analysis of magnetically levitated planar actuators. The results for two different commutation methods are presented.
... Further, when mounting permanent magnets side-by-side, i.e. on the same back-plane, knowledge of their interaction behavior is essential. Examples are found in the assembly force calculations of planar magnet arrays [162] or in glue strength calculations in beam insertion devices [66,170]. ...
This thesis researches the analytical surface charge modeling technique which provides a fast, mesh-free and accurate description of complex unbound electromagnetic problems. To date, it has scarcely been used to design passive and active permanent-magnet devices, since ready-to-use equations were still limited to a few domain areas. Although publications available in the literature have demonstrated the surface-charge modeling potential, they have only scratched the surface of its application domain.
The research that is presented in this thesis proposes ready-to-use novel analytical equations for force, stiffness and torque. The analytical force equations for cuboidal permanent magnets are now applicable to any magnetization vector combination and any relative position. Symbolically derived stiffness equations directly provide the analytical 3 x 3 stiffness matrix solution. Furthermore, analytical torque equations
are introduced that allow for an arbitrary reference point, hence a direct torque calculation on any assembly of cuboidal permanent magnets. Some topics, such as the analytical calculation of the force and torque for rotated magnets and extensions to the field description of unconventionally shaped magnets, are outside the scope of this thesis are recommended for further research.
A worldwide first permanent-magnet-based, high-force and low-stiffness vibration isolation system has been researched and developed using this advanced modeling technique. This one-of-a-kind 6-DoF vibration isolation system consumes a minimal amount of energy (<1W) and exploits its electromagnetic nature by maximizing the isolation bandwidth (>700Hz). The resulting system has its resonance <1Hz with a -2dB per decade acceleration slope. It behaves near-linear throughout its entire 6-DoF working range, which allows for uncomplicated control structures. Its position accuracy is around 4um, which is in close proximity to the sensor’s theoretical noise level of 1um.
The extensively researched passive (no energy consumption) permanent-magnet based gravity compensator forms the magnetic heart of this vibration isolation system. It combines a 7.1kN vertical force with <10kN/m stiffness in all six degrees of freedom. These contradictory requirements are extremely challenging and require the extensive research into gravity compensator topologies that is presented in this thesis.
The resulting cross-shaped topology with vertical airgaps has been filed as a European patent. Experiments have illustrated the influence of the ambient temperature on the magnetic behavior, 1.7h/K or 12N/K, respectively. The gravity compensator has two integrated voice coil actuators that are designed to exhibit a high force and low power consumption (a steepness of 625 N2/W and a force constant of 31N/A) within
the given current and voltage constraints. Three of these vibration isolators, each with a passive 6-DoF gravity compensator and integrated 2-DoF actuation, are able to stabilize the six degrees of freedom.
The experimental results demonstrate the feasibility of passive magnet-based gravity compensation for an advanced, high-force vibration isolation system. Its modular topology enables an easy force and stiffness scaling. Overall, the research presented in this thesis shows the high potential of this new class of electromagnetic devices for vibration isolation purposes or other applications that are demanding in terms of
force, stiffness and energy consumption. As for any new class of devices, there are still some topics that require further study before this design can be implemented in the next generation of vibration isolation systems. Examples of these topics are the tunability of the gravity compensator’s force and a reduction of magnetic flux leakage.
... Examples of applications where an accurate force calculation between cuboidal magnets is required are passive lifting and holding devices [2], certain types of rotating permanent magnet bearings and couplings [4][5][6][7][8][9][10][11][12] (Fig. 1(a)), magnetic gears [13][14][15] and magnetic suspensions [16][17][18][19]. Also when magnets are mounted side-by-side, i.e. on the same back-plane, knowledge of their interaction behavior may be necessary, such as in structures where the force between permanent magnets mounted on the same frame needs to be known, e.g. for mechanical calculations [20] or in glue strength calculations in beam insertion devices [2,3]. Section 2 discusses the main differences between numerical and the analytical methods, and which design and practical parameters should be carefully considered when going through the process of selecting a suitable modeling method for a specific problem. ...
... A solution for this configuration is especially useful when calculating the force between adjacent permanent magnets, e.g. when they are placed in a closed array. Such problem was encountered but not fully solved in [20] to obtain the internal forces of a planar magnet array. The expressions in [29] are unclear on this matter, however further investigation of these expressions shows that the zero limit of all three variables u, v and w leads to an expression which equals zero, hence ...
... Therefore, expressions describing the force acting on each PM in the array are needed, since these forces cause the deformation. In previous research, the static forces among the permanent magnets in a Halbach array were determined [6]. A next step is to determine the force resulting from the coils in the stator. ...
... The Lorentz force equation is (6) where is the volume of the wire segment, is the current density in the wire segment, and is the external magnetic field. Applying this integral to the current carrying volume and permanent magnet in Fig. 2 results in (7) where is the magnetic flux density due to node and . ...
This paper concerns the analysis of the force between a rectangular coil and a cuboidal permanent magnet. The magnetic flux density distribution due to the permanent magnet is determined using the surface charge method, and an analytical equation is obtained for the Lorentz force on a cuboidal current carrying volume. These analytical results are used to calculate the Lorentz force on a rectangular coil modeled using four current carrying volumes. The calculations are verified both by measurements and numerical integration. These results can be used in the design and real-time control of planar actuators for industrial levitation/positioning platforms or other (ironless) actuators.
... A classical multipole or Halbach array is a linear array of magnets stacked to approximate a single magnet with sinusoidal magnetisation, first analysed in the '70s (Halbach 1981;Shute et al. 2000). Multipole arrays have been analysed for a variety of force-producing applications; only a small selection are included in the bibliography here (Lee, Lee, and Gweon 2004;Robertson, Cazzolato, and Zander 2005;Rovers et al. 2009). One advantage of using multipole arrays is to focus the magnetic field on one side of the array, such to increase the forces exerted by which magnetic field on one side of the array and to reduce or eliminate any need for magnetic shielding on the reverse side. ...
... Such a system, called here a 'planar Halbach' array, is shown in Figure 8, with five magnets per side and 90°magnetisation rotation in both the x-z and y-z planes between successive magnets. Moser et al. (2002), Rovers et al. (2009), andLomonova (2009a) (the latter two with associated publications) have examined the idea of a 'quasi-Halbach' planar multipole array in which all magnetisation directions are restricted to one of the orthogonal directions of the axes (i.e., no diagonal magnetisations). Another planar multipole array, simpler again, is the 'patchwork' array, which alternates between positive and negative vertical magnetisation between successive magnets in both directions. ...
In this paper we present a public framework in Matlab for calculating forces between magnets and between multipole arrays of magnets. The multipole arrays are situated as pairs in opposition and the force vs. displacement characteristics calculated based on a forces superposition of the magnets that compose the arrays. The software framework provides a convenient method of calculating magnetic forces and allows useful comparisons between a variety of multipole configurations. Design optimisation may also be performed as the abstractions of the software permit easy iteration of high-level variables of systems of interest. The code is public and may be re-used and modified by anyone, hence providing a useful service to researchers in the field who wish to perform such calculations as described above. Collaboration is encouraged to improve and extend the software, yielding benefits to the research community as a whole.
This research combines simulation technology, a surrogate model, and a dung beetle optimizer (DBO) to propose a structural optimization design method for lightweight adhesive modules. The structure of the wall-climbing robot is introduced, and its adhesion stability is analyzed. Through simulation comparison of four typical Halbach array magnetic circuit modes, it was determined that the adhesion generated by the three-magnetic circuit structure mode is more advantageous. Determine the parameters that need to be optimized through sensitivity analysis. The Chebyshev model of adhesion force and parameters was established. An optimization model aimed at lightweight and the constraints of adhesion stability and structural parameters was set. The penalty function combined with DBO was used to solve the optimization model. Compared with before optimization, the weight of the adhesive module is reduced by 11.7 %. The experiments verified the adhesive module’s adhesion force and the robot’s load capacity.
