Jun 27, 2016 - the vehicle will start to rollaway for the drum/lining friction coefficient lower than .... Force reaction at (a) secondary brake shoe, (b) strut, and.
UNIVERSITI TEKNOLOGI MALAYSIA DECLARATION OF THESIS / UNDERGRADUATE PROJECT PAPER AND COPYRIGHT Author’s full name :
MUHAMMAD HADI BIN HASHIM
Date of birth :
26th JULY 1993
Title :
PREDICTION OF PARKING BRAKE TORQUE USING MATHEMATICAL MODELLING
Academic Session :
2015/2016
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If the thesis is CONFIDENTIAL or RESTRICTED, please attach with the letter from the organization with period and reasons for confidentiality or restriction.
ii
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Signature
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Name of Supervisor I
:
DR. ABD RAHIM BIN ABU BAKAR
Date
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27th JUNE 2016
PREDICTION OF PARKING BRAKE TORQUE USING MATHEMATICAL MODELLING
MUHAMMAD HADI BIN HASHIM
A report submitted in partially fulfillment of the requirements for the award of the degree of Bachelor of Engineering (Mechanical-Automotive)
Faculty of Mechanical Engineering Universiti Teknologi Malaysia
JUNE 2016
ii
DECLARATION
I declare that this thesis entitled “PREDICTION OF PARKING BRAKE TORQUE USING MATHEMATICAL MODELLING" is the result of my own research except as cited in the references. The thesis has not been accepted for any degree and is not concurrently submitted in candidature of any other degree.
Signature
:.…………………………….
Name
: MUHAMMAD HADI BIN HASHIM
Date
: 27th JUNE 2016
iii
DEDICATION
To the people who have encouraged me to grow professionally and spiritually over the semester, especially my family. Included also are my great fellow educators, supervisor, friends and people that I am indebted to.
iv
ACKNOWLEDGEMENTS
All praises and thanks are due to Allah, who guided and helped me throughout my degree journey. Glory be to Allah who has given me the strength, patience and knowledge to continue and complete my study.
I would like to express my deepest gratitude to my supervisor, Dr Abd Rahim Bin Abu Bakar his constant guidance, encouragement and patience. I owe them heartfelt thanks for her time and effort in providing me assistance throughout my project undertaking.
I would like to express my sincere thanks to all technicians of Automotive Lab, especially to Mr. Shamsuri for helping me in getting some parameters for my project. Special thanks go to Nurul Husna Binti Ramli and all my friends. They are place for me to ask for advice and support when I am feeling down.
My special thanks go to my mother, father, and my family for their endless support and prayers. They provided me with love, care, and motivation, without which, I cannot survive to undertake all the challenges in my study.
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ABSTRACT
Insufficient force applied by the driver on parking brake system may cause a vehicle to rollaway. This occurs when the torque generated by the parking brake is less than the torque required to hold the vehicle. This project attempts to assess parking brake performance using a static parameter model of leading-trailing drumtype parking brake system. The theoretical results are validated against existing experimental data. Leading-trailing with pivot drum brake design has been selected because the physical similarities of Proton Saga BLM drum brake. Two types of loading have been considered, i.e. concentrated and distributed loads.
This
theoretical result shows that the parking brake system comply FMVSS 135, EEC and ECE R13H standards.
The torque predicted by the theoretical analysis almost
identical with the experimental results where the average difference is below 2 percent. Parametric studies are also performed at various road slopes, coefficients of friction, vehicle weights and distance of drum pivot. Theoretical results shows that the vehicle will start to rollaway for the drum/lining friction coefficient lower than 0.3. It is interesting to see that the parking brake design is capable of holding the vehicle stationary at maximum road slope of 19°. The vehicle remains stationary with a maximum of five (5) passengers and distance of drum pivot less than 0.036 m.
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ABSTRAK
Ketidakcukupan daya yang dikenakan oleh pemandu terhadap sistem brek parkir boleh menyebabkan sesebuah kenderaan itu bergerak. Ini berlaku apabila daya kilas yang dihasilkan oleh unit brek parkir lebih rendah daripada daya kilas yang diperlukan untuk mengekalkan kenderaan dalam keadaaan pegun. Projek ini berusaha untuk menilai keupayaan brek parkir menggunakan model parameter statik untuk sistem brek parkir jenis gelendong pembawa-pengikut. Keputusan teoritikal ini diujisahkan menggunakan data ujikaji sedia ada. Gelendong pembawa-pengikut dengan pivot dipilih kerana mempunyai persamaan fizikal dengan brek gelendong Proton Saga BLM. Keputusan teoritikal menunjukkan sistem brek parkir yang sedia ada menepati piawaian FMVSS 135, EEC dan ECE R13H. Jangkaan daya kilas oleh analisis teoritikal hampir sama dengan keputusan ujikaji di mana purata perbezaan adalah kurang daripada 2 peratus. Kajian parametrik juga telah dilakukan pada pelbagai kecerunan jalan, pekali geseran, berat kenderaan dan jarak pangsi gelendong. Daripada keputusan teoritikal ia menunjukkan bahawa kenderaan akan mula bergerak apabila pekali geseran gelendong/kekasut lebih rendah daripada 0.3. Sesuatu yang menarik untuk diperhatikan adalah rekabentuk brek parkir berupaya mengekalkan kenderaan dalam keadaan pegun pada kecerunan maksima iaitu 19°. Kenderaan tersebut juga berupaya kekal pegun dengan lima (5) penumpang dan jarak pangsi gelendong mestilah kurang daripada 0.036 m.
vii
TABLE OF CONTENTS
CHAPTER
TITLE
DECLARATION
ii
DEDICATION
iii
ACKNOWLEDGEMENTS
iv
ABSTARCT
v
ABSTRAK
vi
TABLE OF CONTENTS
1
2
PAGE
vii
LIST OF TABLES
x
LIST OF FIGURES
xi
LIST OF ABBREVIATIONS
xiii
LIST OF SYMBOLS
xiv
INTRODUCTION
1
1.1
Introduction
1
1.2
Statement of Problems
2
1.3
Objectives of Research
4
1.4
Scope of Research
4
1.5
Significant of Research
4
1.5
Thesis Organization
5
LITERATURE REVIEW
7
2.1
7
Introduction
viii
2.2
Regulation of Parking Brake Systems
8
2.3
Working Principle of Parking Brake System
8
2.3.1
Conventional Drum Brake System
9
2.3.2
Active Parking Brake System
15
2.3.3.1 Electro-Mechanical Parking Brake
16
2.3.3.1 Electric Parking Brake
18
2.4
Analysis of Service Brake and Parking Brake
19
System 2.4.1
2.4.2
2.5
3
Service Brake
19
2.4.1.1 Theoretical Approach
20
2.4.1.1 Numerical Approach
26
2.4.1.1 Experimental Approach
26
Parking Brake System
27
2.4.2.1 Theoretical Approach
28
2.4.2.2 Numerical Approach
29
2.4.2.3 Experimental Approach
30
Summary
33
METHODOLODGY
35
3.1
Introduction
35
3.2
Theoretical Modeling of Parking Brake Unit
37
3.2.1
Torque Required to Hold a Vehicle
38
3.2.2
Torque Generated by the Parking Brake
38
3.2.2.1 Parking Brake Model Parameter
42
3.2.2.2 Concentrated load Leading-Trailing
44
with pivot 3.2.2.3 Distributed load Leading-Trailing
46
with pivot
4
3.3
Parametric Study
49
3.4
Summary
51
RESULTS AND DISCUSSION
52
4.1
52
Introduction
ix
4.2
Validation of Parking Brake Model
52
4.3
Parametric Studies
54
4.3.1
Vehicle Weight
55
4.3.2
Lining Coefficient of Friction
56
4.3.3
Road Slope
57
4.3.4
Distance from Pivot to center, a’’
58
4.4
5
Summary
59
CONCLUSIONS AND RECOMMENDATIONS
60
5.1
Conclusions
60
5.2
Recommendations
62
REFERENCES APPENDIXES
64
x
LIST OF TABLE
TABLE NO.
TITLE
PAGE
3.1
Parameter For Proton Saga BLM Drum Brake Model
43
3.2
Different Vehicle Weight
50
3.3
Lining Coefficient Of Friction
50
3.4
Slope Road
50
3.5
Distance From Pivot To Center, A’
50
xi
LIST OF FIGURE
FIGURE NO.
TITLE
PAGE
2.1
Example of parking brake system.
10
2.2
Type of parking brake controllers (levers).
11
2.3
Centre lever type.
11
2.4
Pedal-operated parking brake.
12
2.5
Typical cable and conduit.
13
2.6
Example of linkage components.
14
2.7
Components of drum parking brake.
15
2.8
Electro-mechanical parking brake unit.
17
2.9
Schematic of Electro-mechanical parking brake.
17
2.10
Parts of electrical parking brake.
18
2.11
Self-energizing in a drum.
21
2.12
Leading shoe with pivot.
22
2.13
Leading shoe with parallel sliding abutment.
23
2.14
Leading shoe with inclined abutment.
24
2.15
Due-servo brake with sliding abutment.
25
2.16
One-dimensional model of parking drum brake system.
29
2.17
Work program of electrical controlled system
30
2.18
Schematic diagram of torque measurement.
31
2.19
Parking brake test bench.
32
3.1
Overall research flow.
36
3.2
Diagram of vehicle which is parked at a slope.
38
3.3
Typical parking brake system.
39
xii
3.4
Force reaction at handbrake lever
39
3.5
Parts in drum brake component. Force reaction at (a) secondary brake shoe, (b) strut, and (c) parking brake lever. Concentrated loading Leading-Trailing with pivot drum brake. Distributed Load Leading-Tailing with pivot Generated torque by theoretical analysis and experimental data. Percentage of difference of generated torque between theoretical and experimental data. Torque required at different weight of the vehicle Torque generated at variable drum/lining coefficient of friction with torque required by the vehicle
40
4.5
Torque required at different slope road
58
4.6
Torque generated with different parameter of a’
59
3.6 3.7 3.8 4.1 4.2 4.3 4.4
41 45 47 53 54 56 57
xiii
LIST OF ABBREVIATION
AC
-
Alternate Current
FMVSS
-
Federal motor vehicle safety standard
EEC
-
European Economic Safety
EPB
-
Electric parking brake
EMPB
-
Electro-mechanical parking brake
xiv
LIST OF SYMBOL
m
-
Mass of vehicle
Fa
-
Handbrake ratio
r
-
Radius of drum brake
Rwheel
-
Radius of tire
µ
-
lining coefficient of friction
a
-
brake length dimension, mm
ˆ 0
-
Arc of the angle a0, rad
1
-
Angle between the beginning of the lining and straight lines
connecting centre and pivot point, deg w
-
Width of drum lining
1
CHAPTER 1
1
1.1
INTRODUCTION
Introduction
A brake is a mechanical device that inhibits motion, slowing or blocking of a moving object or preventing its motion. The general uses of the brakes can be divided into three basic functions such as. to decelerate a vehicle, to maintain vehicle speed and to hold a vehicle stationary on the hill [1]. Most brakes commonly use friction between two surfaces pressed together to change the kinetic energy of the moving object into high temperature, though other methods of energy transition may be applied. For example, regenerative braking converts much of the energy to electrical energy, which may be stashed away for later usage [2]. Brakes can be classified into two components, i.e. as service brakes, used for normal brake and the secondary or emergency brake are used during partial brake system failure and parking brakes [1]. The basic principle of the brakes system is to provide clamping force that generated between the disc/pad and drum/lining. Insufficient clamping force may cause the vehicle fail to decelerate or stop as intended.
2 There are typically three types of parking brake systems available in today’s vehicle such as fully mechanical (conventional), electro-mechanical (EMPB) and electric (EPB) parking brake systems. Fully mechanical parking brake control may be operated either by hand through center lever or by foot through foot pedal. When the hand-brake is applied, the brake cable passes through an intermediate lever, to increase the force of your poll, this force is then split evenly between your brakes by an equalizer. In a foot-operated and pedal parking brake system, their release mechanisms are mounted on a bracket under the left and on the instrument panel. The parking brake is applied through pressing the pedal downward by the driver’s foot.
When the electromechanical parking brake is to be applied, the electric motor is actuated by the electromechanical parking brake control unit. The spindle is driven by the electric motor via the belt and swash plate gear mechanism. Through the rotary movement of the spindle, the thrust nut moves forwards on the spindle thread. The thrust nut comes into contact with the brake piston and presses it against the brake pads. The brake pads will press against the brake disc. Electric Parking Brake (EPB) is a system for a vehicle that comprises the temporary brake of driving and long-term parking brake, operated by an electronic button and release parking brake at the appropriate time by electrical control unit according accelerate pedal, clutch and engine speed signals.
1.2
Statement of Problem
Brake is a mechanism used to absorb the kinetic energy of the vehicle with the aim to stop the motion. Brakes transform kinetic energy into heat. Since the acceleration required during an emergency brake maneuver is much higher than the
3
acceleration during normal operation, the brake power must be much higher than the motor power of the vehicle.
The standard of braking systems fitted to vehicles is a key safety issue and a sufficient braking capability is one of the most important qualities a vehicle must have European Economic Community (EEC) regulation. The first federal safety regulating the braking performance is FMVSS 105 then being replaced by FMVSS 135 that requires stringent stability performance under a variety of braking conditions.
The parking brake must hold the fully loaded and empty vehicle
stationary on an incline of 30% for standard transmission vehicles, or 20% for vehicle with automatic transmission.
The numerous methods can be used to predict parking brake performance in the design stage and they are analytical, multibody, and finite element method. The advantages of these approaches are more economical because it just uses a parking brake model for taking a measurement that will reduce the cost. Besides that, these methods save time from having an analysis in the lab and wait for the data. After that, the brake model design can easily be changed if it’s not complying with FMVSS. The main problem of these approaches is to obtain a good prediction that close to the experimental result.
This project seeks to identify and build a mathematical model of parking brake that can predict brake torque with good accuracy against test data.
4
1.3
Objective of Research
The primary objectives of the research are: 1.
To identify and select the suitable parking brake model of Proton Saga BLM.
2.
To validate parking brake model against existing test data.
3.
To perform parametric studies in order to assess performance of the parking brake system.
1.4
Scope of Research
The scopes of the research are as follows: 1.
The study is carried out on hand-brake lever type with, leadingtrailing drum-type of Proton Saga BLM.
2.
The validation is made using an existing experimental data [3]
3.
The parametric studies are performed at various road slopes, vehicle weight, drum/lining friction coefficients and distance of the drum pivot.
1.5
Significance of the Research
Evolution of technology has brought along the progression of parking brake system has brought an innovation of an advanced parking brake system which functions more electronically than mechanically.
However, mechanical parking
5
brake system is still preferred to be used widely. While improving in technology factors, there still have concepts that focus on improving the performance of initial braking torque only and the sustainability of vehicle torque is given less priority. Typically drum brake acquire higher chances of vehicle rollaway than disc brake mainly because of heat dissipation. Disc brake allows more heat to escape because of the larger surface area of disc brake directly in contact with air than drum brake. It is crucial to park a vehicle in a hot brake condition because there is a high tendency for the vehicle to rollaway [3]. This work is being taken out in order to collapse information and estimations for the auto engineers to design a parking brake system which play along the measures drafted by certain agencies and institutions and also to ensure vehicle standstill by avoiding any rollaway of the vehicle at all times. In summation, the engineers also can easily predict and design parking brake performance that related to many factors just using simple calculation in quickest way.
1.6
Thesis Organization
This thesis consists of five chapters as summarized in the following:
Chapter Two offers some background and relevant data that is necessary in the current field. It starts with regulations on, parking brake systems and is accompanied by a brief description of the functioning principle of a conventional, electromechanical and fully electronic parking brake system. After that, Performance analysis of service and parking brake systems by theoretical, numerical and experimental approaches is also presented. At the end of chapter, the summary on literature review is provided.
6
Chapter Three explains the research methodology which encompasses of the development of mathematical modeling on for torque generated and required for parking brake system.
Chapter Four discusses theoretical results that has been validated using experimental data [3] Parametric studies are carried out by varying the gradient of road, lining coefficient of friction, vehicle weight and dimension of drum brake.
Chapter Five provides conclusion of this study and recommendations for future study.
7
CHAPTER 2
LITERATURE REVIEW
2.1 Introduction
In cars, the parking brake, also called hand brake or emergency brake is usually used to keep the vehicle stationary. It is sometimes also used to prevent a vehicle from rolling when the operator needs both feet to operate the clutch and throttle pedals. There are some factors that will make a failure due to parking brake such as slope, force and friction. Failing in parking brake with not enough force to remain the vehicle will make it to rollaway. Therefore, a study should be carried out in evaluating parking brake performance including vehicle rollaway.
This chapter presents some background and relevant data that is necessary in the current work is to measure functioning of a fully mechanical parking brake system. The review takes up with regulations on, parking brake systems and is accompanied by a brief description of the functioning principle of conventional, electromechanical and fully electronic parking brake systems. This chapter also gives a performance analysis of service and parking brake systems that covers
8
theoretical, mathematical and experimental advances made by former researchers. In the end this chapter ends with a summary of the literature review.
2.2 Regulations on Parking Brake Systems
The standard of braking systems fitted to vehicles is a key safety issue and a sufficient braking capability is one of the most important qualities a vehicle must have. One of the working forums that discussed the regulation on braking system is World Forum for Harmonization of Vehicle Regulations. The first federal safety regulating the braking performance are FMVSS 105 then been replaced by FMVSS 135 that requires streingent stability performance under variety of braking conditions [1]. Besides that, European Economic Community (EEC) regulation specified that the handbrake system of a laden vehicle in class M1 (passenger cars comprising no more than eight seats in addition to the driver's seat) must be able to hold vehicle in 20% gradient. The brakes must produce a certain vehicle deceleration when the brakes are heated through repeated brake applications and recover in prescribed manner. The parking brake must hold the fully loaded and empty vehicle stationary on a incline of 30% for standard transmission vehicles, or 20% for vehicle with automatic transmission [1].
2.3 Working Principle of Parking Brake Systems
There are typically three types of parking brake systems available in today’s vehicle namely, fully mechanical (conventional) [1], electro-mechanical (EMPB) [4] and electric (EPB) [4] parking brake systems. The mechanical parking brake, used to
9
prevent the vehicle from rolling when parked, acts on the rear wheels. It is engaged with an auxiliary foot pedal or a hand pull and is held by a ratchet until released. Drum brakes, once used extensively on all four wheels, are now used primarily on rear wheels. Typically, disk brakes are used on the front wheels, but sometimes on the rear wheels [2].
Each of the design types has its own advantages and
disadvantages. For example, the conventional parking brake system is comparatively inexpensive compared to EMPB and EPB.
Nevertheless, the disadvantages of
conventional parking brake over EMPB and EPB are the driver has to use a significant measure of force to tear or to press the stick and center lever types or pedal type, respectively, the system cannot prevent vehicle rollaway that due to insufficient clamping force when park on a side road, and the system cannot alert the driver if he/she forgot to engage or disengage the parking brake system. However, the conventional parking brake is still favored by most car manufacturers due to its uncomplicated purpose and low fabrication costs.
2.3.1
Conventional Drum Brake System
The primary use of the parking brake is to prevent the vehicle from moving while it is stationary. The parking brake can also be used as an emergency brake to bring a vehicle to rest if the primary brake fails, as it operates independently from the primary system's hydraulic circuit. The mechanical parking brake can be actuated by a lever, pedal or push rod located inside the cabin. An example of each of these mechanisms is shown in Figures 2.1. Cables connect the lever, pedal or push rod to the brake which is generally located at the rear wheels of the vehicle.
10
Figure 2.1 :Example of parking brake system [5].
Parking brake control may be operated either by hand through center lever or stick lever or by foot through foot pedal [5] as shown in Figure 2.2. The center lever parking brake usually positioned in the center of the cabin and is connected to the parking brake cables. It is operated when the driver applies a displacement to the lever, which results in a displacement being applied to the cable. The lever has an integrated ratchet mechanism that allows the displacement to be fixed at intervals specified by the vehicle manufacturer. The ratchet mechanism has a release button which allows the parking brake to be disengaged. An example of a center lever parking brake lever is shown in Figure 2.3. Pedal and pull rod parking brakes are attached to the cables using a similar lever mechanism that also employs a ratchet mechanism to fix the displacement.
11
Figure 2.2: Type of parking brake controllers (levers) [5].
Figure 2.3: Center lever type [6].
Stick-type lever is a handle that locates under instrument panel. This lever still exists in many older vehicles and a few current light- and medium- duty trucks. It works quite similar with center lever which the lever has to be pulled to apply the
12
parking brake.
Ratchet mechanism is used to hold the lever returns to release
position, so that the tension of the cables is maintained when the brake is applied. To release the brake, the lever should be turned about 90° and then pushed forward.
A conventional pedal-operated parking brake in Figure 2.4 is generally mounted on a fixed portion of the vehicle in the vehicle driver's foot space. When the parking brake is to be applied, the foot brake pedal is depressed by the driver's foot so that a tension force on a brake cable is exerted through a brake pedal lever and transmitted to cables to actuate brakes on the vehicle wheels. At the same time, the brake pedal lever is engaged with a pawl and prevented from pivoting, so that it is held in a locked condition. When the vehicle is started, a release knob is pulled for actuating a release lever to disengage the pawl from the brake pedal lever. Eventually, the brake pedal lever is unlocked.
Released parking brake handle
Pedal-operated parking brake Figure 2.4: Pedal-operated parking brake.
Parking brake system needs linkages to transmit force from the brake controller to the rear brake unit. Generally, typical linkages consist of cables, rods, levers, and equalizers [5]. The linkage consists of supporting components such as cable retainers and hooks to maintain the cable position on the rear axle, frame, and
13
under-body of the vehicle. These retainers allow the cable moves flexibly at their point of body attachment and help the equalizer to provide equalizing action. Most of the parking brakes use cables as a medium to connect the brake controller to the rear brake unit. The cables are made of high-strength strands of steel wire that are tightly twisted together so that the cables could functioning without failure even though they are being pulled with a huge amount of force [5]. There are various methods for the cables to be connected to the other parts of linkage. Some of the cables have threaded rods or clevises at their ends and some of them have ball or thimble-shaped connectors that fit into holes and slots on other parts of the linkage [5]. Most of cables are partially covered with a flexible metal conduit that able to protect the cables from chaffing and rubbing against the underside of the vehicle in Figure 2.5. To make the conduit stay still and did not move together with the cable, it was fastened to a bracket on underside of the vehicle at one end and another is attached to the backing plate. To increase the performance of the system, most of cables are coated with nylon or plastic so that it could move smoothly through the conduit. Besides, the coatings also help to reduce corrosion and contamination to the cable [5].
Figure 2.5: Typical cable and conduit [5].
Commonly, cables from brake controllers are directly connected to an equalizer. However, some of brake systems have an intermediate lever to increase the pulling force. The main function of the equalizer is to distribute the force equally through two cables that transmitting the forces to rear brake unit. The equalizer can
14
be a lever mounted on a pivot or U-shaped grooved guide that will moves together with the cables when it is pulled by a front cable as illustrated in Figure 2.6. Owen [5] stated that some of the center lever type parking brakes do not use an equalizer because it had separated cable for each brake units that attached directly to the brake controller.
Figure 2.6: Example of linkage components [1].
In a modern passenger car, the rear brake unit can be either a disc or a drum. A rear drum parking brake unit uses service brake shoes as actuator to lock the wheels. It is commonly used in the parking brake system because it is a relatively simple, cost effective, and the self-energizing action of the brake shoes provides more holding torque to the brake unit.
Parking brake systems like the one in the illustration on Figure 2.7 use a cable, a lever, and a strut to apply the parking brakes. A cable is attached to the lower portion of the brake lever and the upper portion is attached to a swivel or pivot on the secondary shoe. The lever pushes the rear shoe outward and the brake strut forces the forward shoe into the drum. The secondary shoe is the shoe facing the rear of the vehicle. These two parts cable and strut work together to apply both the secondary and the primary shoes with equal force. As the driver engages the parking
15
brake the cable pulls the lever forcing the secondary shoe in one direction and primary in the other. It does this by acting upon the brake strut located between the two. It is because this strut is located further down the lever arm than the swivel that its leverage forces the primary shoe outward.
Figure 2.7: Components of drum parking brake [3].
2.3.2
Active Parking Brake Systems
A regenerative brake is a technology to recover the vehicle kinetic energy as electric energy when the brake is applied. This helps to improve the fuel efficiency (electric cost) and its use has been expanding mainly with the hybrid and electric vehicles which use motors for driving power [4, 7, 8]. In order to overcome the weaknesses of a conventional parking brake system, an active parking brake system. There are two types of an active parking brake system namely electro-mechanical (EMPB) and electric parking brake (EPB) systems. The main advantages of this system are given as follows:
16
i.
Greater freedom in designing the interior.The handbrake lever has been discontinued and replaced by a button. This allows greater freedom in the interior layout and in designing the centre console and footwell area.
ii.
Greater functionality for the customer. The use of an electronic control system and CAN network, the electromechanical parking brake offers additional helpful functions for the customer (such as AUTOHOLD or dynamic drive off assistant) and a higher level of comfort.
iii.
Rollaway prevention mode. After the vehicle is parked, the system will monitor and control the parking brake force to ensure the vehicle in safe parking condition even the rear brake unit is overheated or the friction level is low.
2.3.2.1 Electro-Mechanical Parking Brake System
Figure 2.8 shows an appearance of an electromechanical brake unit developed for the rear wheels of 1500cc-class vehicles and Figure 2.9 shows its internal structure [4]. This unit has a linear motion mechanism and a motor equipped in parallel and parallel shaft gears are used for the transmission of torque between them. The torque produced by the motor is transmitted to the linear motion mechanism via the gears and converted to the load to press the pad onto the disc. The motor is important to generate displacement for the organization. A set of gears is used as speed reduction in the actuator. The conversion mechanism commonly consists of screw shaft and nut which the beam is only rotated and the nut is moveable depended on the rotation of the light beam. Release of the electromechanical parking brake is only possible with the ignition switched on. The electromechanical parking brake is released by pressing the brake pedal and actuating the electromechanical parking brake button at the same time [7] .
17
When the driver puts the seat belt on, closes the door and starts the engine, the electromechanical parking brake is released automatically when the accelerator pedal is pressed to drive off. The release point is thereby calculated depending on the tilt angle and engine torque. The warning lamps in the buttons and in the dash panel insert go out.
Figure 2.8: Electro-mechanical parking brake unit [4].
Figure 2.9: Schematic of Electro-mechanical parking brake[4].
18
2.3.2.2 Electric Parking Brake System
An electronic parking brake eliminates the bulky parking brake lever or pedal inside the car [8]. Electrical Parking Brake (EPB) is a system for vehicle that comprises the temporary brake of driving and long-term parking brake, operated by an electronic button, and releases parking brake an appropriate time by electrical control unit according accelerate pedal, clutch and engine speed signals [8].
It
requires no effort, and no thought in the case of a smart EPB system. The smart EPB system automatically uses the brake without driver input when the vehicle stops moving or the transmission is placed in Park [8]. An electronic parking brake can also deliver a built-in intelligent release function so that it is not accidentally left on when the transmission is put in power train. The EPB can also be employed to offer a "hill holding" capability for vehicles with manual transmissions so the car doesn't roll backwards, or forwards, when arrested along a slope [8]. EPB can also serve as an emergency brake should the hydraulic brakes fail, although when using the rear brakes only, it can require a long time to slow down or block off a speeding vehicle. As shown in Figure 2.10.
Figure 2.10: Parts of electrical parking brake[8]
19
Parts of electrical parking brake: 1. DC-motor 2. Gear box 3. Cable 4. Screw transmission 5. Emergency release device 6. Tension sensor 7. Housing
2.4
Analysis of Service Brake and Parking Brake System
In this section, analyses of service brake and parking brake system are described in detail based on either brake factor or clamping force or brake torque. It can be analyzed using theoretical, numerical and experimental approaches.
2.4.1
Service Brake
Service brake system is one of the crucial safety systems in a vehicle. Its main functions are to maintain vehicles’ speed when travelling downhill and to reduce the speed until the vehicle stop completely. The service brake typically uses hydraulic system as a medium to transfer force from brake pedal to front and rear brake units by producing clamping force and later generate braking torque.
20
In order to predict or assess service brake performance, there are numerous studies have been carried out by previous researchers either by theoretical [1] or numerical or experimental approaches.
2.4.1.1 Theoretical Approach
A number of service brake models have been developed to analyze the brake performance. For example, [1] has developed both disc and drum brake models to estimate brake factor. The brake factor is defined as ration between friction force and force application to the brake unit. The brake factor for standard caliper disc brake as given in Equation (2.1); Brake factor, BF
Fd 2 L Fa
(2.1)
Where; Fd is drag force of the disc Fa is application force from the piston µL is lining coefficient of friction
Figure 2.11 illustrates forces reaction in a drum due to self-energizing effect for leading shoe. In order to apply brake force to drum, piston force, Fa is applied to brake shoe when the driver pressed the brake pedal. The force pushes the brake shoe against the drum which the brake shoe was pivoted at another end that was marked as A in the diagram. Consequently, normal force, Fn is formed as the reaction force due to the pushing force of the piston. At the same location, friction force of known as drag force, Fd is formed as a result of drum rotation. Brake factor for self-energizing effect in a drum brake was derived as Equation (2.2) which µL was the lining coefficient of friction, and h, b and c were the length dimension as shown in Figure 2.11.
21
Drum rotational direction
Figure 2.11: Self energizing in a drum [1].
Fd Lh Fa b L c
(2.2)
For reversed direction of the drum, brake factor of the trailing shoe is shown in Equation (2.3). For a complete set of leading-trailing-type of the brake shoes, the brake factors was given in Equation (2.4)
Fd Lh Fa b L c
(2.3)
Fd 2 L h b Fa 1 L c b 2
(2.4)
The brake factor for brake shoe with pivot on each shoe in Figure 2.12 could be derived as Equation (2.5).
22
Figure 2.12: Leading shoe with pivot [1].
Fd Fa
Lh r
ˆ a r 0 sin 0 cos 3 L 1 a cos 0 2cos 3 2 r 4 sin 0 2sin 3 2
(2.5)
where;
a = brake length dimension, mm
ˆ 0 = arc of the angle a0, rad 1 = angle between beginning of the lining and straight line connecting center and pivot point, deg
2 1 arc angle, deg
3 1 2 , deg
Whereas, the brake factor for trailing-type brake shoe, minus sign in the denominator of Equation (2.6) is replaced by plus sign. Its equation was shown in Equation (2.7). Fd Fa
Lh r
ˆ a r 0 sin 0 cos 3 L 1 a cos 0 2cos 3 2 r 4 sin 0 2sin 3 2
(2.6)
23
The total brake factor for leading-trailing-type brake shoes is the summation of the individual leading shoe, BF1 and trailing shoe, BF2 brake factors as shown in Equation (2.7). BF = BF1 + BF2 = Fd1 / Fa + Fd2 / Fa
(2.7)
where; Fd1 = drag force on leading shoe, N Fd2 = drag force on trailing shoe, N
The schematic of leading shoe with parallel sliding abutment is illustrated in Figure 2.13. Its brake factor is derived as in Equation (2.8). Whereas, the brake factor for trailing shoe was shown in Equation (2.9).
Figure 2.13: Leading shoe with parallel sliding abutment [1].
F
F
Fd 1 L DB L2 E B Fa
Fd 2 L DB L2 E B Fa
B
B
L GB L2 H B
(2.8)
L GB L2 H B
(2.9)
where;
c a o c DB S cos S sin r r r r
24
c a c o E B S cos S sin r r r r FB
ˆ 0 sin 0 a o S 4 sin 0 2 r r
GB cos S sin H B FB S cos sin
o = brake dimension, mm r = drum radius, mm
0 = arc angle of lining, deg β = angle between center of the arc angle and horizontal center line, deg γ = angle between beginning of lining and horizontal center line, deg
S = friction coefficient at shoe tip and abutment
For brake shoes with inclined abutment as shown in Figure 2.14, their actions are the same as the Equations (2.8) and (2.9), but the remainder is only µs which should be replaced with (µs + tan ψ) where ψ was the leaning angle of the abutment. The sum of those equations would form the total brake factor for leading-trailing brake shoe with inclined abutment.
Figure 2.14: Leading shoe with inclined abutment [1].
25
The schematic of the brake duo servo brake with sliding abutment is illustrated in Figure 2.15. The relationship shown earlier can be used to determine the brake factor.
In this case, the internal application force of primary shoe
designated by 1, becomes the actuation force by 2. The equation for leading shoe for due-servo is the same as the leading trailing type.
Where,
F
LGB L2 H B
F
LGB L2 EB
Fd 1 L DB L2 EB Fa Fd 2 L DB L2 EB Fa
B
B
(2.10)
(2.11)
Fax F c / a d 1 (r / a) Fa Fa
(2.12)
Figure 2.15: Due-servo brake with sliding abutment [1].
There are also a number of drum brake types available apart from leadingtrailing-type of brake shoe such as two-leading, duo-servo, s-cam and wedge brake. Analysis of these types of brake is not given here because of different in design geometry and details of analysis can be found in [1].
26
2.4.1.2 Numerical Approach
In numerical approach, a mathematical analysis using iteration and reputation has been utilized to resolve the trouble. This type of process needs a longer time to complete the analysis. This approaches provided some of errors due to assumption and rounding of number. Hence, the conception of the computer in this advanced era is truly helpful to obtain a solution using this method. The most preferred method to the analysis is using finite element method. The finite element method has become a potent instrument for the numerical answers for a broad scope of engineering problems. Two type of numerical approach to analysis the brake performance [10]. The first approach is a finite element modal analysis of the brake system and being used to identify its eigenvalues and to relate them to the squeal occurrence. The second approach is a specific finite element programme, the appropriate for nonlinear dynamic analyses in the time domain and is particularly addressed to study contact problems with friction between deformable bodies.
2.4.1.3 Experimental Approach
Experimental approach is essential to quantify the nature of a phenomenon and the various operating conditions affecting. Thus, it also important in verification of the result obtained from theoretical and numerical approaches. An experimental research conducted in friction behavior and squeal vibration generation for four different disc brake pads at low speeds [11]. They studied the influence of speed and pressure variation. The test included braking with continuously increasing and decreasing brake pressure at constant speed and vice versa. Their results indicated that no squeal vibrations were generated below a coefficient of friction of 0.4. Braking conditions with a high coefficient of friction were related to more frequent squeal vibration generation.
27
The carried out theoretical and experimental work on a floating caliper disc brake system using a linear, lumped, and distributed parameter model [12]. They investigated the dynamic stability by using the complex eigenvalues. Their work was done with a constant friction coefficient. Their results indicated that a small error between experimental and theoretical natural frequencies, the mode of the disc is significantly responsible of the frequencies of the disc brake vibration and noise as well as the mode of each component. The system is more unstable if the friction coefficient, lining stiffness, length of the pads, and thickness of the lining are large, the system is more stable if the Young’s modulus and the mass of the disc and pads are large.
An experimental work conducted on brake and clutch facing samples in sliding motion at different levels of loading, slip speed and sliding acceleration [13]. They performed short time experiments using Pin-on-disc sliding contact in the laboratory test stand. They obtained a comprehensive view of the influence of the main sliding parameters by means of an artificial neural network. They investigated the not weak influence of the sliding acceleration to improve the friction coefficient prediction during transient operations. They concluded that the higher the sliding acceleration, the higher the friction coefficient. The materials have exhibited nearly linear dependence of the friction coefficient on the pressure contact in the studied ranges.
2.4.2
Parking Brake System
The principal role of parking brake system is to provide sufficient clamping force and subsequently brake torque to the brake unit in order to obtain a vehicle stationary either on a plane or slope road condition. This can be accomplished when the brake torque generated by the rear brake unit is equal or more than the brake
28
torque required by the vehicle. Otherwise, the vehicle will rollaway. Therefore, it is important to assess parking brake performance for those two road conditions to assure safety of the vehicle. The parking brake performance can be asses through theoretical or numerical or experimental approaches.
2.4.2.1 Theoretical Approach
In this approach, parking brake performance is typically predicted by developing mathematical equations of the parking brake system. To date, there is a number of studies has been done using this approach. Some research has developed one-dimensional disc type conventional parking brake model using a number of linear springs as shown in Figure 2.16 [3]. In the study, performance of the parking brake model was assessed by predicting clamping force as given in Equation (2.10) then the result was then validated against test data. In addition, they also investigated vehicle rollaway with temperature effects. The parametric studies are performed at various road slopes, vehicle weight, thermal expansion coefficients of the drum and the lining, and drum/lining friction coefficients. These parameters are seen to be significant and often varied during vehicle operation.
29
Figure 2.16: One-dimensional model of parking drum brake system [3].
F C Al El hb Al El l Tlol Ad Ed d Tlol R Fi Ad E d d T Al El lod Ad Ed lol
(2.13)
where ΔT is given by the following equation; T T t To
and Tt To Tamb e
(2.14) hAt pcpV
Tamb
(2.15)
2.4.2.2 Numerical Approach
Primary input parameters are engine speed signal, parking brake switch signal, ignition switch signal, accelerator pedal position signal, clutch pedal position
30
signal, brake staff position signal. Former three are digital signals, others are analog signals. Control principle is shown as Figure 2.17.
Figure 2.17: Work program of electrical controlled system [8].
2.4.2.3 Experimental Approach
Experimental approach is conducted in two-test conditions, namely onvehicle and on-bench tests. The following subsection will describe the result for both experiments. On-vehicle test is important in setting up the bench. Test is conducted to obtain force required to pull the handbrake and torque generated by braking system in terms of handbrake ratchet teeth.
An experiment has been conducted for drum type parking brakes that involving on-vehicle and on-bench test [3]. On-vehicle test was conducted on three varying gradients from 2° until 8°. The level of ratchet teeth is recorded and the torque required is calculated based on the slope of the road. In the test rig, the vehicle brake will be test in two temperature condition that is 25° in room
31
temperature and variable temperature when heated. In order to study the effect of force applied, the handbrake is engaged based on two methods, either specified force applied to the handbrake lever or position of ratchet teeth. In the experiments, it was found that the brake torque in the downward direction is about 30 percent higher than the upward direction.
Figure 2.18: Schematic diagram of torque measurement [3].
The brake torque predicted with the lump parameter model is reasonably close to the experimental results, with an average difference of not more than 12 percent. From the experimental analysis shows that, torque generated by drum brake linearly increases when surface temperature of drum brake increases. The test rig that been set up as shown in Figure 2.18 and Figure 2.19.
32
Figure 2.19: Parking brake test bench [3].
Recently [14], carried out a study to investigate the friction coefficient of copper and aluminum with sliding velocity. In the experiments it was found that as the sliding velocity increases from 1 to 2 m/s, friction coefficient of copper increases from
0.25 to 0.36. On the other hand, friction coefficient of aluminum
increases from 0.46 to 0.60 as the sliding velocity increases from 1 to 2 m/s. Due to the interaction of the asperities of two contact surfaces, frictional heat generation occurs and hence temperature increases at the contact surfaces. Due to more adhesion of pin material on the disc with the increase in sliding velocity, friction increases.
33
2.5 Summary
The designing of parking brake system must meet the requirement by many of regulation. The component and system must be following the targets as stated in FMVSS 105, FMVSS 135 and ECE. In FMVSS 135, the parking brake must hold the fully loaded and empty vehicle stationary on a incline of 30% for standard transmission vehicles, or 20% for vehicle with automatic transmission.
Nowadays, parking brake system having many type of improvement concerning to the safety and comfortable usage. The technologies have improved parking brake systems into three types that is fully mechanical (conventional), electro-mechanical (EMPB) and electric (EPB) parking brake system.
The
conventional parking brake system consists of three major parts, namely parking brake controller, parking brake linkages and rear parking brake unit.
The
conventional parking brake system can be manipulated through three methods, those are centered lever, stick lever and foot pedal. To transmit and multiply or equalize the force applied, parking brake linkages which consist of cables, rods, levers, and equalizers or adjusted are used. Actuation of parking brake system is carried out by the rear parking brake unit which can be used the drum unit or disc unit. The active parking brake system (electro-mechanical (EMPB) and electric (EPB) parking brake) commonly uses switch button to engage or disengage the brake system. The electrical signal is sent to parking brake actuator which located before parking brake linkage for EMPB or at rear brake unit for EPB. Brake actuators are the devices that convert the compressed air force into a mechanical force, which activates the brake. Between parts use in the active parking brake’s actuator are motor, gear reduction, conversion mechanism, locking mechanism, and sensor.
From the literature review, it testifies that most of the researchers emphasize on the theoretical, numerical and experimental method. However, the study on parking brake systems has not been well addressed despite its importance. For
34
instance, [3] assessed conventional parking brake performance including vehicle rollaway by analytical and experimental methods. However, the assessment was made based on a drum brake of parking brake system. For the theoretical method, his use a lump parameter that excesses the dynamic analysis of parking brake system that include of deflection on the drum brake. Therefore, it is significant in current research to fill the gap by assessing performance of conventional drum-type of parking brake system using mathematical modeling of a static analysis of drum brake. This is due to most of current passenger cars are still using drum brake systems in the rear axle.
35
CHAPTER 3
METHODOLOGY OF PRESENT STUDY
3.1
Introduction
This chapter provides a detailed description of research methodology to undertake the research. It can be divided into two main stages as follows:
1.
Theoretical modelling of parking brake system and its validation, and
2.
Evaluation of parking brake performance for various input parameters.
The overall work flow of research can be seen in Figure 3.1.
36
Figure 3.1: Overall research flow.
3.2
Theoretical Modeling of a Parking Brake Unit
In parking brake system, the prediction value must be calculated are at the torque. There are two type of torque that’s important in applications, which are torque required to hold the vehicle and torque generated by parking brake. When certain force is applied to handbrake lever, the force will be divided into two then be delivered to rear parking brake unit through cable. Then it will develop certain braking force on rear drum unit to stop the drum from moving. Product of the reaction force, coefficient of friction and radius of the drum is known as torque generated. To measure the torque, Saga BLM parking brake model have been used to develop mathematical modeling that carried out through static analysis. For
37
measuring torque required, free body diagram of vehicle on the road slope is developed and force reaction against the vehicle is carried out through static analysis .
3.2.1
Torque Required to Hold a Vehicle
From static analysis, torque required by the vehicle to maintain stationery is varies by the slope. Figure 3.2 shows a vehicle with certain mass m that applied at the vehicle is parked on gradient θ. Reaction force due to mass of the vehicle located at centre of gravity of the vehicle to gravity direction. The reaction force is resolved against road slope axis x’ and proportional axis to the road slope y’. There are reaction for acting on the tire of vehicle which is normal to the road slope because of mass of the vehicle. When parking brake is applied at the rear tyres, the friction force only produced at that tyres. Thus, force equation at x’ direction is Ffric r Fr mg sin
(3.1)
To convert the friction force in term of torque, the force must been times it the radius of the wheel. Treq r Fr Rwheel mg sin Rwheel
Where;
r
= friction coefficient between road and tyre
Fr
= normal force at rear tyre and
Rwheel = radius of tyre.
(3.2)
38
Figure 3.2: Diagram of vehicle which is parked at a slope [3].
3.2.2
Torque Generated by the Parking Brake
In typical parking brake system in Figure 3.3, torque generated at drum/lining interface when force is applied at handbrake lever. When the force at the handbrake, it will be deliver to the rear drum brake which located on the left and the right. The force will be split into two by the cable equalizer. When the force reaches the drum brake, the cable that attach on the mechanism in drum brake will generated torque to stop the drum. It needs analysis of on handbrake lever and the mechanism on the parking brake type.
39
Figure 3.3: Typical parking brake system [1].
Torque which generates at drum brake is applied from handbrake lever and it is transmitted through brake cable. When handbrake lever had been pulled, it will cause tension Fc to the cable. Using principle of moment, the cable tension Fc which always depends on the force applied Fhb to the lever and the lengths of force reaction point to the lever pivot. The handbrake ratio is depends on the lengths from the force point to the pivot. Figure 3.4 illustrates forces on handbrake lever and the lengths of the force reaction to the pivot where dl is the length from handrail to pivot and ds is length from cable to pivot. The tension Fc must be must time by two because it must been split to the left and right rear drum brake.
Figure 3.4: Force reaction at handbrake lever.
40
Moment of parking brake lever at the pivot,
Fhbd l 2 Fc d s
(3.3)
Displacement at the force reaction point depends on distance of the point to the pivot point l. Force ration at handbrake lever is
Fc
Fhb d l 2 ds
(3.4)
With the assumption brake cable is very efficient without any losses force and elongation of the cable, the displacement of cable, which is tied at the handbrake lever and parking brake lever, is approximately same. In order to apply force against drum brake, parking brake lever is attached at the primary brake shoe. Once cable is pulled, the lever is pushed inward and strut is pushed against the secondary brake shoe. The primary shoe will not been used for the parking brake system. The piston will neglected because it never been used in conventional parking brake. The parts that involve in parking brake mechanism on the Figure 4 and the free body diagram on the Figure 3.5.
Figure 3.5: Parts in drum brake component [3].
41
(c) Figure 3.6: Force reaction at (a) Secondary brake shoe, (b) Strut, and (c) Parking brake lever.
When handbrake lever is pulled, force generates at parking brake lever is assumed proportional with ratio of reaction force on the component. As shown in Figure 3.6, Fc represents the pulling force from cable against parking lever when handbrake lever force applied, and Fa1 represents reaction force at the strut and Fa2 at the pivot point when the parking lever is pulled. Moment equation at the lever is,
Fa1la Fc lb
(3.5)
Ratio between forces reaction on strut against cable force is,
Fa1 Fc
la lb
(3.6)
Consequently from reaction of parking lever pushes the brake strut, secondary brake shoe is pushed by the strut outward against drum as illustrated in Figure 3.6 (b). Resolve the force for x-axis,
42 Fa 2 Fc Fa1 0
(3.7)
After simultaneous the Equation (3.6) with Equation (3.7), we got the second lever brake ratio,
Fa 2 Fc (
la 1) lb
(3.8)
To analyze the parking brake torque, the ratio of the handbrake and brake lever must be combined, thus:
Fa1
Fhb l a d l 2 lb d s
(3.9)
And
Fa 2
Fhb d l l a ( 1) 2 d s lb
(3.10)
For interaction between the strut and the drum brake, the equation have been developed.
3.2.2.1 Parking Brake Model Parameter
As previously mentioned, leading trailing with pivot mathematical model has been developed by two type of system that are for distributed loading and concentrated loading on the drum shoes. These models have been choosing because the mechanisms are similar like parking brake for Saga BLM. These parking brake
43
also have a pivot at the bottom of brake shoes same as the drum brake that been studying. This pivot point became the center of movement for the drum brake shoes. This theoretical approach are using a static model are been different with a dynamic model. This theoretical approach did not consider of the deflection or temperature changer by the drum brake system. It does just consider of value of the parameter by the model itself. When the parameter and dimension are been change, the value are being change a lot because these theoretical approach are depending on the parameter and dimension of the model. Those parameters and dimension are being taken from the brake component itself. Those parameters are shown in Table 3.1.
The parking brake system to be analysed is leading trailing with pivot drum brake type. This is because of the similarities for the force applied to the drum brake and the point of moment at the pivoted location. From this model type drum brake, it has been separated into two loading system that are concentrated and distributed loading.
Table 3.1: Parameter for Proton Saga BLM drum brake model Parameter
symbol
value
Drum brake dimension Drum radius, m
r
0.105
Brake dimension,m
c
0.04
Brake dimension,m
a
0.076
Brake dimension,m
o
0.025
Drum thickness,m
t
0.005
Brake dimension,m
h
0.116
Brake dimension,m
b
0.076
Arc of lining angle, rad
a.
1.83
Brake dimension,m
a'
0.077
44
alpha/2,°
α/2
52.5
alpha,°
α.
105
alpha1,°
α1
50
alpha3,°
α3
205
µ
0.37
Lining-drum friction coefficient, dimensionless
Distance for brake ratio at drum brake Cable tension to the pivot, m
La
0.125
pivot to the strut, m
Lb
0.03
Distance for brake ratio at hand lever Force applied to pivot, m
dl
0.11
Tension of cable to the pivot, m
ds
0.03
3.2.2.2 Concentrated loading Leading-Trailing with pivot
This model is just considering single lining segment for the load being applied on it. The force normal is being applied at the center of the drum brake. This model did not consider a friction force that applied through along the lining. Figure 3.7 illustrates forces reaction in a drum due to concentrated loading Leading-Trailing with pivot. The force pushes the brake shoe against the drum which the brake shoe was pivoted at the other end that was marked as A in the diagram. Consequently, normal force, Fn is formed as the reaction force due to the pushing force of the piston. At the same location, friction force of known as drag force, Fd is formed as a result of drum rotation. Brake factor for self-energizing effect in a drum brake was derived as Equation (2.2) which µL was the lining coefficient of friction, and h, b and c were the length dimension as shown in Figure 3.7.
45
Figure 3.7: Concentrated loading Leading-Trailing with pivot drum brake.
For the concentrated loading on downward direction, the reaction force is occurring at the small spot of parking brake shoe. To analyzed this parking brake system, we must taking a moment at the point A, M a 0 ;
Fd ( L b c) Fa1h
Fd Lh Fa1 b L c
(3.11)
(3.12)
To find a torque the force at the drum brake must multiply with the radius of the wheel.
T Fd r
Fhb l a d l Lh r 2 lb d s b Lc
(3.13)
For trailings shoes equation are bit different by changing the denominator with (+) sign and change the force applied to it became Fa2.
46
T Fd r (
Fhb d l l a Lh ( 1) r) 2 d s lb b Lc
(3.14)
For torque generated on both shoe should be
Tgen
Fhb d l l a Fhb d l l a Lh r ( 1) L h r 2 d s lb 2 d s lb Fd r b Lc b Lc
(3.15)
Torque generated at the leading shoes drum brake using concentrated loading model can be calculated using this Equation 3.15.
3.2.2.3 Distributed Load Leading-trailing with pivot
This model is just considering multiple of lining segment for the load being applied on it. The force normal is being applied at the whole of the brake lining.. Figure 3.8 shown the concentrated load applied at the brake lining of the drum brake. The brake factor of the trailing shoe is determined by using the plus sign in. The dimension a' has a major effect on the brake factor of the leading-trailing shoe brake for a given lining coefficient of friction as illustrated in Fig. 3.8.
47
Figure 3.8: Distributed Load Leading-Tailing with pivot.
For the leading-trailing with pivot parking brake, the pressure distribute at the whole brake shoe area that attach to the drum for leading shoes on downward direction. To analyzed this parking brake system, we must taking a moment at the pivot point, M a 0 ; Fd h M1 M 2 0
(3.16)
Due to friction force Fd , max
M1
Prwda sin
(3.17)
Due to normal force in term of Fd , max
M2
P
rwd (r a cos )
After resolving the integrated function for look like this;
(3.18)
and
, the final equation will
48
Fd Fa1
Lh r
ˆ a r 0 sin 0 cos 3 L 1 a cos 0 2cos 3 2 r 4 sin 0 2sin 3 2
(3.19)
To find the torque, Equation 3.19 will be multiplied by the radius of the wheel. Hence, Equation 3.20 had been created.
Fhb d l l a L ha 2 d s lb T Fd r ˆ a 0 sin 0 cos 3 L r a cos 0 2cos 3 2 4 sin 0 2sin 3 2
(3.20)
Where;
a = brake length dimension, mm
ˆ 0 = arc of the angle a0, rad 1 = angle between the beginning of the lining and straight lines connecting center and pivot point, deg
2 1 arc angle, deg
3 1 2 , deg w width of drum lining
For the trailing shows for the downward direction, equation are bit different by changing the denominator with (+) sign and change the force applied to it became Fa2.
Fhb d l l a ( 1) L ha 2 d s lb T Fd r ˆ a 0 sin 0 cos 3 L r a cos 0 2cos 3 2 4 sin 0 2sin 3 2
(3.21)
49
Then, the equations for torque generated by distributed load of leading and trailing with pivot are:
Fhb d l l a L ha r 2 d s lb T Fd r ˆ 0 sin 0 cos 3 a L r a cos 0 2 cos 3 2 4 sin 0 2sin 3 2 Fhb d l l a ( 1) L ha r 2 d s lb ˆ a 0 sin 0 cos 3 L r a cos 0 2 cos 3 2 4 sin 0 2sin 3 2
3.3
(3.22)
Parametric Studies
In analytical analysis, various parameters such as vehicle weight, geometry of brake model and lining coefficient of friction are varied to investigate their effect to torque generated. The slope of the road will be varied on the torque required to hold the vehicle. Besides, the effect of the torque generated when the vehicle is parked in different facing direction also be studied to determine the best position of the vehicle should be parked. Table 3.2 shows table for different vehicle weight with maximum passenger. Table 3.3 shows table for torque generated for different coefficient of friction. Table 3.4 and Table 3.5 show torque required for different slope and torque generated when changing the parameter of the drum brake a’.
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Table 3.2 Different vehicle weight Torque required
Weight 1250 1329 1408 1487 1566 -
Treq(Mcar) Treq(Mcar +1p) Treq(Mcar +2p) Treq(Mcar +3p) Treq(Mcar +4p) Treq (11.3°)
Table 3.3 Lining Coefficient of Friction Coefficient of friction Tgen (µ=0.1) Tgen (µ=0.2) Tgen (µ=0.3) Tgen (µ=0.35) Tgen (µ=0.39) Treq (11.3°)
Torque generated
Table 3.4 Slope road SLOPE 4.17-Malaysia average 11.3° 12.3° 13.3° 14.3° 15.3° 16.3° 19°-World
Torque required
Table 3.5 Distance from pivot to center, a’ Distance a’ 0.6 0.65 0.7 0.75 0.8
Torque required
51
3.4
Summary
Different force applied in drum brake, it type produce a different parking brake torque. The geometry of the model provides a different equation that must be solved in same principle of moment. The torque required to hold the vehicle will be involved with the weight of the vehicle, coefficient of friction and the slope of the road. Torque generated at the drum brake must be higher than the torque required holding the vehicle in order to prevent the rollaway phenomenon.
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CHAPTER 4
RESULTS AND DISCUSSION RESULTS AND DISCUSSION
4.1
Introduction
This chapter shows validation for parking brake model while having parametric studies and this chapter ends with a short summary.
4.2
Validation of Parking Brake Model
The result for two theoretical analysis are using the parameter that have been measured can been compared with the experimental data [3]. For the both theoretical analysis have been analysis with a leading shoes analysis and a leading and trailing shoes analysis. The value provided for single shoe a little bit smaller than a both shoes. To get a suitable value of generated torque, the lining coefficient of friction is being adjusted to meet nearer value as experimental data [3]. The acceptable value for lining coefficient of friction is within a range of 0.29-0.38 [26]. The lining coefficient of friction that been using for these analysis are 0.37 because the
53
generated torque on distributed load for leading and trailing are being nearer to the experimental value. Trend for this generated torque have been shown in Figure 4.1. Generated torque by distributed load is being selected because of the nearer value with the experimental data.
Figure 4.1: Generated torque by theoretical analysis and experimental data.
The data for the Distributed load of Leading-Tailing with pivot are been taken for compared with the experimental data. In Figure 4.2, the maximum and average differences in torque between theoretical and experimental are 5.97 percents and 2.16 percents, respectively. Compared with the test conducted by [18] which produced the maximum difference of 35 percent, hence it can be said that current result is acceptable. Therefore, the behavior of the handbrake system on the test bench can be used to represent the actual hand brake. With the verification, further studies on hand brake can be carried out on the test bench [3]. The maximum and average different of the test bench are 25 percents and the 15 percents. Hence, it can be said that the current result is acceptable. Therefore, the distributed load of leading-trailing with pivot theoretical analysis can be used to represent the actual parking brake system of Proton Saga BLM. With the verification, further studies on parking brake model with this type of analysis can be carried out by parametric study.
54
14
Percentage Difference %
12 10 8 6 4 2 0 114
151
193
241
333
400
500
Force applied, Fhb (N)
Figure 4.2: Percentage of difference of generated torque between theoretical and experimental data.
4.3
Parametric Studies
Some of parameters can be manipulated to study the effect of the parameters against braking performance in terms of braking torque. The parametric study can be done using the brake model that had been validated. The parameter can be manipulated consist of vehicle weight, sloped road, drum/lining coefficient of friction and geometry of brake model. In order to indicate the vehicle rollaway phenomenon, parametric study is simulated based on two types of torque, namely, torque generated and torque required as specified in Equations (3.21) and (3.23). The vehicle is assumed rollaway when torque generated lower than torque required.
55
From the regulation that had been discussed on Section 1.1, parking brake system must to hold or make it stationary of a passenger car on gradient road at 11.3° with applied force of 400N. This regulation can be used as requirement or benchmark in order to indicate the critical state of parking brake.
4.3.1
Vehicle Weight
Figure 4.3 shows torque required when the vehicle is parked with two different weights. The vehicle weight is 1250kg and passenger weight is assumed to be 79kg per person. The figure 4.3 shows, when the weight became higher, the torque required also increase. The increasing in force applied to the brake pedal, the torque generated became bigger due to the increasing weight on the vehicle. The minimum torque generated by the mass of the vehicle only is about 347 Nm. The average torque required for one passenger that will be added in the vehicle are 21.93 Nm. When increasing the weight of the vehicle with a maximum passenger of 5 people with average weight, the torque required to hold the vehicle stationery should be 434 Nm. From the graph on Figure 4.3 that shows, the torque that provide from model of distributed load of leading trailing of Saga BLM still can hold the vehicle stationery at a maximum load in the vehicle of 5 passenger.
56
700
Tgen < Treq = rollaway Tgen > Treq = not rollaway
Torque,Tq (Nm)
600 500 400
Tgen (downward)
300
Tgen (upward) Treq(M = 1250)
200
Treq(M = 1566)
100 0 0
200
400
600
Force applied,Fhb (N)
Figure 4.3: Torque required at different weight of the vehicle
4.3.2
Lining Coefficient of friction
Figure 4.4 shows torque generated with different of drum/lining coefficient of friction. The applied force is increase to the different of drum/lining coefficient of friction that produce torque generated at drum brake. It can be seen that, the vehicle can remain stationery on the road slope of 11.3° when the friction coefficient between drum and lining above than 0.2. Based on thesis [26] the value of lining coefficient of friction for normal vehicle around 0.29- 0.38. From Figure 4.4 the value for lining coefficient must above the 0.2 to meet the regulation by the ECER13 and FMVSS 135 that can been that show its meet with the range of value of lining coefficient [26].
57
800
Tgen < Treq = rollaway Tgen > Treq = not rollaway
700 600 Torque,Tq (Nm)
Treq (11.3°)
500
Tgen (µ=0.1)
400
Tgen (µ=0.2)
300
Tgen (µ=0.3) Tgen (µ=0.35)
200
Tgen (µ=0.38)
100 0 0
200 400 Force applied,Fhb (N)
600
Figure 4.4: Torque generated at variable drum/lining coefficient of friction with torque required by the vehicle.
4.3.3
Road Slope
Figure 4.5 shows torque generated with different of slope due to the road gradient. The regulation from ECE R13H, the drum brake model should generate torque to hold stationery at 20% of slope equal to 11.3° .Extracted from [15], the gradient of feeder road are less than are equal to 20%, but will follow natural benches and topography features when using existing roads and when newly specified by forest engineer. For average slope for Malaysia road are about 7.3% that is 4.17° that is lower than a regulation thus this parking brake model are suitable for Malaysia road standard. Baldwin Street is the steepest street in the world. It is located in the suburbs of Dunedin, New Zealand South [28]. According to records, the steepness of the road is 19 degrees or 35 % steepness recorded. From the graph presented, the torque that provide from model of distributed load of leading trailing
58
of Saga BLM still can hold the vehicle stationery at a world steepest road with a slope of 19°.
700
Tgen < Treq = rollaway Tgen > Treq = not rollaway
600
Torque,Tq (Nm)
500 Tgen (downward)
400
Tgen (upward) 300
Treq 11.3°
200
World Max=19° Malaysia Avrge=4.17°
100 0 0
200
400
600
Force applied,Fhb (N)
Figure 4.5: Torque required at different slope road
4.3.4 Distance from Pivot to Center, a’
Figure 4.6 shows torque generated with changing in parameter on the drum brake model. The dimension being changes are the value of distance between the pivot to the center of the drum, a'. This parameter provide a huge different when the value are being changing.
Regarding to the equation of distributed loading of
leading-trailing, the parameter a' positioned at the denominator. Hence, increasing in dimension of a' will reduce the torque generated by the parking brake. From Figure 4.6, the dimension of a' must be below 0.075m to stop it from rollaway at sloped 20% due to regulation.
59
1200.00
Tgen < Treq = rollaway Tgen > Treq = not rollaway
Torque,Tq (Nm)
1000.00
Treq (11.3°)
800.00
Tgen (a'=0.6)
600.00
Tgen (a'=0.65) Tgen (a'=0.7)
400.00
Tgen (a'=0.75)
200.00
Tgen (a'=0.8)
0.00 0
200
400
600
Force applied,Fhb (N)
Figure 4.6: Torque generated with different parameter of a’
4.4
Summary
Theoretical results show that predicted torque is quite close to the measured torque on the distributed loading. Then, it was compared to experimental data [3] that shown the mathematical model are acceptable. Then, parametric studies are carried out at lining coefficient of friction from 0.1 until 0.39, distance from pivot to center, a’ from 0.6 till 0.8, vehicle weight from vehicle without passenger till loaded with 5 passengers and slope of the hill from 4.17° till 19°. It is found that the vehicle did not rollaway at all range of conditions except when lining coefficient of friction is below than 0.3 and the dimension of a' upper than 0.075m.
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CHAPTER 5
CONCLUSION AND RECOMMENDATIONS
5.1
Conclusion
Parking brake system is a very important part for vehicle safety. It is used to assure that a vehicle does not travel or roll off when the vehicle is parked on a level or slope road. For the safety procedure, the parking brake system must comply with some regulation to be impost with relevant authorities. For example, FMVSS 135 stated that the parking brakes must be capable to hold the vehicle stationary for 5 minutes on 20% grade in both the forward and reverse directions with maximum applied force of 400N for hand-operated parking brakes. FMVSS 135 regulation shows similarities with ECE R13H and EEC regulation that requires vehicle to stay stationary on 20% for forward and reverse direction. Therefore, it is important for car manufacturers and brake engineers to plan and design a good parking brake unit using a proper approach to measure its performance prior to the batch production.
This research is intended to introduce methodology to assess parking brake performance for Proton Saga BLM drum brake using theoretical approach. The theoretical approach been developed by mathematical model of drum brake system.
61
According to the physical similarities of Proton Saga BLM drum brake type, leadingtrailing with pivot type drum brake has been chosen. For further analysis, leadingtrailing with pivot drum brake has been separate into two different types that consist of different force applied to the drum brake. Distributed load and concentrated load produce a different mathematical model analysis. To ensure validity of those mathematical models, it is then compared with the experimental test data at bench test [3].
For mathematical model analysis, the derivation of equation starts with the hand brake ratio that must been divided into two for separating the force for each drum brake system. After that, we must analyse lever ratio at the drum brake that consist of two ratios whether for the leading or trailing shoes. The lever ratio can be manipulated for both shoes according the direction of the vehicle whether going upward or downward. Lastly, the hand brake ratio and lever brake ratio can be transferring it into concentrated and distributed load system. Before getting some result, the parameter of the drum brake must be measured precisely for providing an accurate result. The torques generated by the drum brake is different between going downward and going upward. From the mathematical analysis, parking brake system which is parked with front side towards downhill direction requires high braking torque than front side towards uphill direction. This is due to design mechanisms of the leading-trailing type of the drum brake unit. For having a maximum torque generated, mathematical model of downward direction are being validated with the data from experimental result.
Comparing torque generated between both loading types, the distributed loading provides a higher torque than the concentrated loading. The lining coefficient of friction are being set-up for 0.37 because the torque generated closer to the experimental data and it’s still in a range of lining coefficient of friction for normal vehicle by [26] After comparing the result between the distributed loadings drum brake unit with the experimental, the maximum and average differences in torque between theoretical and experimental are 5.97 percents and 2.16 percents,
62
respectively. This percentage of difference still below 10 percent, hence it can be said that the current result is acceptable. Therefore, the distributed load of leadingtrailing with pivot theoretical analysis can be used to represent the actual parking brake system of Proton Saga BLM. With the verification, further studies on parking brake model with this type of analysis can be carried out by parametric study.
Some of parameters can be manipulated to study the effect of the parameters against braking performance in terms of braking torque. The parameters that can be manipulated consist of vehicle weight, sloped road, drum/lining coefficient of friction and geometry of brake model. In order to indicate the vehicle rollaway phenomenon, parametric study is simulated based on two types of torque, namely, torque generated and torque required. After analysed all the parametric study, some of conclusion can been provided. Firstly, increasing the vehicle weight to 1566kg still will not make the vehicle to roll away. Secondly, Proton Saga BLM parking brake model still can still stationery on the world maximum slope of 19°. Thirdly, lining coefficient for this model should upper than 0.2 in order to prevent it from rollaway at 11.3°. Lastly, the distance between pivot to the centre of drum brake, a' must be below 0.075m to stop it from rollaway at sloped 20% due to regulation.
5.2
Recommendations for Future Work
The following suggestions are made for future work:
1.
Inclusion of installation gap in model
There are will be good results when Inclusion of installation gap in model. The gap will change the value of the torque generated to the parking brake system. The gaps provide a distance for the shoes move toward it. For the future analysis, this may close the gap between experimental and theoretical results.
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2.
Temperature effect
Current study assumes that there is no effect of the temperature because it’s static modeling. The temperature of drum brake may contribute some effect on the clamping force to the drum brake. This may also contribute to the gap between predicted and test data. In order to reduce the difference of the result, some study can been done by including temperature effect.
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REFERENCE [1] Limpert, R. Brake Design and Safety. 2nd ed. Edited by Warrendale. Pa:Society of Automotive Engineers, 1999. [2] Maliye, S. Regenerative and Anti-lock braking system in electric car. Master Thesis, India: National Institute of Technology, Rourkela, 2014. [3] Ishak. Performance of fully mechanical parking brake system. Master Thesis, Malaysia: Universiti Teknologi Malaysia, 2013. [4] Yamsani, Akilesh. "Gradeability for Automobiles." ournal of Mechanical and Civil Engineering, 2014: 35-41. [5] Owen. Automotive Brake system, Classroom Manual. 5th ed. Clifton Park, New York: Cengage Learning, 2011. [6] McKinlay, A.J. The phenomenon of vehicle park brake rollaway. PhD Thesis, University of Leeds, 2007. [7] Volkswagen AG, Wolfsburg. Wolfsburg Patent 000.2811.60.20. 2005. [8] Xiaoci. "An electrical parking brake system design and analysis." Advanced Materials Research (Trans Tech Publications, Switzerland), 2011: 2254-2257. [9] V, Manjunath T. "Structural and Thermal Analysis of Rotor Disc of Disc Brake." International Journal of Innovative Research in Science,Engineering and Technology, 2013: 7741-7750. [10]Aldo. "Linear and Nonlinear Numerical Approaches to Brake Squeal Noise." Mechanical system and signal processing, 2013: 228-235. [11]Eriksson, M. "Friction behaviour and squeal generation of disc brake sat low speeds,." Proc Instn Mech Engrs, 2011: 225-230. [12]Joe. "Analysis Of Disc Brake Instability Due To Friction-Induced Vibration Using A Distributed Parameter Model." International Journal of Automotive Technology, 2008: 161-171. [13]Senatore, A. "Experimental investigation and neural network prediction of brakes and clutch material frictional behavior considering the sliding acceleration influence,." Tribology International,, 2003: 195-202. [14]Dewan. Friction coefficient and Wear Rate of Copper and AluminiumSliding against Mild stee. Malaysia: Universiti Malaysia Pahang, 2013.
65 [15]Abdullah, C.H. ROAD SLOPE MANAGEMENT IN MALAYSIA. Slope Engineering Branch, 2010. [16]Agudelo, CarlosE. "Technical overview of brake performance testing for Original Equipment and Aftermarket industries in the US and European markets." Technical Report , n.d.: 1-27. [17]AVIATION, CIVIL. "STANDARD PASSENGER AND BAGGAGE WEIGHTS." CIVIL AVIATION SAFETY AUTHORITY (CIVIL AVIATION), 1990. [18]Chrysler, Daimler. "An introduction to Brake system." SAE Brake Colloquium,. [19]Coast Guard. Passenger Weight and Inspected Vessel. DEPARTMENT OF HOMELAND, 2010. [20]Franco Gamero, M.S. PCBRAKE FACTOR . 2012. [21]Hill, Primrose. "BRAKING REGULATIONS." The Road Safety , 2009. [22]International, The Union Cyclist. "TOP 10 CYCLE CLIMBS IN ASIA PACIFIC." 2014. [23]K.Gopinath, Prof. Machine Design II. Indian Institute of Technology Madras, 2012. [24]KHAIRNAR, H. P. "Estimation of automotive brake drum−shoe interface frictioncoefficient under varying conditions of longitudinal forces using Simulink." Research article, 2015: 215-219. [25]M.Ghazaly, Nouby. "Experimental Investigation Of Drum brake Performance for Passanger Car." The IIER-Science Plus International Coference. Kuala Lumpur, 2014. 35-38. [26]Muhammad Najib Bin Abdul Hamid. "Analysis Of Drum Brake Squeal." Universiti Sains Malaysia, 2015. [27]Phamphlet, Darcom. ANALYSIS AND DESIGN OF AUTOMOTIVE BRAKE SYSTEMS. Virginia: U.S. DEPARTMENT OF COMMERCE, 1976. [28]record, Guiness World. "Guiness World recordh." ttp://www.guinnessworldrecords.com/search?term=steepest+road. 2016. [29]TRANSPORTATION, U.S. DEPARTMENT OF. LABORATORY TEST PROCEDURE FOR FMVSS 135 Light Vehicle Brake Systems. U.S Patent TP-135-01. December 2, 2005. [30]Yamasaki, T. "Electro-mechanical Brake Unit with Parking Brake." NTN Technical Review 81 (2013).
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APPENDIXES
ANALYSIS DATA
Table A: Torque generated for concentrated load Leading- Trailing with Concentrated load Torque (Fhb) Torque(L) Torque(T) (L+T) Experimental (1 axle) 114 64.13 32.85 96.97 142.50 151 84.94 43.51 128.45 192.00 193 108.56 55.61 164.18 261.00 241 135.56 69.44 205.01 321.50 333.00 187.31 95.95 283.27 447.00 400.00 225.00 115.26 340.26 530.00 500.00 281.25 144.07 425.32 668.00 Average Percentage
Percentage Difference -31.95 -33.10 -37.10 -36.23 -36.63 -35.80 -36.33 35.31
Table B: Torque generated for distributed load Leading- Trailing with Distributed loading (Fhb) Torque(L) Torque(T) Torque (L+T) Experimental (1 axle) 114 121.87 29.13 151.00 142.50 151 161.43 38.59 200.01 192.00 193 206.33 49.32 255.65 261.00 241 257.64 61.58 319.23 321.50 333.00 356.00 85.09 441.09 447.00 400.00 427.63 102.21 529.84 530.00 500.00 534.53 127.77 662.30 668.00 Average Percentage
PercentageDifference 5.97 4.17 -2.05 -0.71 -1.32 -0.03 -0.85 2.16
67
Table C: Torque required at different weight of the vehicle
(Fhb) 0 114 151 193 241 333 400 500
Tgen Tgen (downward) (upward) 0 0 151.00 126.97 200.01 168.17 255.65 214.95 319.23 268.41 441.09 370.87 529.84 445.49 662.30 556.86
1250 Treq(M = 1250) 347.01 347.01 347.01 347.01 347.01 347.01 347.01 347.01
1329 Treq(M = 1329) 368.94 368.94 368.94 368.94 368.94 368.94 368.94 368.94
Weight 1408 Treq(M = 1408) 390.87 390.87 390.87 390.87 390.87 390.87 390.87 390.87
1487 Treq(M = 1478) 412.80 412.80 412.80 412.80 412.80 412.80 412.80 412.80
1566 Treq(M = 1566) 434.73 434.73 434.73 434.73 434.73 434.73 434.73 434.73
Table D: Torque required at different slope road World Malaysia max average slope
regulation Tgen Tgen (Fhb) (downward) (upward) 0
0
0
11.3° Treq 11.3°
13.3° Treq 13.3°
15.3° Treq 15.3°
16.3° Treq 16.3°
4.17° Treq 4.17°
19° Treq 19°
347.01
407.40 467.30 497.04
128.78
576.56
114
151.00
126.97
347.01
407.40 467.30 497.04
128.78
576.56
151
200.01
168.17
347.01
407.40 467.30 497.04
128.78
576.56
193
255.65
214.95
347.01
407.40 467.30 497.04
128.78
576.56
241
319.23
268.41
347.01
407.40 467.30 497.04
128.78
576.56
333
441.09
370.87
347.01
407.40 467.30 497.04
128.78
576.56
400
529.84
445.49
347.01
407.40 467.30 497.04
128.78
576.56
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Table E: Torque generated at variable drum/lining coefficient of friction with torque required by the vehicle.
(Fhb) 0 114 151 193 241 333 400 500
Treq (11.3°) 347.01 347.01 347.01 347.01 347.01 347.01 347.01 347.01
0.1 Tgen (µ=0.1) 0 28.85 38.21 48.84 60.98 84.26 101.22 126.52
0.2 Tgen (µ=0.2) 0 62.59 82.90 105.96 132.31 182.82 219.60 274.50
Coefficient of friction 0.3 0.35 Tgen Tgen (µ=0.3) (µ=0.35) 0 0 107.18 151.00 141.97 200.01 181.46 255.65 226.59 319.23 313.09 441.09 376.08 529.84 470.10 662.30
0.38 Tgen (µ=0.38) 0 158.61 210.09 268.52 335.31 463.31 556.52 695.65
Table F: Torque generated with different parameter of a’
(Fhb) 0 114 151 193 241 333 400 500
Treq (11.3°) 347.01 347.01 347.01 347.01 347.01 347.01 347.01 347.01
0.6 Tgen (a'=0.6) 0 221.81 293.80 375.52 468.92 647.92 778.29 972.86
0.65 Tgen (a'=0.65) 0 185.86 246.19 314.67 392.92 542.92 652.16 815.20
0.7 Tgen (a'=0.7) 0 162.95 215.84 275.88 344.49 476.00 571.77 714.71
0.75 Tgen (a'=0.75) 0 151.00 200.01 255.65 319.23 441.09 529.84 662.30
0.8 Tgen (a'=0.8) 0 69.38 91.90 117.46 146.67 202.66 243.44 304.30
