Category AIRCRAF DESIGN

From the discussion on aircraft behavior in a small disturbance, it is clear that both aircraft geometry and mass distribution are important in the design of an aircraft with satisfactory flying qualities. The position of the CG is obtained by arranging the

aircraft components relative to one another to suit good in-flight static stability and on-ground stability for all operational envelopes. The full aircraft and its component moments are estimated semi-empirically (e. g., DATCOM and RAE data sheets) as soon as drawings are available and followed through during the next phase; the pre­diction is improved through wind-tunnel tests and CFD analyses. In the conceptual design stage, the control area on the wing and empennage (i. e., flap, aileron, rud­der, and elevator) are sized empirically from past experience (and DATCOM and RAE data sheets). However, the CG position relative to the aircraft NP is tuned afterwards.

Chapter 6 describes the aerodynamic design of major aircraft components. Chapter 11 considers the sizing of the wing and empennage and also establishes the matched-engine size. Whereas statistics of past designs proved vital for config­uring the empennage, the placement of components relative to one another is based on a designer’s experience, which forms a starting point for the conceptual design phase.

The important points affecting aircraft configuration are reviewed as follows:

1. Fuselage. The fuselage has a destabilizing effect – the fuselage lift (although minimal) and moment add to instability – and its minimization is preferred. In addition to keeping costs down, the fuselage may be kept straight (with the least camber). Mass distribution should keep inertia close to the fuselage centerline. A BWB requires special considerations.

The fuselage length and width are determined from the payload specifica­tions. The length-to-average-diameter ratio for the baseline aircraft version may be around 10. The closure angles are important, especially the gradual closure of the aft end, which should not have an upsweep of more than what is neces­sary – even for a rear-loading door arrangement that must have an upsweep. The front closure is blunter and must provide adequate vision polar without excessive upper-profile curvature.

For a pressurized cabin, the cross-section should be maintained close to the circular shape. Vertical elongation of the cross-section should be at a min­imum to accommodate the below-floorspace requirements. For small aircraft, fuselage-depth elongation may be due to placement of the wing box; for larger aircraft, it may be due to the container size. Care must be taken so that the wing box does not interfere with the interior cabin space. Generous fairing at the wing-body junction and for the fuselage-mounted undercarriage bulge is recommended. An unpressurized fuselage may have straight sides (i. e., a rect­angular cross-section) to reduce the production costs. In general, a rectangular fuselage cross-section is used in conjunction with a high wing. The undercar­riage for a high-wing aircraft has a fuselage bulge.

2. Wing. Typically, an isolated wing has a destabilizing effect unless it has a reflex at the trailing edge (i. e., the tail is integrated into the wing such as all-wing aircraft like the delta wing and BWB). The larger the wing camber, the more sig­nificant is the destabilizing effect. Optimizing an aerofoil with a high L/D ratio and with the least Cmwing is a difficult task not discussed herein. Wind-tunnel tests and CFD analyses are the ways to compromise. It is assumed that aerody – namicists have found a suitable aerofoil with the least destabilizing moment for

the best L/D ratio. The coursework worked-out example uses an aerofoil from the proven NACA series.

Sizing of an aircraft, as described in Chapter 11, determines the wing refer­ence area. The structures philosophy settles the aspect ratio; that is, maximizing the wing aspect ratio is the aim but at the conceptual design stage, it starts with improving on past statistics on which a designer can be confident of its struc­tural integrity under load. The wing sweep is obtained from the design maxi­mum cruise speed. It has been found that, in general, a wing-taper ratio from 0.4 to 0.5 is satisfactory. The twist and dihedral in the conceptual design stage are based on past experience and data sheets.

Positioning of the wing relative to the fuselage depends on the mission role, but it is sometimes influenced by a customer’s preference. A high – or low-wing position affects stability in opposite ways (see Figure 12.6). The wing dihedral is established in conjunction with the sweep and position relative to the fuse­lage. Typically, a high-wing aircraft has an anhedral and a low-wing aircraft has a dihedral, which also assist in ground clearance of the wing tips. In extreme design situations, a low-wing aircraft can have an anhedral (see Figure 12.7) and a high-wing aircraft can have a dihedral. There are case-based “gull-wing” designs, which are typically for “flying boats.” Passenger-carrying aircraft are predominantly low-winged but there is no reason why they should not have high wings; a few successful designs exist. Wing-mounted, propeller-driven aircraft favor a high wing for ground clearance, but there are low-wing, propeller-driven aircraft with longer undercarriage struts. Military transport aircraft invariably have a high wing to facilitate the rear-loading of bulky items.

3. Nacelle. The stability effects of a nacelle are similar to those of a fuselage. An isolated nacelle is destabilizing but, when integrated to the aircraft, its position relative to the aircraft CG determines its effect on the aircraft. That is, an aft – mounted nacelle increases stability and a forward-mounted nacelle on a wing decreases stability. The stability contribution of a nacelle also may be throttle- dependent (i. e., engine-power effects).

The position of the nacelle on an aircraft is dictated by the aircraft size. The best position is on the wing, thereby providing bending relief during flight. The large forward overhang of a nacelle decreases air-flow interference with the wing. For smaller aircraft, ground clearance mitigates against wing-mounting; for these aircraft, nacelles are mounted on the aft fuselage. An over-wing nacelle mount for smaller aircraft is feasible – a practice yet to gain credence. Even a fuselage-mounted nacelle must adjust its position relative to how close the vertical height is from the aircraft CG without jet efflux interfering with the empennage in proximity.

4. Fuselage, Wing, and Nacelle. It is good practice to assemble these three com­ponents without the empennage in order to verify the total moment in all three planes of reference. The CG position is established with the empennage installed; then it is removed for a stability assessment. This helps to design the empennage as discussed herein. Figure 12.10 shows the typical trends of pitch­ing moments of the isolated components; together, they will have a destabilizing effect (i. e., positive slope). The aim is to minimize the slope – that is, the least destabilizing moment.

Equation 12.2 provides insight to the pitching-moment contribution from the geometrical arrangement. It shows that minimizing the vertical distance of the components from the aircraft CG also minimizes their pitching-moment contributions.

5. Empennage. The empennage configuration is of primary importance in an air­craft design. The reference sizes are established by using statistical values of tail – volume coefficients, but the positioning and shaping of the empennage require considerable study. This is another opportunity to check whether the statistical values are adequate. The sweeping of the empennage increases the tail arm and may also enhance the appearance; even low-speed, smaller aircraft incorporate sweep. Chart 4.2 and Figures 4.24 and 4.25 show several possible empennage configurations.

A conventional aircraft H-tail has a negative camber, the extent depending on the moment produced by an aircraft’s tail-less configuration, as described previously. For larger, wing-mounted turbofan aircraft, the best position is a low H-tail mounted on the fuselage, the robust structure of which can accommo­date the tail load. A T-tail on a swept V-tail increases the tail arm but should be avoided unless it is essential, such as when dictated by an aft-fuselage-mounted engine. T-tail drag is destabilizing and requires a larger area if it is in the wing wake at nearly stalled attitudes. The V-tail requires a heavier structure to sup­port the T-tail load. Smaller turbofan aircraft are constrained with aft-fuselage – mounted engines, which force the H-tail to be raised up from the middle to the top of the V-tail. The canard configuration affords more choices for the aircraft CG location. In general, if an aircraft has all three surfaces (i. e., canard, wing, and H-tail), then they can provide lift with a positive camber of their sectional characteristics. It is feasible that future civil aircraft designs of all sizes may fea­ture a canard.

Typically, a V-tail has a symmetric aerofoil but for propeller-driven air­planes, it may be offset by 1 or 2 deg to counter the skewed flow around the fuselage (as well as gyroscopic torque).

The discussion is the basis for the design of any other type of empennage configuration, as outlined in Table 4.2. If a designer chooses a twin-boom fuse­lage, the empennage design must address the structural considerations of twin booms. (Tail-less aircraft are less maneuverable.)

An H-tail also can be dihedral or adhedral, not necessarily for stability rea­sons but rather to facilitate positional clearances, such as to avoid jet efflux.

6. Undercarriage. A retracted undercarriage does not contribute to the aerody­namic load but when it is extended, it generates substantial drag, creating a nose-down moment. To address this situation, there should be sufficient eleva­tor nose-up authority at a near-stall, touch-down attitude, which is most critical at the forwardmost CG position. Designers must ensure that there is adequate trim authority (i. e., the trim should not run out) in this condition.

7. Use of Any Other Surface. It is clear how stability considerations affect air­craft configurations. Despite careful design, an aircraft prototype may show unsatisfactory flying qualities when it is flight-tested. Then, additional sur­faces (e. g., ventral fin and delta fin) may be added to alleviate the problem. Figure 12.15 shows two examples of these modifications. It is preferable to avoid the need for additional surfaces, which add penalties in both weight and drag.

12.7 Military Aircraft: Nonlinear Effects

A discussion on military aircraft nonlinear effects is found at the Cambridge Uni­versity Web site.

Figure 12.16. Typical modern fighter aircraft

Spinning of an aircraft is a post-stall phenomenon (see [5]). An aircraft stall occurs in the longitudinal plane. Unavoidable manufacturing asymmetry in geometry and/or asymmetric load application makes one wing stall before the other. This creates a rolling moment and causes an aircraft to spin around the vertical axis, following a helical trajectory while losing height – even though the elevator has maintained in an up position. The vertical velocity is relatively high (i. e., descent speed on the order of 30 to 60 m/sec), which maintains adequate rudder authority, whereas the wings have stalled, losing aileron authority. Therefore, recovery from a spin is by the use of the rudder, provided it is not shielded by the H-tail (see Section 4.9). After straightening the aircraft with the rudder, the elevator authority is required to bring the aircraft nose down in order to gain speed and exit the stall.

Spinning is different than spiraling; it occurs in a helical path and not in a spiral. In a spiral motion, there is a large bank angle; in spinning, there is only a small bank angle. In a spiral, the aircraft velocity is sufficiently high and recovery is primarily achieved by using opposite ailerons. Spin recovery is achieved using the rudder and then the elevator.

There are two types of spin: a steep and a flat-pitch attitude of an aircraft. The type of spin depends on the aircraft inertia distribution. Most general-aviation air­craft have a steep spin with the aircraft nose pointing down at a higher speed, making recovery easy – in fact, the best aircraft recover on their own when the controls are released (i. e., hands off). Conversely, the rudder authority in a flat spin may be low. A military aircraft with a wider inertia distribution can enter into a flat spin from which recovery is difficult and, in some cases, impossible. A flat spin for transport aircraft is unacceptable. Records show that the loss of aircraft in a flat spin is pri­marily from not having sufficient empennage authority in the post-stall wake of the wing.

The prediction of aircraft-spinning characteristics is still not accurate. Although theories can establish the governing equations, theoretical calculations are not nec­essarily reliable because too many variables are involved that require accurate val­ues not easily obtainable. Spin tunnels are used to predict spin characteristics, but the proper modeling on a small scale raises questions about its accuracy. In partic­ular, the initiation of the spin (i. e., the throwing technique of the model into the tunnel) is a questionable art subjected to different techniques. On many occasions, spin-tunnel predictions did not agree with flight tests; there are only a few spin tun­nels in the world.

The best method to evaluate aircraft spinning is in the flight test. This is a rel­atively dangerous task for which adequate safety measures are required. One safe method is to drop a large “dummy” model from a flying “mother” aircraft. The model has onboard, real-time instrumentation with remote-control activation. This is an expensive method. Another method is to use a drag chute as a safety measure during the flight test of the piloted aircraft. Spin tests are initiated at a high altitude; if a test pilot finds it difficult to recover, the drag chute is deployed to pull the air­craft out of a spin. The parachute is then jettisoned to resume flying. If a test pilot is under a high g-strain, the drag chute can be deployed by ground command, where the ground crew maintains real-time monitoring of the aircraft during the test. Some types of military aircraft may not recover from a spin once it has been established. If a pilot does not take corrective measures in the incipient stage, then ejection is the routine procedure. FBW technology avoids entering spins because air data rec­ognize the incipient stage and automatic-recovery measures take place.

The diagrams in Figure 12.12 show an exaggerated aircraft flight path (i. e., altitude changes in the pitch plane). In the pitch plane, there are two different types of aircraft dynamics that result from the damping experienced when an aircraft has a small perturbation. The two longitudinal modes of motion are as follows:

1. Short-period oscillation (SPO) is associated with pitch change (a change) in which the H-tail plane acts as a powerful damper (see Figure 12.1). If a dis­turbance (e. g., a sharp flick of the elevator and return) causes the aircraft to enter this mode, then recovery is also quick for a stable aircraft. The H-tail acts

like an aerodynamic spring that naturally returns to equilibrium. The restor­ing moment comes from the force imbalance generated by the angle of attack, a, created by the disturbance. Damping (i. e., resistance to change) comes as a force generated by the tail plane, and the stiffness (i. e., force required) comes from the stability margin. The heavy damping of the H-tail resists changes to make a quick recovery.

The bottom diagram of a short period in Figure 12.12 plots the variation of the angle of attack, a, with time. All aircraft have a short-period mode and it is not problematic for pilots. A well-designed aircraft oscillatory motion is almost unnoticeable because it damps out in about one cycle. Although aircraft velocity is only slightly affected, the angle of attack, a, and the vertical height are related. Minimum a occurs at maximum vertical displacement and maxi­mum a occurs at about the original equilibrium height. The damping action offered by the H-tail quickly smooths out the oscillation; that is, one oscilla­tion takes a few seconds (typically, 1 to 5 s). The exact magnitude of the period depends on the size of the aircraft and its static margin. If the H-tail plane area is small, then damping is minimal and the aircraft requires more oscillations to recover.

2. Phugoid motion is the slow oscillatory aircraft motion in the pitch plane, as shown in the bottom diagram in Figure 12.12. It is known as the long-period oscillation (LPO) – the period can last from 30 s to more than 1 min. Typi­cally, a pilot causes the LPO by a slow up and down movement of the ele­vator. In this case, the angle of attack, a, remains almost unchanged while in the oscillatory motion. The aircraft exchanges altitude gain (i. e., increases in potential energy [PE]) for decreases in velocity (i. e., decreases in kinetic energy [KE]). The phugoid motion has a long period, during which time the KE and PE exchange. Because there is practically no change in the angle of attack, a, the H-tail is insignificant in the spring-mass system. Here, another set of spring – mass is activated but is not shown in schematic form (it results from the aircraft configuration and inertia distribution – typically, it has low damping charac­teristics). These oscillations can continue for a considerable time and fade out comparatively slowly.

The frequency of a phugoid oscillation is inversely proportional to an aircraft’s speed. Its damping also is inversely proportional to the aircraft L/D ratio. A high L/D ratio is a measure of aircraft performance efficiency. Reducing the L/D ratio to increase damping is not preferred; modern designs with a high L/D ratio incorporate automatic active control (e. g., FBW) dampers to minimize a pilot’s workload. Con­ventional designs may have a dedicated automatic damper at a low cost. Automatic active control dampers are essential if the phugoid motion has undamped charac­teristics.

All aircraft have an inherent phugoid motion. In general, the slow motion does not bother a pilot – it is easily controlled by attending to it early. The initial onset, because it is in slow motion, sometimes can escape a pilot’s attention (particularly when instrument-flying), which requires corrective action and contributes to pilot fatigue.

directional stability = Cnp lateral stability =

directional

inadequate Cno

12.6.2 Directional and Lateral Modes of Motion

Aircraft motion in the directional (i. e., yaw) and the lateral (i. e., roll) planes is cou­pled with sideslip and roll; therefore, it is convenient to address the lateral and direc­tional stability together. These modes of motion are relatively complex in nature. FAR 23, Sections 23-143 to 23.181, address airworthiness aspects of these modes of motion. Spinning is perceived as a post-stall phenomenon and is discussed sepa­rately in Section 12.7.

The four typical modes of motion are (1) directional divergence, (2) spiral, (3) Dutch roll, and (4) roll subsidence. The limiting situation of directional and lateral stability produces two types of motion. When yaw stability is less than roll stability, the aircraft can enter directional divergence. When roll stability is less than yaw sta­bility, the aircraft can enter spiral divergence. Figure 12.13 shows the two extremes of directional and spiral divergence. The Dutch roll occurs along the straight initial path, as shown in Figure 12.14.

The wing acts as a strong damper to the roll motion; its extent depends on the wing aspect ratio. A large V-tail is a strong damper to the yaw motion. It is important to understand the role of damping in stability. When configuring an aircraft, design­ers need to optimize the relationship between the wing and V-tail geometries. The four modes of motion are as follows:

1. Directional Divergence. This results from directional (i. e, yaw) instability. The fuselage is a destabilizing body, and if an aircraft does not have a sufficiently large V-tail to provide stability, then sideslip increases accompanied by some roll, with the extent depending on the roll stability. The condition can continue until the aircraft is broadside to the relative wind, as shown in Figure 12.13.

2. Spiral. However, if the aircraft has a large V-tail with a high degree of direc­tional (i. e., yaw) stability but is not very stable laterally (i. e., roll) (e. g., a low – wing aircraft with no dihedral or sweep), then the aircraft banks as a result of rolling while sideslipping.

This is a nonoscillatory motion with characteristics that are determined by the balance of directional and lateral stability. In this case, when an aircraft is in a bank and sideslipping, the side force tends to turn the plane into the rela­tive wind. However, the outer wing is traveling faster, generating more lift, and the aircraft rolls to a still higher bank angle. If poor lateral stability is avail­able to negate the roll, the bank angle increases and the aircraft continues to turn into the sideslip in an ever-increasing (i. e., tighter) steeper spiral, which is spiral divergence (see Figure 12.13). In other words, spiral divergence is strongly affected by Clr.

The initiation of a spiral is typically very slow and is known as a slow spiral. The time taken to double the amplitude from the initial state is long – 20 s or more. The slow buildup of a spiral-mode motion can cause high bank angles before a pilot notices an increase in the g-force. If a pilot does not notice the change in horizon, this motion may become dangerous. Night-flying without proper experience in instrument-flying has cost many lives due to spiral diver­gence. Trained pilots should not experience the spiral mode as dangerous – they would have adequate time to initiate recovery actions. A 747 has a nonoscilla­tory spiral mode that damps to half amplitude in 95 s under typical conditions; many other aircraft have unstable spiral modes that require occassional pilot input to maintain a proper heading.

3. Dutch Roll. A dutch roll is a combination of yawing and rolling motions, as shown in Figure 12.14. It can happen at any speed, developing from the use of the stick (i. e., aileron) and rudder, which generate a rolling action when in yaw. If a sideslip disturbance occurs, the aircraft yaws in one direction and, with strong roll stability, then rolls away in a countermotion. The aircraft “wags its tail” from side to side, so to speak. The term Dutch roll derives from the rhyth­mic motion of Dutch iceskaters swinging their arms and bodies from side to side as they skate over wide frozen areas.

When an aircraft is disturbed in yaw, the V-tail performs a role analogous to the H-tail in SPO; that is, it generates both a restoring moment proportional to the yaw angle and a resisting, damping moment proportional to the rate of yaw. Thus, one component of the Dutch roll is a damped oscillation in yaw. However, lateral stability responds to the yaw angle and the yaw rate by rolling the wings of the aircraft. Hence, the second component of a Dutch roll is an

oscillation in a roll. The Dutch-roll period is short – on the order of a few seconds.

In other words, the main contributors to the Dutch roll are two forms of static stability: the directional stability provided by the V-tail and the lateral stability provided by the effective dihedral and sweep of the wings – both forms offer damping. In response to an initial disturbance in a roll or yaw, the motion consists of a combined lateral-directional oscillation. The rolling and yawing frequencies are equal but slightly out of phase, with the roll motion leading the yawing motion.

Snaking is a pilot term for a Dutch roll, used particularly at approach and landing when a pilot has difficulty aligning with the runway using the rudder and ailerons. Automatic control using yaw dampers is useful in avoiding the snaking/Dutch roll. Today, all modern transport aircraft have some form of yaw damper. The FBW control architecture serves the purpose well.

All aircraft experience the Dutch-roll mode when the ratio of static direc­tional stability and dihedral effect (i. e., roll stability) lies between the limiting conditions for spiral and directional divergences. A Dutch roll is acceptable as long as the damping is high; otherwise, it becomes undesirable. The character­istics of a Dutch roll and the slow spiral are both determined by the effects of directional and lateral stability; a compromise is usually required. Because the slow-spiral mode can be controlled relatively easily, slow-spiral stability is typi­cally sacrificed to obtain satisfactory Dutch-roll characteristics.

High directional stability (C„p) tends to stabilize the Dutch-roll mode but reduces the stability of the slow-spiral mode. Conversely, a large, effective dihe­dral (rolling moment due to sideslip, Cy) stabilizes the spiral mode but desta­bilizes the Dutch-roll motion. Because sweep produces an effective dihedral and because low-wing aircraft often have excessive dihedral to improve ground clearance, Dutch-roll motions often are poorly damped on swept-wing aircraft.

4. Roll Subsidence. The fourth lateral mode is also nonoscillatory. A pilot com­mands the roll rate by application of the aileron. Deflection of the ailerons gen­erates a rolling moment, but the aircraft has a roll inertia and the roll rate builds up. Very quickly, a steady roll rate is achieved when the rolling moment gener­ated by the ailerons is balanced by an equal and opposite moment proportional to the roll rate. When a pilot has achieved the desired bank angle, the ailerons are neutralized and the resisting rolling moment very rapidly damps out the roll rate. The damping effect of the wings is called roll subsidence.

Once an aircraft is built, its flying qualities are the result of the effects of its mass (i. e., inertia), CG location, static margin, wing geometry, empennage areas, and control areas. Flying qualities are based on a pilot’s assessment of how an aircraft behaves under applied forces and moments. The level of ease or difficulty in control­ling an aircraft is a subjective assessment by a pilot. In a marginal situation, recorded test data may satisfy airworthiness regulations yet may not prove satisfactory to the pilot. Typically, several pilots evaluate aircraft flying qualities to resolve any debat­able points.

It is important that the design maintain flying qualities within preferred levels by shaping the aircraft appropriately. Whereas theoretical analyses help to minimize discrepancies, flying qualities can be determined only by actual flight tests. Like any other system analysis, control characteristics are rarely amenable to the precise the­ory due to a lack of exact information about the system. Therefore, accurate design information is required to make predictions with minimal error. It is cost-intensive to generate accurate design information, such as the related design coefficients and derivatives required to make theoretical analyses, which are conducted more inten­sively during Phase 2 of a project. Practically all modern aircraft incorporate active control technology (ACT) to improve flying qualities. This is a routine design exer­cise and provides considerable advantage in overcoming any undesirable behavior, which is automatically and continuously corrected.

Described herein are six important flight dynamics of particular design inter­est. They are based on fixed responses associated with small disturbances, making

the rigid-body aircraft motion linearized. Military aircraft have additional consider­ations as a result of nonlinear, hard maneuvers, which are discussed in Section 12.9. The six flight dynamics are as follows:

• short-period oscillation

• phugoid motion (long-period oscillation)

• Dutch roll

• slow spiral

• roll subsidence

• spin

During Phase 1 (conceptual design) of an aircraft design project, the initial empen­nage is sized using statistical data. Section 3.20 provides preliminary statistics of the empennage tail-volume coefficients. Figure 12.11 provides additional statis­tics for current aircraft (twenty-one civil and nine military aircraft types), plotted
separately for the H-tail and the V-tail. Statistics for aircraft using FBW are included in the figure. It is advised that readers create separate plots to generate their own air­craft statistics for the particular aircraft class in which they are interested to obtain an appropriate average value.

For civil aircraft designs, the typical H-tail area is about a quarter of the wing reference area. The V-tail area varies from 12% of the wing reference area, SW, for large, long aircraft to 20% for smaller, short aircraft. There may be minor changes in empennage sizing when more detailed analyses are carried out in Phase 2 of the design.

Military aircraft require more control authority for greater maneuverability and they have shorter tail arms that require larger tail areas. The H-tail area is typically about 30 to 40% of the wing reference area. The V-tail area varies from 20 to 25% of the wing reference area. Supersonic aircraft have a movable tail for control. If a V-tail is too large, then it is divided in two halves.

Modern aircraft with FBW technology can operate with more relaxed stabil­ity margins, especially for military aircraft designs; therefore, they require smaller empennage areas compared to older conventional designs (see Figure 12.18).

In this book, trim surfaces are earmarked and not sized. Designers must ensure that there is adequate trim authority (i. e., the trim should not run out) in any condition. This is typically accomplished in Phase 2 after the configuration is finalized.

As explained previously, roll stability derives primarily from the following three aircraft features:

low-wing Г, it is typically between 1 and 3 deg, depending on the wing sweep. For a straight-wing aircraft, the maximum dihedral rarely exceeds 5 deg. For a high-wing sweep, it may require an anhedral, as discussed herein.

2. Wing Position Relative to the Fuselage (see Figure 12.7). Section 12.3.3 explains the contribution to the rolling moment caused by different wing positions rela­tive to the fuselage. Semi-empirical methods are used to determine the extent of the rolling-moment contribution.

3. Wing Sweep at Quarter-Chord, Л/ (see Figure 12.8). The lift produced by a swept wing is a function of the component of velocity, Vn, normal to the c/ line; that is, in steady rectilinear flight:

Vn = V cos Л

When an aircraft sideslips with angle в, the component of velocity normal to the c/ line becomes (small в):

Vn = V cos^1/4 – в) = V(cos Л1/4 + в sin Д1/4в)

For the leeward wing:

VnJw = Vcos^1/4 + в) = V(cos Л1/4 – в sin Л1/4в)

The windward wing has V’n > Vn and vice versa; therefore, it provides ДLift as the restoring moment in conjunction with the lift decrease on the leeward wing. As Л1/4 increases, the restoring moment becomes powerful enough that it must be compensated for by the use of the wing anhedral.

The equation of motion in the yaw plane can be set up similarly to the pitch plane. The weathercock stability of the V-tail contributes to the restoring moment.

Figure 12.4 depicts moments in the yaw plane. In the diagram, the aircraft is yawing to the left with a positive yaw angle в. This generates a destabilizing moment by the fuselage with the moment (NF = YF x lf), where YF is the resultant side force by the fuselage and lf is the distance of Lf from the CG. Contributions by the wing, H-tail, and nacelle are small (i. e., small projected areas and/or shielded by the fuselage projected area). The restoring moment is positive when it tends to turn the nose to the right to realign with the airflow. The weathercock stability of the V-tail causes the restoring moment (NVT = YT x lt), where YT is the resultant side force on the V-tail (for small angles of (в + a), it can be approximated as the lift generated by the V-tail, LVT) and lt is the distance of LT from the CG. Therefore, the total aircraft yaw moment, N (for conventional aircraft), is the summation of NF and NVT, as given in Equation 12.9:

Nacc = NF + NVT

At equilibrium flight:

Nacc = 0; i-e-, NVT = ~NF

In coefficient form, the fuselage contribution can be written as:

C„f = – k„kRlNF [(Sf[25]f )/(Sw Ь)]в

where kn = empirical wing-body interference factor kR = empirical correction factor Sf = projected side area of the fuselage lf = fuselage length b = wing span

In coefficient form, the V-tail contribution can be written as in Equation 12.11 (LVT is in the coefficient form CLVT):

CnVT = [(lt/Svt)/(Swc)]vvtClvt = Lvt Vv vvtClvt (12.12)

where

VV = V-tail volume coefficient = (lt/Svt)/(Swc) (12.13)

(introduced in Section 3.20, derived here).

Equation 12.9 in coefficient form becomes:

Cn_cg = – knkmNF[(Sflf)/(SwЬ)]в + LvtVv wtClvt (12.14)

In equilibrium, J^force = 0, when drag = thrust and lift = weight (see Figure 3.9). An imbalance in drag and thrust changes the aircraft speed until equilibrium is reached. Drag and thrust act nearly collinearly; if they did not, pitch trim would be required to balance out the small amount of pitching moment it can develop. The same is true with the wing lift and weight, which are rarely collinear and gen­erate a pitching moment. This scenario also must be trimmed, with the resultant lift and weight acting collinearly.

Together in equilibrium, moment = 0. Any imbalance results in an aircraft rotating about the Y-axis. Figure 12.9 shows the generalized forces and moments (including the canard) that act in the pitch plane. The forces are shown normal and parallel to the aircraft reference lines (i. e., body axes) and abnormal to air­craft velocity. Lift and drag are obtained by resolving the forces of Figure 12.9 into

Geometry, Velocity, and Force Details

^taBifiy ^rmargin

Force and Moment Details

Figure 12.9. Generalized force and moment in the pitch plane

perpendicular and parallel directions to the free-stream velocity vector (i. e., aircraft velocity). The forces can be expressed as lift and drag coefficients, dividing by qSw, where q is the dynamic head and Sw is the wing reference area. Subscripts identify the contribution made by the respective components. The arrowhead directions of component moments are arbitrary – they must be assessed properly for the com­ponents. With its analysis, Figure 12.9 gives a good idea of where to place aircraft components relative to the aircraft CG and NP. The static margin is the distance between the NP and the CG.

The generalized expression for the moment equation can be written as in Equa­tion 12.1, which sums up all the moments about the aircraft CG. In the trimmed condition, the aircraft moment about the aircraft CG must be zero (Maccg = 0):

Mac_cg = (Nc X lc + Cc X Zc + Mc)canard + (Nw X lw + Cw X Zw + Mw)wing

+ (Nt X lt + Ct X zt + Mt )tail

+ Mfus + Mnac + (thrust X zth + nacdrag x zth) + any other item (12.1)

In the conceptual design stage, the forces and moments of each component are estimated semi-empirically (i. e., US DATCOM and RAE data sheets [now ESDU]) from the drawings. When assembled together as an aircraft, each component is influ­enced by the flow field of the others (e. g., the flow over the H-tail is affected by the wing flow). Therefore, a correction factor, n, is applied. This is shown in Equa­tion 12.2 written in coefficient form, dividing by qSwc, where q is the dynamic head,

c is the wing MAC, and Sw is the wing reference area. Subscripts identify the contri­bution of the respective components. The moment coefficients of the components are computed initially as isolated bodies and then converted to the reference wing area:

Cmcg — CNc(Sc/Sw Wc / c) + CCc(Sc/Sw )(zc/c) + Cmc (Sc/Sw )]forcanard

+ CNw(1a/c) nw + CCw (za/c)nw + Cmw nw]forwmg

+ CN( (St/Sw)(1t/c)nt + CCt (St/Sw)(zt/c)nt + Cmt (St/Sw)nt]fortail

+ Cmfus + Cnac + (thrust x zth + nacudrag X zth)/qSwc (12.2)

where:

1. n(— q /q<x>) represents the wake effect of lifting surfaces behind another lifting surface producing downwash, qt is the incident dynamic head, and is the free-stream dynamic head.

2. The vertical distances (z) of each component can be above or below the CG, depending on the configuration, described as follows:

(a) For fuselage-mounted engines, zth is likely to be above the aircraft CG and its thrust generates a nose-down moment. In underslung wings, engines have the zth below the CG, generating a nose-up moment. For most mili­tary aircraft, the thrust line is very close to the CG; therefore, for a prelimi­nary analysis, the zth term can be ignored (i. e., no moment is generated with thrust unless it is vectored).

(b) The drag of a low wing below the CG (za) has a nose-down moment and vice versa for a high wing. For midwing positions, which side of the CG must be noted; the (za) may be small enough to be ignored in the preliminary analysis.

(c) The position of the H-tail shows the same effect as for the wing but is invari­ably above the CG. For a low H-tail, zt can be ignored. In general, the drag generated by the H-tail is small and can be ignored; this is also true for a T-tail design.

(d) For the same reason, the contribution of the canard vertical distance, zc, also can be ignored.

In summary, Equation 12.2 can be further simplified by comparing the order of mag­nitude of the contributions of the various terms. Initially, the following simplifica­tions are suggested:

1. The vertical z distance of the canard and the wing from the CG is small. There­fore, the terms with zc and za can be omitted.

2. The canard and H-tail reference areas are much smaller that the wing reference area and their C («drag) component force is less than a tenth of their lifting forces. Therefore, the terms with CCc (Sc/Sw) and CCt (St/Sw) can be omitted – even for a T-tail, but it is best to check its overall contribution.

3. A high or low wing has za with opposite signs. For a midwing, za may be small enough to be ignored.

Equation 12.2 can be simplified as follows:

Cmcg — [CNc(Sc/Sw)(lc/c) + Cmc(Sc/Sw)] + [CNw(la/c)^w + Cmw ^w]

+ [CNt (St/Sw)(lt/c) nt + Cmt (St/Sw) nt]

+ Cmfus + Cnac + (thrust x zth + na^drag X zth)/qSwc (12.3)

A conventional aircraft does not have a canard. Then, Equation 12.3 can be fur­ther simplified to Equation 12.4. The conventional aircraft CG is possibly ahead of the wing MAC. In this case, the H-tail must have negative lift to trim the moment generated by the wing and the body:

Cmcg — [CNw (la /c) + Cmw ] + [Cm (St/Sw )(lt/c)nt + Cmt (St/Sw )nt ]

+ Cmfus + Cnac + (thrust x zth + nac-drag X zth)/qSwc (12.4)

Normal forces now can be resolved in terms of lift and drag; for small angles of a, the cosine of the angle is 1. The drag components of all the CN are very small and can be neglected.

Then, the first term:

CNw(la/ c) + Cmw ^ CLw(la/c) + Cm

and the second term:

Cm (St/Sw)(lt/c)nt + Cmt(St/Sw)nt ^ Cl (St/Sw)(lt/c)nt + Cmt (St/Sw)nt

where

Cmt (St/Sw)nt < Cl (St/Sw)(lt /c)nt

Hence, the moment contribution by the H-tail is represented as Cm_HT — Cut (Sh/Sw )(lt /c)nHT.

Then, Equation 12.4 is rewritten (note the sign) as:

Cmcg — [CLw (la /c) + Cmw] + Cm_HT + Cmfus + Cm

+ (thrust x zth + na^drag x zth)/qSwc

where

Cm_HT — —ClHT X [(lt/Sht)/(Swc)]nHT – — — Vh VHtClHT (12.6)

For conventional aircraft, CLHT has a downward direction to keep the nose up; therefore, it has a negative sign. Here,

VH — H-tail volume coefficient — (lt /SHT)/(Swc) (12.7)

(introduced in Section 3.20, derived here).

Then, Equation 12.5 without engines becomes:

Cmcg — [Cl w (la /c) + Cmw] Cm_HT + Cmfus

A convenient method is to analyze the effects on aircraft pitching moments of isolated aircraft components. Next, the airplane less the empennage is esti­mated, thereby determining the appropriate H-tail moment required to balance the moments at the cruise condition. Figure 12.10 shows the pitching moment contribu­tion by components of a conventional aircraft (considered mass-less to examine only

continent pitch stability ptch dab Ity van at on vrth CG variation

the aerodynamic characteristics). The wing and fuselage have destabilizing moments (i. e., nose up), which must be compensated for by the tail to counter the wing and fuselage moments; hence, Equation 12.6 has negative sign.

The second diagram in Figure 12.10 shows the stability effects of different CG positions on a conventional aircraft. The stability margin is the distance between the aircraft CG and the NP (i. e., a point through which the resultant force of the aircraft passes). When the CG is forward of the NP, then the static margin has a positive sign and the aircraft is statically stable. The stability increases as the CG moves farther ahead of the NP.

There is a convenient range from the CG margin in which the aircraft design exhibits the most favorable situation. In Figure 12.10, the position B is where the CG coincides with the NP and shows neutral stability (i. e., at a zero stability mar­gin) – the aircraft can still be flown with the pilot’s efforts controlling the aircraft attitude. In fact, an aircraft with relaxed stability can have a small negative margin that requires little force to make rapid maneuvers – these aircraft invariably have a FBW control architecture (see Section 12.10) in which the aircraft is flown continu­ously controlled by a computer.

Engine thrust has a powerful effect on stability. If it is placed above and behind the CG such as in an aft-fuselage-mounted nacelle, it causes an aircraft nose-down pitching moment with thrust application. For an underslung wing nacelle ahead of the CG, the pitching moment is with the aircraft nose up. It is advisable for the thrust line to be as close as possible to the aircraft CG (i. e., a small ze to keep the moment small). High-lift devices also affect aircraft pitching moments and it is better that these devices be a small arm’s-length from the CG.

In summary, designers must carefully consider where to place components to minimize the pitching-moment contribution, which must be balanced by the tail at the expense of some drag – this is unavoidable but can be minimized.

Roll stability is more difficult to analyze compared to longitudinal and lateral sta­bilities. A banked aircraft attitude through a pure roll keeps the aircraft motion in

the plane of symmetry and does not provide any restoring moment. However, roll is always coupled with a yawed motion, as explained previously. As a roll is initiated, the sideslip velocity, v, is triggered by the weight component toward the down-wing side, as shown in Figure 12.5. Then (see the previous section), the sideslip angle is в = tan-1(v/u). The positive angle of roll, Ф, is when the right wing drops as shown in the figure (the aircraft is seen from the rear showing the V-tail but no windscreen). A positive roll angle Ф generates a positive sideslip angle, в. The angle of attack increases the sideslip.

Recovery from a roll is possible as a result of the accompanying yaw (i. e., cou­pled motion) with the restoring moment contributed by increasing the lift acting on the wing that has dropped. Roll static-stability criteria require that an increase in the roll angle, Ф, creates a restoring moment coefficient, Cl (not to be confused with the sectional aerofoil-lift coefficient). The restoring moment has a negative sign.

Having a coupled motion with the sideslip, Figure 12.6 shows that Cl is plotted against the sideslip angle в, not against the roll angle Ф because it is в that generates the roll stability. The sign convention for restoring the rolling moment with respect

to в must have Qp negative; that is, with an increase of roll angle Ф, the sideslip angle в increases to provide the restoring moment. An increase in в generates a restoring roll moment due to the dihedral. At zero Ф, there is no в; hence, the zero rolling moment (Cl = 0).

The wing dihedral angle, Г, is one way to increase roll stability, as shown in Figure 12.5. The dropped wing has an airflow component from below the wing gen­erating lift, while at the other side, the airflow component is from the upper side of the wing that reduces the angle of attack (i. e., the lift reduction creates a restoring moment).

The position of the wing relative to the aircraft fuselage has a role in lateral stability, as shown in Figure 12.6. At yaw, the relative airflow about the low wing has a component that reduces the angle of attack; that is, the reduction of lift and the other side act in opposite ways: a destabilizing effect that must be compensated for by the dihedral, as explained previously. Conversely, a high-wing aircraft has an inherent roll stability that acts opposite to a low-wing design. If it has too much stability, then the anhedral (-ve dihedral) is required to compensate it. Many high – wing aircraft have an anhedral (e. g., the Harrier and the BAe RJ series).

An interesting situation occurs with a wing sweepback on a high-speed air­craft, as explained in Figure 12.8. At sideslip, the windward wing has an effectively reduced sweep; that is, the normal component of air velocity increases, creating a lift increment, whereas the leeward wing has an effectively increased sweep with a slower normal velocity component, thereby losing lift. This effect generates a rolling moment, which can be quite powerful for high-swept wings; even for low – wing aircraft, it may require some anhedral to reduce the excessive roll stability (i. e., stiffness) – especially for military aircraft, which require a quick response in a roll. Tu-104 in Figure 12.7 is a good example of a low-wing military aircraft with a high sweep coupled with an anhedral.

The side force by the fuselage and V-tail contributes to the rolling moment, as shown in Figure 12.8. If the V-tail area is large and the fuselage has a relatively smaller side projection, then the aircraft CG is likely to be below the resultant side force, thus increasing the stability. Conversely, if the CG is above the side force, then there is a destabilizing effect.

Figure 12.8. V-tail contribution to roll

12.3.1 Summary of Forces, Moments, and Their Sign Conventions

12.4 Theory

Forces and moments affect aircraft motion. In a steady level flight (in equilibrium), the summation of all forces is zero; the same applies to the summation of moments. When not in equilibrium, the resultant forces and moments cause the aircraft to maneuver. The following sections provide the related equations for each of the three aircraft planes. A sense of these equations helps in configuring aircraft in the con­ceptual design phase.

Directional stability can be compared to longitudinal stability but it occurs in the yaw plane (i. e., the XY plane about the Z-axis), as shown in Figure 12.4. By defi­nition, the angle of sideslip, в, is positive when the free-stream velocity vector, V, relative to aircraft is from the right (i. e., the aircraft nose is to the left of the veloc­ity). V has component aircraft velocities u along the X-axis and v along the Y-axis, subtending the sideslip angle в = tan-1(v/u).

The V-tail is subjected to an angle of incidence (в + a), where a is the sidewash angle generated by the wing vortices (like the downwash angle in longitudinal sta­bility). Static stability criteria require that an increase in the sideslip angle, в, should generate a restoring moment, Cn, that is positive when turning the nose to the right. The moment curve slope of Cn is positive for stability. At zero в, there is no yawing moment (i. e., Cn = 0).

Yaw motion invariably couples with roll motion because neither is in the plane of symmetry. In yaw, the windward wing works more to create a lift increase while the lift decreases on the other wing, thereby generating a rolling moment. Therefore, a pure yaw motion is achieved by the use of compensating, opposite ailerons. The use of an aileron is discussed in the next section.