Content uploaded by Jean LE BESNERAIS
Author content
All content in this area was uploaded by Jean LE BESNERAIS on Dec 02, 2016
Content may be subject to copyright.
Abstract—This article deals with the numerical simulation
methodology of electromagnetic vibrations in electrical machines.
Three methodologies are compared to evaluate vibrations due to
Maxwell forces in the Finite Element Analysis (FEA) software
Optistruct: the direct method based on Fourier series expansion of
nodal magnetic forces, and the unit rotating wave or tooth
excitation methods using vibration synthesis.
These three methods have been implemented in MANATEE
electromagnetic and vibroacoustic software in a fully automated
way. The different coupling strategies applied to a simplified
electrical machine give same results but with different calculation
times and Optistruct model complexities. It is concluded that the
tooth excitation method is the preferred one to carry variable
speed calculations or vibroacoustic optimization.
Index Terms—Finite Element Analysis, electromagnetic forces,
vibrations, multiphysic simulation, coupling, electrical machines,
TCL
I. I
NTRODUCTION
He assessment of structural-borne vibration and acoustic
noise (SBN) due to electromagnetic forces at an early
design stage is important in an increasing number of electrical
machine applications. Based on analytical or semi-analytical
multiphysics models, MANATEE
®
software [1] allows to carry
this task during early design iterations in a very short computing
time (typically a few seconds on a 2.6 GHz single CPU laptop).
However, its built-in semi-analytical vibroacoustic models
cannot account for complex transfer paths through stator frame
and footings. In this case, a structural Finite Element Analysis
(FEA) model of the electrical machine in its real environment
should be used for an accurate estimation of SBN. MANATEE
has been already coupled to the open source GetDP finite
element software [2], but it can hardly handle industrial CAD
geometries. A new coupling between MANATEE and ALTAIR
OptiStruct structural FEA software [3] has therefore been
developed.
The coupling between electromagnetic models and structural
FEA models is generally tedious due to the complexity of
M. Régniez, Q. Souron, P. Bonneel and J. Le Besnerais are with
EOMYS ENGINEERING, 121, rue de Chanzy, 59260 Lille-Hellemmes,
FRANCE (website: www.eomys.com, email: contact@eomys.com)
magnetic force distribution. Therefore, some automated
coupling tools must be developed to speed up the vibroacoustic
design process (definition of loads, resolution and post-
processing). This article presents and compares three different
methods of coupling using MANATEE
®
software [1] and
ALTAIR HyperWorks
®
(pre-processing and solver).
Electromagnetic forces distribution are calculated using fast
subdomain electromagnetic models of MANATEE, but they
could be computed with other software such as Flux
®
from
ALTAIR, Maxwell from ANSYS or Jmag from JSOL.
In a first part, the electromagnetic to structural mechanics
coupling principle and the three different methods are
explained. Then, the structural FEA model is described and a
global automated process including the definition of the
geometry and meshing, definition of dynamic loads and post-
processing is proposed. Finally, the different coupling methods
are validated, compared and discussed.
II. E
LECTROMAGNETIC TO MECHANICAL COUPLING
A. Coupling assumptions
The main assumptions of the electromagnetic to mechanical
coupling are the following ones:
1. A weak coupling is assumed between structural
mechanics and electromagnetics: airgap deflections do
not change the electromagnetic flux distribution and
resulting Maxwell stress;
2. Computed electromagnetic forces are supposed to be
only due to Maxwell stress: magnetostriction forces
are not considered;
3. Electromagnetic forces are calculated on the stator
lamination, and applied at the middle of tooth tips as
an equivalent force including radial and tangential
components.
The first assumption is easily fulfilled, as structural deflections
(a few micrometers) are much smaller than the magnetic airgap
width (typically 0.3 to 30 mm).
Numerical simulation of structural-borne
vibrations due to electromagnetic forces in
electric machines – coupling between Altair
Optistruct and Manatee software
M. Régniez, Q. Souron, P. Bonneel and J. Le Besnerais
T
The second one relies on EOMYS ENGINEERING experience,
during which the high noise and vibration level observed on
more than 50 electrical machines (both synchronous and
induction machines) from W to MW, from 5 to 100 000 rpm,
was analyzed, explained and reduced based on Maxwell forces
only.
The last assumption assumes that the electromagnetic
simulation results have been post processed to determine the
equivalent load vector to be applied on the tooth tips at each
time step (see
Fig. 1
). Indeed, some of flux may enter the tooth
edge or may vary a lot along the tooth tips.
Fig. 1: illustration of the equivalent load vector (bottom) of a set of nodal
forces (top) on a simplified geometry
This post processing may be simplified in applications with a
small airgap and a high number of teeth (for instance, high
torque induction motors used in naval propulsion) where the
Maxwell stress does not strongly vary along the tooth tip and
no significant flux enters through tooth edges.
The derivation of the local load vector from the airgap Maxwell
stress is not in the scope of this paper and authors may refer to
[4] on this topic. The projection method is assumed to be the
same for the three simulation workflows presented in this paper.
B. Magnetic vibration generation process
Magnetic forces responsible of electromagnetic torque
production and the so-called electromagnetic acoustic noise can
be expressed as resulting from an electromagnetic stress called
Maxwell stress, in
. Along the tangential direction it can
be written in polar coordinates as
(1)
where
is the permeability of vacuum,
and
are
respectively the radial and tangential magnetic fields. This
tangential stress averaged along the circumference give the
constant torque experienced by both stator and rotor.
A significant part of magnetic forces occurs along the radial
direction, pulling the stator towards the rotor and reciprocally:
(2)
Due to time and airgap periodicities, both components of the
airgap magnetic stress can be decomposed using Fourier
transform:
!"#$
% &
(3)
where refers to the time vector and
to the angle position on
the stator. The term
is the amplitude of the Fourier
transform of the airgap magnetic stress and &
is its phase.
According to (3), the overall magnetic stress can be
decomposed as an infinite sum of progressive rotating stress
waves of wavenumber and frequency #$
with $
the
fundamental electrical frequency. The term can also be called
circumferential wavenumber or spatial frequency and
corresponds to half the number of nodes of the force wave. The
mechanical rotation frequency of the force wave is #$
.
A radial force wave of wavenumber gives stator radial
deflections of wavenumber with same phase as shown on
Fig.
2
.
Fig. 2 Examples of radial forces of wavenumber 0, 1, 3 and 4
A tangential force wave of wavenumber ' gives a torsional
deflection of the yoke with same phase whereas a tangential
force wave of wavenumber ( ) gives a radial deflection of
the yoke of wavenumber with different phases due to the teeth
bending moment. This can be seen in
Fig. 3
.
Fig. 3 Effect of tangential forces of wavenumber 0, 1, 3 and 4
Pulsating waves are characterized by fixed nodes in the stator
frame, and can be decomposed as two waves propagating at the
same frequency but with opposite wavenumber (for
example ! and !), the frequency being taken
positive by convention.
The stator modes of a lamination can also be described using
axial and circumferential wavenumbers, and one can show that
there is a strong resonance if both the force spatial pattern
(wavenumber) and frequency matches with the modal shape
and natural frequency [5].
The overall magnetic vibration level due to Maxwell stress
rotating waves can be calculated by three ways:
• a direct coupling based on the application of nodal
magnetic forces “as is”;
• a rotating wave excitation method used for instance in
[4][7][9];
• a tooth excitation method used for instance in [9].
Some of the methods are based in the frequency domain using
frequency sweeps or in the time domain using impulse
response. This paper only considers frequency-based methods.
C. Principle of the direct method
This first method called “direct coupling method” aims at
directly applying the input data of forces in Fourier series, given
in equation (3), to the nodes of the structural models. Dynamic
loads with different magnitudes and phases at several
frequencies and at each node are applied.
On the contrary, the principle of the vibration synthesis is to
inject elementary electromagnetic forces in a structural model
to fully characterize it separately from the calculation of
electromagnetic loads. A synthetization algorithm calculates
the real vibration level using superposition principle.
One strong advantage of this method is to avoid redoing
structural calculation when changing only electromagnetic
loadings. For instance, the structural characterization does not
need to be rerun when doing variable speed simulations; if a
design optimization is carried on some variables that do not
significantly change the structural response of the stator and
frame (e.g. magnet pole shaping, current injection, rotor
skewing, etc), the structural characterization is only run once.
This leads to a decreasing of computation time for calculations
of new operating points or design iterations.
D. Principle of the rotating waves excitation method
As seen in (3)Erreur ! Source du renvoi introuvable.,
Maxwell magnetic forces on the stator can be decomposed in
Fourier series as a sum of rotating force waves characterized by
a rotation frequency and a wavenumber . The most natural
vibration synthesis method therefore consists in applying a
rotating excitation with a unit magnitude and a variable
frequency. Examples of elementary rotating waves are shown
in
Fig. 4
.
Fig. 4 Elementary rotating force wave under OptiStruct
®
: force injection on
the left, stator displacement on the right
The 2D spectrum of magnetic forces is generally sparse. In
particular, the wavenumbers present in induction machines are
proportional to GCD(Zs,Zr,2p) while they are proportional to
GCD(Zs,2p) in most permanent synchronous machines, where
GCD stands for the Greatest Common Divider. Besides that, the
larger is the wavenumber, the stiffer is the structural response;
depending on the application, the maximum wavenumber to be
considered might vary from 4 to 20. In healthy machines (no
eccentricity) with distributed windings, all the wavenumbers
are necessary even. This means that only a limited number of
magnetic wavenumbers can excite the stator and the rotor
structures. MANATEE software has a built-in analysis of the
theoretical content of magnetic force harmonics which allows
to identify the relevant rotating waves automatically, as well as
their associated frequency range. This allows to calculate the
useful elementary Frequency Response Functions.
To apply this principle on the FEA structural model, tooth tip
nodes are identified and stored in tables. Then, rotating force
waves are projected on these nodes. If a node is located at
angular position
*
, it sees the following forces of pulsation#+:
,
*
,
-
*
*
.+ % &
/01,
-
*
0
23
4
56789
:;
<
(4)
Using a single node at tooth tip the delay
*
is simply
= )!">?. This load is defined for both tangential and
radial direction. Illustration of tooth tip nodes and force vector
of rotating waves applied at the center node are represented in
Fig. 5
.
In this paper, the wavenumber is chosen varying from -4 to
4. Negative wavenumbers are necessary to account for
pulsating force waves and rotating direction of travelling
waves. Pulse-Width Modulation is known to modulate currents
and create rotating force waves of wavenumber 2p (number of
poles) travelling in opposite directions [11].
Fig. 5 Example of rotating force waves - force vector of the center node is
represented (from MANATEE)
E. Principle of the tooth excitation method
The second method consists in exciting the stator tooth by
tooth. It is a unit magnitude excitation at variable frequency.
In a stator with cyclic symmetry, all the tooth Frequency
Response Function (FRF) should be identical with a simple
phase shift – however, in a real geometry, the symmetry is
generally broken and all FRF need to be calculated.
After this calculation, the unit-magnitude tooth complex
FRFs are modulated and summed up using the superposition
principle.
F. Description of the coupling algorithm
The same algorithm applies for the three coupling methods
studied in this paper.
Fig. 6
sums up the necessary input for the electromechanical
coupling. It describes electromagnetic loads (calculated from
the radial and tangential time and angular flux distribution in
the middle of the airgap at each lamination axial slice), mesh
parameters (mainly number of layers for lamination and
number of nodes for tooth tip) and structural mechanics solver
parameters (number of modes and frequency range of the study,
as dynamic vibrations are computed using modal expansion
method).
Fig. 7
shows the principle of the coupling tool algorithm.
Nodal force calculation is first carried with MANATEE
®
software. Studied machines can be either designed with a
Computer-Aided Design (CAD) software and directly imported
or, optionally, created with a script using Tool Command
Language (TCL). TCL is an open source interpreted language,
which is known to have a simple and programmable syntax. It
can be used as a standalone application or embedded in
programs. The automation of ALTAIR
®
geometry, meshing,
and pre-processing is done with TCL language.
Fig. 6 Coupling tool inputs
Fig. 7 Coupling tool algorithm
The mechanical solver used is ALTAIR OptiStruct
®
. It is a
structural analysis solver for linear and nonlinear simulation
under static or dynamic loadings. It is based on finite-element
and multi-body dynamics technology, through advanced
analysis and optimization algorithms. In mechanical
engineering, and more specifically in Noise, Vibration and
Harshness (NVH) domain, OptiStruct
®
gives a fully featured
NVH solver containing computation of normal modes, complex
eigenvalues, frequency response analysis, random response,
transient and acoustic analysis.
Furthermore, pre-processing to create a readable file for
OptiStruct
®
uses Bulk Data syntax. This means the automated
pre-processing can easily be adapted to other mechanical
solvers using this syntax, as MSC.Nastran
®
for example. The
OptiStruct
®
process is shown in
Fig. 8
. The pre-processing is
realized with ALTAIR HyperMesh
®
and consists in definition
of nodes and elements of meshing, material properties, loads,
constraints and result files. The computation is solved with
ALTAIR Optistruct
®
. The results that can be visualized with
ALTAIR HyperView
®
are created in a binary file (H3d file).
Other output files are generated, for example Frequency
Response Function (FRF) files (obtained with HGFREQ
option), containing complex displacement of each node of the
mesh, induced by electromagnetic loads.
Fig. 8 OptiStruct
®
process
III. S
TRUCTURAL
FEA
M
ODEL
A. Parameters
1) Geometry
To illustrate these different coupling methods, the
electromagnetic forces have been computed in a Squirrel Cage
Induction Machine (SCIM) of 200 kW, designed for railway
application, using MANATEE
®
software. The choice has been
made to simplify the real machine’s geometry to compare more
quickly the different coupling methods (e.g. removal of slot
isthmus to have a lighter structural model). The main geometric
parameters of the studied machine’s stator core and slots
represented on
Fig. 9
are summarized in
T
ABLE
1
.
T
ABLE
1:
M
ACHINE
'
S GEOMETRY
STATOR CORE
@ABCDE
FG
HDDHI
J
K
L
M
36
FAHDE
NOPBDHDE
Q
KR
L
B
M
0.400
O@@DE
NOPBDHDE
Q
KCF
L
B
M
0.270
SD@THI
U
KHV
L
B
M
0.350
RFWD
IDOTIH
X
KR
L
B
M
0.035
STATOR SLOT
YSFH
IDOTIH
X
Z
L
B
M
0.030
YSFH
[ONHI
\
Z
L
B
M
0.012
Fig. 9 Studied stator lamination under MANATEE
2) Physical properties
T
ABLE
2
shows physical properties of the machine used for the
simulation. The material is assumed to be isotropic to ease the
set-up of the model. However, MANATEE allows to define
more realistic orthotropic materials in Optistruct.
T
ABLE
2:
M
ACHINE
'
S MECHANICAL PROPERTIES
QUANTITY VALUE
]FA@T
BFNASAK
^
L
_`P
M
210
`FOKKF
@
a
KbFDGGObOD@H
c
L
M
0.3
QD@KOHR
d
L
WT
B
e
M
7900
bPSbASPHDN
BPKK
B
L
WT
M
159.1
3) Meshing
The 3D meshing is built by an extrusion of the 2D mesh of
a tooth (see
Fig. 11
) using a Solid Map Mesh tool. Meshing is
done with hexahedral elements which are solid elements
extracted from 2D quad elements.
TABLE
shows the retained 2D meshing information of a
tooth, from meshing represented on
Fig. 11
.
TABLE 3: 2D MESHING INFORMATION (see Fig. 11)
QUANTITY VALUE
7ff7
g
7fh
4
7ff7
g
*ij
16
kf7
7fh
4
lfmj
7fh
15
lfmj
*ij
16
knlj
16
In order to apply the equivalent load vector, the mesh must
include a node in the middle of the tooth tip. To achieve this,
the number of elements on the tooth tip must be even.
T
ABLE
3
shows the meshing information of the final 3D model.
T
ABLE
3:
3D
M
ESHING INFORMATION
Finite element model data information
Total of Elements 221 184
Total of Degrees of Freedom
703 296
Finite element model data information
Total of Elements 221 184
Element Type Information
CHEXA Elements
213 696
CPENTA Elements
7 488
Fig. 9
and
Fig. 10
show the 2D lamination geometry and the final
3D meshed model.
Fig. 10 Stator structural mesh automatically built using MANATEE coupling
to Optistruct
B. Automated process
The process of creating a geometry, meshing, defining material
properties and applying dynamic loads with the different
methods is implemented using TCL language in ALTAIR
HyperMesh
®
environment. This process is described in the next
paragraphs.
1) Geometry and meshing
This first part deals with the steps for the automated creation
of the geometry and meshing. This part can be skipped if the
structural model of the lamination and frame is already
available, and if node sets at tooth tips have been already
defined. Industrial CAD file sometimes include unnecessary
details such as slot isthmus, slot wedges, bolt holes, airduct
spacers, etc. It is therefore interesting to be able to
automatically define an ideal lamination geometry and
integrate it back into the real CAD model to lighten the
structural model, especially for laminations several hundreds
of slots.
The first step involves the creation of the drawing of one
tooth in 2D. The second step consists in meshing this 2D tooth
with a certain number of nodes, given by the user, all along
the simplified tooth as it is shown in
Fig. 11
. More complex
tooth shapes are available in MANATEE.
Fig. 11 Parametrization of the meshing - 2D tooth
Then, the surface is extruded and the new volume is meshed
using the Solid Map Mesh function of ALTAIR HyperMesh
®
which allows to do a 3D meshing from a 2D meshed surface.
The number of mesh layers along the axial direction can also
be defined by the user with a variable named
knlj
.
Once the elementary tooth is created in 3D, 3D meshing of
the single tooth is then copied and rotated by o
pq
r
s
t
with >
the number of teeth. Finally, interface nodes have to be
merged as shown in
Fig. 12
in order to create one whole solid
and not >
independent parts.
Sets of nodes are then defined for different parts of the
stator. For example, nodes of the lamination bottom part are
brought in the Lam_Bottom node set. In the same way, nodes
of the lamination top part are brought in Lam_Top node set.
These sets are used to define boundary conditions of the
structure. Here, the stator is clamped at the bottom and free at
the top. A node set for the external yoke surface is also
defined. FRF are calculated at these nodes as electromagnetic
acoustic noise comes from the radial deflections of the
external yoke in this case. In this example, 186 node sets are
automatically defined.
Fig. 12 Interface merged nodes
Physical properties are then applied on the created structure.
All the steps described (creation of the geometry and meshing,
rotation, interface nodes merging, sets and physical properties
definition) are automated using one single TCL script.
The next step is the dynamic loads definition using the three
different methods described in part II. This part is also
automated using TCL.
2) Direct coupling method
To practically implement this method into ALTAIR
OptiStruct
®
, dynamic loads are implemented with several Load
Collectors.
Each wavenumber is first set and for each tooth node ii, two
TABLED1 Load Collectors are defined:
u
**
v w
vx x
xyx**
r
s
% z
v (named for
instance, C_Tooth_2_r_0 for the second tooth and ')
{
**
v w
vx |. x
xyx**
r
s
% z
}
v (named for
instance, D_Tooth_2_r_0 for the second tooth and ')
where w
v and z
v are respectively the amplitude and
phase of the injected wave. They actually corresponds to inputs
of the method and v is the frequency variable.
Then, the Bulk Data keyword RLOAD1 is used and allows to
define a frequency-dependent dynamic load$v, following
$v wLuvM % ={vM0
*5y~•
(5)
Fig. 13 represents an example of the use of RLOAD1.
Fig. 13 Example of RLOAD1 for tooth 14 and wavenumber r=-3
In this example, it is important to notice that the ID of the
excited tooth (Const_T_14 (16): radial or tangential), the
magnitude of the cosine (C_Tooth_14_r_2 oru
€
v), as well
as the magnitude of the sine (D_Tooth_14_r_2 or{
€
v) are
Load Collectors.
Therefore, the number of Load Collectors needed for the
direct method is given by
#
k•
‚ x #
x >
% >
% ƒ (7)
with, #
k•
„ Number of Load Collectors,
#
: Number of wavenumber,
>
„ Number of teeth and
ƒ„ Number of miscellaneous Load Collectors.
The variable ƒ contains various type of Load Collectors: the
damping table (TABDMP1, as a function of frequency), the
frequency table (FREQi), the simulation setup (EIGRL, $
…*
$
…n†
number of modes, etc.), the Load Collector containing all
dynamic loads (DLOAD), the Load Collector containing
boundary conditions. In the case of the direct coupling method,
ƒ ‡
In equation (7), there are #
x >
(x sine corresponding to
multiplication) cosine magnitudes, #
x >
sine magnitudes,
#
x >
RLOAD1 dynamic loads, >
constant ID representing
the support vector (radial and/or tangential) for each tooth and
ƒ various type of Load Collectors.
Then, those Load Collectors are gathered using DLOAD
Bulk Data keyword. DLOAD defines a dynamic loading
condition for frequency response analysis of load sets defined
via RLOAD1 or RLOAD2 (not used in this method) entries.
DLOAD is define into a Load Step definition. Also, Load Step
numbers depend on the method used.
For the direct coupling method, there is only one Load Step
containing all dynamic loads previously defined. #
k
is noted as
the number of load steps:
#
k
)(8)
3) Vibration synthesis rotating wave method
The implementation of the vibration synthesis with the
rotating wave method is very close to the direct coupling one.
The only difference is that a DLOAD Load Collector is created
to gather all the dynamic load RLOAD1 for each wavenumber
(#
DLOAD instead of one). Therefore, the number of Load
Collectors needed is:
#
k•
‚ x #
x >
% >
% ƒ
% #
(12)
The variable ƒ
contains the same Load Collectors as the direct
coupling method excluding the one containing all dynamic
loads (DLOAD). For the vibration synthesis by rotating wave,
ƒ
ˆ
As for previous methods, those Load Collectors are used
into series of Load Steps. In this method, the loading is done
with rotating wave. It is to say that there is as much as Load
Steps as wavenumbers wanted (including negative and positive
wavenumbers):
#
k
#
(13)
4) Vibration synthesis by tooth method
To achieve the tooth excitation method in ALTAIR
HyperMesh
®
, the Bulk Data keyword RLOAD2 is used. This
defines a frequency-dependent dynamic load ‰$ using:
1‰$< Šw x $0
*19‹85y‹7<
Œ(9)
where $ sets the amplitude of the load (constant input),
&$ is set to zero and o is the delay (set to zero).
With this method the same dynamic load is applied at each
tooth, independently of the wavenumber. The number of Load
Collectors is then strongly reduced as shown on equation (10):
#
k•
! x >
% ƒ
(10)
Here, ƒ
contains various type of Load Collectors: the
damping table (TABDMP1, as a function of frequency), the
frequency table (FREQi), the simulation setup
(EIGRL,$
…*
,$
…n†
, number of modes, etc.), the Load
Collector containing boundary conditions and the table
containing unit magnitude at each frequency (TABLED1). For
the vibration synthesis by tooth method, ƒ
‡ One can note
that the number of Load Collectors does not depend on the
number of wavenumbers.
In equation (10), there are >
Load Collectors representing
RLOAD2 dynamic load for each tooth and >
constant ID
representing the support vector (radial and/or tangential).
To practically implement this method into ALTAIR
OptiStruct
®
, one dynamic load is created for each tooth.
Fig. 14
represents an example of the use of RLOAD2 for this particular
method.
Fig. 14: Example of RLOAD2 for tooth 34
As for the previous method, those Load Collectors are used
in Load Steps assembly. The loading is done “by tooth”, it is to
say that the number of load steps #
k
equals the number of teeth:
#
k
>
(11)
IV. R
ESULTS AND COMPARISON
A. Results
Methods presented in II are tested with an automatized and
simplified model. Electromagnetic forces are computed with
MANATEE
®
software but can obviously be calculated with any
electromagnetic computation software, as JMAG
®
or Flux
®
.
Two particular electromagnetic forces are considered here:
- An electromagnetic force occurring at 550 Hz and for
!; this force is actually close to a structural mode of the
stator considered.
- An electromagnetic force of wavenumber ˆ at 2500
Hz; this force is not close to a mode of the structure.
In this example, residual vectors are not computed in order to
reduce computation time. Practically, in Altair HyperMesh
®
,
for each Load Step the RESVEC function is set to NO. In order
to counterbalance the fact that static modes are not taken into
account through the residual vector, the frequency range of the
analysis is enlarged to [0-5000] Hz. The maximum frequency
is set to twice the maximum exciting frequency.
Although this excitation is rather simple, all the frequency
response functions are calculated.
Fig. 15
shows the quadratic
sum in the plane of the lamination of the tooth excitation FRFs
for several teeth. It is possible to see on this figure that the
yoke’s response to tooth excitation has a resonance around 1267
Hz. It corresponds to the first bending mode of the stack; no
other mode can be seen in the FRF. It can be noted that the real
response does not depend on the excited tooth as expected due
to the cyclic symmetry.
Fig. 15: Quadratic sum of the frequency responses for the tooth excitation
method
Fig. 16
represents the same FRF for the rotating wave excitation
method for ! and !. Two peaks around 950 Hz and
1267 Hz are observed in the frequency response of the external
yoke. The first peak corresponds to the mode 2 of the yoke and
the second peak is the first bending mode of the stack. As
expected the FRF of r=2 and r=-2 wavenumbers are the same.
Fig. 16: Quadratic sum of the frequency responses for the rotating wave
method for ! and !
B. Comparison between the three methods
1) Optistruct model complexity
The quadratic velocity of the external yoke of the clamped-free
stator is directly linked to the acoustic radiation of the machine.
Then, the maximum quadratic velocity is calculated using FRF
results from ALTAIR Optistruct
®
for the three methods. It is
calculated at the frequencies 550 and 2500 Hz. The number of
Load Collectors needed for each coupling method is displayed
in the following table.
TABLE 1: NUMBER OF LOAD COLLECTORS
Direct
method
Vibration
synthesis for
tooth
method
Vibration
synthesis for
rotating wave
method
Number of teeth
>
‚•
Number of
wavenumber
#
!
ˆ
!
Not used
#
Ž
ˆ
ˆ
Number of
miscellaneous
Load Collectors
ƒ
‡
ƒ
‡
ƒ
ˆ
Total number of
load
collectors
•
•‘
257 77 1021
For this application, the number of Load Collectors is higher for
the third method because 9 wavenumbers are considered,
against 2 for the direct method. In the second method,
wavenumber is not considered as the excitation is done tooth by
tooth.
2) Vibration response
Fig. 17
represents the logarithmic maximum quadratic velocity
of the external yoke, calculated with the three methods, for the
two frequencies considered by the excitation. Results are the
same for the three methods. Differences observed in the linear
values are due to numerical error from FEA computation.
Fig. 17: Overall averaged vibration level – Method comparison
It can also be observed that the three methods give similar
results for the two electromagnetic forces considered, which are
respectively close to a structural mode at 550 Hz and far from a
mode at 2500 Hz.
3) Computation time
TABLE 2
lists computation times of solver and post-processing
for the three methods. Computation times for the solving step
are similar for the three methods. These computation times
could be decreased choosing lower number of slices
knlj
for the lamination.
The computation time relative to post-processing is longer for
the tooth excitation method. Indeed, 36 FRF corresponding to
each tooth excitation, are to be synthetized in Matlab
®
, against
1 FRF for the direct method and 9 FRF for the third method.
TABLE 2: COMPUTATION TIME COMPARISON
Methods Solver elapsed time
(h:min:s) Post-
processing Total
Direct coupling 00:11:18 00:00:09 00:11:27
Excitation by
tooth 00:25:54 00:03:19 00:29:13
Excitation by
rotating wave 00:16:08 00:00:58 00:17:06
In this example the rotating wave excitation method is faster
than the tooth excitation method. However, the resolution time
depends on several parameters such as:
• number of different wavenumbers;
• number of frequency harmonics;
• number of stator teeth.
The modal analysis of the stator, calculated in all three methods,
takes around 8 min of calculation. This means that if the
electromagnetic loads must be updated, one can reuse the modal
basis and only have 3 min of calculation per load update using
the direct method – taking the example of 100 operating points,
this would mean 300 min of calculation in total. The post
processing scripts of vibration synthesis are based on a matrix
multiplication and are therefore highly robust to the update of
electromagnetic load or increase of tooth number and
wavenumbers. This means that even if the modal basis of the
structure is stored using the direct method, one should use the
vibration synthesis method to significantly reduce calculation
time in case of variable speed calculations or electromagnetic
design optimizations loops (e.g. pole shaping, current
injection).
V. C
ONCLUSION
This article presents three different methods to couple
electromagnetic loadings with structural mechanics within
ALTAIR Optistruct. Those methods have been implemented in
MANATEE
®
as an automated process, and compared in terms
of accuracy, computation times and number of Load Collectors.
It is shown that the vibration synthesis method is more
efficient than the direct method when doing optimization or
variable speed analysis. Depending on the electrical machine,
the use of rotating excitation instead of tooth excitation might
be much faster.
Although the method has been applied on a simplified
geometry using MANATEE electromagnetic loads, it can take
other electromagnetic FEA results as an input and it can apply
to complex CAD geometries, considering both radial and
tangential loadings. As an example, it has been successfully
applied to this 240-slot stator of induction machine with
complex frame and footings:
Figure 18 : Full Optistruct model of a 240-slot stator connected to its base
frame
VI. F
UTURE
W
ORK
Future work aims at providing some experimental
measurements to compare Optistruct simulations to
experimental data and to the semi-analytical models of
MANATEE.
The method presented in this paper generalized to the excitation
of outer rotor permanent magnet machines will be also
presented.
R
EFERENCES
[1] MANATEE software, Magnetic Acoustic Noise Analysis Tool for
Electrical Engineering, Version 1.05, http://www.eomys.com, EOMYS
ENGINEERING, 2016
[2] GetDP, P. Dular and C. Geuzaine, University of Liège,
http://www.getdp.info
[3] ALTAIR HyperWorks software, www.altairhyperworks.com, Version
12.0, ALTAIR, 2016
[4] Boesing, M. (2013). Noise and Vibration Synthesis based on Force
Response Superposition. Fakultät für Elektrotechnik und
Informationstechnik der Rheinisch-Westfälischen Technischen
Hochschule (RWTH), Aachen.
[5] Soedel, W. (1993). Vibrations of shells and plates. Marcel Dekker.
[6] E. Devillers, J. Le Besnerais, Q. Souron and M. Hecquet,
“Characterization of acoustic noise and vibrations due to magnetic forces
in induction machines for transport applications using MANATEE
software”, ISMA 2016
[7] Roivainen, J. (2009). Unit-wave response-based modeling of
electromechanical noise and vibration of electrical machines. Helsinki
University of Technology.
[8] Boesing, M., Schoenen, T., Kasper, K. a, & Doncker, R. W. De. (2010).
Vibration Synthesis for Electrical Machines Based on Force Response
Superposition. IEEE Transactions on Magnetics, 46(8), 2986–2989.
[9] Akira Saito, Hiromitsu Suzuki, Masakatsu Kuroishi, Hideo Nakai,
Efficient forced vibration reanalysis method for rotating electric
machines, Journal of Sound and Vibration, Volume 334, 6 January 2015,
Pages 388-403
[10] Torregrossa, D., Fahimi, B., Member, S., Peyraut, F., & Miraoui, A.
(2012). Fast Computation of Electromagnetic Vibrations in Electrical
Machines via Field Reconstruction Method and Knowledge of
Mechanical Impulse Response, 59(2), 839–847.
[11] J. Le Besnerais, V. Lanfranchi, M. Hecquet and P. Brochet,
"Characterization and Reduction of Audible Magnetic Noise Due to
PWM Supply in Induction Machines," in IEEE Transactions on Industrial
Electronics, vol. 57, no. 4, pp. 1288-1295, April 2010
B
IOGRAPHIES
Margaux Régniez currently works in EOMYS ENGINEERING as
an R&D engineer on the analysis of acoustic noise and vibrations in electrical
systems. Following a Fd. Sc. specialized in Vibration, Acoustics and Signal
processing (Université du Maine, France) in 2008 and a M. Sc. specialized in
Vibration and Acoustics (Ecole Nationale Supérieure d’Ingénieurs du Mans) in
2012, she made an in industrial PhD thesis in Acoustics at LAUM laboratory
of the Université du Maine, on the vibrations damping of antenna’s reflectors
of satellites using micro-perforations, with the CNES and the company Thales
Alenia Space. She did a post-doc on friction-induced noise in electrical valve
placed in an exhaust pipe with Tenneco.
Quentin SOURON graduated in 2013 from the University of
Technology of Compiègne (UTC) in Mechanical Engineering, specialized in
acoustics and industrial vibrations. After an experience in an acoustic
engineering office where he was in charge of acoustics missions in several
domains (environment, buildings, industry), he made a research project in an
academic laboratory on new types of noise barriers made of phononic crystals.
In 2014, he joined EOMYS ENGINEERING as an R&D engineer in charge of
vibro-acoustic aspects of electrical systems (rotating machines and passive
components).
Pierre Bonneel graduated in 2014 from the “Ecole Nationale
Supérieure des Sciences Appliquées et de Technologie” of Lannion (ENSSAT)
in signal analysis and electronics. After a first experience in software
development in speech synthesis at Voxygen, he currently works in EOMYS
ENGINEERING as an R&D engineer in informatics. He is responsible of the
development and support of MANATEE software.
Jean Le Besnerais currently works in EOMYS ENGINEERING as
an R&D engineer on the analysis of acoustic noise and vibrations in electrical
systems. Following a M. Sc. specialized in Applied Mathematics (Ecole
Centrale Paris, France) in 2005, he made an industrial PhD thesis in Electrical
Engineering at the L2EP laboratory of the Ecole Centrale de Lille, North of
France, on the reduction of electromagnetic noise and vibrations in traction
induction machines with ALSTOM Transport. He worked from 2008 to 2013
as an engineer in the railway and wind industries (Alstom, Siemens Wind
power, Nenuphar Wind) on some multiphysic design and optimization tasks at
system level (thermics, acoustics noise and vibrations, electromagnetics,
structural mechanics and aerodynamics). In 2013, he founded EOMYS
ENGINEERING, a company providing applied research and development
services including modeling and simulation, scientific software development
and experimental measurements.
