chapter 3 design and optimization of brake...
TRANSCRIPT
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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
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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
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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,
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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.
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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.
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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)
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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)
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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)
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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)
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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
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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)
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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
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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
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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.
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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
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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
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)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
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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)
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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
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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)
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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.
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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)
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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.
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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
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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
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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
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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
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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
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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
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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
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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.
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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.