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Axial Magnetic Bearing Using Permanent Magnets
Introduction
Permanent magnet bearings are used in turbomachinery, 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 Moving Mesh features and a Sensitivity interface to calculate the axial magnetic stiffness coefficient as defined by Equation 1. With the Moving Mesh features 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, G. Lemarquand and V. Lemarquand, “Halbach Structures for Permanent Magnets Bearings,” Progress In Electromagnetics Research M, vol. 14, pp. 236–277, 2010.
Application Library path: ACDC_Module/Devices,_Transducers_and_Actuators/axial_magnetic_bearing
Modeling Instructions
From the File menu, choose New.
New
In the New window, click  Model Wizard.
Model Wizard
1
In the Model Wizard window, click  2D Axisymmetric.
2
In the Select Physics tree, select AC/DC > Magnetic Fields, No Currents > Magnetic Fields, No Currents (mfnc).
3
Click Add.
4
Click  Study.
5
In the Select Study tree, select General Studies > Stationary.
6
Click  Done.
Geometry 1
Define all the necessary parameters here.
Global Definitions
Parameters 1
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, locate the Parameters section.
3
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 Parametric Sweep node in order to compute the axial magnetic stiffness.
Geometry 1
Follow the instructions below to construct the model geometry.
1
In the Model Builder window, under Component 1 (comp1) click Geometry 1.
2
In the Settings window for Geometry, locate the Units section.
3
From the Length unit list, choose mm.
Rectangle 1 (r1)
1
In the Geometry toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type R2-R1.
4
In the Height text field, type h0*3.
5
Locate the Position section. In the r text field, type R1.
6
In the z text field, type -h0/2-h0+dZ.
7
Click  Build Selected.
8
Click to expand the Layers section. In the table, enter the following settings:
Layer name
Thickness (mm)
Layer 1
h0
9
Select the Layers on top checkbox.
10
Click  Build Selected.
Rectangle 2 (r2)
1
Right-click Rectangle 1 (r1) and choose Duplicate.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type R4-R3.
4
Locate the Position section. In the r text field, type R3.
5
In the z text field, type -h0/2-h0.
6
Click  Build Selected.
Rectangle 3 (r3)
1
In the Geometry toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type 70.
4
In the Height text field, type 160.
5
Locate the Position section. In the z text field, type -80.
6
Locate the Layers section. In the table, enter the following settings:
Layer name
Thickness (mm)
Layer 1
5
7
Select the Layers to the right checkbox.
8
Select the Layers on top checkbox.
9
Click  Build Selected.
10
Click the  Zoom Extents button in the Graphics toolbar.
Fillet 1 (fil1)
1
In the Geometry toolbar, click  Fillet.
2
On the object r1, select Points 1, 4, 5, and 8 only.
3
On the object r2, select Points 1, 4, 5, and 8 only.
4
In the Settings window for Fillet, locate the Radius section.
5
In the Radius text field, type 2.
6
Click  Build All Objects.
7
Click the  Zoom Extents button in the Graphics toolbar.
The model geometry should look like the one shown in the figure above.
Enclose the inner air domain by an Infinite Element Domain to model the surrounding space.
Definitions
Infinite Element Domain 1 (ie1)
1
In the Definitions toolbar, click  Infinite Element Domain.
2
Select Domains 1, 3, and 10–12 only.
3
In the Settings window for Infinite Element Domain, locate the Geometry section.
4
From the Type list, choose Cylindrical.
Now set up the physics for the magnetic field: use the Magnetic Flux Conservation in Solids node for the magnets (one per magnetization direction).
Magnetic Fields, No Currents (mfnc)
Magnetic Flux Conservation in Solids 1
1
In the Physics toolbar, click  Domains and choose Magnetic Flux Conservation in Solids.
2
Select Domains 6 and 9 only.
3
In the Settings window for Magnetic Flux Conservation in Solids, locate the Constitutive Relation B-H section.
4
From the Magnetization model list, choose Remanent flux density.
5
Specify the e vector as
0
r
0
phi
-1
z
Magnetic Flux Conservation in Solids 2
1
In the Physics toolbar, click  Domains and choose Magnetic Flux Conservation in Solids.
2
Select Domains 4 and 7 only.
3
In the Settings window for Magnetic Flux Conservation in Solids, locate the Constitutive Relation B-H section.
4
From the Magnetization model list, choose Remanent flux density.
5
Specify the e vector as
0
r
0
phi
1
z
Magnetic Flux Conservation in Solids 3
1
In the Physics toolbar, click  Domains and choose Magnetic Flux Conservation in Solids.
2
Select Domain 5 only.
3
In the Settings window for Magnetic Flux Conservation in Solids, locate the Constitutive Relation B-H section.
4
From the Magnetization model list, choose Remanent flux density.
Magnetic Flux Conservation in Solids 4
1
In the Physics toolbar, click  Domains and choose Magnetic Flux Conservation in Solids.
2
Select Domain 8 only.
3
In the Settings window for Magnetic Flux Conservation in Solids, locate the Constitutive Relation B-H section.
4
From the Magnetization model list, choose Remanent flux density.
5
Specify the e vector as
-1
r
0
phi
0
z
Materials
In the magnets, use a material that will be created by modifying an existing entry from the library.
Add Material
1
In the Materials toolbar, click  Add Material to open the Add Material window.
2
Go to the Add Material window.
3
In the tree, select AC/DC > Hard Magnetic Materials > Sintered NdFeB Grades (Chinese Standard) > N50 (Sintered NdFeB).
4
Right-click and choose Add to Component 1 (comp1).
5
In the Materials toolbar, click  Add Material to close the Add Material window.
Materials
Generic Magnet
1
Select Domains 4–9 only.
2
In the Settings window for Material, locate the Material Contents section.
3
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
4
In the Label text field, type Generic Magnet.
Magnetic Fields, No Currents (mfnc)
Add a Force Calculation feature to compute the total magnetic force on the inner magnets.
Force Calculation 1
1
In the Physics toolbar, click  Domains and choose Force Calculation.
2
Select Domains 4–9 only.
3
In the Settings window for Force Calculation, locate the Force Calculation section.
4
In the Force name text field, type 0.
Setting the force name to 0, the axial force component can be accessed as mfnc.Forcez_0, where mfnc is the identifier for the Magnetic Fields, No Currents interfaces.
By using the split by connectivity the inner and the out magnets sets are separated in different features. Outer is automatically identified with name 1. Labels are changed according to this naming.
Force Calculation, Object 1
1
Right-click Force Calculation 1 and choose Split by Connectivity.
2
In the Settings window for Force Calculation, type Force Calculation, Object 1 in the Label text field.
Force Calculation, Object 0
1
In the Model Builder window, under Component 1 (comp1) > Magnetic Fields, No Currents (mfnc) click Force Calculation 1.
2
In the Settings window for Force Calculation, type Force Calculation, Object 0 in the Label text field.
Specify a reference level for the magnetic scalar potential by constraining its value in one point.
Zero Magnetic Scalar Potential 1
1
In the Physics toolbar, click  Points and choose Zero Magnetic Scalar Potential.
2
Select Point 30 only.
Next, add the Moving Mesh and Sensitivity interfaces to use for calculating the axial magnetic stiffness coefficient as defined by Equation 1 of the Model Definition section.
Add Physics
1
In the Physics toolbar, click  Add Physics to open the Add Physics window.
2
Go to the Add Physics window.
3
In the tree, select Mathematics > Optimization and Sensitivity > Sensitivity (sens).
4
Click the Add to Component 1 button in the window toolbar.
5
In the Physics toolbar, click  Add Physics to close the Add Physics window.
Sensitivity (sens)
With the Sensitivity 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
1
In the Physics toolbar, click  Global and choose Global Control Variables.
2
In the Settings window for Global Control Variables, locate the Control Variables section.
3
In the Control variables table, enter the following settings:
Variable
Initial value (1)
dZ
0
Global Objective 1
1
In the Physics toolbar, click  Global and choose Global Objective.
2
In the Settings window for Global Objective, locate the Global Objective section.
3
In the q text field, type mfnc.Forcez_0.
Component 1 (comp1)
Deforming Domain 1
1
In the Physics toolbar, click  Moving Mesh and choose Free Deformation.
2
Select Domain 2 only.
3
In the Settings window for Deforming Domain, locate the Smoothing section.
4
From the Mesh smoothing type list, choose Laplace.
Prescribed Deformation 1
1
In the Moving Mesh toolbar, click  Prescribed Deformation.
2
Click the  Select All button in the Graphics toolbar.
3
In the Settings window for Prescribed Deformation, locate the Geometric Entity Selection section.
4
Click  Clear Selection.
5
Click the  Select Box button in the Graphics toolbar.
6
Select Domains 4–6 only.
7
Locate the Prescribed Deformation section. Specify the dx vector as
0
R
dZ
Z
Prescribed Mesh Displacement 1
1
In the Moving Mesh toolbar, click  Prescribed Mesh Displacement.
2
Select Boundaries 3, 4, 6, 18, 19, 21, 23–27, 30, and 42–45 only.
Mesh 1
1
In the Model Builder window, under Component 1 (comp1) right-click Mesh 1 and choose Build All.
2
Click the  Zoom Extents button in the Graphics toolbar.
The mesh should look like the one shown in the figure above. Note that the physics controlled meshing automatically produced a mapped mesh in the infinite elements.
Study 1
Add a Parametric Sweep 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
1
In the Study toolbar, click  Parametric Sweep.
2
In the Settings window for Parametric Sweep, locate the Study Settings section.
3
Click  Add.
4
From the list in the Parameter name column choose dZ (Axial displacement), and set the Parameter unit to mm.
5
Click  Range.
6
In the Range dialog, type -40 in the Start text field.
7
In the Step text field, type 2.
8
In the Stop text field, type 40.
9
Click Replace.
10
In the Model Builder window, click Study 1.
11
In the Settings window for Study, locate the Study Settings section.
12
Clear the Generate default plots checkbox.
Solution 1 (sol1)
1
In the Study toolbar, click  Show Default Solver.
2
In the Model Builder window, expand the Solution 1 (sol1) node.
3
Right-click Study 1 > Solver Configurations > Solution 1 (sol1) > Stationary Solver 1 and choose Sensitivity.
4
In the Study toolbar, click  Compute.
Results
In the Model Builder window, expand the Results node.
Study 1/Parametric Solutions 1 (sol2)
Create datasets for result visualization in specific domains.
Study 1/Parametric Solutions 1 (3) (sol2)
1
In the Model Builder window, expand the Results > Datasets node.
2
Right-click Results > Datasets > Study 1/Parametric Solutions 1 (sol2) and choose Duplicate.
3
In the Settings window for Solution, locate the Solution section.
4
From the Solution list, choose dZ=8 (sol27).
Selection
1
In the Results toolbar, click  Attributes and choose Selection.
2
In the Settings window for Selection, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Domain.
4
Select Domains 4–9 only.
Revolution 2D 1
1
In the Results toolbar, click  More Datasets and choose Revolution 2D.
2
In the Settings window for Revolution 2D, locate the Data section.
3
From the Dataset list, choose Study 1/dZ=8 (sol27).
4
Click to expand the Revolution Layers section. In the Start angle text field, type -100.
5
In the Revolution angle text field, type 280.
Use the following instructions to get the plots shown in Figure 2 to Figure 5.
Magnetic Flux Density, 2D Visualization
1
In the Results toolbar, click  2D Plot Group.
2
In the Settings window for 2D Plot Group, type Magnetic Flux Density, 2D Visualization in the Label text field.
Surface 1
1
Right-click Magnetic Flux Density, 2D Visualization and choose Surface.
2
In the Settings window for Surface, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1) > Magnetic Fields, No Currents > Magnetic > mfnc.normB - Magnetic flux density norm - T.
3
In the Magnetic Flux Density, 2D Visualization toolbar, click  Plot.
Streamline 1
1
In the Model Builder window, right-click Magnetic Flux Density, 2D Visualization and choose Streamline.
2
In the Settings window for Streamline, locate the Streamline Positioning section.
3
From the Positioning list, choose Uniform density.
4
In the Density level text field, type 7.77.
5
Locate the Coloring and Style section. Find the Point style subsection. From the Color list, choose Gray.
6
In the Magnetic Flux Density, 2D Visualization toolbar, click  Plot.
7
Click the  Zoom Extents button in the Graphics toolbar.
Compare this plot with Figure 2.
Force vs dZ and Force Balance
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Force vs dZ and Force Balance in the Label text field.
Global 1
1
Right-click Force vs dZ and Force Balance and choose Global.
2
In the Settings window for Global, locate the Data section.
3
From the Dataset list, choose Study 1/Parametric Solutions 1 (sol2).
4
Locate the y-Axis Data section. Click  Clear Table.
5
Click Replace Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Component 1 (comp1) > Sensitivity > sens.gobj1 - Objective value - N.
Add to this plot also the force balance.
6
Click Add Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Component 1 (comp1) > Magnetic Fields, No Currents > Mechanical > Total electromagnetic force, object 0 (spatial frame) - N > mfnc.Forcez_0 - Total electromagnetic force, object 0, component z.
7
Locate the y-Axis Data section. In the table, enter the following settings:
Expression
Unit
Description
sens.gobj1
N
Total electromagnetic force, object 0
mfnc.Forcez_0+mfnc.Forcez_1
N
Force Balance
8
In the Force vs dZ and Force Balance toolbar, click  Plot.
Compare the plot just created with that shown in Figure 3.
9
Click  Plot.
Magnetic Stiffness Coefficient
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Magnetic Stiffness Coefficient in the Label text field.
Global 1
1
Right-click Magnetic Stiffness Coefficient and choose Global.
2
In the Settings window for Global, locate the Data section.
3
From the Dataset list, choose Study 1/Parametric Solutions 1 (sol2).
4
Locate the y-Axis Data section. Click  Clear Table.
5
In the table, enter the following settings:
Expression
Unit
Description
fsens(dZ)
N/m
Sensitivity of Fz w.r.t. dZ
6
In the Magnetic Stiffness Coefficient toolbar, click  Plot.
Compare this plot with Figure 4.
Magnetic Fields, Revolution Visualization
Finally, reproduce Figure 5 using the following steps.
1
In the Results toolbar, click  3D Plot Group.
2
In the Settings window for 3D Plot Group, type Magnetic Fields, Revolution Visualization in the Label text field.
Volume 1
1
Right-click Magnetic Fields, Revolution Visualization and choose Volume.
2
In the Settings window for Volume, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose mfnc.normB - Magnetic flux density norm - T.
3
In the Magnetic Fields, Revolution Visualization toolbar, click  Plot.