Finite Element Analysis [FEA] Simulations for Comparison to Thermal Data

Obtained from Vehicle Testing on the Grossglocker High Alpine Road

STEWART WREN - RICHARD SIMS

EURAC Poole Ltd - Production Engineering Dept.

1 Introduction

Brake discs are designed to suit the individual requirementsof the particular vehicle application. There are many designcriteria that shall not be discussed at any great length here.Ultimately the brake disc has to meet the thermal require-ments of the braking application, whereby the kinetic energy(KE) and potential energy (PE) of the vehicle motion is con-verted largely into heat at the frictional interface of brake andpads. The temperature increase as a result of this heat inputmust not adversely affect the operational capability of thedisc and thus it must be designed to manage such changes.

Secondly the disc must meet the geometric requirementsof the vehicle, ensuring that the shape and size are suitablefor the wheel positioning, mounting and calliper alignment.The fluid flow schemes around the disc that provide the cool-ing are also of particular importance and shall dictate cer-tain geometries of the disc, in particular the topology of thevanes. Certain driving scenarios such as an emergency stop

Figure 1: Front ventilated disc of the VW Tiguan showing a cut-sectionthrough the vanes. Mounting holes are omitted for preliminary studies.

or sustained braking on an incline will subject the disc tolarger changes in temperature, along with increased subse-quent stresses and deformations.

The use of Finite Element Analysis (FEA) simulations cou-pled with Computational Fluid Dynamics (CFD) analysis,can assist in the design process and validate or disregard de-sign iterations. This will take the final component throughcomprehensive testing and approval stages with substantiallyreduced costs and lead times. It is not something that shouldbe relied upon solely but in accompaniment with experimen-tal failure testing, a suitable product should emerge. It isbecoming increasingly common for customers to request pre-

liminary simulation work to be carried out at the quotationstage, necessitating the use of FEA simulations.

2 Theory

The mathematical braking model that is employed here is asimplification of a hugely complex process in which many vari-ables and phenomena are present. However, it is importantto accurately calculate the amount of energy that is added tothe braking system during deceleration; the validity of subse-quent results will be critically dependent upon these values.

In reality the vehicle motion is governed by braking, cor-nering, engine, changes in road camber and variations in roadincline. The energy from the motion of the vehicle is not en-tirely converted to heat at the frictional interface upon brak-ing. The following incomprehensive list includes some sourcesof energy loss.

1. Loss of tyre adhesion with road surface.2. Tyre deformation along with the associated pressure

change in the tyre.3. The elastic deformation within the springs of the suspen-

sion system.4. The displacement of the shock absorbers.5. The minimal but present deformation in the structure of

the vehicle.6. Even in modern fuel tanks that incorporate the use of

baffles there will be some degree of fluid displacement7. The force of aerodynamic drag upon the vehicle which is

subject to change in accordance with vehicle velocity.8. The small amount of passenger displacement within the

vehicle under severe braking instances.9. Losses in the compressibility of hydraulic fluid

10. Noise and vibrations generated from frictional interac-tions.

The energy that is converted to heat from the frictionalmechanism will begin by increasing the temperature of thedisc and pad. This heat energy will be transmitted to contigu-ous or neighbouring components within the braking systemand take heat away from the brake disc. These sub assemblycomponents include:

1. brake pad2. pad backing plate3. calliper piston(s)4. calliper5. hydraulic fluid6. mounting hub7. wheel section that is in contact with the disc/hub

The primary mechanisms that transfer heat away from thedisc are convective and radiative cooling, the former of which

S. Wren FEA Simulations for Comparison to Thermal Data Obtained from Vehicle Testing on the Grossglocker High Alpine Road 2

uses the circulating air flow within the disc vents and wheelarch to remove energy from the disc.

Values of emissivity obtained experimentally by Tirovic in(Day 2005) were found to range from 0.2 to 0.9. Recommen-dations were made to use an average value of 0.55. Valuesof emissivity taken from (Modest 2013) for cast iron that hasbeen newly turned as 0.44. Values ranging from 0.6-0.7 werealso given for turned and heated cast iron. An emissivityvalue of 0.55 will be chosen for preliminary studies.

The mathematical model that is employed here assumes astraight road of a constant incline, with an equal distributionof mass over the left and right sides of the car. Corners areonly considered for their reduction in velocity and not theasymmetrical distribution of mass over left and right sides ofthe car. Forces of aerodynamic drag are neglected currentlybut will slightly assist in deceleration of the vehicle.

The disc is modelled as an individual component in a waywhich does not consider the calliper assembly, pads or wheel.The disc is constrained via the hub mounting face and heat isapplied to the frictional faces as shown in Fig. 2. The radialsymmetry of the component lends itself to the use of a geo-metric symmetry condition, in which only a sector of the discis modelled, whilst retaining the characteristics of the entiredisc. Fig. 3 shows the geometry that was used. The heat fluxis applied to a non-rotating disc, such that the thermal inputis not concentrated to any one area. Instead the heat flux isapplied with a uniform distribution over the entire swept area.In reality an non-uniform distribution of heat flux would bepresent and located within an area closely representing thepad contact envelope. It would also be rotationally dynamicin nature. The braking instances are modelled over a finite

q′′(t)

Cradiative

Cconvective

q′′(t)

T∞Tinitial

Figure 2: Free Body Diagram of the simplified braking model. The under-side surface is constrained and a uniform heat flux is applied to both frictionplates.

time period designated by the time between t1 and t2. Theapproximation is made that the deceleration due to brakingab is constant over this time. The act of braking will reducethe velocity of the vehicle from v1 to v2. The time-evolutionof the flux is represented by a ramp change from zero heatflux at t1 to a maximum heat flux q′′max at a time tmax, whichis calculated as 10 percent of the total braking duration. Af-ter which, there is a ramp change down to zero heat flux att2 as shown in Fig. 4. No single brake disc is responsi-ble for the deceleration of the entire vehicle mass, nor is itresponsible for mass of only one quarter. For deceleration it

Figure 3: Eighteen degree sector of brake disc geometry.

is necessary to determine the effective mass for which eachrespective brake is responsible. This apportionment of massover the front and rear axles is critical in determining suitablevalues of flux for any particular braking instance. With theassumption of straight roads and no corners, it follows thatthe left and right discs receive equal amounts of heat energy.

t0 t1 t2

Braking

Braking

q′′(t)

q′′max

t

v(t)

0

v1

v2

t1 t2

Figure 4: Comparisons between the heat flux q′′(t) and velocity v(t) withrespect to time.

2.1 Basic methodology for calculating the equivalentmass braked by a single disc

Firstly the horizontal position of the centre of gravity (COG)must be found. The static mass over each axle is known fromthe provided experimental data. The horizontal distance a1

from the front axle to the COG is

a1 = (lmr)mf (1)

Where l is the wheelbase, mv is the vehicle mass, mr is themass over the rear axle and mf is the mass over the frontaxle.

The horizontal distance a2 from the rear axle to the COG is

a2 = l − a1 (2)

20/04/2015 S-R001

S. Wren FEA Simulations for Comparison to Thermal Data Obtained from Vehicle Testing on the Grossglocker High Alpine Road 3

The maximum deceleration abmax possible without tyre slip-page. Where the tyre adhesion coefficient ψ = 0.8 (Guiggiani2014)

abmax = ψg (3)

The vertical height h of the COG

h ' 0.2l (4)

Braking force Fbf over the front axle

Fbf =(mvabmaxh) + (mvga2)

l(5)

Braking force Fbr over the rear axle

Fbr = mvg − Fbr (6)

Dynamic weight balance ξf over the front axle

ξf = 0.213(Fbf

Fbr+ 1) (7)

Dynamic weight balance ξr over the rear axle

ξr = 1− ξf (8)

Dynamic braking force Fbfd over the front axle

Fbfd = 0.8Fbf (9)

Dynamic braking force Fbrd over the rear axle

Fbrd = 0.8Fbr (10)

Mass braked mbf by front axle

mbf = mvξf (11)

Mass braked mbr by rear axle

mbr = mvξr (12)

Mass braked by a single front disc

mbfdisc = mbf/2 (13)

Mass braked by a single rear disc

mbrdisc = mbr/2 (14)

Now that the equivalent mass braked by a single disc is deter-mined, it can be used to calculate the braking energy involvedin a single braking instance. This study will focus on the frontbrake only. The distance travelled along the slope s is foundby

s = v1tb + 0.5at2b (15)

The vertical component is found from simple trigonometryusing the angle of incline. Road incline 5.5824◦ over entirelength.Braking energy Eb consists of the following components,

Eb = GPE +KE (16)

The change in height δh during any braking instance is foundusing the values of v1 and v2 at times t1 and t2 respectively.

Using the basic equations of motion and assuming brakingdeceleration to be constant.

Eb = (mbfdiscgδh) + 0.5(kmbfdisc)(v21 − v2

2) (17)

Where the rotational inertia factor k = 1.1 (Limpert 2011).It is assumed that for the apportioned mass used for the GPEcomponent is the same as used in the KE component. Brakeduration tb

tb = t2 − t1 (18)

Braking power Pb

Pb =Eb

tb(19)

The heat flux is the rate of heat energy transfer into thedisc over the swept area of the disc. This region bound bythe inner and outer diameter swept by the rotating disc isthe frictional annulus and only half of the area in a dual-padsystem.Frictional area Af

Af = 2π(R2 − r2) (20)

The heat flux q′′

q′′ = Eb/Af (21)

The q′′ values are calculated for each braking event through-out the driving scenario, this data along with t1, t2 and tmax

can be tabulated and used to make the final time-amplitudehistory of heat flux, as shown in Listing 1. The heat fluxinput is taken to be almost entirely into the disc as opposedto the pad. A figure of 98.5 percent is used (Day 2005).Where the time of maximum heat flux tmax is

tmax = t1 + ((t2 − t1)/10) (22)

Listing 1: Example of a partial heat flux amplitude for only four brakinginstances.

Time Heat Flux-------------------0 017.05 017.55 0.12928441822.01 030.97 031.417 0.16919089735.44 044.62 044.747 0.24780507145.89 045.9 046.117 0.16584355948.16 0... ...

3 Historical Work

Prior to the work with Grossglockner, attempts were madeto suitably simulate an emergency stop and a fifteen-stopbraking case. The emergency stop case considered a fullyladen vehicle travelling at top speed along a flat surface. Thefifteen-stop braking case considered the comparisons betweendynamometer testing conducted by Roulunds Braking andsimulations that aimed to replicate similar vehicle dynam-ics. The experimental test use a dynamometer to consider 15successive instances of maximum vehicle acceleration up to avelocity of 100km/hr, followed by a constant deceleration of0.4g to 0km/hr.

20/04/2015 S-R001

S. Wren FEA Simulations for Comparison to Thermal Data Obtained from Vehicle Testing on the Grossglocker High Alpine Road 4

4 Grossglockner

Grossglockner is a long alpine road in Austria that is routinelyused for the testing of brake discs amongst other transmissionand braking components. The large altitudes and frequentbends provide manufacturers the opportunity to test underextreme driving conditions and for continual heat input tothe disc. A range of different driving styles were adoptedacross a series of tests, whilst allowing for sufficient coolingin between.

The particular descent route (GG3) that is the focus of thisdocument, as shown in Fig. 5, begins at Edelweißspitze [co-ordinates 47.123890, 12.831235] and takes the first right turnonto the road Taxenbacher Fusch and ends just before thetoll bridge at [47.142790, 12.814140] having travelled 14.4km.The altitude at the start of the descent is 2558m and ends at1118m. Altitude data was obtained using from Google Earth.The resolution and accuracy of the data is not known, but as-sumed to be accurate enough for what is required. The angleof inclination, taken from this data is 5.5824◦degrees.

In descent GG3 there were 107 instances of braking overa total descent time of 954s. The run was completed in an“aggressive” manner and reached speeds of 80kmhr−1.

Figure 5: View of Grossglockner track - descent route marked out in lightblue. Source Google Earth.

5 Vehicle Data

Property Value Source

Vehicle mass 2263 kg Experimental dataVehicle weight 22,200 NMass over the front axle 1167 kg RoulundsMass over the rear axle 1096 kg RoulundsPercent mass at front 52Wheelbase 2.604 m (Tiguan 2015)Vehicle height 1.686 m (Tiguan 2015)

Table 1: VW Tiguan vehicle data

6 Experimental Data

Roulunds Braking were responsible for the experimental pro-cedures conducted at Grossglockner. For each descent an on-board computer recorded the following:

1. Temperature of all four disc as a function of time,recorded from the frictional surface using a thermocou-ple.

2. The temperature of the hydraulic fluid in the subsystemsfor each respective disc.

3. Ambient initial temperatures are recorded.4. Velocity of the vehicle with respect to time.5. Braking acceleration w.r.t time6. Brake pressure w.r.t time7. Brake pedal travel w.r.t time

It is from this data that an attempt is made to map asuitable time-amplitude history for heat flux. This coupledwith suitable initial conditions and material data should yieldsimulation results that are comparable to the experimentaldata.

For each braking event the start and end time t1 and t2were taken from spikes in hydraulic fluid pressure. Thesetimes were used to identify the corresponding values of veloc-ity at the start v1 and end v2 of each braking event. Lineardeceleration over the braking event can be calculated fromfrom these values.

This value can then be used in the calculation of distancetravelled during the braking event. Which if coupled with theassumption of a constant incline, a drop in elevation can becalculated for each braking event.

7 Simulation Procedure

ABAQUS the commercial FEA code is utilised to simulatethe thermal-mechanical behaviour of the brake disc duringoperation. It can be used to apply the necessary thermaland mechanical loadings to the geometry of the brake discand to review the resulting behaviour. ABAQUS is a unit-less computational environment and so great care must beapplied to the initialisation and analysis phases of simulation.SI units are adopted for simplicity along with a temperaturescale in degrees Kelvin.

7.1 Part

The 3D geometry of the brake disc for the VW Tiguan is mod-elled on SOLIDWORKS according to an approved machiningdrawing. To increase computational efficiency a portion orsector of this disc is cut away from the whole. In this par-ticular case the sector was of an 18◦ angle. This is basedupon the degree of rotational symmetry or number of vanesin total. The geometry must be understood by ABAQUS andso the proprietary file format of SOLIDWORKS .sldprt isconverted to a .STEP file. The part is designated as a 3Ddeformable part.

7.2 Properties

The relevant material data was assigned to the geometry.Temperature dependent material data was used where pos-sible.

7.3 Model Properties

Absolute zero is defined as 0K and the Stefan-Boltzmann con-stant is set to 5.56E-08 Wm−2K−4 in accordance with SIunits.

7.4 Step

A transient computational “step” is created which offerscoupled-temperature displacement behaviour. The total time

20/04/2015 S-R001

S. Wren FEA Simulations for Comparison to Thermal Data Obtained from Vehicle Testing on the Grossglocker High Alpine Road 5

period for the relevant descent is set, in this case being 954seconds. The maximum number of time steps is set to 10E6,the initial time step is set to 0.5s and both maximum andminimum time-steps are set to 3E-5 in order to provide simu-lation control. The maximum allowable temperature increaseper increment is set to 100 K.

7.5 Loads

The thermal loading applied to the braking surfaces is charac-terised by a surface heat-flux and governed a time-amplitudehistory. The flux values for each braking instance are cal-culated prior to the simulation. These values are tabulatedas decimal equivalents using a maximum flux value, whichis entered alongside the time-amplitude history. The associ-ated times of each flux event are also tabulated in a mannerrepresented by Listing 1. This assigns a non-zero K value oftemperature to the disc.

7.6 A Predefined Field

A constant nodal temperature is set for the initial tempera-ture conditions of the disc according to the experimental data.

7.7 Interaction Properties

The modes of convective and radiative cooling are definedusing this interaction module. The radiative cooling is de-fined using surface radiation with and emissivity of 0.55 anda constant ambient sink temperature of 293.15K. This ra-diative cooling is applied to all surfaces of the disc with theexception of the symmetry condition faces and the mountingfaces. The convective cooling of the disc is approximated witha surface film condition with a convective cooling coefficientof 5 Wm−2K−1. The sink temperature and surface selectionchoices are the same as per the radiative setup.

7.8 Boundary Conditions

The surface of the disc that is in mating contact with the hubof the car is constrained using the Encastre function whereU1 = U2 = U3 = UR1 = UR2 = UR3 = 0. This dictatesthat the nodes circumscribed by the mounting surface can-not move or rotate about any axis. The second boundarycondition constrains the disc from deformations that wouldinvalidate the scheme of symmetry. The two surfaces thatcut out the sector of the disc are constrained in the outwardsdirection in a cylindrical polar coordinate system. This co-ordinate system is defined using the radial centre of the disc,and the nodes are only free to move in the U2 direction.

7.9 Mesh

To make the algorithmic mesh generation easier and lessprone to failure a feature called virtual topology is used. Thisreduces the number of separate surfaces defined within the.STEP file, into larger surfaces that still reflect the 3D geom-etry but can be more readily divided up into mesh elements.Edges shorter than 5E-05m are absorbed into larger topolo-gies. The mesh is created using a global seed value of 0.002m,which dictates the resolution of the overall mesh, by dividingup the geometry into cells that have a characteristic lengthof 0.002m. The element type is tetrahedral and the nodes areassigned as coupled-temperature displacement.

Elements 52,142Nodes 11,113Global Seed Size 0.002

Table 2: Mesh Details

Figure 6: View of the meshed geometry with a global seed size of 2mm.

7.10 Job

A job file is created that controls the simulation of the model.The parallelisation of cores is used to distribute the computa-tional load across multiple cores. These jobs are saved in theform of .CAE files that can be re-run at a later point. Typi-cal simulation times for this particular procedure are around13 hours. For each time increment, 44,454 equations are re-quired with the current mesh, collectively computing a to-tal of 2.31E+10 floating point operations. The results arerecorded in output-database files .ODB are typically around3 Gigabyte in size. It is important to create history outputsin order to extract the required data from the .ODB file. Thefollowing history outputs are set:

1. NT11 - nodal temperature2. U - displacement for all axes3. S Mises – von Mises stress for all axes4. E Principal – Principal strains for all axes

8 Preliminary Results

0 100 200 300 400 500 600 700 800 900 1,000 1,100 1,200300

400

500

600

700

800

900

Time s

Tem

peratureK

Experimental T (t)

Simulation T (t)

Figure 7: Comparisons between experimental and simulated temperaturesat the centre of the outboard friction face as a function of time. Additionalprofiles describe the simulated temperatures at differing locations upon thedisc from the same simulation.

A multitude of simulations have been carried out so far,many of which only a stepping stone to the next. These it-erative stages have provided insight into creating an optimal

20/04/2015 S-R001

S. Wren FEA Simulations for Comparison to Thermal Data Obtained from Vehicle Testing on the Grossglocker High Alpine Road 6

simulation process for the Grossglockner driving route. Theresults of the latest simulation are shown in Fig. 7. The un-even cooling of the braking face can clearly be seen in Fig. 8.The absence of material within the vane cavity could reducethe extent of conductive heat transfer away from the frictionalsurface to the surrounding disc.

Fig. 9 appears to demonstrate an asymmetric cooling ofbraking surfaces by way of conductive heat transfer to thetop hat section of the disc.

Figure 8: A view of the contours of nodal temperature (NT11) values at atime of 655 seconds and time increment of 1310.

Figure 9: Additional view of the contours of nodal temperature (NT11)values at a time of 655 seconds and time increment of 1310.

9 Future Work and Improvements

Experimental data sets for other driving styles were alsorecorded during the alpine testing. This data can be used tocreate additional simulations for use in back-to-back compar-isons. Current proposals for simulation improvement include:

1. Suitably modify the dynamic loading of axles can be eas-ily modified for the condition of inclines. Using the dy-namic equilibrium equations and a suitable free body di-agram (FBD) of a of a vehicle decelerating on a decline,it shall be possible to solve for the reactionary forces andthus the apportionment of vehicle weight over each re-spective axle. This improved dynamic weight balancewill be a function of the vehicle deceleration and will dif-fer with each braking instance. This may yield a moresuitable time-amplitude history of flux input to the brakeas it does not use a set value for the front-to-rear massratio.

2. For larger deflections and stresses it may be appropriateto investigate the use of the non-linear geometry moduleNLGeom.

3. Heat flux could be defined to an area the size and shapeof the brake pad that rotates about the disc axis in ac-cordance with the vehicle velocity. Ultimately a LumpedElement Modelling approach could be employed for dy-namic simulations that integrate pad rotation and heatflow and frictional contact between surfaces.

The cooling of the brakes is largely governed by the airflowcharacteristics through the disc vents and within the wheelarch of the car. Without knowledge of these flow speeds andpressures it is difficult to accurately simulate the convectivecooling of the disc. This is of particular importance to longerscenarios which incorporate many instances of braking. Ex-cessive or insufficient cooling over the course of the scenariocan have a significant effect upon the accuracy of the result-ing data. It may be possible to formulate an approximatetime varying convective cooling mechanism that uses an am-bient sink temperature that acts as a function of disc tem-perature. Additionally, there is potential to apply a velocitybased mechanism for convective cooling that approximatesthe aerodynamic cooling and resulting transitions betweenlaminar and turbulent flow schemes. This may prove diffi-cult to verify without CFD simulations, or a larger pool ofexperimental data.

Results from CFD studies carried out by M Tirovic in (Day2005) indicate that the convective heat transfer values differfrom region to region upon the disc. Convective heat trans-fer coefficients ranged from 4 to 40Wm−2K−1. It is possibleto configure the ABAQUS model so that the mechanism ofcooling is graduated across relevant surfaces. Heat transfercoefficients were recorded for multiple rotational velocities.These values would be a good starting point for a coolingscheme that uses the vehicle velocity to approximate suitablevalues for cooling coefficients. The current values for the con-vective cooling coefficient could be altered from a constant toa dynamic input.

References

Day, A. J. (2005). Braking of Road Vehicles. University ofBradford. isbn: 1-851-43230-2.

Guiggiani, M (2014). The Science of Vehicle Dynamics.Springer. isbn: 978-94-017-8532-7.

Limpert, Rudolf (2011). Brake Design and Safety. 3rd ed.SAE International. isbn: 978-0-7680-3438-7.

Modest, M. (2013). Radiative Heat Transfer. Elsevier. isbn:978-0-12-386944-9.

Tiguan (2015). url: http://www.volkswagen.co.uk/new/tiguan-gp/which-model.

20/04/2015 S-R001