American Flyers is happy to provide Chapter 2 from the Instrument Flying Handbook, an FAA Publication

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Chapter 2

Factors

Introduction

This chapter outlines the factors affecting aircraft perfor-mance as a result of aerodynamics, including a review of basic aerodynamics, the atmosphere, and the effects of icing. Pilots need an understanding of these factors for a sound basis for prediction of aircraft response to control inputs, especially with regard to instrument approaches, while holding, and when operating at reduced airspeed in instrument meteorological conditions (IMC). Although these factors are important to the visual flight rules (VFR) pilot, they must be even more thoroughly understood by the pilot operating under instrument flight rules (IFR). Instrument pilots rely strictly on instrument indications to precisely control the aircraft; therefore, they must have a solid understanding of basic aerodynamic principles in order to make accurate judgments regarding aircraft control inputs.

Review of Basic Aerodynamics

As an instrument pilot, you must understand the relationship and differences between the aircraft’s flightpath, angle of attack, and pitch attitude. Also, it is crucial to understand how the aircraft will react to various control and power changes because the environment in which instrument pilots fly has inherent hazards not found in visual flying. The basis for this understanding is found in the four forces and Newton’s laws.

Flightpath: The line, course, or Angle of attack: The acute angle track along which an aircraft is formed between the chord line of an flying or is intended to be flown. airfoil and the direction of the air

that strikes the airfoil.

The four basic forces acting upon an aircraft in flight are: lift, weight, thrust, and drag. The aerodynamic forces produced by the wing create lift. A byproduct of lift is induced drag. Induced drag combined with parasite drag (which is the sum of form drag, skin friction, and interference drag) produce the total drag on the aircraft. Thrust must equal total drag in order to maintain speed.

Lift must overcome the total weight of the aircraft, which is comprised of the actual weight of the aircraft plus the tail-down force used to control the aircraft’s pitch attitude. Understanding how the aircraft’s thrust/drag and lift/weight relationships affect its flightpath and airspeed is essential to proper interpretation of the aircraft’s instruments, and to making proper control inputs.

Newton’s First Law

Newton’s First Law of Motion is the Law of Inertia, which states that a body in motion will remain in motion, in a straight line, unless acted upon by an outside force. Two outside forces are always present on an aircraft in flight: gravity and drag. Pilots use pitch and thrust controls to counter these forces to maintain the desired flightpath. If a pilot reduces power while in straight-and-level flight, the aircraft will slow. A reduction of lift will cause the aircraft to begin a descent. [Figure 2-1]

Induced drag: Caused by the same Parasite drag: Caused by the friction factors that produce lift, its amount of air moving over the structure, its varies inversely with airspeed. As amount varies directly with the airspeed decreases, the angle of airspeed. The higher the airspeed, the attack must increase, and this greater the parasite drag. increases induced drag.

Newton’s Second Law

Newton’s Second Law of Motion is the Law of Momentum, which states that a body will accelerate in the same direction as the force acting upon that body, and the acceleration will be directly proportional to the net force and inversely proportional to the mass of the body. This law governs the aircraft’s ability to change flightpath and speed, which are controlled by attitude (both pitch and bank) and thrust inputs. Speeding up, slowing down, entering climbs or descents, and turning are examples of accelerations that pilots control in everyday flight. [Figure 2-2]

Newton’s Third Law

Newton’s Third Law of Motion is the Law of Reaction, which states that for every action there is an equal and opposite reaction. As shown in figure 2-3, the action of the jet engine’s thrust or the pull of the propeller lead to the reaction of the aircraft’s forward motion. This law is also responsible for a portion of the lift that is produced by a wing, by the downward deflection of the airflow around it. This downward force of the relative wind results in an equal but opposite (upward) lifting force created by the airflow over the wing.

Relative wind: The direction from which the wind meets an airfoil.

Atmosphere

Air density is a result of the relationship between temperature and pressure. This relationship is such that density is inversely related to temperature and directly related to pressure. For a constant pressure to be maintained as temperature increases, density must decrease, and vice versa. For a constant temperature to be maintained as pressure increases, density must increase, and vice versa. These relationships provide a basis for understanding instrument indications and aircraft performance.

Standard Atmosphere

The International Civil Aviation Organization (ICAO) established the ICAO Standard Atmosphere as a way of creating an international standard for reference and compu-tations. Instrument indications and aircraft performance specifications are derived using this standard as a reference. Because the standard atmosphere is a derived set of conditions that rarely exist in reality, pilots need to understand how devi-ations from the standard affect both instrument indications and aircraft performance.

In the standard atmosphere, sea level pressure is 29.92" Hg and the temperature is 15 °C (59 °F). The standard lapse rate for pressure is approximately a 1" Hg decrease per 1,000 feet increase in altitude. The standard lapse rate for temperature is a 2 °C (3.6 °F) decrease per 1,000 feet increase, up to the tropopause. Since all aircraft performance is compared and evaluated in the environment of the standard atmosphere, all aircraft performance instrumentation is calibrated for the standard atmosphere. Because the actual operating conditions rarely, if ever, fit the standard atmos-phere, certain corrections must apply to the instrumentation and aircraft performance.

Pressure Altitude

There are two measurements of the atmosphere that pilots must understand: pressure altitude and density altitude. Pressure altitude is the height above the standard datum pressure (29.92" Hg) and is used for standardizing altitudes for flight levels (FL) and for calculations involving aircraft performance. If the altimeter is set for 29.92" Hg, the altitude indicated is the pressure altitude.

Standard Atmosphere

ICAO Standard Atmosphere at sea level is 15 °C and 29.92" Hg. Most small aircraft manuals use this as a reference for their perfor-mance charts.

Flight level (FL): A measure of altitude used by aircraft flying above 18,000 feet.

Density Altitude

Density altitude is pressure altitude corrected for nonstandard temperatures, and is used for determining aerodynamic performance in the nonstandard atmosphere. Density altitude increases as the density decreases. Since density varies directly with pressure, and inversely with temperature, a wide range of temperatures may exist with a given pressure altitude, which allows the density to vary. However, a known density occurs for any one temperature and pressure altitude combination. The density of the air has a significant effect on aircraft and engine performance. Regardless of the actual altitude an aircraft is operating, its performance will be as though it were operating at an altitude equal to the existing density altitude.

Lift

Lift always acts in a direction perpendicular to the relative wind and to the lateral axis of the aircraft. The fact that lift is referenced to the wing, not to the Earth’s surface, is the source of many errors in learning flight control. Lift is not always “up.” Its direction relative to the Earth’s surface changes as you maneuver the aircraft.

The magnitude of the force of lift is directly proportional to the density of the air, the area of the wings, and the airspeed. It also depends upon the type of wing and the angle of attack. Lift increases with an increase in angle of attack up to the stalling angle, at which point it decreases with any further increase in angle of attack. In conventional aircraft, lift is therefore controlled by varying the angle of attack (attitude) and thrust.

Pitch/Power Relationship

An examination of figure 2-4 provides insight into the relationship between pitch and power when it comes to controlling flightpath and airspeed. In order to maintain a constant lift, when the airspeed is reduced, the pitch must be increased. The pilot controls pitch through the elevators, which in effect controls the angle of attack. When back pressure is applied on the elevator control, the tail lowers and the nose rises, thus increasing the wing’s angle of attack and lift.

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Safety Reminder

As density altitude increases, performance decreases—be aware on hot days at high altitudes.

Thrust is controlled by using the throttle to establish or maintain desired airspeeds. The most precise method of controlling flightpath is to use pitch control while simul-taneously using power (thrust) to control airspeed. In order to maintain a constant lift, a change in pitch will require a change in power, and vice versa.

If you want the aircraft to accelerate while maintaining altitude, thrust must be increased to overcome drag. As the aircraft speeds up, lift is increased. To keep from gaining altitude, you must lower the pitch to reduce the angle of attack. If you want the aircraft to decelerate while maintaining altitude, thrust must be decreased. As the aircraft slows down, lift is reduced. Then you must increase the pitch in order to increase the angle of attack and maintain altitude.

Drag Curves

When induced drag and parasite drag are plotted on a graph, the total drag on the aircraft appears in the form of a “drag curve.” [Figure 2-5] Graph A of figure 2-5 shows a curve based on thrust versus drag, which is primarily used for jet aircraft. Graph B of figure 2-5 is based on power versus drag, and it is used for propeller-driven aircraft. This chapter focuses on power versus drag charts for propeller-driven aircraft.

Understanding the drag curve can provide valuable insight into the various performance parameters and limitations of the aircraft. Because power must equal drag to maintain a steady airspeed, the curve can be either a drag curve or a “power-required curve.” The power-required curve represents the amount of power needed to overcome drag in order to maintain a steady speed in level flight.

The propellers used on most reciprocating engines achieve peak propeller efficiencies in the range of 80 to 88 percent. As airspeed increases, the propeller efficiency will increase until it reaches its maximum. Any airspeed above this maximum point will cause a reduction in propeller efficiency. An engine that produces 160 horsepower will have only about 80 percent of that power converted into available horsepower, approximately 128 horsepower. This is the reason the thrust-and power-available curves change with speed.

Regions of Command

The drag curve also illustrates the two regions of command: the region of normal command, and the region of reversed command. The term “region of command” refers to the relationship between speed and the power required to maintain or change that speed. “Command” refers to the input the pilot must give in terms of power or thrust to maintain a new speed.

The “region of normal command” occurs where power must be added to increase speed. This region exists at speeds higher than the minimum drag point primarily as a result of parasite drag. The “region of reversed command” occurs where additional power is needed to maintain a slower airspeed. This region exists at speeds slower than the minimum drag point (L/DMAX on the thrust-required curve, figure 2-5) and is primarily due to induced drag. Figure 2-6 shows how one power setting can yield two speeds, points 1 and 2. This is because at point 1 there is high induced drag and low parasite drag, while at point 2 there is high parasite drag and low induced drag.

Control Characteristics

Most flying is conducted in the region of normal command: for example, cruise, climb, and maneuvers. The region of reversed command may be encountered in the slow-speed phases of flight during takeoff and landing; however, for most

Regions of command: The Static longitudinal stability: The relationship between speed and the aerodynamic pitching moments power required to maintain or change required to return the aircraft to the that speed in flight. equilibrium angle of attack.

general aviation aircraft, this region is very small and is below normal approach speeds.

Flight in the region of normal command is characterized by a relatively strong tendency of the aircraft to maintain the trim speed. Flight in the region of reversed command is characterized by a relatively weak tendency of the aircraft to maintain the trim speed. In fact, it is likely the aircraft will exhibit no inherent tendency to maintain the trim speed in this area. For this reason, you must give particular attention to precise control of airspeed when operating in the slow-speed phases of the region of reversed command.

Operation in the region of reversed command does not imply that great control difficulty and dangerous conditions will exist. However, it does amplify errors of basic flying technique—making proper flying technique and precise control of the aircraft very important.

Speed Stability

Normal Command

The characteristics of flight in the region of normal command are illustrated at point A on the curve in figure 2-7. If the aircraft is established in steady, level flight at point A, lift is equal to weight, and the power available is set equal to the power required. If the airspeed is increased with no changes to the power setting, a power deficiency exists. The aircraft will have the natural tendency to return to the initial speed to balance power and drag. If the airspeed is reduced with no changes to the power setting, an excess of power exists. The aircraft will have the natural tendency to speed up to regain the balance between power and drag. Keeping the aircraft in proper trim enhances this natural tendency. The static longitudinal stability of the aircraft tends to return the aircraft to the original trimmed condition.

An aircraft flying in steady, level flight at point C is in equilibrium. [Figure 2-7] If the speed were increased or decreased slightly, the aircraft would tend to remain at that speed. This is because the curve is relatively flat and a slight change in speed will not produce any significant excess or deficiency in power. It has the characteristic of neutral stability; the aircraft’s tendency is to remain at the new speed.

Slow Airspeed Safety Hint

Be sure to add power before pitching up while at slow airspeeds to prevent losing airspeed.

Reversed Command

The characteristics of flight in the region of reversed command are illustrated at point B on the curve in figure 2-7. If the aircraft is established in steady, level flight at point B, lift is equal to weight, and the power available is set equal to the power required. When the airspeed is increased greater than point B, an excess of power exists. This causes the aircraft to accelerate to an even higher speed. When the aircraft is slowed to some airspeed lower than point B, a deficiency of power exists. The natural tendency of the aircraft is to continue to slow to an even lower airspeed.

This tendency toward instability happens because the variation of excess power to either side of point B magnifies the original change in speed. Although the static longitudinal stability of the aircraft tries to maintain the original trimmed condition, this instability is more of an influence because of the increased induced drag due to the higher angles of attack in slow-speed flight.

Trim

A trim tab is a small, adjustable hinged surface, located on the trailing edge of the aileron, rudder, or elevator control surface. It is used to maintain balance in straight-and-level flight and during other prolonged flight conditions so the pilot does not have to hold pressure on the controls. This is accomplished by deflecting the tab in the direction opposite to that in which the primary control surface must be held.

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Reversed Logic

In the region of reversed command, as you slow down you require more power.

The force of the airflow striking the tab causes the main control surface to be deflected to a position that will correct the unbalanced condition of the aircraft.

Because the trim tabs use airflow to function, trim is a function of speed. Any change in speed will result in the need to retrim the aircraft. A properly trimmed aircraft seeks to return to the original speed before the change. Therefore, it is very important for instrument pilots to keep the aircraft in constant trim. This will reduce the workload significantly and allow pilots to tend to other duties without compromising aircraft control.

Slow-Speed Flight

Anytime you are flying near the stalling speed or the region of reversed command, such as in final approach for a normal landing, the initial part of a go-around, or maneuvering in slow flight, you are operating in what is called slow-speed flight. It is characterized by high angles of attack and in many cases, the need for flaps or other high-lift devices.

Small Airplanes

Most small airplanes maintain a speed well in excess of 1.3 times VSO on an instrument approach. An airplane with a stall speed of 50 knots (VSO) has a normal approach speed of 65 knots. However, this same airplane may maintain 90 knots

(1.8 VSO) while on the final segment of an instrument approach. The landing gear will most likely be extended at the beginning of the descent to the minimum descent altitude, or upon intercepting the glide slope of the instrument landing system. The pilot may also select an intermediate flap setting for this phase of the approach. The airplane at this speed will have good positive speed stability, as represented by point A on figure 2-7. Flying at this point, you can make slight pitch changes without changing power settings, and accept minor speed changes knowing that when the pitch is returned to the initial setting, the speed will return to the original setting. This reduces your workload.

You would usually slow down to a normal landing speed when on a relatively short final. When you slow the airplane to 65 knots, 1.3 VSO, the airplane will be close to point C. [Figure 2-7] At this point, precise control of the pitch and power becomes more crucial for maintaining the correct speed. Pitch and power coordination is necessary because the speed stability is relatively neutral—the speed tends to remain at the new value and not return to the original setting. In addition to the need for more precise airspeed control, you would normally change the aircraft’s configuration by adding landing flaps. This configuration change means you must guard against unwanted pitch changes at a low altitude.

If you allow the speed to slow several knots, the airplane could enter the region of reversed command. At this point, the airplane could develop an unsafe sink rate and continue to lose speed if you do not take prompt, corrective action. Proper pitch and power coordination is critical in this region due to speed instability and the tendency of increased divergence from the desired speed.

Large Airplanes

Pilots of larger airplanes with higher stall speeds may find the speed they maintain on the instrument approach is near

1.3 VSO, putting them near point C (in figure 2-7) the entire time the airplane is on the final approach segment. In this case, precise speed control is necessary throughout the approach. It may be necessary to overpower or underpower in relation to the target power setting in order to quickly correct for airspeed deviations.

For example, a pilot is on an instrument approach at 1.3 VSO, a speed near L/DMAX, and knows that a certain power setting will maintain that speed. The airplane slows several knots below the desired speed because of a slight reduction in the power setting. The pilot increases the power slightly, and the airplane begins to accelerate, but at a slow rate. Because the airplane is still in the “flat part” of the drag curve, this slight increase in power will not cause a rapid return to the desired speed. The pilot may need to increase the power higher than normally needed to maintain the new speed, allow the airplane to accelerate, then reduce the power to the setting that will maintain the desired speed.

Climbs

The ability for an aircraft to climb depends upon an excess power or thrust over what it takes to maintain equilibrium. Excess power is the available power over and above that required to maintain horizontal flight at a given speed. Although the terms power and thrust are sometimes used interchangeably (erroneously implying they are synonymous), distinguishing between the two is important when considering climb performance. Work is the product of a force moving through a distance and is usually independent of time. Power implies work rate or units of work per unit of time, and as such is a function of the speed at which the force is developed. Thrust also a function of work, means the force which imparts a change in the velocity of a mass.

For a given weight of the aircraft, the angle of climb depends on the difference between thrust and drag, or the excess thrust. When the excess thrust is zero, the inclination of the flightpath is zero, and the aircraft will be in steady, level flight. When thrust is greater than drag, the excess thrust will allow a climb angle depending on the amount of excess thrust. When thrust is less than drag, the deficiency of thrust will induce an angle of descent.

Acceleration in Cruise Flight

Aircraft accelerate in level flight because of an excess of power over what is required to maintain a steady speed. This is the same excess power used to climb. When you reach the desired altitude and lower the pitch to maintain that altitude, the excess power can now accelerate the aircraft to its cruise speed. Reducing power too soon after level-off will result in a longer period of time to accelerate.

Turns

Like any moving object, an aircraft requires a sideward force to make it turn. In a normal turn, this force is supplied by banking the aircraft in order to exert lift inward as well as upward. The force of lift is separated into two components at right angles to each other. [Figure 2-8] The upward-acting lift and the opposing weight together become the vertical lift component. The horizontally-acting lift and its opposing centrifugal force are the horizontal lift component, or centripetal force. This horizontal lift component is the sideward force that causes an aircraft to turn. The equal and opposite reaction to this sideward force is centrifugal force, which is merely an apparent force as a result of inertia.

The relationship between the aircraft’s speed and bank angle to the rate and radius of turns is important for instrument pilots to understand. You can use this knowledge to properly estimate bank angles needed for certain rates of turn, or for figuring how much to lead when intercepting a course.

Work: A physical measurement of Thrust (aerodynamic force): The force used to produce movement. forward aerodynamic force produced

by a propeller, fan, or turbojet engine Power (mechanical): Work done in as it forces a mass of air to the rear, a period of time. behind the aircraft.

Rate of Turn

The rate of turn, normally measured in degrees per second, is based upon a set bank angle at a set speed. If either one of these elements changes, then the rate of turn will change. If the aircraft increases its speed without changing the bank angle, then the rate of turn will decrease. Likewise, if the speed decreases without changing the bank angle, the rate of turn will increase.

Changing the bank angle without changing speed will also cause the rate of turn to change. Increasing the bank angle without changing speed will increase the rate of turn, while decreasing the bank angle will reduce the rate of turn.

The standard rate of turn, 3° per second, is used as the main reference for bank angle. Therefore, you must understand how the angle of bank will vary with speed changes, such as slowing down for holding or an instrument approach. Figure 2-9 shows the turn relationship with reference to a constant bank angle or a constant airspeed, and the effects on rate of turn and radius of turn.

Radius of Turn

The radius of turn will vary with changes in either speed or bank. If the speed is increased without changing the bank angle, the radius of turn will increase, and vice versa. If the speed is constant, increasing the bank angle will reduce the radius of turn, while decreasing the bank angle will increase the radius of turn. This means that intercepting a course at a higher speed will require more distance, and therefore, require a longer lead. If the speed is slowed considerably in preparation for holding or an approach, a shorter lead is needed than that required for cruise flight.

Coordination of Rudder and Aileron Controls

Anytime ailerons are used, adverse yaw is produced. This yaw causes the nose of the aircraft to initially move in the direction opposite of the turn. Correcting for this yaw with rudder, when entering and exiting turns is necessary for precise control of the aiplane when flying on instruments. You can tell if the turn is coordinated by checking the ball in the turn-and-slip indicator or the turn coordinator.

As you bank the wings to enter the turn, a portion of the wing’s vertical lift becomes the horizontal component; therefore, without an increase in back pressure, the aircraft will lose altitude during the turn. The loss of vertical lift can be offset by increasing the pitch in one-half bar width increments. Trim may be used to relieve the control pressures; however, if used, it will have to be removed once the turn is complete.

In a slipping turn, the aircraft is not turning at the rate appropriate to the bank being used, and the aircraft falls to the inside of the turn. The aircraft is banked too much for the rate of turn, so the horizontal lift component is greater than the centrifugal force. A skidding turn results from excess of centrifugal force over the horizontal lift component, pulling the aircraft toward the outside of the turn. The rate of turn is too great for the angle of bank, so the horizontal lift component is less than the centrifugal force.

The ball instrument indicates the quality of the turn, and should be centered when the wings are banked. If the ball is out of its cage on the side toward the turn, the aircraft is slipping and you should add rudder pressure on that side to increase the rate of turn, and adjust the bank angle as required. If the ball is out of its cage on the side away from the turn, the aircraft is skidding and rudder pressure toward the turn should be relaxed or the bank angle increased. If the aircraft is properly rigged, the ball should be in the center when the wings are level; use rudder and/or aileron trim if available.

Adverse yaw: A flight condition at Cage: The black markings on the the beginning of a turn in which the ball instrument indicating its neutral nose of the aircraft starts to move in position. the direction opposite the direction the turn is being made.

The increase in induced drag (caused by the increase in angle of attack necessary to maintain altitude) will result in a minor loss of airspeed if the power setting is not changed. Accept this loss of airspeed—an attempt to maintain airspeed may divert your attention at a critical time.

Load Factor

Any force applied to an aircraft to deflect its flight from a straight line produces a stress on its structure; the amount of this force is termed load factor. A load factor is the ratio of the aerodynamic force on the aircraft to the gross weight of the aircraft (e.g., lift/weight). For example, a load factor of 3 means the total load on an aircraft’s structure is three times its gross weight. When designing an aircraft, it is necessary to determine the highest load factors that can be expected in normal operation under various operational situations. These “highest” load factors are called “limit load factors.”

Aircraft are placed in various categories, i.e., normal, utility, and acrobatic, depending upon the load factors they are designed to take. For reasons of safety, the aircraft must be designed to withstand certain maximum load factors without any structural damage.

The specified load may be expected in terms of aerodynamic forces, as in turns. In level flight in undisturbed air, the wings are supporting not only the weight of the aircraft, but centrifugal force as well. As the bank steepens, the horizontal

Load factor: Lift to weight ratio.

lift component increases, centrifugal force increases, and the load factor increases. If the load factor becomes so great that an increase in angle of attack cannot provide enough lift to support the load, the wing stalls. Since the stalling speed increases directly with the square root of the load factor, you should be aware of the flight conditions during which the load factor can become critical. Steep turns at slow airspeed, structural ice accumulation, and vertical gusts in turbulent air can increase the load factor to a critical level.

Effects of Icing

One of the hazards to flight is aircraft icing. Pilots should be aware of the conditions conducive to icing, the types of icing, the effects of icing on aircraft control and performance, and the use and limitations of aircraft deice and anti-ice equipment.

Structural icing refers to the accumulation of ice on the exterior of the aircraft; induction icing affects the powerplant operation. Significant structural icing on an aircraft can cause aircraft control and performance problems. To reduce the probability of ice buildup on the unprotected areas of the aircraft, you should maintain at least the minimum airspeed for flight in sustained icing conditions. This airspeed will be listed in the Pilot’s Operating Handbook/Airplane Flight Manual (POH/AFM).

The most hazardous aspect of structural icing is its aerodynamic effects. [Figure 2-10] Ice can alter the shape of an airfoil, which can cause control problems, change the angle of attack at which the aircraft stalls, and cause the aircraft to stall at a significantly higher airspeed. Ice can reduce the amount of lift an airfoil will produce and greatly increase drag. It can partially block or limit control surfaces which will limit or make control movements ineffective. Also, if the extra weight caused by ice accumulation is too great, the aircraft may not be able to become airborne and, if in flight, the aircraft may not be able to maintain altitude. Any accumulation of ice or frost should be removed before attempting flight.

Another hazard of structural icing is the possible uncommanded and uncontrolled roll phenomenon, referred to as roll upset, associated with severe in-flight icing. Pilots flying aircraft certificated for flight in known icing conditions should be aware that severe icing is a condition outside of the aircraft’s certification icing envelope. Roll upset may be caused by airflow separation (aerodynamic stall) which induces self-deflection of the ailerons and loss of or degraded roll-handling characteristics. These phenomena can result from severe icing conditions without the usual symptoms of ice accumulation or a perceived aerodynamic stall.

Negative pressure

 

 

 

 

 

 

 

 

 

 

 

 

American Flyers is happy to provide Chapter 2 from the Instrument Flying Handbook, an FAA Publication

Chapter 1Chapter 2Chapter 3Chapter 4Chapter 5Chapter 6Chapter 7Chapter 8Chapter 9Chapter 10Chapter 11

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