Maneuvers & Gusts

J. Gordon Leishman

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

42 Maneuvers & Gusts

Introduction

To design and engineer an aircraft, it is necessary to know what it is expected to do regarding its flight characteristics, including its maneuver performance. All aircraft types must be able to maneuver to some degree, some more than others. Airliners, for example, are only expected to perform gentle maneuvers such as banked turns. Examples of more extreme flight maneuvers are high-rate turns, rapid pull-ups, and various aerobatics, as shown in the photograph below, or any situation where an aircraft may follow a curvilinear path under non-steady, accelerated flight conditions.

Formation aerobatic maneuvers as performed by a flight team.

However, any flight maneuver will have some limits, and the aircraft’s actual maneuver performance will be constrained by its aerodynamic capabilities and the structural strength of its airframe. For example, the ability to perform various high load factors or high “g” maneuvers is expected of many military airplanes, especially fighter airplanes. Therefore, not only must the airframe be sufficiently strong to carry the loads in these maneuvers, but the aerodynamic performance of the aircraft must be carefully considered in that aerodynamics is not a prematurely limiting factor. In this regard, the desired maneuver capability is not limited too early by the onset of wing stall and/or buffeting.

Natural gusts in the atmosphere will also cause loads on the aircraft over and above 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 normal flight envelope. For this reason, the aircraft’s needed maneuver performance and gust loading capabilities must be precisely defined, and the aerodynamics 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 airspeed-load factor or “ $V$ – $n$ ” diagram, for both maneuvers and atmospheric gusts.
Be able to interpret the $V$ – $n$ diagram and identify critical airspeeds.
Appreciate why the airspeed at which wing stall occurs will increase in a maneuver or with the application of a load factor.

Aircraft Equations of Motion

As previously derived, the following general equations can be used to describe the motion of an aircraft, 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 are accelerated flight conditions, the lift on the wing will not equal the weight of the aircraft because of the need for the wing to create whatever lift value is needed to produce the needed accelerations to follow the required flight path. This lift force may be greater or less than the airplane’s weight, so during flight, the load factor can be positive (i.e., an upward acceleration) or negative (downward acceleration).

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 forgoing 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 a circular flight path of radius $R$ = constant and at a constant airspeed $V_{\infty}$ , as shown in the figure below. 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.

Vertical equilibrium in the maneuver requires that

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

Therefore, the lift required on the wing 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 $L = nW$ , i.e., the number of effective “g’s.” So it can be seen that for a given radius of the flight path, 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 and/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.

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 the figure below.

The balance of forces on an airplane in turning maneuver.

Vertical equilibrium requires that

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

and horizontal equilibrium requires

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

where $R$ is the 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 aircraft but also to produce the inward radial force to create the needed centripetal acceleration to execute the turn.

Solving for the lift required gives

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

and so the load factor is

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

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 operating envelope for an aircraft. The figure below shows a representative $V$ – $n$ diagram for an airplane as a function of indicated airspeed, although sometimes equivalent airspeed (which is related to actual dynamic pressure) or Mach number will be used on the “airspeed” axis.

Representative maneuver envelope for an airplane in terms of load factor versus airspeed. Each type of aircraft will have its own specific envelope.

Aerodynamic Limits

As previously discussed, airplanes will be limited aerodynamically by the onset of stall and/or buffet. The stall speed with a steady load factor $n$ is

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

This latter result shows that when pulling any “g” loading with $n > 1$ , the aircraft will stall at a higher airspeed. Notice from the $V$ – $n$ diagram that as the load factor increases, the stall airspeed follows a curve defined by Eq. 11, i.e., its value increases, and so it traces out one part of the operating 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 aircraft will be operating at the edge of stall and also pulling the maximum load factor, i.e.,

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

The maximum maneuvering airspeed is often referred to as $V_A$ (or sometimes $V_a$ ).

Equation 12 defines an aerodynamic limitation on overall flight performance because of the attainment of the maximum lift coefficient $C_{L_{\rm max}}$ on the wing and also 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 will need to operate the aircraft at its maximum maneuvering equivalent airspeed $V_A$ , which is marked on the airspeed indicator (white arc). This approach is required so unexpected turbulent gusts will not create load factors that could potentially overload the airframe.

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

Normal Category Airplanes

For transport category airplanes (e.g., airliners), $n_{\rm max}$ ranges from -1g to +2.5g or up to +3.8g, depending on the design takeoff weight. To the limit load factors, a value of 50% is added for structural design purposes (i.e., an extra margin for extra safety), i.e., the ultimate structural strength needs to be at least 150% of the design limit load, which then becomes known as the ultimate load.

For normal category and commuter category airplanes, limit load factors range from $-1.52$ g to $+3.8$ g. The maximum specified load factor is only 2.5g for airplanes with a gross weight of less than 50,000 lb but increases linearly with weight up to a maximum of 3.8g. The relevant equation defined by the FARs is $n = 2.1 + 24,000/(W +10,000)$ up to a maximum of 3.8g where $W$ is measured in lb. For utility category airplanes (or those configured to operate in this category) range from $-1.76$ g to $+4.4$ g, which often allows the aircraft to perform limited aerobatics, at least within a specific center of gravity range.

Aerobatic Airplanes

For fully acrobatic or aerobatic category airplanes, the load factor range from $-3.0$ g to $+6.0$ g. However, in particular for aerobatic and military fighter airplanes, many aircraft types are designed to tolerate load factors much higher than the minimum required by regulations, often between $-10$ g and $+12$ g. Aerobatic aircraft 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 aircraft, i.e., $W_{\rm GTOW}$ .
The area inside the “Normal flight envelope,” as marked out by the $V$ – $n$ diagram, is the combination of airspeeds and load factors where the aircraft can be safely flown without stalling or suffering structural failure.
At a load factor unity ( $n = 1$ ), which is level flight, the stall limit can be easily identified on the $V$ – $n$ diagram.
The corner airspeed where the aircraft 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 $V_A$ (it is an equivalent airspeed and the limit of the “white arc” on the airspeed indicator).
Notice that both positive and negative load limits are identified. The airplane is usually unable to withstand as much negative loading and positive loading as will 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., this represents a “redline” or never-exceed airspeed, and is usually identified as $V_{\rm NE}$ and marked on the pilot’s airspeed indicator. If exceeded, even by some narrow margin, then structural failure can be expected. Therefore, a “yellow arc” zone will also be marked on the airspeed indicator so the pilot knows that the aircraft is operating near its maximum airloads and structural capability.

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 can simulate the magnitude and distribution of the airloads encountered during the actual flight, as shown in the photograph below. Naturally, this sort of test to failure would not be done in the air! The airplane will be randomly selected “as built” on the regular production line, so no special considerations are given to the aircraft structure.

Test on the ground can be used verify the ultimate load factor of a wing.

Other tests performed in this type of ground rig include simulations of extreme negative loads on the wing to validate the lower part of the $V$ – $n$ diagram, maneuver loads, and emergency landing loads. For a military aircraft, ballistic damage to the wing may also be a consideration, i.e., various types of damage may result in structural stress and a loading limitation. Eventually, the wing will be tested to destruction to verify the predicted ultimate loads on the aircraft can indeed be sustained. 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 one for the gust loads in the atmosphere that are produced when flying along in straight-and-level flight. The diagrams are basically the same, but the information being presented has different interpretations. 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 the figure below, have the most pronounced effects on the airplane regarding 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 aircraft.

The primary effect of a vertical gust is to increase the angle-of-attack of the wing and so increase the wing lift, the principle being shown in the figure below. While there will also be an effect on drag, the effects are small because of the angles typically involved.

The change in angle of attack on the wing will be

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

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

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

and the change in the lift is

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

The change in the load factor is then

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

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

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

There are two interesting observations from Eq. 17:

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

2. The load factor for a given gust intensity varies linearly with airspeed. So lines with different values of $W_g$ can be plotted on the $V$ – $n$ diagram, an example being shown in the figure below. These sets of straight lines give 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 type of aircraft will have its own specific envelope in terms of gust magnitudes and airspeeds.

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

If the gust is sufficiently severe, the airplane’s resulting load factor may cross over the allowable limit load. However, it cannot exceed the ultimate load without significant structural damage or even structural failure. Even if the limit load factor is exceeded in flight, there is a possibility that minor damage may still have occurred (e.g., wrinkled skin panels from buckling), and the aircraft must be carefully inspected before further flight.

Corner Airspeed

Notice that if the airplane is flying slower than the corner airspeed (remember this is marked as $V_A$ or $V_a$ on the $V$ – $n$ diagram), any large gust will cause an angle-of-attack to exceed the value for maximum lift coefficient and so cause the aircraft to stall. Suppose the aircraft is flown at or below its maximum maneuvering speed. In that case, neither an atmospheric gust nor abrupt control movements (such as a rapid pull-up maneuver using full-up elevator) will be sufficient to cause the aircraft to exceed its maximum structural load factor. Consequently, when encountering very turbulent air, the aircraft should be flown at or below the maximum maneuvering speed so that any gusts and/or any control inputs cannot exceed its structural limit loads.

Regulations and the FARs

The FARs define the values of the vertical gust velocities, as shown in the table below. $V_B$ is the design speed for maximum gust intensity, which assumes that the aircraft 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 also are based on atmospheric gust statistics. They are used in airplane design to ensure that the airplane is strong enough to withstand all anticipated structural loads in 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 aircraft’s maneuver and gust envelope so that it can be suitably designed to carry all expected flight loads, plus a margin of safety. Aircraft cannot be infinitely strong, so the $V-n$ diagram must be consistent with the aircraft’s intended purpose. The final performance capabilities of the aircraft will be limited by its aerodynamic capabilities and/or the structural strength of its airframe. Gusts in the atmosphere cause loads on the aircraft that are over and above those produced in smooth air. All aircraft must be designed to be strong enough to carry the normally expected flight loads and the extra loads induced from encounters with turbulent air. The FARs define these gust conditions depending on the aircraft type. It is reassuring that the structural margins built into certified aircraft designs are significant. Even when encountering the most severe turbulence, one can be confident that the aircraft will be strong enough to withstand all of the anticipated flight loads.

5-Question Self-Assessment Quickquiz

For Further Thought or Discussion

Think about some of the structural issues in designing a wing for a fully aerobatic aircraft. 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 aircraft: glider, single-engine general aviation aircraft, and a small business jet.

Other Useful Online Resources

To understand more about an aircraft’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 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