Introduction

            At Kitty Hawk, North Carolina on December 17, 1903, Wilbur and Orville Wright flew the first powered aircraft in the history of humankind.  They designed their airplane using physical principles developed by such great scientists as Isaac Newton, an Englishman who lived from 1643 to 1727 and Daniel Bernoulli, a Dutchman who lived from 1700 to 1782.

            The scientific principles used by the Wright brothers still apply today – physics hasn’t changed – only our understanding of physics and the tools we have at our disposal have changed.  But the greatest tool we have, our minds – your minds – are the same now as in the 18^th century.

            In this short course, you will learn how to design an airplane capable of flying in the atmosphere on the planet Mars.

Some Basic Scientific Principles

Scientific Principle No. 1 – Newton’s First Law applied to Linear Motion

            The portion of Newton’s First Law of motion we will deal with is: 

A body in motion continues to move at constant speed along a straight line, unless an unbalanced force acts upon the body.

            The example of this law is an airplane traveling horizontally at constant speed.  When the thrust of the engine is exactly equal to the drag of the airplane, and the combined lift or the wing and the horizontal stabilizer is exactly equal to the airplane’s weight, the airplane will continue to fly in a straight line at constant velocity.

            Scientific Principle No. 2 – Newton’s Second Law applied to weight and mass

            Newton’s second law is usually stated:

                        The relationship between an object's mass m,

its acceleration a, and the applied force F is

 F = m * a.

            With regard to weight and mass, we can rewrite the second law as:

                        The relationship between an object’s mass m,

                        The local gravity g, and the local weight is

W = m * g

            Scientific Principle No. 3 - Lift

            Air flowing over the top of an airplane wing travels faster than air flowing under the wing.  According to the Bernoulli Principle for subsonic airfoils (V² + P = Constant) this leads to lower air pressure on the top of the wing than on the bottom of the wing.  The difference between the two pressures times the area of the wing is the wing’s lift.

            Scientific Principle No. 4 – Drag

            There are many sources of drag on an airplane.  For our purposes, we will limit the exhibition to that portion of the drag caused by air impacting upon the aircraft, i.e. the dynamic pressure multiplied by the coefficient of drag.

            Scientific Principle No. 5 – Pitch Moment

            As a consequence of air flowing over and under a wing at different speeds, there is a relative circulation of air around the wing that causes a pitching moment, i.e. a rotational force or torque trying to cause an airplane, in level flight, to pitch nose down.

            Scientific Principle No. 6 – Torque equals Force times Distance

            When a force is applied perpendicularly to a moment arm, the Force (F) times the Length (L) of the lever arm is called the Torque (T)

T = F * L

            Scientific Principle No. 7 -- Newton’s First Law applied to Rotational Motion

            With regard to rotational motion, Newton’s First Law may be stated as:

A non-rotating body will not begin to rotate unless an unbalanced moment acts upon the body. 

            The example is again the airplane traveling horizontally at constant speed.  The pitch moment of the airplane’s wing must be exactly balanced by an equal and opposite torque created by a downward force (negative lift) generated by the airplane’s horizontal tail acting through a moment arm (the distance between the wing and the horizontal stabilizer).  When these two moments are in exact balance, the airplane will not change its pitch as it flies.

Preliminary Design

            The airplane designer’s first task is to figure out what he or she wants the airplane to do.  Before designing the airplane, there are hundreds and perhaps thousands of details to think about, but we’ll start with only four questions:

            Where do you want your airplane to fly?

How much will your airplane weigh?

            How fast will your airplane go?

            How high will your airplane fly?

            The first question is easy – you want to design an airplane that can fly on the planet Mars.  You’ll have to answer the other three questions yourself.

            Lift, Drag, & Pitch Moment

            Lift and drag and pitch moment are like three peas in a pod.  When an airplane is flying horizontally, lift is a vertical force (usually up) and drag is a horizontal force (usually pulling backwards) and pitch moment is a rotational force (usually trying to cause the airplane to pitch nose down).

To compute any of these forces we have to know the dynamic pressure caused by moving through the Martian atmosphere.  Dynamic pressure is that pressure you feel when moving through an atmosphere.  The wind that is hard to walk into causes a pressure on your body – no wind, no pressure – therefore we call the pressure dynamic because the velocity of the atmosphere past an object creates the pressure. 

P_d = ½ * r * V²

Where:

            P_d = Dynamic Pressure (lbs/sq ft)

            r  = Mass density (lb – sec²/ft⁴)

            V  = Velocity (ft/sec)  Þ  V² = Velocity² (ft²/sec²)

Examine the units of Mass Density and Newton’s Second Law applied to weight and mass.  Weight Density is weight per unit volume and is sometimes represented by the Greek letter g.  Air, at sea level here on earth, has a Weight Density of 0.075 lbs/ft³.  The Gravitational Constant, g, at sea level here on earth is 32.2 ft/sec².

The Mass Density is therefore:

r = g/g  Þ  (0.075 lbs/ft³/32.2 ft/sec²) = 0.002329 lb – sec²/ft⁴

            Let’s take a low flight speed like 60 miles/hour Þ 88 ft/sec

P_d = ½ * r * V² Þ  0.5 * 0.002329 * 88² = 0.9018 lbs/sq ft

            Once you know the dynamic pressure of the air moving past a wing, you can figure out the Lift, Drag, & Pitch Moment of a wing.

            All forces on a moving wing are given by one basic equation

Force = C * A * P_d  (lbs)

The moment on a wing moving through an atmosphere is given by a similar equation that also includes the chord (c) of the wing (the distance from the leading edge of an airfoil to the trailing edge).

Moment = C * A * c * P_d (ft-lbs)

Where C is some coefficient that is determined experimentally or analytically, A is the area the force is acting on.

Aeronautical Engineers use three basic coefficients:  C_L (Lift Coefficient), C_D (Drag Coefficient), and C _M (Moment Coefficient).

Where do Aeronautical Engineers get these coefficients?  The answer is:  text books, experiments they run, fluid dynamic theory, and the Internet. 

The National Advisory Committee for Aeronautics (NACA) published some wonderful scientific papers on aerodynamic coefficients.  The National Aeronautics and Space Administration (NASA) replaced the NACA in 1958.  Fortunately, NASA has a wonderful on-line library at its Langley Research Center (LARC) and you can easily go there on the Internet to find the old reports.  One of the best papers is NACA Report 460 that was published in 1933.  Its URL is:

http://naca.larc.nasa.gov/reports/1933/naca-report-460

            The figures on the next two pages show the type of data that Aeronautical Engineers AND YOU can use to design an airplane.  There are two different airfoil shapes shown here.  There are hundreds and hundreds of different shapes available.  Report 460 shows 78 different shapes.  You may be able to find others in the library, on the Internet, or elsewhere.  Your job is to select a wing that you think will work on Mars.

            Inspect each diagram.  Notice that there is a parameter called Angle of Attack, a.  As the wing moves through the air, the airfoil is inclined to the flight direction at an angle. The angle between the chord line and the flight direction is called the angle of attack. Angle of attack has a large effect on the lift generated by a wing.  In general, if the angle of attack increases, the increases.  However, look at the C_D.  An increase in the C_L usually means an increase in the C_D.

            Also look at the shape of the two airfoils.  One is called symmetrical and the other cambered.  A cambered airfoil simply means that a line equally spaced between the upper and lower surfaces of the airfoil drawn from the leading edge of the airfoil to the trailing edge of the airfoil, is curved.  For a symmetrical airfoil the chord and the camber line are identical.  This isn’t so for a cambered airfoil.

            Note the C_L at a = 0.  The symmetrical airfoil’s C_L = 0.  The cambered    airfoil’s C_L = 0.13

            So, if we had an NACA 2306 wing with a wingspan of 10 feet and a chord of 2 feet flying here on earth at 120 miles per hour (176 ft/sec), what are the aerodynamic forces when the angle of attack is = 10 degrees?  Assume that the plane is flying at sea level.

            Without going through the calculation, PD = 3.607 lbs/sq ft.  If the wingspan is 10 feet and the chord is 2 feet, the wing area (A) is 20 sq ft.

            The lift equation is:

Lift = C_L * A * P_d = 0.88 * 20 * 3.607 = 63.48 lbs

            The drag equation is:

Drag = C_D * A * P_d = 0.06 * 20 * 3.607 = 4.33 lbs

            The moment equation is:

Pitch Moment = C_M * A * c * P_d = -0.04 * 20 * 2 * 3.607 = - 5.77 ft-lbs

Why an Aeroplane needs a Tail

            First, aeroplane isn’t spelled wrong.  Aeronautical Engineers like to spell things in an old fashioned way.  Aeronautical Engineers even spell tail differently.  The tail of an aeroplane, to an Aeronautical Engineer, is an empennage.  You can look that up in a dictionary also.

            Let’s go back to Newton’s First Law.  The law states that an airplane needs to be completely balanced to fly at constant speed in a straight line.

            If out airplane weighed exactly 64.48 lbs and had an engine producing a thrust of exactly 4.33 lbs, would the airplane be balanced?

            NOPE!!!

            The unbalanced pitch moment of – 5.77 ft-lbs would cause the airplane’s nose to pitch down and the airplane would crash.  To counteract this unbalanced pitch moment we need to add an empennage – every airplane you’ve ever seen probably has one.  The tail is usually much smaller that the main wing and far behind the main wing.  The horizontal portion of the tail is called the horizontal stabilizer.  To counter the negative moment, the horizontal stabilizer actually has negative lift – it pushes the back of the airplane down so that the nose of the airplane comes back up.

            So, to balance our airplane, let’s say we put a little tail 10 feet back from the main wing.  For now, assume it has no drag and no pitch moment of its own.  If it had a negative lift of 0.577 lbs, then using the Torque Equation,

T = F * L Þ 0.577 * 10 = 5.77 ft-lbs

            The total lift of the airplane is now reduced by 0.577 lbs

Airplane Weight =  63.48 – 0.58 = 62.9 lbs

Everything is now in balance.  The airplane characteristics are:

                        Weight = 62.9 lbs

                        Engine Thrust = 4.33 lbs

                        Airplane Speed at sea level = 120 mph

            So far this is good.  But, what’s missing?  What about an engine?  Does your car have an engine?  What makes you think an airplane doesn’t need one?

            Go to the Internet and research airplane engines (Hint:  if you are designing a small airplane, look under model airplane engines).

Select an engine for your airplane.  Find out its weight and its thrust.

The list below is all the things you have to know about your airplane:

            Your airplane weighs AA pounds. 

The wing you have selected has BB pounds of lift.

The tail you have selected has CC pounds of negative lift.

Your airplane’s wing has a positive moment of DD foot-pounds.

            Your airplane’s tail has a positive moment of EE foot pounds.

            The negative lift of the tail causes a negative moment of FF foot-pounds.

            The total drag of your airplane is GG pounds.

            The thrust of the engine is HH pounds.

            Final point......

Aeroplane Balance

            In order for a real airplane to stay balanced, the center of gravity (c.g.) of the airplane should be located just about at 25% back on the main wing from the leading edge.  This is called the quarter chord point.  So, on our sample, the main wing has a chord of 2 feet so the c.g. should be 6 inches back from the main wing’s leading edge.

            There are good sites on the Internet that explain how to calculate c.g.

http://www.grc.nasa.gov/WWW/K-12/airplane/acg.html

Ladies & Gentlemen – Start Your Design

Basic Steps

Choose an airplane weight (remember, you’re on Mars now), choose the height at which you want you airplane to fly, and choose the speed at which you want it to go at.

Do the research.  What is the gravity constant, g on Mars?  What is the density of the Martian atmosphere at the height you want your airplane to fly?

Calculate P_D.

Think about how much things weigh on your airplane.  What do you think the wing weighs?  What about the fuel?  What about the engine?  What about the body and the empennage?  What about a payload?  Does all this stuff equal the weight that you think your total airplane will weigh?

If you think you have a good handle on the weight, start looking for an airfoil design for the main wing.  Decide on a foil shape and an angle of attack.  A good design usually uses coefficients that come from a linear (almost straight) portion of the graph. 

Select a wingspan and chord.  A good rule of thumb is keeping the aspect ratio (the ratio or the wingspan to the chord) above 6.

            Work out the Lift, Drag, and Pitch Moment of the main wing.

             The combined Lift of both the main wing and the horizontal stabilizer must equal the airplane’s weight;

             The combined Drag of both the main wing and the horizontal stabilizer must equal the airplane’s thrust;

             The Pitch Moment of both wings, taken about the quarter chord point must equal zero.

                        There’s a lot more to it than this, but if you’ve gotten this far you’re off to a great start.  You ought to consider an Engineering career.  Today, most colleges no longer give Aeronautical Engineering degrees.  Today they call them Aerospace Engineering.  Regardless of what it is called, engineering is a good career and there are good engineering colleges right here on Long Island.

             There are many other engineering specialties – Electrical, mechanical, civil, chemical, computer, ceramic, just to name a few.  If you found this exercise to be fun, think about going into engineering when it’s time to think about college.....