Calculus Aviation Project

J

Jay_E

Filing Flight Plan
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Gainesvile, FL
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Display name:
Jake
Hello Pilots, I’m not a pilot although I’d love to be one someday. I’m currently a college student studying engineering. I’m taking Calculus this semester and have been assigned to do a semester project about aviation. Specifically, I’ll be using calculus to model descent paths using polynomials and working with vertical decent rates and acceleration. The assignment prompt talks about three different mathematical conditions that determine approach path, namely cruising altitude, horizontal distance from touchdown and decent rate. The second part of the project asks me to contact a real pilot and find out from them what other factors influence descent from a pilot’s perspective. I was wondering if anyone here would be willing to share what you know about descent from your experience as a pilot. Thanks in advance.
 
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IDK about the math, the FAA determines most of it :)
 
Well, I usually think airspeed is purity darned important. I'll also pay a bit of 'tention to ground speed and position...
 
Many things determine descent path. I assume you mean a constant straight in approach from cruising altitude? As hinted above, ATC determines some of it, obstacle clearance is needed too can change the descent path.

Wind, power, flap position ( if equipped) play a role. I further assume fixed wing aircraft, rather than helicopters, powered aircraft?
 
Sounds like you'll be plagiarizing something already published decades ago in "Physics for Naval Aviators". :)

As far as a "real pilot" view of descents... Pull power back until desired descent rate that won't hit anything is achieved. Desired descent rate is simple time/speed/distance algebra. I want to land over there X distance away and I have Y altitude to lose. A right triangle and Geometry and Pythagorean Theorem covers it.

Air traffic control restrictions and airspace factor more heavily than the triangle.
 
Jay_E said:
The second part of the project asks me to contact a real pilot and find out from them what other factors influence descent from a pilot’s perspective. I was wondering if anyone here would be willing to share what you know about descent from your experience as a pilot. Thanks in advance.
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If in controlled airspace it's determined by ATC, unless you are cleared at your discretion. If flying GA in uncontrolled airspace, it's based on published pattern altitude of the airport you are landing at, factoring in your airspeed, groundspeed, wind direction, and distance from the airport. In the aircraft I fly under VFR conditions, it's usually no more than 500 FPM. This is for many factors, including airspeed, aircraft performance limits, and comfort. Other factors include terrain and other obstacles and traffic in your area.

IFR flights are more consistent as you are flying published approaches and patterns. Altitude, speed, and descent are usually controlled by ATC based on traffic at that airport at that time.

I think your questions pertain more to the physics of flight, which is a different topic in itself. Most of what you are asking otherwise is dictated by ATC and the FAA.
 
Most of us use the provided tool to gauge our descent.
Read about the PAPI or VASI (google or youtube those)
This is a set of lights on the runway that tell the pilot of he is on the correct angle to the runway.

Weather, obstacles, time of day, runway length are non physics factors that come in to play.

At night if I have a really long runway, I tend to come in a bit higher just to ensure if the engine fails, I can glide to the runway. During the day it is not as critical since I can see the ground and choose a field.

Sometimes poor planning is a factor. If for whatever reason, the pilot has too much speed or is too high, he/ she needs to bleed off the airspeed or descend. The pilot can put the plane into a slip and descend at a very steep angle without increasing the speed. Think of it as sort of a sideways dive. Look up "slip to landing"

Go to a local air field and take a discovery flight.
Film the landing and debrief w/ the pilot after.
 
Head/Tailwinds will change the decent angle.
Approach systems will change the decent angle (although normally ~3 degrees)
ATC can change the decent angle/approach path.
500fpm is a "normal" comfortable descent for small airplanes.
If I have a choice, I like to come in high enough that a power failure doesn't jeopardize my ability to glide in.
 
I think it all comes down to calculations involving pi.

Basically, how fast do you want to get on the ground and get to the airport restaurant before they run out of blueberry. Oh, that's pie. Never mind.
 
Also account for passengers.
If there are passengers (Especially if one is also a pilot), the angle will be miscalculated every time :)
 
SixPapaCharlie said:
Also account for passengers.
If there are passengers (Especially if one is also a pilot), the angle will be miscalculated every time :)
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And they start screaming and crying when you point the nose straight down and they're hanging from their seat belts in the descent.

Such whiners. We haven't hit Vne yet.

Be quiet. We will get to the blueberry pie faster. Sheesh.
 
SkyDog58 said:
I think it all comes down to calculations involving pi.

Basically, how fast do you want to get on the ground and get to the airport restaurant before they run out of blueberry. Oh, that's pie. Never mind.
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This is absolutely wrong. It all comes down to calculations involving pee.
 
Power back,drop the nose keep the vasi the right colors,land
 
You might find the "track log and graph" feature of flightaware.com useful. The attached picture shows 4 landings. They each have different profiles; some were dictated by environmental factors (i.e. a very steep and slow descent into a very short field, or a very shallow descent over a bay at night to minimize risk) others were dictated by Air Traffic Control and published instrument procedures (for use in bad weather). That website has thousands of these graphs. You could study one airport and correlate descent paths to weather patterns. I would bet that a detailed study of arrivals would show that individual approach controllers can be identified by their descent patterns.

One other note about the graphs, keep in mind that the speed shown is the ground speed, which is the vector sum of true airspeed and windspeed vectors, which of course change constantly.
DescentProfiles_zpseynsybdk.jpg
 
Jay_E said:
Hello Pilots, I’m not a pilot although I’d love to be one someday. I’m currently a college student studying engineering. I’m taking Calculus this semester and have been assigned to do a semester project about aviation. Specifically, I’ll be using calculus to model descent paths using polynomials and working with vertical decent rates and acceleration. The assignment prompt talks about three different mathematical conditions that determine approach path, namely cruising altitude, horizontal distance from touchdown and decent rate. The second part of the project asks me to contact a real pilot and find out from them what other factors influence descent from a pilot’s perspective. I was wondering if anyone here would be willing to share what you know about descent from your experience as a pilot. Thanks in advance.
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Change in pressure, the rate of and the corresponding effect it has on people's ears is one thing that may influence my maximum rate of descent. Sometimes when I have a really clogged up person, I can't manage more than about 200' per minute descent. There is also the increase in true airspeed and reduction in fuel consumption at altitude that influences me to stay high as long as possible and use a maximum structural speed, idle power descent to the runway. Then there is also instructions and limitations issued by ATC, especially when flying IFR, or under instrument flight rules, where ATC determines your operational parameters for you, and you comply with their instructions. I may also be restricted in my maximum and minimum altitudes and profile on approach by over and underlying air space restrictions.

Now sometimes it's so cold up high, I say screw it and fly down low taking the fuel and speed penalty down low to stay warm, rather than fire up my gas heater, plus the view is much better.
 
There is a physics problem of the maximum speed of a falling sphere from very high. The air resistance puts a limit on it. Figure that first. Its simpler. This one is not too hard. The air resisteance is e^-kt or something like that. Multiply this times the speed it falls in a vacuum. Figure a k that gives a top speed of about 100mph. Or so. Not sure about that hint, but this problem is not too hard.

With an airplane, "best glide" is the speed at which the airplane will glide the furthest. In small trainers its usually 70knots and the descent rate is about 500 feet per minute. See if you can use Bernoillis equation to calculate best glide.
Good luck. This one is harder.

Dont blame me if something in the above is wrong. This is all from memory from a long time ago.
 
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Pretty sure he is looking for physical influences on the flight path versus having to get to the bathroom or worrying about plugged ears.

As you might guess there are many outside influences on an aircraft during flight. Some have already been alluded to and I may be repeating them. Speed, wind speed, wind speed variations, updrafts, down drafts,cross winds, control inputs, aircraft reaction times, engine response, aircraft weight and probably many other things influence how the aircraft performs. Pilots use visual cues either outside or on instruments to compensate for the perturbations that normally occur while flying, reactions that might be difficult to model, but certainly possible depending on how involved in this you want to get.
 
PaulS said:
Pretty sure he is looking for physical influences on the flight path versus having to get to the bathroom or worrying about plugged ears.

As you might guess there are many outside influences on an aircraft during flight. Some have already been alluded to and I may be repeating them. Speed, wind speed, wind speed variations, updrafts, down drafts,cross winds, control inputs, aircraft reaction times, engine response, aircraft weight and probably many other things influence how the aircraft performs. Pilots use visual cues either outside or on instruments to compensate for the perturbations that normally occur while flying, reactions that might be difficult to model, but certainly possible depending on how involved in this you want to get.
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The second part of the project asks me to contact a real pilot and find out from them what other factors influence descent from a pilot’s perspective.
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You assume that limitation why?:dunno:
 
Dunno - he asked about vertical descent rates and acceleration. And since it's normally a constant rate (say 200 - 500 fpm) there's no acceleration. But, there is at the top and bottom. When you leave cruise altitude you accel from zero fpm descent and when you hit the bottom you decel back to zero fpm. It's an S-curve.
 
PaulS said:
Which limitation?
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That he doesn't want to consider human factors. There are no parameters to the paper except an analysis on descent. Plenty of ways you can use human factors to play into the analysis to produce something more interesting than the average.
 
Henning said:
That he doesn't want to consider human factors. There are no parameters to the paper except an analysis on descent. Plenty of ways you can use human factors to play into the analysis to produce something more interesting than the average.
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And those could be eliminated if a pressurized plane is assumed.
 
gasfiltered said:
You might find the "track log and graph" feature of flightaware.com useful. The attached picture shows 4 landings. They each have different profiles; some were dictated by environmental factors (i.e. a very steep and slow descent into a very short field, or a very shallow descent over a bay at night to minimize risk) others were dictated by Air Traffic Control and published instrument procedures (for use in bad weather). That website has thousands of these graphs. You could study one airport and correlate descent paths to weather patterns. I would bet that a detailed study of arrivals would show that individual approach controllers can be identified by their descent patterns.

One other note about the graphs, keep in mind that the speed shown is the ground speed, which is the vector sum of true airspeed and windspeed vectors, which of course change constantly.
DescentProfiles_zpseynsybdk.jpg
Click to expand...

This is probably the most useful advice the OP could get that I have seen so far. The OP doesn't care about throttle positions, or airspace, or FAR's, the OP wants to know what the vertical profile line should look like, as he has to replicate it with a polynomial function.
 
Sac Arrow said:
This is probably the most useful advice the OP could get that I have seen so far. The OP doesn't care about throttle positions, or airspace, or FAR's, the OP wants to know what the vertical profile line should look like, as he has to replicate it with a polynomial function.
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I think the instructor is looking for more than just that, I may be wrong, but I suspect due to the seeking of pilot information of other influences. The OP didn't say what he was trying to solve for. If it was just to solve for profile, what result from the profile is being sought?:dunno: "Sick person profile?" "Maximum efficiency profile?", "Plane on fire profile?" "General standard comfort profile?":dunno:
 
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Henning said:
I think the instructor is looking for more than just that, I may be wrong, but I suspect due to the seeking of pilot information of other influences. The OP didn't say what he was trying to solve for.
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Well, what the instructor is looking for is real life scenarios for the vertical track profile, for a good reason. It may throw a wrinkle in to the mathematical modeling.
 
Sac Arrow said:
Well, what the instructor is looking for is real life scenarios for the vertical track profile, for a good reason. It may throw a wrinkle in to the mathematical modeling.
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Read my edit, that's what I was getting at with the human factors variable. "Best descent profile" is dependent on the most critical factor for that flight.
 
Henning said:
That he doesn't want to consider human factors. There are no parameters to the paper except an analysis on descent. Plenty of ways you can use human factors to play into the analysis to produce something more interesting than the average.
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I'm not saying those aren't important, but those are very difficult to model mathematically and generally aren't an aircraft issue. Things like wind gusts and hysteresis of controls, for lack of a better term, and other things can be modeled if he wants to go there.
 
Henning said:
Read my edit, that's what I was getting at with the human factors variable. "Best descent profile" is dependent on the most critical factor for that flight.
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Fine, beer's on me.
 
Sac Arrow said:
Well, what the instructor is looking for is real life scenarios for the vertical track profile, for a good reason. It may throw a wrinkle in to the mathematical modeling.
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I read this:
Jay_E said:
The second part of the project asks me to contact a real pilot and find out from them what other factors influence descent from a pilot’s perspective. I was wondering if anyone here would be willing to share what you know about descent from your experience as a pilot. Thanks in advance.
Click to expand...

maybe too simply. Weather might make you choose a different descent rate, an engine fire, a passenger or pilot with a sinus or ear problem, ATC instructions, ...
 
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PaulS said:
I'm not saying those aren't important, but those are very difficult to model mathematically and generally aren't an aircraft issue. Things like wind gusts and hysteresis of controls, for lack of a better term, and other things can be modeled if he wants to go there.
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in a calculus class, demonstrating the development of this piece of knowledge alone makes a professor smile. There is nothing in the OP that said this project was limited to an aircraft descent profile, it said it incorporates it. This isn't an aerodynamics or engineering class, it's a math class that develops complex logic skills. I'm not assuming what his goal is or what information he doesn't need to consider.
 
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Henning said:
in a calculus class, demonstrating the development of this piece of knowledge alone makes a professor smile. There is nothing in the OP that said this project was limited to an aircraft descent profile, it said it incorporates it. This isn't an aerodynamics or engine needing class, it's a math class that develops complex logic skills. I'm not assuming what his goal is or what information he doesn't need to consider.
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It's an engineering class, not a human psychology or physiology class. He certainly can try to model a passenger with an earache, or a pilot with his pants on fire, but it really won't matter. A steeper or shallower final approach, faster or slower, that's all fair game.
 
Henning said:
Yes, which opens up the door to produce an efficiency value analysis of pressurized vs unpressurized aviation.
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That might be beyond the scope of the initial project. Once one starts talking "value" that brings in economics.

Besides, limiting the descent profile for human factors can be modeled after the model is developed by limiting the change in pressure over time to some value. That is assuming the op isn't modeling a glider where there is a minimum sink rate.
 
Matthew said:
I read this:


maybe too simply. Weather might make you choose a different descent rate, an engine fire, a passenger or pilot with a sinus or ear problem, ATC instructions, ...
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Yes of course. The OP has been assigned two sequential tasks: 1) Determine what the elevation profile could possibly look like, taking in to account variables, and 2) model it.

Henning said:
in a calculus class, demonstrating the development of this piece of knowledge alone makes a professor smile. There is nothing in the OP that said this project was limited to an aircraft descent profile, it said it incorporates it. This isn't an aerodynamics or engineering class, it's a math class that develops complex logic skills. I'm not assuming what his goal is or what information he doesn't need to consider.
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Well that's right, and what he might (or more accurately will) find is that you can't simply model the entire elevation (or decent even) profile with a single polynomial equation. The recognition of this, if not the solution to, is what the instructor is looking for in my opinion.
 
He might also be after some real world numbers.

In my world: Cruise altitude (start of descent) might be around 5000 AGL, end of descent about 1000 AGL, and rate of descent about 500 fpm. Airspeeds might be constant, or close to it, all the way at around 120kts.

Others, with higher or lower performance or pressurized aircraft, will have different numbers.
 
Matthew said:
He might also be after some real world numbers.

In my world: Cruise altitude (start of descent) might be around 5000 AGL, end of descent about 1000 AGL, and rate of descent about 500 fpm. Airspeeds might be constant, or close to it, all the way at around 120kts.

Others, with higher or lower performance or pressurized aircraft, will have different numbers.
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Perhaps, but I doubt he is being asked to model flight dynamics (I might be wrong). I suspect he is only looking to replicate a zig zagged line. Flight simulation models (essentially real time CFD analysis) are probably well above his level.
 
Hey Everyone, Just wanted to say thanks for all the feedback and ideas. This is very helpful and It will give me a lot to research and incorporate in the project. It's greatly appreciated. Honestly, the project was quite open ended so although I think it was mostly about descent paths I'll look at everything and see what else I can bring in.
 
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