Esgandari - Finite Elementsin Analysis and Design

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

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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

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

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

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.


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.


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.


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


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.


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.

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Fig. 8. Application of pressure.

Fig. 9. Co-simulation common region is a node set as shown below.

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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.


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.

References

[1] A. Abubakar, Modelling and Simulation of Disc Brake Contact Analysis andSqueal (Ph.D. thesis), Department of Mechanical Engineering, Faculty ofEngineering. University of Liverpool, Liverpool, United Kingdom: Liverpool,United Kingdom, 2005.

[2] E. Duni, G. Toniato, R. Saponaro, et al., Vehicle dynamic solution based on niteelement tire/road interaction implemented through implicit/explicit sequen-tial and co-simulation approach, Training 2013 (2010) 1118.

[3] M. Esgandari, O. Olatunbosun, Computer aided engineering prediction ofbrake noise: modeling of brake shims, J. Vib. Control (2014).

[4] M. Esgandari, O. Olatunbosun, R. Taulbut, Effect of Damping in Complex EigenvalueAnalysis of Brake Noise to Control Over-Prediction of Instabilities: An ExperimentalStudy, SAE Brake Colloquium, SAE International, Jacksonville, FL, 2013.

[5] M. Esgandari, O. Olatunbosun, R. Taulbut, Frictional Coefcient Distribution Patternon Brake Disk-pad Contact Interface to Reduce Susceptibility of Brake NoiseInstabilities Causing Brake Squeal, EuroBrake, Dresden, Germany: FISITA, 2013.

[6] Y. Goto, H. Saomoto, N. Sugiura, et al., Struct. Des. Technol. Brake SquealReduct. Using Sensit. Anal. (2010).

[7] A. Gravouil, A. Combescure, Multi-time-step explicit-implicit method for non-linear structural dynamics, Int. J. Numer. Methods Eng. 50 (2001) 199225.

[8] F. Harewood, P. McHugh, Comparison of the implicit and explicit nite elementmethods using crystal plasticity, Comput. Mater. Sci. 39 (2007) 481494.

[9] E.L.W. Klaus-Jrgen Bathe, Large eigenvalue problems in dynamic analysis,J. Eng. Mech. Div. 98 (1972) 14.

[10] S.W. Kung, K.B. Dunlap, R.S. Ballinger, Complex eigenvalue analysis forreducing low frequency brake squeal, Soc. Automot. Eng. 109 (2000) 559565.

[11] P. Lamarque, C. Williams, Brake Squeak: The Experiences of Manufacturers andOperators and Some Preliminary Experiments, Institution of AutomobileEngineers, 1938.

[12] P. Liu, H. Zheng, C. Cai, et al., Analysis of disc brake squeal using the complexeigenvalue method, Appl. Acoust. 68 (2007) 603615.

[13] F. Massi, L. Baillet, O. Giannini, et al., Brake squeal: linear and nonlinearnumerical approaches, Mech. Syst. Signal Process. 21 (2007) 23742393.

[14] F. Massi, O. Giannini, L. Baillet, Brake squeal as dynamic instability: anexperimental investigation, J. Acoust. Soc. Am. 120 (2006) 1388.

[15] H. Mills, Brake Squeak, The Institution of Automobile Engineers, 1939.[16] D.V. Murthy, R.T. Haftka, Derivatives of eigenvalues and eigenvectors of a

general complex matrix, Int. J. Numer. Methods Eng. 26 (1988) 293311.[17] M. Nouby, K. Srinivasan, Parametric studies of disk brake squeal using nite

element approach, J. Mek. 29 (2009) 5266.[18] H. Ouyang, A. AbuBakar, Complex eigenvalue analysis and dynamic transient

analysis in predicting disc brake squeal, Int. J. Veh. Noise Vib. 2 (2006) 143155.[19] H. Ouyang, W. Nack, Y. Yuan, et al., Numerical analysis of automotive disc

brake squeal: a review, Int. J. Veh. Noise Vib. 1 (2005) 207231.[20] B.N. Parlett, The Symmetric Eigenvalue Problem. The Symmetric Eigenvalue

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solution of large sparse generalized symmetric eigenvalue problems, Math.Comput. 35 (1980) 18.

[22] J. Shaw, S. Jayasuriya, Modal sensitivities for repeated eigenvalues andeigenvalue derivatives, AIAA J. 30 (1992) 850852.

[23] M. Trcka, J.L.M. Hensen, M. Wetter, Co-simulation of innovative integratedHVAC systems in buildings, J. Build. Perform. Simul. 2 (2009) 21.

[24] N.-M. Wang, B. Budiansky, Analysis of sheet metal stamping by a nite-element method, J. Appl. Mech. 45 (1978) 7382.

[25] D. Yang, D. Jung, I. Song, et al., Comparative investigation into implicit, explicit,and iterative implicit/explicit schemes for the simulation of sheet-metalforming processes, J. Mater. Process. Technol. 50 (1995) 3953.

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Fig. 11. Acceleration vs. frequency co-simulation.


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

Documents

Esgandari - Finite Elementsin Analysis and Design