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The spring stiffness expressions for zy-magnetization are derived using Eq. (14) and Appendix A. The matrix Q(u, v, w) is in the discontinuous points of the expressions given by The stiffness expressions for zz-magnetization is derived using Eq. 14 and Appendix A. The matrix Q(u, v, w) is in the discontinuous points of the expressions given by

Figure 6 The spring stiffness expressions for zy-magnetization are derived using Eq. (14) and Appendix A. The matrix Q(u, v, w) is in the discontinuous points of the expressions given by The stiffness expressions for zz-magnetization is derived using Eq. 14 and Appendix A. The matrix Q(u, v, w) is in the discontinuous points of the expressions given by

Related Figures (9)

Figure 1. Impression of a high-precision lithographic machine (courtesy of ASML, Veldhoven, the Nether- lands) (a), the static magnetic bearing within the machine (b) and a schematic view of a repelling magnetic bearing (c).
Figure 1. Impression of a high-precision lithographic machine (courtesy of ASML, Veldhoven, the Nether- lands) (a), the static magnetic bearing within the machine (b) and a schematic view of a repelling magnetic bearing (c).
Figure 2. Half cross-section of (a) and quarter top view (b) of a magnetic bearing with a horizontal airgap and planar 7-by-7 Halbach array from [2] with both arrays vertically aligned (dashed lines). The variables are used in Section 4.
Figure 2. Half cross-section of (a) and quarter top view (b) of a magnetic bearing with a horizontal airgap and planar 7-by-7 Halbach array from [2] with both arrays vertically aligned (dashed lines). The variables are used in Section 4.
Figure 3. Definition of the dimension and displacement variables (a) and alignment of the side surfaces of two permanent magnets (b).
Figure 3. Definition of the dimension and displacement variables (a) and alignment of the side surfaces of two permanent magnets (b).
The vector E (u, v, w) for zy-magnetization was derived in [7,8] and is in the dis- continuous points given by
The vector E (u, v, w) for zy-magnetization was derived in [7,8] and is in the dis- continuous points given by
By inserting these equations in the appropriate cases, e.g. the one described in Eqs. (3)-(6), a fully analytical and continuous force and stiffness calculation method for cuboidal permanent magnets is obtained. Integrating this result in the regular analytical expressions makes it extremely suitable for an optimization routine to achieve an opti- mized static planar magnetic bearing.
By inserting these equations in the appropriate cases, e.g. the one described in Eqs. (3)-(6), a fully analytical and continuous force and stiffness calculation method for cuboidal permanent magnets is obtained. Integrating this result in the regular analytical expressions makes it extremely suitable for an optimization routine to achieve an opti- mized static planar magnetic bearing.
Like the force expressions above, these expressions enable fully analytical and con- tinuous stiffness results for cuboidal permanent magnets for any position.
Like the force expressions above, these expressions enable fully analytical and con- tinuous stiffness results for cuboidal permanent magnets for any position.
Table 1. Dimensions of the 7-by-7 Halbach array (Ref. [2]) used in the simulation. Figure 4, The three force components F,, Fy and Fz, respectively, for the 7-by-7 Halbach array from [2] with horizontal movement along the x-axis. The displacement along y and z is zero in this simulation. The solid lines are analytically obtained and the circles by the Maxwell Stress method.
Table 1. Dimensions of the 7-by-7 Halbach array (Ref. [2]) used in the simulation. Figure 4, The three force components F,, Fy and Fz, respectively, for the 7-by-7 Halbach array from [2] with horizontal movement along the x-axis. The displacement along y and z is zero in this simulation. The solid lines are analytically obtained and the circles by the Maxwell Stress method.
Figure 5. The stiffness matrix represented by the nine stiffness components Ky to Kzz, respectively in (a)-(i), for the 7-by-7 Halbach array from [2] with horizontal movement along the x-axis. The displacement along y and z is zero in this simulation. The solid lines are analytically obtained and the circles by the Maxwell Stress method. Due My ine movement along x only the left column of the symmetrical stiffness matrix (containing aFy OFy “Ox? Ox, and % Fy =) is obtained by numerical derivation of the force obtained using the Maxwell Stress method.
Figure 5. The stiffness matrix represented by the nine stiffness components Ky to Kzz, respectively in (a)-(i), for the 7-by-7 Halbach array from [2] with horizontal movement along the x-axis. The displacement along y and z is zero in this simulation. The solid lines are analytically obtained and the circles by the Maxwell Stress method. Due My ine movement along x only the left column of the symmetrical stiffness matrix (containing aFy OFy “Ox? Ox, and % Fy =) is obtained by numerical derivation of the force obtained using the Maxwell Stress method.
Intermediate variable Q which is used in analytical expressions of the stiffness matrix K in the case that both magnets are magnetized along the z-axis: Intermediate variable Q which is used in analytical expressions of the stiffness matrix K in the case that the first magnet is magnetized along the z-axis and the second along the y-axis: [ntermediate variable Q which is used in analytical expressions of the stiffness matrix K
Intermediate variable Q which is used in analytical expressions of the stiffness matrix K in the case that both magnets are magnetized along the z-axis: Intermediate variable Q which is used in analytical expressions of the stiffness matrix K in the case that the first magnet is magnetized along the z-axis and the second along the y-axis: [ntermediate variable Q which is used in analytical expressions of the stiffness matrix K
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