Mathematics-in-Action - Airplane Design Workshop - Dr. Ian Halliwell Senior Research Scientist...

Mathematics-in-Action- Airplane Design Workshop -

Dr. Ian Halliwell

Senior Research Scientist

Avetec Inc, Springfield, Ohio

July 21, 2008

Ian Halliwell, Avetec Inc 2

Contents

1. The Role of Mathematics: From Molecules to Moon Landings– Mathematics, Science & Engineering– Simple Calculations– Complex Calculations

2. How an Airplane Flies– Airplane Configurations– Major Features– Gas Turbine Engines

3. Airplane Design Workshop


1. The Role of Mathematics: From Molecules to Moon Landings


The Role of Mathematics: From Molecules to Moon Landings (1)

• We all use mathematics everyday for a variety of mundane purposes– Making change, telling time

• On a much higher level, mathematics forms the fabric of today’s technology-based society

– Science & technology

– Commerce

– Defense

– Computers

– Electronics

– Design & manufacturing processes

– Medicine

• We use mathematics to model concepts that we cannot grasp in any other way– Small scale objects, such as atoms & molecules

– Very large-scale objects, such as our solar system and universe

– Very complex systems, such as a gas turbine engine cycle or the flow field inside a compressor


The Role of Mathematics: From Molecules to Moon Landings (2)

• Small stuff! • Bigger stuff!

CFD Grid of a Stator & Rotor in a Turbine

Prediction of Rotor Pressure Distribution

Mars – Earth Transfer Orbit


Mathematics, Science & Engineering

• While mathematics may be considered as a separate subject in its own right, it also plays a vital role as a tool in science & engineering.

• Scientists discover and explain natural phenomena, often expressing them in theoretical or mathematical terms.

• Mathematicians describe scientific phenomena and help us to understand them using mathematical theories. Some examples are:

– Electricity

–

–

–

• Engineers turn concepts into hardware that is able to be used for the benefit of everyone.


Simple Calculations• In aircraft design we can calculate the thrust needed from the engines to accelerate down the runway

and take off

• Balancing forces horizontally:

• Balancing forces vertically:

• Combining the two gives:

Where M = aircraft mass

dV/dt = aircraft acceleration

μ = runway coefficient of friction

R = undercarriage reaction force = Mg – L

g = gravitational acceleration

RdtdVMDT

LMgR

LMgDTdtdVM


Complex CalculationsFlow around Aircraft & Inside Engines (Computational Fluid Dynamics)

• For the last 20 years or so, powerful computers have enabled us to calculate the behavior of the flow fields around the blade rows inside an engine and around complete aircraft.

• This has led to improved designs and better use of expensive test facilities.

CFD Results from a Turbine Stage CFD Grid around an F-16A Aircraft


2. How an Airplane Flies


How an Airplane Flies

• There are four forces acting on an airplane in flight:

1. Lift acts upwards

2. Weight acts downwards

3. Thrust acts forwards

4. Drag acts backwards

• At a constant altitude the lift balances the weight

• At a steady speed the thrust balances the drag

Forces on an Airplane in Flight


How an Airplane Flies

• Lift is produced by air flowing over the wings

The air moves faster over the upper surface than over the lower surface This results in a lower pressure on the upper surface (the suction surface) than that on the lower surface (the pressure surface)

• Thrust is produced by the engine(s)

• Drag is produced by friction of the air over the airplane

• Weight is made up of the airframe, the engines, the fuel & the payload

An Airplane Wing – Streamlines & Forces


Airplane Configurations


Airplane Configurations (1):

Boeing 777-200LR Commercial Airliner



Boeing 777-200LR & -300ER Commercial Airliners



Embraer 145 Regional Jet


Airplane Configurations (3)

Aerospatiale-BAC Concorde Supersonic Transport



Lockheed Martin F-35A Lightening II Joint Strike Fighter



Apache Longbow (Boeing AH-64D) Scout/Attack Helicopter


Airplane Features


Wings• The pressure generated on the wings by the air flowing over them provides lift to keep the

airplane in the sky. Lift is caused by forward motion of the airplane, and the forward motion is provided by thrust from the engines. If the wings are big, with a lot of area, they generate a lot of lift and less speed is needed to keep the plane up. However, the wings would be very heavy and also expensive to make. We might be better off to use small wings but fly faster to generate the same lift, by using bigger, more powerful engines - perhaps three engines instead of two.

• As the air passes over the wings, it rubs against the wing surface and the friction causes drag, slowing the plane down or forcing the pilot to use more power from the engines to maintain speed if he wants to get to his destination on time. If we sweep the wings back we can reduce the drag at high speeds. Some aircraft have forward sweep - the air doesn’t know the difference!

• Drag is also affected by the aspect ratio of the wing. A long skinny wing has a high aspect ratio. A short stubby wing has a low aspect ratio. A high aspect ratio wing produces more lift and is more stable at low speeds than a high aspect ratio wing, but will cause more drag if we want to fly fast.

• The control surfaces on the wings are called ailerons. These are used to cause the aircraft to roll in a turn, which keeps it stable. Flaps can also be extended from the wing trailing edge. These increase the wing area temporarily and are used at low speed to produce lift while taking off or landing.


Tail

• The tail is normally located at the rear of the aircraft and is made up of horizontal and vertical surfaces. The horizontal parts are called the tailplane. They are like small wings and act to stabilize the airplane in level flight by trimming it about its center of gravity. By rotating the horizontal control surfaces the pilot can make the aircraft climb or dive. Movement of the vertical stabilizer or rudder is used to turn the aircraft. This is coordinated with movement of the ailerons. The size of the tail determines how well it does its job and how fast the airplane can respond to the pilot’s requirements, but it should be remembered that a large tailplane is also heavy and expensive.

• A conventional tail is at the rear of the aircraft and protrudes from the sides of the fuselage. Many modern airplanes have what is known as a “T” tail. Some very special airplanes have “tails” near the front, known as canards. These provide special aerodynamic stability characteristics, although their location could also be partly dictated by the aircraft structure.

• For tails, the same comments apply to area and aspect ratio as for wings.


Seating Layout

• The size and shape of the aircraft fuselage depends on how many passengers are to be carried and how the seats are arranged. The seats are distributed between the different fare classes but their distribution is also affected by the need to protect the passengers and to ensure their safe and rapid evacuation in an emergency. The passenger compartment must also blend with the front and tail sections of the airplane, and accommodate the massive frames that support the wing and tail surfaces.

• The width of the aircraft or cross section is very important because once it is fixed it is impossible to change. If necessary, the capacity of the cabin can be stretched by adding a “plug”. The overall cross section should be kept small to reduce the aircraft weight and drag, yet the resulting shape must provide sufficient comfort and necessary aisle space. More seats per row will lead to a wide fuselage and produce a drag penalty but also will offer potential for stretching and growing the aircraft in future.

• Based on a fixed number of passengers, the program will adjust the fuselage length and width automatically according to the seat layout that you select.


Cruise Conditions

• The speed and altitude at which the mission is flown both affect the fuel consumption and the fuel consumption affects the operating cost of the airplane. So while all of us would like to travel very fast, especially on long trips, we know that this will be expensive. High speed and high altitude are not so important on short trips because much of the time in the air is spent either accelerating and climbing or decelerating and descending. A lot of thrust is needed to fly fast and therefore the engines will be big, heavy and expensive and a lot more fuel will need to be carried. Flying fast also generates a great deal of drag, which means that the engines become even bigger and burn a lot more fuel.

• In the design program, you will be asked to select a cruise speed in knots. (A knot is a speed of one nautical mile per hour, and this is equivalent to 1.15 regular miles per hour) You will also be asked to select the altitude at which your airplane will cruise (in thousands of feet). The design program will calculate the cruise Mach number of your aircraft at the altitude selected. (By definition, sound travels at a Mach number of one, but the speed of sound gets less as we climb higher.) You are also asked to choose how much fuel your aircraft will carry at take-off. After it has run, the program will tell you if you have sufficient fuel on board to make the trip. Remember – in an airplane you can’t just pull over to the side of the road and call AAA!


Engines• Most modern aircraft engines are gas turbines. They come in many forms. A turbojet

produces thrust by taking a relatively small amount of air and expelling it backwards at a very high speed. These engines are very noisy, especially at take off. A turbofan engine takes in a much greater volume of air than a turbojet, but pushes it backwards at a much lower speed. This produces the same thrust and is much quieter. A gas turbine engine can also be used to turn a propeller to produce thrust. This is called a turboprop. A modern version of a turboprop engine has the propeller at the rear. The propeller blades are curved backwards so that they can perform well at high speeds, just like swept wings. They are also very quiet and very efficient engines.

• We can choose to have two large engines or three or four smaller ones to generate the thrust we need. We also must decide how to mount them on the aircraft. Some airplanes have the engines suspended from the wings by pylons. On others the engines are clustered near the tail. Tri-jets use a combination of wing- and tail-mounted engines.

• In the design program you will be asked how much thrust your airplane needs at its cruise conditions. This will then be divided equally between however many engines you have installed. Three or four engines were used for long distance flights until just recently. This meant that the mission could still be completed quite safely in the event of an engine failure. However, many modern aircraft (like the Boeing 767 and 777) have only two engines, even to fly over the Atlantic and Pacific Oceans. This is because two large engines are cheaper to manufacture and operate than four small ones, and more importantly, because engines are continually becoming more reliable and are much less likely to fail.


Engine Configurations


Gas Turbine Engine Configurations (1)

Turbojet with Two Compressors


Example of a Simple Turbojet

Teledyne Continental J402 Turbojet Missile Engine (640 lbf Thrust)



A Turboprop


Example of a Turboprop

TPE 331 Turboprop



A Twin-Spool Unmixed High Bypass Ratio Turbofan


Example of a High Bypass Ratio Turbofan

CFM56 High Bypass Ratio Turbofan (30,000 lbf thrust)


3. Airplane Design Workshop


Introduction.

• You will be given access to computer software that will enable you to design a complete airplane. The design of an airplane is extremely complex and many decisions must be taken concerning its size, the wings, the tail, and the type and number of engines. We also must know about the mission that it is going to be used for. Where is it going to fly from? Where is it going to fly to? How high should it fly? How fast should it fly? All of these factors affect how heavy it will be, what its dimensions will be, and how much it will cost to build and operate.

• The price of the aircraft and its cost of operation will affect how much a passenger will have to pay to fly on it. If no one can afford to fly on it the airlines will not buy it, and if the airlines will not buy it, the aircraft companies will not build it. Can you design an airplane that Boeing would want to build?

• Your job has been made easier because we have already decided that you should consider the aircraft mission to be – Washington D.C. to Sydney, Australia. We have also decided that the aircraft should carry 200 passengers. But you still have many choices to make.


Running the Design Program

• You will be given the chance to choose between a number of options on 5 major design parameters – wings, tail, engines, seating layout, and aircraft cruise conditions. When you have decided how each of these characteristics should be specified, you tell the design code to build the airplane. It will then draw the aircraft and make some comments on your design.


Destination• The aircraft mission starts in Washington D.C. and ends in Sidney, Australia.

Brush up on your geography, because you will be presented with a map of the world and you will have to select the second city when it is needed for definition of your aircraft mission. The altitude and length of the runway from which you are operating both affect the performance required from your design. The program will print out the runway data. The range and time of flight are also calculated for you and printed out.

Building the Aircraft• When you have selected all of the design parameters you hit the “build” button

and the design code will construct your airplane and produce a picture of it.

You will be judged on flight time and ticket price.

Good luck with your design!!


Airplane Design Workshop

Mathematics-in-Action - Airplane Design Workshop - Dr. Ian Halliwell Senior Research Scientist...

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Transcript of Mathematics-in-Action - Airplane Design Workshop - Dr. Ian Halliwell Senior Research Scientist...