Why is math important?
Math is important because it is able to explain many things that we observe but are unable to quantify.
While engineering may be simplified down to being little more than applied Math, there are some forms of math that may not necessarily make the full implementation.
"A function which gives the slope of a curve; that is, the slope of the line tangent to a function. "
"It is well known that the first derivative of position (symbol x) with respect to time is velocity (symbol v) and the second is acceleration (symbol a). It is a little less well known that the third derivative, i.e. the rate of change of acceleration, is technically known as jerk or jolt (symbol j). Jerk is a vector but may also be used loosely as a scalar quantity because there is not a separate term for the magnitude of jerk analogous to speed for magnitude of velocity. "
"An integral is a mathematical object that can be interpreted as an area or a generalization of area. Integrals, together with derivatives, are the fundamental objects of calculus. Other words for integral include antiderivative and primitive. The Riemann integral is the simplest integral definition and the only one usually encountered in physics and elementary calculus. "
Those are the simplest definitions that I could find across the internet. Taking the derivatives of Position gives us Velocity, Acceleration, and Jerk/Jolt. However, I do not necessarily have position at my disposal from the onboard data recorder - I have velocities and acceleration. This is where Integrals come into play: being able to convert acceleration to velocity and velocity to position.
The 'magic', per se, of integrals is that they evaluate the area below a curve. While that may not mean much, the area below a curve is also the definite value consumed or quantified by that curve.
Integrals work off the same theory as the Reimann Sum: If you slice the area below a curve into infinitely small rectangles, you can add up the area of those rectangles.
In my case, the data that the onboard data recorder saves is recorded in 0.01 second intervals - or 100hz. As such, that is the width of the smallest rectangles that I have to work with, resulting in an area of .01 (width) times the Y value provided by the recorder for that timestamp. However, it is a tedious act to sum over 400 data points for a 1/8 mile run or over 600 points for a 1/4 mile run. It is far easier to retrieve a best-fit trendline of the graph, and integrate it. Due to the shape of the curve, integration is not quite as accurate, however is is within 3%, which is a fairly tight tolerance for so many data points.
See the examples below:
Given the input of Engine Rotational Velocity, expressed in Rotations per Minute, I can apply two operations:
- Derivative: To find the rates of acceleration of the engine during the 4.10 second run (The red Graph)
- Integral: To find the number of times the engine rotated in those 4.10 seconds (Shown in text using the best-fit trend line in light blue)
Using the Reimann Sum on the raw data, a total revolution count of 481.46 is resulted. Meanwhile, the best-fit line's integral evaluates to 478.575 revolutions, less than 1% difference. This is sufficient for most needs.
Observe that there are two spots where the engine's rotational acceleration (the red line) is nearly zero, meaning the velocity is constant (as shown in the blue line). Each of the two spots have different causes.
I have commanded the engine's control computer to take power away from 0.5 to 0.75 seconds to keep the tire stuck to the track by traction, and not overpower the available traction.
When the transmission is in low gear, the hydraulic coupling slips to allow the engine to gain speed. Once in high gear, this hydraulic coupling begins to stop slipping, and the engine RPM becomes much more controlled. This process of progressive lockup begins around 2.5 seconds until the end of the run when the engine is shut down.
Given the input of Driveshaft Rotational Velocity, expressed in Rotations per Minute, I can apply two operations:
- Derivative: To find the rates of acceleration of the driveshaft during the 4.10 second run (The red Graph)
- Integral: To find the number of times the driveshaft rotated in those 4.10 seconds (Shown in text using the best-fit trend line in light blue)
Using the Reimann Sum on the raw data, a total revolution count of 256.93 is resulted. Meanwhile, the best-fit line's integral evaluates to 251.28 revolutions, a difference of 2.25%. Due to the nature of what I may use this calculation for, while the integral is good for a best-guess or for general reference, the Reimann sum is better for my use.
Given the input of Fuel Flow, expressed in Gallons per Minute (a velocity), I can apply one useful operation:
- Integral: To find the total amount of fuel that went into the engine in those 4.10 seconds (Shown in text using the best-fit trend line in light blue)
Note that for this graph, I have also graphed the Reimann sum of the data points in red.
Using the Reimann Sum on the raw data, a total fuel consumption of 63.05 ounces is resulted. Meanwhile, the best-fit line's integral evaluates to 63.264 ounces, less than 1/3 of a percent difference. This is sufficient for even my biggest needs of calculation.