Unit 4: Maneuver Performance - Subjective Questions

Turning performance describes an aircraft's ability to change its flight direction while satisfying aerodynamic, structural, propulsive, and operational constraints.

The principal parameters are:

• Turn radius, : Radius of the circular flight path. A smaller radius represents a tighter turn.
• Turn rate, : Rate at which the aircraft changes heading, expressed in or :
  
• Load factor, : Ratio of lift to aircraft weight:
  
• Bank angle, : Inclination of the lift vector from the vertical during a level turn.
• Airspeed, : Strongly influences both turn radius and available aerodynamic load factor.

Turning performance may be classified as instantaneous, where stored kinetic energy may be lost, or sustained, where speed and altitude can be maintained throughout the turn.

Consider an aircraft in a coordinated level turn at bank angle . Resolve lift into vertical and horizontal components.

For vertical equilibrium:

The horizontal component supplies centripetal force:

Dividing the second equation by the first gives:

Therefore, the turn radius is:

Since , the turn rate is:

Using and , these may also be written as:

Thus, at a given speed, increasing bank angle or load factor decreases turn radius and increases turn rate.

During a coordinated level turn, the vertical component of lift must balance the aircraft weight:

Since load factor is defined as :

Important consequences are:

• At , .
• At , .
• As approaches , approaches zero and the theoretical load factor approaches infinity.
• The stall speed rises with load factor according to:
  

Therefore, steeply banked level turns require substantially greater lift, increase structural loading, and raise the stall speed. In practice, aerodynamic and structural limitations prevent the bank angle from reaching in a level turn.

The achievable load factor is restricted by several constraints:

• Aerodynamic constraint: Maximum lift coefficient limits the available lift:
  
  At low speed, this is usually the dominant restriction.
• Structural constraint: The airframe has specified positive and negative limit load factors. Exceeding the ultimate load may cause structural failure.
• Stall constraint: When the required angle of attack exceeds the stalling angle, lift cannot increase further.
• Physiological constraint: High positive load factors can cause grey-out or blackout, while negative load factors can cause red-out and discomfort.
• Propulsive constraint: In a sustained maneuver, thrust must balance the increased drag associated with high lift.
• Control constraint: Elevator authority, control-surface deflection, and stability characteristics may limit attainable load factor.
• Speed constraint: Dynamic pressure, flutter, compressibility, and the design dive speed restrict high-speed maneuvers.

The permissible load factor is therefore the lowest limit imposed by these constraints at the flight condition considered.

For a coordinated level turn:

At a specified speed, radius decreases as load factor increases. Therefore, the minimum radius occurs at the maximum permissible load factor :

The corresponding bank angle is:

This result assumes:

• A coordinated and level turn
• Constant speed
• Sufficient lift and control authority
• No sideslip
• The specified load factor can be produced without stalling

If the aircraft is aerodynamically limited, must be replaced by:

If both aerodynamic and structural restrictions apply, the usable load factor is:

The corner velocity is the speed at which the positive aerodynamic stall boundary intersects the positive structural load-factor limit. It is the lowest speed at which the maximum structural load factor can be developed.

For the positive stall boundary:

At the corner point, . Therefore:

The level-turn radius is:

Substituting the corner conditions gives:

Hence:

Below , the aircraft stalls before reaching the structural limit. Above , the structural limit is reached first and radius generally increases with . Thus, under the idealized aerodynamic and structural model, the tightest level turn occurs at or near corner velocity.

For a coordinated level turn, the turn rate is:

At a fixed speed, turn rate increases with load factor. Thus, the maximum turn rate at that speed is:

When stall and structural limits are considered together, the usable load factor is:

In the idealized instantaneous-turn model, the maximum turn rate normally occurs at the corner velocity:

Therefore:

The corresponding turn rate in degrees per second is:

For sustained turns, available thrust and drag may reduce the usable load factor, so the actual maximum sustained turn rate may occur at a different speed.

Instantaneous turning performance describes the turn capability available at a particular instant, even if the aircraft cannot maintain its speed or altitude.

• It is mainly restricted by stall, structural load factor, and control authority.
• The aircraft may use stored kinetic or potential energy.
• Speed may decrease rapidly because drag exceeds thrust.

Sustained turning performance requires the aircraft to maintain the maneuver without losing speed or altitude.

• Thrust must equal drag:
  
• Lift must support the required load factor.
• Available power or thrust is often the principal restriction.
• Sustained load factor is generally lower than instantaneous load factor.

Therefore, an aircraft may briefly achieve a high turn rate at the structural or aerodynamic limit but may be unable to sustain that rate because of the large induced drag generated during the maneuver.

The aerodynamic load factor available at a given true airspeed is:

From this relation:

• Increasing altitude: Air density decreases, so less lift and load factor are available at the same true airspeed. A higher true airspeed is required to produce the same load factor.
• Increasing wing loading : Reduces available load factor and increases stall speed. This generally increases minimum turn radius and reduces low-speed maneuverability.
• Increasing : Increases available lift, lowers stall speed, and improves low-speed turning performance.

The stall speed is:

Thus, lower wing loading, greater maximum lift coefficient, and greater air density improve aerodynamic turning capability. Actual sustained performance also depends on engine thrust and the drag generated at high lift coefficients.

During a pull-up from level flight, the center of curvature lies above the aircraft. Lift acts toward the center, while weight acts away from it. The normal-force equation is:

Dividing by and using :

Hence, the pull-up radius is:

The angular rate of flight-path rotation is:

Therefore:

Important observations are:

• A pull-up requires at the level-flight entry point.
• Increasing decreases radius and increases angular rate.
• At , the flight path is locally straight.
• The equations describe the instantaneous condition; speed generally changes during a finite pull-up because gravity and drag affect tangential motion.

For a downward-curving maneuver initiated from level flight, take upward lift as positive and let the center of curvature be below the aircraft. The inward normal force is then:

Using and :

Therefore, the radius is:

The magnitude of the downward flight-path angular rate is:

If upward rotation is defined as positive, the signed rate is:

Key cases are:

• : Locally straight level flight
• : Ballistic or zero-lift push-over, with
• : Negative lift produces a tighter downward curvature

Negative structural limits and physiological tolerance usually make severe push-over maneuvers more restrictive than positive pull-ups.

At the same speed , the governing expressions are:

• Coordinated level turn:
  
• Pull-up from level flight:
  
• Push-over from level flight:
  

The difference results from the role of weight:

• In a level turn, weight is balanced by the vertical component of lift.
• In a pull-up, weight opposes the inward force supplied by lift.
• In a push-over, weight assists the downward curvature.

Consequently, the same numerical lift load factor does not produce the same curvature in all three maneuvers. The sign convention for and angular rate must always be stated when comparing vertical-plane maneuvers.

Let the flight-path angle be measured upward from the horizontal, and take the normal direction toward increasing . The component of weight opposite this normal direction is . Therefore:

Using and :

Thus:

Since normal acceleration is , the radius magnitude is:

At level-flight entry, and , giving:

Therefore:

• gives , representing a pull-up.
• gives zero instantaneous curvature.
• gives , representing a push-over.

This general equation shows that both lift and the normal component of weight determine vertical-plane curvature.

For a coordinated level turn:

When :

Therefore:

For a pull-up:

When , , so the same approximations result:

Thus, at very high positive load factor, the direct contribution of gravity becomes small compared with lift-induced acceleration, and horizontal and vertical-plane maneuver equations approach the same limiting form. This is a mathematical approximation; structural, physiological, and aerodynamic limits prevent indefinitely large load factors.

A maneuvering - diagram plots airspeed on the horizontal axis and load factor on the vertical axis.

Its principal boundaries are:

• Positive stall boundary:
  
• Negative stall boundary:
  
  where is negative.
• Positive structural limit: A horizontal line at .
• Negative structural limit: A horizontal line at .
• Maximum permissible speed: Commonly represented by a vertical boundary at the design dive speed or another specified limiting speed.

To construct the diagram:

1. Calculate the positive and negative stall curves over the speed range.
2. Draw the structural limit-load-factor lines.
3. Mark their intersections with the stall curves.
4. Apply the high-speed boundary.
5. Identify the permitted maneuver envelope enclosed by these limits.

Points outside the envelope correspond to stall, structural overload, or excessive speed.

Lift is given by:

Since :

At the positive stall angle, , giving:

If is the positive stall speed:

At the negative stall angle, :

Both boundaries are parabolic because varies with .

The maneuvering or corner speed is the intersection of the positive stall boundary and positive structural limit:

Below , an abrupt increase in angle of attack ideally causes a stall before the positive structural limit is exceeded. Above , the structural limit can be exceeded before the wing stalls. Maneuvering speed is therefore not a universal guarantee against all structural damage, especially under rapid, repeated, or combined control inputs.

Each point on a - diagram represents a combination of airspeed and normal load factor.

• Inside the envelope: The combination is permitted under the assumptions used to construct the diagram.
• Above the positive stall curve at low speed: The required positive lift cannot be produced; the aircraft stalls.
• Below the negative stall curve: The wing reaches its negative angle-of-attack limit.
• Above the positive structural boundary: Positive structural overload may occur.
• Below the negative structural boundary: Negative structural overload may occur.
• To the right of the speed boundary: Dynamic pressure, flutter, compressibility, or other high-speed limits may be exceeded.
• At : The aircraft may be in steady level flight.
• At : The aircraft follows an instantaneous zero-normal-force or ballistic condition.
• At the corner point: The stall and positive structural limits occur simultaneously.

The diagram defines allowable instantaneous combinations. It does not by itself prove that a maneuver can be sustained with the available thrust.

A pull-up is restricted by the following factors:

• Positive stall: The required lift may exceed .
• Structural limit: Excessive positive load factor may permanently deform or fail the airframe.
• Pilot tolerance: High positive acceleration can cause grey-out, blackout, or loss of consciousness.
• Elevator authority: The tail and elevator must generate sufficient pitching moment to reach the commanded load factor.
• Energy loss: High induced drag reduces speed unless sufficient thrust is available.
• Speed variation: As the aircraft climbs during the pull-up, kinetic energy is converted into potential energy.
• Buffet and compressibility: At high speed, shock waves, buffet, or control-force changes may restrict the maneuver.
• Ground clearance: A pull-up initiated too late may not provide sufficient trajectory clearance despite an adequate load factor.

At level entry, the radius is:

Although raising reduces radius, the usable value is limited by the most restrictive aerodynamic, structural, physiological, or control boundary.

A push-over is initiated by reducing lift below weight or by producing negative lift. Its principal limitations include:

• Negative structural limit: Aircraft generally tolerate a smaller magnitude of negative load factor than positive load factor.
• Negative stall: The wing may reach its minimum negative lift coefficient.
• Pilot tolerance: Negative acceleration causes blood to move toward the head and may result in red-out or injury.
• Restraint and cabin limits: Occupants and unsecured equipment tend to move upward during low or negative .
• Engine-system limits: Some fuel and lubrication systems cannot operate for long under negative acceleration.
• Control authority: Elevator or stabilizer capability may limit the attainable negative load factor.
• Speed increase: During a descending maneuver, gravity can rapidly increase speed and lead to high dynamic pressure or overspeed.

At level entry:

A positive pull-up normally has , whereas a push-over has and may have . Push-over capability is commonly more restricted because negative structural and physiological limits are less permissive.

Given:

1. Corner velocity

2. Minimum level-turn radius

3. Maximum instantaneous level-turn rate

Converting to degrees per second:

4. Pull-up radius at the same speed and load factor

Thus, the results are: