Three-Dimensional Analytical Calculation of the Torque Between Permanent Magnets in Magnetic Bearings

Abstract

In the high-precision industry, accurate vibration isolation and magnetic levitation are extremely important. As a result, high-performance vibration isolation and magnetic bearings based on permanent magnets are increasingly considered. This paper proposes improved analytical expressions for the torque on cuboidal permanent magnets applied to a magnetic bearing. These novel expressions are valid for any relative magnet position, especially when surfaces of the different magnets are in the same plane. Further, the torque can be obtained with respect to any reference point. Although these equations seem rather complicated, they enable an extremely fast and accurate calculation of the torque on a permanent magnet in the presence of a magnetic field of another permanent magnet. These properties enable a fast design and optimization process of such bearings using fully analytical expressions.

3-D Analytical Calculation of the Torque Between Perpendicular Magnetized Magnets in Magnetic Suspensions

“…In this paper, the torque on PM2 is derived for the case that PM1 is magnetized along the -axis and PM2 is magnetized along the -axis. This completes the torque expressions that were given for parallel magnetization in [16]. For other perpendicular magnetization directions, a rotation of the coordinate system suffices to obtain the torque.…”

“…Fig. 3 shows such a situation in which (12) and (14) reduce to (16) (17) and the expressions (8)- (10) become undefined. A fully analytical, hence computationally inexpensive, equation in these situations is the most accurate and fast solution to this problem and Fig.…”

“…has been derived for the force, stiffness, and torque of parallel magnetized magnets in [13] and [16].…”

“…Reference [15] used Virtual work to obtain analytical expressions for the torque around one of the permanent magnet centers. On the other hand, in [16], the Lorentz force was used to obtain expressions that describe the torque around any given point, and additionally, the necessary complementary expressions for the special case of aligned edges were derived. This paper completes these torque computations with the computation of torque between perpendicular magnetized permanent magnets for any given relative magnet position and torque reference.…”

Passive Permanent Magnet Bearing for Rotating Shaft : Analytical Calculation

“…Indeed, the force and the stiffness are the values of importance for the design and optimization of a bearing. So, authors have tried to work out analytical expressions of the force exerted between two ring permanent magnets (Kim et al, 1997) (Lang, 2002) (Samanta & Hirani, 2008) (Janssen et al, 2010) (Azukizawa et al, 2008). This chapter intends to give a detailed description of the modelling and approach used to calculate analytically the force and the stiffness between two ring permanent magnets with axial or radial polarizations (Ravaud, Lemarquand & Lemarquand, 2009a) (Ravaud, Lemarquand & Lemarquand, 2009b).…”