Engineering
How are Engineering and Math used in my environment?
The better question is, "How are they not?"
Applied Math = Physics, Applied Physics = Engineering.
I will admit that even with my upbringing, I was not a big fan of Mathematics in primary or secondary school, or even the early stages of university. A large part of this, as I found out later, was due to learning disabilities and imperfect instructors.
When I started applying mathematics to motorsports, a number of different theories finally clicked in my mind. As much as I had loathed Algebra 2, Calculus 1, and Calculus 2, it finally all made sense.
Since I had access to the raw data from the onboard data recorder, I started applying mathematical processing to it. I was actually quite surprised at how much it simplifies the learning of the formulas when applied to an otherwise pre-understood data set.
Using computer programming skills, I was able to apply the Riemann Sum formula to select data points, providing new information that was not previously available.
Where the Math meets the Road: Part 1 - Friction and Torque (Suspension in a Nutshell)
An area where math becomes highly involved is that of suspension setup. Taking an object at rest, and accelerating it to 2 G's within half of a second is no small feat, more-so when said object is 2900 pounds (1900 pounds is slightly more manageable). It requires a significant amount of traction - also known as friction.
The events that take place during the first second of a pass dictates everything that happens for the next few seconds afterward, with a snowball effect. It is not quite a binary matter- "Either it works or it doesn't" does not apply here. It is more of a range between "Not Working....Working acceptably.....Working ideally".
This video is from a cold day on February 19, 2012. The track was cold, and we had a very experimental shock absorber package. A longer than usual Burnout was used to infuse some extra heat into the track and tires.
The way we had the shocks tuned for this pass, the extension was very loose, allowing the full launching force to be applied to the tires. However, the compression was set very firm.
Theory: The tires would be held pressed down into the track, maximizing traction
Actual: The track was cold enough that the tires did not gain initial traction, rather, the force of the axle pushing down turned the tires into basketballs, bouncing them (and the rest of the car) back up. Once this occurred, traction was further broken, causing a moment in which we call Smoking the Tires. Jump to 3:30 in the video for the slow motion segment.
During the initial movement of the launch, the driveshaft begins to turn while the tires are sitting still on the ground. What then happens next, is that the pinion gear starts to "climb" up on the ring gear. This action, per Newton's Laws, creates two equal and opposite forces: The body rising up away from the axle, and the axle being pushed down away from the body and into the ground. At the same time, the tire is twisting from being stationary while the drivetrain is starting to rotate. Depending on the frictional force between the tire and ground, the tire may rotate with the ground, or it may slip and spin.
The high frame rate video shown here allows you to see how 2 different vehicles react at the launch in slow motion.
Notice for the brief second that as the drivetrain rotates, the tire twists and squats until the tension in the sidewall of the tire is high enough to break the frictional force between the tire and ground.
In this video, you can see the acts of physics taking place as about 2500 horsepower is unleashed upon the track surface.
You can also see excess unburned fuel being emitted from the exhaust just before launch takes place.
An axle assembly is NOT supposed to look like this! However, such a violent structure failure enables a full view of how the gears are designed. The pinion gear on the left meshes with the ring gear on the right. The ring gear is attached to the tires, and thus the side exposed here would normally be rotating downward. When the rotating pinion climbs the stationary ring gear, it causes the rest of the vehicle to lift away from the axle assembly. There soon becomes a point where structural fatigue takes place and wins....typically in a rather messy form. In addition to bits of metal scattered about, there is also 2 quarts of very thick oil within the assembly which will splatter and spill out.
Where the Math meets the Road: Part 2a - Aerodynamics
Aerodynamics are an active property that many people interact with on a daily basis. It is one of the determining factors of vehicle fuel mileage, is the reason that you can duck behind a corner to escape the cold wind, and how we are able to generate electricity (and other forces) from the wind.
The subject of aerodynamics brings with it the properties of drag and with the use of wings, can create downforce and lift.
If you have ever extended your hand out of the window of a moving car, you have experience drag. It is the force of the wind trying to push your hand towards the rear of the car. This drag force can be easily calculated using a set of observed and pre-determined factors. [Source: Nasa.gov]
Drag Force = (coefficient of Friction) x (density of air) x (Frontal Area) x [ ( Velocity ^2 ) / 2 ]
Drag Force = ( 0.22, an arbitrary value ) x ( 1.225 kg/m^3 ) x ( [1m x 1m] + [0.25m x 0.15m] ) x [ ( 62 m/s ^ 2 ) / 2 ]
Drag Force = ( 0.22 ) x (1.225 kg/m^3 ) x ( 1.0375 m^2 ) x ( 1922 m^2/s^2 )
Drag Force = 537 Newtons = 120 pounds at 139 MPH
More likely than not, you have probably seen a car or truck with a wing raised up 4-6" off of the rear decklid. Sometimes, these wings are practical, other times they are only for aesthetic or even marketing purposes.
With that said, some wings do become practical at a given speed. Depending on the design, they can become influential at speeds as low as 40 MPH (like the ones used on various 70's and 80's Volkswagen and Porsche-esque vehicles to counteract the design of the rear engine cover).
NASA has developed a great piece of software, FoilSim III, which allows the virtual manipulation of wind designs and reports estimations of the resulting airflow properties.
We use wings to generate downforce, which helps push the tires against the track and increase traction. Generating downforce comes with drag, and setting the angle of a wing is a compromise between forces and drag.
Increasing the angle of a wing also increases the frontal area, which increases drag. The shape of the wing dictates how much angle is required to generate a specific amount of downforce.
Using measurements made from the rear wing, I was able to provide the inputs requested by FoilSimIII. Keeping in mind that FSIII is at best a guesstimation, it still provides an idea of what real-world results are occurring. The "lift" calculations are specified as negative numbers because downforce is being produced.
Where the Math meets the Road: Part 2b - Aerodynamics
Another application of aerodynamics is through controlling the airflow, not with wings, but through ducting.
An inlet scoop is used to direct airflow into the engine. A ram-air effect is created in doing so, creating a slightly higher-than-ambient air pressure.
In the video to the left, the scoop is sealed off to the engine internally, so that air is unable to leak out. However, when the throttle is closed, the air has nowhere to go. As a result, the entire hood lifts up until the seal from the scoop to the engine is broken, allowing the air to leak out around the bottom.
Another use of aerodynamic control is for secondary deceleration. While race vehicles are equipped with hydraulic brakes just as road cars, many are also equipped with parachutes.
Parachutes deploy into the airstream to collect air, slowing the vehicle down until the airspeed is no longer sufficient to support the parachute's weight.
The use of a parachute enables the vehicle to decelerate from 230+ miles per hour to a complete stop in about 2500 feet, experiencing a peak of negative 2.5 G's. All vehicles that exceed 200 MPH are required to have a backup parachute to be used in the event that the primary parachute fails to deploy.
Different parachutes are made for different vehicle weights and speeds. This is because a smaller parachute for a light vehicle would not collect enough air to properly slow a heavy vehicle. Likewise, a large parachute for a heavy vehicle would deploy too aggressively for a lightweight vehicle. Additionally, at higher speeds, a large parachute can deploy with too much force for either case. A parachute that is too large is capable of lifting the back tires off the ground, as shown in this photo of another competitor (Lower Left). Deployment force is reduced on larger parachutes by implementing vents, which allows air to escape the parachute (Lower Right).
Where the Math meets the Road: Part 3 - Centripetal Forces
One of the forces in play which may initially go unnoticed is the centripetal force applied to the rear tires. From the typical spectator view, this force is not often recognized, however upon closer inspection it is very easy to see. Drag racing tires are specifically design to very easily balloon, such that their diameter grows while their width shrinks. Tire Growth is possible because the tread (13.5 to 16" wide depending on the tire) is very heavily weighted, and the entire construction is rather soft. At high speed, the centripetal forces "throw" the tread outward, and the soft sidewalls go from curved to straight to permit the extra growth.
Assuming the tread of the tire weighs 6.8 kilograms (15 pounds), is traveling at 101 meters per second (227 MPH), and has a radius of 0.45 meters (17.75 inches), then the following holds true:
Centripetal Force = ( Mass * Velocity^2 ) / Radius
Centripetal Force = ( 6.8 kg * [101 m/s]^2 ) / 0.40 m
Centripetal Force = ( 6.8 * 10,201 m^2/s^2 ) / 0.45 = 154,148 N = 34,653 Pounds at 227 MPH
In this photo taken during a burnout, the growth of the tire can be clearly seen. The tire grows from approximately 33 inches tall to more than 38 inches tall. During the burnout, the wheel speed approaches 200 MPH.
In this photo taken at about 0.25 seconds after launch, notice the torque of the engine lifting the left front wheel off the ground.
Notice that as the tire grows taller, it becomes narrower. The sides of the tire are clamped by a beadlock wheel. This permits the tire to go as narrow as 12 inches, despite being installed on a 16 inch wide wheel.
In this photo taken at about 1.25 seconds after launch, the tires have visibly began deforming and gaining height. Notice that the left front tire is still being lifted off the ground by the torque of the engine.
From this close-up view, the narrowing of the tire is quite easy to see. In fact, it appears as if the tire is trying to become skinnier than the wheel it is attached to. If you look closely, you will see 8 screws in the side of the wheel. These go into the innermost part of the tire, in attempt to keep it from coming loose. The tires used in this video have a contact patch 16 inches wide.
From this more distant (and partially obscured) view, the physical distortion of the tire can be observed. Depending on the speed, size, and construction of a given tire, it can grow between 2 and 8 inches. The tires in this video place at the bottom of that scale, as they are a very specific "no-growth" design, and have a contact patch only 13 inches wide.