Axial Magnetic Bearing Using Permanent Magnets

Permanent magnet bearings are used in turbo machinery, pumps, motors, generators, and flywheel energy storage systems, to mention a few application areas; contactless operation, low maintenance, and the ability to operate without lubrication are some key benefits compared to conventional mechanical bearings. This example illustrates how to calculate design parameters like magnetic forces and stiffness for an axial permanent magnet bearing.

Figure 1: Model illustration of an axial magnetic bearing using permanent magnets. The black arrows show the magnetization direction of the permanent magnets.

Model Definition

Set up the problem in a 2D axisymmetric modeling space. Figure 1 shows a 3D view of the model with the magnetization directions of the magnets indicated. COMSOL Multiphysics calculates the total magnetic force on an object by integrating the vector expression

where n is the outward normal vector and T is the Maxwell stress tensor, over the object’s outer boundaries.

The negative of the derivative of the total magnetic force with respect to the position is referred to as the magnetic stiffness. By this definition, the axial magnetic stiffness of the bearing is

(1)

where Fz is the total axial magnetic force on the bearing. This model calculates the magnetic stiffness in the axial direction only; calculating the magnetic stiffness in the radial direction as well as the coupled stiffness coefficients requires a complete 3D model.

The model parameters are taken from Ref. 1.

Results

A steady-state study is performed to calculate the magnetic forces and the axial magnetic stiffness coefficient. Figure 2 shows the magnetic flux density norm and the magnetic vector potential for an axial displacement of the inner magnets of z = 40 mm. Figure 3 illustrates the axial component of the magnetic force on the inner magnets as a function of axial displacement. Figure 4 displays the sensitivity of the axial magnetic force with respect to the axial displacement. The negative of this plot is the axial magnetic stiffness coefficient. Finally, Figure 5 shows the magnetic flux density norm in 3D at an axial displacement of 8 mm.

Figure 2: Magnetic flux density norm and magnetic vector potential for an axial displacement of the inner magnets of 40 mm.

Figure 3: Axial component of the magnetic force versus axial displacement.

Figure 4: Sensitivity of the axial magnetic force with respect to axial displacement versus axial displacement. The negative of this quantity is the axial magnetic stiffness coefficient.

Figure 5: Magnetic flux density norm at an axial displacement of 8 mm.

Notes About the COMSOL Implementation

In this model, use the Magnetic Fields interface to model the magnetic field. Also, add an Infinite Element Domain to model the open region of free space surrounding the magnets. You can then calculate the total magnetic force components with the Maxwell stress tensor method by adding a Force Calculation node in the inner magnet domains. Also, add Deformed Geometry and Sensitivity interfaces to calculate the axial magnetic stiffness coefficient as defined by Equation 1. With the Deformed Geometry interface you parameterize the axial displacement of the inner magnets. Then, use the axial component of the magnetic force as a global objective and the axial displacement parameter as a global control variable for the Sensitivity interface to obtain the derivative dFz/dz. Using a Parametric Sweep study node, you finally compute the axial magnetic stiffness as a function of the axial displacement.

Reference

1. R. Ravaud and G. Lemarquand, “Halbach Structures for Permanent Magnet Bearings”, Progress In Electromagnetics Research M, vol. 14, pp. 236–277, 2010.

ACDC_Module/Motors_and_Actuators/axial_magnetic_bearing

Modeling Instructions

From the menu, choose .

New

In the window, click .

Model Wizard

In the window, click .

In the tree, select .

Click .

In the tree, select .

Click .

Geometry 1

Define all the necessary parameters here.

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
R1	10[mm]	0.01 m	Inner radius of inner magnet
R2	20[mm]	0.02 m	Outer radius of inner magnet
R3	22[mm]	0.022 m	Inner radius of outer magnet
R4	32[mm]	0.032 m	Outer radius of outer magnet
h0	10[mm]	0.01 m	Magnet height
Br	1[T]	1 T	Remanent flux density of magnet
dZ	0[mm]	0 m	Axial displacement

Later, dZ will be used as a global control variable for a sensitivity analysis and the parameter of a node in order to compute the axial magnetic stiffness.

Geometry 1

Follow the instructions below to construct the model geometry.

In the window, under click .

In the window for , locate the section.

From the list, choose .

Rectangle 1 (r1)

In the toolbar, click .

In the window for , locate the section.

In the text field, type R2-R1.

In the text field, type h0*3.

Locate the section. In the text field, type R1.

In the text field, type -h0/2-h0+dZ.

Click .

Click to expand the section. In the table, enter the following settings:


Layer name	Thickness (mm)
Layer 1	h0

Select the check box.

Click .

Rectangle 2 (r2)

Right-click and choose .

In the window for , locate the section.

In the text field, type R4-R3.

Locate the section. In the text field, type R3.

In the text field, type -h0/2-h0.

Click .

Rectangle 3 (r3)

In the toolbar, click .

In the window for , locate the section.

In the text field, type 70.

In the text field, type 160.

Locate the section. In the text field, type -80.

Locate the section. In the table, enter the following settings:


Layer name	Thickness (mm)
Layer 1	5

Select the check box.

Click .

Click the button in the toolbar.

Fillet 1 (fil1)

In the toolbar, click .

On the object , select Points 1, 4, 5, and 8 only.

In the window for , locate the section.

In the text field, type 2.

Click .

Click the button in the toolbar.

The model geometry should look like the one shown in the figure above.

Enclose the inner air domain by an to model the surrounding space.

Infinite Element Domain 1 (ie1)

In the toolbar, click .

Select Domains 1, 3, and 10–12 only.

In the window for , locate the section.

From the list, choose .

Materials

Use air as the material for all domains and override it in permanent magnets with a material that will be created based on a modification of an existing entry from library.

In the toolbar, click and choose .

Add Material

Go to the window.

In the tree, select .

Click in the window toolbar.

In the tree, select .

Click .

In the toolbar, click to close the window.

Materials

N50 (Sintered NdFeB) (mat2)

Select Domains 4–9 only.

In the window for , locate the section.

In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Recoil permeability	murec_iso ; murecii = murec_iso, murecij = 0	1	1	Remanent flux density
Remanent flux density norm	normBr	Br	T	Remanent flux density

In the text field, type Generic magnet.

Now set up the physics for the magnetic field: use the default node with default settings for the air domains and add separate nodes for the magnets (one per magnetization direction).

Magnetic Fields, No Currents (mfnc)

Magnetic Flux Conservation 2

In the window, under right-click and choose .

Select Domains 6 and 9 only.

In the window for , locate the section.

From the list, choose .

Specify the e vector as


0	r
0	phi
-1	z

Magnetic Flux Conservation 3

In the toolbar, click and choose .

Select Domains 4 and 7 only.

In the window for , locate the section.

From the list, choose .

Specify the e vector as


0	r
0	phi
1	z

Magnetic Flux Conservation 4

In the toolbar, click and choose .

Select Domain 5 only.

In the window for , locate the section.

From the list, choose .

Magnetic Flux Conservation 5

In the toolbar, click and choose .

Select Domain 8 only.

In the window for , locate the section.

From the list, choose .

Specify the e vector as


-1	r
0	phi
0	z

Force Calculation 1

Add a feature to compute the total magnetic force on the inner magnets.

In the toolbar, click and choose .

Select Domains 4–6 only.

Keeping the default force name, 0, the axial force component can be accessed as mfnc.Forcez_0, where mfnc is the identifier for the interfaces.

Specify a reference level for the magnetic scalar potential by constraining its value in one point.

Zero Magnetic Scalar Potential 1

In the toolbar, click and choose .

Select Point 30 only.

Next, add the and interfaces to use for calculating the axial magnetic stiffness coefficient as defined by Equation 1 of the Model Definition section.

Add Physics

In the toolbar, click to open the window.

Go to the window.

In the tree, select .

Click in the window toolbar.

In the toolbar, click to close the window.

Sensitivity (sens)

With the interface you can compute the right-hand side of Equation 1 as the derivative of the global objective defined as the axial force component mfnc.Forcez_0 with respect to the global control variable defined as the axial displacement dZ.

Global Control Variables 1

Right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Variable	Initial value
dZ	0

Global Objective 1

In the toolbar, click and choose .

In the window for , locate the section.

In the q text field, type mfnc.Forcez_0.

Definitions

Deforming Domain 1

In the toolbar, click and choose .

Select Domain 2 only.

Prescribed Deformation 1

In the toolbar, click and choose .

Deforming Domain 1

In the window, click .

In the window for , locate the section.

From the list, choose .

Prescribed Deformation 1

Click the button in the toolbar.

In the window, click .

In the window for , locate the section.

Click .

Click the button in the toolbar.

Select Domains 4–6 only.

Locate the section. Specify the dx vector as


0	R
dZ	Z

Prescribed Mesh Displacement 1

In the toolbar, click and choose .

Select Boundaries 3, 4, 6, 18, 19, 21, 23–27, 30, and 42–45 only.

Mesh 1

In the window, under click .

In the window for , click .

Click the button in the toolbar.

The mesh should look like the one shown in the figure above. Notice that the physics controlled meshing automatically produced a mapped mesh in the infinite elements.

Study 1

Add a study step to calculate the axial and radial force components for different axial positions of the inner magnets. Vary the axial displacement from z = −40 mm to z = 40 mm.

Parametric Sweep

In the toolbar, click .