# Mathematics in aircraft design

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#### FrankenWompus

##### Member
I did a quick search, but didn't come up with anything. I did find the thread with books and resources, but it didn't answer my question, so here it goes

Like many here I suspect, I have dreams of designing and building my own airplane. I've got some criteria I'd like to meet, and I plan to build from plans or a kit (or both) before tackling my own design... but it's never too early to start the design process. I've designed and built several of my own RC aircraft, but that's been mostly using the tried and true German standard of Gudenuff and eyeballed aerodynamics. I have designed a few RC parts with Fusion360 and seen them through to the real world with a 3d printer, but they've never involved any real math-based strength assurances that they'll work. Essentially they're just overbuilt and they do the trick.

I'm not ashamed to say that math was never my strong suit in High School. I did go back to college in my mid 20's and got through to Calculus pretty well, but Calc 2 had my number and that was that. I've looked though a few books on design and they all have a bit (or a lot) of math in them. Maybe it's because it's been 10 years since I've looked at an equation, but it looks a bit foreign to me.

I guess my question is, how steep of a learning curve do I have to get up? What level of math competency is enough to design an aircraft that is safe and isn't going to have a rapid unplanned disassembly? I don't need the equations themselves, but what kind of math should I be comfortable with before I start to tackle this? I'd hate to get deep into it, put some numbers on paper, and figure out my wing won't be strong enough, or big enough, or something.

#### wsimpso1

##### Super Moderator
Staff member
Log Member
Algebra and first class in integral and differential calculus is about all that you will actually use. Simple numerical integration works pretty darned well for doing the work. This WILL be used to find spanwise loads on the wing and tailplanes, and then get up to shear, moment, and deflection diagrams. This is beam theory from Mechanics of Materials, and is also applicable to the fuselage as a structure. Much of this also applied by numerical integration.

Stability calculations require use of first derivatives, as that is what is happening. Not terribly sophisticated mathematically, but you do need to know what is going on with differential calculus.

For fluid flows, both internal and external, the first course in fluids and Bernoulli's equation is just about it. First course in calculus does fine but mostly it is algebra.

The rest of airplane design is going to show you integral signs but know that you can do much of it numerically and be fine. Want to know if you can do the math? Get out TOWS, and grind through it. Chapter 2 is kind of abstract, but if the rest works for you, you are there.

Quite frankly, the mathematics is the easiest part. The hard part is all of the technical concepts that have to be balanced and kept in mind. Airplanes are both easy mathematically and VERY hard in terms of creation of a suitable total result.

Billski

#### plncraze

##### Well-Known Member
HBA Supporter
Richard Hiscocks wrote a book called "Design of Light Aircraft." He retired as head of design for DeHavilland of Canada. The intro to his book states that everything inside it can be done with a pocket calculator. Chris Heintz wrote the book "Flying on your own Wings" and he says the same thing.
My opinion is correcting the math mistakes in some books is a pain and debugging spreadsheets is a close second. The most important thing is a willingness to learn.

#### FrankenWompus

##### Member
Thank you for the response! That's actually great to hear. I'm not quite up to snuff on some of these things, but nothing you've mentioned is completely foreign and there are quite a few places I can get a refresher online before I get going.

Get out TOWS, and grind through it.
I'm not familiar with TOWS... ?

Richard Hiscocks wrote a book called "Design of Light Aircraft." He retired as head of design for DeHavilland of Canada. The intro to his book states that everything inside it can be done with a pocket calculator. Chris Heintz wrote the book "Flying on your own Wings" and he says the same thing.
My opinion is correcting the math mistakes in some books is a pain and debugging spreadsheets is a close second. The most important thing is a willingness to learn.
These look like great book suggestions. Design of Light Aircraft is about $300 on Amazon, and Flying on Your Own Wings is$25... so Flying on Your Own Wings it is!

#### BJC

##### Well-Known Member
HBA Supporter
I'm not familiar with TOWS... ?

BJC

#### FrankenWompus

##### Member

BJC
On it's way, thank you!

#### wsimpso1

##### Super Moderator
Staff member
Log Member
All of these books are listed in:

TOWS covers the very basics of airfoils, wings, flaps/control surfaces including calculations. It also has a catalog of the NACA foils including design process, velocity fields, etc. Start there, then add Harry Riblett's GA Airfoils, Pazmany's Light Airplane Design, and Evans Light Plane Designer's Handbook and you have a pretty good look at the topic.

Do these cover everything? No. Neutral Point and Stability is not covered, nor is Wing Structure, but we can sure help with that right here.

Billski

HBA Supporter

#### plncraze

##### Well-Known Member
HBA Supporter
Don't get intimidated by Theory of Wing Sections. It is a thick book but most of it is data, not text. If you know someone with Kindle Unlimited see if you can find Pazmany's book for free. It is a neat book but not for 75 bucks. Also Join the EAA and get a Kitplanes digital subscription. Then you have access to some good stuff.

#### rdj

##### Well-Known Member
The design process can be broken into two parts: conceptual design and detailed design. Like Billski said, you won't need much more than algebra to do conceptual design and performance characterization. Much of that is even empirical; you can simply read it off charts or compare similar aircraft. General design books will give you plenty to ponder in terms of wing area, aspect ratio, taper, horizontal stabilizer effectiveness, power loading, etc. etc. etc. The general stability calculations are the only ones that really push you much into calculus, unless you're planning to do CFD analysis or some similar high level optimization technique.

The math gets more intense when you begin detailed design. A lot depends at that point on what material your plane is being constructed from. Aluminum, wood, tube, glass, carbon rod? Each method has its own unique 'formulas for success', where the generic Mechanics of Materials Stress/Strain formulas meet real material with real manufacturing variations. It also helps to have a working knowledge of aircraft fasteners, fabrication techniques (welding, layup, riveting, etc.) and practical vs. calculated material characteristics, among other topics. For example, as another forum member has noted before, 'airshow morons' can poke a hole right through the thickness of aluminum skin that calculations show is adequate for many wings.

In general, my point is to not worry too much about the math. That will be the least of your design hurdles

#### drgondog

##### Active Member
I believe RDJ reduced your question succinctly - namely "In general, my point is to not worry too much about the math. That will be the least of your design hurdles".

As I reflect on me - at 23 with a MS in Aero, match through Calculus of Variations, steeped in Navier-Stokes and annointed in Chaos Theory - I didn't know diddly about designing an airframe from a blank sheet of paper. A couple of years later after immersion in Airframe Design, Airframe Structures and Advanced Design I acquired some measure of 'diddly'.

The math, based on your background isn't the 'problem to be solved'. The problem is that aircraft design at a high performance/complexity isn't so different from designing and building something like say a Bonanza or Stearman or any 1930s' base technology airframe of wood, metal and fabric. Performance expectations introduce complexity in envelope design, production methods, weight reductions, etc.

From a Personal viewpoint you stated correctly that you were not striving to build something that prematurely and unexpectedly dis-assembled in flight.

So before you even get to analysis per se (Aero, Structures, Stab & Control, Performance) have you got beyond 'general knowledge) of preliminary layout design, conceptual space and equipment and fuel layout, control linkages to empennage, wing, engine; calculate Cg based on above; place wing based on Cg; size and place empennage; electrical layout and design for instruments, engine, etc;

From an Analytical point of view, the above is the easiest set of tasks. The skin and connective structural tissue of the airframe is where the detailed design EXPERIENCE comes into play. Airframe structures, for example is both analytics and an Art - to visualize loadpaths and safely reduce complex geometry for parts into safe chunks of shapes reducible to analysis.

And so on and on.

I found that several books combine to present much practical detail for the Designer to absorb.

One such is Fundamentals of Aircraft Design by Leland Nicolai, Aerospace Engineering - University of Dayton

Aircraft Design: A conceptual Approach by Daniel P. Raymer, AIAA Education Series, and

The latter is as close as I have found to a tutorial for airframe structures ranging from practical to theoretical - no calculus required.

#### Map

##### Active Member
I am going through the whole airplane design process in my videos "Airplane Design Tutorials". Here is the link to the first one, the rest come up if you search for my name on Youtube.

#### Vigilant1

##### Well-Known Member
I found that several books combine to present much practical detail for the Designer to absorb.

One such is Fundamentals of Aircraft Design by Leland Nicolai, Aerospace Engineering - University of Dayton

Aircraft Design: A conceptual Approach by Daniel P. Raymer, AIAA Education Series, and

The latter is as close as I have found to a tutorial for airframe structures ranging from practical to theoretical - no calculus required.
Like the OP, my math skills are limited (algebra is fine, trig I can hack through, what little calculus I truly understood left me long ago). Even with this meager numerical toolset, I've found it possible to do some basic conceptual design tasks (Raymer is my hero, and there are other good sources). I'm finding that design of structures is a much tougher nut. I'm glad you found Peery to be useful, but in my opinion his prose is very obtuse. I suppose this was a common way of writing when he wrote the book (1950), and it is an engineering textbook and not targeted at a "lay" audience, but that doesn't make it easier to understand. Thankfully, it does have good drawings and some exercises (no answers ). I'm sure it is a great text for aspiring engineers who have a good math foundation and who need to be exposed to a lot of different situations.
While it is quite different, I found Richard Hiscocks' "Design of Light Aircraft" to be useful and remarkably straightforward. It is clearly targeted toward amateurs, and gets right to the point. Unfortunately, it doesn't cover structural design, but appears to do a good job of explaining how to calculate loads (which is a very important first step, and a logical one after the conceptual design phase). Loaded with concrete examples of every concept, which I find to be the best way to explain things.
Regarding structures, Billski has been very generous with his time in explaining many typical calculations (see the HBA "stickies." Some "greatest hits" include Beam Theory Explained---How Spars Work, Mechanics of Composite Plates, Beams, and Bigger Structures, etc). "Stress without Tears" has been recommended by mcrae0104, so that will be the next stop in my search for a basic explanation of aircraft structures.

Thanks again,

Mark

#### plncraze

##### Well-Known Member
HBA Supporter
Kitplanes magazine has a digital subscription which allows access to some of their magazine archives. Check out David Paule's Stressing Structures series for good descriptions of structural concepts.

#### Lendo

##### Well-Known Member
F.W. Have a look at John Roncz articles for sport Aviation, ( still available on the Net) those papers are very complete with the Math's examples. Bottom line you need the Maths for everything, it isn't hard but can be sometimes complex. I have many Text books and take what I can from all of them. I would like to say who has help me the most on HBA, but don't want to commit anyone for anything.
It does boil down to research. I've found the Stall Speed (SS) has defined my Light Sport approach, for other SS isn't so important, or for others extremely low landing speed for STOL aircraft is the most important, for others it speed. Horses for courses so they say, pick your mission and proceed.
George

#### rbarnes

##### Well-Known Member
I have found Dr. Gudmundsson's book "General Aviation Aircraft Design" to be an excellent resource as well. Lots of college level algebra. I plan on using OpenVSP Aero to test stability.

Dr. Gudmundsson book repeatedly references other designs as starting points which I like a lot. No need to reinvent the wheel. The structures, control layouts, et al. have probably already been figured out in a similar design. No sense in not "borrowing" from other's tried and true methods.

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#### Lendo

##### Well-Known Member
rbarnes - agreed, some example errors. I think it's a step beyond Roncz, but Roncz's basics are good.
George