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Modeling Vibration in an Induction Motor
Introduction
This model simulates the electromechanical effects in a three phase induction motor. The assembly consists of stator, rotor and housing. In this example, the eddy currents are induced in the rotor by the time harmonic currents on the stator windings and the rotor’s rotation. The air gap between the rotor and stator is assumed asymmetric and the resulting vibrations in the motor are analyzed.
The electromagnetic simulation of an induction motor is performed in 2D whereas the multibody dynamics simulation is performed in 3D. The rotational torque, when an alternating current is passed through the stator windings, is calculated as a function of time. It is used in the multibody dynamics model in order to compute the angular speed of the rotor.
Note: This model requires the AC/DC Module and the Multibody Dynamics Module.
Model Definition
The three phase induction motor-housing assembly is shown in Figure 1.
Figure 1: The geometry of a three phase induction motor-housing assembly.
The induction motor modeled in this example has the following parts:
•
Stator
•
Rotor
•
Shaft
•
Housing
•
Bearings
•
Foundation
All parts, except the bearings and foundation, are physically modeled. The latter are modeled as massless springs.
The model uses two different physics interfaces:
•
Rotating Machinery, Magnetic interface to simulate the electromagnetic fields in the motor.
•
Multibody Dynamics interface to simulate the motion of the rotor and vibration in the housing.
Figure 2: Cross sectional view of three phase induction motor showing stator, rotor, and the housing. The figure shows three different coil regions A, B, and C in the stator representing three phases of the motor. The air gap between the stator and rotor is also highlighted.
Electromagnetic model description
The electromagnetic field equations are solved in a transverse section of the induction motor as shown in Figure 2. In the electromagnetic simulation, only the following parts are considered:
•
Stator
•
Rotor
•
Air gap
The stator is made of steel. The stator steel is laminated and its conductivity is zero. The inner part of the rotor is made of steel and outer part is made of aluminum and they have the conductivity of 1.6 × 106 S/m and 3.74 × 107 S/m respectively. There exists a slight misalignment between the rotor and stator, making the air gap asymmetric.
Each of the three phases of the stator winding spans 45 degrees and they are separated by 120 electrical degrees. The alternating current (60 Hz) through the stator windings is applied using a homogenized multiturn coil feature with 2045 turns. The geometrical dimensions for this model are obtained from Ref. 1. More details about the electromagnetic model can be found in Ref. 2.
Multibody Dynamics Model Description
In the multibody dynamics model, the stator, rotor, and shaft are considered to be rigid. The rotor is rigidly mounted on the shaft. The bearings are modeled by elastic hinge joints between the rotor and housing. The bearings are responsible for supporting the rotor and in turn transmitting the forces to the housing. The housing is assumed elastic and made of structural steel. The housing is supported on the foundation using elastic fixed joints. The bearing and foundation stiffness parameters are given in Table 1 below.
Table 1: Bearing and foundation Parameters.
Parameter
Value
Bearing translational stiffness (kb)
1 × 106 N/m
Bearing rotational stiffness (kbr)
1 × 104 Nm/rad
Foundation translational stiffness (kf)
1 × 106 N/m
Foundation rotational stiffness (kfr)
1 × 104 Nm/rad
Electromagnetic–multibody Dynamics coupling
Electromagnetic Torque
The electromagnetic torque calculated in the Rotating Machinery, Magnetic interface is applied on the rotor as well as on the stator in the Multibody Dynamics interface. As the rotor is free to rotate about its own axis, it starts accelerating by overcoming the inertial resistance. The reaction torque on the stator bends the motor-housing assembly in the direction opposite to the rotor rotation.
Rotor Speed
The rotor speed calculated using the angular motion of the hinge joint in the Multibody Dynamics interface is transferred to the Rotating Machinery, Magnetic interface.
Electromagnetic Forces
In addition to the above, the misalignment between the stator and rotor also causes nonzero resultant electromagnetic forces. These forces calculated in the Rotating Machinery, Magnetic interface are also applied to the rotor and stator in the Multibody Dynamics interface. Since these forces are oscillating in nature, the cause vibrations in the motor.
Results and Discussion
A transient analysis is performed to calculate the torque, speed, and vibration in an induction motor. The magnetic flux density norm in the rotor and stator coils at a particular instant is shown in Figure 3.
Figure 3: Magnetic flux density in the rotor and stator at t = 0.7 s.
The von Mises stress distribution in the housing together with the velocity of the rotor is shown in Figure 4. It can be seen that the stress values are higher near the bearing and at the connection between the housing and the foundation.
Figure 4: von Mises stress distribution in the housing and the rotor velocity profile at t = 0.7 s.
Figure 5 illustrates the electromagnetic torque as a function of time. It can be seen that in the absence of any loading torque on the shaft, the electromagnetic torque goes to zero as the rotor achieves the speed equal to the stator electrical frequency (60 Hz).
The angular speed of the rotor as a function of time is shown in Figure 6. It can be seen that it takes 0.7 seconds for the rotor to reach the steady state speed. The time delay for the rotor speed to reach the stator electrical frequency depends on the inertia of the rotor.
Figure 5: Electromagnetic torque in the rotor as a function of time.
Figure 6: Rotor angular speed as a function of time.
Figure 7 shows the electromagnetic forces experienced by the rotor as a function of time. These vibrating forces in the transverse direction of the rotor are caused by the misalignment between the stator and rotor.
The frequency spectrum of electromagnetic forces can be seen in Figure 8. It is clear that the frequency of these forces is 120 Hz which is twice the stator electrical frequency.
Figure 7: Electromagnetic forces in the rotor in transverse and axial directions.
Figure 8: Frequency spectrum of electromagnetic forces in the rotor.
The forces experienced by the bearings on both sides as a function of time are shown in Figure 9 and Figure 10. These forces are transmitted to the foundation through elastic housing. The foundation forces as a function of time at one of the locations are shown in Figure 11. The frequency spectrum of the same forces can be seen in Figure 12.
Figure 9: Forces in bearing-1 in transverse and axial directions.
Figure 10: Forces in bearing-2 in transverse and axial directions.
Note that the dominant frequency contribution is centered around 60 Hz in spite of the electromagnetic forces having a frequency of 120 Hz. Small peaks can also be seen around 83 Hz which is the first natural frequency of the induction motor-housing assembly.
Figure 11: Forces in the housing-foundation connection at one of the locations.
Figure 12: Frequency spectrum of forces in the housing-foundation connection.
As the electromagnetic forces are acting on the rotor in the transverse direction, and the bearings supporting the rotor at the ends have finite stiffness, rotor starts vibrating in the transverse direction with respect to the stator. This orbital motion, a combination of rotation and vibration, of the rotor at both the bearing locations is shown in Figure 13. Note that the orbits are not concentric because of the asymmetry in the inertia of the rotor in the axial direction.
Figure 13: Orbital motion of the rotor at bearing locations.
Notes About the COMSOL Implementation
•
The 2D geometry of the induction motor is the cross section of the 3D geometry. The length of the 2D geometry in the out-of-plane direction in the Rotating Machinery, Magnetic interface is set to the length of the rotor in the 3D geometry.
•
The 2D geometry has stationary and rotating domains. As it is finalized with the assembly option, an identity pair forms automatically between the stationary and rotating domains.
•
For faster computation, linear shape functions are used instead of the default quadratic shape functions for the magnetic vector potential.
•
Some parts having comparatively higher structural stiffness are modeled as rigid elements using the Rigid Material model. The remaining parts are assumed flexible and modeled using Linear Elastic Material.
References
1. K. Davey, “Induction Motor Analysis: International TEAM Workshop Problem 30,” www.compumag.org/jsite/images/stories/TEAM/problem30a.pdf.
2. COMSOL Application Gallery, Induction Motor in 2D, www.comsol.com/model/induction-motor-in-2d-14487.
Application Library path: Multibody_Dynamics_Module/Electrical_Machinery/induction_motor_vibration/
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.
2
In the Select Physics tree, select AC/DC > Electromagnetics and Mechanics > Rotating Machinery, Magnetic (rmm).
3
Click Add.
4
Click  Study.
5
In the Select Study tree, select General Studies > Time Dependent.
6
Click  Done.
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
Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file induction_motor_vibration_parameters.txt.
Import the 3D geometry of an induction motor in a 3D Component and then create 2D geometry using Cross Section functionality.
Add Component
In the Model Builder window, right-click the root node and choose Add Component > 3D.
Geometry 2
Import 1 (imp1)
1
In the Geometry toolbar, click  Import.
2
In the Settings window for Import, locate the Source section.
3
Click  Browse.
4
Browse to the model’s Application Libraries folder and double-click the file induction_motor_vibration.mphbin.
Work Plane 1 (wp1)
1
In the Geometry toolbar, click  Work Plane.
2
In the Settings window for Work Plane, locate the Plane Definition section.
3
From the Plane list, choose yz-plane.
Form Union (fin)
1
In the Model Builder window, under Component 2 (comp2) > Geometry 2 click Form Union (fin).
2
In the Settings window for Form Union/Assembly, locate the Form Union/Assembly section.
3
From the Action list, choose Form an assembly.
4
Clear the Create pairs checkbox.
5
In the Geometry toolbar, click  Build All.
Geometry 1
In the Model Builder window, under Component 1 (comp1) click Geometry 1.
Cross Section 1 (cro1)
In the Geometry toolbar, click  Cross Section.
Form Union (fin)
1
In the Model Builder window, under Component 1 (comp1) > Geometry 1 click Form Union (fin).
2
In the Settings window for Form Union/Assembly, locate the Form Union/Assembly section.
3
From the Action list, choose Form an assembly.
4
In the Geometry toolbar, click  Build All.
Definitions (comp1)
Disable the analysis of the geometry as the remaining small geometric details can be kept.
Geometry 2
1
In the Model Builder window, expand the Component 1 (comp1) > Definitions node, then click Component 2 (comp2) > Geometry 2.
2
In the Settings window for Geometry, locate the Cleanup section.
3
Clear the Automatic detection of small details checkbox.
Definitions (comp1)
Variables 1
1
In the Model Builder window, under Component 1 (comp1) right-click Definitions and choose Variables.
2
In the Settings window for Variables, locate the Variables section.
3
In the table, enter the following settings:
Name
Expression
Unit
Description
Ia
1[A]*sqrt(2)*cos(w0*t)
A
Current on phase A
Ib
1[A]*sqrt(2)*cos(w0*t+120[deg])
A
Current on phase B
Ic
1[A]*sqrt(2)*cos(w0*t-120[deg])
A
Current on phase C
Assign materials to the different parts of the motor.
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 Built-in > Air.
4
Click the Add to Component button in the window toolbar.
5
In the tree, select Built-in > Aluminum.
6
Click the Add to Component button in the window toolbar.
7
In the Materials toolbar, click  Add Material to close the Add Material window.
Materials
Aluminum (mat2)
Select Domain 18 only.
Steel: Rotor
1
In the Model Builder window, right-click Materials and choose Blank Material.
2
In the Settings window for Material, type Steel: Rotor in the Label text field.
3
Select Domains 19 and 20 only.
4
Locate the Material Contents section. In the table, enter the following settings:
Property
Variable
Value
Unit
Property group
Relative permeability
mur
30
1
Basic
Electric conductivity
sigma
1.6e6[S/m]
S/m
Basic
Relative permittivity
epsilonr
1
1
Basic
Steel: Stator
1
Right-click Steel: Rotor and choose Duplicate.
2
In the Settings window for Material, type Steel: Stator in the Label text field.
3
Locate the Geometric Entity Selection section. Click  Clear Selection.
4
Select Domain 3 only.
5
Locate the Material Contents section. In the table, enter the following settings:
Property
Variable
Value
Unit
Property group
Relative permeability
mur
30
1
Basic
Electric conductivity
sigma
0[S/m]
S/m
Basic
Relative permittivity
epsilonr
1
1
Basic
Rotating Machinery, Magnetic (rmm)
1
In the Model Builder window, under Component 1 (comp1) click Rotating Machinery, Magnetic (rmm).
2
Select Domains 3–20 only.
3
In the Settings window for Rotating Machinery, Magnetic, locate the Thickness section.
4
In the d text field, type L.
5
Click to expand the Discretization section. From the Magnetic vector potential list, choose Linear.
6
From the Magnetic scalar potential list, choose Quadratic.
Coil: Phase A
1
In the Physics toolbar, click  Domains and choose Domain Coil.
2
In the Settings window for Domain Coil, type Coil: Phase A in the Label text field.
3
Select Domains 5 and 15 only.
4
Locate the Coil section. From the Conductor model list, choose Homogenized multiturn.
5
Select the Coil group checkbox.
6
In the Icoil text field, type Ia.
7
Locate the Homogenized Conductor section. In the N text field, type n0.
8
From the list, choose User defined.
9
Find the High-frequency effective loss subsection. Clear the Include harmonic loss checkbox.
Reversed Current Direction 1
1
In the Physics toolbar, click  Attributes and choose Reversed Current Direction.
2
Select Domain 15 only.
Coil: Phase B
1
In the Model Builder window, right-click Coil: Phase A and choose Duplicate.
2
In the Settings window for Domain Coil, type Coil: Phase B in the Label text field.
3
Locate the Domain Selection section. Click  Clear Selection.
4
Select Domains 9 and 11 only.
5
Locate the Coil section. In the Icoil text field, type Ib.
Reversed Current Direction 1
1
In the Model Builder window, expand the Coil: Phase B node, then click Reversed Current Direction 1.
2
In the Settings window for Reversed Current Direction, locate the Domain Selection section.
3
Click  Clear Selection.
4
Select Domain 9 only.
Coil: Phase C
1
In the Model Builder window, right-click Coil: Phase B and choose Duplicate.
2
In the Settings window for Domain Coil, type Coil: Phase C in the Label text field.
3
Locate the Domain Selection section. Click  Clear Selection.
4
Select Domains 6 and 13 only.
5
Locate the Coil section. In the Icoil text field, type Ic.
Reversed Current Direction 1
1
In the Model Builder window, expand the Coil: Phase C node, then click Reversed Current Direction 1.
2
In the Settings window for Reversed Current Direction, locate the Domain Selection section.
3
Click  Clear Selection.
4
Select Domain 6 only.
Magnetic Continuity 1a
1
In the Physics toolbar, click  Pairs and choose Magnetic Continuity.
2
In the Settings window for Magnetic Continuity, locate the Pair Selection section.
3
Click  Add.
4
In the Add dialog, select Identity Boundary Pair 1 (ap1) in the Pairs list.
5
Click OK.
Force Calculation: Rotor
1
In the Physics toolbar, click  Domains and choose Force Calculation.
2
In the Settings window for Force Calculation, type Force Calculation: Rotor in the Label text field.
3
Locate the Force Calculation section. In the Force name text field, type Rotor.
4
Specify the r0 vector as
0
x
dy
y
5
Select Domains 18–20 only.
Force Calculation: Stator
1
Right-click Force Calculation: Rotor and choose Duplicate.
2
In the Settings window for Force Calculation, type Force Calculation: Stator in the Label text field.
3
Locate the Force Calculation section. In the Force name text field, type Stator.
4
Specify the r0 vector as
0
x
0
y
5
Locate the Domain Selection section. Click  Clear Selection.
6
Select Domains 3–7 and 9–16 only.
Mesh 1
1
In the Model Builder window, under Component 1 (comp1) click Mesh 1.
2
In the Settings window for Mesh, locate the Sequence Type section.
3
From the list, choose User-controlled mesh.
Size
1
In the Model Builder window, under Component 1 (comp1) > Mesh 1 click Size.
2
In the Settings window for Size, locate the Element Size section.
3
From the Predefined list, choose Extra coarse.
Size 1
1
In the Model Builder window, right-click Free Triangular 1 and choose Size.
2
In the Settings window for Size, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Domain.
4
Select Domains 18–20 only.
5
Locate the Element Size section. From the Predefined list, choose Finer.
Size 2
1
Right-click Size 1 and choose Duplicate.
2
In the Settings window for Size, locate the Geometric Entity Selection section.
3
Click  Clear Selection.
4
Click  Paste Selection.
5
In the Paste Selection dialog, type 17 in the Selection text field.
6
Click OK.
7
In the Settings window for Size, locate the Element Size section.
8
Click the Custom button.
9
Locate the Element Size Parameters section.
10
Select the Maximum element size checkbox. In the associated text field, type 0.00075.
Size 3
1
Right-click Size 2 and choose Duplicate.
2
In the Settings window for Size, locate the Geometric Entity Selection section.
3
Click  Clear Selection.
4
Click  Paste Selection.
5
In the Paste Selection dialog, type 8 in the Selection text field.
6
Click OK.
7
In the Settings window for Size, locate the Element Size Parameters section.
8
In the Maximum element size text field, type 0.00125.
Boundary Layers 1
1
In the Mesh toolbar, click  Boundary Layers.
2
In the Settings window for Boundary Layers, locate the Domain Selection section.
3
From the Geometric entity level list, choose Domain.
4
Select Domain 19 only.
Boundary Layer Properties
1
In the Model Builder window, click Boundary Layer Properties.
2
In the Settings window for Boundary Layer Properties, locate the Boundary Selection section.
3
Click  Paste Selection.
4
In the Paste Selection dialog, type 163 164 in the Selection text field.
5
Click OK.
Boundary Layers 2
1
In the Mesh toolbar, click  Boundary Layers.
2
In the Settings window for Boundary Layers, locate the Domain Selection section.
3
From the Geometric entity level list, choose Domain.
4
Select Domain 18 only.
Boundary Layer Properties
1
In the Model Builder window, click Boundary Layer Properties.
2
In the Settings window for Boundary Layer Properties, locate the Boundary Selection section.
3
Click  Paste Selection.
4
In the Paste Selection dialog, type 161 162 in the Selection text field.
5
Click OK.
Next, add a Multibody Dynamics interface to the 3D component.
Component 2 (comp2)
In the Model Builder window, click Component 2 (comp2).
Add Physics
1
In the Home toolbar, click  Add Physics to open the Add Physics window.
2
Go to the Add Physics window.
3
In the tree, select Structural Mechanics > Multibody Dynamics (mbd).
4
Click the Add to Component 2 button in the window toolbar.
5
In the Home toolbar, click  Add Physics to close the Add Physics window.
Multibody Dynamics (mbd)
1
Click the  Click and Hide button in the Graphics toolbar.
2
Select Domain 14 only.
3
Click the  Click and Hide button in the Graphics toolbar.
4
Select Domains 1–4, 6–15, 17, 18, 21, 23, 25, 27, and 29–36 only.
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 Built-in > Structural steel.
4
Click the Add to Component button in the window toolbar.
5
In the tree, select Built-in > Aluminum.
6
Click the Add to Component button in the window toolbar.
7
In the Materials toolbar, click  Add Material to close the Add Material window.
Materials
Aluminum (mat6)
1
In the Settings window for Material, locate the Geometric Entity Selection section.
2
Click  Paste Selection.
3
In the Paste Selection dialog, type 6 in the Selection text field.
4
Click OK.
Multibody Dynamics (mbd)
Add Rigid Material nodes for modeling the rotor and stator. Also apply electromagnetic torque and forces on both parts.
Rigid Material: Rotor with Shaft
1
In the Physics toolbar, click  Domains and choose Rigid Material.
2
In the Settings window for Rigid Material, type Rigid Material: Rotor with Shaft in the Label text field.
3
Locate the Domain Selection section. Click  Paste Selection.
4
In the Paste Selection dialog, type 1-4, 6-12 in the Selection text field.
5
Click OK.
Applied Force 1
1
In the Physics toolbar, click  Attributes and choose Applied Force.
2
In the Settings window for Applied Force, locate the Location section.
3
From the list, choose Centroid of selected entities.
Location: Boundary 1
1
In the Model Builder window, expand the Applied Force 1 node, then click Location: Boundary 1.
2
In the Settings window for Location: Boundary, locate the Boundary Selection section.
3
Click  Paste Selection.
4
In the Paste Selection dialog, type 45 53 in the Selection text field.
5
Click OK.
Applied Force 1
1
In the Model Builder window, click Applied Force 1.
2
In the Settings window for Applied Force, locate the Applied Force section.
3
Specify the F vector as
0
x
comp1.rmm.Forcex_Rotor
y
comp1.rmm.Forcey_Rotor
z
Rigid Material: Rotor with Shaft
In the Model Builder window, click Rigid Material: Rotor with Shaft.
Applied Moment 1
1
In the Physics toolbar, click  Attributes and choose Applied Moment.
2
In the Settings window for Applied Moment, locate the Applied Moment section.
3
Specify the M vector as
comp1.rmm.Tax_Rotor
x
0
y
0
z
Rigid Material: Stator
1
In the Physics toolbar, click  Domains and choose Rigid Material.
2
In the Settings window for Rigid Material, type Rigid Material: Stator in the Label text field.
3
Locate the Domain Selection section. Click  Paste Selection.
4
In the Paste Selection dialog, type 15, 17-18, 21, 23, 25, 27 in the Selection text field.
5
Click OK.
Applied Force 1
1
In the Physics toolbar, click  Attributes and choose Applied Force.
2
In the Settings window for Applied Force, locate the Applied Force section.
3
Specify the F vector as
0
x
comp1.rmm.Forcex_Stator
y
comp1.rmm.Forcey_Stator
z
Rigid Material: Stator
In the Model Builder window, click Rigid Material: Stator.
Applied Moment 1
1
In the Physics toolbar, click  Attributes and choose Applied Moment.
2
In the Settings window for Applied Moment, locate the Applied Moment section.
3
Specify the M vector as
comp1.rmm.Tax_Stator
x
0
y
0
z
Next is to define the rotor-housing and housing-foundation connections using elastic Hinge Joints and Fixed Joints respectively.
Attachment: Bearing 1
1
In the Physics toolbar, click  Boundaries and choose Attachment.
2
In the Settings window for Attachment, type Attachment: Bearing 1 in the Label text field.
3
Locate the Boundary Selection section. Click  Paste Selection.
4
In the Paste Selection dialog, type 183-184, 186-187 in the Selection text field.
5
Click OK.
Attachment: Bearing 2
1
In the Physics toolbar, click  Boundaries and choose Attachment.
2
In the Settings window for Attachment, type Attachment: Bearing 2 in the Label text field.
3
Locate the Boundary Selection section. Click  Paste Selection.
4
In the Paste Selection dialog, type 498-501 in the Selection text field.
5
Click OK.
Hinge Joint: Bearing 1
1
In the Physics toolbar, click  Global and choose Hinge Joint.
2
In the Settings window for Hinge Joint, type Hinge Joint: Bearing 1 in the Label text field.
3
Locate the Attachment Selection section. From the Source list, choose Attachment: Bearing 1.
4
From the Destination list, choose Rigid Material: Rotor with Shaft.
5
Locate the Joint Elasticity section. From the list, choose Elastic joint.
Joint Elasticity 1
1
In the Model Builder window, expand the Hinge Joint: Bearing 1 node, then click Joint Elasticity 1.
2
In the Settings window for Joint Elasticity, locate the Spring section.
3
In the ku text field, type kb.
4
In the kθ text field, type kbr.
Hinge Joint: Bearing 2
1
In the Model Builder window, right-click Hinge Joint: Bearing 1 and choose Duplicate.
2
In the Settings window for Hinge Joint, type Hinge Joint: Bearing 2 in the Label text field.
3
Locate the Attachment Selection section. From the Source list, choose Attachment: Bearing 2.
Attachment: Foundation 1
1
In the Model Builder window, expand the Hinge Joint: Bearing 2 node.
2
Right-click Multibody Dynamics (mbd) and choose Attachment.
3
In the Settings window for Attachment, type Attachment: Foundation 1 in the Label text field.
4
Locate the Boundary Selection section. Click  Paste Selection.
5
In the Paste Selection dialog, type 390-391 in the Selection text field.
6
Click OK.
Attachment: Foundation 2
1
Right-click Attachment: Foundation 1 and choose Duplicate.
2
In the Settings window for Attachment, type Attachment: Foundation 2 in the Label text field.
3
Locate the Boundary Selection section. Click  Clear Selection.
4
Click  Paste Selection.
5
In the Paste Selection dialog, type 470-471 in the Selection text field.
6
Click OK.
Attachment: Foundation 3
1
Right-click Attachment: Foundation 2 and choose Duplicate.
2
In the Settings window for Attachment, type Attachment: Foundation 3 in the Label text field.
3
Locate the Boundary Selection section. Click  Clear Selection.
4
Click  Paste Selection.
5
In the Paste Selection dialog, type 472-473 in the Selection text field.
6
Click OK.
Attachment: Foundation 4
1
Right-click Attachment: Foundation 3 and choose Duplicate.
2
In the Settings window for Attachment, type Attachment: Foundation 4 in the Label text field.
3
Locate the Boundary Selection section. Click  Clear Selection.
4
Click  Paste Selection.
5
In the Paste Selection dialog, type 392-393 in the Selection text field.
6
Click OK.
Fixed Joint: Foundation 1
1
In the Physics toolbar, click  Global and choose Fixed Joint.
2
In the Settings window for Fixed Joint, type Fixed Joint: Foundation 1 in the Label text field.
3
Locate the Attachment Selection section. From the Source list, choose Fixed.
4
From the Destination list, choose Attachment: Foundation 1.
5
Locate the Joint Elasticity section. From the list, choose Elastic joint.
Joint Elasticity 1
1
In the Model Builder window, expand the Fixed Joint: Foundation 1 node, then click Joint Elasticity 1.
2
In the Settings window for Joint Elasticity, locate the Spring section.
3
In the ku text field, type kf.
4
In the kθ text field, type kfr.
Fixed Joint: Foundation 2
1
In the Model Builder window, right-click Fixed Joint: Foundation 1 and choose Duplicate.
2
In the Settings window for Fixed Joint, type Fixed Joint: Foundation 2 in the Label text field.
3
Locate the Attachment Selection section. From the Destination list, choose Attachment: Foundation 2.
Fixed Joint: Foundation 3
1
Right-click Fixed Joint: Foundation 2 and choose Duplicate.
2
In the Settings window for Fixed Joint, type Fixed Joint: Foundation 3 in the Label text field.
3
Locate the Attachment Selection section. From the Destination list, choose Attachment: Foundation 3.
Fixed Joint: Foundation 4
1
Right-click Fixed Joint: Foundation 3 and choose Duplicate.
2
In the Settings window for Fixed Joint, type Fixed Joint: Foundation 4 in the Label text field.
3
Locate the Attachment Selection section. From the Destination list, choose Attachment: Foundation 4.
Component 1 (comp1)
In the Model Builder window, click Component 1 (comp1).
Rotating Domain 1
1
In the Physics toolbar, click  Moving Mesh and choose Rotating Domain.
2
Select Domains 17–20 only.
3
In the Settings window for Rotating Domain, locate the Rotation section.
4
In the α text field, type comp2.mbd.hgj1.th.
5
Locate the Axis section. Specify the rax vector as
0
X
dy
Y
Mesh 2
1
In the Model Builder window, under Component 2 (comp2) click Mesh 2.
2
In the Settings window for Mesh, locate the Physics-Controlled Mesh section.
3
In the table, clear the Use checkbox for Geometric Analysis, Detail Size.
4
Click  Build All.
Study 1
Step 1: Time Dependent
1
In the Settings window for Time Dependent, locate the Study Settings section.
2
In the Output times text field, type range(0,0.001,0.7).
3
From the Tolerance list, choose User controlled.
4
In the Relative tolerance text field, type 0.001.
Solution 1 (sol1)
1
In the Study toolbar, click  Show Default Solver.
2
In the Model Builder window, expand the Solution 1 (sol1) node, then click Time-Dependent Solver 1.
3
In the Settings window for Time-Dependent Solver, click to expand the Time Stepping section.
4
From the Steps taken by solver list, choose Intermediate.
5
From the Maximum step constraint list, choose Constant.
6
In the Maximum step text field, type 0.0002.
7
From the Maximum BDF order list, choose 2.
8
In the Model Builder window, expand the Study 1 > Solver Configurations > Solution 1 (sol1) > Time-Dependent Solver 1 node.
9
In the Model Builder window, expand the Study 1 > Solver Configurations > Solution 1 (sol1) > Time-Dependent Solver 1 node, then click Direct.
10
In the Settings window for Direct, locate the General section.
11
From the Solver list, choose PARDISO.
12
In the Model Builder window, expand the Study 1 > Solver Configurations > Solution 1 (sol1) > Time-Dependent Solver 1 > Segregated 1 node, then click Multibody Dynamics.
13
In the Settings window for Segregated Step, click to expand the Method and Termination section.
14
From the Jacobian update list, choose Once per time step.
15
From the Termination technique list, choose Tolerance.
16
In the Tolerance factor text field, type 1.
17
In the Model Builder window, click Study 1.
18
In the Settings window for Study, locate the Study Settings section.
19
Clear the Generate default plots checkbox.
20
In the Study toolbar, click  Compute.
Results
Follow the instructions below to plot the magnetic flux density, electromagnetic torque, and the angular speed of the rotor as shown in Figure 3, Figure 5 and Figure 6 respectively.
Magnetic Flux Density Norm
1
In the Results toolbar, click  2D Plot Group.
2
In the Settings window for 2D Plot Group, type Magnetic Flux Density Norm in the Label text field.
3
Locate the Plot Settings section. From the Frame list, choose Spatial  (x, y, z).
Surface 1
Right-click Magnetic Flux Density Norm and choose Surface.
Arrow Surface 1
1
In the Model Builder window, right-click Magnetic Flux Density Norm and choose Arrow Surface.
2
In the Settings window for Arrow Surface, locate the Arrow Positioning section.
3
Find the x grid points subsection. In the Points text field, type 50.
4
Find the y grid points subsection. In the Points text field, type 50.
5
Locate the Coloring and Style section.
6
Select the Scale factor checkbox. In the associated text field, type 0.055.
Study 1/Solution 1 (1) (sol1)
In the Model Builder window, expand the Results > Datasets node, then click Study 1/Solution 1 (1) (sol1).
Selection
1
Right-click Study 1/Solution 1 (1) (sol1) 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
Click  Paste Selection.
5
In the Paste Selection dialog, type 4-20 in the Selection text field.
6
Click OK.
Study 1/Solution 1 (3) (sol1)
In the Model Builder window, under Results > Datasets right-click Study 1/Solution 1 (2) (sol1) and choose Duplicate.
Selection
1
In the Model Builder window, right-click Study 1/Solution 1 (3) (sol1) 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
Click  Paste Selection.
5
In the Paste Selection dialog, type 13, 29-36 in the Selection text field.
6
Click OK.
Study 1/Solution 1 (4) (sol1)
In the Model Builder window, under Results > Datasets right-click Study 1/Solution 1 (3) (sol1) and choose Duplicate.
Selection
1
In the Model Builder window, expand the Study 1/Solution 1 (4) (sol1) node, then click Selection.
2
In the Settings window for Selection, locate the Geometric Entity Selection section.
3
Click  Clear Selection.
4
Click  Paste Selection.
5
In the Paste Selection dialog, type 1-4, 6-12, 17, 21, 25, 27 in the Selection text field.
6
Click OK.
Magnetic Flux Density Norm
1
In the Model Builder window, under Results click Magnetic Flux Density Norm.
2
In the Magnetic Flux Density Norm toolbar, click  Plot.
3
Click the  Zoom Extents button in the Graphics toolbar.
Electromagnetic Torque (Rotor)
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Electromagnetic Torque (Rotor) in the Label text field.
3
Click to expand the Title section. From the Title type list, choose Manual.
4
In the Title text area, type Electromagnetic Torque (Rotor).
Global 1
1
Right-click Electromagnetic Torque (Rotor) and choose Global.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
comp1.rmm.Tax_Rotor
N*m
Torque, z-component
4
Click to expand the Coloring and Style section. From the Width list, choose 2.
Electromagnetic Torque (Rotor)
1
In the Model Builder window, click Electromagnetic Torque (Rotor).
2
In the Settings window for 1D Plot Group, locate the Legend section.
3
Clear the Show legends checkbox.
4
In the Electromagnetic Torque (Rotor) toolbar, click  Plot.
5
Click the  Zoom Extents button in the Graphics toolbar.
Rotor Speed
1
Right-click Electromagnetic Torque (Rotor) and choose Duplicate.
2
In the Settings window for 1D Plot Group, type Rotor Speed in the Label text field.
3
Locate the Title section. In the Title text area, type Rotor Speed.
Global 1
1
In the Model Builder window, expand the Rotor Speed node, then click Global 1.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
comp2.mbd.hgj1.Tht
rad/s
Relative angular velocity
4
In the Rotor Speed toolbar, click  Plot.
5
Click the  Zoom Extents button in the Graphics toolbar.
Follow the instructions below to plot the electromagnetic forces as a function of time and its frequency spectrum as shown in Figure 7 and Figure 8 respectively.
Electromagnetic Forces (Rotor)
1
In the Model Builder window, right-click Rotor Speed and choose Duplicate.
2
In the Settings window for 1D Plot Group, type Electromagnetic Forces (Rotor) in the Label text field.
3
Locate the Title section. In the Title text area, type Electromagnetic Forces (Rotor).
Global 1
1
In the Model Builder window, expand the Electromagnetic Forces (Rotor) node, then click Global 1.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
rmm.Forcex_Rotor
N
Electromagnetic force, x-component
rmm.Forcey_Rotor
N
Electromagnetic force, y-component
Electromagnetic Forces (Rotor)
1
In the Model Builder window, click Electromagnetic Forces (Rotor).
2
In the Settings window for 1D Plot Group, locate the Legend section.
3
Select the Show legends checkbox.
4
From the Position list, choose Lower left.
5
Locate the Plot Settings section.
6
Select the y-axis label checkbox. In the associated text field, type Force (N).
7
In the Electromagnetic Forces (Rotor) toolbar, click  Plot.
8
Click the  Zoom Extents button in the Graphics toolbar.
Electromagnetic Forces (Rotor): Frequency
1
Right-click Electromagnetic Forces (Rotor) and choose Duplicate.
2
In the Settings window for 1D Plot Group, type Electromagnetic Forces (Rotor): Frequency in the Label text field.
3
Locate the Plot Settings section. Clear the y-axis label checkbox.
4
Locate the Legend section. From the Position list, choose Upper right.
Global 1
1
In the Model Builder window, expand the Electromagnetic Forces (Rotor): Frequency node, then click Global 1.
2
In the Settings window for Global, locate the x-Axis Data section.
3
From the Parameter list, choose Discrete Fourier transform.
4
From the Show list, choose Frequency spectrum.
5
Select the Frequency range checkbox.
6
In the Maximum text field, type 300.
7
In the Electromagnetic Forces (Rotor): Frequency toolbar, click  Plot.
8
Click the  Zoom Extents button in the Graphics toolbar.
Follow the instructions below to plot the bearing and foundation forces as shown in Figure 9, Figure 10, Figure 11 and Figure 12 respectively.
Forces (Bearing-1)
1
In the Model Builder window, right-click Electromagnetic Forces (Rotor) and choose Duplicate.
2
In the Settings window for 1D Plot Group, type Forces (Bearing-1) in the Label text field.
3
Locate the Title section. In the Title text area, type Forces (Bearing-1).
Global 1
1
In the Model Builder window, expand the Forces (Bearing-1) node, then click Global 1.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
comp2.mbd.hgj1.Fx
N
Joint force, x-component
comp2.mbd.hgj1.Fy
N
Joint force, y-component
comp2.mbd.hgj1.Fz
N
Joint force, z-component
4
Click to expand the Legends section. From the Legends list, choose Manual.
5
In the table, enter the following settings:
Legends
x-component
y-component
z-component
Forces (Bearing-1)
1
In the Model Builder window, click Forces (Bearing-1).
2
In the Settings window for 1D Plot Group, locate the Legend section.
3
From the Position list, choose Upper left.
4
In the Forces (Bearing-1) toolbar, click  Plot.
5
Click the  Zoom Extents button in the Graphics toolbar.
Forces (Bearing-2)
1
Right-click Forces (Bearing-1) and choose Duplicate.
2
In the Settings window for 1D Plot Group, type Forces (Bearing-2) in the Label text field.
3
Locate the Title section. In the Title text area, type Forces (Bearing-2).
Global 1
1
In the Model Builder window, expand the Forces (Bearing-2) node, then click Global 1.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
comp2.mbd.hgj2.Fx
N
Joint force, x-component
comp2.mbd.hgj2.Fy
N
Joint force, y-component
comp2.mbd.hgj2.Fz
N
Joint force, z-component
4
In the Forces (Bearing-2) toolbar, click  Plot.
5
Click the  Zoom Extents button in the Graphics toolbar.
Forces (Foundation-1)
1
In the Model Builder window, right-click Forces (Bearing-2) and choose Duplicate.
2
In the Settings window for 1D Plot Group, type Forces (Foundation-1) in the Label text field.
3
Locate the Title section. In the Title text area, type Forces (Foundation-1).
Global 1
1
In the Model Builder window, expand the Forces (Foundation-1) node, then click Global 1.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
comp2.mbd.fxj1.Fx
N
Joint force, x-component
comp2.mbd.fxj1.Fy
N
Joint force, y-component
comp2.mbd.fxj1.Fz
N
Joint force, z-component
4
In the Forces (Foundation-1) toolbar, click  Plot.
5
Click the  Zoom Extents button in the Graphics toolbar.
Forces (Foundation-1): Frequency
1
In the Model Builder window, right-click Forces (Foundation-1) and choose Duplicate.
2
In the Settings window for 1D Plot Group, type Forces (Foundation-1): Frequency in the Label text field.
Global 1
1
In the Model Builder window, expand the Forces (Foundation-1) 1 node, then click Results > Forces (Foundation-1): Frequency > Global 1.
2
In the Settings window for Global, locate the x-Axis Data section.
3
From the Parameter list, choose Discrete Fourier transform.
4
From the Show list, choose Frequency spectrum.
5
Select the Frequency range checkbox.
6
In the Maximum text field, type 300.
Forces (Foundation-1): Frequency
1
In the Model Builder window, click Forces (Foundation-1): Frequency.
2
In the Settings window for 1D Plot Group, locate the Legend section.
3
From the Position list, choose Upper right.
4
Locate the Plot Settings section. Clear the y-axis label checkbox.
5
In the Forces (Foundation-1): Frequency toolbar, click  Plot.
6
Click the  Zoom Extents button in the Graphics toolbar.
Follow the instructions below to plot the orbital motion of the rotor as shown in Figure 13.
Rotor Orbits
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Rotor Orbits in the Label text field.
3
Locate the Title section. From the Title type list, choose Manual.
4
In the Title text area, type Rotor Orbits.
5
Locate the Data section. From the Dataset list, choose Study 1/Solution 1 (2) (sol1).
Global 1
1
Right-click Rotor Orbits and choose Global.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
mbd.hgj1.u2
mm
Displacement (y direction)
4
Locate the x-Axis Data section. From the Parameter list, choose Expression.
5
In the Expression text field, type mbd.hgj1.u3.
6
From the Unit list, choose mm.
7
Select the Description checkbox. In the associated text field, type Displacement (z direction).
Color Expression 1
1
Right-click Global 1 and choose Color Expression.
2
In the Settings window for Color Expression, locate the Expression section.
3
In the Expression text field, type t.
Global 2
1
In the Model Builder window, under Results > Rotor Orbits right-click Global 1 and choose Duplicate.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
mbd.hgj2.u2
mm
Displacement (y direction)
4
Locate the x-Axis Data section. In the Expression text field, type mbd.hgj2.u3.
Annotation 1
1
In the Model Builder window, right-click Rotor Orbits and choose Annotation.
2
In the Settings window for Annotation, locate the Annotation section.
3
In the Text text field, type Bearing 1.
4
Locate the Position section. In the X text field, type 1e-3.
5
In the Y text field, type -1e-3.
6
Locate the Coloring and Style section. Clear the Show point checkbox.
Annotation 2
1
Right-click Annotation 1 and choose Duplicate.
2
In the Settings window for Annotation, locate the Annotation section.
3
In the Text text field, type Bearing 2.
4
Locate the Position section. In the X text field, type -3e-3.
5
In the Y text field, type 1.5e-3.
Rotor Orbits
1
In the Model Builder window, click Rotor Orbits.
2
In the Settings window for 1D Plot Group, locate the Legend section.
3
Clear the Show legends checkbox.
4
Locate the Axis section. Select the Preserve aspect ratio checkbox.
5
In the Rotor Orbits toolbar, click  Plot.
6
Click the  Zoom Extents button in the Graphics toolbar.
Housing Stress
1
In the Results toolbar, click  3D Plot Group.
2
In the Settings window for 3D Plot Group, locate the Data section.
3
From the Dataset list, choose Study 1/Solution 1 (3) (sol1).
4
In the Label text field, type Housing Stress.
Surface 1
1
Right-click Housing Stress and choose Surface.
2
In the Settings window for Surface, locate the Expression section.
3
In the Expression text field, type mbd.mises.
4
Click to expand the Range section. Select the Manual color range checkbox.
5
In the Maximum text field, type 5e4.
6
Locate the Coloring and Style section. From the Color table list, choose TrafficLight.
Deformation 1
1
Right-click Surface 1 and choose Deformation.
2
In the Settings window for Deformation, locate the Scale section.
3
Select the Scale factor checkbox. In the associated text field, type 2000.
4
In the Housing Stress toolbar, click  Plot.
5
Click the  Zoom Extents button in the Graphics toolbar.
Housing Stress (with Rotor)
1
In the Model Builder window, right-click Housing Stress and choose Duplicate.
2
In the Settings window for 3D Plot Group, type Housing Stress (with Rotor) in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 1/Solution 1 (2) (sol1).
4
Locate the Plot Settings section. From the Frame list, choose Spatial  (x, y, z).
Surface 1
1
In the Model Builder window, expand the Housing Stress (with Rotor) node, then click Surface 1.
2
In the Settings window for Surface, locate the Data section.
3
From the Dataset list, choose Study 1/Solution 1 (3) (sol1).
4
From the Solution parameters list, choose From parent.
Deformation 1
1
In the Model Builder window, expand the Surface 1 node, then click Deformation 1.
2
In the Settings window for Deformation, locate the Scale section.
3
In the Scale factor text field, type 1.
Contour 1
1
In the Model Builder window, right-click Housing Stress (with Rotor) and choose Contour.
2
In the Settings window for Contour, locate the Data section.
3
From the Dataset list, choose Study 1/Solution 1 (3) (sol1).
4
From the Solution parameters list, choose From parent.
5
Locate the Expression section. In the Expression text field, type mbd.mises.
6
Locate the Levels section. In the Total levels text field, type 10.
7
Locate the Coloring and Style section. From the Coloring list, choose Uniform.
8
From the Color list, choose Black.
9
Click to expand the Title section. From the Title type list, choose None.
10
Locate the Coloring and Style section. Clear the Color legend checkbox.
Deformation 1
1
Right-click Contour 1 and choose Deformation.
2
In the Settings window for Deformation, locate the Scale section.
3
Select the Scale factor checkbox. In the associated text field, type 1.
Surface 2
1
In the Model Builder window, right-click Housing Stress (with Rotor) and choose Surface.
2
In the Settings window for Surface, locate the Data section.
3
From the Dataset list, choose Study 1/Solution 1 (4) (sol1).
4
From the Solution parameters list, choose From parent.
5
Locate the Expression section. In the Expression text field, type mbd.vel.
6
Locate the Coloring and Style section. From the Color table list, choose AuroraBorealis.
Deformation 1
1
Right-click Surface 2 and choose Deformation.
2
In the Settings window for Deformation, locate the Scale section.
3
Select the Scale factor checkbox. In the associated text field, type 1.
Housing Stress (with Rotor)
1
In the Model Builder window, under Results click Housing Stress (with Rotor).
2
In the Settings window for 3D Plot Group, locate the Color Legend section.
3
From the Position list, choose Right double.
4
In the Housing Stress (with Rotor) toolbar, click  Plot.
5
Click the  Zoom Extents button in the Graphics toolbar.
Create animations to visualize the variation of magnetic flux density and stress distribution in different parts of the motor over time.
Animation 1
1
In the Results toolbar, click  Animation and choose Player.
2
In the Settings window for Animation, locate the Frames section.
3
In the Number of frames text field, type 50.
Animation 2
1
Right-click Animation 1 and choose Duplicate.
2
In the Settings window for Animation, locate the Scene section.
3
From the Subject list, choose Housing Stress.
4
Locate the Animation Editing section. From the Time selection list, choose Interpolated.
5
In the Times (s) text field, type range(0,0.01,0.7).
6
Locate the Frames section. In the Number of frames text field, type 70.
Animation 3
1
Right-click Animation 2 and choose Duplicate.
2
In the Settings window for Animation, locate the Scene section.
3
From the Subject list, choose Housing Stress (with Rotor).