Chapter 5 Non-Linear Contact Analysisarahim/Chapter5.pdf · Chapter 5 Non-Linear Contact Analysis...

Chapter 5 Non-Linear Contact Analysis 94

Chapter 5

Non-Linear Contact Analysis

5.1 Introduction

There are three prediction methods available for researchers in studying disc brake

squeal, namely complex eigenvalue analysis, dynamic transient analysis and normal

mode analysis. The former two methods are largely dependent upon contact

interaction between the disc and pad interface while the third does not take into

account the interaction at the disc/pad interface. It has been hypothesised that squeals

can be triggered when there are closer frequencies between the disc brake

components particularly the disc and the brake pad (Mahajan et al, 1999)

The essence of the complex eigenvalue method lies in the inclusion of an asymmetric

friction matrix that may be derived from the results of contact pressure analysis.

Therefore the contact pressure distribution is believed to be an important parameter

in order to examine the stability of the disc brakes. In this chapter contact pressure

distributions at the pads-disc interface are extensively studied. Comparisons are

made particularly in selecting a correct contact algorithm that is available in

ABAQUS. Further simulations on contact pressure distribution are also performed in

terms of assembly models, wear and variations in material properties. These results

are thought to be significant when predicting instabilities in the disc brake.

5.2 Comparisons on contact pressure distribution

5.2.1 Contact Interaction schemes

5.2.1.1 Small sliding versus finite sliding

As discussed in the previous chapter, ABAQUS does provide three schemes in

defining relative surface motions, namely small, finite and infinitesimal sliding. In

infinitesimal sliding, the scheme assumes that both the relative motion of the surfaces

and the absolute motion of the contacting bodies are small. Since large rotation is

necessary for the disc brake problem, this type of scheme is ignored. Small and finite

sliding schemes allow large rotation and include geometric nonlinearities and they


are being considered for comparison. Similarity and differences between small and

finite sliding are given as follows:

• Both formulations allow two bodies to undergo large motions. However

limitation in the small sliding is that it assumes that there will be a

relatively small amount of sliding of one surface along the other

• The slave node can transfer load to any nodes on the master surface in the

finite sliding while it can only transfer load to a limited number of nodes on

the master surface for the small sliding.

• In small sliding analysis every slave node interacts with its own tangent

plane on the master surface and consequently slave nodes are not monitored

for possible contact along the entire master surface. Thus the small sliding

is less expensive computationally than the finite sliding contact.

• The finite sliding contact requires that master surfaces to be smooth which

it must has a unique surface normal at all points while the small sliding

contact does not need master surfaces to be smooth.

In this section, comparisons between the two sliding schemes are made in terms of

contact pressure distribution, contact area and simulation time. In doing so, validated

model that previously developed is employed. A brake-line pressure of 0.85 MPa and

a rotational speed of 6.3 rad/s are imposed to the model. For the contact interface

between the pad and the disc a kinetic friction coefficient of µ = 0.522 is prescribed.

A penalty friction constraint is chosen for comparison. It is believed that the results

will suggest which scheme should be adopted throughout this research.

From figures 5.0a and 5.0b, it is seen that contact pressure distributions are almost

the same for both the piston and finger pads. Maximum contact pressures also nearly

identical for both schemes. Comparisons between small and finite sliding in terms of

contact area, maximum pressure and simulation time are described in Table 5.0. As

mentioned earlier, finite sliding scheme is more computationally expensive than the

small sliding scheme. This is proved to be true as stated in Table 5.0 in which finite

sliding takes about 1502s to complete the simulation while small sliding only takes

about 1385s which is a reduction in 8%. It seems that the two schemes have little

difference in contact analysis particularly for disc brake contact analysis. Based on


the results, small sliding scheme will be adopted for subsequent analyses due to its

computational advantage over the finite sliding in which a similar contact pressure

distribution, contact area and maximum pressure can be obtained for both schemes.

Furthermore, using finite sliding more care should be paid in smoothing the master

contact surfaces whereas nothing need to be done for small sliding.

a) Small sliding

b) Finite sliding

Figure 5.0: Contact pressure distribution between small (top) and finite sliding

(bottom) schemes at the piston (left) and finger (right) pads. Top of the diagrams are

the leading edge


Table 5.0: Comparison between small sliding and finite sliding

Small Sliding Finite Sliding Parameter

Piston Finger Piston Finger

Contact Area (m2) 5.74E-4 6.24E-4 5.74E-4 6.23E-4

Highest Contact

Pressure (MPa) 17.94 9.44 18.00 9.46

Simulation Time (s) 1385 1502

5.2.1.2 Penalty versus Lagrange for friction constraints

By default, ABAQUS uses a penalty scheme to impose friction constraints for

tangential interaction in the contact analysis. The penalty scheme allows some

relative motion of the surfaces when it should be sticking. The magnitude of sliding

is limited to an elastic slip, which is characterised by slip tolerance and contact

surface length. Another method that can enforce the sticking and sliding constraints

more precisely is Lagrange multipliers scheme. This scheme does not allow any

relative motion between two closed surfaces until resultant shear stress is equivalent

to the critical shear stress. Therefore presence of small amounts of microslip is

permitted (Oldfield, 2004). Since Lagrange multipliers add more degrees of freedom

to the model and often increase the number of iterations for a converged solution,

this scheme leads to an increase in computational time. Furthermore, convergence is

also an issue using the Lagrange multipliers compared to the penalty method.

Nevertheless, it is worthwhile to perform simulation using both schemes for the sake

of comparison.

In doing so, results obtained in the previous section, i.e., from small sliding scheme,

are used. By looking at Tables 5.0 and 5.1, particularly at the contact area and

maximum pressure there are no differences between the two schemes. Similarly

contact pressure distributions as shown in figures 5.0a and 5.1 in both schemes are

identical. However in terms of the computational cost, Lagrange multipliers scheme

requires more time for completing the simulation compared to the penalty scheme.

Lagrange multipliers scheme takes about 2040s for a single analysis whereas penalty

method only takes around 1385s, which is an increase of 47% in computational time.

From the results, it is suggested that for disc brake squeal problem exact sticking


condition is not necessary. This is true because most of squeal events occur at quite

high speeds. It has been found in (James, 2003) that most of squeal events occur at a

rotational speed of above 3 rad/s or equivalent to 2.3 km/h of a vehicle speed (or

equivalent to linear speeds of 0.9 km/h and 1.4 km/h at the inner and outer radius of

the brake pad respectively). Furthermore, it has been observed that one of the main

characteristics of squeal is that no apparent sticking state is present at the disc/pad

contact interface. Even though one can argue that this (no apparent sticking) may be

applied at the macroscopic level but not in the microscopic state. As this work only

considers squeal behaviour at macroscopic level, any conditions or behaviour that is

present at microscopic level is not under consideration. Therefore, it is thought that

penalty scheme is most suitable for this work due to its computational cost advantage

over Lagrange multipliers and will be used in subsequent analyses.

Figure 5.1: Contact pressure distribution using Lagrange multipliers formulation

at the piston (left) and finger (right) pads. Top of the diagram is the leading edge

Table 5.1: Simulation results of contact analysis

Lagrange Multipliers Parameter

Piston Finger

Contact Area (m2) 5.74E-4 6.24E-4

Highest Contact

Pressure (MPa) 17.94 9.44

Simulation Time (s) 2040


5.2.2 Brake Pad Surface Topography

Liles (1989) is a pioneer of developing a large finite element model and incorporated

it with the complex eigenvalues analysis to study brakes squeal. In his model, a

perfect and smooth friction material is assumed. This has been followed by many

researchers after him. With limitations of computer technology and experimental

methods at that time, confirmation on the predicted contact pressure distribution is

rarely made. Thus, correlation between predicted and experimental results has never

been investigated.

Nowadays with advances of computer technology, contact pressure distribution can

be simulated much easier and faster. Static contact pressure distribution can also be

captured in the experiments more accurately using current tools and equipments such

as Pressurex

sensor film with Topaq

system (Sensor Products LLC), PET films

(Nitta, 1995), a transferred oil film (Yamaguchi et al, 1997) and the ultrasonic

reflection (Quinn et al, 2002 and Pau et al, 2004). Even though dynamic contact

pressure remains impossible to measure and capture, it is thought that measured

static contact pressure can be used for confirmation of FE analysis. It is also the time

to determine whether the previous assumption on perfect surface model is still valid

or not.

In this section, two different contact surface models are simulated. The first model

assumes the perfect surface while the second model considers real surface profiles

that have been measured in the previous chapter. Comparison is made on both on

static and dynamic contact pressure distributions (defined as when the disc is being

rotated but there is no vibration). However, confirmation against experimental results

is only made on the static contact pressure distribution, because experimental results

of only static contact are possible and available. First the perfect surface model is

simulated under a brake-line pressure of 2.5 MPa and a rotational speed of 6 rad/s.

For the contact interface between the pad and the disc, static and kinetic friction

coefficient of sµ = 0.7 and kµ = 0.522 (taken from experiments) are prescribed

respectively. At a stationary condition, it is found that contact pressures are

distributed symmetrically both at the piston and the finger pads as shown in figures

5.2a and 5.2b. When the disc starts to rotate at a specified rotational speed contact


pressure distributions at the piston and the finger pads are no longer symmetric and

are now shifted towards the leading edge. This is consistent with previous findings

(Tirovic and Day, 1991, Ripin, 1995, Hohmann et al, 1999, Tamari et al, 2000, and

Rumold and Swift, 2002). However, correlation against experimental results in terms

of static contact pressure distributions and contact area are not good. From figure

5.3a, it is seen that predicted contact areas are much higher than those in the

experiments, which give relative errors up to 162% and 136% for the piston and the

finger pads respectively. While for the real surface model, as discussed in the

previous chapter, a good correlation is achieved between predicted and experimental

results with small relative errors in static contact areas as described in figure 5.3a.

For the real surface model, contact pressure distributions are also shifted towards the

leading edge when the disc starts to slide as shown in figures 5.2c and 5.2d.

Since dynamic contact pressure cannot be measured experimentally, predicted results

on the real surface model are used for comparison against the perfect surface model

for sliding disc. By looking at the figures 5.2a and 5.2b (perfect surface model)

against figures 5.2c and 5.2d (real surface model) during sliding condition, there are

huge differences between the two results particularly in contact pressure

distributions, maximum pressures and contact areas. For comparison, contact areas at

the piston and finger pads are considered. From figure 5.3b it shows that contact

areas during disc sliding condition for the perfect surface model are higher than those

obtained for the real surface model (an increase of about 121% for the piston pad and

79% for the finger pad). Based on the results predicted in both models and

comparing to the experimental data, it is suggested that assumption made by previous

researchers on perfect surface is no longer valid. Therefore from this point upwards,

one should consider a real surface topography and the brake pad should be modelled

accordingly.


a) Perfect surface-piston pad

b) Perfect surface- finger pad

c) Real surface- piston pad

Figure 5.2: Contact pressure distributions at Ω =0.0 rad/s (left) and Ω =6.0

rad/s (right). Top of each diagram is the leading edge


d) Real surface- finger pad

Figure 5.2 (cont’d): Contact pressure distributions at Ω =0.0 rad/s (left) and Ω =6.0

rad/s (right). Top of each diagram is the leading edge

a) Ω = 0.0 rad/s

b) Ω = 6.0 rad/s

Figure 5.3: Comparison on contact area at different speeds


5.2.3 Disc Brake FE Modelling

The subject of contact pressure distribution has been studied by a number of people.

Several FE assembly models have been developed and proposed. For instance, Ripin

(1995) used only a rigid surface for the disc and deformable brake pads when

studying effect of several parameters on contact pressure distribution. Lee et al

(1998) included only a deformable disc surface and deformable brake pads. Tirovic

and Day (1991) and Hohmann et al (1999) presented a deformable disc and included

more disc components than those of Ripin (1995) and Lee et al (1998). Among them,

only Tamari et al (2000) and Rumold and Swift (2002) employed a deformable disc

with all disc brake components. The current FE model is referred to the previous

model that has been used earlier in this chapter.

In this section, several levels of a FE assembly model as proposed by the above

researchers are simulated. The main purpose is to see how contact pressure

distribution and contact area vary from one model to another. Comparison is also

made between the proposed models against the current model. The proposed models

are given as follows:

a) Model A1: A rigid disc surface with a piston and a piston pad.

b) Model A2: Rigid disc surfaces with a piston pad

c) Model A3: Deformable disc surface and brake pads

d) Model A4: Deformable disc surface with major disc brake components

5.2.3.1 Model A1

Model A1 consists of the piston, piston pad and rigid surface for the disc as shown in

figure 5.4. There are a number of spring elements between the piston and the pad

back plate with appropriate stiffness values. The outer wall of the piston is

constrained by spring elements in the radial direction. At the trailing abutment the

pad is constrained by spring elements in the radial and circumferential direction

while at leading side only circumferential direction is constrained. Such an assembly

model has never been proposed by any previous researchers. However it is

worthwhile to simulate the model with the presence of the piston and spring elements

as contact interaction as opposed to the models A2 and A3.


In this simulation, a brake-line pressure of 0.85 MPa and a rotational speed of 6.3

rad/s are imposed to the model. For the contact interface between the pad and the

disc, a kinetic friction coefficient of µ = 0.522 is prescribed. From figure 5.5 it is

seen that contact pressure distribution is concentrated at the trailing edge. Contact

area is increased around 17% more than the current (baseline) model as described in

figure 5.12a. However, the maximum pressure is found to be less than the current

model by 21% as illustrated in figure 5.12b.

Figure 5.4: FE model with a pad, a piston and a rigid surface

Figure 5.5: Contact pressure distribution at sliding speed of Ω = 6.3 rad/s of the

piston pad. Top of the diagram is the leading edge


a) Predicted contact area

b) Predicted maximum contact pressure

Figure 5.6: Simulation results of contact area and maximum pressure.

5.2.3.2 Model A2

This assembly model consists of two rigid surfaces of the disc and two brake pads as

shown in figure 5.7. Constraints are applied in the radial and circumferential

directions both at the leading and trailing abutments. Brake-line pressure applies

directly to the top of the back plates. This level of modelling is presented by Ripin

(1995) to obtain contact pressure distribution.


Figure 5.7: FE model with two rigid surfaces and two deformable pads

Brake operating conditions are applied similarly as described in Model A1. At the

piston pad the area in contact is seen much lesser than the current model and Model

A1 as shown in figure 5.8. There are about 81% and 37% of reduction in contact area

and maximum pressure respectively compared to the current model as described in

figure 5.6. While for the finger pad, the contact area and maximum pressure are also

seen to be less than the current model by reduction of 26% and 34% respectively.

Contact pressure distribution, and comparison of contact area and maximum pressure

are shown in figures 5.8 and 5.6 respectively.

Figure 5.8: Contact pressure distributions at rotational speed of Ω =6.3 rad/s of

the piston (left) and the finger (right) pads. Top of the diagram is the leading edge


5.2.3.3 Model A3

For Model A3, a deformable disc and two brake pads, as shown in figure 5.9, are

developed. The model assembly is replicated to the one, which is presented by Lee et

al (1998) in their study. The assembly model is constrained similar to Model A2.

Brake-line pressure is imposed directly onto the top of the back plates.

Figure 5.9: FE model with a deformable disc and brake pads

From figure 5.10 it is shown that area in contact and maximum pressure are almost

identical to Model A2 for the piston pad. However it still has much difference

compared to the current model with differences between the two models being –78%

and –32% for contact area and maximum pressure respectively. While for the finger

pad predicted contact area and maximum pressure seem to have different scenarios,

which it only gives a minor difference with 3% extra in contact area but around 40%

reduction in the maximum pressure.

Figure 5.10: Contact pressure distribution of the piston (left) and the finger (right)

pads at a rotational speed of Ω =6.3 rad/s. Top of the diagram is the leading edge


5.2.3.4 Model A4

This assembly model consists of a disc, two pads, a calliper and a piston as shown in

figure 5.11. Such a model is proposed by Tirovic and Day (1991) and Hohmann et al

(1999). The calliper arms are constrained in the radial and circumferential directions.

The disc is constrained in all three translational directions but it free to rotate in the

normal direction of the disc surface. Spring elements are employed for contact

interactions between the components except at the disc/pad interface.

From figure 5.12 it is shown that contact pressure distribution is almost the same

with the current model, as shown in figure 5.0a, at the piston pad. The contact area

and maximum pressure are different from those of the current model by about +16%

and –4% respectively as described in figure 5.6. However there is a large difference

in contact area at the finger pad, which gives about 46% reduction to that in the

current model. While it is seen from figure 5.6 that the maximum pressure is closest

to that predicted in the current model, by a reduction of 2%.

Figure 5.11: FE model without a carrier and two guide pins


Figure 5.12: Contact pressure distribution of the piston (left) and the finger (right)

pads at a rotational speed of Ω = 6.3 rad/s. Top of the diagram is the leading edge

5.2.4 Effect of Wear

When two solid bodies are rubbed together they will experience material removal

wear. In engineering applications the wear depth is a function of normal pressure,

sliding distance and specific wear coefficient and other factors. Liu et al (1980)

stated that Rhee (1970) in his study had shown the wear rate of most friction

materials could be given as follows:

cbatvkph = (5.0)

where h is the amount of wear , p is the contact force , v is the sliding speed, t is the

time and k is the wear constant which is a function of the material and temperature.

a, b and c are constants that should be determined experimentally and c is usually

close to unity. The effect of the load and the sliding speed on the wear rate in current

work is assumed to be linear and therefore constants, a and b, are taken to be unity.

With this assumption, equation (5.0) now reduces to that of equation (3.14).

In incorporating wear into the FE model, the methodology that was proposes by

Podra et al (1999) and Kim et al (2005) is adopted in this work. Even though Bajer et

al (2004) also performed wear simulation particularly in disc brake their

methodology requires a specific version of ABAQUS, namely v6.5. This version is

not available to the researcher at the moment. In simulating wear, contact analysis is


firstly performed in order to determine normal contact pressure generated at the

piston and the finger pads interface. Using equation (3.14), wear

displacements/depths are calculated based on the following parameters:

1) Predicted contact analysis generates in the contact analysis, p

2) Sliding time, t

3) Specific wear rate coefficient, k

4) Pad effective mean radius, r

5) Rotational speed, Ω (depends on each case)

The sliding time is arbitrary and is set to t = 600s for each simulation in order to

observe any significant changes in wear pattern and contact pressure distribution.

This reflects a braking test in which the disc is running for 600 second under braking

and then stops. After that the test is repeated up to several times. A constant specific

wear rate coefficient is assumed for all braking applications and is set to k = 1.78e-

13m3/Nm (Jang et al, 2004). The effective pad radius is r = 0.11m and rotational

speed is constant and it varies from case to case. Then, based on the calculated wear

displacements at steady state, a new surface profile for the piston and the finger pads

is created. Figure 5.13 shows the overall procedure of wear simulation that has been

proposed in this work. It is worth noting that wear displacements calculated in this

work is not validated with experimental data. Further investigation is required and

this is not considered in this thesis.

For a case study, a brake-line pressure of 0.81 MPa and a rotational speed of 3.2

rad/s are imposed to the model. For contact interface between the pad and the disc a

kinetic friction coefficient of µ = 0.565 is prescribed. Effect of wear on contact

pressure distribution is performed up to six different simulations. It can be seen from

figures 5.14b ~ 5.14g that contact pressure distributions are spreading away from

their initial locations (figure 5.14a). At the piston pad, it is shown that contact

pressure distribution is largely spreading at the trailing edge. While for the finger

pad, contact pressure distribution seems to spread symmetrically at the leading and

trailing edges. It is suggested that given a long period of braking application contact

pressure distribution can spread to cover entire surface of the friction material.


WEAR CALCULATION

- Calculate wear displacements

rtkphw Ω=

Figure 5.13: Flow chart of proposed wear simulation

From the wear simulation, it is found that contact area varies from one simulation to

another. The areas in contact at the piston pad as shown in figure 5.15(a) are larger

than the baseline model by 26%, 25%, 5% and 8% for simulations 1, 2, 3, and 4

respectively. While for simulations 5 and 6 the contact areas are slightly less than the

baseline model by 4% and 13% respectively. At the finger pad, it is observed that

the contact areas are all less than the baseline model except for simulation 4. The

relative contact areas compared to the baseline model are –3%, -9%, -24%, -3% and

–16% for simulations 1, 2, 3, 5 and 6 respectively while + 11% for simulation 4.

INPUT FILE

- Create initial surface

profiles of the brake pads,

hi

CONTACT ANALYSIS

- Determine normal

contact pressure

PROFILE

MEASUREMENTS

- Measure surface profile

of the piston and the

finger pads

WEAR PROFILE

- Change the node geometry

win hhh m=


The maximum pressure predicted in the wear simulation also varies for each wear

simulation. Differences in the maximum contact pressure can be seen from figure

5.15(b). From the results, it shows that at the piston pad most wear conditions give

less maximum contact pressure than the baseline model except for simulation 5

which gives more maximum contact pressure. The relative differences in maximum

contact pressure for simulation 1, 2, 3, 4 and 6 are 52%, 48%, 59%, 43% and 56%

respectively while +1% for simulation 5. At the finger pad, the maximum contact

pressures are found to be less than the baseline model by 11%, 22%, 13% and 25%

for simulations 1, 2, 4 and 5 respectively. While maximum contact pressure is found

to be larger than the baseline model by 38% and 58% for simulations 3 and 6

respectively.

a) Baseline model

b) Simulation 1

Figure 5.14: Predicted contact pressure distributions under wear simulation at the

piston (left) and finger (right) pads. Top of each diagram is the leading edge


c) Simulation 2

d) Simulation 3

e) Simulation 4

Figure 5.14(cont’d): Predicted contact pressure distributions under wear

simulation at the piston (left) and finger (right) pads. Top of each diagram is the

leading edge


f) Simulation 5

g) Simulation 6

Figure 5.14(cont’d): Predicted contact pressure distributions under wear simulation

at the piston (left) and finger (right) pads. Top of each diagram is the leading edge




Figure 5.15: Results of predicted contact area and maximum contact pressure for

several wear simulations

5.2.5 Effect of Elastic Properties

Variations in material properties of disc brake components can have immense

influence on components/assembly dynamic characteristics and squeal generation. In

a recent review Chen et al (2003c) showed that variation of density and Young’s

modulus in the disc and friction material can change dynamic characteristics of the

components and as a result the brake system instabilities are also changed.

Furthermore, they stated that variations in contact pressure might have a significant

effect on excitation or squeal generation. These contact pressure variations might

also be due to variations in material properties of above components. The effect of


friction material modulus on contact pressure distribution has been studied by

Tirovic and Day (1991) and Ripin (1995). In this section, friction material and disc

elastic properties are varied for a particular range and are simulated in order to see

variations on the contact area and maximum contact pressure.

5.2.5.1 Friction material

The friction material used in this work is an orthotropic material, which baseline

elastic constants are described in Table 4.2 in the previous chapter. These constant

will be varied from %5± to %20± . The results for different elastic constants are

shown in figures 5.16a and 5.16b for predicted contact area and maximum contact

pressure. In figure 5.16a it is seen that for softer friction material contact areas at the

piston and the finger pad are slightly higher that the baseline. While for the hard

material, i.e. increased by 5% to 20% it shows that contact areas are decreased by a

small amount. However the trends are shown differently for the maximum contact

pressure where it is slightly lower for softer material and are a bit higher for the hard

material in the piston and the finger pad. These trends are acceptable since in general

as the contact area increases the maximum contact pressure should decrease. From

the results it is found the highest contact areas are 7.0e-4 m2 and 7.5e-4 m

2 for the

piston and the finger pads respectively when the elastic constants are reduced by

20%. While the lowest contact areas are 5.1e-4 m2 and 5.4e-4m

2 for the piston and

the finger pads respectively when the elastic constants are added by 20%. Therefore

it is suggested that softer friction material can increase area in contact while the hard

material reduces it. This finding is consistent with the results of Tirovic and Day

(1991) and Ripin (1995).


0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

Co

nta

ct

Are

a (

1E

-4 m

^2)

-20

-15

-10 -5

Baselin

e 5

10

15

20

Variation (%)

Piston pad

Finger pad


0

2

4

6

8

10

12

14

16

18

20

Maxim

um

Pre

ssu

re (

MP

a)

-20

-15

-10 -5

Baselin

e 5

10

15

20

Variation (%)

Piston pad

Finger pad


Figure 5.16: Results on the effect of Young’s modulus of the friction material

5.2.5.2 Disc

Similar to the friction material, the disc Young’s modulus will be varied from %5±

to %20± . From figures 5.17a, it can be seen that variations in disc Young’s modulus

do not much influence contact area particularly at the piston pad. There is small

increase in the contact area i.e. around 4E-5m2 at the finger pad when the Young’s

modulus is decreased from the baseline. However in figure 5.17b, it is found that the

maximum contact pressure remains the same for the piston and the finger pads for all


range of variations. From those results, it is suggested that the maximum contact

pressure is not sensitive to the variations of disc Young’s modulus.

5.2

5.4

5.6

5.8

6.0

6.2

6.4

6.6

6.8

7.0

7.2

Co

nta

ct

Are

a (

1E

-4 m

^2)

-20

-15

-10 -5

Baselin

e 5

10

15

20

Variation (%)

Piston pad

Finger pad


0

2

4

6

8

10

12

14

16

Maxim

um

Pre

ssu

re (

MP

a)

-20

-15

-10 -5

Basel in

e 5

10

15

20

Variation (%)

Piston pad

Finger pad


Figure 5.17: Results of variations on Young’s modulus of the disc


5.2.5.3 Backing plate

Figures 5.18a and 5.18b show the area in contact and maximum contact pressure at

the piston and finger pads respectively. From the results, it shows that there are no

apparent changes in the maximum contact pressure at the piston and finger pads

compared to the baseline model. There are also no significant changes in the contact

area of the piston pad under a wide range of variation of Young’s modulus.

However, the finger pad shows slight variations in the contact area, which reduces in

Young’s modulus, tend to increase the contact area and vice-versa. By increasing

Young’s modulus, the area in contact tends to decrease. Even though contact area at

the finger pad looks to decrease significantly, the difference is actually quite small

i.e. only around 1E-4m2

(between -20% and +20%) thus it might not affect maximum

contact pressure. Furthermore maximum contact pressure distribution is predicted at

level of 1x106 Pascal. Any small changes in the maximum contact pressure cannot be

clearly seen in the figure. From the predicted results, it seems to suggest that

variation of the Young’s modulus of the backing plate did not much affect overall

behaviour of the contact area of the brake pads and it certainly not influence the

maximum contact pressure.


5.2

5.4

5.6

5.8

6.0

6.2

6.4

6.6

6.8

7.0

7.2

Co

nta

ct

Are

a (

1E

-4 m

^2)

-20

-15

-10 -5

Baselin

e 5

10

15

20

Variation (%)

Piston pad

Finger pad


0

2

4

6

8

10

12

14

16

Maxim

um

Pre

ssu

re (

MP

a)

-20

-15

-10 -5

Baselin

e 5

10

15

20

Variation (%)

Piston pad

Finger pad

b) Predicted maximum contact Pressure

Figure 5.18: Results of variations on Young’s modulus of the backing plate

5.2.5.4 Carrier

Figures 5.19a and 5.19b show predicted contact area and maximum contact pressure

at the piston and finger pads. From the results, it seems to suggest that at the piston

pad the areas in contact for reductions of 5% to 15% are almost identical to the

baseline model. However, when it comes to further reduction, i.e. 20% the area in

contact increases significantly. While when the Young’s modulus is increased by 5%

similar amount of contact area to the baseline model is predicted. The area in contact

is increased slightly when the Young’s modulus increased by 10% to 20%. For the


finger pad, the highest contact area is found at reduction of 15%. While the lowest

contact area is predicted at reduction of 10%. Similar level of contact area is found at

increments of 10% to 20% but is still lower than the baseline model. Slightly higher

contact areas are generated at reductions of 5% and 20%. While at increment of 5%,

the area in contact is almost identical to the baseline model. In the meantime, there

are no significant changes in maximum contact pressure at the piston and finger

pads. This indicates that variations of Young’s modulus of the carrier did not much

influence the maximum contact pressure.

5.4

5.6

5.8

6.0

6.2

6.4

6.6

6.8

7.0

Co

nta

ct

Are

a (

1E

-4 m

^2)

-20

-15

-10 -5

Baselin

e 5

10

15

20

Variation (%)

Piston pad

Finger pad


0

2

4

6

8

10

12

14

16

18

Maxim

um

Pre

ssu

re (

MP

a)

-20

-15

-10 -5

Baselin

e 5

10

15

20

Variation (%)

Piston pad

Finger pad


Figure 5.19: Results of variations on Young’s modulus of the carrier


5.2.5.5 Calliper

Variations of Young’s modulus of the calliper are investigated. Figures 5.20a and

5.20b show predicted contact area and maximum contact pressure of the piston and

finger pads. From figure 5.20a, it illustrates that there are no changes in contact area

for the piston pad under a wide range of Young’s modulus. Similarly, areas in

contact are almost similar to the baseline model at the finger pad for reductions of

5% to 20%. However, there are slightly changes in contact area particularly at

increments of 5% to 20%. For the maximum contact pressure, figure 5.20b seems to

indicate that there are no changes for the whole variations either in the piston pad or

the finger pad.

5.4

5.6

5.8

6

6.2

6.4

6.6

6.8

7

Co

nta

ct

Are

a (

1E

-4 m

^2)

-20

-15

-10 -5

Baselin

e 5

10

15

20

Variation (%)

Piston pad

Finger pad


0

2

4

6

8

10

12

14

16

Maxim

um

Pre

ssu

re (

MP

a)

-20

-15

-10 -5

Baselin

e 5

10

15

20

Variation (%)

Piston pad

Finger pad


Figure 5.20: Results of variations on Young’s modulus of the calliper


5.3 Summary

This chapter has focused on non-linear contact analysis of the disc brake model with

the main objective of determination of contact pressure distribution, contact area and

maximum contact pressure. These three parameters are useful for subsequent work

especially in comparing predicted results from one model to another. The first two

earlier sections describe several potential contact interaction schemes that are

available in ABAQUS. The first comparison is made between small sliding and

finite sliding schemes. It is found that even though the small sliding scheme assumes

the slave node could slide by relatively a small amount at the master surface

compared to the finite sliding scheme the predicted results between the two schemes

are almost identical. The main advantage of the small sliding over the finite sliding

scheme is that it saves about 8% of computational time.

The second comparison is made to examine two friction stiffness constraints, namely

the penalty method and Lagrange multipliers method. The penalty method allows

some relative motion of the surfaces during sticking while Lagrange multipliers do

not allow at all the relative motion during sticking state. In addition, Lagrange

multipliers can enforce more precisely the sticking and sliding constraints than the

penalty method. However, computational cost is an issue as Lagrange method takes

more time for a single analysis. This is proved to be true when Lagrange method

requires 2040s compared to 1385s using the penalty method, which is an increase of

47% in computational time. On the other side, predicted results for both methods are

identical. By looking at those results, the penalty method is more suitable and will be

used together with small sliding scheme for subsequent work.

The third comparison that follows is to determine validity of current methodology

and to compare it with a previous common practice that assumes a perfect surface.

The first comparison is made on static contact pressure between the real surface and

the perfect surface against experimental results. Then the second is made on dynamic

contact pressure between the real and perfect surfaces. The results have shown that

predicted static contact pressure distribution and contact area are more realistic to the

experimental results in the real surface model than those obtain in the perfect surface

model. There are also large differences in the dynamic contact pressure distribution


and contact area between the real surface and the perfect surface models. Therefore

from those comparisons, it is suggested that current methodology adopted in this

work is acceptable and previous assumption is no longer valid.

Different levels of disc brake modelling are also studied. Four disc brake models are

simulated and its effect on contact pressure distribution, contact area and maximum

contact pressure are examined and compared to the baseline model. It is shown from

those four models that none of them have similar predictions with the baseline

model. Therefore, it is suggested that a careful consideration should be taken in

developing a reduced model since boundary conditions and component interactions

have significant effects in contact analysis. Given inaccurate results in contact

pressure distribution may lead discrepancy in subsequent analysis such as the

complex eigenvalue analysis.

The following study is to see variations of contact pressure distribution, contact area

and maximum contact pressure under the effect of wear. In simulating wear effect, a

simplified wear formula is used. Contact pressure distribution, contact area and

maximum contact pressure are used to compare for each wear simulation with the

baseline model. The results have shown that these three parameters vary from one

simulation to another. Contact pressure distribution is spreading away from its initial

locations at the centre of the brake pads towards the leading and/or trailing edge. It

can be predicted that given a long period of a braking application, the whole area of

the brake pads will be in contact.

In the last section variations of contact area and maximum contact pressure due to

changes in elastic properties of selected components were investigated. For friction

material it is observed that softer material could increase contact area while hard

material reduces it. For rest of the components, it is found that changes in its elastic

properties do not much affect the maximum contact pressure even though there are

slightly changes in the contact area. It means that the maximum contact pressure is

not sensitive to any changes in elastic properties of the disc, calliper, backing plate

and carrier.

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