-
Review
Implicitexplicit co-simulation of brake noise
Mohammad Esgandari n, Oluremi OlatunbosunDepartment of
Mechanical Engineering, University of Birmingham, UK
a r t i c l e i n f o
Article history:Received 23 July 2014Received in revised form15
December 2014Accepted 31 January 2015
Keywords:Brake
noiseSquealImplicitExplicitCo-simulationCo-executionFinite Element
AnalysisComplex Eigenvalue Analysis
a b s t r a c t
The Finite Element Analysis (FEA) method has long been used as a
means of reliable simulation of brake noise.In the simulation and
analysis of brake noise, computational time is a signicant factor.
Complex EigenvalueAnalysis (CEA) is the most common brake noise
analysis method since it can provide a quick prediction of
thefrequency of the instability [19]. However, most non-linear
behaviours of the system are simplied in order toachieve a quick
analysis result in the frequency domain. The explicit analysis can
provide a morecomprehensive understanding of the system by taking
into account non-linear variables of the system intime domain.
However, performing an explicit analysis is expensive in terms of
computing time and costs.
This study investigates effectiveness of a hybrid
implicit/explicit FEA method which combines frequencydomain and
time domain solution schemes. The time/frequency domain
co-simulation analysis simulates thebrake unit partly in implicit
and partly in explicit, and combines the results. The aim of the
study is to investigatethe suitability of implicit-explicit
co-simulation for brake Noise, Vibration and Harshness (NVH)
evaluations. Thesimulation results are correlated with the vehicle
test results. The rotation of the disc is simulated in
explicitdomain while the rest of the loadings are based in the
frequency domain using CEA implicit method. Theperformance of the
co-simulation solution scheme is evaluated and compared to the
implicit CEA analysis.
& 2015 Elsevier B.V. All rights reserved.
Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 162. Theory . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1. Implicit solution method. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 172.2. Explicit solution method . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 18
3. Experimental investigation . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 184. Numerical investigation . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 18
4.1. FEA model set-up . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 184.2. Model validation . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 194.3. Complex eigenvalue analysis procedure .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 214.4. Co-simulation analysis procedure . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 21
5. Analysis results and discussion . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 235.1. Complex eigenvalue analysis results. . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 235.2. Co-simulation analysis results. . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 23
6. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 23References . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1. Introduction
Recent advances in the computer technology have enabled
resear-chers to build more complicated and comprehensive numerical
models
to study the brake squeal problem. These models often
providerelatively quick results compared to experimental methods.
Improve-ments in algorithm and formulation of loads and boundary
conditionshave enabled researchers to obtain a more accurate
representation ofthe phenomenon, and consequently improve the
Noise, Vibration andHarshness (NVH) attributes of the brake system
[4,5].
There are numerous experimental, analytical and numericalmethods
to investigate the brake noise problem [3]. Experimental
Contents lists available at ScienceDirect
journal homepage: www.elsevier.com/locate/nel
Finite Elements in Analysis and Design
http://dx.doi.org/10.1016/j.nel.2015.01.0110168-874X/& 2015
Elsevier B.V. All rights reserved.
n Correspondence to: University of Birmingham, Edgbaston,
Birmingham, UKE-mail address: [email protected] (M. Esgandari).
Finite Elements in Analysis and Design 99 (2015) 1623
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approaches were the very rst method of investigating brake
noise.Early examples of this are Lamarque (1938) [11] and Mills
(1939) [15].Nowadays full-vehicle noise search tests are among the
most reliablemethods for investigating the problem, as well as
conrming thendings of numerical investigations. Experimental
methods areusually very useful for conrming results from other
studies, as theydemonstrate a complete presentation of the NVH
performance ofbrakes.
There has been a long debate in choosing the right
numericalmethod. Several studies have been published comparing the
timedomain and the frequency domain analyses, discussing
theirrespective virtues [18]. The outcome of this debate in the
literaturerecommends that probably the most reliable, accurate and
com-prehensive solution could be performing two different
numericalapproaches to identify the squeal mechanism; one being the
FiniteElement Analysis (FEA) modal analysis of the disc brake
system todene its eigenvalues, and relate them to the squeal
occurrence.Another one being a nonlinear analysis in the time
domain, with afocus on the contact problem with the friction
between deform-able bodies, namely disc and friction materials
(pads). Then thetwo approaches are compared, and the onset of
squeal is predictedboth in the frequency domain by the linear model
and in the timedomain by the nonlinear one [13].
FEA method is a common simulation technique for academicand
industrial investigations of brake noise. FEA solution methodsfor
brake noise investigation are typically either implicit or
explicitand solved incrementally. In the implicit approach a
solution tothe set of Complex Eigenvalue Analysis (CEA) equations
is obtainedby iterations for each time increment, until a
convergence criterionis obtained. The frequency domain results
present the behaviour ofthe system over the range of frequency, in
one step of time (t ti).The system of equations in the explicit
approach is reformulated toa dynamic system and is solved directly
to determine the solutionat the end of the time increment without
iterations.
The equation solution phase of an FEA can use an implicit
orexplicit technique. The implicit method employs a more reliable
andrigorous scheme in considering the equilibrium at each step of
timeincrement. However, in the case of analyses involving large
elementdeformation, highly non-linear plasticity or contact between
sur-faces the implicit solver is known to face problems in
converging toa correct solution.
The implicit technique is ideally suited for long duration
pro-blems where the response is moderately nonlinear. While each
timeincrement associated with the implicit solver is relatively
large, it iscomputationally expensive and may pose convergence
challenges.
On the other hand, the explicit technique is very robust
andideal for modelling short duration, highly nonlinear events
invol-ving rapid changes in contact state and large material
deformation.In the case of the explicit FEA method the system
equations aresolved directly to determine the solution without
iteration provid-ing a more robust alternative method. In the
explicit method non-linear properties of the system are taken into
account at the cost ofcomputing time.
Implicit or explicit, each of them has specic advantages
anddisadvantages. However, there is possibility of combining these
twointo a hybrid analysis, called co-simulation. Co-simulation is
thecoupling of different simulation systems where different
substructuresof a model exchange data during the integration time.
Co-simulationtechnique allows combining heterogeneous solvers and
is less timeconsuming when different load and model cases require
differentamount of time for the solution between two different
solvers.
In the case of brake noise analysis, co-simulation
capabilitycombines the unique strengths of Abaqus-Standard and
Abaqus-Explicit to more effectively perform complex simulations.
Implicit-explicit co-simulation allows the FEA model to be
strategicallydivided into two parts, one to be solved in the
frequency domain
and the other in the time domain. The two parts are solved
asindependent problems and coupled together to ensure continuityof
the global solution across the interface boundaries [23].
Appli-cation of this analysis technique can provide signicant
savings oftime and money in the automobile development cycle
[2,7].
2. Theory
2.1. Implicit solution method
The equation of motion of a vibrating system is
repeatedlyreviewed in the literature [18,10,19,12,17]. Also, the
concept andformulation of the complex eigenvalue problem is an
establishedtopic in various publications [9,21,16,20,22,6].
Furthermore, theapplication of the CEA problem to the modal
analysis of a model isintegrated in most FEA packages. The
eigenvalue problem fornatural modes of small vibration of a FEA
model is:
2 M C K fg 0 1where M : mass matrix, symmetric and positive; C :
dampingmatrix; K: stiffness matrix; : eigenvalue and fg:
eigenvector(mode of vibration). Also, the equation of motion for a
basicsystem of single degree of freedom is:
mqc _qkq 0 2where m: mass, c: damping, k: stiffness, and q:
modal amplitude.The solution will be in the format of q Aexpt,
assuming A is aconstant and
c2m7
c2
4m2 km
r3
The solution may have real and imaginary parts, and that
isbecause the terms under the square root can become negative.When
c, damping, is negative, the real part of the solution(C=2M) is
positive. A negative damping causes system oscilla-tions to grow
rather than decaying them which is normallyexpected from damping
phenomenon. Hence a positive real partis just another way of
indicating potentiality of observing aninstability [1].
In the implicit method the state of the FEA model is updatedfrom
time t to tt. A fully implicit procedure means that thestate at tt
is determined based on information at time tt.On the other hand,
the explicit method solves for tt based oninformation at time t
[8]. There are different solution proceduresused by implicit FEA
solvers. A form of the NewtonRaphsonmethod is the most common and
is employed by Abaqus. Whensolving a quasi-static boundary value
problem, a set of non-linearequations is assembled as:
G u ZvBT u dV
ZSNT t dS 0 4
where G is a set of non-linear equations in u, which is the
vector ofnodal displacements. B is the matrix relating the strain
vector todisplacement. The product of BT and the stress vector, ,
isintegrated over a volume, V . N is the matrix of element
shapefunctions and is integrated over a surface, S. The surface
tractionvector is denoted by t.
Eq. (4) is solved by incremental methods where loads
ordisplacements are applied in time steps of t to an ultimate
timeof t. The state of the analysis is updated incrementally from
time tto time tt. An estimation of the roots of Eq. (4) is made,
suchthat for the ith iteration:
ui1 utti1 utti Gutti
u
" #1Gutti 5
M. Esgandari, O. Olatunbosun / Finite Elements in Analysis and
Design 99 (2015) 1623 17
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where utti is the vector of nodal displacements for the
ithiteration at time tt. The partial derivative on the
right-handside of the equation is the Jacobian matrix of the
governingequations which is referred to as the global stiffness
matrix, K .Eq. (5) is manipulated and inverted to produce a system
of linearequations:
K utti
ui1 Gutti 6
Eq. (6) is solved for each iteration for the change in
incrementaldisplacements, ui1. In order to solve for ui1 the global
stiffnessmatrix K , is inverted. Although, this is a
computationally expensiveoperation, iteration ensures that a
relatively large time incrementcan be used while maintaining
accuracy of solution [8]. Followingthe iteration i, ui1 is
determined and a better approximation ofthe solution is made as
utti1 , through Eq. (5). This is used as thecurrent approximation
to the solution of the subsequent iteration i.
The accuracy of the solution is conrmed by the
convergencecriterionwhere the updated value for Gmust be less than
a tolerancevalue. For a complex job it can be difcult to predict
how long it willtake to solve or even if convergence will
occur.
Abaqus-Standard uses a form of the NewtonRaphson
iterativesolution method to solve for the incremental set of
equations.Formulating and solving the Jacobian matrix is the most
compu-tationally expensive process. The modied NewtonRaphsonmethod
is the most commonly used alternative and is suitablefor non-linear
problems. The Jacobian is only recalculated occa-sionally and in
cases where the Jacobian is unsymmetrical it is notnecessary to
calculate an exact value for it [8].
2.2. Explicit solution method
The explicit method is best applicable to solve dynamic
problemswhere deformable bodies are interacting. The explicit
method isideally suited for analysing high-speed dynamic events
where accel-erations and velocities at a particular point in time
are assumed to beconstant during a time increment and are used to
solve for the nextpoint in time [25,24]. The explicit solver in
Abaqus uses a forwardEuler integration scheme as follows:
ui1 ui ti1 _u i 12 7
_u i12 _u i 12 t
i1 ti2
u 8
where u is the displacement and the superscripts refer to the
timeincrement. Explicit method is when the state of the analysis
isadvanced by assuming constant values for the velocities _u and
theaccelerations u across half-time intervals. The accelerations
arecomputed at the start of the increment by:
uM1 UF i Ii 9where F is the vector of externally applied forces,
I is the vector ofinternal element forces (Q in some notations) and
M is the lumpedmass matrix [8]. The explicit integration operator
is conditionallystable; it can be shown that the stability limit
for the operator is givenin terms of the highest eigenvalue in the
system:
tr 2max
10
where max is the maximum element eigenvalue. The
explicitprocedure is ideally suited for analysing high-speed
dynamic events[25].
3. Experimental investigation
In development of numerical methods for investigating brake
NVHattributes experimental approaches are signicant as an
indication of
accuracy of the proposed numerical method. The vehicle test is
themost accurate representation of the NVH performance of the brake
inreal life since some brake noise initiation mechanisms only exist
onthe car and are dependent on the actual manoeuvre performed.
To validate the FEA simulation using co-simulation
technique,initially a full vehicle test was conducted. This is
aimed atreplicating the SAE J2521 dynamometer noise search
procedure.The vehicle test is performed by installing various
sensors atdifferent parts of the vehicle, to record the required
data. In orderto record the brake noise, microphones were installed
inside thecar, approximately near the driver or passenger's ear. To
obtain arobust and reliable result from the vehicle noise search
tests, theSAE J2521 test procedure manoeuvres were repeated to
identifythe recurrent noises. These manoeuvres are described in
Table 1.
Fig. 1 previews the vehicle test results, based on the
mostfrequent noise occurrences obtained from a set of tests.
As seen in Fig. 1, there are occurrences of brake noise
infrequency ranges of 1.9 kHz and 5.2 kHz, which are repeated
indifferent manoeuvres. Therefore, the numerical analysis of
thesame brake unit is expected to represent instabilities in
thesefrequencies which indicates likelihood of noise
occurrence.
4. Numerical investigation
4.1. FEA model set-up
The FEA model of the brake system consists of all components
ofthe corner unit, excluding the suspension links and
connectionswhere the brake unit is connected to the chassis. Fig. 2
represents this.
Major components included in the model are disc, hub,
calliper(and anchor), pad assembly. Brake uid is also simulated
toreplicate the transfer of pressure which is more realistic
thanapplication of force on the pistons. Each component is assigned
itscorresponding material properties. The material properties of
theFEA model are provided in Table 2.
Different iterations of CEA form a set of squeal analysis.
Variablesdifferentiating each iteration of the CEA in this study
are the level ofdisc-pad contact interface Coefcient of Friction
(COF) and brakepressure level. Different levels of COF are
simulated and analysed as
Table 1SAE J2521 test manoeuvres.
Module Repetition Velocity (km/h) Pressure (bar) IBT (1C)
Drag 266 drags 3 and 10 530 5030050Deceleration 108 stops 50 530
5025050Forward & reverse 50 drags 3 and 3 020 15050
707580859095100105110115120
0 1000 2000 3000 4000 5000 6000 7000 8000
Soun
d Pr
essu
re L
evel
[dB
(A)]
Frequency [Hz]
Vehicle Noise TestDrag Reverse Decceleration Forward
Fig. 1. Vehicle brake noise test result SAE J2521 test
procedure.
M. Esgandari, O. Olatunbosun / Finite Elements in Analysis and
Design 99 (2015) 162318
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a safety margin for the analysis, to ensure every possible
instabilityis captured. Although there are slight variations on the
level of COFfrom one stop to the next (due to the surface
undergoing differenttemperatures), this is assumed to be minimal
and the overall COFdoes not vary signicantly. The piston pressure
variations simulatedifferent levels of application of the brake.
This level of pressuretakes the pressure increase of the booster
into account. The analysisprocedure is repeated for different
combinations of friction and
pressure. The set of results obtained from all different
iterationsmentioned forms the squeal analysis results.
FEA model of the disc is constructed using rst order hexahe-dral
and pentahedral elements. The pad assembly consists of thefriction
material (pad), back-plate and the shim. The back-platesand the
shims are made of steel and the pads are made ofanisotropic
friction material. The COF used for the friction materialis in the
range of [0.35, 0.7]. Different parts of the pad assembly
arepresented in Fig. 3. All parts are modelled using the
hexahedraland pentahedral elements. All parts are meshed
individually andtied together.
The outer mesh of the piston is in contact with the inner
surfaceof the bore of the calliper and the brake uid there in
between ismodelled with uid elements. Calliper body and anchor are
mod-elled with second order tetrahedral elements. Anchor spring
whichholds the calliper and anchor together are modelled as 1D
elementswith a directional stiffness to replicate the spring. The
hub is anindependent component from the rest of the assembly and it
spinsalong with the disc. Consequently it is modelled as a rigid
body. Fig. 4shows a cut-away view of the disc brake assembly
showing theinteraction of all components.
4.2. Model validation
The FEA model is required to be the most accurate
representa-tion of the individual components of the system.
Geometry of thesystem components and their material properties are
expected to
Fig. 2. FEA model of the brake unit.
Table 2Material properties assigned to the brake FEA model.
Component Material Density (kg/m3) Young's Modulus (MPa)
Poisson's ratio Section type Element type
Disc Grey Iron 7100 107,000 0.26 Solid C3D8/C3D6Pads Friction
Material 2800 Solid C3D8/C3D6Anchor body SG Iron 7100 170,000 0.275
Solid C3D10Anchor Spring Spring SPRING2Caliper SG Iron 7100 170,000
0.275 Solid C3D10Piston Steel 7900 207,000 0.3 Solid C3D10Piston
spring Steel 7900 207,000 0.3 Solid C3D10Hub Rigid R3D3Back-plate
Steel 7820 206,800 0.29 Solid C3D8/C3D6Shim Aluminium 2750 71,000
0.33 Solid C3D8/C3D6Brake uid Fluid F3D3
Fig. 3. Brake pad assembly.
M. Esgandari, O. Olatunbosun / Finite Elements in Analysis and
Design 99 (2015) 1623 19
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signicantly contribute to this. In order to validate the FEA
modelof the brake, individual components are tested using
impacthammer test or shaker test methods. Resonant frequencies
ofthe component are correlated with the FEA model and the
RelativeFrequency Difference (RFD) [13] is obtained for each mode.
Fig. 5shows the layout of the response points on the disc, for the
shaker
Fig. 4. Brake assembly cross section.
Fig. 5. Modal test response recording points brake disc.
Table 3FEA model validation disc modal correlation.
Mode number Test frequency (Hz) FEA frequency (Hz) RFD (%) MAC
(%)
1 711 737 3.6 982 920 944 2.7 853 1244 1270 2.1 734 1247 1270
1.8 795 1325 1365 3 426 1335 1367 2.4 717 1971 1993 1.1 848 1975
1991 0.8 869 2490 2508 0.7 8810 2907 2933 0.9 9111 2915 2933 0.6
8812 3443 3513 2 7213 4043 4071 0.7 6514 4056 4071 0.4 5115 4301
4355 1.3 3116 4620 4740 2.6 8217 4684 4655 -0.6 8818 5346 5383 0.7
8619 5362 5383 0.4 6320 5497 5505 0.2 8021 6032 6083 0.8 9722 6655
6825 2.5 6623 6813 6851 0.6 7124 6828 6851 0.3 5825 6963 7130 2.4
41
Table 4FEA model validation caliper modal correlation.
Mode number Test frequency (Hz) FEA frequency (Hz) RFD (%) MAC
(%)
1 1960 2071 5.7 31.22 2151 2116 1.6 71.53 3747 3981 6.2 69.14
4331 5 4479 4460 0.4 42.16 4951 5200 5.0 30.67 5629 6048 7.4 37.98
5950 6462 8.6 72.69 6554 6514 0.6 16.8
Table 5FEA model validation anchor (bracket) modal
correlation.
Mode number Test frequency (Hz) FEA frequency (Hz) RFD (%) MAC
(%)
1 755 729 3.4 86.32 863 845 2.1 88.43 1077 1028 4.5 86.74 1431
1443 0.8 91.35 2054 2138 4.1 91.26 2724 2834 4.0 81.37 2841 2988
5.2 63.88 3058 2965 3.0 43.99 3575 3224 9.8 62.610 4117 4054 1.5
26.711 4543 4262 6.2 38.712 4918 4818 2.0 41.613 5380 5045 6.2
51.914 5472 5252 4.0 24.715 5845 5621 3.8 48.716 7191 6891 4.2
49.417 7555 7293 3.5 64.318 7726 7655 0.9 44.019 8279 8573 3.5
47.920 8892 8912 0.2 40.021 9557 9770 2.2 53.1
M. Esgandari, O. Olatunbosun / Finite Elements in Analysis and
Design 99 (2015) 162320
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test application. To ensure accuracy of the results, the
accelerationresponse is recorded from more than 200 points.
An RFD of less than 5% is assumed to be acceptable in this
study.Also, the Modal Assurance Criteria (MAC) [14] is evaluated
for eachmode, to conrm the modal behaviour of the part is matching
thatof the FEA model. Table 3 presents the experimental and
FEAfrequencies of the disc, as well as the RFD and MAC
percentageswhich show an acceptable correlation.
The modal correlation for the disc shows a very good agree-ment
in the natural frequencies and the predicted mode shapes.Table 4
shows the modal correlation for the brake caliper.
Table 5 compares the experimental and numerical modalbehaviours
of the anchor (bracket).
The whole pad assembly was also hammer tested, and Table 6shows
the modal correlation for the pad assembly.
Once all components of the model are correlated with
theexperimental results for the modal behaviour, the model
isexpected to give accurate prediction of the unstable
frequencies.
4.3. Complex eigenvalue analysis procedure
The CEA procedure occurs based on different boundary
condi-tions, loadings and analyses steps assigned to the FEA model.
Asthe analysis starts the central spring undergoes a tension, and
thisis performed to give more realistic contact behaviour between
thecentral spring and the back-plate. Then the initial volume
ofvarious components is calculated, and there is no loading
involvedyet. This is performed using nSTEP, PERTURBATION. In the
thirdstep, an initial displacement of 0.1 mm is applied to all the
boltpretention reference nodes to eliminate any rigid body motion
inthe system and also establish the required contacts. This
willeliminate any convergence issues which may arise in preload
ofbolts in the next step. In the bolt clamp-up step the disc-hub
andcaliper-knuckle pretension nodes are preloaded. There is a 42
KNload on each bolt connecting the disc to the hub, and this value
is32 KN for caliper-knuckle bolts. These levels of pretention
areaimed at simulating the actual level of torque applied on
therespective bolts on the vehicle. Then the brake uid pistons
arepressurised to push shims and consequently the pad assembly.This
pressure can vary for different stages of the analysis. The nalstep
is when the disc is rotated. The required rotational velocity
isgiven to all the nodes of the disc and hub assembly to simulate
therotating wheel using nMOTION step. Also friction between
therotor and pads is specied by nCHANGEFRICTION step. The
rota-tional velocity assumed for the disc in the CEA is 3.68 rad/s.
Themodal analysis is performed next. This includes extraction
ofnatural frequencies using nFREQUENCY step as a requisite
toperform mode-based CEA in the next step. In the CEA unstablemodes
are identied. The step is performed by nCOMPLEXFRE-QUENCY. The
unstable modes can be identied during complexeigenvalue extraction.
The eigenvectors represent the mode shape.The unstable mode is
identied during complex eigenvalueextraction as the real part of
the eigenvalue corresponding to anunstable mode is positive.
Although the software reports thisinformation in the format of
negative damping ratio, it is commonpractice to consider the
absolute value of the negative dampingratio and report it as a
percentage. In this study the percentage ofthe damping ratio is
assumed as the strength of instability [4].
4.4. Co-simulation analysis procedure
A co-simulation (also referred to as co-execution) is the
simulta-neous execution of two analyses that are executed in
Abaqus-CAE insynchronization with one another using the same
functionality withtwo different solvers. In the co-simulation
analysis the FEA model issplit into two sections, the implicit
section which is solved in thefrequency domain and the explicit
section which is solved in the timedomain. The analysis procedure
in the implicit section is very similarto the normal CEA
procedure.
For a co-simulation analysis comprising of Abaqus-Standardand
Abaqus-Explicit, the interface region and coupling schemes forthe
co-simulation need to be specied. A common region isdened for both
implicit and explicit models, referred to as theinterface region.
Interface region is the section for exchanging databetween the two
sections of the model and the coupling scheme isthe time
incrementation process and frequency of data exchange.An interface
region can be either node sets or surfaces whencoupling
Abaqus-Standard to Abaqus-Explicit. The implicit regionprovides the
boundary condition and loadings to the explicitregion through these
nodes.
Table 6FEA model validation pad modal correlation.
Mode number Test frequency (Hz) FEA frequency (Hz) RFD (%)
1 2433 2491 2.382 3784 3789 0.133 5009 5014 0.104 5915 5932
0.295 7769 7698 0.916 9712 9825 1.16
Fig. 6. Co-simulation model implicit.
M. Esgandari, O. Olatunbosun / Finite Elements in Analysis and
Design 99 (2015) 1623 21
-
In the FEA model, a Standard-Explicit co-simulation interaction
iscreated to dene the co-simulation behaviour. Only one
Standard-Explicit co-simulation interaction can be active in a
model. Thesettings in each co-simulation interaction must be the
same in theAbaqus-Standard model and the Abaqus-Explicit model. The
solutionstability and accuracy can be improved by ensuring the
presence ofmatching nodes at the interface. A model can have
dissimilar meshesin regions shared in the Abaqus-Standard and
Abaqus-Explicit modeldenitions. However, there are known
limitations associated withdissimilar mesh in the co-simulation
region. When the Abaqus-Standard and Abaqus-Explicit co-simulation
region meshes differ,the solution accuracy may be affected.
In the co-simulation, the Abaqus-Standard can be forced to
usethe same increment size as Abaqus-Explicit, or use different
incre-ment sizes in Abaqus-Standard from those in Abaqus-Explicit.
This iscalled sub-cycling. The chosen time incrementation scheme
forcoupling affects the solution computational cost and accuracy
but
not the solution stability. The sub-cycling scheme is frequently
themost cost effective since Abaqus-Standard time increments
arecommonly much longer than Abaqus-Explicit time increments.
The implicit model of the brake includes the same componentsas
the CEA model and the explicit model consists of the brake
disc,friction materials and hub. The side surfaces of the friction
materialsare the implicitexplicit common nodes. The anchor mount
isconstrained in all degrees of freedom. The brake hub is
modelledas a rigid body which is tied to the disc and rotates with
it. The discis constrained to the hub and is limited to only
rotation about itscentral axis which resembles normal rotation of
the brake disc.Rotational velocity is not supplied to the disc in
this standardanalysis. Fig. 6 shows the implicit model of the
co-simulation.
The explicit model consists of the brake disc, hub and
frictionmaterials where the assembly is subject to rotation applied
to thedisc through the rigid hub. An analytical rigid surface has
beencreated and tied to the disc. Angular velocity is applied to
the rigidsurface reference node, which will drive the disc.
Application ofangular velocity is simulated in Abaqus-Explicit.
Fig. 7 shows theexplicit model of the co-simulation analysis.
The disc is rotated with the initial velocity of 10 km/h and
bydeceleration it reached a full stop. This replicates the brake
systemduring drag and stopping in forward or reverse direction,
based onthe direction of the velocity. Simulation of deceleration
is achievedby gradual increase in the level of applied pressure.
Fig. 8 presentsthis gradual application of pressure, reaching 10
bar in 0.1 s andmaintaining that level of pressure for another 0.1
s.
The implicit and explicit sections of the model are not
completelyindependent. The analysis results in the implicit section
are transferredto the explicit section. This happens in time
increments and eithersections of the model can transfer data in
predened time increments.Abaqus uses a timing logic of the data
transfer called subcycle. This
Fig. 7. Co-simulation model explicit.
0
2
4
6
8
10
12
0 0.05 0.1 0.15 0.2 0.25
Pres
sure
(Bar
)
Time (Sec)
Application of Pressure
Fig. 8. Application of pressure.
Fig. 9. Co-simulation common region is a node set as shown
below.
0.00.51.01.52.02.53.03.54.04.5
0 1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000
Dam
ping
Rat
io (%
)
Frequency (Hz)
CEA Instability Prediction0.35 0.45 0.55 0.7
Fig. 10. CEA squeal analysis results baseline model friction
levels of 0.35, 0.45,0.55 and 0.7 and pressures of 2, 5, 10 and 25
bars in forward direction.
M. Esgandari, O. Olatunbosun / Finite Elements in Analysis and
Design 99 (2015) 162322
-
means at predened number of cycles of one solver, data is
transferredto the other. Fig. 9 shows the common nodes for data
transfer onthe pads.
In the co-simulation analysis, the analytical surface on the
disc isassumed as the major output point. This is based on the
hypothesisthat for all major noise initiation mechanisms in the
brake system,the brake disc acts as the amplier to the noise and it
is consideredas the radiator of the noise. Therefore, the
co-simulation analysis isbased on the hypothesis that acceleration
at the disc surface, wherethe maximum displacement takes place, is
an indication of noise-related instabilities. This is commonly on
the far end of the discwhere the structural support from the hub
connections and theswan neck of the disc are minimal.
5. Analysis results and discussion
5.1. Complex eigenvalue analysis results
Fig. 10 illustrates squeal analysis results for pressure
variations of 2,5, 10 and 25 bars and COF of 0.35, 0.45, 0.55 and
0.7. It is repeatedlymentioned in the literature that not all
instabilities predicted by CEAmay occur in reality as the CEA
always overestimates the instabilities[5]. Analysing different COF
levels helps repetition of instabilities tooccur from different
combinations of friction, pressure and operatingdirection. This can
highlight the actual unstable frequency comparedto the other
over-predicted instabilities by CEA.
The squeal analysis results show that instabilities occur
potentiallyin the frequency ranges of 1.9, 2.6, 3.8 and 6.2 KHz.
Comparison of theCEA results with the vehicle test results also
reveal that among twonoisy frequencies only 1.9 kHz is reported,
although the number ofinstability occurrences (only two) is not
signicant.
5.2. Co-simulation analysis results
By performing the co-simulation analysis, the explicit section
ofthe model provides time domain results. The node with themaximum
displacement is identied on the disc outer layer, andacceleration
is obtained as the output. Once the accelerationversus time data is
obtained in Abaqus-Explicit, it is transformedinto acceleration
versus frequency. Peaks in the accelerationfrequency graph are
assumed as an indication of squeal frequency.
Fig. 11 presents the co-simulation analysis results.
Unstablefrequencies are predicted at 1.9 kHz and 5.1 kHz at the
frictionlevel of 0.45 and pressure of 2 bar. These are in good
correlationwith the noisy frequencies of the vehicle test presented
in Fig. 1.
6. Conclusions
A co-simulation analysis for brake noise investigation is
devel-oped using Abaqus FEA package. The brake FEA
co-simulationmodel comprising of the disc and pads is developed. In
Abaqus-Standard to Abaqus-Explicit co-simulation technique the
twosolvers are simultaneously running on corresponding sections
ofthe model. The co-simulation model is capable of accurately
pre-dicting the unstable frequency by providing time-domain
resultsin a lower computing time.
Co-simulation is a powerful tool to efciently analyse the
responseof a full brake model. Co-simulation technique is capable
of perform-ing squeal analysis by simulating the rotation of the
disc and the disc-pad contact interface in Abaqus-Explicit and the
rest of the loadings inAbaqus-Standard.
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0
2
4
6
8
10
12
0 1,000 2,000 3,000 4,000 5,000 6,000 7,000
Acc
eler
atio
n (m
/s^2
)
Frequency (Hz)
Co - simulation - Explicit Result
Fig. 11. Acceleration vs. frequency co-simulation.
M. Esgandari, O. Olatunbosun / Finite Elements in Analysis and
Design 99 (2015) 1623 23
Implicitexplicit co-simulation of brake
noiseIntroductionTheoryImplicit solution methodExplicit solution
method
Experimental investigationNumerical investigationFEA model
set-upModel validationComplex eigenvalue analysis
procedureCo-simulation analysis procedure
Analysis results and discussionComplex eigenvalue analysis
resultsCo-simulation analysis results
ConclusionsReferences