Math section

[Autodidact]

Since mathematics is so intrinsic to everything about aircraft, from flying them and building them to, most of all, designing them, I thought it might be useful to have a section where you could ask an aircraft related math question and get an engineers answer (which would tend toward the succinct and practical) as opposed to a theoretical mathematicians answer (which may or may not tend toward the glib and obtuse ).
It probably won't be the most heavily trod section, but might prove to be of assistance to those who burn the midnight oil, trying to get their head around some seemingly obscure (to regular folk, anyway) mathematical formula, concept, or technique.

[Tom Nalevanko]

Math is math.... There is no difference between engineering math and any other kind of math. Various physics and engineering concepts are described by math. Understanding the math helps in understanding the description but it doesn't work the other way. I think that you are looking for a good math teacher or book. The following site is run by a friend who answers questions about math. It might help: Numericana - Numbers, Numerical Arcana, Numerical Facts...

In general, there is a wealth of math teaching efforts on the web. Just have to look a little...

Blue skies,

Tom

[mz-]

To understand things like force, acceleration, speed, location, mass or energy, it massively helps to think in SI units where the conversion factor between all these is 1.

Let's look at energy for example:

To lift 1000 kg 1000 meters in the Earth's gravity field of 9.81 m/s^2 takes E = m*g*y = 1000 kg * 9.81 m/s^2 * 1000 m = 9,810,00 kgm^2/s^2 = 9.81 Megajoules or 9.81 MJ.

To accelerate 1000 kg from 0 to 20 m/s takes E = 0.5*m*v^2 = 1000 kg * 0.5 * (20 m/s)^2 = 200,000 kgm^2/s^2 = 0.2 MJ

Power and Force:
If you want to do the former lift/elevator* thing in 20 seconds, it uses an average power (because of physics it's easy to make it constant at constant rising speed) of P = E/t = 9.81 MJ/20 s = 490.5 kW. So about 500 kW. That's very large piston engine, about 700 hp.

You can also think about this way: Speed:1 km/20 s = 50 m/s. Force: 1000 kg*9.81 m/s^2 = 9810 N. Power = speed*force = 50 m/s * 9810 N = about 500 kW.

The latter acceleration example does not mean constant power. If acceleration is constant then F=M*a means a constant force. But since power is dependant on speed also, P = F*v, that means that the power needed to maintain constant acceleration increases towards the end. If the thing is done in 10 s then the acceleration is 2 m/s^2 and this a force of 2000 kgm/s^2 = 2000 N.
At the endpoint at 20 m/s where the power needed is largest, it's 20 m/s * 2000 N = 40,000 W = 40 kW. A car engine can do this handily. (And actually, this is what cars do, accelerate a 1000 kg mass from 0 to 20 m/s.)


Avgas has an energy content of about 47 MJ / kg. If you use a 20% efficiency to mechanical work number, it's roughly 10 MJ/kg, so about 1 kg of avgas for the former elevator/lift* example, 0.02 kg or 20 grams for the latter catapult or winch example.

*damn the English language's homonyms. This elevator or lift here means the thing in buildings that goes up and down between storeys.


This would be more complex with a mix of lb in mass, ft in height, knots or mph or ft/s of speed, dunnowhat unit of force, BTU of energy etc.

[autoreply]

To understand things like force, acceleration, speed, location, mass or energy, it massively helps to think in SI units where the conversion factor between all these is 1.

Not to mention that most conversion factors are pretty simple as well. Meters to feet is *3/10, kg to lbs is *2 +10% and so on. Metric really is the easier way to go, even if it involves converting units twice

[Mac790]

Metric really is the easier way to go, even if it involves converting units twice

Yes I agree with you, the metric system is much easier to work with, almost entire world use it.

Countries which have officially adopted the metric system. Only three nations have not officially adopted the International System of Units as their primary or sole system of measurement: Burma, Liberia, and the United States.

Countries by date of metrication. Green represents 1800, yellow 1900, red 1980, and black identifies countries that have not adopted the metric system as the primary measurement system.

Seb

[Autodidact]

Tom, yes math is math. I have plenty of math books and most importantly an intense desire to read and work through them, therefore my request was mostly for the benefit of others. Some people are fairly intimidated by math and make the assumption that it is too difficult for them and a math section here on the forum would represent an efficiency, in that you wouldn't have to go searching for something that is unfamiliar, i.e., you could talk to people you are familiar with already. It would make it easier to find the answer to questions like "How do you calculate a static margin?" since you wouldn't have to do a search - you'ld know that it is almost certainly in the "Math" section. It could serve as an "encouragement" and a "stepping stone", and a general demonstrative promotion to the use of the more in depth resources, a resource for the best books to use, websites (as you provided above, thanks!), etc.
To avoid OT discussion, a requirement for questions to be aviation related might be necessary ( all of mathematics can be aviation related, I imagine, but there's really no reason to have discussions about String Theory and what not). Could be called "Aeronautics Math".
I won't harp incessantly about this, I just thought it would be a neat feature to have. If, however, there is a concern for not providing assistance to bad people, I can certainly understand that. It's would be a shame if the world has come to that, though.

mz-, yep, SI is a much more convenient system to use, I think, but as an American, I feel compelled to learn both SI and whatever it is we call pounds/slugs, feet/inches ( the BS system? ). I don't mind, I love the history of it all as much as I do the actual stuff (math and engineering) itself.

[autoreply]

mz-, yep, SI is a much more convenient system to use, I think, but as an American, I feel compelled to learn both SI and whatever it is we call pounds/slugs, feet/inches ( the BS system? ). I don't mind, I love the history of it all as much as I do the actual stuff (math and engineering) itself.

Well, you might find it offensive, but the rest of the world calls your system the imperial system. Think of that Great Empire you were once part of

While it might be a good idea, especially math of aero questions can usually very well be solved using either google or wikipedia. Simply typing in the keywords of your question is usually sufficient to have a clear answer in seconds.

[Autodidact]

Ah, yes. The Imperial System. I knew it was something like that. Someday there will be no more empires, and I hope it's a good day.

[litespeed]

Yes,

the system that was used in the Empire where the sun never sets.
Once the sun did start setting it made a lot of sense to cooperate rather than been insular and expecting everyone (95%of the planets population) to see it your way or else. Or is the earth still flat?

Now think of that and the list of the remaining countries to use Imperial- the stuff empire builders use.

USA, Burma and Liberia- Antartica doesn't count, the penguins don't give a **** what dumb humans do.
USA would find it much easier selling stuff to the world and save a lot of problems, if SI was used. And the rest of the world might feel a bit more comfortable with Uncle Sam at the same time.

Maybe Iraq did have weapons of mass destruction, but no-one could understand the map numbers -because it was written in Imperial?



Just the view from this side of the planet.

[bmcj]

Either system is fine, and valid, for doing engineering. As long as you have the understanding and the proper tables, the answer will be correct regardless of which system is used. Metric's big advantage is the ease of the calculation (being a base 10 system). The biggest point of confusion for the Imperial system is the difference of pound-mass and pound-weight.

[btravis]

My standard Rant when people start saying the metric system is better:

I am a mechanical engineer, former designer of solid fuel fired power plants, both waste to energy, and fossil fuel fired. I have been part owner of a joint venture between a US, German, and Swedish firmsand was fortunate enough to have worked all over the world, using both systems. And the statement that the metric system is easier or makes more sense is just plain false, and drives me crAAAAZY. It is foisted on us (in the USA) by an academic system based on tenure and not competence.

Both systems have thier strengths and drawbacks in certain situations. The only thing that is truly easier in the metric system from an engineering standpoint is length. Unless, of course you need to divide the length by 3. i.e. 10m/3 is 3.333333333333 ... Divide 10 feet by 3 and you get 40 inches, a nice round number.

And then, now that you got me started, a 10" sch 40 pipe is a 250 DN pipe in the metric system. It is actually 10.75 OD and approximately 10 in I D, but 273 mm OD. And someone payed someone a lot of money to develop a parellel standard in the metric system that is identical in every way to the long standing british system. You can order it either way, but you still get a 10" sch 40 pipe. If converting to metric is so important, why does Europe or most of it still use English units in their pipe threads (or at least did when I lived there)? Rip out the plumbing! Lug nut threads on foreign cars? Lots of entertainment can be had in this topic.

And what about wire sizes? AWG No. 2, or 32 mm^2? neither means much to me. I still have to look up how many amps each will carry.

And once you get weight involved, the acceleration of gravity is 32.2 ft/s^2 or 9.81 m/s^2... How much does your airplane weight in Newtons? It's mass is not the same number like it is in US units! I could go on and on, but you get the point.

And then there is the little matter of metric vs SAE fasteners. I was once asked, when working an oil refinery in California why we didn't replace all the fasteners in the place with a standard metric bolts, studs, nuts, etc. I had to explain, that, well when the refinery was built, there was no metric standard fasteners. And that since that refinery, alone, would cost about $2.5 billion to build in today's dollars, and there are probably something close to a million fasteners there. Oh, and by the way, many of those have to have custom torquing procedures because they are handling things like 2500 psi hydrogen, and, by the way, what do you really want to pay for a gallon of gas?

What were we talking about?

Oh. The one thing that does make some sense is nautical miles, since it actually relates to something on the ground, the distance between the equator and the pole, divided by 60 degrees, divided by 60 minutes, and actually appears on the vertical longitude lines on the maps we use. Kilometers? I have no idea.

Now why we divided an hour and a direction by 60 or 360, I have no idea.
__________________

[lr27]

I started to go on about units, but that's not what this thread is for.

I suggest, rather than calling it a math section, you call it a calculation section. I think that might convey your intent more clearly.

[autoreply]

Oh. The one thing that does make some sense is nautical miles, since it actually relates to something on the ground, the distance between the equator and the pole, divided by 60 degrees, divided by 60 minutes, and actually appears on the vertical longitude lines on the maps we use. Kilometers? I have no idea.

But it's very easy. While a nautical mile is exactly one arcminute(and thus 1/21600 of the earth's circumference at the equator) 10,000 km is exactly the distance from pole to equator. Sounds easier to me doesn't it?

Everything in SI is related to as few units as possible and without any weird conversion factors (be it 1/12, 1/60th or 1/21600th), except for zeroes. That is the big benefit of the SI, compared to the imperial system, not to mention one unit, instead of 10 different miles and 3 different gallons

Doing aerodynamic calculations or calculating power takes at least two calculations less in SI and I've done those frequently

[Autodidact]

I like the metric system - the english system works just fine as well. What is bothersome is having to go back and forth between them. If I had to pick one I'd probably pick metric because decimals and fractions go together so well in SI (I think that's what I meant to say). Since our number system is base ten, it seems to make sense for our units to be as well, but that's not terribly important, it's what your used to I think.

A math or calculation section would be up to Jman to implement, and since this thread is not accomplishing anything else, we might as well commiserate/argue about systems of measurement!

[orion]

I started to go on about units, but that's not what this thread is for. I suggest, rather than calling it a math section, you call it a calculation section. I think that might convey your intent more clearly.

OK, enough about units - work with whatever you feel comfortable and then convert if others want to see the answers in a different form.

I like the original intent of this thread since it could be of use to demonstrate some of the techniques used to do the design stuff. Obviously we can't do all in the discussion forum form since that would be a rather significant dissertation but I think a lot can be covered without getting too bogged down in minutia.