26

CHAPTER 3

DESIGN AND OPTIMIZATION OF BRAKE DRUM

3.1 INTRODUCTION

The commercial brake system uses disc brake for front wheels and

drum brake for the rear wheels. Gray cast iron is the conventional material

used for making brake drums of light and heavy motor vehicle. The problems

encountered in the cast iron material are described in the second chapter. An

Al MMC brake drum has been designed to replace the heavy cast iron brake

drum of a typical passenger vehicle. The design parameters such as inner

radius, outer radius, and the width of drum, load and the allowable

deformation are kept same for both cast iron and MMC brake drum. The

theoretical formulation for the evaluation of stress, deformation and

temperature rise has been described in this chapter. The finite element

analysis of the cast iron and MMC brake drum has been also presented in this

chapter.

3.2 BRAKE DRUM FOR THE ANALYSIS

A brake drum assembly of a light passenger vehicle which is used

for the analysis is shown in Figure 3.1. The drum brake consists of backing

plate, brake shoes, brake drum, wheel cylinder, return springs and a self

adjusting system. Brake shoes consist of a steel shoe with the friction material

lining riveted or bonded to it. The brake drum has holes in the hub which are

27

used to mount the brake drum on the hub of the wheel and the design

parameters are listed in the Table 3.1.

Figure 3.1 Brake drum assembly

Table 3.1 Parameters of the brake drum

Parameter Value

Inner diameter (mm) 180

Outer diameter (mm) 205

Outer width (mm) 12

Inner width (mm) 40

Contact angle per shoe 113

Width of shoe (mm) 30

Number of shoes 2

The properties of the cast iron and MMC materials used for the

analysis are given in Table 3.2.

c

h b

28

3.3 METAL MATRIX COMPOSITE BRAKE DRUM

Metal Matrix Composites (MMCs) have emerged as a class of

advanced materials capable of advanced structural, aerospace and automotive

applications. MMCs have matured during the last two decades and are

currently used as structural components subjected to cyclic loads.

3.3.1 Selection of Metal Matrix Composite

There has been interest in using aluminum based metal matrix

composites (MMCs) for brake disc and drum materials in recent years. While

much lighter than cast iron, they are not as resistant to high temperatures and

are sometimes only used on the rear axles of automobiles because the energy

dissipation requirements are not as severe compared with the front axle.

While applying brake, the brakes convert the kinetic energy of the moving

vehicle into thermal energy. This thermal energy diffuses through conduction

within the brake drum and dissipates by convection and radiation from the

outer surface of the brake drum. The material used for the brake drum should

have the required physical, mechanical and thermal properties apart from

being light weight. The failure of the brake drum is due to the high

temperature generated inside the drum and also due to the high stresses

applied on the drum.

3.3.2 Selection of Matrix

Matrix is a continuous phase in which the reinforcement is

uniformly distributed. The advantages of metallic matrices as compared to

polymer matrices are their higher tensile strength, and shear modulus, high

melting point, low coefficient of thermal expansion, resistance to moisture,

dimensional stability, high ductility and toughness. For the matrix,

29

characteristics such as density, strength, high temperature strength, ductility

and toughness are to be considered. Generally Al, Ti, Mg, Ni, Cu, Pb, Fe, Ag,

Zn have been used as the matrix material. The main focus is given to

aluminium matrix because of its unique combination of good corrosion

resistance and excellent mechanical properties. The combination of light

weight, high thermal conductivity, and low cost makes the aluminium matrix

well suited for use as a matrix metal. The melting point of aluminum is high

enough to satisfy the application requirements. Also aluminium can

accommodate a variety of reinforcing agents including continuous boron,

aluminium oxide, SiC, and graphite fibers and various particles, short fibers

and whiskers.

3.3.3 Selection of Reinforcement

Reinforcement increases strength, stiffness and temperature

resistance capacities of MMCs. The reinforcement has different sizes, shapes

and volume fractions in the composite. The reinforcement may be continuous

fibers, whiskers or particles. The continuous fiber reinforcement has superior

properties than the discontinuous reinforcements, but suffers from the

disadvantage of anisotropic properties added to the need of adopting near net

shape forming techniques. The discontinuous reinforcement offers isotropic

properties and amenability to be processed by conventional secondary metal

forming techniques. Particulates are most common and least costly

reinforcement materials. The SiC particulate reinforced Al MMCs have good

potential for use as wear resistance materials. Actually particulates lead to a

favorable effect on properties such as hardness, wear resistance and

compressive strength. Selection criteria for the ceramic reinforcement include

factors like elastic modulus, tensile strength, density, melting temperature,

thermal stability, compatibility with matrix material and cost effectiveness.

30

The silicon carbide particulate is selected since it satisfies all the above

requirements.

3.3.4 Properties of A356/SiCp MMC

Aluminium alloy reinforced with SiC particles exhibit increased

strength and stiffness as compared to non-reinforced aluminium alloy. In

contrast to the base metal, the composite retains its room temperature tensile

strength at higher temperatures. Discontinuous silicon carbide/aluminium

MMCs are being developed by the aerospace industry for use in airplane

skins, intercoastal ribs, and electrical equipment. In the liquid metal

processing technique, the molten aluminium has the tendency to react with the

reinforcing materials. The severity of the reaction is based on the kinetics and

the prevailing thermodynamic conditions. The presence of alloying elements

in the matrix has the influence on viscosity, contact angle and reaction rate

with the dispersed particles. In the case of SiC, the following reaction is

observed with the molten aluminium alloy (Pai 2001).

3SiC + 2Al Al2 C3 + 3Si G = -51.3KJ mol-1

However, the above reaction can be prevented by the presence of

about 8% Silicon in the matrix, wherein the higher chemical potential of SiC

retards reaction. In A356/LM25 cast aluminium alloys, the above reaction is

not observed. So, this combination of the A356 and SiC is more suitable for

making MMCs with good mechanical properties. The mechanical properties

of cast iron and A356/SiCP is given in Table 3.2 (Limpert 1999, Foltz 1999.

31

Table 3.2 Material properties of cast iron and A356/SiCP

Property Cast Iron A356/SiCP

Tensile strength (MPa) 276 300

Youngs modulus (GPa) 100 112

Poissons ratio 0.3 0.3

Density (kg/m3) 7228 2828

Thermal conductivity (Nm/hm K) 174465 374400

Specific heat (Nm/kg K) 419 970

3.3.5 Requirements of MMC Brake Drum

The current investigation is aimed at the development of MMC

brake drum to replace the heavy cast iron brake drum used in the light

passenger vehicles. The dimensional parameters of the brake drum which are

used for the analysis is listed in the Table 3.1. Hence, the MMC brake drum

has to satisfy the braking and thermal requirements of the conventional brake

drum.

3.4 POWER ABSORBED IN THE BRAKE DRUM

During braking, the kinetic and the potential energies of a moving

vehicle are converted into thermal energy through friction in the brakes. For a

vehicle decelerating on a level surface from a higher velocity V1 to a lower

velocity V2 the braking energy Eb is given by

22212221 22

IVVmEb (3.1)

32

If the vehicle comes to a stop, then V2 = 2 = 0 and equation (3.1)

becomes

22

21

21 ImVEb (3.2)

When all rotating parts are expressed relative to the revolutions of

the wheel, then with V= R, equation (3.2) becomes

2Vkm

VmR

I12mE

21

212b

(3.3)

mRIk 21

Typical values of k for passenger cars range from 1.05 to 1.15 in

high gear to 1.3 to 1.5 in low gear. Corresponding values for trucks are 1.03

to 1.06 for high gear and 1.25 for low gear (Limpert 1999).

Braking power Pb is equal to braking energy divided by the time

t during which braking occurs, or

dtEdP bb (3.4)

If the deceleration a is constant, then the velocity V(t) is given by

atVV t 1)( (3.5)

33

Equations (3.3) through (3.5) yield the brake power as

).(.. 1 atVamkPb (3.6)

From the above equation (3.6) it is observed that the braking power

is not constant during the braking process. At the beginning of braking (t=0),

brake power is maximum, decelerating to zero when the vehicle stops.

The time ts for the vehicle come to a stop is

a

Vts 1 (3.7)

The average braking power Pbav excluding tire slip over the

braking time for a vehicle coming to a stop is

2

... 1VamkPbavg (3.8)

For a vehicle traveling downhill while decelerating, the brakes have

to absorb kinetic and potential energy as illustrated in Figure 3.2. Using

energy balance, the braking energy is

22212 VVkmWhEb

(3.9)

For a continued braking at constant speed, equation (3.9) becomes

with V1 = V2

WhEb (3.10)

34

Figure 3.2 Kinetic and potential energy on grade

Braking power during continued braking is obtained by

differentiating energy with respect to time, or

sNmdtdh

dhEd

dtEdP bbb /,

(3.11)

With the gradient expressed by angle and the actual distance

traveled on the highway expressed by l (Figure 3.2), the change in height

and road distance is related to the slope and is given as

dldh

sin

and equation (3.11) may be rewritten as

sinWVPb (3.12)

The average braking power qo may be expressed as

2)1( 1aWVskqo

(3.13)

35

The tire slip accounts for the energy absorbed by the tire/roadway

due to partial slipping of the tire. In the extreme, when the brake is locked, no

energy will be absorbed by the brake, i.e., s =1.

For the general case of deceleration on a downgrade, Equation

(3.13) may be modified by adding sin () to deceleration a. For uphill

braking, the slope effect is subtracted from the actual deceleration because the

brakes do not have to absorb all of the kinetic energy, since a portion is

transformed into potential energy.

The maximum brake power Pmax produced at the onset of braking is

equal to

bavgPP 2max (3.14)

as shown in Figure 3.3.

Figure 3.3 Variation of brake power with time

qo=constant, P

q(t) q(o), Pb(o)

0

Brak

e po

wer

Time ts

36

3.4.1 Temperature Rise in Single Stop Braking

In a single stop with high deceleration levels, the braking time may

be less than the time required for the heat to penetrate through the drum or

rotor material. Under these conditions, no convective brake cooling occurs

and all braking energy is assumed to be absorbed by the brake drum and the

lining.

For brake drums, the heat penetration time tb to reach the outer

drum surface is given by

5

2Ltb (3.15)

If a linearly decreasing braking power is assumed as shown in

Figure 3.3, the surface temperature as a function of time may be expressed as

si t

ttk

qTtLT

321

45, 2/10

2/1

(3.16)

Differentiation of equation (3.16) with respect to the time indicates

a maximum of the surface temperature at t = ts/2. Thus, the maximum surface

temperature Tmax, in a single stop brake application neglecting the ambient

cooling may be expressed as

21

21

)0(2

1

max, 185

ck

tqTT siL

(3.17)

37

The temperature rise in brake drums with braking time at 50 km /h

is shown in Figure 3.4.

0

50

100

150

200

250

0 0.5 1 1.5 2 2.5 3 3.5

Braking time (s)

Surfa

ce te

mpe

ratu

re (

OC)

MMC Brake drumCI Brake drum

Figure 3.4 Temperature rise in brake drums with time

3.4.2 Temperature Rise During Repeated Braking

During repeated brake applications, the vehicle is decelerated at a

given deceleration from 50kmph to a lower or zero speed, after which the

vehicle is accelerated again to test speed and next braking cycle is carried out.

Brake pumping involves repeated brake application from one single speed

until the vehicle stops. Brake temperature attained during brake pumping will

be less than those achieved during repeated braking because the braking

power is lower.

The brake temperature attained during repeated braking may be

computed from simple analytical solutions, provided the braking power,

cooling intervals, and braking times remain unchanged during the braking

test. Under these conditions, the equations for computing the temperature

38

increase during repeated brake applications may be expressed in simple form.

Assumptions are that the drum or disc can be treated as a lumped system, and

that the heat transfer coefficient and thermal properties are constant. In the

lumped system analysis, the temperature is assumed to be uniform throughout

the drum or rotor, making it a function of time only and not of space.

If the braking time is considerably less than the cooling time, then

the cooling during braking may not be significant. Hence, the temperature of

the brake drum will increase uniformly as given by the Equation (3.18).

cvtqT so

(3.18)

Also, the lumped formulation results in a differential equation

describing the cooling of the brake after a brake application as shown in

Equation (3.19).

)( TThAdtdTcv i (3.19)

With an initial temperature of Ti, integration of equation (3.19)

yields a cooling temperature response given as

vctAhi

i eTT

TtT

(3.20)

An analysis combining heating by means of Equation (3.18) and

cooling by means of Equation (3.20) may be developed to derive the

temperatures of a brake after the first, second, third or nth brake application.

The relative brake temperature before the nth brake application is

39

vctAh

vctAhvctAhn

b eeeTtT

a

1

1 1

(3.21)

The relative brake temperature after the nth application will

therefore become

vctAh

vctAhn

b eTeTtT

a

1

1 (3.22)

The limit values of the temperature before and after braking for a

large number of cycles an may be obtained from Equations (3.21) and (3.22) by dropping the term involving the factor na.

0255075

100125150175

0 250 500 750 1000 1250 1500Time (s)

Tem

pera

ture

(oC

)

Cast ironMMC

Figure 3.5 Temperature rise in Brake Drums during repeated braking

The temperature rise in the brake drum during braking from a speed

of 50 km/h for a braking time of 2.12 seconds and a cooling time of 180

seconds is shown in Figure 3.5.

40

3.4.3 Temperature Rise During Continues Braking

When the brakes are applied during a long downhill descend, the

cooling effect while braking must be considered. Similar to the lumped

temperature formulation, the temperature response of a brake drum during

continued braking is computed by (Limpert 1999).

AhqTe

hAqTTtT ovctAhoi

(3.23)

Substituting the properties of cast iron and Al MMC in Equation

(3.23) the average brake drum temperature rise is computed for the continued

braking condition. Figure 3.6 shows the comparison of temperature rise in

cast iron and Al MMC brake drum during continuous braking.

050

100150200250300

0 2 4 6 8 10Time (s)

Tem

pera

ture

( o C

)

Cast iron MMC

Figure 3.6 Temperature rise in brake drums during continuous braking

41

3.5 EFFECT OF TEMPERATURE RISE IN BRAKE DRUM

For the lining material, high temperature results in reduced

coefficient of friction, increased wear rate and burning (Eyre 1992). The high

braking forces at elevated temperatures results in bell mouth effect on the

brake drum resulting in a reduced braking effect (Rohatgi 1992). In addition,

because of the unequal coefficient of thermal expansion of lining and drum

material, the drum/shoe contact area is reduced resulting in reduced braking

force. Thermal shock is induced in the drum because of rapid and frequent

application of the brake (Kwok 1999). High temperature also increases the

rate of wear of drum resulting in reduced drum life. As the maximum

operating temperature of aluminium MMC brake drum is less, a reduction of

the temperature rise is essential to retain the braking effect at high

deceleration levels and also the to reduce wear of drum and lining materials.

3.6 HEAT GENERATED INSIDE THE BRAKE DRUM

The kinetic energy of a moving vehicle is converted into thermal

energy during braking through friction in the brakes. The average brake

power is given by

2

1kmaVPbavg

The maximum brake power Pmax produced at the onset of braking is

1max kmaVP

Assuming 50% of brake power for rear wheels, the maximum brake

power per hour for one rear brake drum excluding tire slip is given by

42

)3600)(5.0)(5.0(1max" kmaVP

= 900 kmaV1 (3.24)

The above power is converted into heat at the drum surface. The

maximum heat flux into flowing in to the brake drum is given by

LR

kmaVq1

1max 2

900"

(3.25)

3.6.1 Relative Brake Power Absorbed by the Brake Drum

The analysis of brake drum temperature rise requires an accurate

determination of both the total power generated in the brake and also how this

power is distributed to the drum and the shoe. The thermal power distribution

is directly related to thermal resistance associated with both sides of interface

where the heat is generated. The flow of heat into the brake drum and the shoe

are shown in Figure 3.7.

Figure 3.7 Heat flow in brake drum and shoe

43

The heat transfer into the drum and the shoe is determined from the

equivalent resistant network.

d

S

S

d

RR

qq

(3.26)

The total heat generation is equal to the heat absorbed by the drum

and the shoe.

2/1

s

d

s

d

s

d

S

d

kk

cc

qq

(3.27)

The relative power absorbed by the drum is given by (Limpert

1999)

2/11

1

ddd

sss

kckc

(3.28)

3.6.2 Temperature Rise in Brake Drum

With a high deceleration level, resulting high heat generation in a

single stop, the braking time may be less than the time required for the heat to

penetrate through the drum and shoe which will lead to temperature rise at the

interface. The maximum surface temperature rise Tmax in a single stop

without ambient cooling may be expressed as (Limpert 1999).

2/12/1

max2/1

max"

185

ddd

s

kctqT

(3.29)

44

3.7 FORMULATION OF OBJECTIVE FUNCTION

The objective of this optimization is to minimize the temperature

rise inside the drum surface without sacrificing the strength and rigidity of the

brake drum. The objective function is arrived by substituting the value of

heat flux from Equation (3.25) in the Equation (3.29). The objective function

is expressed as

211

2/115.75)(

ddd

s

kcLRtkmaVTF

(3.30)

3.7.1 Design Variables

The objective function usually has many parameters, out of those,

some are highly sensitive. These are called design parameters and for this

case, inner radius and width of the brake drum are considered as design

parameters. The limiting values for the inside radius of the drum is based on

the brake torque, space limitation and the minimum thickness required for the

drum to satisfy the required strength characteristics. The length of the brake

shoe limits the width of the shoe. The lower and upper limiting values of the

variables are

R1 min = 0.085m R1 max = 0.097m

Lmin = 0.030m Lmax = 0.047m

3.7.2 Design Parameters

Design parameters are constant in relation to design variables.

Here, the design parameters are k, m, V1, a, ts, d , Cd , kd, and E. The mass,

initial velocity and the deceleration are selected for a typical passenger as

45

shown in the Table1. The density, the specific heat and thermal conductivity

for cast iron are selected from reference (Limpert 1999). The density, specific

heat and the thermal conductivity of MMC are calculated from the available

data. The density, the specific heat and the thermal conductivity of the brake

shoe are selected from reference (Limpert 1999). From the equation given in

reference (Limpert 1999) for temperature rise, it is observed that the

temperature rise depends on the amount of heat flux into drum. The input

parameters for cast iron and MMC brake drum are shown in Table 3.3.

3.7.3 Design constraints

The functional relationships between the design variables and other

design parameters are known as design constraints. They should be calculated

to fall within the allowable limits. In this optimisation, the constraints are the

stress and the deformation. The braking force is applied on either side of the

drum through the brake shoes. This induces the stress in the drum and it

should be less than the half of the yield stress of the material. The strain in the

drum should also be limited since more strain will reduce the braking effect.

The stress induced in the drum because of braking (Sb) is given by

1122 LRRF

S Nb (3.31)

The circumferential strain in the brake drum during braking is

given by

)(1 rcc E (3.32)

46

Table 3.3 Input parameters

Input parameters Cast iron

brake drum Al MMC

brake drum

Laden mass of Vehicle (kg)

Initial velocity (km/h)

Stopping distance (m)

Correction factor (k)

Relative brake power to drum ()

Modulus of Elasticity (GPa)

Allowable stress (MPa)

Density of brake drum (kg/m3)

Density of shoe lining (kg/m3)

Specific heat of brake drum (Nm/kg K)

Specific heat of shoe lining (Nm/kg K)

Thermal Conductivity (Nm/hmK)

Thermal Conductivity of shoe lining (Nm/hmK)

1000

90

40

1.4

0.875

100

276

7228

4174

419

2034

174465

4174

1000

90

40

1.4

0.907

112

300

2828

4174

970

2034

374400

4174

3.8 GENETIC ALGORITHM

Genetic Algorithm (GA) mimics the principles of natural genetics

and natural selection that can be used to obtain near global and robust

solutions for optimization problems. This optimization technique uses a

simple genetic algorithm consisting of reproduction, cross-over and mutation

operators. Computer program is used for random number generation, string

copying and bit conversion. The objective function is used for selection,

cross-over and mutation operations.

47

3.8.1 Fitness Function

GAs are suitable to solve maximization problems. The

minimization problems are transformed into maximization problems by

suitable transformations. The fitness function is derived from the objective

function and is used in successive genetic operations. The minimization

problem is equivalent to the maximization problem such that the optimum

point remains unchanged. The fitness function used is given as

)(1

1)(xf

xF

(3.33)

The present problem is a constrained optimization. Since GAs are

ideal for the unconstrained optimization, it is necessary to transform it into an

unconstrained problem. Penality function method has been proposed to handle

the constraints (Deb 1991). A formulation based on violation of normalized

constraints is proposed in this work.

3.8.2 Violation Parameter

The design variables are checked for constraint violation. If the

design variables violate the constraints, then a higher value will be assigned

for the violation parameter. If not, a lower value will be assigned. In this

process, the design variables which violate the constraints will give very high

objective function value. When evaluated, this results in a very low fitness

function which reduces the probability of selection of this particular set in the

next generation and so on.

)(1

1)(xobj

xF

(3.34)

48

3.8.3 Total String Length

The operating parameters for the GA are established based on the

convergence of the problem as well as solution time. Based on the accuracy of

the solution desired and the data available the lengths of strings are

determined. If a four bit binary string is used to code a variable, then the

string (0000) is decoded to the lower limit of variables. The upper limit is

(1111) and any other string to a value in the range between the lower and

upper limits. There can be only 24 (16) different strings possible, because

each bit position can take a value of 0 or 1. Using a four bit binary substring

16 available sections can be represented. In this work, the total string length is

taken as 20 to make more number of sections available for the problem. Each

substring has 10 bits. In the present work, the substring lengths of all

variables are assumed as equal.

3.8.4 Maximum Generation

The termination process of a loop was carried out by fixing the

maximum number of generations. In this work, the maximum number of

generation is fixed as 300 after making number of trial runs.

3.8.5 Crossover and Mutation Probability

Higher value is preferred for the crossover probability whereas

lower value is ideal for the mutation probability. The crossover probability is

selected as 0.9 and the mutation probability as 0.001.

49

3.8.6 String Length

The string length of the two design variables is assigned as 10. The

strings representing individuals in the initial population are generated

randomly, and the binary strings are decoded for further evaluation.

Depending on the evaluation results of the first generation and the GA

parameters, population for the next generation is created. Generation of

population for the subsequent generation depends on the selection operator as

well as on the crossover and mutation probability. The algorithm repeats the

same process by generating new population, and evaluating its fitness as well

as constraint violation.

3.8.7 Computer Program

A tailor made computer program using C language has been

developed to operate the GA, and to obtain the best possible solution. The

approach consists of minimization of temperature rise in the brake drum

surface without sacrificing the strength and rigidity.

3.9 RESULTS AND DISCUSSIONS

The rapid and frequent application of brakes increases the heat flux

into the drum which leads to temperature rise in brake drums. The variation of

temperature rise with drum inner radius and the width for a cast iron brake

drum is shown in Figure 3.8. The temperature rise reduces with increase of

drum/shoe contact area. The increase in inner radius leads to increase in stress

and deformation as indicated in Equations (3.31) and (3.32). Increase in shoe

width is also limited to the effective brake drum width. The temperature rise

is maximum (666C) for 0.085mm inner radius and 0.03mm shoe width. The

temperature rise is minimum (372C) for 0.097mm inner radius and 0.047mm

50

width. The variation of temperature rise with drum inner radius and the

width for MMC brake drum is shown in Figure 3.9. The maximum and the

minimum temperature rise are 509C and 285C respectively. The

nontraditional optimization technique Genetic Algorithm is used to optimize

the dimensions of the brake drum and the shoe width to reduce the rise in the

brake drum.

Width

250

350

450

550

650

750

0.083 0.088 0.093 0.098Inner radius (m)

Tem

pera

ture

rise

(oC

) 0.03 m

0.036 m

0.042 m

0.047 m

Figure 3.8 Variation of temperature in cast iron brake drum

Width

250

350

450

550

650

0.083 0.088 0.093 0.098

Inner radius (m)

Tem

pera

ture

rise

( oC

) 0.03m0.036m0.042m0.047m

Figure 3.9. Variation of temperature in MMC brake drum

51

The temperature rise is optimized for cast iron and MMC brake drum of a

passenger car by having the same design variables. The input parameters are

given in the Table 3.3. The outer radius and drum width are also kept same

for both the drums. The input parameters for the GA are given in Table 3.4.

The variation of inner radius with number of generations for cast iron brake

drum is shown in Figure 3.10. More fluctuations are observed up to 200

generations then it reaches the optimum value with smaller fluctuations. The

variation of the width with the number of generations for cast iron drum is

shown in Figure 3.11. More fluctuations are observed up to 250 generations

and it reaches the optimum value after 260 generations. The temperature rise

with number of generations is shown in Figure 3.12. The temperature rise is

reduced with number of generations and it reaches its optimum value after

225 generations. The variation of inner radius with number of generations for

MMC brake drum is shown in Figure 3.13. More fluctuations are observed

up to 35 generations then it reaches the optimum value with smaller

fluctuations.

Table 3.4 GA parameters

GA Parameters Cast iron brake drum MMC brake

drum Number of parameters Total string length Population size Maximum generations Crossover probability Mutation probability Minimum limit for inner radius (m) Maximum for inner radius (m) Minimum limit for width (m) Maximum limit for width (m)

2 20 20

300 0.9

0.001 0.085 0.097 0.03

0.047

2 20 20

300 0.9

0.001 0.085 0.097 0.03 0.047

52

86

88

90

92

94

96

98

0 50 100 150 200 250 300

Generations

Inne

r Rad

ius

(mm

)

Figure 3.10 Variation of inner radius for cast iron brake drum

25

30

35

40

45

50

55

0 50 100 150 200 250 300

Generations

Sho

e W

idth

(mm

)

Figure 3.11 Variation of width for cast iron brake drum

53

0

50

100

150

200

0 50 100 150 200 250 300Generations

Tem

pera

ture

rise

(oC

)

Figure 3.12 Variation of temperature rise in cast iron drum

80

85

90

95

100

0 50 100 150 200 250 300

Generations

Inne

r Rad

ius

(mm

)

Figure 3.13 Variation of inner radius for MMC brake drum

The variation of the width with the number of generations for

MMC brake drum is shown in Figure 3.14. More fluctuations are observed up

to 30 generations and it reaches the optimum value after 125 generations. The

temperature rise with number of generations is shown in Figure 3.15. The

54

temperature rise is reduced with number of generations and it reaches its

optimum value after 105 generations. The results of the optimum values for

the cast iron and MMC brake drum are shown in the table 3.5. The stress and

the deformation are within the allowable limit. The temperature rise in cast

iron and MMC brake drum before and after optimisation are shown in Figure

3.16. The optimized values of temperature rise for cast iron and MMC brake

drum are 101.97 C and 21.3 C respectively.

25

30

35

40

45

50

55

0 50 100 150 200 250 300

Generations

Sho

e W

idth

(mm

)

Figure 3.14 Variation of width for MMC brake drum

0

10

20

30

40

50

0 50 100 150 200 250 300

Generations

Tem

pera

ture

rise

(oC

)

Figure 3.15 Variation of temperature rise in MMC brake drum

55

Table 3.5 Optimized values

S.No. Parameters Cast iron brake drum MMC brake

drum 1 2 3 4 5 6 7

Inner radius (mm) Width (mm) Maximum Stress (MPa) Maximum Strain Temperature rise ( C) Mass (kg) Factor of Safety

96.5 46.9

67.16 0.00034 101.97

3.2 4.1

93.99 47

38.77 0.00017

21.3 1.38 4.5

The temperature rise in cast iron brake drum is reduced by 254.6 C

by this optimization. For the MMC brake drum, a reduction of temperature

rise is observed as 187.2 C. The mass of cast iron and MMC brake drum

before and after optimization are shown in Figure 3.17. By this optimization,

the mass of the cast iron and MMC brake drum are also reduced by 25% and

18% respectively.

356.6

208.5

101.97

21.3

0 100 200 300 400

CI

MMC

CI

MMC

Temperature rise ( T) ( C)

After Optimization

BeforeOptimization

Figure 3.16 Temperature rise in brake drums before and after

optimization

56

4.3

1.68

3.2

1.38

0 1 2 3 4 5

CI

MMC

CI

MMC

Mass (kg)

After Optimization

BeforeOptimization

Figure 3.17 Mass of brake drums before and after optimization

3.10 CONCLUSIONS

a. The inner radius and the width of cast iron and MMC brake

drums are optimized using Genetic Algorithm, a

nontraditional optimization technique.

b. The temperature rise in cast iron brake drum before

optimization is observed as 356.6 C. After optimization, the

temperature rise is reduced to 101.97 C. So, a net reduction

in temperature rise of 255 C is achieved.

c. The temperature rise in MMC brake drum before

optimization is observed as 208.5C. After optimization, the

temperature rise is reduced to 21.3C. So, a net reduction in

temperature rise of 187.5 C is achieved.

57

d. The mass of the cast iron brake drum before optimization is

4.3 kg. After optimization, the mass of the drum is 3.2 kg.

So, a weight reduction of 24% is achieved for the cast iron

brake drum.

e. The mass of the MMC brake drum before optimization is

1.68 kg. After optimization, the mass of the brake drum is

1.38 kg. So, a weight reduction of 20% is achieved for the

MMC brake drum.

f. If the conventional cast iron brake drum brake drum is

replaced by the MMC brake drum the temperature rise is

reduced to 208.5C. If Genetic algorithm is applied, the

temperature rise is still reduced to 21.3C.

g. It is observed that the GA leads to larger reduction in

temperature rise due to its search for global optimum as

against the local optimum in traditional search methods.

Since these results are encouraging, the GA can be

effectively used for other complex and realistic designs often

encountered in engineering applications.