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SUBMISSION NUMBER (EB2022-TSD-009)
A Structural Dynamics Modification Strategy based on Expanded Squeal
Operational Deflection Shapes
*Guillaume Martin1 , Etienne Balmes1,2 , Thierry Chancelier3 , Sylvain Thouviot3 , Rémi Lemaire3
1SDTools
44 Rue Vergniaud, 75013 Paris, France (E-mail: [martin;balmes]@mail.sdtools.com)
2Processes and Engineering in Mechanics and Materials laboratory (PIMM) ENSAM, CNRS, CNAM, HESAM Université
151 Boulevard de l’Hôpital, 75013 Paris, France (E-mail: balmes@ensam.eu)
3Hitachi Astemo France S.A.S.
126 rue de Stalingrad, 93700 Drancy, France (E-mail: [thierry.chancelier;sylvain.thouviot;remi.lemaire]@hitachiastemo.com)
DOI (FISITA USE ONLY)
ABSTRACT: To analyze brake squeal, measurements are performed to extract Operational Deflection Shapes (ODS)
characteristic of the limit cycle. The advantage of this strategy is that the real system behavior is captured, but measurements
suffer from a low spatial distribution and hidden surfaces, so that interpretation is sometimes difficult. It is even more difficult
to propose system modifications from test alone. Historical Structural Dynamics Modification (SDM) techniques need mass
normalized shapes which is not available from an ODS measurement. Furthermore, it is very difficult to translate mass,
damping or stiffness modification between sensors into physical modifications of the real system. On the model side, FEM
methodology gives access to fine geometric details, continuous field over the whole system. Simple simulation of the impact
of modifications is possible, one typical strategy for squeal being to avoid unstable poles. Nevertheless, to ensure accurate
predictions, test/FEM correlation must be checked and model updating may be necessary despite high cost and absence of
guarantee on results. To combine both strategies, expansion techniques seek to estimate the ODS on all FEM DOF using a
multi-objective optimization combining test and model errors. The high number of sensors compensates for modeling errors,
while allowing imperfect test. The Minimum Dynamics Residual Expansion (MDRE) method used here, ensures that the
complex ODS expanded shapes are close enough to the measured motion but have smooth, physically representative, stress
field, which is mandatory for further analysis. From the expanded ODS and using the model, the two underlying real shapes
are mass-orthonormalized and stiffness-orthogonalized resulting in a reduced modal model with two modes defined at all
model DOFs. Sensitivity analysis is then possible and the impact of thickness modifications on frequencies is estimated. This
provides a novel structural modification strategy where the parameters are thickness distributions and the objective is to
separate the frequencies associated with the two shapes found by expansion of the experimental ODS.
The methodology will be illustrated for a recent disk brake test and model.
KEY WORDS: Structural Dynamics Modifications, Operational Deflection Shapes, Minimum Dynamic Residual Expansion,
Sensitivity analysis, Squeal
1. INTRODUCTION
To analyze brake squeal, measurements are performed to extract
two Operational Deflection Shape (ODS) characteristic of the limit
cycle. These shapes have the advantage to capture the real system
behavior but suffer from low spatial distribution of sensors.
Historical Structural Dynamics Modification (SDM) techniques
allow to work directly on test data need but scaled modes are
needed, which is not available from an ODS. Moreover, classical
SDM strategies consider modifications that are point masses or
relative springs/dampers between sensors that are very difficult to
translate into physical modifications of the real system.
On the model side, the Finite Element Method (FEM) gives access
to fine geometric details, continuous field over the whole system.
Simple simulation of the impact of modifications is possible, one
typical strategy for squeal handling being to separate modal
frequencies to avoid interactions involved in squeal. However, to
New modes
Topology
correlation
ODS
FEM
Minimum
Dynamics
Residual
Expansion
Modal model
defined at model
DOFs
Thickness
sensitivity map
Thickness
distribution map
Brake
design
constraints
Figure 1 SDM after expansion procedure
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ensure accurate predictions, test/FEM correlation must be checked
and model updating may be necessary despite high cost and
absence of guarantee on results.
To combine the fine spatial description of FEM with the ability to
work directly on test data provided by standard SDM strategies, the
procedure summarized in Figure 1 has been developed. It relies on
three main steps:
- expand ODS (defined at sensors) to estimate shapes at all model
DOFs using the Minimum Dynamics Residual Expansion
(MDRE)
- build an approximate modal model with the ODS shapes at the
limit cycle frequency
- use a novel SDM strategy, where the parameters are thickness
distributions on the model surface, to predict mode frequency
evolutions
Section 2 first presents the test case used as illustration throughout
this paper: test geometry, measured ODS, FEM and topology
correlation (model observation through sensors). ODS expansion
and modal model estimation is then explained in section 3.
Section 4 details the new SDM strategy and a consistency check is
performed. The full procedure is finally applied in section 5 to the
test case: a thickness distribution map is proposed that should
reduce the squeal occurrence.
2. TEST CASE DESCRIPTION
The test case used is a Hitachi Astemo disc brake presenting a
squeal occurrence near 4050Hz. To understand the behavior, an
ODS measurement is performed using a 3D Scanning Laser
Doppler Vibrometer (3D-SLDV). The test wireframe composed of
1293 sensors (431 points) is superposed over the FEM in Figure 2.
Individual sensors are represented as red arrows on figure left. The
gap between each wireframe node and the closest model surface is
shown on figure right to evaluate the topology correlation.
Figure 2 Test case topology correlation: sensors as red
arrows (left) and node to surface distance map (right)
Squeal is reproduced on a dyno test bench with constant brake
pressure and constant disc rotational speed. When squeal occurs,
the 3D-SLDV sequentially scans all points. Using reference
accelerometers placed on the brake system, the sequential
measurements are then combined using the procedure developed in
[1] which deals with the variation of the limit cycle in frequency,
amplitude and shape due to the wheel spin. A complex shape is
obtained, which contains the two main real shapes that interact
throughout the limit cycle. This complex shape is shown in Figure
5 left, highlighting a classical lobe motion of the disc and a
deformation localized at the center and right part of the bracket.
On the model side, a non-linear static analysis is first performed to
obtain the contact stiffness distributions engendered by braking
pressure and torque. The model is linearized around this static state,
and a Complex Eigenvalue Analysis is performed [2]. The
correlation between complex mode shapes and the ODS is
evaluated using the Modal Assurance Criterion [3]. At this stage,
the best scenario is when there is good agreement between one
numerical mode close to the squeal frequency and the ODS. The
model is thus deemed representative and used with confidence to
propose squeal countermeasures.
Often, the correlation is not satisfying, which comes either from
test, model or both. The model may not be representative enough
of the true system dynamics. The test may lead to a limit cycle
where non-linearities are high enough to modify shapes notably
from a combination of two nearby modes that are the classical
origin of instabilities [4], [5]. Other test error sources such as noise,
bad sensor location,… can also contribute the low correlation [6].
For this case, with no satisfying correlation between test and
analysis, the MDRE method presented in the next section gives a
appropriate methodology to estimate the model response from
measurement and a not fully wrong model.
3. EXPANSION AND MODAL MODEL
One difficulty with vibration measurements is the low spatial
resolution. Even for 3D-SLDV sensors, motion of hidden or
unreachable parts cannot be measured. The first objective of
expansion methods is the estimation of the motion at all DOFs of a
model from an experimental shape defined at sensors. Several
expansion methods have been developed such as static [7],
dynamic [8], Error in Constitutive Relation [9] and MDRE [10].
The two last methods introduce the idea that both modeling and test
errors can be defined and combined into a multi-objective cost
function.
Section 3.1 quickly reminds the MDRE theory. It is applied to the
test case in section 3.2 with a fine tuning of the balance between
modeling and test error to ensure that the expanded ODS is close
enough to the measured motion but has smooth stress field. This is
mandatory to obtain a physically representative modal model in
section 3.3 on which the proposed novel SDM strategy can be
applied.
3.1. Theory of Minimum Dynamic Residual Expansion
MDRE combines a modeling error and a test error . The
two errors depend on the expanded shape defined at all
model DOF which corresponds to the shape minimizing the multi-
objective cost function
(1)
where the weighting coefficient leads to verification of model
equations for low values and exact observation of the test for high
values.
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To measure the difference between the expanded shape and
the test measurement, the model is first observed at sensors
using an observation matrix built from the test/FEM
superposition. The test error is then defined by
(2)
To evaluate how the model is forced to follow the test, dynamic
residual loads and the associated energy norm define the
modeling error
(3)
with , , the stiffness and mass matrices and the expansion
pulsation. One notes that is equivalent to say that
is a normal mode and the associated modal frequency.
More details on the implementation, norm definitions and
examples can be found in [1], [11]. One just reminds here that
model reduction is mandatory to solve the expansion in an
acceptable time. This reduction is performed on the basis composed
by the normal mode shapes and the enrichment with static
loads at sensors
. The reduced coordinates can thus be split
in two
(4)
with the part linked to the model modes and the part linked
to the enrichment at sensors.
The test case used has 1.7 million model DOFs and 1293
vibrometer sensors. The offline phase consisting in computing 100
normal modes and the enrichment with 1293 static loads at sensors
takes about an hour. Computing the expanded shape associated to
a single only takes a second.
3.2. Expansion result
The ODS shown in Figure 5 left is expanded using MDRE for
several values of the weight equally spaced on a logarithmic
scale. Increasing gives more and more weight to the test error on
the multi-objective cost function (1): the test error decreases up to
0 while the modeling increases to its maximum. To display
modeling and test error evolution whose values are not in the same
range, the relative error is defined
(5)
and the relative model error is split in a part linked to the normal
modes and a part linked to the sensor enrichment using (4).
Figure 3 shows the evolution of the relative test and modeling
errors with . An interesting value is found at . For lower
values, modeling error was mainly due to the modal part. Higher
values lead to a quick increase of the modeling error due to the
enrichment part, which can occur because noise is transferred from
the test error to the modeling error or because dynamics are missing
in the normal mode part to better match the measurement.
Figure 3 Evolution with of relative test and modeling
errors
Display of the spatial repartition of modeling and test errors is
helpful to go deeper in the analysis of sources of bad correlation as
can be seen in [1], [11], [12] which is helpful if model updating
must be performed. This is not the main topic for this paper so it
will not be developed here. Authors want to mention that figures
similar to Figure 3 were wrong in previous papers [9], [10], model
errors linked to mode and enrichment parts being inverted: this is
corrected here with a more convincing interpretation.
The choice of has a high influence on the stress field of the
expanded shape. In Figure 4 right, a too high value of leads to
stress concentration at sensors which is clearly not physical and
thus cannot be used for further model error localization or
sensitivity studies.
Figure 4 Stress field of expanded shape for (left)
and (right)
is thus finally chosen and is quite satisfying: the relative
test error of 15% is reasonable since the test shape contains noise
and the relative modeling error is very low at about 5%. This
answers the question raised at the end of section 2: despite the
absence of correlation between mode shapes and ODS, relatively
small residual loads are needed to reproduce the ODS so that the
model accuracy is acceptable. Figure 5 shows the measured ODS
on the left and the retained expanded shape on the right.
Figure 5 Squeal ODS (left) and expanded shape with
(right)
When the modeling error is too high, model updating is necessary
to gain confidence in the prediction of system evolution in response
to modifications. Even if model updating is always useful, it often
takes long time and there is no guarantee to reach satisfying results.
In the meantime, the expansion gives access to a shape defined at
all model DOFs with good agreement with the real system. This
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can be exploited to build a simple modal model on which SDM can
be applied.
3.3. Modal model from ODS expansion
To perform SDM, a modal model is needed. Standard modal
scaling uses collocated measurements (input load and output
motion at the same location and direction) which is not available
for ODS because the system excitation is neither controlled nor
measured. To still obtain a modal model, the real and imaginary
parts of the complex expanded ODS are extracted and the two are
mass-orthonormalized and stiffness-orthogonalized with the
nominal model. This leads to reduced mass matrix, stiffness matrix
and basis
;
(6)
Figure 6 shows these two shapes for the illustrated application.
Because these are obtained from expansion allowing modeling
error, the two corresponding frequencies and
are higher than the expansion frequency (4050Hz): the first
shape is at 4119Hz ( ) and the second at 4217Hz
( ).
Figure 6 Modal model shapes built from ODS expansion
To correct these frequency shifts, diagonal values of the reduced
stiffness matrix are modified to correspond to the expected limit
cycle frequency. Reduced matrices actually used for further SDM
are thus
;
(7)
with
4. STRUCTURAL DYNAMICS MODIFICATION
From a modal model, SDM techniques allow to predict the
evolution of modes in response to modifications. SDM theory in
section 4.1 and limitations of the historical use of SDM lead to the
proposition of a new strategy in section 4.2 which relies on
sensitivity analysis to thickness modifications. Consistency is
finally checked in section 4.3.
4.1. SDM theory
The objective of SDM is to estimate the influence of a structural
modification (typically modification of mass and stiffness) from a
second order reduced modal model
(8)
with ; and the
reduction basis defined at specific output locations.
Even without a full knowledge of , it is possible to introduce
modifications of
- mass at output locations
- stiffness between outputs
With the hypothesis that the reduced basis subspace remains
representative enough, these modifications are projected on this
basis and introduced in the model
(9)
with and
New modes associated to this modified reduced model allow to
evaluate frequency shifts and eventually shape evolution
(combination of reduced basis vectors).
Historically [13], [14], the modal model is built from experimental
modal analysis with the basis being composed of identified
mode shapes defined at sensors. These modes are normalized using
colocalized transfers so that and the
diagonal matrix of squared mode pulsations. The main drawback
of this strategy is that model outputs are sensors and thus mass and
stiffness modifications are only possible at sensor locations:
add/remove mass at sensor locations or add/remove stiffness
between sensors. It is then very difficult to interpret these very
coarse modifications as real physical modifications of the real
system.
To allow more realistic modifications, the use of expansion
technique was proposed in [15] : the reduction basis is extended
from outputs at sensors to outputs at all model DOFs. An example
shown in Figure 7 left highlights that after expansion it is possible
to connect a stiffener whereas interface nodes do not correspond to
measured points. After expansion, we have a direct access to the
mass and stiffness assembly matrices corresponding to each
element: mass and stiffness modifications in equation
(9) corresponding to the removal of some elements is easily tested.
Figure 7 Modification adding stiffener [15] and topology
optimization example [16]
Another objective is to obtain realistic modifications. For this
purpose, the first thought would be addition/removal of elements
as in topology optimization techniques [16] illustrated in Figure 7
right. While quite precise spatially, this strategy still ends up with
a rather coarse description of the final surface and is limited to
element removal only. A new SDM strategy that tackles both issues
is thus proposed in the next section.
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4.2. A SDM using thickness modification
The proposed SDM strategy is based on the sensitivity analysis of
the reduced model modal frequencies to thickness modification at
the component surfaces. Without entering into the details which
can be found in [17], the sensitivity of a modal frequency to a
parameter is related the sensitivity of the mass and stiffness
matrices to this parameter projected on the mode
(10)
with and the pulsation and shape of mode .
Rearrangement of this equation gives an energy interpretation: the
evolution of the squared modal frequency is equal to twice the
difference between the strain and kinetic energy evolution.
(11)
The parameter for this study is a small thickness increase at the
component surfaces. A thin layer of finite elements is built by
extruding the external surface of the structure in the normal
direction. This is illustrated with the test case in which thickness
modifications are evaluated for the caliper and bracket components
shown on Figure 8 left. The map of normals to the external surface
shown on top right is used to extrude the thin layer shown at bottom
right (thickness is increased to 1mm on the image for visualization
purpose, but the actual thickness for computation is 0.01mm)
Figure 8 Caliper yellow and bracket green component
(left), normal map (top right) and thin layer (bottom
right)
On this thin layer, motion is only known at the initial surface (set
of nodes ). To estimate the motion of extruded nodes (set of nodes
), dynamic expansion is performed. Motion known at DOFs
is enforced and motion sought at DOFs is computed by
allowing external forces at known DOFs only
(12)
(13)
with the dynamic stiffness
As illustration, Figure 9 shows the estimated motion of the thin
layer for a mode of interest.
Figure 9 Dynamic expansion of thickness layer
Strain and kinetic energy are then computed at Gauss points of the
thickness layer and energy density maps on the surface are obtained
as shown in Figure 10. The difference between strain and kinetic
energy density maps gives the thickness sensitivity map (or “shift”
energy density map): it is directly related to the frequency shift
from the sensitivity equation (11).
Figure 10 Energy density maps: strain energy (top left),
kinetic energy (top right) and shift energy (bottom)
In areas with positive shift energy density, thickness increase adds
more strain energy than kinetic energy thus increasing the modal
frequency. To also increase the modal frequency in areas with
negative shift energy, the extrusion should be in a negative
direction corresponding to removing material.
From the thickness sensitivity map, a thickness distribution map is
derived. Note that this step is generally not straightforward as many
constraints must be respected such as
- Do not deform functional surfaces
- Avoid removing too much material to ensure brake
performances and durability
- Avoid adding too much material to limit material costs and
brake weight
- Keep smooth surface curvature
- …
For the sake of simplicity, these difficulties are not addressed in
this paper. The thickness distribution map is simply the thickness
sensitivity map with maximum amplitude normalized to one, thus
leading to a maximum thickness modification of 1mm. Because the
thickness distribution map can be positive or negative, special
attention must be paid to elements whose nodes are not all
positively or negatively extruded. Figure 11 left shows three
scenarii for each model element:
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- Red element: all nodes are associated to a positive thickness
value (material addition). Extrude outside the surface to get
positive volume and count energy positively
- Blue element: all nodes are associated to a negative thickness
value (material removal). Extrude inside the surface to get
positive volume but count energy negatively
- Grey element: nodes are associated to positive and negative
thickness values. The element volume is shrinked, thus not
reliable and finally removed from the thickness layer.
Nevertheless, it would have contained very low energy content
(around zero thickness modification)
Figure 11 right shows the element kept in the layer. Like previously,
dynamic expansion from known displacement at the nominal
surface is performed to get the motion of the whole layer.
Figure 11 Extrusion direction and warped element (left),
dynamic expansion on retained elements (right)
Using the equation (11), the evolution of the modal frequency in
response to this modification is obtained from the nominal
frequency , the strain energy and the kinetic energy
contained in the thickness layer.
(14)
This strategy which looks at the frequency evolution of one mode
at a time will be called SDM-1Mode. If more than one mode is
available in the reduction basis, it is more interesting to come back
to the SDM equation (9) to take into account the impact of the
modification on all modes at once.
The mesh is split in two parts
- one containing elements of the nominal model whose reduced
mass and stiffness matrices are those of equation (7)
- another with new elements for the modification. Since new nodes
are added, the response of on those nodes is estimated by
dynamic expansion and the and were defined in
equation (9).
Solving equation (9) then gives new modes for which both
frequency and shape may vary. This second strategy which looks
at the evolution of all modes at once will be called SDM-AllModes.
Note that all these steps are quite fast. In our example, the thin layer
is about 250.000 DOFs: from a thickness distribution, extrusion,
dynamic expansion and new mode computation only take 15s.
4.3. Consistency analysis
To evaluate the proposed SDM strategy, a consistency analysis
must be performed. Because further application will be done on the
ODS at 4050Hz, the closest nominal model mode (at 4005Hz) is
chosen and one seeks to increase its frequency.
The thickness sensitivity map of this mode is shown in Figure 10.
This map is directly applied as signed thickness distribution map to
build the thickness layer, whose motion after dynamic expansion
was shown on Figure 11 right. SDM-1Mode and SDM-AllModes
are finally computed to evaluate mode evolution.
To obtain a reference, the same signed thickness distribution map
is applied to the model using a morphing strategy [18]: surface
nodes are moved and volume interior nodes smoothly follow to
keep good element quality. Computation of mode shapes on the full
morphed models performed: these modes are the reference ones
(hereafter called true model modes).
Figure 12 shows, for thickness variations up to 5 mm, the
comparison between true mode frequencies (solid line), SDM-
1Mode predictions (line with x crosses) and SDM-AllModes values
(dashed line). For all modes in this frequency band, the two SDM
strategies are quite accurate even if SDM-AllModes is always
closer to the true model mode evolution.
Figure 12 Comparison of SDM-1Mode and SDM-
AllModes with true model frequency evolution in the
frequency band of interest for the ODS
The major interest of using SDM-AllModes is found when close
modes interact with each other. This is shown in Figure 13 where
two modes cross around frequency 5575Hz and scale 2.5. SDM-
AllModes is still very close to the true model evolution, but SDM-
1Mode clearly fails in predicting the modal frequency evolution
because shape evolution is not possible.
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Figure 13 Comparison of SDM-1Mode and SDM-
AllModes with true model frequency evolution in a
frequency band where mode crossing occurs
5. SQUEAL REDUCTION OBJECTIVE:
SEPARATE MODE FREQUENCIES
The final objective is to modify the system in order to reduce squeal
occurrences. As explained in [12], brake instability can be analyzed
as the interaction of real mode shapes (often mainly two) leading
to a complex mode with negative damping. Making the hypothesis
that the ODS complex shape contains the two main real shapes
responsible of the instability, the idea is to separate them in
frequency.
The SDM strategy detailed in section 4.2 is thus applied on the
modal model obtained after expansion of the ODS in section 3.3.
Figure 14 shows that for the first mode, areas with the highest
thickness sensitivity are located on the top and left of the bracket
and on top of the caliper. For the second mode, most sensitive areas
are mainly on top of the caliper cylinder.
Figure 14 Thickness sensitivity map for the two modes of
the modal model after ODS expansion
The objective being to separate frequencies, the map shown in
Figure 15 is the difference between the two previous ones. In red
areas, the gap between the first and the second mode frequencies
increases whereas it is the contrary in blue areas.
Figure 15 Sensitivity map to separate the frequencies
displayed on the two mode shapes
One difficulty with this thickness sensitivity map is that few nodes
have a very high value that would lead to very localized thickness
increase. To smooth the map, it is first normalized with the
maximum value set to one and then square root is applied to the
values, reducing the gap between high and medium values. The
resulting map is the thickness distribution finally applied with
several scales going from -5 to 5 mm.
Figure 16 shows the evolution of the frequency in response to the
thickness distribution maps. Both SDM-1Mode and SDM-
AllModes give very close predictions.
Figure 16 Absolute frequency evolution with the thickness
modification
One mode is more sensitive than the other to the modification but
the main focus is the relative frequency shift between the two
modes
shown in Figure 17. Using positive or negative scale is almost
equivalent even if frequency shift is a bit higher with negative scale
(second mode frequency gets lower than the first mode frequency).
Figure 17 Relative frequency shift evolution with the
thickness modification
If a frequency shift higher than 3% is for instance required, the
chosen scale is -3 and the corresponding thickness modification
map is shown in Figure 18.
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Figure 18 Final thickness distribution map to reach 3% of
frequency shift
6. CONCLUSION
This study defines a new strategy to propose surface thickness
modifications with the objective to shift frequencies associated
with shapes measured during squeal occurrences. From ODS, it has
been shown that a fine tuning of MDRE algorithm leads to shape
estimates at all model DOFs with physically representative stress
distribution. A modal model is extracted from the expanded shapes
and a frequency correction is applied. Thickness sensitivity maps,
computed for each mode with SDM technique, help determining
areas where material addition/removal has a strong influence on the
modal frequencies. A final thickness distribution map is then
derived. Consistency analysis confirmed that this strategy can
predict true mode shifts with accuracy. The procedure has finally
been applied to an industrial brake system with the objective to
separate two modal frequencies: a thickness varaiation map to get
a 3% frequency shift has been proposed.
The proposed strategy has been developed in the Structural
Dynamics Toolbox (for use with MATLAB) [19] as it provides all
the necessary background for test analysis, correlation and SDM.
Using the models typically considered for CEA, one avoids the
need to perform correlation and updating by considering expansion
and assuming that the limit cycle frequency corresponds to modal
frequencies of the expanded shapes. Thickness sensitivity maps
immediately give insight to propose system modifications.
Of course, as the model is improved through correlation and update
processes, the sensitivity studies should be recomputed and this is
quite realistic as the computation time is not huge, typically a few
hours. It is worth noting that the expansion gives information about
model errors and can help updating.
The next step is obviously to test the proposed thickness
modifications experimentally. Other future developments are
considered:
- Optimization of thickness distribution maps from thickness
sensitivity maps to respect design constraints
- Extension of the SDM strategy after MDRE to other
modifications such as chamfers, local remeshing, …
- Evaluate the robustness of stress field estimation in expanded
shapes to model error and sensor location
- Instead of using ODS, try to identify modes of the brake system
in operating conditions close to instability but still stable. This
would give access to the root cause of the squeal; damping
could be introduced (not available with ODS) and more
confidence would be gained because of a higher knowledge of
the system dynamics in the modal model.
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ACKNOWLEDGEMENT
The main authors would like to thank all partners from Hitachi
Astemo France for their support providing experimental data and
nominal model. Their feedback throughout the whole project was
greatly appreciated.
