Introduction to aeronautics a design perspective pdf download

The most exciting moment for an aeronautical engineer is when his or her design becomes a working aircraft, the endpoint of a journey that begins in the classroom. Add to Wishlist. They learn through the use of specific analytical principles, practical examples, and case studies, with corresponding problems to solve. Notify your administrator of your interest. Stiles, And J. The wind, according to our anemometers at this time was blowing a little over 20 miles; 27 miles according to the government anemometer at Kitty Hawk.

The phenomenon was not well understood at the time, but it was observed that fighter planes with thinner airfoils could fly faster before encountering the problem. Two similar fighter aircraft produced by the same company exemplify the effect. The Hawker Typhoon and Tempest fighters had the same engine and fuselage, but the Typhoon had a smaller wing with a greater thickness-to-chord ratio. As maximum speeds of fighter aircraft have continued to increase, their airfoils have gotten progressively thinner, so that the thin, highly cambered airfoil sections of the outer wing panel of the F are similar though far from identical to the airfoils of the World War I Sopwith Camel!

To understand how this works, consider the untapered, swept wing in Fig. Sweeping the wing without changing its shape increases the effective chord length.

Increasing the airfoil chord length without changing thickness lowers the airfoil's thickness-to-chord ratio. This in turn reduces the amount the flow must speed up to get past the airfoil.

Mcrit therefore increases because, if the flow does not accelerate as much, the freestream Mach number can be higher before flow over the airfoil reaches the speed of sound. Practical supersonic airfoil shapes also generate minimum drag at zero lift and zero angle of attack. For well-designed supersonic aircraft, the drag-due-to-lift parameter k increases smoothly to very high supersonic values, sometimes exceeding 0.

This practice reduces the aircraft's wave drag because Shock Cone Fig. Also, if a wing tip or other component projects outside the shock cone it will generate an additional shock wave. This further increases total wave drag on the aircraft and, where the two shock-waves intersect or interfere, causes additional disruption to the flow.

In the case of very high Mach numbers, this shock-wave interaction contributes to additional heating of the aircraft's skin, which can lead to structural damage or failure.

Example 4. Its maximum speed was a carefully kept secret. Yet, the practice of designing every part of an aircraft to fit inside the shock cone generated by its nose is especially important for aircraft like the SR that fly at very high Mach numbers.

At those speeds, impingement of a shock wave on a wing leading edge could cause excessive heating i. The SR designers must have obeyed this rule.

So, with a picture of the SR like Fig. Assuming this is the same as the Mach angle at its maximum speed, we solve Eq. Effective free stream direction in vicinity of the wing Downwash Downwash leads to less lift because angle of attack is reduced and more drag because lift vector tilts.

The phenomenon of sound is caused by the propagation of pressure waves through a medium. The speed of propagation depends on the medium air, water, etc. The abbreviation used is a. Because temperature decreases in the standard atmosphere up to about 37, ft, the speed of sound decreases as you ascend in this region.

This partially explains why an aircraft can attain a higher Mach number at higher altitudes. The speed of sound at sea level is about kn, whereas it is only about kn at 40, ft. Mcrjt is always less than 1. As the aircraft accelerates beyond Mcrit, shock waves form on the wings, fuselage, and other surfaces. A shock wave is a thin, highly viscous region that sharply increases the flow's static pressure but decreases the flow's velocity and total pressure. Because cambered airfoils generate more wave drag in supersonic flight than do symmetrical airfoils, the additional term in Eq.

The flapped area for the trailingedge flaps, as defined in Fig. Aspect ratios of the wing and horizontal tail are S ' , ' S, Then, using Eq. But this is before the effect of the strakes on the wing is included. Most of the additional drag in a turn results from additional induced drag caused by the additional lift required to turn.

Recall from Eq. This can make it very difficult for many aircraft to sustain turns at high load factors. Example 5. How do their load factors, turn radii, and rates of turn compare?

The turn rates are calculated using Eq. The simplest of these is a turn made purely in a vertical plane. Such a maneuver completed through deg of turn to the original flight conditions is called a loop. Figure 5. The maneuver is started with a pull-up, a vertical turn from initial straight and level conditions. At the top of the loop, the aircraft is performing a pull-down from the inverted flight condition.

A pull-down can also be entered from straight-and-level flight by rolling changing the bank angle 0 the aircraft until it is inverted. A vertical turn initiated by rolling inverted and then pulling down, completing the turn to level flight headed in the opposite direction is known as a split-S.

For modern fighter aircraft that routinely use load factors of nine, Eqs. As the nose comes up, the speed begins to bleed off, and at about KIAS or a little less you start to feel a tickle in the stick. This tickle is the first signs of stall, and it occurs before the drag due to lift wing-tip vortices and flow separation gets too large.

If you pull into heavy buffet at this point, the excessive drag will cause the aircraft to slow down too much, and you will not make it over the top. But, if you pull any less than "on the tickle" your load factor will not be enough, and you also will not make it over the top. So, you pull on the tickle, relaxing backpressure on the stick as the aircraft slows down. As the pitch attitude exceeds 90 deg, you throw your head back and look to find the horizon, and then pull down to it being careful to keep the wings level.

If you do everything correctly, the speed over the top is about kn, and the load factor is about 1. Then, as you continue to pull down inverted the speed builds up, as does the backpressure and load factor required to stay on the tickle.

When the load factor reaches 4, you hold that until you complete the maneuver. From the cockpit, a T loop flown as just described looks like a circle, as shown in Fig. But what would it look like from the ground or from another aircraft.

Use the information on Fig. Solution: Equation 5. Now, with a nearly ft radius the aircraft will climb nearly ft as it executes the first quarter of the loop, so that its altitude when it is going straight vertical will be close to 18, ft.

Now, the radius decreased so much in this first part of the loop that we predict the aircraft will only climb another ft or less as it executes the second quarter of the loop, so that its altitude when it is going over the top will be close to 19, ft. So, we predict that a T loop would look like Fig. These limitations are often summarized on a chart known as a V-n diagram. The maximum positive and negative load factors that the aircraft structure can sustain are shown as horizontal lines on the chart because for this particular aircraft these structural limits are not functions of velocity.

Equation 5. The vertical line on Fig. In this case, the maximum structural airspeed is not a function of load factor. On many aircraft, maximum structural speed decreases at high positive and negative load factors. The feature of the aircraft that sets the maximum speed varies. For the aircraft of Fig. Flight above this speed is prohibited because shock-induced separation causes control difficulties. For other aircraft, the limit is set by the structural strength required by the wings, windscreen, etc.

For many high-speed aircraft the maximum speed is actually a temperature limit because at faster speeds skin friction and shock waves generate so much heat that the aircraft skin will melt!

For the F, the q limit is set by maximum engine inlet temperature. From Eq. The velocity labeled corner velocity on Fig. This velocity satisfies the conditions for quickest, tightest turn because a faster velocity would not see an increase in n as a result of the structural limit, and a slower velocity would see n limited to less than its maximum value by the stall. The term "corner velocity" might also have been chosen to reflect the fact that the aircraft makes its sharpest corner at that speed.

What is its corner velocity at sea level? Solution: Corner velocity is calculated using Eq. Specific excess power is a measure of an aircraft's ability to increase its specific energy He, the sum of its kinetic and potential energy divided by its weight. Energy is changed by doing work, raising an object against the pull of gravity to a higher altitude, accelerating an object to a faster velocity, or both.

The rate of doing work is power, but only part of an aircraft's power can be used for this. A portion of an aircraft's power available must be used to balance the power required to overcome the aircraft's drag. The work done by this portion of the aircraft's power is converted by air viscosity into heat and air turbulence. The aircraft's power available that is in excess of its power required is its excess power. It can be used to verify that a given aircraft design meets a specific Ps requirement, such as the one listed in Table 1.

It is more common for Ps to be calculated and plotted for an aircraft for a range of altitudes and Mach numbers to create a Ps diagram. Note that the Ps values are indicated by contour lines, lines connecting points with equal values of Ps. Lines of constant energy height are also plotted on the diagram.

For all combinations of altitude and Mach number inside this envelope, the aircraft has sufficient thrust to sustain level flight. This can be done either by diving so that, as in a glide, a component of weight acts opposite the drag to reach airspeeds above its maximum level-flight speed or by performing a zoom or zoom climb as in "zooming to meet our thunder" in the U.

Air Force Song. A zoom climb occurs when an aircraft climbs so as to convert airspeed into altitude. If the aircraft whose Ps diagram is shown in Fig. The maximum rate of climb at any given altitude is achieved at the speed where Ps is maximum. However, because the aircraft can be zoomed at the end of its climb to get to a particular altitude faster, the minimum time to climb is achieved by changing energy height, not just height, as fast as possible.

A trajectory is shown on Fig. Note that Ps has decreased everywhere on the diagram. This is because more induced drag and therefore more power required result from the five times greater lift required to generate a load factor n — 5. What causes this limit? Solution: All of the answers to these questions can be determined by looking at the Ps diagrams.

On Fig. The speed depicted on Fig. In reality, this limit could be caused by engine inlet limitations, aircraft skin temperature limits, or even just the fact that the aircraft has not been flight tested beyond this limit. When test pilots "push the edge of the envelope," they are demonstrating that the aircraft is safe to fly in areas of the flight envelope that are achievable but have not been demonstrated yet. Note that the q limit in Fig. Insufficient thrust prevents sustaining 5 g at a higher speed.

This speed is limited by stall, buffet, or maximum usable angle of attack. It was created by calculating the differences between the Ps values of two different fighter aircraft, aircraft A and aircraft B, at each point on a Ps diagram.

Regions of the diagram with similar values of Ps differences are shaded the same. The level flight envelope for aircraft A is shown in black and for aircraft B in gray. At very high Mach numbers, aircraft A has exclusive use of a range of velocities and altitudes that are outside aircraft B's level flight envelope.

Likewise, at low speeds aircraft B has an advantage. Aircraft B has exclusive use of a range of very low speeds and high altitudes. Comparative Ps diagrams such as this are very useful to fighter pilots as they plan how to conduct an aerial battle against an adversary aircraft of a particular type.

In the case shown in Fig. At the same time, the pilot of aircraft B would attempt to keep the fight high and slow to lower speeds, where that aircraft has the advantage. Similar diagrams made for higher load factors are also used because most extended aerial fights involve a great deal of turning. The following notes hit the very "high points" of performance. As you review them, if the meanings of the symbols and short comments do not immediately come to your mind, go to the appropriate section in this chapter, and read more details.

The diagram is made for a fixed aircraft weight, configuration, and altitude, and so load factor is a variable. The major axes of the chart are airspeed or Mach number and turn rate. Maneuverability diagrams for different aircraft are often compared in much the same way Ps diagrams are compared.

An interesting result clearly shown on Fig. For aircraft with corner velocities well below their critical Mach number, so that Cimax is constant all along the stall boundary, minimum turn radius occurs at corner velocity along with maximum turn rate.

The minimum instantaneous turn radius actually occurs at a much lower velocity, although the approximation made by saying it also occurs at corner velocity is quite good. The actual minimum instantaneous turn radius occurs where the aerodynamic limit line extends farthest toward the upper-left corner of the diagram and crosses the lowest-valued constant turn radius line.

The methods discussed up to this point in this text enable the engineer to take an existing aircraft design, estimate its aerodynamics and thrust characteristics, and predict its performance capabilities. The challenge for aircraft designers is to turn this process around and use the analysis methods to design an aircraft that will have the desired performance capabilities.

Table 1. But how does an aircraft designer know how to design an aircraft to meet those requirements. There are so many interrelated variables to control and choices to make that aircraft designers use an analysis method called constraint analysis to narrow down the choices and help them focus on the most promising concepts.

In many cases, constraint analysis will eliminate some aircraft concepts from further consideration. In other instances, constraint analysis will identify two conflicting design requirements that no single aircraft configuration can satisfy. All other variables in Eq. For instance, one of the design requirements in Table 1. For this constraint, the climb and acceleration terms in Eq. Another requirement in Table 1.

In this case, the two right-hand terms of Eq. For each requirement, Eq. When several constraint lines are plotted on a single set of axes, a constraint diagram like Fig. Performing a constraint analysis allows an aircraft designer to make much more intelligent choices about aircraft configuration, engine size, etc.

Constraint analysis is always an approximation because it depends so heavily on accurate predictions of the aerodynamic characteristics of an aircraft that is not yet built!

The wise designer will choose a design point that is a small amount above or away from all of the constraint lines, so that the final product will still meet all the requirements even if its aerodynamics differ from the original predictions. In Fig. The solution space can be more correctly described as lying "inside" all of the boundaries set by the various constraint lines.

For the takeoff constraint, the takeoff distance equation, Eq. Thrust lapse and weight fraction are included in Eq. The discussion of takeoff performance in Sec. The assumption of zero lift prior to rotation is a conservative one, especially on soft or grassy fields. The actual takeoff time and distance can be slightly reduced to less than what is assumed in Eq. The predicted CLmaK — 1. When the nosewheel is lowered to the runway, the lift generated by the aircraft is effectively reduced to zero.

Using the predicted CLmax for landing of 1. Stinton, D. Based on your knowledge of the relationship between induced drag and aspect ratio and the relative importance of induced and parasite drag at low speeds, would you suggest a high or low aspect ratio for the wing of this plane? What problems do you expect this aircraft to have with takeoff and landing? Brainstorm five concepts for overcoming these problems. If the aircraft is flying at 5-deg angle of attack, calculate CL and CD.

Its wing area is Calculate the thrust required to fly at a velocity of KTAS at a standard sea level and b an altitude of 5 km. Given an lb T flying at 10, ft, determine the following: a thrust required TR at Mach 0.

Compare the result to the 30,ft data in Appendix B using the same Mach number. Its wing span is Assume thrust acts along the flight path. Using maximum thrust? What is its maximum endurance at 20, ft standard day? The T's wing area is ft 2 , and the coefficient of rolling friction for braking on the dry runway is 0. Answer the following questions: a What is the T's stall speed? Assume the stall speed is the same as for clean configuration. How will this affect the T's landing distance?

Its maximum lift coefficient with fullspan blown flaps fully extended is 2. After touchdown, the pilot lowers the nose immediately to the runway and retracts the flaps to achieve zero lift for the braking roll. What would happen to the aircraft load factor? Under the same flight conditions what would the 3-g stall speed be? What causes each of the limits? Can any point inside the V-n diagram be sustained?

Assume standard day. Assume you are both at 10, ft and kn. What do you think? You've just analyzed the F! See Fig. Purchased from American Institute of Aeronautics and Astronautics 6 Stability and Control "The balancing of a gliding or flying machine is very simple in theory. It merely consists in causing the center of pressure to coincide with the center of gravity. But in actual practice there seems to be an almost boundless incompatibility of temper which prevents their remaining peaceably together for a single instant, so that the operator, who in this case acts as peacemaker, often suffers injury to himself while attempting to bring them together.

Whereas performance analysis sums the forces on an aircraft, stability and control analysis requires summing the moments acting on it as a result of surface pressure and shear-stress distributions, engine thrust, etc. Stability analysis also deals with the changes in moments on the aircraft when it is disturbed from equilibrium, or in other words from the condition when all forces and moments on it sum to zero.

An aircraft that tends to drift away from its desired equilibrium condition, or that oscillates wildly about the equilibrium condition, is said to lack sufficient stability. The Wright brothers intentionally built their aircraft to be unstable because this made them more maneuverable.

As the preceding quotation from Wilbur Wright suggests, such an aircraft can be very difficult and dangerous to fly. Control analysis determines how the aircraft should be designed so that sufficient control authority sufficiently large moments generated when controls are used is available to allow the aircraft to fly all maneuvers and at all speeds required by the design specifications.

Good stability and control characteristics are as essential to the success of an aircraft as are good lift, drag, and propulsion characteristics. Anyone who has flown a toy glider that is out of balance or that has lost its tail surfaces, or who has shot an arrow or thrown a dart with missing tail feathers, knows how disastrous poor stability can be to flying.

Understanding stability and control and knowing how to design good stability characteristics into an aircraft are essential skills for an aircraft designer. This process must begin by defining quite a number of axes, angles, forces, moments, displacements, and rotations. Note that the vertical z axis is defined as positive downward! The reason for this choice is a desire to have consistent and convenient definitions for positive moments. Positive moment directions are defined consistent with the right-hand rule used in vector mathematics, physics, and mechanics.

This rule states that if the thumb of a person's right hand is placed parallel to an axis of a coordinate system, then the fingers of that hand will point in the positive direction of the moment about that axis.

Because the moment about the aerodynamic center of an airfoil or wing was defined in Chapter 3 as being positive in a nose-up direction, the right-hand rule requires that the lateral spanwise axis of the aircraft coordinate system be positive in the direction from the right wing root to the right wing tip. A natural starting point for the coordinate system is the aircraft's center of gravity because it will rotate about this point as it moves through the air.

The aircraft's longitudinal axis down its centerline is chosen parallel to and usually coincident with its aircraft reference line defined in Chapter 4 , but positive toward the aircraft's nose so that a moment tending to raise the left wing and lower the right wing is positive.

This axis is chosen as the jc axis to be consistent with performance analysis. Making x positive toward the front allows the aircraft's thrust and velocity to be taken as positive quantities. Because a rotation about the longitudinal axis to the right or clockwise is positive, for consistency it is desired that a moment or rotation about the aircraft's vertical axis such that the nose moves to the right be considered positive.

This requires that the vertical axis be positive downward so that the right-hand rule is satisfied. The y axis is generally taken as vertical in performance analysis, but an x, v, z coordinate system must satisfy another right-hand rule in order to be consistent with conventional vector mathematics. The right-hand rule for threedimensional orthogonal each axis perpendicular to the others coordinate systems requires that if the thumb of a person's right hand is placed along the coordinate system's x axis the fingers point in the shortest direction from the system's y axis to its z axis try this on Fig.

To satisfy this right-hand rule as well as all of the previous choices for positive directions, the coordinate system's y axis must be the aircraft's lateral axis positive out the right wing , and the z axis must be the vertical axis positive down. A coordinate system such as this, which has its origin at the aircraft center of gravity and is aligned with the aircraft reference line and lateral axis, is referred to as a body axis system.

For consistency with aerodynamic analysis, the nose-up moment is labeled M. Because M is the moment about the y axis, the moment about the x axis is labeled C and the moment about the z axis is labeled N, to make them easier to remember. Note that the symbol C is used instead of L to avoid confusion with aircraft lift. Forces on the aircraft can be broken into components along the jc, v, and z axes. These force components are labeled X, 7, and Z, respectively.

Note that the axis and moment symbol convention used in this book are by no means universal. The choice of letters signifying the different axes and moments is nearly universal, but the use of capitals and lower case, as well as script or italic fonts, varies significantly.

One common variation is to use upper-case letters for the three moments, but then use lower-case subscripts when transforming them into moment coefficients. The student must determine the particular convention used when reading any stability and control book or paper in order to avoid confusion. It has three degrees of freedom in translation linear motion , which are orthogonal to each other. Components of its velocity along the x, y, and z axes are labeled u, v, and w.

Note that lower case is used to avoid confusion with V, which typically has both u and w components. The aircraft also has three degrees of freedom in rotation, also orthogonal to each other. Figure 6. This motion is called rolling, and the maneuver is called a roll Control surfaces on the aircraft's wings called ailerons deflect differentially one trailing edge up and one trailing edge down to create more lift on one wing, less on the other, and therefore a net rolling moment.

Rotation of the aircraft about the lateral axis is called pitching. An elevator is a movable surface attached to a fixed immovable horizontal stabilizer, a small horizontal surface near the tail of the aircraft, which acts like the feathers of an arrow to help keep the aircraft pointed in the right direction. A stabilator combines the functions of the horizontal stabilizer and the elevator. The stabilator does not have a fixed portion.

It is said to be allmoving. A movable surface called a rudder, which is attached to the aircraft's fixed vertical stabilizer deflects to generate a lift force in a sideways direction. Because the vertical stabilizer and rudder are toward the rear of the aircraft, some distance from its center of gravity, the lift force they generate produces a moment about the vertical axis that causes the aircraft to yaw. For example, the surface on the F in Fig. Pitch control for this aircraft is provided by canards, stabilators placed forward of rather than behind the wings, and elevons, control surfaces at the rear of the wings.

Elevens move together to function as elevators and also move differentially like ailerons to provide roll control. The Vought F7U Cutlass twin-jet flying-wing fighter of the s and s used control surfaces exactly like elevons, but the manufacturer called them "ailevators! The moveable control surfaces attached to the fixed surfaces of the V-tail are called ruddervators, because they function as elevators when moving together and rudders when moving differentially.

The act of adjusting the control surfaces of an aircraft so they generate just enough force to make the sum of the moments zero is called trimming the aircraft. The trim condition is an equilibrium condition in terms of moments. Strictly speaking, the sum of the forces acting on an aircraft does not have to be zero for it to be trimmed. For instance, an aircraft in a steady, level turn would be considered trimmed if the sum of the moments acting on it is zero, even though the sum of the forces is not.

There are two types of stability that must be achieved in order to consider a system stable. The first is static stability, the initial tendency or response of a system when it is disturbed from equilibrium. If the initial response of the system when disturbed is to move back toward equilibrium, then the system is said to have positive static stability. When the ball is displaced from the bottom of the a Positive static stability b Negative static stability 2 1 c Neutral static stability Fig.

The system is described as statically stable. When centered on the dome, the ball is in equilibrium. However, if it is disturbed from the equilibrium condition, then the slope of the dome causes the ball to continue rolling away from its initial position.

This is called negative static stability because the system's initial response to a disturbance from equilibrium is away from equilibrium. The system is described as statically unstable. Figure 6Ac shows neutral static stability. The ball on the flat surface, when displaced from equilibrium, is once again in equilibrium at its new position, so it has no tendency to move toward or away from its initial condition.

Dynamic stability refers to response of the system over time. Note that the system also has positive static stability because its initial tendency when displaced from the zero displacement or equilibrium axis is to move back toward that axis. This in turn generates forces that, because the system is stable, tend to return it to equilibrium again.

These restoring forces overcome the momentum of the overshoot and generate momentum toward equilibrium, which causes another overshoot when equilibrium is reached, and so on.

This process of moving toward equilibrium, overshooting, then moving toward equilibrium again is called an oscillation. If the time history of the oscillation is such that the magnitude of each successive overshoot of equilibrium is smaller, as in Fig. Note that the second graph in Fig. The springs and shock absorbers on an automobile are familiar examples of systems with positive static and dynamic stability.

When the shock absorbers are new, the system does not oscillate when the car hits a bump. The system is said to be highly damped. As the shock absorbers wear out, the car begins to oscillate when it hits a bump, and the oscillations get worse and take longer to die out as the shock absorbers get more worn out.

The system is then said to be lightly damped. A system that has positive static stability but no damping at all continues to oscillate without ever decreasing the magnitude or amplitude of the oscillation. It is said to have neutral dynamic stability because over time the system does not get any closer to or farther from equilibrium. The time history of a system with positive static stability but neutral dynamic stability is shown on the left-hand graph of Fig.

On the right side of Fig. The time histories in Fig. The one on the left has negative static stability as well, so that it initially moves away from equilibrium and keeps going. The time history on the right is for a system that is statically stable, so that it initially moves toward equilibrium, but the amplitude of each overshoot is greater than the previous one. Over time, the system gets further and further from equilibrium, even though it moves through equilibrium twice during each complete oscillation.

The term "longitudinal" is used for this analysis because the moment arms for the pitch control surfaces are primarily distances along the aircraft's longitudinal axis. Also, the conditions required for longitudinal trim the case where moments about the lateral axis sum to zero are affected by the airplane's velocity, which is primarily in the longitudinal direction. The complete analysis of the static and dynamic stability and control of an aircraft in all six degrees of freedom is a broad and complex subject requiring an entire book to treat properly.

A sense of how such problems are framed and analyzed can be obtained from studying the analysis of the longitudinal static stability and control problem. The longitudinal problem involves two degrees of translational freedom, the x and z directions, and one degree of freedom in rotation about they axis. The aircraft's center of gravity is marked by the circle with alternating black and white quarters.

The lift forces of the wing and horizontal tail are shown acting at their respective aerodynamic centers. The moment about the wing's aerodynamic center due to the shape of its airfoil is also shown. The upper-case symbols L, Lt, and Mac are used as in Chapter 4 for wing lift, tail lift, and wing moment respectively to indicate that they are forces and moments produced by three-dimensional surfaces, not airfoils.

The horizontal tail is assumed to have a symmetrical airfoil, so that the moment about its aerodynamic center is zero. For consistency with the way two-dimensional airfoil data are presented, the locations of the wing's aerodynamic center ;cac and the whole aircraft's center of gravity jtcg are measured relative to the leading edge of the wing root.

The distance of the Fig. Summing the moments shown in Fig. For steady flight, the forces must also sum to zero. In practice, the elevator attached to the horizontal tail is deflected to provide the necessary lift from the tail so that the sum of the moments is zero when the aircraft is at the angle of attack required to make the sum of the forces zero. Note that for aircraft configurations such as the one shown in Fig.

Note also that Eqs. Similar relations can be derived for flying-wing aircraft, airplanes with canards, etc. If any of the required CL values are greater than CLmax for their respective surfaces, then the aircraft does not have sufficient control authority to trim in that maneuver for those conditions. To remedy this situation, the aircraft designer must either increase the size of the deficient control surface or add high-lift devices to it to increase its Cz,max.

Example 6. Both the wing and the canard of this aircraft have rectangular planforms. At this speed, its cambered main wing generates — N m of pitching moment about its aerodynamic center.

If the maximum lift coefficient for its canard is 1. Solution: Note that the pitching moment about the aerodynamic center is drawn nose up in Fig. The actual moment is nose down because its value is given as a negative. To trim at the specified minimum speed, the canard must generate sufficient lift so that the net moment on the aircraft measured about the center of gravity is zero.

If the disturbance causes the aircraft's angle of attack to increase, a statically stable aircraft would generate a negative pitching moment that would tend to return it to the trim condition. Likewise, if the disturbance reduced a, a statically stable aircraft would generate a positive pitching moment. This stability criterion is a condition that must be satisfied in order for an aircraft to be stable. This is typical at low angles of attack. Because aircraft must produce lift in most cases for equilibrium, only positive absolute angles of attack are useful as trim angles of attack.

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