Models.acdc.Axial Magnetic Bearing

8/10/2019 Models.acdc.Axial Magnetic Bearing

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Solved with COMSOL Multiphysics 5.0

1 | A X I A L M A G N E T I C B E A R I N G U S I N G P E R M A N E N T M A G N E T S

Ax i a l Magn e t i c B e a r i n g U s i n g

P e rmanen t Magn e t s

Introduction

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 somekey benefits compared to conventional mechanical bearings. This model illustrates

how to calculate design parameters like magnetic forces and stiffness for an axial

permanent magnet bearing.

Outer magnets

Inner magnets

r

z

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

Model Definition

Set up the problem in a 2D axisymmetric modeling space. Figure 1shows 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


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where nis the outward normal vector and Tis the Maxwell stress tensor, over the

objects 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 Fzis 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 2shows the magnetic flux density norm and the

magnetic vector potential for an axial displacement of the inner magnets of z= 40 mm.

Figure 3and Figure 4illustrate the radial and axial component, respectively, of the

magnetic force on the inner magnets as a function of axial displacement. Figure 5displays 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 6shows the magnetic flux density norm in 3D at an axial displacement

of 8 mm.

f n T1

2---n H B( ) n H( )B

T+= =

kz

dFz

dz----------=


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

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


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Figure 4: Axial component of the magnetic force versus axial displacement.

Figure 5: Sensitivity of the axial magnetic force with respect to axial displacement versusaxial displacement. The negative of this quantity is the axial magnetic stiffness coefficient.


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Figure 6: Magnetic flux density norm at an axial displacement of 8 mm.

Notes About the COMSOL Implementation

Use the Magnetic Fields physics to model the magnetic field. Add an Infinite Element

Domain to model the open region of free space surrounding the magnets. Calculate

the total magnetic force components with the Maxwell stress tensor method by addinga 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, you 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 derivativedFz/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. 236277, 2010.


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Model Library path:ACDC_Module/Motors_and_Actuators/

axial_magnetic_bearing

Modeling Instructions

From the Filemenu, choose New.

N E W

1 In the Newwindow, click Model Wizard.

M O D E L W I Z A R D

1 In the Model Wizardwindow, click 2D Axisymmetric.

2 In the Select physicstree, select AC/DC>Magnetic Fields (mf).

3 Click Add.

4 Click Study.

5 In the Select studytree, select Preset Studies>Stationary.

6 Click Done.

D E F I N I T I O N S

Define all the necessary parameters here.

Parameters1 On the Modeltoolbar, click Parameters.

2 In the Settingswindow for Parameters, locate the Parameterssection.


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3 In the table, enter the following settings:

Later, you will use dZas a global control variable for a Sensitivity physics and the

parameter of a Parametric Sweep node in order to compute the axial magnetic

stiffness.

G E O M E T R Y 1

1 In the Model Builderwindow, under Component 1 (comp1)click Geometry 1.2 In the Settingswindow for Geometry, locate the Unitssection.

3 From the Length unitlist, choose mm.

Follow the instructions below to construct the model geometry.

Rectangle 1 (r1)

1 On the Geometrytoolbar, click Primitivesand choose Rectangle.

2 In the Settingswindow for Rectangle, locate the Sizesection.3 In the Widthtext field, type R2-R1.

4 In the Heighttext field, type h0*3.

5 Locate the Positionsection. In the rtext field, type R1.

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

7 Click the Build Selectedbutton.

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

9 Select the Layers on topcheck box.


Name Expression Value DescriptionR1 10[mm] 0.010000 m Inner radius of inner magnet

R2 20[mm] 0.020000 m Outer radius of inner magnet

R3 22[mm] 0.022000 m Inner radius of outer magnet

R4 32[mm] 0.032000 m Outer radius of outer magnet

h0 10[mm] 0.010000 m Magnet height

Br 1[T] 1.0000 T Remanent flux density of magnet

dZ 0 0.0000 Axial displacement

Layer name Thickness (mm)

Layer 1 h0


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Rectangle 2 (r2)

1 Right-click Component 1 (comp1)>Geometry 1>Rectangle 1 (r1)and choose Duplicate.

2 In the Settingswindow for Rectangle, locate the Sizesection.

3 In the Widthtext field, type R4-R3.

4 Locate the Positionsection. In the rtext field, type R3.

5 In the ztext field, type -h0/2-h0.


Rectangle 3 (r3)

1 On the Geometrytoolbar, click Primitivesand choose Rectangle.

2 In the Settingswindow for Rectangle, locate the Sizesection.

3 In the Widthtext field, type 70.

4 In the Heighttext field, type 160.

5 Locate the Positionsection. In the ztext field, type -80.

6 Locate the Layerssection. In the table, enter the following settings:

7 Select the Layers to the rightcheck box.

8 Select the Layers on topcheck box.


10 Click the Zoom Extentsbutton on the Graphicstoolbar.

Fillet 1 (fil1)

1 On the Geometrytoolbar, 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 Settingswindow for Fillet, locate the Radiussection.

5 In the Radiustext field, type 2.

Layer name Thickness (mm)

Layer 1 5


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6 Click the Build All Objectsbutton.

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

D E F I N I T I O N S

Enclose the inner air domain by an Infinite Element Domain to model the

surrounding space.

Infinite Element Domain 1 (ie1)

1 On the Definitionstoolbar, click Infinite Element Domain.2 Select Domains 1, 3, and 1012 only.

3 In the Settingswindow for Infinite Element Domain, locate the Geometrysection.


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4 From the Typelist, choose Cylindrical.

M A T E R I A L S

Use air as the material for all domains.

A D D M A T E R I A L

1 On the Modeltoolbar, click Add Materialto open the Add Materialwindow.

2 Go to the Add Materialwindow.

3 In the tree, select Built-In>Air.

4 Click Add to Componentin the window toolbar.

5 On the Modeltoolbar, click Add Materialto close the Add Materialwindow.

M A G N E T I C F I E L D S ( M F )

Now set up the physics for the magnetic field. Use the default Ampre's Law node with

default settings for the air domains and add separate nodes (one per magnetization

direction) for the magnets.

Ampre's Law 2

1 On the Physicstoolbar, click Domainsand choose Ampre's Law.

2 Select Domains 6 and 9 only.

3 In the Settingswindow for Ampre's Law, locate the Magnetic Fieldsection.


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4 From the Constitutive relationlist, choose Remanent flux density.

5 Specify the Brvector as

Ampre' s Law 3


2 Select Domains 4 and 7 only.3 In the Settingswindow for Ampre's Law, locate the Magnetic Fieldsection.



Ampre' s Law 4


2 Select Domain 5 only.




Ampre' s Law 5


2 Select Domain 8 only.



0 r

0 phi

-Br z

0 r

0 phi

Br z

Br r

0 phi

0 z


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Add a Force Calculation feature to compute the total magnetic force on the inner

magnets.

Force Calculation 1

1 On the Physicstoolbar, click Domainsand choose Force Calculation.2 Select Domains 46 only.

Keeping the default Force name, 0, the axial force component can be accessed as

mf.Forcez_0, where 'mf' is the identifier for the Magnetic Fields physics.

Next, add Deformed Geometry and Sensitivity interfaces to use for calculating the

axial magnetic stiffness coefficient as defined by Equation 1of the Model Definition

section.

A D D P H Y S I C S

1 On the Physicstoolbar, click Add Physicsto open the Add Physicswindow.

2 Go to the Add Physicswindow.

3 In the Add physicstree, select Mathematics>Deformed Mesh>Deformed Geometry (dg).


5 In the Add physicstree, select Mathematics.

A D D P H Y S I C S


2 In the Add physicstree, select Mathematics>Optimization and Sensitivity>Sensitivity

(sens).


A D D P H Y S I C S


2 In the Add physicstree, select Mathematics.

3 On the Physicstoolbar, click Add Physicsto close the Add Physicswindow.

-Br r0 phi

0 z


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D E F O R M E D G E O M E T R Y ( D G )

On the Physicstoolbar, click Sensitivity (sens)and choose Deformed Geometry (dg).

1 In the Model Builderwindow, under Component 1 (comp1)click Deformed Geometry

(dg).

2 In the Settingswindow for Deformed Geometry, locate the Domain Selectionsection.

3 Click Clear Selection.

4 Click Paste Selection.

5 In the Paste Selectiondialog box, type 2,4-9in the Selectiontext field.

6 Click OK.

Free Deformation 1

1 On the Physicstoolbar, click Domainsand choose Free Deformation.


Override the default zero mesh displacement on the outer boundaries of the inner

magnets, which are allowed to move in the axial direction.

Prescribed Mesh Displacement 2

1 On the Physicstoolbar, click Boundariesand choose Prescribed Mesh Displacement.

2 In the Settingswindow for Prescribed Mesh Displacement, locate the Boundary

Selectionsection.


4 In the Paste Selectiondialog box, type 8-17,38-41in the Selectiontext field.

5 Click OK.

6 In the Settingswindow for Prescribed Mesh Displacement, locate the Prescribed

Mesh Displacementsection.

7 In the dztext field, type dZ.

S E N S I T I V I T Y ( S E N S )

With the Sensitivity interface you can compute the right-hand side of Equation 1as

the derivative of the global objective defined as the axial force componentmf.Forcez_0with respect to the global control variable defined as the axial

displacement dZ.

Global Control Variables 1

1 On the Physicstoolbar, click Globaland choose Global Control Variables.


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2 In the Settingswindow for Global Control Variables, locate the Control Variables

section.

3 In the Control variablestable, enter the following settings:

Global Objective 1

1 On the Physicstoolbar, click Globaland choose Global Objective.

2 In the Settingswindow for Global Objective, locate the Global Objectivesection.

3 In the Objective expressiontext field, type mf.Forcez_0.

M E S H 1

1 In the Model Builderwindow, under Component 1 (comp1)click Mesh 1.

2 In the Settingswindow for Mesh, locate the Mesh Settingssection.

3 From the Element sizelist, choose Finer.

Use an even finer mesh in the magnet domains.

Size 1

1 Right-click Component 1 (comp1)>Mesh 1and choose Size.

2 In the Settingswindow for Size, locate the Geometric Entity Selectionsection.

3 From the Geometric entity levellist, choose Domain.


5 In the Paste Selectiondialog box, type 4-9in the Selectiontext field.

Variable Initial value

dZ 0


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6 Click OK.

7 In the Settingswindow for Size, locate the Element Sizesection.

8 From the Predefinedlist, choose Extremely fine.

Free Triangular 1

1 In the Model Builderwindow, right-click Mesh 1and choose Free Triangular.

2 In the Settingswindow for Free Triangular, locate the Domain Selectionsection.



Mapped 1

Right-click Mesh 1and choose Mapped.

Distribution 1

1 In the Model Builderwindow, under Component 1 (comp1)>Mesh 1right-click Mapped

1and choose Distribution.

2 Select Boundaries 28, 32, and 33 only.


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3 Click the Build Allbutton.

The mesh should look like that shown in the figure above.

S T U D Y 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 Sweep1 On the Studytoolbar, click Parametric Sweep.

2 In the Settingswindow for Parametric Sweep, locate the Study Settingssection.

3 Click Add.

4 In the table, enter the following settings:

5 Click Range.

6 In the Rangedialog box, type -40in the Starttext field.

7 In the Steptext field, type 2.

8 In the Stoptext field, type 40.

Parameter name Parameter value list Parameter unit

dZ


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9 Click Replace.

10 In the Model Builderwindow, click Study 1.

11 In the Settingswindow for Study, locate the Study Settingssection.

12 Clear the Generate default plotscheck box.

Solution 1

1 On the Studytoolbar, click Show Default Solver.

2 In the Model Builderwindow, expand the Solution 1node.

3 Right-click Stationary Solver 1and choose Sensitivity.

4 On the Studytoolbar, click Compute.

R E S U L T S

In the Model Builderwindow, expand the Resultsnode.

Data Sets

Create data sets for result visualization in specific domains.

1 In the Model Builderwindow, expand the Results>Data Setsnode.

2 Right-click Study 1/Parametric Solutions 1and choose Duplicate.

3 In the Settingswindow for Solution, locate the Solutionsection.

4 From the Solutionlist, choose dZ=8.0000.

5 On the Resultstoolbar, click Selection.

6 In the Settingswindow for Selection, locate the Geometric Entity Selectionsection.


8 Select Domains 49 only.

9 On the Resultstoolbar, click More Data Setsand choose Revolution 2D.

10 In the Settingswindow for Revolution 2D, locate the Datasection.

11 From the Data setlist, choose Study 1/Parametric Solutions 1 (2).

12 Click to expand the Revolution layerssection. Locate the Revolution Layerssection.

In the Start angletext field, type -100.

13 In the Revolution angletext field, type 280.

Use the following instructions to get the plots shown in Figure 2through Figure 6

2D Plot Group 1

1 On the Resultstoolbar, click 2D Plot Group.


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2 In the Model Builderwindow, under Resultsright-click 2D Plot Group 1and choose

Surface.

3 On the 2D plot grouptoolbar, click Plot.

4 In the Model Builderwindow, right-click 2D Plot Group 1and choose Contour.

5 In the Settingswindow for Contour, click Replace Expressionin the upper-right

corner of the Expressionsection. From the menu, choose Model>Component

1>Magnetic Fields>Magnetic>Magnetic vector potential (Material)>Aphi - Magnetic

vector potential, phi component.


7 Click the Zoom Extentsbutton on the Graphicstoolbar.

Compare this plot with Figure 2.

1D Plot Group 2

1 On the Modeltoolbar, click Add Plot Groupand choose 1D Plot Group.

2 On the 1D plot grouptoolbar, click Global.

3 In the Settingswindow for Global, locate the Datasection.

4 From the Data setlist, choose Study 1/Parametric Solutions 1.

5 Click Replace Expressionin the upper-right corner of the y-axis datasection. From

the menu, choose Model>Component 1>Magnetic Fields>Mechanical>Electromagnetic

force (Material)>mf.Forcer_0 - Electromagnetic force, r component.


7 Click to expand the Legendssection. Clear the Show legendscheck box.

The plot should be as shown in Figure 3.

1D Plot Group 3





5 Click Replace Expressionin the upper-right corner of the y-axis datasection. From

the menu, choose Model>Component 1>Sensitivity>sens.gobj1 - Objective value.

6 Locate the y-Axis Datasection. In the table, enter the following settings:

Expression Unit Description

sens.gobj1 N Fz


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Compare the plot you just generated with that shown in Figure 4.

1D Plot Group 4





5 Locate the y-Axis Datasection. In the table, enter the following settings:

6 Click on any cell in the 2nd table row and then click the Deletebutton below the

table.

7 On the 1D plot grouptoolbar, click Plot. Compare this plot with Figure 5.

Finally, reproduce Figure 6using the following steps.

3D Plot Group 5


2 In the Model Builderwindow, under Resultsright-click 3D Plot Group 5and choose

Volume.


Expression Unit Description

fsens(dZ) Sensitivity of Fz w.r.t. dZ [N/m]


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Models.acdc.Axial Magnetic Bearing

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