thermal analysis of vented and normal disc brake rotors-doc

8/13/2019 Thermal Analysis of Vented and Normal Disc Brake Rotors-doc

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THERMAL AND STRUCTURAL ANALYSIS OF

VENTED AND NORMAL DISC BRAKE ROTORS

APROJECTREPORT

SUBMITTEDINPARTIALFULLFILMENT

FOR

THEAWARDOFDEGREEOFBACHELOROFTECHNOLOGY

IN

MECHANICALENGINEERING

BY

CH.KRISHNACHAITANYAVARMA(07241A0309)

PADMANABHDAS(07241A0313)

PUNEETKUMAR.J(07241A0315)

DEPARMENTOFMECHANICALENGINEERING

GOKARAJURANGARAJUINSTITUTEOFENGINEERINGANDTECHNOLOGY

(AFFLIATEDTOJAWAHARLALNEHRUTECHNOLOGICALUNIVERSITY)

HYDERABAD

20072011

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DEPARTMENT OF MECHANICAL ENGINEERING

JNTUCOLLEGE OF ENGINEERING

CERTIFICATE

Certificate

ThisistocertifythattheprojectentitledTHERMAL AND STRUCTURAL ANALYSIS OF VENTED

AND NORMAL DISC BRAKE ROTORS being submitted by Mr. ch Krishna chaitanya varma,

Mr. puneet kumar.j, Padmanabh das, in partial fulfillment for the award of degree of bachelor o

technology in mechanical engineering to the jawaharlal Nehru technological university is a record

of bonafide work carried out under my guidance and supervision. The results embodied in thisproject have not been submitted to any other university or institute for the award of degree

external guide internal guide

Mr. pradeep reddy. Mr. v. ratna kiran.

Proprietor. asst. professor.

orange technologies. griet.

Ameerpet, bachupally,

Hyderabad. Hyderabad.

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ACKNOWLEDGEMENT

I express my gratitude to chairman, project Review Committee, JNTU College of

Engineering, for their valuable recommendations and for accepting this project work.

I express my deep sense of gratitude towards my able and acknowledge guide, Mr. Ratna

kiran, Asst. Professor, Mechanical Engineering , GRIET, Hyderabad, to whom I owe the credit

of being the moving spirit behind this project, whose guidance and constant inspiration led me

towards its completion.

I convey my sincere thanks to Mr.K.G.K MURTHY, Head of the Mechanical Engineering

Department & Mr.P.S.V.KURMA RAO Professor, GOKARAJU RANGARAJU INSTITUE OF

ENGINEERING AND TECNOLOGY, HYDERABAD for his kind cooperation in the

completion of the project.

At this juncture, I feel that, I am grateful to Mr.PRADEEP, ORANGE TECHNOLOGIES,

AMEERPET, HYDERABAD, for assistance in completion of project work.

Finally, I extend my sense of gratitude to all my friends, teaching and non teaching staff,

who directly or indirectly help me in this endeavor.

CH, Krishana chaitanya varma (07241A0309)

Padmanabh Das (07241A0313)

Puneet kumar. J (07241A0315)

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ABSTRACT

Safety aspect in automotive engineering has been considered as a number one

priority in development of new vehicle. Each single system has been studied anddeveloped in order to meet safety requirement. Instead of having air bag, good

suspension systems, good handling and safe cornering, there is one most critical system

in the vehicle which is brake systems.

Without brake system in the vehicle will put a passenger in un safe position. Therefore,

it is a must for all vehicles to have proper brake system. Due to critical system in thevehicle, many of researchers have conducted a study on brake system and its entire

component. In this project, the author has conducted a study on ventilated and normaldisc brake rotor of normal passenger vehicle with full load of capacity. The study is

more likely concern of heat and temperature distribution on disc brake rotor.

Steady state and transient response has been conducted through the heat transfer analysiswhere to predict the worse case scenario and temperature behaviors of disc brake rotor.

In this study, finite element analysis approached has been conducted in order to

identify the temperature distributions and behaviors of disc brake rotor in steady stateand transient responses. Ansysis has been used as finite elements software to perform

the thermal analysis on both responses. Both results have been compared for better

justification. Thus, both results provide better understanding on the thermal

characteristic of disc brake rotor and assist the automotive industry in developingoptimum and effective disc brake rotor.

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NOMENCLATURE

c Specific heat (J/kg K)

C Capacity matrix

D Elasticity matrix

E Young.s modulus (N/mm)

h Heat convection coefficient (W/m K)

k Thermal conductivity (W/m K)

K Stiffness matrix

K Conductivity matrix

P Normal pressure (N/mm)

P Force vector

Ph Hydraulic pressure (N/mm)

q Heat flux (W/m)

r, , z Cylindrical coordinates

R Heat source vector

T Temperature (K)

T Ambient temperature (K)

U Displacement vector normal component of displacements

Thermal expansion coefficient ( /C)

Strain vector

Coefficient of friction

Poisson.s ratio

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Density (Kg/m)

Stress vector

Angular velocity (rad/s)

Subscripts

f Body force

i,j Sub regions i and j, respectively

n Normal direction surface traction

t Temperature rise (K)

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1.

Contents

1.

Introduction 91.1 Introduction 9

2. Statement of problem 10

3. objective of scope 11

4. research methodology 12

5. thesis outline 13

6. literature review 14

6.1introduction 14

7. brake system review 15-16

8. vehicle brake system 17-18

9. parts of disc brake 19

9.1disc calipers 19

9.2disc pad 20

9.3brake disc rotor 20

9.3.1 disc brake rotor description 21

9.4brake pads 21-22

10.modeling software 23

11.catia v5 24

11.1 introduction 24

11.2 part modeling 24

11.3 general modeling for each part 24-25

11.4 fundamental 25

12.finite element analysis 26

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12.1 introduction 26

12.2 procedure for ansys analysis 27

12.3 built the model 27

12.4 material properties 27

12.5 solution 28

13.finite element generation 29

13.1 boundary conditions and loading 29

13.2 model display 29

13.3 material defection 29-30

13.4 solution 30

13.5 post processor 30

14.finite element formulation for heat conduction 31-32

15.thermal analysis 33

15.1 types of thermal analysis 33

15.2 planning the analysis 33

16.structural analysis 34

16.1 types of structural analysis 34

17.structural and static analysis 35

18.modeling analysis 36

19.definition of problem domain 37

20.modeling and analysis 38-40

21.creating a finite element mesh 41

21.1 solid90 element description 40-42

22.applying the boundary conditions 33

22.1 thermal boundary conditions 43-44

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22.2 result 45-46

23.structural analysis of normal disc brake rotor 47

23.1 structural analysis of boundary conditions 47

23.2 results 48-54

24.creating a finite element mesh for ventilated disc brake rotor 55-56

25.applying the boundary conditions 56-59

26.structural analysis of ventilated disc brake rotor 60

26.1 structural boundary conditions 60-61

26.2 result 61-65

27.conclusion 66

28.references 67-68

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1. INTRODUCTION

1.1. Introduction

Brakes are most important safety parts in the vehicles. Generally all of the vehicleshave their own safety devices to stop their car. Brakes function to slow and stop the

rotation of the wheel. To stop the wheel, braking pads are forced mechanically against

the rotorldisc on both surfaces. They are compulsory for all of the modern vehicles and

the safe operation of vehicles. In short, brakes transform the kinetic energy of the carinto heat energy, thus slowing its speed.

Brakes have been retuned and improved ever since their invention. The increases intravelling speeds as well as the growing weights of cars have made these improvements

essential. The faster a car goes and the heavier it is, the harder it is to stop. An effective

braking system is needed to accomplish this task with challenging term where materialneed to be lighter than before and performance of the brakes must be improved. Today's

cars often use a combination of disc brakes and drum brakes. For normal sedan car,

normally disc brakes are located on the front two wheels and drum brakes on the backtwo wheels. Clearly shows that, together with the steering components and tyres

represent the most important accident avoidance systems present on a motor vehicle

which must reliably operate under various conditions. However, the effectiveness of

braking system depends on the design itself and also the right selection of material.systems than follow with some improvements. In order to understand the behaviors of

braking system, there are three functions that must be complied for all the time (Smith,2002);

a) The braking system must be decelerate a vehicle in a controlled and repeatablefashion and when appropriate cause the vehicle to stop.

b) The braking should permit the vehicle to maintain a constant speed when traveling

downhill.c) The braking system must hold the vehicle stationary when on the flat or on a

gradient.

Nowadays, there are lot of software has been developed in order to cater the

modeling and the finite element analysis on the vehicle component such as (Automatic

Dynamic of Mechanical Systems), CATIA, ANSYS. There is an advantage of using that

powerful computational analysis software where by using those would make it easier,less cost better accuracy and less computing time. Most of the software is used in the

wide range of industries such as automotive, oil and gas, aerospace, marine, heavy duty

engineering ,construction, electro-mechanical and general mechanical industries. In thisproject, design package CATIA and finite element package will be used to generate

model and run analysis on the chosen component.

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2.STATEMENT OF PROBLEM

If looking on the overall automotive parts, besides engines, there are more crucial

parts that engineers need to look into consideration. Suspension, brake, electrical,hydraulic and gear are all the crucial systems in the automotive areas. Each of all system

has their own functionality which brings life to the automation industries. Brakes is such

a crucial system in stopping the vehicle on all moving stages including braking duringhigh speed, sharp cornering, traffic jam and downhill. All of those braking moments

give a different value of temperature distribution and thermal stress. Good performance

of disc brake rotor comes from good material with better mechanical and thermal

properties. Good designs of disc brake rotor are varying across the range of the vehicles.There are different design and performance of disc brake rotor if compared between

passenger, commercial and heavy duty vehicle. There are also other constraints such ascost, weight, manufacturing capability, robustness and reliability, packaging,maintenance and servicing.

For example, heavy duty vehicle need large size of disc brake rotor if compared topassenger vehicle. Due to that, it will increased total weight of vehicle as well as fuel

consumption and reduces performances of the vehicle. Moreover, high weight of vehicle

induces to high temperature increased during braking where the higher value oftemperature during braking could lead to braking failure and cracking of disc brake

rotor.

This project concerns of the temperature distribution and constraint of the discbrake rotor. Most of the passenger cars today have disc brake rotors that are made of

grey cast iron (Mackin, 2002). Grey cast iron is chosen for its relatively high thermal

conductivity, high thermal diffusivity and low cost (Mackin, 2002). In this project, theauthor will investigate on the thermal issues of normal passenger vehicle disc brake

rotor, where the investigation are to determine the temperature behaviour of the disc

brake rotor due to severe braking of the disc brake rotor by using Finite ElementAnalysis (FEA).

According to (Valvano and Lee, 2000), braking performance of a vehicle can be

significantly affected by the temperature rise in the brake components. High temperature

during braking will caused to:

Brake fade, Premature wear, Brake fluid vaporization, Bearing failure, Thermal cracks,

Thermally-excited vibration.

Therefore, it is important to study and predict the temperature rise of a given brake

component and assess its thermal performance in the early design stage. Finite element

analysis (FEA) has been preferred and chosen method to investigate some of the above

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concerns such as disc brake rotor temperature rise and thermal cracks (Valvano and Lee,

3.OBJECTIVE AND SCOPE

The aim at the end of this project is to predict the temperature rise and the

temperature behaviour of ventilated disc brake rotor with full passenger in the vehicle.In

achieving this aim, project objectives are set as below:

To understand the working principles, components, standards and theories through

a literature study.

To understand the working principle of FEA Software (ANSYS)

To understand the fundamental of heat transfer through thermal analysis of disc

brake rotor.

To clearly justifL the result and conclusion.

The knowledge gained from this project is to be able to understand the steps needed

in thermal analysis of disc brake rotor by using FEA method. The methods used in thisproject can later be used in future as reference for similar research and development.

There is the wide range of study on the disc brake rotor. The disc brake rotor could

be studied on the various areas such as material improvement on the disc brake rotor,vibration on the disc brake, noise and squeal of the disc brake and thermal stress analysis

on the disc brake rotor. However, on this project, the author will intend to emphasize

details on the thermal analysis on the disc brake rotor of normal passenger vehicle with

fullcapacity of passenger.

The scopes of the projects are:

Literature review on the working principles, components, standards and theories.Construction of 2D and 3D model of disc brake rotor.

FE model (Meshing of Geometry model)

Finite element analysis on steady state and transient analysis which shows thetemperature distribution of disc brake rotor.

Final justification and conclusion.

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4.RESEARCH METHODOLOGY

Begin with a literature review, alot of paper and journal has been read up and a part of ithas been considered in this project. Meanwhile, Coordinate Measuring Machine (CMM)

has been used to measure the major coordinate of real disc brake rotor. CMM has beenused in order to get accurate dimension of disc brake rotor. Later, the precise dimensions

have been used to translate in 2D and 3D drawing by using CATIA.

In the second stage, load analysis has been done where the heat flux and convectional

heat transfer coefficients has been calculated. Load analysis calculated basedon full load of passenger in the normal passenger vehicle. Later, value of load analysis

has been applied on finite element analysis.

Next, the fractional 3D model of disc brake rotor has been transfer to finite element

software which is ANSYS. Thermal analysis has been done on steady state and transient

responses. Assigning material properties, load and meshing of the model has been donein this stages. Then, completed meshing model has been submitted for analysis. Finally

an expected result from the steady state and transient responses of thermal analysis has

been obtained. A flow chart below shows a better understanding of overall contents ofthis project.

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5.THESIS OUTLINE

This project consist total of seven chapters as Chapter 1 represent as an

introduction to this whole project. Chapter 1 described the problem statements,objectives and scope of the project, research methodology and the overall outline of the

contents in this project.

This chapter also described a description of ventilated disc brake rotor and its

components. A few published papers were reviewed and discussed in this chapter. Alsoa few theory of finite element analysis has been reviewed and introduction to finite

elements sohares are presented.

A few basic functions, operations and procedure of using CMM have been presented inthis chapter. Meanwhile, this chapter also discussed on material justification of disc

brake rotor. The material justifications described the material properties and

manufacturing process of disc brake rotor.

Load analysis has been done in Chapter 4. A few assumptions have been made in

order to reduce the complexity of the analysis. In load analysis, heat flux andconvectional heat transfer coefficient has been calculated. Then, the value of the load

analysis has been applied in the finite element analysis.

This chapter explained the development of 3D model and transferred process to finite

elements softwares. Then, the finite element analysis are clarified step by step fromassigning material properties till submitted the model for analysis.

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6.LITERATURE REVIEW

6.1 Introduction

Normally, thermal stress analysis has been performed to any of material related to

thermal process in order to oversee the behaviour and character of material. Anyabnormality regards to thermal input will give the high values on the stress magnitude of

the studied materials.

The high values of stress magnitude will shows deformation on

certain areas which load has been applied on it. Design and analysis of certain parts orcomponent will took much time and it is costly. Therefore, without any analysis or

design tools, it would be limitations on repeated analysis. For decades, finite element

analysis (FEA) has been a preferred method to address some of the above concerns. Itcan be used to compare the design alternatives and hence, optimize the brake rotor

design prior to production of prototype components (Valvano and Lee, 2000).

A literature review was conducted to investigate the past research that has been done inmany areas related to this work. In addition, description, histories, functions and theory

of disc brake rotor will be discussed in this chapter. Furthermore, theory of finite

element method related to thermal analysis will be presented as well in this chapter.

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7.BRAKE SYSTEM REVIEW

7.1 HISTORY OF BRAKE SYSTEM DEVELOPMENT

In the early days of the automobile, drum brakes were standard. Drum brakes

offered several advantages over other types of brakes. One of these was that the drum

could keep out water and dust, materials that could damage disc brakes which were outin the open.

Major advancement in brake technology came in 191 8 with the invention of fourwheelhydraulic brake systems by Malcolm Loughead. The hydraulic brake system

replaced the mechanical brake system that was in use at this time. The mechanicalsystem had numerous disadvantages. It made it difficult to brake all the wheels evenly,often causing a loss of control. In addition, it required drivers to exert tremendous

amounts of force on the brake pedal to slow the car. The hydraulic brake system

multiplied the force that was applied to the brake, lessening the amount of force needed

to be applied to the brake pedal by the driver.

This system was first used in the 1918 Duesenberg. Its advantages quickly caught on

and by 1929, four wheel hydraulic braking systems were standard equipment on higher

priced cars. The main problem with drum brakes is that the heat is not efficiently

disbursed. The heat that is produced inside the drum does not escape easily since the

drum prevents wind from drawing it away. However, disc brakes killed the issues whenit allowed the heat to be carried away which increased the efficiency of the brake.

However, their use was limited up until the 1950's since their efficiency was notrequired and they required more pedal pressure to operate. The reason for the higher

pedal pressure is that disc brakes have no self-servo effect or no self-energizing capacity

that the drum brakes have.

The self-servo effect is caused by the forward motion of the car. This forward motion

helps pull the brake shoe into contact with the drum. This helped lower the required

pedal pressure. Now that their efficiency was needed and the hydraulic brake systemmultiplied the force applied to the brake pedal, disc brakes seemed to be the better

alternative. Chrysler was the first to widely introduce the disc brake in its cars in theearly 1950's. The system did not have much success till automaker Studebaker toreintroduce the system in 1964. This time it saw much more success and in a few years,

disc brakes were common on most new cars. One of the reasons that disc brakes were a

success with the Studebaker and not theChrysler was due to the development of the power braking system. Power brakes

became

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common in the 195Ots, after Chrysler had developed and dropped its disc brake

program.The system assisted the movement of the piston in the master cylinder which meant that

the driver needed to apply less peddle pressure to get the same braking effectiveness.

Therefore, since ease of braking was no longer an issue, the adoption of the more

efficientdisc brake became widespread

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8. VEHICLE BRAKE SYSTEMAs we know, the basic functions of the brake system are to slow down a vehiclespeed to the point we need. It's also help to maintain acceleration during moving

downhill and keep the vehicle on static conditions. Brakes operate by converting the

kinetic energy (motion) of an automobile into heat energy(Source: Halderman J.D, 1996)

Driver exerts a force on brake pedal which is further amplified by power booster.The force on brake pedal pressurizes brake fluid in a master cylinder; brake fluid is

designed for extreme conditions, generally a silicone based DOT5 brake fluid is

recommended. The hydraulic force developed by brake fluid is transmitted to a wheelcylinder or calliper at each wheel which is used to force friction material against thedrum or rotor. The friction between the friction material and rotating drum or rotor

causes the rotating part to slow and eventually stop.

In the passenger or commercial vehicle, there are always two main types of brakes

assemblies that have been used. Those types of brakes are drum and disc brakes whichhave been described as below.

Drum brakes have their pads located inside of a drum.Like the disc in disc brakes, drum brakes also are attached to the wheels. Usually, main

components of drum brake for passenger or commercial vehicle consist of brake shoes,backing plate, parking brake cable and wheel cylinder. When the brake pedal is pressedthe curved brake shoes (pads) are pushed outward so that they make contact with the

rotating drum. Retracting spring is used in this type of brake (BOSCH, 1996). Just as

with disc brakes, this causes friction which turns kinetic energy into heat energy, thus

slowing and stopping the car to the right point. There is an advantage of using drumbrakes, where there is low cost of common parts. However, there are also some

disadvantages, such as the drum heats up and expands away from the lining material

which increasing fading. It is also have lower efficiency in wet braking action.

When the brake pedal is pushed, the pads (often called brake shoes) push up against thewheel disc. The wheel that attached with the rotor will affected by force from pads and

makes the wheel stop rotate. Those for both of disc and drum brakes are refer to

mechanical, hydraulic and power brake systems in order to make the brake systemsfunction smoothly. According to many researchers, disc brake system has many

advantages over drum brakes. The major part of rotor is exposed to air; therefore there is

sufficient air flow over brakes to dissipate the heat generated resulting in cooling down

of

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rotor temperature easily. The rotor expands in the direction of the friction material in

discbrakes as opposed to drum brakes. The pressure applied on the rotor is more uniform

resulting in even braking action as compared with drum brakes. It is also possible on wet

stopping when water slide off the rotor surface.WHEEL

)

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9. Parts of disc brake

9.1 DISC CALLIPERS

There are two types of disc callipers where further classified as floating and fixed

calliper. shows a type of floating calliper. This type of brake uses only a single piston tosqueeze the brake pad against the rotor (BOSCH, 1996). The reactive force shifts the

calliper housing and presses opposite side of braking pad against rotor. Referring to

Figure the brake fluid pushes the piston when the brake is applied to the left of thepiston and immediately pushes the inner pads and presses it against the rotorldisc, the

sliding calliper housing reacts by shifting towards right pushing the left pad against the

disc.

Floating Calliper Design

(Source: BOSCH Automobile Handbook, 1996)Other type of disc callipers is a fixed calliper. shows a type of fixedcalliper. In these types of brakes, the caliper body is fixed and uses two or more pistons

on each side of the rotor. The pistons are located in each half section of the fixed

calliper.

Hydraulic pressure is applied during braking to each of the piston. Each of the pistons

has a function to press against the brake pads of the brake disc. Shaped piston seals willretract the piston when the brakes are released. Referring t, the brake fluid pushes the

both piston when the brake is applied to the left and the right of the piston and

immediately pushes the both inner pads and presses it against the rotorldisc. Normally,

these types of brake calliper are used in high performance and heavy duty vehicle due tohigh physical strength.

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9.2BRAKE PADS

As shown in Figure brake pads consist of steel carrier which the pad arebonded to the steel carrier. According to (Gerschler, 1980), organically bonded pads

consist of metallic, ceramic or organic friction materials in a bonded mass such asrubber or synthetic resin. The bonded friction materials can withstand temperatures up to

750c, with short term peaks-up to 950'~ where the friction coefficient is between 0.25

and 0.5.

There is an advantage of brake pads, where most of them are poor to thermal

conductivity which protects the hydraulic actuating elements from overheating. It is alsoease to manufacture and low cost. However, the pads needs to inspect frequently due to

rapid wear as result from higher temperatures and contact pressures associated with theoperation of abrake disc.

9.3 BRAKE DISC / DISC BRAKE ROTOR

The heat generated on the surfaces of disc brake rotor when brake

applied. Materials of disc brake rotor usually are made from cast iron, spheroidal-

graphite cast iron or cast steel. It is chosen as a rotor material due to low cost of materialand performs high thermal resistance. This type of material normally suit to normal

passenger vehicle but not for high performance car. Once brake pads contacts to rotating

rotor, there will be huge amount of heat generated to stop or slow down the vehicle. Therotor temperature can exceed 350'~ for normal cars and 1500'~ for race cars (Halderman,

1996).

Disc brake rotor is a crucial part in the brake system where the main role of the rotor isto reduce the heat generated by dissipates all of the heat. In that case, ventilated disc

brake rotor is much better than solid rotor where more airflow from the surrounding area

to dissipate produced heat. Figure 2.9, shows the internal vanes allow air to circulatebetween two friction surfaces of the rotors

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9.3.1. DISC BRAKE ROTOR DESCRIPTION

Overall idea on vehicle brake system and disc brake theories has been described asabove. As similar to the type disc brake described above, the author used the disc brake

rotor from normal passenger vehicle. The disc brake rotor was taken from normal

passenger vehicle which having type of ventilated disc (Figure 2.1 0). Basically, discbrake rotor consists of rotating circular plate and cylinder disc (hat) attached and rotated

to wheel hub.

The rotating circular plate which also call annular disc has two flat surfaces separated

by 32 internal vanes. Figure 2.1 1 shows the cross sectional view of ventilated disc brake

rotor with outer diameter measured as 250 mm, 4.5 mm thickness of plate and havingmass approximately 4 kg

9.4. BRAKE PADS

As shown in Figure 2.7, brake pads consist of steel carrier which the pad are

bonded to the steel carrier. According to (Gerschler, 1980), organically bonded pads

consist of metallic, ceramic or organic friction materials in a bonded mass such asrubber or synthetic resin. The bonded friction materials can withstand temperatures up to

750c, with short term peaks-up to 950'~ where the friction coefficient is between 0.25

and 0.5.

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There is an advantage of brake pads, where most of them are poor to thermal

conductivity which protects the hydraulic actuating elements from overheating. It is alsoease to

manufacture and low cost. However, the pads needs to inspect frequently due to rapid

wear as result from higher temperatures and contact pressures associated with the

operation of a brake disc.

A Sample of Brake Pads

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10. MODELLING SOFTWARE

There are different softwares available for modeling some of them are:

1. Solid works

2. Pro-E

3. Ideas

4. Inventor

5. Mechanical desktop

6. Unigraphics

7. Catia v5

CATIA V5 (computer aided thre dimensional interactive application)a multi platform

CAD/CAM/CAE is used as the modeling tool in this project

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11.CATIAV511.1 INTRODUCTION

CATIA V5 provides the power of parametric design. With parametric, we define the

modal according to the size and positional relationship of its parts.

11.2PARTMODELLINGMany techinacal designs consists of complex assemblies made from angular shaped

parts. This type of design work can be made asier by part and assembly modeling

capabilities that are well integrated. The CATIA V5 is a 3-D parametric solid modeler

with both part and assembly modeling capabilities. You can see the CATIA V5 to model

piece parts and then combine them into more complex assemblies. With CATIA V5 a

part is designed by sketching its components shapes and defining their size shape and

inter relationships. By succesfuly creating these features you counstruct the part in a

building block fashion. Since CATIA V5 has parametric features, you can change one

feature and all related features are automatically updated to reflectthe change and its

effects throughout the part. It can be used to create angular shaped part, to which 3Dsurface can be applied to create hybrid parts consisting of mixture of angular and curved

shapes. This provides the ability to create model designs with shapes of varying types.

11.3 GENERAL MODELING PROCESS FOR EACH PART

Plan the part

Create the base feature

Create the remaining features

Analyze the part

Modify the features as necessary

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Assembly modeling

Assemblies can be created from parts, either combined individually or grouped in

subassemblies. The CATIA V5 builds these individualparts and subassemblies into anassembly in a hierachica manner according to relationships defined cy constrains. As in

part modeling , the parametric relationship allows you to quickly update an entire

assembly based on a change in one of its parts.

The general process for assemblies and subassemblies is similar to that of building parts:

Lay out the assembly

Create the base part

Create and attach the remaining parts

Analyze the assembly

Modify the assembly as necessary

11.4 FUNDAMENTALS

CATIA V5 employs two operating modes for part modeling, model made formodeling 3Dparametric parts and drawing mode for creating 2D drawings of them.

These modes operate independently but share the same design data. Part modeling

requires beginning the design work in model mode where a model of the part is

immediately built. Then the drawing mode can be used at any point to document the

design. In traditional CUMPUTER AIDED DESIGN, a 2D drawing is created at the

beginning and then 3D model is built to analyze, and verify the initial concept.

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12. FINITE ELEMENT ANALYSIS

12.1 INTRODUCTION

The finite element method is numerical analysis technique for obtaining approximate

solutions to a wide variety of engineering problems. Because of its diversity andflexibility as an analysis tool, it is receiving much attention in almost every industry. In

more and more engineering situations today, we find that it is necessary to obtain

approximate solutions to problem rather than exact closed form solution.

It is not possible to obtain analytical mathematical solutions for many engineering

problems. An analytical solutions is a mathematical expression that gives the values of

the desired unknown quantity at any location in the body, as consequence it is valid forinfinite number of location in the body. For problems involving complex materialproperties and boundary conditions, the engineer resorts to numerical methods that

provide approximate, but acceptable solutions.

The finite element method has become a powerful tool for the numerical solutionsof a wide range of engineering problems. It has been developed simultaneously with the

increasing use of the high- speed electronic digital computers and with the growing

emphasis on numerical methods for engineering analysis. This method started as ageneralization of the structural idea to some problems of elastic continuum problem,

started in terms of different equations

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.

12.2 PROCEDURE FOR ANSYS ANALYSIS

Static analysis is used to determine the displacements stresses, stains and forces in

structures or components due to loads that do not induce significant inertia and damping

effects. Steady loading in response conditions are assumed. The kinds of loading thatcan be applied in a static analysis include externally applied forces and pressures, steady

state inertial forces such as gravity or rotational velocity imposed (non-zero)

displacements, temperatures (for thermal strain).

A static analysis can be either linear or non linear. In our present work weconsider linear static analysis.

The procedure for static analysis consists of these main steps Building the model

Obtaining the solution Reviewing the results.

12.3 BUILD THE MODEL

In this step we specify the job name and analysis title use PREP7 to define the elementtypes, element real constants, material properties and model geometry element type both

linear and non- linear structural elements are allowed. The ANSYS elements library

contains over 80 different element types. A unique number and prefix identify each

element type.

E.g. BEAM 94, PLAN 71, SOLID 96 and PIPE 16E

12.4 MATERIAL PROPERTIES

Young.s modulus (EX) must be defined for a static analysis. If we plan to apply inertia

loads (such as gravity) we define mass properties such as density (DENS). Similarly ifwe plan to apply thermal loads (temperatures) we define coefficient of thermal

expansion

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12.5 SOLUTION

In this step we define the analysis type and options, apply loads and initiate the finite

element solution. This involves three phases:

Pre-processor phaseSolution phasePost-processor phase

Pre-processor:

Pre processor has been developed so that the same program is available on micro, mini,

super-mini and mainframe computer system. This slows easy transfer of models onesystem to other.

The following Table 3.1 shows the brief description of steps followed in each phase:

TABLE .1

PRE PROCESSOR

PHASE

SOLUTION PHASE POST PROCESSOR

GEOMETRY

DEFINITION

ELEMENT MATRIX

FORMATION

POST SOLUTION

OPERATION

MESH GENERATION OVERALL MATRIXTRIANGULARIZATION

POST DATA PRINTOUT FOR REPORTS

MATERIAL WAVE FRONT POST DATA

DEFINITIONS SCANING POST DATA

DISPLAY

CONSTRAIN

DEFINITIONS

DISPLACEMENT,

STRESS,ET.,

LOAD DEFINITIONS CALCULATION

MODEL DISPLAY

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13. FINITE ELEMENT GENERATION:

The maximum amount of time in a finite element analysis is spent on generating

elements and nodal data. Pre processor allows the user to generate nodes and elementsautomatically at the same time allowing control over size and number of elements. There

are various types of elements that can be mapped or generated on various geometric

entities.

The elements developed by various automatic element generation capabilities of

pre processor can be checked element characteristics that may need to be verified beforethe finite element analysis for connectivity, distortion-index etc.

Generally, automatic mesh generating capabilities of pre processor are used ratherthan defining the nodes individually. If required nodes can be defined easily by defining

the allocations or by translating the existing nodes. Also on one can plot, delete, or

search nodes.

13.1 BOUNDARY CONDITIONS AND LOADING:

After completion of the finite element model it has to constrain and load has to be

applied to the model. User can define constraints and loads in various ways. Allconstraints and loads are assigned set ID. This helps the user to keep track of load cases.

13.2 MODEL DISPLAY:

During the construction and verification stages of the model it may be necessary to view

it from different angles. It is useful to rotate the model with respect to the global systemand view it from different angles. Pre processor offers this capabilities. By windowing

feature pre processor allows the user to enlarge a specific area of the model for clarity

and details. Pre processor also provides features like smoothness, scaling, regions, active

set, etc for efficient model viewing and editing.

13.3 MATERIAL DEFECTIONS:

All elements are defined by nodes, which have only their location defined. In the case of

plate and shell elements there is no indication of thickness. This thickness can be given

as element property. Property tables for a particular property set 1-D have to be input.Different types of elements have different properties for e.g.

Beams: Cross sectional area, moment of inertia etc

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Shell: Thickness

Springs: StiffnessSolids: None

The user also needs to define material properties of the elements. For linear static

analysis, modules of elasticity and Poisson.s ratio need to be provided. For heat transfer,coefficient of thermal expansion, densities etc. are required. They can be given to the

elements by the material property set to 1-D.

13.4 SOLUTION:

The solution phase deals with the solution of the problem according to the problem

definitions. All the tedious work of formulating and assembling of matrices are done by

the computer and finally displacements are stress values are given as output. Some of

the capabilities of the ANSYS are linear static analysis, non linear static analysis,transient dynamic analysis, etc.

13.5 POST- PROCESSOR:

It is a powerful user- friendly post- processing program using interactive colour

graphics.It has extensive plotting features for displaying the results obtained from the finite

element analysis. One picture of the analysis results (i.e. the results in a visual form) can

often reveal in seconds what would take an engineer hour to assess from a numericaloutput, say in tabular form. The engineer may also see the important aspects of the

results that could be easily missed in a stack of numerical data.

Employing state of art image enhancement techniques, facilities viewing of:

Contours of stresses, displacements, temperatures etc.Deform geometric plotsAnimated deformed shapes

Time-history plots

Solid sectioning

Hidden line plotLight source shaded plotBoundary line plot etc.

The entire range of post processing options of different types of analysis can be

accessed through the command/menu mode there by giving the user added flexibility

and convenience.

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14. FINITE ELEMENT FORMULATION FOR HEAT

CONDUCTIOIN

The unsteady heat conduction equation of each body for an axis-Symmetric

problem described in the cylindrical coordinate system is given as follows:1r z

T T T

c r k k

t r r r z z

= + (3.1)

With the boundary conditions and initial condition*

0 T = T o n (3.2)1 ( - ) n q h T T o n = (3.3)*

n n 2 q = q o n (3.4)0 T = T a t t i m e = 0 (3.5)

Where , c , r k and z k are the density, specific heat ant thermal conductivities in rand z direction of the material, respectively. Also ,T* is the prescribed temperature, h

the heat transfer coefficient,*n q the heat flux at each contact interface due to friction, T

the ambient temperature, 0 T the initial temperature and 0 , 1 and 2 are theboundaries on which temperature, convection and heat flux are imposed, respectively.

Using Galerkin.s approach, a finite element formulation of unsteady heat Eq.

(3.1) can be written in the following matrix form as

C T T + K H T T =R (3.6)

Where T C is the capacity matrix, T KH is the conductivity matrix. T and Rand are the nodal temperature and heat source vector, respectively.

The most commonly used method for solving Eq. (3.6) is the direct integration

method based on the assumption that temperature t T at time t and temperature t t T + at

time t + t have the following relation:

( ) . .t t t 1 t t T + T T T + t= + +

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(3.7)

Eq.(3.7) can be used to reduce the ordinary differential Eq.(3.6) to the

following implicit algebraic equation:

( ) ( )T 1 T t t T 2 T t 2 t 1 t t C b K H T C b K H T b R b R + + + = + +

(3.8)

Where the variable 1 b and 2 b are given by

1 b = t , ( ) 2 b = 1 t(3.9)

For different values of , the well-known numerical integration schemecan be obtained [23].in this study, 0.5 1.0 was used, which is anunconditionally stable scheme.

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15 THERMAL ANALYSIS

A thermal analysis calculates the temperature distribution and related thermalquantities in brake disk. Typical thermal quantities are:

1. The temperature distribution

2. The amount of heat lost or gained3. Thermal fluxes

15.1 Types of thermal analysis:

1. A steady state thermal analysis determines the temperature

distribution and other thermal quantities under steady state loadingconditions. A steady state loading condition is a situation where

heat storage effects varying over a period of time can be ignored.

2. A transient thermal analysis determines the temperature distributionand other thermal quantities under conditions that varying over a

period of time.

15.2 PLANNING THE ANALYSIS:

In this step a compromise between the computer time and accuracy of the analysis ismade. The various parameters set in analysis are given below:

Thermal modeling Analysis type . thermal h-method.

Steady state or Transient? Transient Thermal or Structural? Thermal

Properties of the material? Isotropic Objective of analysis- to find out the temperature distribution in the brake disk

when the process of braking is done. Units- SI

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16. STRUCTURAL ANALYSIS

Structural analysis is the most common application of the finite element analysis. Theterm structural implies civil engineering structure such as bridge and building, but also

naval, aeronautical and mechanical structure such as ship hulls, aircraft bodies and

machine housing as well as mechanical components such as piston, machine parts andtools.

16.1 Types of structural analysis:

The seven types of structural analyses in ANSYS. One can perform the following typesof structural analysis. Each of these analysis types are discussed as follows:

Static analysis

Modal analysis Harmonic analysis

Transient dynamic analysis Spectrum analysis Buckling analysis Explicit dynamic analysis

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17. STRUCTURAL STATIC ANALYSIS:

A static analysis calculates the effects of steady loading conditions on a structure,while ignoring inertia and damping effects such as those caused by time varying loads.

A

static analysis can, however include steady inertia loads (such as gravity and rotationalvelocity), and time varying loads that can be approximated as static equivalent loads

(such as static equivalent wind and seismic loads).

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18. MODELING AND ANALYSIS

It is very difficult to exactly model the brake disk, in which there are stillresearches are going on to find out transient thermo elastic behavior of disk brake

during braking applications. There is always a need of some assumptions to

model any complex geometry. These assumptions are made, keeping in mind thedifficulties involved in the theoretical calculation and the importance of the

parameters that are taken and those which are ignored. In modeling we always

ignore the things that are of less importance and have little impact on theanalysis. The assumptions are always made depending upon the details and

accuracy required in modeling.

The assumptions which are made while modeling the process are given below:-

1. The disk material is considered as homogeneous and isotropic.2. The domain is considered as axis-symmetric.

3. Inertia and body force effects are negligible during the analysis.

4. The disk is stress free before the application of brake.5. Brakes are applied on the entire four wheels.

6. The analysis is based on pure thermal loading and vibration and

thus only stress level due to the above said is done. The analysisdoes not determine the life of the disk brake.

7. Only ambient air-cooling is taken into account and no forced

Convection is taken.8. The kinetic energy of the vehicle is lost through the brake disks i.e.

no heat loss between the tyre and the road surface and deceleration

is uniform.

9. The disk brake model used is of solid type and not ventilated one.10. The thermal conductivity of the material used for the analysis is

uniform throughout.11. The specific heat of the material used is constant throughout and

does not change with temperature.

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20.MODELING AND ANALYSIS

It is very difficult to exactly model the brake disk, in which there are still researches are

going on to find out transient thermo elastic behavior of disk brake during braking

applications. There is always a need of some assumptions to model any complex

geometry. These assumptions are made, keeping in mind the difficulties involved in thetheoretical calculation and the importance of the parameters that are taken and those

which are ignored. In modeling we always ignore the things that are of less importance

and have little impact on the analysis. The assumptions are always made depending

upon the details and accuracy required in modeling.

The assumptions which are made while modeling the process are given below:-

1. The disk material is considered as homogeneous and isotropic.

2. The domain is considered as axis-symmetric.

3. Inertia and body force effects are negligible during the analysis.

4. The disk is stress free before the application of brake.

5. Brakes are applied on the entire four wheels.

6. The analysis is based on pure thermal loading and thus only stress level due to the

above said is done the analysis does not determine the life of the disk brake.

7. Only ambient air-cooling is taken into account and no forced Convection is taken.

8. The kinetic energy of the vehicle is lost through the brake disks i.e. no heat loss

between the tyre and the road surface and deceleration is uniform.

9. The disk brake model used is of solid type and ventilated one.

10. The thermal conductivity of the material used for the analysis is uniform throughout.

11. The specific heat of the material used is constant throughout and does not change

with temperature.

DEFINITION OF PROBLEM DOMAIN

Due to the application of brakes on the car disk brake rotor, heat generation takes place

due to friction and this thermal flux has to be conducted and dispersed across the disk

rotor cross section. The condition of braking is very much severe and thus the thermal

analysis has to be carried out. The thermal loading as well as structure is axis-

symmetric. Hence axis-symmetric analysis can be performed, but in this study we

performed 3-D analysis, which is an exact representation for this thermal analysis.

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Thermal analysis is carried out and with the above load structural analysis is also

performed for analyzing the stability of the structure.

The 3d model of the solid type brake is done in CATIA and converted into parasolid

file.

Fig. solid type disk brake 3D model isometric view

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Fig. solid type disk brake 3D model front view

Fig. solid type disk brake 3D model wireframe

21. CREATING A FINITE ELEMENT MESH

According to given specifications the element type chosen is solid 90.Solid 90 is higher

order version of the 3-D eight node thermal element (Solid 70). The element has 20

nodes with single degree of freedom, temperature, at each node. The 20-node elementshave compatible temperature shape and are well suited to model curved boundaries. The

20-node thermal element is applicable to a 3-D, steady state or transient thermal

analysis. If the model containing this element is also to be analyzed structurally, the

element should be replaced by the equivalent structural element (Solid 95).

The parasolid file is imported into ansys and is meshed with 20 node thermal solid 90

element type. The structure, number of nodes and input summary of the element is given

below.

21.1 SOLID90 Element Description

SOLID90 is a higher order version of the 3-D eight node thermal element (SOLID70).

The element has 20 nodes with a single degree of freedom, temperature, at each node.

The 20-node elements have compatible temperature shapes and are well suited to model

curved boundaries. The 20-node thermal element is applicable to a 3-D, steady-state or

transient thermal analysis

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The geometry, node locations, and the coordinate system for this element are shown in

Figure "SOLID90 Geometry". The element is defined by 20 node points and the material

properties. A prism-shaped element may be formed by defining duplicate K, L, and S; A

and B; and O, P, and W node numbers.

SOLID90InputSummary

Nodes

I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z,A,B

DegreesofFreedom

TEMP

MaterialProperties

KXX,KYY,KZZ,DENS,C,ENTH

SurfaceLoads

ConvectionorHeatFlux(butnotboth)andRadiation(usingLab=RDSF)

face1(JILK),face2(IJNM),face3(JKON),

face4(KLPO),face5(LIMP),face6(MNOP)

BodyLoads

HeatGenerations

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HG(I),HG(J),HG(K),HG(L),HG(M),HG(N),HG(O),HG(P),HG(Q),HG(R),

HG(S),HG(T),HG(U),HG(V),HG(W),HG(X),HG(Y),HG(Z),HG(A),HG(B)

Fig. solid type disk brake mesh model

Fig. solid type disk brake mesh model isometric view

Total number of elements = 39800

Total number of nodes = 98104

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22. APPLYING THE BOUNDARY CONDITIONS

In thermal and structural analysis of disk brake, we have to apply thermal and boundary

conditions on 3D disk model of disk brake.

22.1 THERMAL BOUNDARY CONDITIONS

As shown in Fig. a model presents a three dimensional solid disk squeezed by two finite-

width friction material called pads. The entire surface, S, of the disk has three different

regions including S1 and S2. On S1 heat flux is specified due to the frictional heating

between the pads and disk, and S2 is defined for the convection boundary. The rest of

the region, except S1 U S2, is either temperature specified or assumed to be insulated:

the inner and outer rim area of disk.

Fig. Thermal model of Disk brake

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Fig. Temperature boundary condition of 77degrees C applied on solid type Disk brake

Fig. Convection boundary condition applied on solid type Disk brake

Material Properties on Pad and DiskThermal conductivity, K (w/m k) -

Density, (kg/m3) - 1800Specific heat, c (J/Kg k) - 1.88

Poissons ratio, v - 0.3Thermal expansion, (106 / k ) - 0.3Elastic modulus, E (GPa) - 50.2

Coefficient of friction, - 0.2

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Fig. Temperature distribution on solid type Disk brake along the thickness

Fig. Graphical representation of Temperature distribution on solid type Disk brake along

the thickness

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23. STRUCTURAL ANALYSIS NORMAL DISC BRAKE ROTOR

23.1STRUCTURAL BOUNDARY CONDITIONS

Since the axis-symmetric model is considered all the nodes on the hub radius are fixed.

So the nodal displacements in the hub become zero i.e. in radial, axial and angulardirections

Fig. Structural boundary condition applied on solid type Disk brake

Fig. Temperature distribution is applied as Thermal loads on solid type Disk brake from

the thermal analysis

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23.2 RESULTS

Fig. Total deflection of solid type Disk brake

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Fig. Deflection in X-dir of solid type Disk brake

Fig. Deflection in Y-dir of solid type Disk brake

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Fig. Deflection in Z-dir of solid type Disk brake

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Fig. VonMises stress on solid type Disk brake

Fig. X-dir stress on solid type Disk brake

Fig. Y-dir stress on solid type Disk brake

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Fig. Z-dir stress on solid type Disk brake

To optimize the above disk brake a complicated model of ventilated disk brake is taken

and there by forced convection is considered in the analysis.

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Fig. Ventilated type disk brake 3D model isometric view

Fig. Ventilated type disk brake 3D model isometric view on the rear side

Fig. Ventilated type disk brake 3D model front view

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Fig. Ventilated type disk brake mesh model in isometric view

Fig. Ventilated type disk brake 3D model in showing the vents

25. APPLYING THE BOUNDARY CONDITIONS

In thermal and structural analysis of disk brake, we have to apply thermal and boundary

conditions on 3D disk model of disk brake.

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Fig. Temperature boundary condition of 77degrees C applied on Vent type Disk brake

Fig. Convection boundary condition applied on Vent type Disk brake

Results

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Fig. Temperature distribution on Vent type Disk brake on the front side

Fig. Temperature distribution on Vent type Disk brake on the rear side

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Fig. Temperature distribution on Vent type Disk brake along the thickness

Fig. Graphical representation of Temperature distribution on vent type Disk brake along

the thickness

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26. STRUCTURAL ANALYSIS FOR VENTED DISC BRAKE ROTOR

26.1STRUCTURAL BOUNDARY CONDITIONS

Since the axis-symmetric model is considered all the nodes on the hub radius are fixed.

So the nodal displacements in the hub become zero i.e. in radial, axial and angulardirections

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Fig. Structural boundary condition applied on Vent type Disk brake

Fig. Temperature distribution is applied as Thermal loads on Vent type Disk brake from

the thermal analysis

26.2 Results

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Fig. Deflection in Z-dir of Vent type Disk brake

Fig. VonMises stress on Vent type Disk brake

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Fig. X-dir stress on Vent type Disk brake

Fig. Y-dir stress on Vent type Disk brake

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Fig. Z-dir stress on Vent type Disk brake

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27. CONCLUSIONS

The present study can provide a useful design tool and improve the brake performance

of disk brake system. From the below Table we can say that all the values obtained from

the analysis are less than their allowable values. Hence the brake disk design is safe

based on the strength and rigidity criteria. Comparing the different results obtained from

analysis. It is concluded that ventilated type disk brake is the best possible for the

present application.

SolidType VentilatedTypeTotalDeflectionin(mm) 2.351 0.248VonmisesStress 2.26E+12 2.17E+06

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28. REFRENCES

1. KENNEDY, F. E., COLIN, F. FLOQUET, A. AND GLOVSKY, R. Improved

Techniques for Finite Element Analysis of Sliding Surface Temperatures.Westbury House page 138-150, (1984).

2. LIN , J. -Y. AND CHEN, H. -T. Radial Axis symmetric Transient HeatConduction in Composite Hollow Cylinders with Variable Thermal Conductivity,vol. 10, page 2- 33, (1992).

3. BRILLA, J. Laplace Transform and New Mathematical Theory of Visco elasticity,

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4. TSINOPOULOS, S. V, AGNANTIARIS, J. P. AND POLYZOS, D. An AdvancedBoundary Element/Fast Fourier Transform Axis symmetric Formulation for

Acoustic Radiation and Wave Scattering Problems, J.ACOUST. SOC. AMER.,

vol 105, page 1517-1526, (1999).5. WANG, H. -C. AND BANERJEE, P. K.. Generalized Axis symmetric

Elastodynamic Analysis by Boundary Element Method, vol. 30, page 115-131,

(1990).6. FLOQUET, A. AND DUBOURG, M.-C. Non axis symmetric effects for three

dimensional Analyses of a Brake, ASME J. Tribology, vol. 116, page 401-407,

(1994).7. BURTON, R. A. Thermal Deformation in Frictionally Heated Contact, Wear, vol.

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8. ANDERSON, A. E. AND KNAPP, R. A. Hot Spotting in Automotive Friction

System Wear, vol. 135, page 319-337, (1990).9. COMNINOU, M. AND DUNDURS, J. On the Barber Boundary Conditions for

Thermo elastic Contact, ASME J, vol. 46, page 849-853, (1979).

10. BARBER, J. R. Contact Problems Involving a Cooled Punch, J. Elasticity, vol. 8,

page 409- 423, (1978).11. BARBER, J. R. Stability of Thermo elastic Contact, Proc. International

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in the absence of Wear, Wear, vol. 19, page 315-328, (1972).

13. LEE, K. AND BARBER, J.R. Frictionally-Excited Thermo elastic Instability inAutomotive Disk Brakes, ASME J. Tribology, vol. 115, page 607-614, (1993).

14. LEE, K. AND BARBER, J. R. An Experimental Investigation of Frictionallyexcited

Thermo elastic Instability in Automotive Disk Brakes under a Drag BrakeApplication, ASME J. Tribology, vol. 116,page 409-414, (1994).

15. LEE, K. AND BARBER, J. R. Effect of Intermittent Contact on the Stability ofThermo elastic Contact, ASME J. Tribology, vol. 198, page 27- 32, (1995).16. ZAGRODZKE, S, BARBER, J. R. AND HULBERT, G.M. Finite Element

Analysis of Frictionally Excited Thermo elastic Instability, J. Thermal Stresses,

vol. 20, page 185-201, (1997).17. LEROY, J. M., FLOQUET, A. AND VILLECHAISE, B. Thermo-mechanical

Behavior of Multilayered Media: Theory, ASME J. Tribology, vol. 111, page

530-544, (1989).

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18. GONSKA, H. W. AND KOLBINGER, H. J. Temperature and Deformation

Calculation of Passenger Car Brake Disks, Proc. ABAQUS User.s Conference,

Aachen, Germany, page 21- 232, (1993).

19. AKIN, J. E. Application and Implementation of Finite Element Methods,

Academic Press, Orlando, FL, page 318-323, (1982).

20. ZAGRODZKI, P. Analysis of thermo mechanical phenomena in multi diskclutches and brakes, Wear 140, page 291-308, (1990).

21. COOK, R. D. Concept and Applications of Finite Element Analysis, Wiley,Canada, (1981).

22 ZIENKIEWICZ, O. C. The Finite Element method, McGraw-Hill, New York,

(1977).23. BEEKER, A.A. The Boundary Element Method in Engineering, McGraw-Hill,

New York, (1992).

thermal analysis of vented and normal disc brake rotors-doc

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