# An Introduction to Aerodynamics

## CONTENTS OF CURRICULUM UNIT 88.06.07

- Table of Content
- 1. Rationale for the Unit
- 1. General Objectives
- 3. Content and Specific Objectives for the Unit
- Navigation and Spherical Geometry
- The Earth’s Surface and Mapping
- 4. Symbols, Diagrams and Definitions Used
- 5. Sample Items Illustrating Some Of The Specific Objectives
- 6. Sample Lesson Plans: #1
- Lesson plan #2
- Lesson plan #3
- References

### Unit Guide

## Aerodynamics: The Mathematical Implications

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## 3. Content and Specific Objectives for the Unit

### Content Specific Objectives

Mathematics for Airplane Mechanic

- 1. History of Aerodynamics
- ____ a. The students will be able to tell the main developments in the history of flight.
- 2. How airplanes fly
- ____ a. The student will become familiar Forces in Flight the vocabulary of aerodynamical terms
- ____ ____ i. airfoil
- ____ ____ ii. Leading edge
- ____ ____ iii. trailing edge
- ____ b. Students will be able to calculate resultant force
- ____ c. To understand how airfoil produce lift
- ____ d. To calculate the ratio of lift and drag
- ____ e. To calculate (C.F.) centrifugal force

- 3. The Steel Rule
- ____ a. Students will be able to use the steel rule to measure length
- ____ b. To measure different degrees of accuracy
- ____ c. To perform the basic operations on ruler fractions.
- 4. Decimal in Aviation. Students will be able to
- ____ a. Reed decimal numbers
- ____ b. Check dimensions with decimals
- ____ c. Perform the basic operation on decimals
- ____ d. Read a decimal equivalent chart
- ____ e. Determine tolerance and limits
- 5. Measuring Lengths. Students will be able to
- ____ a. Determine the units of length
- ____ b. Calculate perimeter of different shapes
- ____ c. Calculate the circumference of a circle
- 6. The Areas of Simple Figures. The Student will be able to
- ____ a. Determine units of area
- ____ b. Calculate the area of rectangle and squares
- ____ c. Calculate the area of a triangle
- ____ d. Calculate the area of a trapezoid
- 7. Volume and Weight. Students will be able to:
- ____ a. Determine units of volume
- ____ b. To use formulae for volume
- ____ c. To calculate weights of Material
- ____ d. To calculate Board feet
- 8. Angles and Construction. Students will be able to:
- ____ a. Use a protractor
- ____ b. Construct angles
- ____ c. Determine the units of measure of an angle
- ____ d. Calculate angles in aviation
- ____ e. Bisect angles and lines
- ____ f. Construct perpendicular and parallel lines
- 9. Graphic Representation of Airplane Data. Students will be able to
- ____ a. Construct and read a Bar chart
- ____ b. Construct and read a pictograph
- ____ c. To construct a broken line graph
- ____ d. Construct a line graph
- 10. Spherical Geometry. Students will be able to:
- ____ a. Use the globe to measure distances
- ____ b. calculate the distances between two airports
- ____ c. To read polar coordinates
- ____ d. To represent direction using vectors
- ____ e. Pilot the path of flight between cities
- 11. Other Topics—The Weight of the Airplane. Students will be able to
- ____ a. Calculate the wing area
- ____ b. Find the mean chord of a tapered wing
- ____ c. Calculate Aspect Ratio
- ____ d. Calculate the gross weight of an airplane
- ____ e. Calculate pay load
- ____ f. Calculate wing loading
- ____ g. Calculate Power loading
- 12. Airfoils and Wing Ribs. The student will be able to
- ____ a. Draw a scale of the upper camber
- ____ b. Draw a scale of the lower camber
- ____ c. Draw upper and lower camber when data is given as per cent of chord
- ____ d. Draw the nosepiece and foil sections
- ____ e. To calculate the thickness of airfoils
- ____ f. To draw airfoils with negative numbers
- 13. Mathematics of Materials. Students will be able to
- ____ a. Define and demonstrate the tension of different materials
- ____ b. Calculate compressive strength
- ____ c. Calculate shear strength of a material
- ____ d. Calculate Bearing strength
- ____ e. Define and calculate cross-sectional area
- ____ f. Calculate bend allowance
- 14. Aircraft Engine Mathematics. Students will be able to
- ____ a. Define horse power and pressure
- ____ b. Calculate the area of piston head
- ____ c. Calculate different horse power
- ____ d. Calculate fuel cost
- ____ e. Calculate cruising time
- 15. Scale Drawing. Students will be able to
- ____ a. Read information from a blue print
- ____ b. Reproduce a diagram using a scale
- ____ c. Calculate length and distance using a scale

### Forces in Flight

The performance of an airplane depends on five forces that act up on it during flight. In some maneuvers all five of these forces are acting on the airplane. When we see an airplane overhead we cannot help but wonder how it is possible to sustain so much weight in a medium as light as air. The airplane wings must produce a force equal to its weight. This sustaining force that opposes the weight is called lift.*(figure available in print form)*

**
Lift is affected by
**

2. Ascending or descending current

- 1. Density of air due to
- ____ a. High or low barometric pressure
- ____ b. Temperature
- ____ c. Varying moisture content of the air

*
How an Airfoil produces Lift
*

The action of the air on the wing of an airplane during flight is similar to the action of the air on the kite. The air exerts an upward and backward force on the wing. This resultant force can be resolved into the horizontal drag force and the vertical lift force.

*(figure available in print form)*

*Airfoil*: This term refers to a surface designed to produce an efficient lifting force; it is also used in reference to the shape of a cross section of a wing.

*
Angle of Attack
*
: The angle between the chord of the airfoil and the direction in which the airfoil is moving through the air. This direction is often referred to as the flight path of the airplane. The angle of attack can also be defined as the angle between the chord and the direction in which the air is flowing over the wing.

*
Leading edge
*
: The front or forward edge of an airfoil

*
Trailing edge
*
: The rear edge of an airfoil. A typical wing tapers off towards the rear with the edge as sharp as possible.

*
Chord
*
: The straight line from the leading to the trailing edge of an airfoil.

*
Air flow
*
: The fuselage of the airplane is designed so that it will pass through the air with the lease possible resistance. It should be shaped so that the air will follow its surface and flow over it smoothly. The shape of the airfoil must be such that the air will not only flow over it, but will develop a definite force in a given direction. This force is called lift. An airplane can stay in the air only as long as its maintains sufficient relative motion between these two pressures.

*
Drag
*
: Air resists the movement of any object through it. This resistance is called drag. The drag set up by an airplane in steady level flight absorbs all the power development by the engine.

The drag of an airfoil can be attributed to two main causes. The part of the total drag caused by eddying currents and turbulence set up in the air is called
*
profile drag
*
. The skin friction is causes by the friction between the object and the air.

When airfoil is adjusted so that the chord is parallel to the airfoil or at zero angle of attack considerable lift
*
is produced
*
.

*(figure available in print form)*

*
Lower Surface Lift
*
: The leading edge of the wing is slightly higher than the trailing edge so that the air staking the under surface is deflated downward.

*(figure available in print form)*

Newton’s Third Law states that for every action there must be an equal and opposite reaction thus the force required to deflect the air downwards imparts and equal and opposite force that pushes the wing upward.

The lift action of an airfoil can also be explained by Bernoulli’s theorem which states that the pressure on any fluid is least where the velocity is greatest and the pressure is greatest where the velocity is least.

The air moving over the upper surface of an airfoil is forced too travel farther, therefore velocity is increased. The increase in velocity caused a decrease in pressure. This causes lift in the upper surfaces. The air that passes beneath the airfoil has less distance to travel, this results in increase in air pressure on the lower surfaces and in decrease in air pressure on the upper surfaces. The total lift produced by the airfoil is equal to the difference downwards. The wing is said to have a relative angle of attack.

*(figure available in print form)*

The lift drops off at high angles of attack because the air instead of flowing smoothly over the upper surface breaks away from it and forms eddying currents.

When the lift and drag values of an airfoil are plotted on the same graph, the relationship ratio between them at the different angles of attack is shown. The angle of attack that results in the lowest drag is not the same angle that produces the least lift but is usually near the zero angle of attack. When high angles of attacks are reached.

Discussed were forces that affected the airfoil in flight lift and drag. The forces acting on the wings must be considerable in conjunction with the other forces that affect the airplane.

*
Gravity
*

Gravity is the force that plays an important part in the performance of flight maneuver. The direction of gravity is always constant. It is exerted directly towards the center of the earth or downwards.

In straight and level flight the direction of the lift force is exactly opposite to the pull of gravity. The resultant of the lift force and gravity can be determined by completing a vector diagram.

*(figure available in print form)*

*
Thrust
*

The forward force that the propeller develops is called thrust. It pulls or pushes the airplane forward through the air, overcoming the drag or the resultant of lift and gravity plus the drag.

The direction of the thrust is always in line with the crankshaft of the engine which is generally parallel to the longitudinal axis of the plane.

The thrust force developed by a propeller depends on the propeller design, the density of the air; and the speed at which it is turning. When the engine is idling by the thrust is negligible. The maximum thrust is limited by the highest power output of the engine.

*(figure available in print form)*

The total thrust (t, plus t2) must equal the Drag d1 plus the left gravity resultant (d2).

When the airplane is flying at a constant rate of speed in level flight, the thrust pulling it forward must be exactly equal to the drag holding it back.

*
Centrifugal Force
*

Newton’s first Law; a moving object will travel in a straight line indefinitely unless acted upon by an outside force. The law indicates that a force is required to compel any object to travel in a curved path. Centrifugal force can be mathematically determined by the application
*
of C.F.= mv2/f
*

*(figure available in print form)*

*
Mathematical concepts underlying the force in flight
*

The lift and the drag of an airfoil section have a definite relation for any angle of attack. In addition the lift and drag depend on

Where 1 and d are the left and drag in pounds c1 and cd are the lift and drag coefficient which depends on the wing contour used and the angle of attack, is the air density in slugs per cubit foot is, the area of the wing in square feet, and V is velocity or airspeed in ft per second.

- a. the angle of attack
- b. the contour of the wing
- c. the density of the air
- d. the area of the wing
- e. the square of the airspeed
- ____ 1 = c1P/2 s.v2
- ____ D = c3P/2 s.v2

### Vectors and Forces

A vector has both magnitude and direction. In describing force, not only the magnitude but also the direction must be stated. Force is therefore a vector quantity.Speed is the distance travelled in unit time, it is not a vector, but velocity in a vector, since it refers to a speed in a given direction. Therefore velocity signifies both speed and a given direction.

Addition of Forces. Vectors can be drawn graphically by representing each force by a line, the direction of the line being the line along which the force acts. The length of the line represents the magnitude of the force. The arrow head is always placed at the end of the line to indicate the direction in which the force acts.

Any action produced on a body by two forces at the same time will be exactly the same as if a single force whose magnitude is the sum of the other two force. This single force is called the
*
resultant
*
of the other two forces. The two forces are called components.

A diagram is used to show the components of a vector.

If two forces act on a point but their lines of action are not identical, then a diagram is constructed so that each of the two forces is acting away from the point representing the point on which the forces are acting. A line parallel to the other force is drawn through the opposite end of each line, forming a parallelogram. The diagonal line is called the resultant. This resultant force has the same effect on a pivot as the two original forces.