Objective of this projectWhether you realize it or not, maths is a fundamental functionof life and we use it on a daily basis. We use math for everythingfrom balancing our check book, to computing fuel mileage, topurchasing aftermarket goodies, to counting the number of carsahead of you at the stoplight. Even more impressive is that itsthe only universal language in the world. So this project will giveus an insight about the coolest things in which maths is involved.
Maths plays a pivotal role in automobiles. Everything abouta car is based on mathematics. From degree of design towind resistance, performance, engine size andadjustment, power outputs , Bolt holes , part sizes , andeven the Pounds of air in the tires. Everything is measuredand torqued to a specific mathematical formula. Fromdrawing board, to assembly line, it all depends on maths.
In cars , mathematics is involved in- 1. Time and distance 2. Geometry 3. Statistics 4. Angles 5. Ratio and proportion 6. Mensuration1.DraftingArtists and car designers who draw models of cars forproduction need to understand perspective to maketheir drawings and blueprints look right. This includesa knowledge of angles and line lengths, as well asdifferent geometric shapes. Car wheels, forinstance, are really circles, hood tops are arcs, andwindows are quadrilaterals.
2.PartsEach part of an automobile has to fit together like a glove, or theautomobile wont work properly or be safe. Math is used to measureevery part and to make sure those parts are the right size to cometogether as designed. This includes everything from the dimensionsof screws to the width of the frame.
3.PricingAuto manufacturers want to make a profit on the cars they sell, sothey have to keep track of the cost of every single part. Math is usedto calculate which parts manufacturer can deliver the best price onneeded parts and materials. It also is used to determine the final costof the vehicle. If parts cost x dollars and labour costs y, and thecompany wants to make a profit percentage of z, for example, thenthe company would use the following formula to determine the sellprice: total cost = (x+y)z+(x+y)
4.Production RatioMath is used to determine how many cars can be produced anhour, day, week or month. If an auto manufacturer receives an orderfrom corporate to increase production by x cars a day, for example, thespeed of the assembly line has to be adjusted by a particular percentageto accommodate the total number of cars needed. All of the robots ofthe assembly line would need to have their speeds adjusted throughtheir programming or manually by the same percentage.
5.AssemblyMuch of the assembly of cars now is done with the help of robots andother technology. The robots are controlled by specialized computerprograms, and these programs must specify exact parameters foroperation. They must tell the robot, for instance, to hoist a part x numberof feet, apply x pounds of pressure, and distribute x gallons of paint persquare inch. Additionally, the assembly line must be built under exactdimensions, so there is ample room for assembly to occur safely andefficiently---if a robot arm needs to swing back and forth, forexample, the robot needs to be positioned to have a certain number offeet in clearance.
6.Production TimeCertain aspects of auto manufacturing must occur under specific timeparameters. Paint, for instance, has to cure for a specific amount oftime. Math is used to determine how long that time needs to be under aspecific temperature given the chemical composition of the paint.
7.Horsepower & TorqueMaths is used to calculate the horsepower and torque of a car . Ifwe know either the horsepower or torque figures at a givenrpm, its easy to calculate the missing figure by simply plugging inthe numbers.For example-Torque x RPM / 5,252 = Horsepower415 x 4,000 / 5,252 = 316Horsepower x 5,252 / RPM = Torque316 x 5,252 / 4,000 = 415
8.Selecting a CarburetorMaths is also involved for selecting a carburetor. There is a simpleformula that makes the selection process easier. We need to take themaximum rpm and multiply it by the engine displacement. Next dividethat number by 3,456 and multiply it by 0.85. For example, if you plugin a maximum 6,000 rpm and a 350ci displacement, you end up with516 cfm.Street 350 ci6,000 rpm x 350 / 3,456 x 0.85 = 516 cfm
9.Measuring DisplacementTo calculate cubic-inch displacement, we need to know the bore andstroke of the engine, the number of cylinders, and the handy constant of0.7854, which is a shortcut representing a portion of the volumeequation 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,then-Displacement = 4.00 x 4.00 x 3.48 x 0.7854 x 8 = 349.84 ci
10.Tire Diameter and Gear RatioBig tires may be cool, but swapping taller or shorter tires affect the finaldrive ratio. Taller tires effectively change the rear gear ratio, making it"taller" or numerically less. Shorter tires create the opposite effect.Imagine youre building a Pro Street show car and it is already set up with3.08 gears based on a typical 26-inch-tall tire. Obviously, the car will lookkiller with a set of monster 33-inch-tall Mickey Thompson tires, but thisswap to the much taller tires instantly transforms the final drive ratio from3.08 to 2.42!Effective Gear Ratio = (original tire diameter / new tire diameter) x gearratio= 26/33 x 3.08= 2.42
11. Different tyres in different carsMaths is used for making tyres of different shapes. A cars tyres aredesigned to grip the road surface while supporting the car’s weight andalso to stabilize the ride and help steering. Different driving conditionsand different vehicles call for a wide range of tyre designs and all this isdone 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 oftheir oversized treads. It also gives them a different shape and makesthem thinner than normal tyres.5.Special Tyres for driving on snow- Maths is used to give large andboxy treads to car’s tyres. Also the tyres are made up of specialrubber that stay flexible in the cold . For this , automobile engineersneed to use maths to calculate how much rubber in a proper ratioshould 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 acar) that records distance travelled by the car. Thus maths is involved inthe functioning of the odometer.2.Speedometer- A speedometer or a speed meter is a gauge thatmeasures and displays the instantaneous speed of a land vehicle .Without maths we would not get an idea of the distance travelled orspeed of the car. Thus maths plays a vital role here too.3.Radios -Signal processing is one of the most important mathematicalfields 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 amuch more powerful engine than a passage car. The increased powercomes from the engine’s greater capacity- that is the total volume ofthe combustion chambers in its cylinders.In a passage car , enginecapacity may be 1000 cubic centimeters or else. F1 cars have 3 timesthat capacity and develop 500 horsepower , which is 4 or 5 times thehorsepower of an ordinary car.
To make the additional horsepower of F1 cars more effective , the car’sbody is aerodynamically designed(use of shapes) to minimize airresistance. Racing cars need to be extra wide for secure road contact andtraction. It also needs to be given a special racing suspension which addsstability 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 carbonfiber is designed to make use of the airflow the car creates at high speeds. Thesloped front end and rear spoilers make the air press down on the car andkeep it from becoming airborne.2.Use of maths in the cockpitMaths is also used in the cockpit of F1 cars. It is 850 mm long, 350 mm wide atthe 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 , oilpressure and other information which appears at the gauges in the cockpit.
Bernoulli principleThe Bernoulli principle has a big role in the operation of theaerodynamic surfaces of an F1 car. The Bernoulli principle is expressedby an equation, which states that for a given volume of fluid, the totalenergy remains constant. This means that when a fluid is in relativemotion, the energy is split into the ‘parts’. The sum of these parts willnot exceed a certain value, which will remain constant as long as theexternal 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 constantp = 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 inelevation to make the potential energy a variable, so we take this potentialenergy as a ‘constant ’and therefore are able to remove it from the equation.This leaves us with:p + 1/2 r v2 = some (other) constantWe can rewrite this as:p+q=Hp = static Pressure q = 1/2 rv2 = dynamic pressure H = some (other) constantThis basically means that if the dynamic pressure increases, the static pressurehas to decrease and if the dynamic pressure decreases, the static pressure willincrease. This means that if we speed up a fluid, the pressure will fall.
We would like to acknowledge many people without whom thisproject would not me possible-1.Mr.Richard Davies and Mrs.Kamalika Bose for coming out with theidea of ‘Jugaad’ and providing us with a platform for showcasingour talents.2.Mrs.Priya Madan, our maths teacher for constantly helping us.3.Kieron Williams and Katie Vince , our team mates for constantlycommunicating 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 findingtime 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 ResearchersTeam members-Raunak Das(digital engineer , alsoresearcher),Sanjay Banerjee(chief researcher),KatieVince(Group Leader), Kieron Williams(communicationdirector.)