JP4994047B2 - Magnetic bearing device - Google Patents

Description

  The present invention relates to a magnetic bearing device that supports a rotating body that is magnetically levitated by a magnetic attraction force, and more particularly to a magnetic bearing device that promotes damping of vibration with respect to vibration in the radial direction of the rotating body.

  As a conventional magnetic bearing device, a damper ring made of a good electrical conductor material is incorporated in the stator side magnetic pole, and the tip angle of the teeth of the rotor side magnetic pole integrated with the rotating shaft is set to an angle different from 90 degrees. By tilting the sides on both sides with respect to the rotation axis, when vibration occurs on the rotation axis, the eddy current generated inside the damper ring based on the change in magnetic flux absorbs vibration energy and attenuates vibration. Some have been promoted (see, for example, Patent Document 1). FIG. 25 shows a conventional magnetic bearing device described in Patent Document 1. In FIG.

  In FIG. 25, there are a magnetic rotor side magnetic pole 102 integrated with the rotating shaft 101 and a stator side magnetic pole 103 of a bearing facing the magnetic rotor side magnetic pole 102. The magnetic poles 102 and 103 have grooves 107a, 107b, 107c,. 108b, 108c,..., Magnetic flux is induced by the current of the excitation winding 104 to form a magnetic circuit (hereinafter referred to as a magnetic path). This magnetic flux generates a thrust based on the magnetic attractive force in the axial direction between the magnetic poles 102 and 103, and the teeth 111a, 111b... And the teeth 112a, 112b. A radial force against the decrease in magnetic energy based on the decrease in magnetoresistance associated with the decrease in opposing area is generated to try to push back the movement of the shaft 101 in the radial direction.

  Further, damper rings 105 and 106 made of a good electric conductor material and acting as a ring-shaped braking winding are incorporated in the tooth portions 112a, 112b. This is because the eddy current generated inside the damper rings 105 and 106 based on the change of magnetic flux absorbs the vibration energy when the shaft 101 vibrates, thereby promoting the damping of the vibration, that is, the function of the damper effect. It is what you have.

  In FIG. 25 for explaining the damper effect, if the magnetic flux passing from the teeth 112b to 111b per unit length in the circumferential direction, which is a direction perpendicular to the paper surface, is φ, the radial magnetic force acting on the teeth 112b is , Which is proportional to the derivative (−dφ / dr) of the magnetic flux φ with respect to the coordinate r.

  The damper action appears due to the Joule loss of the eddy current flowing in the damper ring caused by the induced electromotive force proportional to the time change (dφ / dt) of the magnetic flux. The change of the magnetic flux can be expressed by the time change of the coordinate r due to the vibration in the radial direction, and is expressed by the equation (Equation 1).

ダンパリング内に流れる渦電流はこの磁束の時間変化で表されるダンパリング内の誘起電圧に基づいて流れるので、この渦電流に基づくジュール損失Ｗは（数２）の式にて表される。ここで、Ｋは比例定数である。

Since the eddy current flowing in the damper ring flows based on the induced voltage in the damper ring represented by the time change of the magnetic flux, the Joule loss W based on the eddy current is expressed by the equation (Equation 2). Here, K is a proportionality constant.

そして、ダンパ効果を表すダンパ定数Ｋｄは（数３）の式にて表される。

A damper constant Kd representing the damper effect is expressed by the equation (Equation 3).

すなわち、ダンパ効果は磁束φの座標ｒに関する導関数（ｄφ／ｄｒ）の２乗に比例している。

That is, the damper effect is proportional to the square of the derivative (dφ / dr) with respect to the coordinate r of the magnetic flux φ.

The diameters of the teeth 111b and the teeth 112b are slightly different from each other in advance, and the side surfaces on both sides of the teeth are inclined with respect to the rotation axis, thereby increasing the value of − (dφ / dt) and increasing the damper effect. It was a magnetic bearing device that can.
Japanese Patent Publication No.52-26293

  For example, when a magnetic bearing device is used as a main spindle for machining, a bearing rigidity capable of withstanding the machining load and a damping characteristic that attenuates vibration caused by machining are required. However, since the conventional magnetic bearing device obtains damping force only by the eddy current generated inside the magnetic damper based on the change of magnetic flux due to the change in the radial direction, it has sufficient damping characteristics to attenuate the vibration caused by machining. It had the subject that it was difficult to ensure the damper effect of.

  The present invention is directed to solving the problems of the prior art, and an object thereof is to provide a magnetic bearing device in which the damper effect is dramatically improved.

In order to achieve the above-described object, the magnetic bearing device according to the present invention is configured so that the first bearing magnetic poles on the stator side and the rotor side are opposed to each other in the thrust direction through the air gap, and the first bearings facing the air gap are provided. The stator magnetic pole has a plurality of concentric grooves centered on the rotation center of the rotor in the bearing magnetic pole, and an electromagnetic winding is installed in at least one of the grooves of the first bearing magnetic pole on the stator side. A radial thrust bearing having a magnetic circuit formed thereon and a second bearing magnetic pole on the stator side and the rotor side are opposed to each other in the thrust direction through a gap, and the rotor is placed on each second bearing magnetic pole facing the gap. A magnetic circuit is formed on the stator and the rotor by having a plurality of grooves concentrically centered on the center of rotation and installing a permanent magnet in at least one of the grooves of the second bearing magnetic pole on the stator side. Radial bearings and radial Al direction so as to face the rotor with a gap, by placing a permanent magnet in the groove with the damper poles, and a magnetic damper which is formed a magnetic circuit in the stator and the rotor, as well as constituted by the radial The minimum magnetic pole tooth width among the plurality of magnetic pole teeth in the thrust bearing is modeled as rg ₁ , and the plurality of magnetic pole teeth are modeled as four magnetic pole teeth, and each magnetic pole tooth has an inner diameter rf ₁ and an outer diameter from the rotation center of the rotor. rf ₂ , width rg ₁ , inner diameter rf ₃ , outer diameter rf ₄ , width rg ₁ , inner diameter rf ₅ , outer diameter rf ₆ , width rg ₁ , inner diameter rf ₇ , outer diameter rf ₈ , width rg ₁ , radial thrust bearing The gap between the rotor and the stator in the thrust direction is g ₁ , the current flowing through the winding of the electromagnetic winding is I ₁ , and the number of turns of the winding is N _1, and the smallest magnetic pole among the plurality of magnetic pole teeth in the radial bearing tooth The rg _2, by modeling the plurality of magnetic pole teeth as four magnetic pole teeth, each of the pole teeth inside diameter rr ₁ from the rotational center of the rotor, the outer diameter rr _2, the width rg _2, inner diameter rr _3, the outer diameter rr ₄ , Width rg ₂ , inner diameter rr ₅ , outer diameter rr ₆ , width rg ₂ , inner diameter rr ₇ , outer diameter rr ₈ , width rg ₂ , a radial bearing with a gap in the thrust direction between the rotor and stator g ₂ , permanent The length of the magnet in the magnetic circuit direction is lp ₂ , the inner diameter of the permanent magnet is Re ₂ , the height of the permanent magnet in the thrust direction is re ₂ , and the holding force of the permanent magnet is Hc _2. R ₀ radius from the center _of, g ₀ gap damper pole and the rotor in the radial _direction, h ₀ in the thrust direction of the height of the damper _pole, lp 0 the length of the magnetic circuit direction of the permanent _magnet, the permanent magnet The radial width of ₀ , the outer diameter of the permanent magnet is Re ₀ , the holding force of the permanent magnet is Hc ₀ , the displacement of the rotor in the radial direction is x, the circle having the radius of the outer diameter rf ₈ and the circle of the radius rf ₈ are in the radial direction When the displacement x is moved to the circle of the original radius rf ₈ , the circle with the angle at the center of rotation α ₁ and the outer diameter rr ₈ as the radius and the circle with the radius rr ₈ are displaced x in the radial direction. The angle at the center of rotation at the intersection with the circle of the original radius rr ₈ when α ₂ , the rotation angle around the thrust axis θ, m the number of radial thrust bearings installed, n the number of radial bearings installed, magnetic When the number of dampers is k, (Equation 4)

を満たすことを特徴とする。

It is characterized by satisfying.

According to a second aspect of the present invention, there is provided the magnetic bearing device according to the first aspect , wherein the bearing rigidity and damping characteristics are adjusted by changing a current flowing through the electromagnet winding.

  According to the above configuration, when the rotor is displaced in the radial direction, the radial damper generates a resultant force of the restoring force in the radial direction opposite to the radial thrust generated in the radial thrust bearing and the radial bearing by the magnetic damper. By increasing the suction force in the direction, the rigidity can be increased and the damping performance can be increased. When the radial bearing has higher rigidity and vibrations due to processing in the radial direction occur, Damping performance can be secured by the damper.

  According to the present invention, it is possible to provide a magnetic bearing device that can quickly reduce vibration when radial vibration is generated in the rotor.

  Hereinafter, embodiments of the present invention will be described in detail with reference to the drawings.

(Embodiment 1)
FIG. 1 is a structural diagram showing a magnetic bearing device according to Embodiment 1 of the present invention. In FIG. 1, reference numeral 17 denotes a rotor which is a rotating body, and a tool 16 is attached to bearing magnetic poles 11 and 12 provided with a plurality of magnetic pole teeth (hereinafter referred to as magnetic teeth). Thrust direction of the rotor 17 at a minute distance from the bearing magnetic poles 11, 12, 12 at the stator side and a distance from the rotor 17 at a minute distance from the bearing magnetic poles 11, 12 at the rotor 17 side Displacement sensors 13 and 14 for detecting the displacement of the rotor 17 are disposed, and are supported in a non-contact manner with the rotor 17.

  The stator-side bearing magnetic poles 1, 3, 5, 7 are provided with a plurality of magnetic teeth corresponding to the bearing magnetic poles 11, 12 on the rotor 17 side, and are arranged in a ring shape surrounding the rotor 17. The stator-side bearing magnetic poles 1 and 3 have a plurality of concentric grooves centering on the rotation center of the rotor 17, and at least one of the electromagnet windings 2 and 4 is installed.

  Ring-shaped permanent magnets 6 and 8 centering on the rotation center of the rotor 17 are provided on the bearing magnetic poles 5 and 7 on the stator side. The permanent magnets 6 and 8 use rare earth iron-based magnets or the like in order to obtain a high attractive force.

  Further, a damper magnetic pole 9 of a magnetic damper is provided at a small distance in the radial direction from the rotor 17, and is arranged in a ring shape surrounding the rotor 17. A ring-shaped permanent magnet 10 is installed in the groove of the damper magnetic pole 9 of the magnetic damper. The permanent magnet 10 also uses a rare earth iron magnet or the like in order to obtain a high magnetic flux.

  As the displacement sensors 13 and 14, well-known eddy current type sensors, capacitance type sensors, optical sensors, and the like are used. Reference numeral 19 denotes a casing, to which stator side members such as stator side bearing magnetic poles 1, 3, 5, and 7 and displacement sensors 13 and 14 are attached. Reference numeral 15 denotes an actuator, for which a motor for rotating the rotor 17 is used.

  The support by magnetic levitation in the radial direction will be described. A current is passed through the electromagnet windings 2 and 4 attached to the stator-side bearing magnetic poles 1 and 3 to generate a magnetic flux, and a magnetic path is formed by the opposing bearing-side bearing magnetic poles 11. This magnetic flux generates a magnetic attractive force in the thrust direction between the bearing magnetic poles 1 and 3 on the stator side and the bearing magnetic pole 11 on the rotor 17 side. Further, the radial displacement is countered by a decrease in magnetic energy accompanying a decrease in the facing area between the magnetic teeth formed on the stator-side bearing magnetic poles 1 and 3 and the rotor-side bearing magnetic pole 11. A radial restoring force is generated to push back the movement of the rotor 17 in the radial direction. This bearing will be referred to as a radial thrust bearing.

  In the rotor-side bearing magnetic pole 12 facing the stator-side bearing magnetic poles 5, 7, the opposing rotation is caused by the magnetic flux generated by the permanent magnets 6, 8 attached to the stator-side bearing magnetic poles 5, 7. A magnetic path is formed by the bearing magnetic pole 12 on the child 17 side. This magnetic flux generates a magnetic attractive force in the thrust direction between the bearing magnetic poles 5 and 7 on the stator side and the bearing magnetic pole 12 on the rotor 17 side. Further, with respect to the displacement in the radial direction, the magnetic energy is reduced due to the reduction of the facing area between the magnetic teeth formed on the stator magnetic poles 5 and 7 and the bearing magnetic pole 12 on the rotor 17 side. A counter radial force is generated to push back the movement of the rotor 17 in the radial direction. This bearing will be referred to as a radial bearing.

  The support by magnetic levitation in the thrust direction will be described. Although the configuration of the control circuit is not shown, a deviation between the signals of the displacement sensors 13 and 14 and the target position is obtained, a current is passed through the electromagnet windings 2 and 4 so as to eliminate the deviation, and the bearing magnetic pole 1 on the stator side. , 3 and the magnetic attraction force generated between the bearing magnetic pole 11 on the rotor 17 side and the position of the rotor 17 in the thrust direction is controlled to fly.

  The magnetic damper will be described. When the rotor 17 is displaced in the radial direction, an eddy current is generated in the rotor 17 and the damper magnetic pole 9 of the magnetic damper in a direction to cancel the change of the magnetic flux, and a force for braking the motion is generated by the eddy current. When this magnetic damper deviates from the equilibrium point, the rotor 17 is attracted to the damper magnetic pole 9 of the magnetic damper, and therefore it is necessary to support the rotor 17 with a spring element.

  In the first embodiment, the spring element is configured in a non-contact manner by the magnetic thrust force of the radial thrust bearing and the radial bearing. Therefore, in order to make non-contact levitation, it is necessary to make the sum of the radial restoring force of the radial thrust bearing and the radial bearing larger than the attractive force of the magnetic damper. The restoring force of the radial thrust bearing and the radial bearing is opposite to the direction in which the rotor 17 is attracted to the damper magnetic pole 9 of the magnetic damper when shifted from the center of rotation in the radial direction.

  Next, the radial thrust bearing, the restoring force of the radial bearing and the force with which the rotor 17 is attracted to the damper magnetic pole 9 of the magnetic damper are formulated. FIG. 2 shows the structure of a radial thrust bearing.

The cross-sectional area of the first magnetic tooth 51 shown in FIG. 2 is S ₁ , the cross-sectional area of the second magnetic tooth 52 is S ₂ , the cross-sectional area of the third magnetic tooth 53 is S ₃ , and the cross-sectional area of the fourth magnetic tooth 54 is S _{4. If} the number of turns of the electromagnet winding is N ₁ , the current flowing through the winding is I ₁ , and the gap between the rotor and the stator of the bearing magnetic pole is g ₁ , the magnetic flux Φ of the magnetic path is (Equation 9)

にて表される。μ_０は真空の透磁率である。

It is represented by μ ₀ is the vacuum permeability.

Since the cross-sectional areas S ₁ , S ₂ , S ₃ , S ₄ are functions of the displacement x in the radial direction, the magnetic flux Φ is differentiated by the displacement x (Equation 10)

にて表される。

It is represented by

  Further, when the distance rθ from the center of the original circle when the circle with the radius r shown in FIG. 3 is displaced by x in the radial direction is obtained at the angle θ, (Equation 11)

にて表されて近似できる。

It can be approximated by

A plurality of magnetic teeth are modeled as four magnetic teeth. The width of the smallest magnetic tooth among the plurality of magnetic teeth of the radial thrust bearing is rg ₁ , the inner diameter of the first magnetic tooth 51 is rf ₁ , the outer diameter is rf ₂ , the inner diameter of the second magnetic tooth 52 is rf ₃ , The diameter is rf ₄ , the inner diameter of the third magnetic tooth 53 is rf ₅ , the outer diameter is rf ₆ , the inner diameter of the fourth magnetic tooth 54 is rf ₇ , and the outer diameter is rf ₈ .

As shown in FIG. 3, the angle at the rotation center at the intersection of the circle having the radius of the outer diameter rf ₈ and the circle of the radius rf ₈ displaced by x in the radial direction and the original circle of the radius rf ₈ is α1, The widths of the first to fourth magnetic teeth 51 to 54 are equal to rg ₁ . As shown in FIG. 4, for example, when a magnetic tooth having an inner diameter of rf ₁ and an outer diameter of rf ₂ is displaced by a displacement x in the radial direction, the differential of the area dS and the displacement x of the area in the minute section dθ is (Equation 12)

にて表される。
また（数１０），（数１２）の式より、（数１３）

It is represented by
From the formulas (10) and (12), (13)

で表される。

It is represented by

When the magnetic energy of the magnetic path in this section is W _m , the restoring force Ff1 with respect to the displacement x is (Expression 14)

にて表される。

It is represented by

  Further, in the range from the angle α1 to πrad, the differential of the area dS and the displacement x of the area in the minute section dθ is (Equation 15)

にて表される。

It is represented by

  If the restoring force for the displacement x in this section is Ff2, (Equation 16)

にて表される。

It is represented by

Next, the restoring force of the radial bearing is obtained. FIG. 5 is a structural diagram of a radial bearing. In FIG. 5, the sectional area of the fifth magnetic tooth 55 is S ₁ , the sectional area of the sixth magnetic tooth 56 is S ₂ , the sectional area of the seventh magnetic tooth 57 is S ₃ , and the sectional area of the eighth magnetic tooth 58 is S. _4. The gap between the rotor 17 and the stator of the bearing magnetic pole is g ₂ , the length of the permanent magnet in the magnetic path direction is lp ₂ , the inner diameter of the permanent magnet is Re ₂ , and the height of the permanent magnet in the thrust direction is re _2. When the holding force of the permanent magnet is Hc ₂ and the cross-sectional area of the permanent magnet is S ₅ , the magnetic flux Φ of the magnetic path is (Equation 17)

にて表される。

It is represented by

Since the cross-sectional areas S ₁ , S ₂ , S ₃ and S ₄ are functions of the displacement x in the radial direction, the magnetic flux Φ is differentiated by the displacement x (Equation 18)

にて表される。

It is represented by

A plurality of magnetic teeth are modeled as four magnetic teeth. The width of the smallest magnetic tooth among the plurality of magnetic teeth of the radial bearing is rg ₂ , the inner diameter of the fifth magnetic tooth 55 is rr ₁ , the outer diameter is rr ₂ , the inner diameter of the sixth magnetic tooth 56 is rr ₃ , and the outer diameter. Rr ₄ , the inner diameter of the seventh magnetic tooth 57 is rr ₅ , the outer diameter is rr ₆ , the inner diameter of the eighth magnetic tooth 58 is rr ₇ , and the outer diameter is rr ₈ . The angle at the rotation center of the intersection of the circle of the original radius rr ₈ when the circle of the circle and the radius rr ₈ in which the outer diameter rr ₈ the radius is displaced x moved in the radial direction and [alpha] 2, fifth to eighth the width of magnetic teeth 55 to 58 and equal rg _2. When the displacement x is shifted in the radial direction, the differential of the area dS and the area displacement x in the minute section dθ is (Equation 19)

にて表される。また図６にｄＳ_５の説明図を示す。
（数１８），（数１９）の式より、（数２０）

It is represented by Further illustrative of the dS ₅ in FIG.
From the equations of (Equation 18) and (Equation 19), (Equation 20)

で表される。

It is represented by

  The restoring force Fr1 with respect to the displacement x in this section is (Expression 21)

にて表される。

It is represented by

  In the range from the angle α2 to πrad, the differential of the area dS and the displacement x of the area in the minute section dθ is (Equation 22)

にて表される。

It is represented by

  When the restoring force for the displacement x in this section is Fr2, (Equation 23)

にて表される。

It is represented by

Next, the attractive force of the magnetic damper is obtained. FIG. 7 shows a structural diagram of the magnetic damper. The area of the damper magnetic pole 9 (magnetic tooth) of the magnetic damper is S ₀ , the radius from the center of the damper magnetic pole 9 and the stator of the magnetic damper is R ₀ , the radius of the rotor 17 is r ₀ , and the stator and rotor 17 The gap in the radial direction is ge ₀ , the height of the damper magnetic pole 9 (magnetic tooth) in the thrust direction is h ₀ , the length of the permanent magnet in the magnetic path direction is lp ₀ , the width of the permanent magnet in the radial direction is re ₀ , When the outer diameter of the magnet is Re ₀ , the holding force of the permanent magnet is Hc ₀ , and the sectional area of the permanent magnet is S ₆ , the attractive force Fd1 of the magnetic damper is (Equation 24)

にて表される。

It is represented by

The area S ₀ of the magnetic teeth of the damper magnetic pole 9, the sectional area S ₆ of the permanent magnet, and the gap ge ₀ are (Equation 25)

にて表される。

It is represented by

Therefore, the force Fd1 _x with respect to the displacement x is (number 26)

にて表される。

It is represented by

  Here, when the number of radial thrust bearings is m, the number of radial bearings is n, and the number of magnetic dampers is k, the radial thrust bearing and the radial bearing are restored in the radial direction in order to achieve non-contact levitation. The condition for making the sum of forces larger than the attractive force of the magnetic damper is (Equation 27)

にて表される。

It is represented by

  Since radial thrust bearings also serve as thrust direction bearings, it is desirable that they be installed facing each other (m = 2) in order to linearize the thrust force in the thrust direction.

Next, a specific design example of the first embodiment will be shown. The minimum magnetic tooth width among the plurality of magnetic teeth of the radial thrust bearing described above is rg ₁ = 0.4 mm, the inner diameter rf ₁ = 9.5 mm, the outer diameter rf ₂ = 9.9 mm from the center of the rotor 17, respectively. Inner diameter rf ₃ = 10.3 mm, outer diameter rf ₄ = 10.7 mm, inner diameter rf ₅ = 11.1 mm, outer diameter rf ₆ = 11.5 mm, inner diameter rf ₇ = 11.9 mm, outer diameter rf ₈ = 12.3 mm The gap in the thrust direction between the radial thrust bearing and the rotor 17 is g ₁ = 0.025 mm, the current passed through the electromagnet winding is I ₁ = 0.15 A, and the number of turns of the winding is N ₁ = 300 turns.

Further, the minimum magnetic tooth width among the plurality of magnetic teeth of the radial bearing is modeled as rg ₁ , and the plurality of magnetic teeth are modeled as four magnetic teeth, and the inner diameter rr ₁ = 9.5 mm from the center of the rotor 17 and the outer diameter. rr ₂ = 9.9 mm, inner diameter rr ₃ = 10.3 mm, outer diameter rr ₄ = 10.7 mm, inner diameter rr ₅ = 11.1 mm, outer diameter rr ₆ = 11.5 mm, inner diameter rr ₇ = 11.9 mm, outer The diameter rr ₈ = 12.3 mm, the radial gap between the radial bearing and the rotor 17 is g ₂ = 0.025 mm, the length of the permanent magnet in the magnetic path direction is lp ₂ = 3 mm, and the inner diameter of the permanent magnet is Re ₂ = 10.7 mm, the height of the permanent magnet in the thrust direction is re ₂ = 1.2 mm, and the holding force of the permanent magnet is Hc ₂ = 891 × 10 ³ A / m.

Then, the radius from the center of the damper magnetic pole and the stator of the magnetic damper is R ₀ = 12.4 mm, the height of the damper magnetic pole (magnetic tooth) in the thrust direction is h ₀ = 1 mm, and the length of the permanent magnet in the magnetic path direction Lp ₀ = 3 mm, the radial width of the permanent magnet is re ₀ = 1.3 mm, the outer diameter is Re ₀ = 15 mm, the holding force of the permanent magnet is Hc ₀ = 891 × 10 ³ A / m, the rotor 17 The radial displacement is x = 0.05 mm, the rotation angle around the thrust axis is θ, the radial thrust bearings are installed m = 1, the radial bearings are installed n = 1, and the magnetic dampers are installed k = 1. FIG. 8 shows the calculation result of (Equation 27) when the radial gap g ₀ between the damper magnetic pole of the magnetic damper and the rotor 17 is changed. The condition is satisfied by the gap of the magnetic damper in which the right side of (Equation 27)> the left side.

(Embodiment 2)
FIG. 9 is a structural diagram showing a magnetic bearing device according to Embodiment 2 of the present invention. In the second embodiment, the magnetic damper of the first embodiment described above is an electromagnet type. FIG. 10 shows a structural diagram of the magnetic damper in the second embodiment. The area of the damper magnetic pole 20 (magnetic tooth) of the magnetic damper is S ₀ , the gap between the damper magnetic pole of the magnetic damper 20 and the rotor 17 is ge ₀ , the radius from the rotation center to the magnetic pole is R ₀ , and the number of turns of the electromagnetic winding is When N ₀ and the current flowing through the winding are I ₀ , the attractive force Fd2 of the electromagnet is (Equation 28)

にて表される。

It is represented by

  Therefore, the force Fd2x with respect to the displacement x is (Equation 29).

にて表される。

It is represented by

  When the number of radial thrust bearings is m, the number of radial bearings is n, and the number of magnetic dampers is k, the sum of the restoring forces in the radial direction of the radial thrust bearing and the radial bearing is used to make contactless levitation. The condition for making the magnetic force larger than the attractive force of the magnetic damper is (Equation 30)

にて表される。

It is represented by

  By using an electromagnet type, the effect of the magnetic damper can be adjusted by changing the current value in accordance with processing.

A specific design example of the second embodiment will be described. The minimum magnetic tooth width among the plurality of magnetic teeth of the radial thrust bearing described above is rg ₁ = 0.4 mm, the inner diameter rf ₁ = 9.5 mm and the outer diameter rf ₂ = 9.9 mm from the rotation center of the rotor 17, respectively. , Inner diameter rf ₃ = 10.3 mm, outer diameter rf ₄ = 10.7 mm, inner diameter rf ₅ = 11.1 mm, outer diameter rf ₆ = 11.5 mm, inner diameter rf ₇ = 11.9 mm, outer diameter rf ₈ = 1.12. The gap in the thrust direction between the radial thrust bearing and the rotor 17 is g ₁ = 0.025 mm, the current flowing through the electromagnet winding is I ₁ = 0.15 A, and the number of turns of the winding is N ₁ = 300 turns. .

Further, the minimum magnetic tooth width among the plurality of magnetic teeth of the radial bearing is modeled as rg ₂ , and the plurality of magnetic teeth are modeled as four magnetic teeth, and the inner diameter rr ₁ = 9.5 mm from the rotation center of the rotor 17, Outer diameter rr ₂ = 9.9 mm, Inner diameter rr ₃ = 10.3 mm, Outer diameter rr ₄ = 10.7 mm, Inner diameter rr ₅ = 11.1 mm, Outer diameter rr ₆ = 11.5 mm, Inner diameter rr ₇ = 11.9 mm , Outer diameter rr ₈ = 12.3 mm, gap in radial direction between radial bearing and rotor 17 g ₂ = 0.025 mm, length of permanent magnet in magnetic path direction lp ₂ = 3 mm, inner diameter of permanent magnet Re ₂ = 10.7 mm, the height of the permanent magnet in the thrust direction is re ₂ = 1.2 mm, and the holding force of the permanent magnet is Hc ₂ = 891 × 10 ³ A / m.

Then, the radius from the center of the damper magnetic pole and the stator of the magnetic damper is R ₀ = 12.4 mm, the height of the damper magnetic pole (magnetic tooth) in the thrust direction is h ₀ = 1 mm, and the current flowing through the electromagnetic winding is I _0. = 0.17 A, the number of turns of the winding is N ₀ = 300 turns, the displacement of the rotor 17 in the radial direction is x = 0.05 mm, the rotational angle around the thrust axis is θ, and the number of radial thrust bearings installed is m = 1. When the number of installed radial bearings is n = 1 and the number of installed magnetic dampers is k = 1, the radial gap g ₀ between the damper magnetic poles of the magnetic damper and the rotor 17 is changed (Equation 30 The calculation result of) is shown in FIG. The condition is satisfied by the gap of the magnetic damper in which the right side of (Equation 30)> the left side.

(Embodiment 3)
FIG. 12 is a structural diagram showing a magnetic bearing device according to Embodiment 3 of the present invention. In the third embodiment, the radial bearing of the second embodiment is an electromagnet type. As in the first embodiment, the width of the smallest magnetic tooth among the plurality of magnetic teeth of the radial bearing is rg ₂ , and the four magnetic teeth have an inner diameter of rr ₁ , an outer diameter of rr ₂ , an inner diameter of rr ₃ , The outer diameter is rr ₄ , the inner diameter is rr ₅ , the outer diameter is rr ₆ , the inner diameter is rr ₇ , and the outer diameter is rr ₈ . N ₂ turns of the electromagnet _coil, and the current flowing through the winding and I _2, the original radius rr when the circle of the circle and the radius rr ₈ in which the outer diameter rr ₈ the radius is displaced x moved in the radial direction the angle at the rotation center of the intersection of the circle of ₈ and [alpha] 2, the width of the four magnetic teeth are equally rg _2. The restoring forces Fr1 and Fr2 with respect to the displacement x are (Equation 31)

にて表される。

It is represented by

  When the number of radial thrust bearings is m, the number of radial bearings is n, and the number of magnetic dampers is k, the sum of the restoring forces in the radial direction of the radial thrust bearing and the radial bearing is used to make contactless levitation. The condition for making the magnetic force larger than the attractive force of the magnetic damper is (Expression 32)

にて表される。

It is represented by

  By using an electromagnet type, the rigidity and the effect of the magnetic damper can be adjusted by changing the current value according to the processing.

Next, a specific design example of the third embodiment is shown. The minimum magnetic tooth width among the plurality of magnetic teeth of the radial thrust bearing described above is rg ₁ = 0.4 mm, the inner diameter rf ₁ = 9.5 mm and the outer diameter rf ₂ = 9.9 mm from the rotation center of the rotor 17, respectively. , Inner diameter rf ₃ = 10.3 mm, outer diameter rf ₄ = 10.7 mm, inner diameter rf ₅ = 11.1 mm, outer diameter rf ₆ = 11.5 mm, inner diameter rf ₇ = 11.9 mm, outer diameter rf ₈ = 1.12. The gap in the thrust direction between the radial thrust bearing and the rotor 17 is g ₁ = 0.025 mm, the current flowing through the electromagnet winding is I ₁ = 0.15 A, and the number of turns of the winding is N ₁ = 300 turns. .

Further, the minimum magnetic tooth width of the plurality of magnetic teeth of the radial bearing is modeled as rg ₂ , and the plurality of magnetic teeth are modeled as four magnetic teeth, and the inner diameter rr ₁ = 9.5 mm from the rotation center of the rotor 17 is diameter _rr 2 = 9.9 mm, inner diameter _rr 3 = 10.3 mm, external diameter _rr 4 = 10.7 mm, inner diameter _rr 5 = 11.1 mm, an outer diameter _rr 6 = 11.5 mm, inner diameter _rr 7 = 11.9 mm, The outer diameter rr ₈ = 12.3 mm, the radial gap between the radial bearing and the rotor 17 is g ₂ = 0.025 mm, the current flowing through the electromagnet winding is I ₂ = 0.15 A, and the number of turns is N ₂ = 300 turns.

Then, the radius from the center of the damper magnetic pole and the stator of the magnetic damper is R ₀ = 12.4 mm, the height of the damper magnetic pole (magnetic tooth) in the thrust direction is h ₀ = 1 mm, and the current flowing through the electromagnetic winding is I _0. = 0.17 A, the number of turns of the winding is N ₀ = 300 turns, the displacement of the rotor 17 in the radial direction is x = 0.05 mm, the rotational angle around the thrust axis is θ, and the number of radial thrust bearings installed is m = 1. When the number of installed radial bearings is n = 1 and the number of installed magnetic dampers is k = 1, the radial gap g ₀ between the damper magnetic poles of the magnetic damper and the rotor 17 is changed (Equation 32 The calculation result of) is shown in FIG. The condition is satisfied by the gap of the magnetic damper in which the right side of (Equation 32)> the left side.

(Embodiment 4)
FIG. 14 is a view showing the structure of a magnetic bearing according to Embodiment 4 of the present invention. In the first to third embodiments described above, the bearing magnetic pole and the damper magnetic pole are configured as separate structures. However, in the fourth embodiment, the magnetic bearing is provided with the bearing magnetic pole and the damper magnetic pole provided in the same magnetic path.

  Since it is not necessary to separately configure a magnetic damper for the damper magnetic pole, the apparatus can be miniaturized and the length of the rotor can be shortened, so that the natural frequency can be increased. Furthermore, the number of parts including a permanent magnet can be reduced, and the cost can be reduced.

  The fourth embodiment will be described with reference to FIG. Reference numeral 17 denotes a rotor, to which a bearing magnetic pole 11 provided with a plurality of magnetic teeth is attached. A stator-side bearing magnetic pole 31 is disposed at a minute distance (gap) in the thrust direction from the bearing magnetic pole 11 on the rotor 17 side. The stator-side bearing magnetic pole 31 is provided with a plurality of magnetic teeth corresponding to the rotor-side bearing magnetic pole 11 and arranged in a ring shape surrounding the rotor 17. The stator-side bearing magnetic pole 31 has a plurality of concentric grooves centering on the rotation center of the rotor 17, and at least one of the electromagnet windings 32 is installed. In addition, the stator-side bearing magnetic pole 31 is provided with a damper magnetic pole 33 for a magnetic damper at a small distance in the radial direction from the bearing magnetic pole 11 of the rotor 17.

  The support by magnetic levitation in the radial direction will be described. A current is passed through the electromagnet winding 32 attached to the bearing magnetic pole 31 on the stator side to generate a magnetic flux, and the magnetic teeth of the bearing magnetic pole 11 on the rotor 17 side facing the magnetic teeth of the bearing magnetic pole 31 on the stator side. A magnetic path is formed by the magnetic teeth of the damper magnetic pole 33 for the magnetic damper. This magnetic flux generates a magnetic attractive force in the thrust direction between the bearing magnetic pole 31 on the stator side and the bearing magnetic pole 11 on the rotor 17 side. Further, with respect to the displacement x in the radial direction, the magnetic energy associated with the decrease in the facing area between the magnetic teeth formed on the stator-side bearing magnetic pole 31 and the magnetic teeth of the bearing magnetic pole 11 on the rotor 17 side is reduced. A restoring force in the radial direction against the decrease is generated to push back the movement of the rotor 17 in the radial direction.

  The operation of the magnetic damper will be described. When the rotor 17 is displaced in the radial direction, an eddy current is generated in the rotor 17 and the damper magnetic pole 33 in a direction to cancel the change of the magnetic flux, and a force for braking the motion is generated by the eddy current. When the magnetic damper deviates from the equilibrium point, the rotor 17 is attracted to the damper magnetic pole 33, so that the rotor 17 needs to be supported by a spring element. In the fourth embodiment, a non-contact spring element is configured by a radial magnetic attractive force of a bearing magnetic pole provided in the same magnetic path as the magnetic damper. Therefore, in order to make non-contact levitation, the restoring force in the radial direction of the bearing magnetic pole needs to be larger than the attractive force of the magnetic damper. The restoring force in the thrust direction of the bearing magnetic pole and the direction in which the rotor 17 is attracted to the damper magnetic pole 33 when shifted in the radial direction from the center of rotation are opposite forces.

  In order to confirm the occurrence of the damper effect, CAE analysis (Computer Aided Engineering) was performed using the model shown in FIG. In order to analyze only the damper effect, the bearing magnetic pole on the rotor 17 side is flattened so that the restoring force in the thrust direction of the bearing magnetic pole does not work. The spring constant in the radial direction was given on the analysis software. The gap between the stator-side magnetic pole and the rotor-side magnetic pole is 0.025 mm in the thrust direction (bearing magnetic pole), 0.1 mm in the radial direction (damper magnetic pole), 0.010 mm in the initial radial direction displacement, and a rare earth permanent magnet is used. FIG. 16 shows the current generated on the stator side at this time, FIG. 17 shows the current generated on the rotor 17 side, and FIG. 18 shows the radial displacement of the rotor 17. As shown in FIG. 18, it can be seen that the displacement of the rotor 17 is attenuated.

Next, the restoring force of the bearing magnetic pole and the attractive force of the damper magnetic pole are formulated. As shown in FIG. 14, a plurality of magnetic teeth are modeled as two magnetic teeth. The cross-sectional area of the ninth magnetic tooth 61 is S ₁ , the cross-sectional area of the tenth magnetic tooth 62 is S ₂ , the area of the damper magnetic pole 33 (magnetic tooth) is S, the number of turns of the electromagnet winding 32 is N, and the current passed through the winding , I ₃ , the gap between the ninth magnetic tooth 61 and the tenth magnetic tooth 62 and the rotor 17 is g ₃ , and the radial gap between the damper magnetic pole 33 and the bearing magnetic pole 11 of the rotor 17 is g. Magnetic flux Φ is (Expression 33)

にて表される。

It is represented by

Since the sectional areas S ₁ and S ₂ are functions of the radial displacement x, the magnetic flux Φ is differentiated by the displacement x (Equation 34).

にて表される。

It is represented by

The inner diameter of the ninth magnetic tooth 61 is r ₁ , the outer diameter is r ₂ , the inner diameter of the tenth magnetic tooth 62 is r ₃ , and the outer diameter is r ₄ . As shown in FIG. 19, the angle at the center of rotation is α at the intersection of the circle having the radius of outer diameter r ₄ and the circle of radius r ₄ displaced x in the radial direction and the original circle of radius r _4. If the widths of the ninth magnetic tooth 61 and the tenth magnetic tooth 62 are equal to rg, the differential of the area of the minute section dθ and the displacement x of the area is given by

にて表すことができる。
（数３４），（数３５）より、（数３６）

Can be expressed as
From (Expression 34) and (Expression 35), (Expression 36)

と表すことができる。

It can be expressed as.

  The restoring force F1 with respect to the displacement x is (Expression 37)

にて表すことができる。

Can be expressed as

  Further, in the range from the angle α to π rad, the differential of the area of the minute section dθ and the displacement x of the area is (Equation 38).

と表すことができる。

It can be expressed as.

  The restoring force F2 with respect to the displacement x in this section is (Equation 39)

と表すことができる。

It can be expressed as.

The suction force Fd3 _x is (number 40) relative to the displacement x of the magnetic damper (number 41) from the equation (42)

と表すことができる。

It can be expressed as.

  Therefore, in order to make the restoring force of the bearing magnetic pole larger than the attractive force of the damper magnetic pole, (Equation 43)

の条件を満たす必要がある。

It is necessary to satisfy the conditions.

  FIG. 20 is a diagram showing an example of a magnetic bearing device constituted by the magnetic bearing of the fourth embodiment. As shown in FIG. 20, the magnetic bearing of the fourth embodiment is installed on the magnetic bearing closer to the tool 16 (front side), and the magnetic bearing farther from the tool 16 (rear side) is the permanent magnet of the first embodiment. It consists of a radial bearing using

Next, a specific design example of the fourth embodiment will be shown. Of the plurality of magnetic teeth on the magnetic pole A surface shown in FIG. 14, the width of the smallest magnetic tooth is rg = 0.4 mm, the inner diameter r ₁ = 9.5 mm and the outer diameter r ₂ = 9. 9 mm, inner diameter r ₃ = 10.3 mm, outer diameter r ₄ = 10.7 mm, the gap in the thrust direction between the magnetic pole A surface and the rotor 17 is g ₃ = 0.050 mm, from the magnetic pole B surface and the center of the stator The radius is R = 12.4 mm, the radial gap between the magnetic pole B surface and the rotor 17 is g = 0.1 mm, the radial displacement of the rotor 17 is x = 0.05 mm, and the rotation angle about the thrust axis is θ. FIG. 21 shows the calculation result of (Equation 43) when the height h of the damper magnetic pole 33 (magnetic tooth) in the thrust direction is changed. The condition is satisfied by the height in the thrust direction of the damper magnetic pole 33 in which the right side of (Equation 43)> the left side.

(Embodiment 5)
FIG. 22 is a structural diagram showing a magnetic bearing device according to Embodiment 5 of the present invention. In the fifth embodiment, the electromagnet winding of the magnetic bearing of the fourth embodiment is a permanent magnet. The cross-sectional area of the ninth magnetic tooth 61 is S ₁ , the cross-sectional area of the tenth magnetic tooth 62 is S ₂ , the area of the damper magnetic pole 37 is S, the length of the permanent magnet in the magnetic path direction is lp, and the radial direction of the permanent magnet is The width is re, the outer diameter of the permanent magnet is Re, the holding force of the permanent magnet is Hc, the sectional area of the permanent magnet is S ₆ , and the gap between the ninth magnetic tooth 61 and the tenth magnetic tooth 62 and the rotor 17 is g _3. When the gap between the damper magnetic pole 37 and the rotor 17 is g, the magnetic flux Φ of the magnetic path is (Equation 44)

で表すことができる。
ここで、永久磁石の断面積Ｓ_６は（数４５）

Can be expressed as
Here, the cross-sectional area _{S 6} of the permanent magnets (number 45)

にて表す。

Represented by

Since the sectional areas S ₁ and S ₂ are functions of the displacement x in the radial direction, the magnetic flux Φ is differentiated by the displacement x (Equation 46)

となる。
（数３５），（数４５）より、（数４７）

It becomes.
From (Equation 35) and (Equation 45), (Equation 47)

と表すことができる。

It can be expressed as.

  Therefore, the restoring force F1 with respect to the displacement x is (Formula 48).

と表すことができる。

It can be expressed as.

  Further, the restoring force F2 with respect to the displacement x in the range from the angle α to π rad is (Equation 49).

と表すことができる。

It can be expressed as.

  Also, the attractive force of the magnetic damper is (Equation 50)

と表すことができる。
（数４１）より、磁気ダンパの吸引力によるラジアル方向の成分は（数５１）

It can be expressed as.
From (Equation 41), the radial component due to the attractive force of the magnetic damper is (Equation 51).

と表すことができる。

It can be expressed as.

  Therefore, in order to make the radial restoring force by the bearing magnetic pole larger than the attractive force of the magnetic damper, (Equation 52)

の条件を満たす必要がある。

It is necessary to satisfy the conditions.

  As shown in FIG. 24, the same effect can be obtained even when the magnetic pole D surface of the damper magnetic pole is arranged on the rotating shaft side of the rotor.

Next, a specific design example of the fifth embodiment will be shown. The minimum magnetic tooth width among the plurality of magnetic teeth on the magnetic pole A surface shown in FIG. 22 is rg = 0.4 mm, the inner diameter r ₁ = 9.5 mm from the rotation center of the rotor 17, and the outer diameter r ₂ = 9. 9 mm, inner diameter r ₃ = 10.3 mm, outer diameter r ₄ = 10.7 mm, the gap in the thrust direction between the magnetic pole A surface and the rotor 17 is g ₃ = 0.025 mm, from the center of the stator on the magnetic pole B surface The radius is R = 12.4 mm, the radial gap between the magnetic pole B surface and the rotor 17 is g = 0.1 mm, the length of the permanent magnet in the magnetic path direction is lp = 3 mm, and the radial width of the permanent magnet is re. = 1.3 mm, outer diameter Re = 15 mm, radial displacement of the rotor 17 is x = 0.05 mm, and the rotation angle about the thrust axis is θ, the damper magnetic pole 37 (magnetic tooth) in the thrust direction Calculation result of (Formula 52) when height h is changed It is shown in Figure 23. The condition is satisfied by the height in the thrust direction of the damper magnetic pole 37 in which the right side of (Expression 52)> the left side.

  Since the magnetic bearing device according to the present invention can increase the damper effect according to the magnitude of the disturbance applied to the rotor, it is useful for machine tools in which the disturbance applied to the rotor changes greatly.

1 is a structural diagram showing a magnetic bearing device according to a first embodiment of the present invention. The figure which shows the structure of the radial thrust bearing in this Embodiment 1. Explanatory drawing of distance r (theta) and angle (alpha) 1 with respect to the displacement x in this Embodiment 1. FIG. Explanatory drawing of micro area dS with respect to displacement x in this Embodiment 1. The figure which shows the structure of the radial bearing in this Embodiment 1. Illustration of small area dS ₅ in the embodiment 1 The figure which shows the structure of the magnetic damper in this Embodiment 1. The figure which shows the calculation result of the magnetic bearing in this Embodiment 1. Structural drawing showing a magnetic bearing device in Embodiment 2 of the present invention The figure which shows the structure of the magnetic damper in this Embodiment 2. The figure which shows the calculation result of the magnetic bearing in this Embodiment 2. Structural drawing showing a magnetic bearing device in Embodiment 3 of the present invention The figure which shows the calculation result of the magnetic bearing in this Embodiment 3. Structural diagram of magnetic bearing in embodiment 4 of the present invention The figure which shows the analysis model of the magnetic bearing in this Embodiment 4. The figure which shows the analysis result (the electric current of the stator side) of the magnetic bearing in this Embodiment 4. The figure which shows the analysis result (current on the rotor side) of the magnetic bearing in this Embodiment 4. The figure which shows the analysis result (displacement of radial direction) of the magnetic bearing in this Embodiment 4. Explanatory drawing of angle (alpha) with respect to the displacement x in this Embodiment 4. The figure which shows the example of the magnetic bearing apparatus comprised with the magnetic bearing of this Embodiment 4. The figure which shows the calculation result of the magnetic bearing in this Embodiment 4. Structural diagram showing a magnetic bearing device according to Embodiment 5 of the present invention The figure which shows the calculation result of the magnetic bearing in this Embodiment 5. The figure which shows the structure of another magnetic bearing in this Embodiment 5. A diagram showing a conventional magnetic bearing device

Explanation of symbols

1, 3, 5, 7, 25, 27, 31, 35 Stator-side bearing magnetic poles 2, 4, 21, 26, 28, 32 Electromagnet windings 6, 8, 10, 36 Permanent magnets 9, 20, 33, 37 Damper magnetic poles 11 and 12 on the stator side Bearing magnetic poles 13 and 14 on the rotor side Displacement sensor 15 Actuator 16 Tool 17 Rotor 19 Casing 51 First magnetic teeth 52 Second magnetic teeth 53 Third magnetic teeth 54 Fourth magnetic teeth 55 5th magnetic tooth 56 6th magnetic tooth 57 7th magnetic tooth 58 8th magnetic tooth 61 9th magnetic tooth 62 10th magnetic tooth

Claims (2)

A plurality of first bearing magnetic poles on the stator side and the rotor side are opposed to each other in the thrust direction through a gap, and each of the first bearing magnetic poles facing the gap is concentrically centered on the rotation center of the rotor. A radial thrust bearing in which a magnetic circuit is formed in the stator and the rotor by installing an electromagnetic winding in at least one of the grooves of the first bearing magnetic pole on the stator side,
The second bearing magnetic poles on the stator side and the rotor side are opposed to each other in the thrust direction through a gap, and each second bearing magnetic pole facing the gap has a concentric shape centered on the rotation center of the rotor. A radial bearing in which a magnetic circuit is formed in the stator and the rotor by installing a permanent magnet in at least one of the grooves of the second bearing magnetic pole on the stator side,
A magnetic circuit is formed between the stator and the rotor by placing a permanent magnet in a groove provided in the damper magnetic pole so as to face the rotor via a gap in the radial direction as a damper magnetic pole on the stator side. And a magnetic damper
The minimum magnetic pole tooth width of the plurality of magnetic pole teeth in the radial thrust bearing is modeled as rg ₁ , and the plurality of magnetic pole teeth are modeled as four magnetic pole teeth, and each magnetic pole tooth has an inner diameter rf ₁ from the rotation center of the rotor. , Outer diameter rf ₂ , width rg ₁ , inner diameter rf ₃ , outer diameter rf ₄ , width rg ₁ , inner diameter rf ₅ , outer diameter rf ₆ , width rg ₁ , inner diameter rf ₇ , outer diameter rf ₈ , width rg ₁ , In the radial thrust bearing, the gap in the thrust direction between the rotor and the stator is g ₁ , the current flowing through the winding of the electromagnetic winding is I ₁ , and the number of turns of the winding is N ₁ .
The minimum magnetic pole tooth width of the plurality of magnetic pole teeth in the radial bearing is modeled as rg ₂ , and the plurality of magnetic pole teeth are modeled as four magnetic pole teeth, and each magnetic pole tooth has an inner diameter rr ₁ from the rotation center of the rotor. , Outer diameter rr ₂ , width rg ₂ , inner diameter rr ₃ , outer diameter rr ₄ , width rg ₂ , inner diameter rr ₅ , outer diameter rr ₆ , width rg ₂ , inner diameter rr ₇ , outer diameter rr ₈ , width rg ₂ , In the radial bearing, the gap in the thrust direction between the rotor and the stator is g ₂ , the length of the permanent magnet in the magnetic circuit direction is lp ₂ , the inner diameter of the permanent magnet is Re ₂ , and the height in the thrust direction of the permanent magnet is high. Re ₂ , and the holding force of the permanent magnet is Hc ₂ ,
The radius from the center of the damper magnetic pole and the stator of the magnetic damper is R ₀ , the radial gap between the damper magnetic pole and the rotor is g ₀ , the height of the damper magnetic pole in the thrust direction is h ₀ , and the permanent The length of the magnet in the magnetic circuit direction is lp ₀ , the radial width of the permanent magnet is re ₀ , the outer diameter of the permanent magnet is Re ₀ , and the holding force of the permanent magnet is Hc ₀ ,
The radial displacement of the rotor x, the intersection of the original circle of radius rf ₈ when the outer diameter rf ₈ and the circle and the circle of the radius rf ₈ that the radius is the displacement x moves in the radial direction Rotate at the intersection of a circle having an angle at the rotation center α ₁ and a radius of the outer diameter rr ₈ and a circle of the radius rr ₈ and the original radius rr _{8 when} the circle of the radius rr ₈ is moved by the displacement x in the radial direction. When the angle at the center is α ₂ , the rotational angle around the thrust axis is θ, the number of radial thrust bearings is m, the number of radial bearings is n, and the number of magnetic dampers is k, (Equation 1 )

Magnetic bearing device you and satisfies the. It said electromagnetic by varying the current applied to the Ishinomaki Line, bearing rigidity, the magnetic bearing device according to claim 1 Symbol mounting and adjusting the damping characteristics.

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