Flight Maneuvers & Gusts

J. Gordon Leishman

doi:https://doi.org/10.15394/eaglepub.2022.1066.n39

55 Flight Maneuvers & Gusts

Introduction

To design and engineer an aircraft, it is necessary to understand its expected flight characteristics, including maneuverability. All aircraft types must be capable of maneuvering to some extent, albeit with varying degrees. Airliners, for example, are only expected to perform gentle maneuvers, such as shallow climbs, descents, and moderate banked turns. Examples of more extreme flight maneuvers include high-rate turns, rapid pull-ups, and aerobatics, as demonstrated by the aircraft performing aerobatics in Figure 1, or in any situation in which an aircraft follows a curvilinear path under non-steady, accelerated flight conditions.

A flight team performs formation aerobatic maneuvers.

However, any flight maneuver has limits, and an aircraft’s actual maneuver performance is constrained by its aerodynamic capabilities and airframe structural strength. For example, the ability to perform various high-load-factor or high “ $g$ ” maneuvers is expected of many military aircraft, especially jet fighters. Therefore, not only must the airframe be sufficiently strong to carry the loads in these maneuvers, but the airplane’s aerodynamic performance must also be carefully considered, as aerodynamic limitations should not prematurely constrain the airplane’s capabilities. In this regard, the desired maneuver capability must not be limited too early by the onset of wing stall and/or buffeting.

Natural gusts in the atmosphere will also cause loads on the airplane that exceed those produced in steady, level flight in smooth air. In particular, vertical gusts can produce transient loads on an airplane that may be as large, or even more significant, than those expected during maneuvers and those that are otherwise within the standard flight envelope. For this reason, the airplane’s maneuver performance and gust-loading capabilities must be precisely defined, and the aerodynamic and structural requirements must be carefully established during the design process. Naturally, these capabilities also need to be verified by structural testing, which will be undertaken using a systematic combination of flight and ground tests.

Learning Objectives

Understand the meaning and significance of the airspeed/load factor or “ $V$ – $n$ ” diagram for maneuvers and atmospheric gusts.
Interpret the $V$ – $n$ diagram and identify critical airspeeds.
Appreciate why the airspeed at which wing stall occurs will increase during a maneuver or when a load factor is applied.

Equations of Motion

As previously derived, the following general equations can be used to describe the motion of an airplane, i.e.,

(1) $\begin{eqnarray*} \mbox{$\parallel$ to flight path:} \quad && \hspace*{-7mm} \left(\frac{W}{g}\right) \frac{d V_{\infty}}{dt} = T \cos \epsilon - D - W \sin \theta \\[10pt] \mbox{$\perp$ to flight path:} \quad && \hspace*{-7mm} \left(\frac{W}{g}\right) \frac{V_{\infty}^2}{R} = L \cos \phi + T \sin \epsilon \cos \phi - W \cos \theta \\[10pt] \mbox{Horizontal plane:} \quad && \hspace*{-7mm} \left(\frac{W}{g}\right) \frac{(V_{\infty} \cos \theta)^2}{R} = L \sin \phi + T \sin \epsilon \sin \phi \end{eqnarray*}$

The angle $\theta$ can be viewed as the climb or flight path angle, and the bank angle is denoted by $\phi$ . It should be remembered that during maneuvers, which involve accelerated flight conditions, the lift generated by the wing will not equal the weight of the airplane, as the wing must create the necessary lift to produce the required accelerations to follow the specified flight path. This lift force may be greater or less than the airplane’s weight. The load factor may be positive or negative depending on the direction of the aerodynamic normal force relative to the airplane, whereas the resulting flight-path acceleration depends on the combined effects of lift and weight.

In many cases, the line of action of the thrust vector relative to the flight path is small, so it is reasonable to assume that $\epsilon = 0$ in the foregoing equations, i.e.,

(2) $\begin{eqnarray*} \mbox{$\parallel$ to flight path:} \quad && \hspace*{-7mm} \left(\frac{W}{g}\right) \frac{d V_{\infty}}{dt} = T - D - W \sin \theta \\[10pt] \mbox{$\perp$ to flight path:} \quad && \hspace*{-7mm} \left(\frac{W}{g}\right) \frac{V_{\infty}^2}{R} = L \cos \phi - W \cos \theta \\[10pt] \mbox{Horizontal plane:} \quad && \hspace*{-7mm} \left(\frac{W}{g}\right) \frac{(V_{\infty} \cos \theta)^2}{R} = L \sin \phi \end{eqnarray*}$

Maneuvers in a Vertical Plane

Consider first the forces on an airplane maneuvering in a vertical plane with wings level ( $\phi$ = 0) following a circular flight path of instantaneous radius $R$ and at a constant airspeed ${V_{\infty}}$ , as shown in Figure 2. Notice that the airplane would perform a complete loop in a vertical plane when continuing this trajectory.

The balance of forces on an airplane in a pull-up maneuver.

The airplane’s mass is $M = W/g$ . Resolving forces normal to the flight path gives

(3) $\begin{equation*} L - W = \left( \frac{W}{g} \right) \frac{V_{\infty}^2}{R} \end{equation*}$

Therefore, the lift required on the wing to follow the flight path is

(4) $\begin{equation*} L = \left( \frac{W}{g} \right) \frac{V_{\infty}^2}{R} + W = \left( 1 + \frac{V_{\infty}^2}{g R} \right) W = n \, W \end{equation*}$

i.e., the lift must be greater than the weight, where the load factor $n$ is

(5) $\begin{equation*} n = \left( 1 + \frac{V_{\infty}^2}{g R} \right) \end{equation*}$

The excess lift is related to the load factor, $n$ , such that $L = n \, W$ , i.e., the number of effective “ $g$ ‘s.” It can be seen that, for a given flight path radius, the load factor increases with the square of the airspeed. Furthermore, for a given airspeed, the load factor is inversely proportional to the radius; i.e., a faster/or tighter flight path will produce a higher load factor. The radius of curvature $R$ of the flight path, in this case, will be

(6) $\begin{equation*} R = \frac{V_{\infty}^2}{g (n - 1)} \end{equation*}$

So, for a given load factor, the radius of the flight path increases quickly with the square of the airspeed.

Looping Maneuver

Continuing the pull-up will result in a looping motion. It is unlikely that a commercial airliner would perform loops, as suggested in Figure 3, but the point is made. As previously discussed, lift on the wing must be sufficient to overcome the airplane’s weight and produce the centripetal acceleration to execute a circular flight path in a vertical plane. The airplane’s angular position along the flight path must also be considered. The lift required in a loop is needed to overcome the component of the weight acting on the airplane at any given angle $\psi$ , plus the necessary centripetal force, i.e.,

(7) $\begin{equation*} L = \underbrace{ \left( \frac{W}{g} \right) \frac{V_{\infty}^2}{R_v}}_{\begin{tabular}{c} \scriptsize Centripetal \\[-3pt] \scriptsize force \end{tabular}} + \underbrace{W \cos \psi}_{\begin{tabular}{c} \scriptsize Component \\[-3pt] \scriptsize of weight \end{tabular}} = \underbrace{ \left(\frac{V_{\infty}^2}{g \, R_v} + \cos \psi \right)}_{\begin{tabular}{c} \scriptsize Load factor \end{tabular}} W = n W \end{equation*}$

where $\psi = 0^{\circ}$ when the airplane is at the bottom of the loop, as shown in Figure 3. Notice that weight always acts vertically downward. It can be assumed that the pilot modulates the thrust to maintain a constant airspeed.

The balance of forces on an airplane in a looping type of flight maneuver with wings level in a vertical plane.

The corresponding load factor will be

(8) $\begin{equation*} n = \left( \cos\psi + \frac{V_{\infty}^2}{g \, R_v} \right) \end{equation*}$

At the bottom of the loop where $\psi = 0^{\circ}$ , the lift must be greater than the weight to overcome both the weight and create the needed inward centripetal force, so

(9) $\begin{equation*} n = \left(1 + \frac{V_{\infty}^2}{g \, R_v} \right) \end{equation*}$

At the top of the loop, where $\psi = 180^{\circ}$ , the weight helps in the direction of the centripetal force, so the load factor is

(10) $\begin{equation*} n = \left(-1 + \frac{V_{\infty}^2}{g \, R_v} \right) \end{equation*}$

And so this is less than at the bottom of the loop. Notice that the load factor is a function of airspeed ${V_{\infty}}$ and the radius of the loop $R_v$ . If the pilot flies a loop at a higher airspeed or tightens the loop to reduce the radius $R_v$ , the pilot will experience higher load factors.

Consider now a flight maneuver in which an aerobatic airplane is inverted at the top of a perfectly circular loop, performed at constant airspeed, with the pilot modulating engine thrust as needed to maintain it; see Figure 4. Consider the case where the combination of airspeed and flight path radius gives

(11) $\begin{equation*} \frac{V_{\infty}^2}{g \, R_v} = 1 \end{equation*}$

Balance of forces during a “loop-the-loop” maneuver.

The lift required is

(12) $\begin{equation*} L = \left( \frac{W}{g} \right) \frac{V_{\infty}^2}{R_v} + W \cos \psi = \left( \cos \psi + \frac{V_{\infty}^2}{g \, R_v} \right) W = n \, W \end{equation*}$

The corresponding load factor will be

(13) $\begin{equation*} n = \left( \cos\psi + \frac{V_{\infty}^2}{g \, R_v} \right) \end{equation*}$

At the bottom of the loop, the lift must be greater than the weight to overcome both the weight and create the needed centripetal force, so that the load factor will be

(14) $\begin{equation*} n = \left(1 + \frac{V_{\infty}^2}{g \, R_v} \right) \end{equation*}$

At the top of the loop, the weight helps provide the inward centripetal force, so the load factor here is

(15) $\begin{equation*} n = \left(-1 + \frac{V_{\infty}^2}{g \, R_v} \right) \end{equation*}$

and it will be less than at the bottom of the loop. In this case, where

(16) $\begin{equation*} \frac{V_{\infty}^2}{g \, R_v} = 1 \end{equation*}$

then the pilot will feel “weightless,” i.e., $n$ = 0.

Notice that for a given load factor, the radius of the flight path increases quickly with the square of the airspeed. This result explains the development of tactics for combat maneuvers by military aircraft, in which tighter maneuvers are often required at lower airspeeds. From

(17) $\begin{equation*} n = \cos\psi + \frac{V_{\infty}^2}{gR_v} \end{equation*}$

the corresponding angular rate in the maneuver is

(18) $\begin{equation*} \omega = \frac{d\gamma}{dt} = \frac{V_{\infty}}{R_v} = \frac{g\left(n-\cos\psi\right)}{V_{\infty}} \end{equation*}$

At the bottom of the loop, where $\psi=0^\circ$ , this reduces to

(19) $\begin{equation*} \omega = \frac{g(n-1)}{V_{\infty}} \end{equation*}$

which confirms that, for a given load factor at that point in the maneuver, tighter turns are obtained at lower airspeeds.

Zero-g Flight Maneuvers

Zero-g maneuvers in an airplane can simulate the weightlessness of space. One aircraft used for this purpose was the “Weightless Wonder,” operated by NASA and also known as a “Reduced Gravity Aircraft.” In a parabolic “zero-g” flight, the airplane follows a trajectory that provides several seconds of weightlessness for the occupants. To conduct this maneuver, the airplane must have a minimum entry speed, which defines the initial kinetic energy required to complete the maneuver without stalling; the airspeed at which the maneuver can be flown depends on the specific airplane, its in-flight weight, and the number of occupants.

The pilots first initiate a steep pull-up climb, and the pitch attitude may approach 45 degrees. The pilots then “push over” to reduce the angle of attack and unload the airplane, allowing it to follow an approximately parabolic trajectory, as shown in Figure 5. During this portion of the maneuver, the airplane and its occupants accelerate nearly together under gravity, so the occupants experience little or no normal reaction force from the cabin floor. Therefore, the apparent load factor is approximately $n \approx 0$ (i.e., “zero-g“), and the occupants feel “weightless.” It is a pleasant experience, although the conditions preceding and following it may not be.

When the components of lift and weight balance the required centripetal acceleration, the occupants experience a period of “zero-g,” during which they feel “weightless,” thereby simulating conditions in space.

At the apogee of the “zero-g” parabolic (nominally) trajectory, the load factor is

(20) $\begin{equation*} n = \cos \psi + \frac{V_{\infty}^2}{g \, R_v} \end{equation*}$

For the weightless condition, $n = 0$ at $\psi = 180^\circ$ , so

(21) $\begin{equation*} \frac{V_{\infty}^2}{g \, R_v} = 1 \end{equation*}$

which gives the combination of airspeed, ${V_{\infty}}$ , and radius of curvature of the flight path, $R_v$ , for the weightless condition. If, for example, the radius of curvature of the flight path is held constant, then the airspeed to fly for the zero-g condition is

(22) $\begin{equation*} V_{\infty} = \sqrt{g \, R_v} \end{equation*}$

Maintaining the desired reduced-g condition is critical. As the airplane enters the maneuver, the airspeed decreases during the climb and then increases again after apogee. The pilots must therefore adjust pitch attitude, thrust, and, when necessary, drag devices to control the trajectory and keep the normal acceleration as close to zero as practical. After the reduced-g portion is complete, recovery is initiated by smoothly increasing lift and applying thrust as needed. These unique “zero-g” conditions, albeit only lasting 10 to 20 seconds, allow engineers and astronauts to experience weightlessness without traveling into space, thereby preparing them for actual space flights.

The repeated cycles of ascent and descent during “zero-g” flights are akin to a roller-coaster ride, which can be uncomfortable for occupants and even for the most seasoned pilots, especially after twenty or more consecutive trajectories. The sensation of weightlessness can also lead to motion sickness in some individuals; the aircraft used for these “zero-g” flights is nicknamed the “Vomit Comet.” The disorientation that humans experience when inner-ear responses, balance cues, and visual cues are disrupted during weightlessness can produce dizziness, nausea, and spatial disorientation, symptoms commonly associated with motion sickness or, in spaceflight, “space sickness.” No one is immune; even test pilots and astronauts may need to take motion sickness medication.

Maneuvers in a Horizontal Plane

Now consider the forces on an airplane in a pure horizontal turn with a bank angle $\phi$ and when flying at a constant airspeed ${V_{\infty}}$ , as shown in Figure 6.

The balance of forces on an airplane in a turning maneuver.

Vertical equilibrium requires that

(23) $\begin{equation*} L \cos \phi = W \end{equation*}$

and horizontal equilibrium requires

(24) $\begin{equation*} L \sin \phi = \left(\frac{W}{g} \right) \frac{V_{\infty}^2}{R_h} \end{equation*}$

where $R_h$ is the instantaneous radius of curvature of the turn.

It is apparent then that to perform a turn, the lift on the wing must again be greater than the weight of the airplane, i.e., $L > W$ , to create the necessary aerodynamic force not only to balance the weight of the airplane but also to produce the inward radial force to create the needed centripetal acceleration to execute the turn.

Solving for the lift required gives

(25) $\begin{equation*} L = \frac{W}{\cos \phi} \end{equation*}$

and so the load factor is

(26) $\begin{equation*} n = \frac{L}{W} = \frac{W / \cos \phi}{W} = \frac{1}{\cos \phi} \end{equation*}$

The preceding result shows that the load factor must increase with the inverse of the cosine of the bank angle $\phi$ . For example, a 60 $^{\circ}$ banked turn will correspond to $n = 2$ , i.e., a load factor of two.

Airspeed-Load Factor Diagram

An airspeed-load factor, or $V$ – $n$ , diagram is one form of the operating envelope for an airplane. Figure 7 shows a representative $V$ – $n$ diagram for an airplane as a function of indicated airspeed. However, equivalent airspeed (related to actual dynamic pressure) or Mach number will sometimes be used on the “airspeed” axis.

Representative maneuver envelope for an airplane in terms of load factor versus airspeed. Each type of airplane has its own envelope.

Aerodynamic Limits

As previously discussed, aerodynamic performance will be limited by the onset of stall and/or buffet. For a positive steady load factor, $n > 0$ , the stall speed on the positive-lift boundary is

(27) $\begin{equation*} V_{s_{ng}} = \sqrt{\frac{ 2 n W}{\varrho_{\infty} S C_{L_{\rm max}}}} = \sqrt{n} \, V_{s_{1g}} \end{equation*}$

This latter result indicates that the airplane will stall at a higher airspeed when subjected to any “ $g$ ” loading with $n > 1$ . Notice from the $V$ – $n$ diagram that as the load factor increases, the stall airspeed follows a curve defined by Eq. 27, i.e., its value increases, and so it traces out one part of the envelope on the $V$ – $n$ diagram.

Structural Limits

The $V$ – $n$ diagram also reflects that an airplane can only structurally withstand a finite amount of loading (e.g., it is capable of only so much wing stress and/or wing bending and/or skin buckling) until it suffers permanent damage or structural failure; this maximum loading is denoted by $n_{\rm max}$ . At one airspeed, which is called the corner airspeed or the maximum maneuvering airspeed, the airplane will be operating at the edge of the stall and pulling the maximum load factor, i.e.,

(28) $\begin{equation*} V_A = \sqrt{\frac{ 2 n_{\rm max} \, W}{\varrho_{\infty} \, S \, C_{L_{\rm max}}}} \end{equation*}$

The maximum maneuvering airspeed is often called $V_A$ (or sometimes $V_a$ ).

Equation 28 defines an aerodynamic limitation on overall flight performance because of the attainment of the maximum lift coefficient $C_{L_{\rm max}}$ on the wing, as well as a structural limitation in terms of a maximum attainable structural load factor. Therefore, for flight in rough air other than light turbulence or “chop,” the pilot or aircrew should operate the airplane at or below its recommended maneuvering or turbulence penetration airspeed, as specified in the approved operating handbook.

The maneuvering airspeed $V_A$ primarily protects the airplane against structural overload from full, abrupt control inputs under the specified design assumptions; it does not guarantee protection against all gust encounters. Sufficiently severe atmospheric gusts can still produce high loads, which is why gust envelopes are treated separately in aircraft structural design. Note that $V_A$ , which is not explicitly marked on many airspeed indicators, is usually placarded in the cockpit or listed in the operating handbook.

The maximum attainable load factor $n_{\rm max}$ that an airplane is designed to withstand, i.e., its structural limits, depends on the airplane type and what it is intended to do. For civil aircraft, the limiting load factor values will be defined by the appropriate certification authority, e.g., the FARs in the U.S. Under limit-load conditions, the FARs require that the airplane’s structural components support those loads without permanent detrimental structural deformation and that stresses remain below the critical yield point, accounting for unexpected events such as severe gust loads. Another consideration is the possibility of an emergency landing at weights higher than the standard landing weights.

Transport, Normal, Utility, and Aerobatic Category Airplanes

For transport-category airplanes, such as airliners, the positive maneuvering load factor limit is usually taken as $+2.5g$ for large airplanes, with a corresponding negative limit of about $-1.0g$ up to the design cruise speed. Under Part 25, the required positive limit maneuvering load factor is weight-dependent but need not exceed +3.8 g. For limit load factors, a 50% safety factor is applied in structural design (i.e., an additional margin for safety). The ultimate structural strength must be at least 150% of the design limit load; this load is then referred to as the ultimate load.

For normal and commuter category airplanes under the legacy Part 23 requirements, the positive limit maneuvering load factor is

(29) $\begin{equation*} n = 2.1 + \frac{24,000}{W + 10,000} \end{equation*}$

where $W$ is the design maximum takeoff weight in pounds, although the required value need not exceed $3.8g$ . The corresponding negative limit maneuvering load factor for normal-category airplanes is typically $-0.4$ times the positive value, giving a lower bound of about $-1.52g$ when the positive value is $+3.8g$ . For utility-category airplanes, the positive limit maneuvering load factor is $+4.4g$ , with a corresponding negative value of about $-1.76g$ . Utility-category airplanes may be approved for limited aerobatics when operated within the specified weight and center-of-gravity limits.

Aerobatic Airplanes

The load factor ranges from $-3.0 g$ to $+6.0 g$ for fully acrobatic or aerobatic category airplanes. However, many purpose-built aerobatic airplanes are designed to tolerate load factors well above the minimum required by regulation, with some being stressed to about $\pm 10g$ . Military fighter airplanes are also designed for high positive maneuver loads, commonly of the order of $+7.5g$ to $+9g$ , although their allowable negative load factors are usually much smaller in magnitude. Aerobatic airplanes are generally much stronger than the pilot could sustain in terms of “ $g$ ” loadings.

The following additional points identify and describe the nature of the $V$ – $n$ diagram and what it means:

Strictly speaking, the $V$ – $n$ diagram applies to a single flight weight. Usually, a $V$ – $n$ diagram is defined at the maximum gross in-flight weight of the airplane, i.e., $W_{\rm GTOW}$ .
The area within the “Normal flight envelope,” as defined by the $V$ – $n$ diagram, comprises airspeeds and load factors for which the airplane can be safely flown without stalling or structural failure.
At a load factor of unity ( $n = 1$ ), which corresponds to level flight, the stall limit can be easily identified on the $V$ – $n$ diagram.
The corner airspeed at which the airplane is operating at the edge of the stall and pulling the maximum load factor can be easily identified. This point is usually marked on the diagram as the maximum maneuvering airspeed or $V_A$ . On many airplanes, $V_A$ is placarded rather than shown directly on the airspeed indicator; it should not be confused with the upper end of the white arc, which denotes the maximum flap-extended speed.
Notice that both positive and negative load limits are identified. An airplane cannot withstand as much negative loading as positive loading, nor can the pilot and passengers. The exception, of course, is an airplane designed specifically for aerobatics.

Limiting Airspeeds

At higher airspeeds, the airplane reaches an aerodynamic limit based on dynamic pressure, i.e., a “redline” or never-exceed airspeed, usually identified as $V_{\rm NE}$ and marked on the pilot’s airspeed indicator. Exceeding this limit places the airplane outside its demonstrated safe operating envelope and may lead to structural damage, aeroelastic instability, or loss of control. A “yellow arc” on the airspeed indicator identifies the caution range, in which operation is permitted only in smooth air and with caution because gusts or abrupt control inputs may produce excessive structural loads.

Structural Testing

The limit and ultimate failure loads of a new airplane design (and its components) are validated on the ground. To this end, a test article of the entire airplane is mounted in a special rig that simulates the magnitude and distribution of the airloads encountered during actual flight, as shown in Figure 8. Naturally, this type of test-to-failure would not be conducted in the air. The structural test article must conform to the approved design and be representative of the production airplane, so no special strengthening may be incorporated merely to pass the test.

Structural tests on the ground can verify the ultimate load factor of a wing.

Other tests performed on this type of ground rig include simulations of extreme negative wing loads to validate the lower portion of the $V$ – $n$ diagram, maneuver loads, and emergency-landing loads. For a military airplane, ballistic damage to the wing may also be a consideration; various types of damage can lead to structural stress and loading limitations. The wing is tested to the required ultimate-load condition to verify that the predicted ultimate loads can be sustained. Additional loading to structural failure may also be conducted to establish the available strength margin and identify the failure mode. In most cases, wings will fail by compression buckling of the upper wing skins.

Gust-Induced Airloads

There are two primary uses for a $V$ – $n$ diagram: one for maneuvering flight, as already considered, and another for gust loads in the atmosphere encountered during straight-and-level flight. The diagrams are essentially the same, but the information presented is interpreted differently. The atmosphere is never stagnant, and turbulence and gusts occur naturally. A gust can affect the airplane from any direction. However, upward (vertical) gusts, as shown in Figure 9, have the most pronounced effects on the airplane in terms of aerodynamic response and induced load factor.

While gusts in the atmosphere can be produced from any direction, vertical gusts will have the most pronounced aerodynamic effects on the airplane.

An upward vertical gust increases the wing’s angle of attack and lift, whereas a downward vertical gust decreases the angle of attack and lift, as shown in Figure 10. While the drag will also be affected, its contribution is minor because of the small angles typically involved.

A gust will change the angle of attack of the wing.

The change in the angle of attack on the wing will be

(30) $\begin{equation*} \Delta \alpha = \tan^{-1} \left( \frac{W_g}{V_{\infty}}\right) \approx \frac{W_g}{V_{\infty}} \mbox{~for small angles} \end{equation*}$

where $W_g$ is denoted here as the vertical gust velocity. If the lift-curve slope of the wing, $C_{L_{\alpha}}$ , is expressed per radian, then the change in lift coefficient will be

(31) $\begin{equation*} \Delta C_L = C_{L_{\alpha}} \left( \frac{W_g}{V_{\infty}} \right) \end{equation*}$

and the change in the lift is

(32) $\begin{equation*} \Delta L = \frac{1}{2} \varrho_{\infty} V_{\infty}^2 S C_{L_{\alpha}} \left( \frac{W_g}{V_{\infty}} \right) = \frac{1}{2} \varrho_{\infty} V_{\infty} S C_{L_{\alpha}} W_g \end{equation*}$

The change in the load factor is then

(33) $\begin{equation*} \Delta n = \frac{\Delta L}{W} = \frac{1}{2} \varrho_{\infty} V_{\infty} \left( \frac{S}{W} \right) C_{L_{\alpha}} W_g \end{equation*}$

Therefore, the total load factor on the airplane compared to straight-and-level unaccelerated flight is

(34) $\begin{equation*} n = 1 + \Delta n = 1 + \frac{1}{2} \varrho_{\infty} V_{\infty} \left( \frac{S}{W} \right) C_{L_{\alpha}} W_g = 1 + \frac{\frac{1}{2} \varrho_{\infty} V_{\infty} C_{L_{\alpha}} W_g}{W/S} \end{equation*}$

There are two interesting observations from Eq. 34:

1. The load factor for a given gust intensity decreases with increasing wing loading, $W/S$ . This outcome indicates that smaller, lighter airplanes (generally with relatively lower wing loadings) respond more strongly to gusts than larger airplanes, such as jet airliners. In this regard, smaller airplanes must be carefully flown at or below the maneuvering airspeed in turbulent air to prevent structural damage.

2. The load factor for a given gust intensity varies linearly with airspeed. Therefore, lines with different values of $W_g$ can be plotted on the $V$ – $n$ diagram, as shown in Figure 11. These sets of straight lines represent the load factor produced on the airplane at a given airspeed from the effects of gusts. When one of these lines intersects the maximum (or minimum) load factor limit, the corresponding airspeed is the maximum airspeed that can be flown without either stalling the wing or exceeding the maximum allowable structural load factor. Each aircraft type will have its own envelope for gust magnitudes and airspeeds.

A representative gust envelope for an airplane in terms of load factor versus airspeed. The magnitude of the gust to which the airplane must be designed depends on its class and type.

If the gust is sufficiently severe, the airplane’s resulting load factor may cross over the allowable limit load. If the applied load approaches or exceeds the ultimate design load, major structural damage or failure may occur. Even if the limit load factor is exceeded during flight, minor damage may still have occurred (e.g., wrinkled skin panels resulting from buckling), and the aircraft must be carefully inspected before further flight.

Corner Airspeed

If the airplane is flying below the corner airspeed (i.e., $V_A$ or $V_a$ on the $V$ – $n$ diagram), a sufficiently large positive angle-of-attack disturbance may cause the wing to stall before the positive maneuvering limit load factor is reached. This behavior does not guarantee protection against all atmospheric gusts. If the airplane is flown at or below its maximum maneuvering speed, $V_A$ , then a full and abrupt control input will not cause the airplane to exceed its limit load factor; instead, the wing will stall. However, sufficiently strong atmospheric gusts may still produce load factors that approach or exceed the structural limits. Consequently, when encountering very turbulent air, the airplane should be flown at or below its maximum maneuvering speed so that any gusts and/or control inputs do not exceed its structural limit loads.

Regulations and the FARs

A commonly used simplified representation of the design vertical gust velocities is shown in the table below. The exact regulatory values and definitions depend on the certification basis and the applicable version of the FARs, so the table should be interpreted as an instructional gust-envelope model rather than a complete statement of the current regulations. $V_B$ is the design speed for maximum gust intensity, which assumes that the airplane is in straight-and-level flight when it encounters the gust and that the effects of the gust are produced instantaneously. The gust value of “66 ft/s” is based on statistical information gathered about turbulence in the lower atmosphere and is the most extreme case considered representative of all-weather flying. The other values are also based on atmospheric gust statistics. They are used in aircraft design to ensure that the aircraft is sufficiently strong to withstand all anticipated structural loads under turbulent conditions. $V_c$ is the design cruise speed; for airplanes in the transport category (airliners), $V_C$ must not be less than $V_B$ + 43 kts. $V_D$ is based on allowable gusts at the maximum dive speed.

Values of vertical gust velocity specified for airplane design. Between 20,000 ft and 50,000 ft, the gust intensity is linearly interpolated.
Airspeed	Below 20,000 ft	Above 50,000 ft
$V_B$ (rough air gust)	66 ft/s	38 ft/s
$V_C$ (gust at max. design speed)	50 ft/s	25 ft/s
$V_D$ (gust at max. dive speed)	25 ft/s	12.5 ft/s

Summary & Closure

It is essential to establish an airplane’s maneuver and gust envelopes so that it can be designed to carry all expected flight loads, with a margin of safety. Airplanes cannot be infinitely strong, so the $V-n$ diagram must be consistent with the airplane’s intended purpose. The final performance of the airplane will be limited by its aerodynamics and/or the structural strength of its airframe. Gusts in the atmosphere impose additional loads on the airplane beyond those produced in smooth air. All airplanes must be designed to withstand the normal flight loads and the additional loads induced by encounters with turbulent air. The FARs define these gust conditions depending on the type of airplane. It is reassuring that the structural margins built into certified aircraft designs are significant. When operated within its approved flight envelope, a certified aircraft is expected to withstand the prescribed maneuver and gust loads, including the specified safety margins.

5-Question Self-Assessment Quickquiz

For Further Thought or Discussion

Think about some structural issues in designing a wing for a fully aerobatic airplane. Hint: Include both normal and inverted flight.
What factors other than aerodynamic or structural may limit the acrobatic maneuver limits?
Compare the relative load factors in response to the same vertical gust at the same airspeed that would be produced on the following airplanes: a glider, a single-engine general aviation airplane, and a small business jet.

Other Useful Online Resources

To understand more about an airplane’s maneuvering flight envelope and the effects of gusts, then follow up with some of these online resources:

Good video explaining the significance of load factor.
The effects of banking angle on load factor – a DELFT video.
What is load factor and how it relates to a “V-n” diagram – a good video.
A piloting interpretation of the a load factor diagram and critical airspeeds.

License

Icon for the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License

Introduction to Aerospace Flight Vehicles Copyright © 2022–2026 by J. Gordon Leishman is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, except where otherwise noted.

Digital Object Identifier (DOI)

https://doi.org/https://doi.org/10.15394/eaglepub.2022.1066.n39