maths in cars - Project Jugaad

Report
Objective of this project
Whether you realize it or not, maths is a fundamental function
of life and we use it on a daily basis. We use math for everything
from balancing our check book, to computing fuel mileage, to
purchasing aftermarket goodies, to counting the number of cars
ahead of you at the stoplight. Even more impressive is that it's
the only universal language in the world. So this project will give
us an insight about the coolest things in which maths is involved.
Maths plays a pivotal role in automobiles. Everything about
a car is based on mathematics. From degree of design to
wind resistance, performance, engine size and adjustment,
power outputs , Bolt holes , part sizes , and even the Pounds
of air in the tires. Everything is measured and torqued to a
specific mathematical formula. From drawing board, to
assembly line, it all depends on maths.
In cars , mathematics is involved in1. Time and distance
2. Geometry
3. Statistics
4. Angles
5. Ratio and proportion
6. Mensuration
1.Drafting
Artists and car designers who draw models of cars for
production need to understand perspective to make
their drawings and blueprints look right. This includes
a knowledge of angles and line lengths, as well as
different geometric shapes. Car wheels, for instance,
are really circles, hood tops are arcs, and windows are
quadrilaterals.
2.Parts
Each part of an automobile has to fit together like a glove, or the
automobile won't work properly or be safe. Math is used to measure
every part and to make sure those parts are the right size to come
together as designed. This includes everything from the dimensions
of screws to the width of the frame.
3.Pricing
Auto manufacturers want to make a profit on the cars they sell, so
they have to keep track of the cost of every single part. Math is used
to calculate which parts manufacturer can deliver the best price on
needed parts and materials. It also is used to determine the final cost
of the vehicle. If parts cost x dollars and labour costs y, and the
company wants to make a profit percentage of z, for example, then
the company would use the following formula to determine the sell
price: total cost = (x+y)z+(x+y)
4.Production Ratio
Math is used to determine how many cars can be produced an hour,
day, week or month. If an auto manufacturer receives an order from
corporate to increase production by x cars a day, for example, the speed
of the assembly line has to be adjusted by a particular percentage to
accommodate the total number of cars needed. All of the robots of the
assembly line would need to have their speeds adjusted through their
programming or manually by the same percentage.
5.Assembly
Much of the assembly of cars now is done with the help of robots and
other technology. The robots are controlled by specialized computer
programs, and these programs must specify exact parameters for
operation. They must tell the robot, for instance, to hoist a part x number
of feet, apply x pounds of pressure, and distribute x gallons of paint per
square inch. Additionally, the assembly line must be built under exact
dimensions, so there is ample room for assembly to occur safely and
efficiently---if a robot arm needs to swing back and forth, for example,
the robot needs to be positioned to have a certain number of feet in
clearance.
6.Production Time
Certain aspects of auto manufacturing must occur under specific time
parameters. Paint, for instance, has to cure for a specific amount of
time. Math is used to determine how long that time needs to be under a
specific temperature given the chemical composition of the paint.
7.Horsepower & Torque
Maths is used to calculate the horsepower and torque of a car . If
we know either the horsepower or torque figures at a given rpm,
it's easy to calculate the missing figure by simply plugging in the
numbers.
For exampleTorque x RPM / 5,252 = Horsepower
415 x 4,000 / 5,252 = 316
Horsepower x 5,252 / RPM = Torque
316 x 5,252 / 4,000 = 415
8.Selecting a Carburetor
Maths is also involved for selecting a carburetor. There is a simple
formula that makes the selection process easier. We need to take the
maximum rpm and multiply it by the engine displacement. Next divide
that number by 3,456 and multiply it by 0.85. For example, if you plug
in a maximum 6,000 rpm and a 350ci displacement, you end up with
516 cfm.
Street 350 ci
6,000 rpm x 350 / 3,456 x 0.85 = 516 cfm
9.Measuring Displacement
To calculate cubic-inch displacement, we need to know the bore and
stroke of the engine, the number of cylinders, and the handy constant of
0.7854, which is a shortcut representing a portion of the volume
equation of a cylinder-3.1417 (pi) divided by 4.
Displacement = bore x bore x stroke x 0.7854 x number of cylinders.
If the bores are 4,stroke is 3.48 and number of cylinders is 8,thenDisplacement = 4.00 x 4.00 x 3.48 x 0.7854 x 8 = 349.84 ci
10.Tire Diameter and Gear Ratio
Big tires may be cool, but swapping taller or shorter tires affect the final
drive ratio. Taller tires effectively change the rear gear ratio, making it
"taller" or numerically less. Shorter tires create the opposite effect.
Imagine you're building a Pro Street show car and it is already set up with
3.08 gears based on a typical 26-inch-tall tire. Obviously, the car will look
killer with a set of monster 33-inch-tall Mickey Thompson tires, but this
swap to the much taller tires instantly transforms the final drive ratio from
3.08 to 2.42!
Effective Gear Ratio = (original tire diameter / new tire diameter) x gear
ratio= 26/33 x 3.08= 2.42
11. Different tyres in different cars
Maths is used for making tyres of different shapes. A cars tyres are
designed to grip the road surface while supporting the car’s weight and
also to stabilize the ride and help steering. Different driving conditions
and different vehicles call for a wide range of tyre designs and all this is
done through Maths.
1. Bias Tyres- These tyres are used in normal cars and are of normal
shape. They give the tyres a smooth ride and are treaded. Maths
is used to give the tyres their shape and is also used to estimate
how much treading should be done.
2. Bus and lorry tyres- Maths is used to make these tyres thinner
than normal tyres so that they can survive long distance driving.
BIAS TYRES
BUS TYRES
3. Dump trucks- In these cars, the tyres are large and wide.
4.Tractor tyres- Maths is used to create and make an estimation of
their oversized treads. It also gives them a different shape and makes
them thinner than normal tyres.
5.Special Tyres for driving on snow- Maths is used to give large and
boxy treads to car’s tyres. Also the tyres are made up of special
rubber that stay flexible in the cold . For this , automobile engineers
need to use maths to calculate how much rubber in a proper ratio
should be added to cars.This is the same for treads.
Tractor tyres
Tyres for driving on snow
1.Odometer- It is an instrument (usually on the instrument panel of a
car) that records distance travelled by the car. Thus maths is involved in
the functioning of the odometer.
2.Speedometer- A speedometer or a speed meter is a gauge that
measures and displays the instantaneous speed of a land vehicle .
Without maths we would not get an idea of the distance travelled or
speed of the car. Thus maths plays a vital role here too.
3.Radios -Signal processing is one of the most important mathematical
fields that supports the design and effective operation of radio in cars.
Speedometer and Odometer
Radio in cars
A Formula 1 car - named for the special formula fuel that it burns has a
much more powerful engine than a passage car. The increased power
comes from the engine’s greater capacity- that is the total volume of
the combustion chambers in its cylinders.In a passage car , engine
capacity may be 1000 cubic centimeters or else. F1 cars have 3 times
that capacity and develop 500 horsepower , which is 4 or 5 times the
horsepower of an ordinary car.
To make the additional horsepower of F1 cars more effective , the car’s
body is aerodynamically designed(use of shapes) to minimize air
resistance. Racing cars need to be extra wide for secure road contact and
traction. It also needs to be given a special racing suspension which adds
stability and helps the car to grip the road firmly.
1.Body of F1 cars is moulded for excess speed (use of shapes and maths)
The low , wide body of a racing car, made of lightweight but strong carbon
fiber is designed to make use of the airflow the car creates at high speeds. The
sloped front end and rear spoilers make the air press down on the car and
keep it from becoming airborne.
2.Use of maths in the cockpit
Maths is also used in the cockpit of F1 cars. It is 850 mm long, 350 mm wide at
the pendals,450mm wide at the steering wheel and 520mm at the rear half .
Maths is also used to calculate and show the fuel level, water temperature , oil
pressure and other information which appears at the gauges in the cockpit.
Bernoulli principle
The Bernoulli principle has a big role in the operation of the
aerodynamic surfaces of an F1 car. The Bernoulli principle is expressed
by an equation, which states that for a given volume of fluid, the total
energy remains constant. This means that when a fluid is in relative
motion, the energy is split into the ‘parts’. The sum of these parts will
not exceed a certain value, which will remain constant as long as the
external conditions do not change..
The three parts of the total energy are:
1) The pressure energy within the fluid.
2) The movement of the air (kinetic energy)
3) The potential energy of the air (in this case, elevation)
This can be written as:
p + 1/2 r v2+ rgh = some constant
p = Pressure
r = Density of fluid
v = Velocity of fluid
g = Acceleration due to Gravity
h = Height of fluid above some reference point
Our average track is fairly level, so a race car will not have enough change in
elevation to make the potential energy a variable, so we take this potential
energy as a ‘constant ’and therefore are able to remove it from the equation.
This leaves us with:
p + 1/2 r v2 = some (other) constant
We can rewrite this as:
p+q=H
p = static Pressure
q = 1/2 rv2 = dynamic pressure
H = some (other) constant
This basically means that if the dynamic pressure increases, the static pressure
has to decrease and if the dynamic pressure decreases, the static pressure will
increase. This means that if we speed up a fluid, the pressure will fall.
We would like to acknowledge many people without whom this
project would not me possible1.Mr.Richard Davies and Mrs.Kamalika Bose for coming out with the
idea of ‘Jugaad’ and providing us with a platform for showcasing
our talents.
2.Mrs.Priya Madan, our maths teacher for constantly helping us.
3.Kieron Williams and Katie Vince , our team mates for constantly
communicating with us.
4. My pal-Sanjay Banerjee for doing half of the research.
5.Last,but not the least we would like to thank ourselves for finding
time to make this project against all odds.
Conclusion
• Cars play an important role in our life.
• We were able to understand the importance
of maths in cars.
• Maths is an universal language.
• We were able to know about cars through
maths .
• Lastly its all because of project jugaad we
were able to know so much.
A presentation by – Raunak Das and Dynamic Researchers
Team members-Raunak Das(digital engineer , also
researcher),Sanjay Banerjee(chief researcher),Katie
Vince(Group Leader), Kieron Williams(communication
director.)

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