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Magnetic Damping of Vibrating Conducting Solids
Introduction
When a conductive solid material moves through a static magnetic field, an eddy current density is induced. That induced eddy current density interacts with the static magnetic field and the result is a Lorentz force back on the solid that counteracts the motion. Therefore, a conducting solid that is vibrating in a static magnetic field experiences a structural damping.
This example computes the damping effect when a cantilever beam is harmonically excited across a range of frequencies and placed in a strong magnetic field. The approach presented here assumes that the relative magnitude of the structural displacements are small, that the material has isotropic and linear properties, and that the damping Lorentz force can be computed from the static magnetic field and the motion induced AC eddy current density. Second order effects arising from the AC magnetic field generated by the eddy currents are not included in the computation. The AC magnetic field is also computed and found to be 2‑3 orders of magnitude smaller than the DC magnetic field.
Figure 1: A vibrating beam next to a current carrying wire experiences magnetic damping.
Model Definition
For a solid material experiencing a time-harmonic forced excitation, the displacement field is of the form
which can also be written in the frequency domain as a phasor:
Thus, the velocity field is given by
Next, consider the effect of a spatially non-uniform but static magnetic flux density, BDC(r). Under the assumption that the local displacements are small enough so that each moving point in the solid sees only the magnetic flux density in the undeformed state, the velocity induced current density is given by
where σ is the material conductivity. The resulting total AC current density is however different as the metallic cantilever beam is inductive, that is it exhibits skin effect. Thus a second, frequency domain, magnetodynamic, problem has to be solved in order to compute the AC current density.
The body forces experienced by a current-carrying domain moving through a magnetic field are then given by the cross product between the AC current density and the static magnetic flux density.
These body forces are then applied to the frequency domain structural mechanics problem and act as a damping force on the system.
The application first computes the static magnetic field due to a current-carrying wire which is next to an aluminum beam. In the second solution step, the beam experiences a forced harmonic vibration and the resulting mechanical beam displacement field and AC current density are computed, yielding also the damping electromagnetic force. The second step is set as a Frequency Domain Perturbation study. The strength of the magnetic field is varied, and the effect of the magnetic damping on the response of the system is observed.
Results and Discussion
Figure 2 shows the magnetic flux density computed for the structure at rest. Figure 3 displays the magnitude of the displacement of the tip of the beam versus excitation frequency for two different magnetic field intensities for the frequency-domain structural dynamics problem. The magnetic field provides significant additional damping. Figure 4 shows a snapshot of the induced eddy current distribution in the beam.
Figure 2: The magnetic field around a current carrying wire.
Figure 3: Displacement of the tip of the beam versus excitation frequency for differing magnetic field strengths.
Figure 4: The AC current distribution.
Notes About the COMSOL Implementation
Solve this application with two physics interfaces — the Magnetic Fields and Solid Mechanics interfaces. Use a Stationary study for the first Magnetic Fields interface and a Frequency Domain Perturbation study for the Magnetic Fields and Solid Mechanics interfaces. The coupling between the two interfaces is automatically considered by using the multiphysics coupling feature referred to as Lorentz Coupling.
Application Library path: ACDC_Module/Motors_and_Actuators/magnetic_damping
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  3D.
2
In the Select Physics tree, select AC/DC>Electromagnetic Fields>Magnetic Fields (mf).
3
Click Add.
4
In the Select Physics tree, select Structural Mechanics>Solid Mechanics (solid).
5
Click Add.
6
Click  Study.
7
In the Select Study tree, select Empty Study.
8
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
In the table, enter the following settings:
Name
Expression
Value
Description
sigma
3.774e7[S/m]
3.774E7 S/m
Material conductivity
a_c
5e5[A]
5E5 A
Applied current on the wire
The Applied current will be used as a sweep parameter.
Geometry 1
1
In the Model Builder window, under Component 1 (comp1) click Geometry 1.
2
In the Geometry toolbar, click Block to create a block for the simulation domain. Leave the default block size.
3
In the Geometry toolbar, click Block again to create a block for the cantilever beam.
Block 2 (blk2)
1
In the Model Builder window, under Component 1 (comp1)>Geometry 1 click Block 2 (blk2).
2
In the Settings window for Block, locate the Size and Shape section.
3
In the Width text field, type 0.9.
4
In the Depth text field, type 0.025.
5
In the Height text field, type 0.1.
6
Locate the Position section. In the y text field, type 0.575.
7
In the z text field, type 0.45.
Finally, add a cylinder for the wire generating the static magnetic field.
Cylinder 1 (cyl1)
1
In the Geometry toolbar, click  Cylinder.
2
In the Settings window for Cylinder, locate the Size and Shape section.
3
In the Radius text field, type 0.05.
4
Locate the Position section. In the x text field, type 0.5.
5
In the y text field, type 0.5.
6
Click  Build All Objects.
7
Click the  Wireframe Rendering button in the Graphics toolbar.
Add variables for the induced current density and body force on the cantilever beam.
Magnetic Fields (mf)
Coil 1
1
In the Model Builder window, under Component 1 (comp1) right-click Magnetic Fields (mf) and choose the domain setting Coil.
2
Select Domain 3 only.
3
In the Settings window for Coil, locate the Coil section.
4
In the Icoil text field, type a_c.
5
In the Model Builder window, expand the Coil 1 node.
Input 1
1
In the Model Builder window, expand the Component 1 (comp1)>Magnetic Fields (mf)>Coil 1>Geometry Analysis 1 node, then click Input 1.
2
Select Boundary 14 only.
Geometry Analysis 1
In the Model Builder window, click Geometry Analysis 1.
Output 1
1
In the Physics toolbar, click  Attributes and choose Output.
2
Select Boundary 13 only.
Solid Mechanics (solid)
The Solid Mechanics interface is active only on the cantilever beam.
1
In the Model Builder window, under Component 1 (comp1) click Solid Mechanics (solid).
2
Select Domain 2 only.
Linear Elastic Material 1
Add a damping factor on Linear Elastic Material Model 1.
1
In the Model Builder window, under Component 1 (comp1)>Solid Mechanics (solid) click Linear Elastic Material 1.
Damping 1
1
In the Physics toolbar, click  Attributes and choose Damping.
2
In the Settings window for Damping, locate the Damping Settings section.
3
From the Damping type list, choose Isotropic loss factor.
4
From the ηs list, choose User defined. In the associated text field, type 0.1.
Fixed Constraint 1
1
In the Physics toolbar, click  Boundaries and choose Fixed Constraint.
2
Select Boundary 5 only.
Boundary Load 1
1
In the Physics toolbar, click  Boundaries and choose Boundary Load.
2
Select Boundary 17 only.
3
In the Settings window for Boundary Load, locate the Force section.
4
Specify the FA vector as
0
x
1e4
y
0
z
5
Right-click Boundary Load 1 and choose Harmonic Perturbation.
Add Material
1
In the Home toolbar, click  Add Material to open the Add Material window.
2
Go to the Add Material window.
3
In the tree, select Built-in>Aluminum.
4
Click Add to Component in the window toolbar.
5
In the tree, select Built-in>Air.
6
Click Add to Component in the window toolbar.
7
In the Home toolbar, click  Add Material to close the Add Material window.
Materials
Aluminum (mat1)
1
In the Model Builder window, under Component 1 (comp1)>Materials click Aluminum (mat1).
2
Select Domains 2 and 3 only.
Air (mat2)
1
In the Model Builder window, click Air (mat2).
2
Select Domain 1 only.
Some artificial conductivity is needed for numerical stability of the AC magnetic simulation.
3
In the Settings window for Material, locate the Material Contents section.
4
In the table, enter the following settings:
Property
Variable
Value
Unit
Property group
Electrical conductivity
sigma_iso ; sigmaii = sigma_iso, sigmaij = 0
10[S/m]
S/m
Basic
Multiphysics
Lorentz Coupling 1 (ltzc1)
In the Physics toolbar, click  Multiphysics Couplings and choose Domain>Lorentz Coupling.
Mesh 1
1
In the Model Builder window, under Component 1 (comp1) click Mesh 1.
2
In the Settings window for Mesh, locate the Physics-Controlled Mesh section.
3
From the Element size list, choose Coarser.
Study 1
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
In the table, enter the following settings:
Parameter name
Parameter value list
Parameter unit
a_c (Applied current on the wire)
0[A] 500000[A]
A
Add a Coil Geometry Analysis study step as the first step to compute the direction of the current in the wire.
Coil Geometry Analysis
In the Study toolbar, click  Study Steps and choose Other>Coil Geometry Analysis.
Stationary
1
In the Study toolbar, click  Study Steps and choose Stationary>Stationary.
2
In the Settings window for Stationary, locate the Physics and Variables Selection section.
3
In the table, clear the Solve for check box for Solid Mechanics (solid).
Frequency Domain Perturbation
1
In the Study toolbar, click  Study Steps and choose Frequency Domain>Frequency Domain Perturbation.
2
In the Settings window for Frequency Domain Perturbation, locate the Study Settings section.
3
In the Frequencies text field, type range(5,0.5,50).
Solution 1 (sol1)
1
In the Study toolbar, click  Show Default Solver.
Some adjustments to the default solver settings will improve the performance.
2
In the Model Builder window, expand the Solution 1 (sol1) node.
3
In the Model Builder window, expand the Study 1>Solver Configurations>Solution 1 (sol1)>Stationary Solver 3 node, then click Fully Coupled 1.
4
In the Settings window for Fully Coupled, locate the General section.
5
From the Linear solver list, choose Direct.
6
In the Model Builder window, click Direct.
7
In the Settings window for Direct, locate the General section.
8
From the Solver list, choose PARDISO.
9
In the Study toolbar, click  Compute.
Results
Study 1/Parametric Solutions 1 (5) (sol4)
1
In the Model Builder window, expand the Results>Datasets node.
2
Right-click Results>Datasets>Study 1/Parametric Solutions 1 (sol4) and choose Duplicate.
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 Domain 2 only.
DC Magnetic Flux Density Norm
The first default plot group shows the magnetic field around a current carrying wire; compare with Figure 2. Give it a more descriptive name.
1
In the Model Builder window, under Results click Magnetic Flux Density Norm (mf).
2
In the Settings window for 3D Plot Group, type DC Magnetic Flux Density Norm in the Label text field.
Multislice 1
1
In the Model Builder window, expand the DC Magnetic Flux Density Norm node, then click Multislice 1.
2
In the Settings window for Multislice, locate the Expression section.
3
From the Expression evaluated for list, choose Static solution.
4
In the DC Magnetic Flux Density Norm toolbar, click  Plot.
5
Click the  Go to Default View button in the Graphics toolbar.
AC Magnetic Flux Density Norm
1
In the Model Builder window, right-click DC Magnetic Flux Density Norm and choose Duplicate.
2
In the Settings window for 3D Plot Group, type AC Magnetic Flux Density Norm in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 1/Parametric Solutions 1 (5) (sol4).
The second plot group shows the AC magnetic flux density. Improve it by plotting the data in the cantilever beam only.
Multislice 1
1
In the Model Builder window, expand the AC Magnetic Flux Density Norm node, then click Multislice 1.
2
In the Settings window for Multislice, locate the Expression section.
3
From the Expression evaluated for list, choose Harmonic perturbation.
4
In the AC Magnetic Flux Density Norm toolbar, click  Plot.
5
Click the  Zoom Extents button in the Graphics toolbar.
The plot now shows the magnitude of the AC magnetic flux density in the beam only.
Stress (solid)
The third plot shows the von Mises stress.
AC Electric Current Density
1
In the Model Builder window, right-click DC Magnetic Flux Density Norm and choose Duplicate.
Add a plot of the AC currents in the beam.
2
In the Settings window for 3D Plot Group, type AC Electric Current Density in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 1/Parametric Solutions 1 (5) (sol4).
Multislice 1
1
In the Model Builder window, expand the AC Electric Current Density node, then click Multislice 1.
2
In the Settings window for Multislice, locate the Expression section.
3
From the Expression evaluated for list, choose Harmonic perturbation.
4
Click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1)>Magnetic Fields>Currents and charge>mf.normJ - Current density norm - A/m².
Streamline 4
In the Model Builder window, right-click AC Electric Current Density and choose Streamline.
Streamline Surface 1
1
In the Settings window for Streamline Surface, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1)>Magnetic Fields>Currents and charge>mf.Jx,...,mf.Jz - Current density (spatial frame).
2
Locate the Streamline Positioning section. From the Positioning list, choose Starting-point controlled.
3
In the AC Electric Current Density toolbar, click  Plot.
4
Click the  Go to Default View button in the Graphics toolbar.
The AC eddy currents circulate within the beam.
Finish by plotting the RMS displacement of the tip of the beam as a function of frequency (Figure 3).
RMS Displacement vs Frequency
1
In the Home toolbar, click  Add Plot Group and choose 1D Plot Group.
2
In the Settings window for 1D Plot Group, type RMS Displacement vs Frequency in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 1/Parametric Solutions 1 (4) (sol4).
Point Graph 1
1
Right-click RMS Displacement vs Frequency and choose Point Graph.
2
Select Point 18 only.
3
In the Settings window for Point Graph, click Replace Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Component 1 (comp1)>Solid Mechanics>Displacement>solid.disp_rms - Displacement, RMS - m.
4
Click to expand the Legends section. Select the Show legends check box.
5
In the RMS Displacement vs Frequency toolbar, click  Plot.
6
Click the  y-Axis Log Scale button in the Graphics toolbar.
Compare the resulting plot with that shown in Figure 3.