JP2002366878A - Method and device for calculating movement of magnetic object in magnetic field - Google Patents

Abstract

PROBLEM TO BE SOLVED: To provide a method and a device for calculating the movement of a magnetic object in a magnetic field by using a practical calculation method of which an accurate magnetic field is found, variation in magnetic force on the surface of a magnetic body such as a magnetic particle is taken into consideration, and meshes are easily devided again accompanying the movement of the particle. SOLUTION: By the method for calculating the magnetic object in the magnetic field, a plurality of border nodes for mesh divisions are determined on a circumference indicating the border of the magnetic powder and a border element method is used for the calculation of the magnetic field by the border nodes on the circumference.

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a method and an apparatus for calculating the motion of a magnetic object in a magnetic field.

[0002]

2. Description of the Related Art In order to solve the motion of a magnetic object in a magnetic field, it is necessary to update the magnetic field in accordance with the movement of the magnetic object and to consider that the magnetic force acting on the object changes.

Heretofore, as a method for solving the motion of one magnetic particle in a magnetic field, calculation of a magnetic force by a magnetic moment method has been performed. At that time, the magnetic field created by the magnetic particles is usually ignored.

When there are a plurality of magnetic particles, the magnetic moments induced by the respective magnetic particles are superimposed,
It was a method of determining the magnetic field and calculating the magnetic force by iteratively calculating so that the whole magnetic field would be consistent.

For example, in the motion of a large number of spherical (three-dimensional) or circular (two-dimensional) magnetic particles in a magnetic field as shown in FIG. Consider the resulting magnetic moment. The magnetic field generated at other particle positions by this magnetic moment is expressed by equation (2). The magnetic field is determined by superimposing and repeatedly calculating this as shown in equation (3), and calculating the magnetic force by equation (4). How to

[0006]

(Equation 1)

[0007]

(Equation 2)

[0008]

(Equation 3)

[0009]

(Equation 4) The variables in the above equation are: B _i : magnetic flux density at the position of particle i B _i ′: external magnetic field at the position of particle i B _ij : magnetic field _{generated by} the magnetic moment of particle j at the position of particle i _mi : particle center Magnetic moment μ: magnetic permeability of particle μ ₀ : magnetic permeability of vacuum a: particle radius F _i : magnetic force acting on particle i.

In this method, only the magnetic force between the magnetic moments at the center position of the particle is considered, and the magnetic force only acts between the particle centers.

The above is a method in which the magnetic field due to the induced magnetization of the magnetic particles is regarded as a perturbation with a constant external magnetic field. However, this is a unavoidable choice due to the limitations of the operation speed of the computer and the memory.

As a conventional technique relating to the above-described calculation method, a paper (Serizawa: Proceedings of the Japan Society of Mechanical Engineers '99 vol.A)
p.288 (1999)).

However, actually, the magnetic flux density on the particle surface changes as shown in FIG. 3, and the magnetic force is originally expressed as Maxwell stress applied to the surface. In the magnetic moment method, since a bar magnet is set at the center of a particle, a change in magnetic flux density on the surface of the magnetic particle cannot be represented, and accurate magnetic force cannot be calculated. Therefore, since the equation of motion of the particle is solved by adding only the magnetic force acting between the particle centers, the torque cannot be considered, and the rotational motion of the magnetic particle due to the magnetic force cannot be obtained.

However, in order to calculate the entire magnetic field including the particles by the difference method or the finite element method and obtain the magnetic flux density on the surface of the particles, the entire region including the magnetic particles is discretized and divided into meshes. So the number of nodes becomes huge,
Further, since it is necessary to update the magnetic field in accordance with the movement of the magnetic particles, practical calculation is difficult in many cases even with the current computer performance.

For example, in the case of mesh division by the finite element method as shown in FIG. 4, it is necessary to update the mesh division as the particles move. However, performing this manually requires enormous labor. And practically impossible.

Further, as shown in FIG. 5, a method of fixing a regular mesh and changing the material constant (permeability) of the mesh according to the magnetic particles moving on the regular mesh is also conceivable. Although this method cannot accurately represent the curve shape of the particle surface, it is considered that a certain degree of magnetic field calculation accuracy can be obtained. However, if the particles are very small with respect to the size of the magnetic field region, the mesh must be further reduced, and the number of meshes and the number of nodes become enormous. For example, 2
When the size of the magnetic field region is 1000 × 1000 and the diameter of the circular particles is 10 in dimension, the size of the mesh needs to be about 1 or less. Difficult.

[0017]

In the conventional magnetic moment method described above, the calculation of the magnetic field is not accurate. Further, in the method in which the induced magnetization of the magnetic particles is substituted by the magnetic moment at the center, a change in the magnetic force on the surface of the magnetic particles cannot be considered. On the other hand, when the update calculation of the magnetic field is performed by a region division type solution such as the difference method or the finite element method, practical calculation is difficult due to the limitation of the calculation time and the memory. In addition, a problem has arisen that the mesh division must be updated as the particles move.

It is an object of the present invention to calculate a precise magnetic field, consider a change in magnetic force on the surface of a magnetic substance such as a magnetic particle, and perform a practical calculation which can easily redivide a mesh as the particle moves. An object of the present invention is to provide a method and an apparatus for estimating the motion of a magnetic object in magnetism using the method.

[0019]

SUMMARY OF THE INVENTION The object of the present invention is to provide a method for calculating the motion of a magnetic body in a magnetic field, wherein the calculation takes into account that the magnetic field changes with the movement of the magnetic body. This is achieved by using the boundary element method for calculating the magnetic field including the magnetic material.

More specifically, the present invention provides a method for calculating the motion of a magnetic substance such as a magnetic powder in a magnetic field using a boundary element method for calculating the magnetic field. A plurality of boundary nodes are defined in a mesh division on a circular circumference representing the boundary of the magnetic body, a motion calculation method of a magnetic object in a magnetic field using a boundary element method for calculating a magnetic field by the boundary nodes on the circumference and Provide equipment.

[0021]

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS An embodiment of the present invention will be described below with reference to FIG. FIG. 1 is a diagram showing a group of particles in a magnetic field.
In FIG. 1, 1 is a magnetic particle which is a magnetic powder, 2 is a magnetic field region surrounded by the outer periphery, and 3 is a rectangular outer boundary. This boundary can be an infinite boundary. Reference numeral 4 denotes a boundary node discretized at the time of calculation by the boundary element method, and reference numeral 5 denotes a direction of a magnetic flux density of an external magnetic field distributed in a magnetic field region.

When calculating the motion of a group of circular (two-dimensional) magnetic particles, ie, a magnetic solid, in a magnetic field, mesh division by the boundary element method is performed as shown in FIG. On the particle surface, that is, on the circumference representing the particle, for example, the contact point division defines eight points (pieces). Sufficient calculation accuracy is obtained with 8 points,
Also, due to the properties of the boundary element method, sufficient calculation accuracy can be obtained even when the mesh sizes are extremely different. Magnetic field area 1
Even if the particle size is 10 or 1 for 000 × 1000, sufficient calculation accuracy can be obtained. For example, any of eight, six, and four boundary nodes can be employed.

FIG. 6 shows an example of an axially symmetric electrostatic field performed to confirm the calculation accuracy of the boundary element method. The inner and outer circumferences representing particles are divided by eight nodes and four elements (secondary elements), and the inner circle is divided. Upper (r = a
= 30μm) is 5000V, on the outer circle (r = b = 10mm)
Is set to 0V. In the theory of electromagnetism, this electrostatic field is represented by an axially symmetric Laplace equation, and the analytical solution is a potential u (r)
= 5000 × (logr-logb) / (loga-logb) and electric field E (r) = 5000 × (loga-logb) / r, but the calculation results by the boundary element method agree within 0.2%, Even when the element sizes are extremely different, sufficient accuracy can be obtained if the circumference is divided into eight nodes by quadratic elements. In the calculation of the magnetic field, the same calculation accuracy is obtained because the Laplace equation similar to the electrostatic field is solved.

The calculation of the magnetic force on the surface of the magnetic particles can be accurately obtained on the circumference. The calculation points of the magnetic force are not limited to the eight nodes on the circumference, and the accuracy of calculating the magnetic force can be improved by interpolating between the contact points. However, too much interpolation requires a great deal of calculation time, so it is desirable that the number of magnetic force calculation points on the circumference be 32 or less.

When the number of particles is large and the number of nodes on the outer periphery is relatively small (about 200 nodes), the number of nodes is given by (number of particles × 8).
When there are 100 circular particles having a diameter of 10 in 000 × 1000, the number is about 1000 nodes.

When a magnetic field is calculated by a region division type solution such as a difference method or a conventional finite element method, as described above, 1 is used.
Since one million nodes are required, the number of nodes is 1000 times, and the memory required for the nodes is also 1000 times.

The amount of calculation is proportional to the number of unknown components of the matrix of the system of linear equations finally assembled. In the case of the conventional boundary element method, almost all of the matrix components are unknown.
0 × 1000 = 1,000,000 components, but most of the matrix components are 0 in the difference method or the boundary element method. When a grid-like mesh division as shown in FIG. 5 is used, since the surrounding eight nodes and the own node are related to one node, the unknown components of the matrix are 9 × 1 million nodes = 9 million Component, and the amount of calculation is 9 times. Therefore, if the boundary element method of this embodiment is used, both the calculation time and the memory can be significantly reduced.

In the boundary element method of this embodiment, the outer circumference can be set to infinity, so that the physical size of the magnetic field region is not limited. In the case of a circular particle, the unknown is almost the number of particles × 8.
However, in a region division type solution such as the difference method or the conventional boundary element method, when the magnetic field region is large, the number of meshes to be divided increases, and the number of unknowns is proportional thereto.

When the magnetic particles are spherical, the boundary contact on the spherical surface is divided by the following method. Consider a cube inscribed in a spherical surface, and create six boundary elements corresponding to six surfaces on the spherical surface. First, four arcs are formed by projecting four variations of one cube into a spherical shape, and a curved surface surrounded by these arcs is defined as one boundary element. This arc is a part of a curve (circle) formed by the intersection of a spherical surface and a plane that simultaneously includes the sides of the square and the center of the sphere, and is obtained as an arc having both ends of the sides of the square as fulcrums and end points.

The boundary contact points are four points of the square, the midpoint of the four arcs, and one point projected on the sphere so that the center point of the square is on a straight line connecting the center of the sphere, for a total of nine points. .

The boundary element thus formed on the spherical surface has a curved shape along the spherical surface. If corresponding boundary elements are created in the same manner for the remaining surfaces of the cube, a total of six boundary elements and 26 boundary contacts are created, and the spherical surface can be divided evenly. The method of using a regular polyhedron inscribed in the spherical surface is suitable for equally dividing the spherical surface, but using a regular hexahedron is the simplest and preferable.

Next, a case of analyzing the behavior of magnetic particles on a magnetic roll of an electrostatic recording device such as a laser printer will be described.

If the diameter of the magnetic roll is 20 mm, the diameter of the particles is 100 μm, and the number of particles is 500, when calculating the magnetic field by the boundary element method, the number of boundary nodes of the magnetic roll is 200.
The total number of nodes is 500 × 8 + 200 = 4200, the number of unknowns is 4200, and the number of unknown components of the matrix is 4
200 × 4200 = 17.64 million. On the other hand, in the case of the difference method or the conventional finite element method, the magnetic field analysis area needs to have a size surrounding the magnetic roll, that is, about 50 mm square, and a mesh size of 10 μm or less.
0 × 5000 = 25 million, the number of unknown elements in the matrix is 8 ×
25 million = 200 million, and practical calculation is almost difficult. Furthermore, since the curved shape of the particle surface cannot be accurately represented, the accuracy of the magnetic field calculation is poor.

In the boundary element method of the present embodiment, the mesh is re-divided only at the eight nodes on the surface (circumference) of each particle when moving the particles. Being able is also a great advantage.

FIG. 7 shows a schematic configuration according to an embodiment of the present invention. The figure shows a processing device 71 for calculating the motion of a magnetic body in a magnetic field using the boundary element method for calculating the magnetic field, and a motion calculation device 70 for a magnetic object in a magnetic field, which includes a processing device 81. The motion calculation device 70 includes a storage medium 72 for storing a magnetic field calculation program and a storage medium 82 for storing a particle behavior calculation program.

The processing device 71 has a plurality of boundary contact setting means 73 which is divided into meshes on the outer periphery (on the circumference) of the magnetic particles. The magnetic field is calculated by the program. It has a magnetic field calculation means 74 and a magnetic force calculation means 75 for calculating a magnetic field force from the calculated magnetic field.

The processing device 81 includes a magnetic particle contact determining means 83 for determining the contact of the magnetic particles, a contact force calculating means 84 for calculating the contact force, and a resultant force for calculating the resultant force from the calculated contact force and the calculated magnetic force. Calculation means 85, time integration means 86 for performing time integration of the equation of motion, magnetic particle position updating means 87 for updating the position of the magnetic particles from the time integration of the equation of motion
Having. By these calculation means, the position of the magnetic particles is updated, that is, the motion of the magnetic particles is calculated. These determined values are returned to the means 73 and 83 described above.

The method of calculating the motion of the magnetic particles by time sharing with the rotation of the magnetic roll is as follows.

When the position of the magnetic particles is updated, the processing unit 7
The magnetic force and contact force acting on the magnetic particles are newly calculated by 1 and 81, and the total force acting on each magnetic particle to which two external forces such as gravity are further applied is calculated (means 91). Time integration of the equation of motion is performed based on this force (means 92), and the position of the magnetic particle is newly updated and repeated.

FIG. 8 is a flowchart of the calculation method according to the embodiment of the present invention. First, mechanical parameters relating to particles and parameters relating to a magnetic field are set (step 100). The initial position of the particles is set (step 101), and the contact of the particles is determined (step 10).
2) The contact force is calculated (step 103). on the other hand,
The shape data is updated (step 106), the magnetic field is calculated by the method described above (step 107), and the magnetic force is calculated (step 108). The time integration of the equation of motion is performed from step 103 and step 108 (step 104), and the particle position is updated (step 10).
5). Repeat these.

Among the forces acting on the magnetic particles, the contact force is calculated by determining the contact from the positional relationship of the magnetic particles, and the total external force acting on the magnetic particles is obtained by adding the magnetic force and gravity at that time, and the equation of motion is integrated over time. Then, the position of the magnetic particles is updated. The time division width at this time is represented by Δt in the figure.

On the other hand, the update calculation of the magnetic field may be performed every time the position of the magnetic particle changes remarkably. The change amount of the particle position during Δt is as small as 10 ⁻⁶ as compared with the particle diameter, and the arrangement of the magnetic substance hardly changes. While performing field update calculations for each t _B in the figure are updating a magnetic force according to magnetic particles which can be 10000 times or more time intervals Delta] t. For course Δt by calculation object, t _B is changed is a Δt«t _B, the number of calculations of the magnetic field is capable of practical calculations not many.

In order to analyze the behavior of the developer carrier, which is a magnetic particle, entering the developing gap, calculation was performed using a model as shown in FIG. The photoreceptor has a diameter of 100 mm, a peripheral speed of 0.2 m / s, and an MGR of 28 mm, which is the same as that shown in FIG. The developer is conveyed by an annular sleeve roll rotating around a fixed MGR, and has a diameter of 30 mm and a peripheral speed of 0.8.
m / s and the development gap was 1 mm.

In order to secure the developer and the transport amount without increasing the number of particles so much to reduce the calculation time, 98 carriers having a diameter of 200 μm, which is slightly larger than usual, were used.

The number of boundary settings required for magnetic field calculation is
GR boundary, point on the circumference such as Al core 216 + 98 × 8 = 10
00. The carrier had a relative magnetic permeability of 10.

On the other hand, the parameters used in the DEM calculation are summarized in Table 1. The values in Table 1 are not experimental values but expected values, but are not far from actual values.

[Table 1]

The calculation is Δt = 2 × 10 ⁻⁹ s, and the magnetic field update is 15 million steps every 2 × 10 ⁻⁵ s (real time 30 m
s), and the calculation time is the workstation (PA8
(500/440 MHz) for 140 hours.

FIG. 10 shows the magnetic field of the magnetic roll when there is no carrier. The arrow in the figure represents the magnetic flux density B. In FIG. 9, the carrier initially placed in the hollow is attracted to the magnetic pole while dropping by gravity, is held on the surface of the sleeve roll, and is conveyed with the rotation of the sleeve. FIG. 11 shows one scene of this state.

The carrier ears stand up along the magnetic flux density vector of FIG. 10 to form a magnetic brush, and the tips are conveyed while rubbing the photoreceptor. The shape of the magnetic brush in FIG. 11 well represents the actual shape on the magnet, and it can be seen that the carrier behavior can be obtained by this calculation method. When the development parameters such as the peripheral speed of the sleeve roll, development gap, and magnetic pole arrangement are changed, it is used to calculate how the state quantities such as the developer conveyance speed, density, and photoconductor rubbing force will change. It can be used.

As an example, the peripheral speed of the sleeve roll is 0.4
FIG. 12 shows a comparison result of the time change of the average velocity (the two-square root of the velocity) of the carrier particles in the case of m / s and 0.8 m / s described above. Although the peripheral speed of the sleeve roll is twice as large, the difference in the average carrier speed is about 35%, and the amount related to the movement of the carrier is not doubled. This is calculated by calculating the friction coefficient μ (= tan θ) of the sleeve surface as the friction angle θ
It is considered that the carrier slips on the surface of the sleeve roll because the angle is set as small as 30 degrees.

Further, comparing the velocity of the carrier particles near the tip of the ear of the magnetic brush as shown in FIG. 11, it was found that there was almost no change. That is,
From this calculation example, it was found that the speed of the carrier at the tip of the magnetic brush was not largely related to the peripheral speed of the sleeve roll, and was mainly determined by the direction and strength of the magnetic pole.

FIG. 13 is a graph obtained from the result of calculating the motion of the magnetic particles on the magnetic roll of the laser printer by the method of calculating the motion of the magnetic object in the magnetic field according to the above-described embodiment. The horizontal axis of the graph is the diameter of the magnetic particles, and the vertical axis is the average value of the contact force acting on the magnetic particles. In a laser printer, magnetic particles agglomerate on a magnetic roll and form a magnetic brush in succession, and the surface of the photoreceptor is developed by the magnetic brush to develop toner attached to the magnetic brush. At this time, the hardness of the magnetic brush greatly affects the image quality,
Higher resolution images can be obtained with a softer magnetic brush.

The average value of the contact force shown in FIG. 13 is an amount corresponding to the hardness of the magnetic brush, and as shown in FIG. 9, it can be seen from the graph that the smaller the particle size of the magnetic particles, the softer the magnetic brush. Was. In addition, similar graphs can be obtained by changing the strength of the magnetic field, the peripheral speed of the magnetic roll, etc., and finding the design conditions for keeping the hardness of the magnetic brush within an appropriate value from those results. This allows the particle size to be determined.

According to these findings, in setting the hardness of the magnetic brush on the magnetic roll of the electrostatic recording device,
At least the strength of the magnetic field and the peripheral speed of the magnetic roll are varied to determine the size of the magnetic particles and the average value of the contact force acting on the magnetic particles. And a method for setting the hardness of the magnetic brush of the electrostatic recording apparatus, which is set based on the obtained relationship, and a method for setting the particle diameter.

[0055]

As described above, according to the present invention, when obtaining the motion of a magnetic object, particularly a large number of magnetic particles (magnetic powder, etc.) in a magnetic field, the particles are obtained by using a new boundary element method. The accuracy of the magnetic field calculation is improved by performing the update calculation of the magnetic field while performing the automatic mesh division according to the movement of the magnetic field, the change of the magnetic force on the surface of the magnetic particles can be considered, and the calculation time and memory can be reduced. At the same time, it is possible to perform a practical calculation in which updating of the mesh division accompanying the movement of the magnetic particles is very easy.

[Brief description of the drawings]

FIG. 1 is a nodal division diagram for a magnetic field update calculation by a boundary element method.

FIG. 2 is an explanatory diagram of a magnetic field calculation method by a magnetic moment method.

FIG. 3 is an explanatory diagram showing a magnetic flux density around magnetic particles.

FIG. 4 is a mesh division diagram by a finite element method.

FIG. 5 is an explanatory diagram of a magnetic field calculation method using a fixed mesh.

FIG. 6 is a node division diagram for calculating an axisymmetric electrostatic field by a boundary element method.

FIG. 7 is a block diagram showing a schematic configuration of an embodiment of the present invention.

FIG. 8 is a flowchart of a calculation method.

FIG. 9 is a calculation model diagram.

FIG. 10 is a magnetic field diagram.

FIG. 11 is a diagram showing a magnetic behavior on a magnetic roll.

FIG. 12 is a graph showing a comparison result of a time change of an average speed of carrier particles.

FIG. 13 is a graph showing the hardness of the magnetic brush determined by the particle diameter of the magnetic particles and the average contact force.

[Explanation of symbols]

DESCRIPTION OF SYMBOLS 1 ... Magnetic particle, 2 ... Magnetic field area, 3 ... Perimeter boundary, 4 ... Boundary node, 5 ... External magnetic field, 6 ... Magnetic moment, 7 ... Magnetic flux density vector, 8 ... Finite element method mesh, 9 ... Fixed mesh, 10 ... Mesh with different permeability.

Claims (8)

[Claims] 1. A method of calculating the motion of a magnetic body in a magnetic field using a boundary element method for calculating a magnetic field, the method comprising the steps of: Define multiple boundary nodes mesh-divided into
A motion calculation method for a magnetic object in a magnetic field, wherein a boundary element method is used to calculate a magnetic field using boundary nodes on an outer periphery. 2. A method for calculating the motion of a magnetic powder in a magnetic field using a boundary element method for calculating a magnetic field, the method comprising the steps of: A method for calculating the motion of a magnetic object in a magnetic field, comprising: defining a plurality of boundary nodes mesh-divided on the circumference of the magnetic field; and using the boundary element method for calculating the magnetic field based on the boundary nodes on the circumference. 3. A method for calculating the movement of a magnetic powder on a magnetic roll of an electrostatic recording device by using a boundary element method for calculating a magnetic field, comprising the steps of: The motion calculation of a magnetic object in a magnetic field characterized by defining a plurality of boundary nodes automatically mesh-divided on the circumference of a circle representing the boundary and using the boundary element method to calculate the magnetic field based on the boundary nodes on the circumference Method. 4. The method according to claim 2, wherein the plurality of boundary nodes are any one of eight, six and four boundary nodes. 5. The method according to claim 1, wherein the outer shape representing the boundary of the magnetic body is spherical, and the number of boundary elements corresponding to the six surfaces of the cube inscribed on the spherical surface on the spherical surface is six, and A motion calculation method of a magnetic object in a magnetic field, wherein the number of boundary nodes is set to 26. 6. An apparatus for calculating the motion of a magnetic object in a magnetic field, comprising: a processor for calculating the motion of the magnetic substance in a magnetic field using a boundary element method for calculating the magnetic field. Means for determining a boundary node mesh-divided on a circular circumference representing the boundary of the magnetic material with the movement of the magnetic body, a storage medium storing a program for applying the boundary element method, and a boundary node on the circumference And a means for using a boundary element method for calculating a magnetic field using a program stored in a storage medium. 7. A method in which a plurality of boundary nodes automatically mesh-divided are defined on a circular circumference representing a boundary of the magnetic powder with the rotation of the magnetic roll, and the boundary element method is used to calculate a magnetic field. An apparatus for calculating a movement position of a magnetic object on a magnetic field, comprising: means for determining a position of a powder on a magnetic roll. 8. A method for setting the hardness of a magnetic brush on a magnetic roll of an electrostatic recording device, comprising: changing at least the strength of a magnetic field and the convergence of the magnetic roll to change the particle size of the magnetic particles and the contact force acting on the magnetic particles. An electrostatic recording apparatus characterized in that a relationship between the hardness of the magnetic brush and the hardness of the magnetic brush becomes softer as the particle size of the magnetic particles becomes smaller from the average value of the magnetic particles, and the particle size to be used is set from the obtained relationship How to set the orientation of the magnetic brush.