Unit 1: Steady Unaccelerated Flight - Subjective Questions

Steady unaccelerated flight is a flight condition in which the aircraft's velocity remains constant in both magnitude and direction. Therefore, its linear and angular accelerations are zero.

For steady, straight, and level flight:

• Horizontal equilibrium: Thrust balances drag.
  
• Vertical equilibrium: Lift balances weight.
  
• The flight-path angle is zero, so the aircraft neither climbs nor descends.
• The resultant force and resultant moment acting on the aircraft are zero.

Thus,

These conditions allow the aircraft to maintain constant altitude, speed, and heading.

The four fundamental forces acting on an aircraft are:

• Lift (): The aerodynamic force acting approximately perpendicular to the relative airflow. It supports the aircraft against its weight.
• Weight (): The gravitational force acting vertically downward through the aircraft's center of gravity. It is given by .
• Thrust (): The propulsive force produced by the engine-propeller or jet system. It generally acts forward along or close to the aircraft's longitudinal axis.
• Drag (): The aerodynamic resistance acting opposite to the direction of motion through the air.

In steady level flight, the forces are balanced:

Lift and weight form the vertical force pair, while thrust and drag form the horizontal force pair.

Consider thrust and drag acting parallel to the flight path, lift acting perpendicular to it, and weight acting vertically downward. Resolving forces parallel and perpendicular to the flight path gives:

Along the flight path:

Normal to the flight path:

Here:

• is flight speed.
• is the radius of curvature of the flight path.
• is the flight-path angle.
• is aircraft mass.

For steady rectilinear flight, and . Therefore,

For steady level flight, , producing:

For steady flight along a straight path, acceleration is zero. The governing equations are:

Steady climb: When ,


Thrust must overcome both drag and the component of weight acting down the flight path.

Steady glide: With no thrust, , and the descending flight-path angle may be represented by a negative . In terms of the positive glide angle below the horizontal:

Steady level flight: For ,

Thus, weight has no component along the flight path in level flight, assists motion in a glide, and opposes motion during a climb.

A drag polar expresses the relationship between the drag coefficient and lift coefficient of an aircraft.

Aircraft drag is commonly divided into:

• Parasite drag, represented by the zero-lift drag coefficient .
• Induced drag, which varies approximately with .

For a finite wing, the induced-drag coefficient is:

Defining

gives the parabolic drag polar:

Here:

• is Oswald's span-efficiency factor.
• is wing aspect ratio.
• is the induced-drag factor.

The polar shows that total drag increases at both very low lift coefficients, where parasite drag dominates, and high lift coefficients, where induced drag dominates.

Since lift and drag are

the lift-to-drag ratio is

Using the parabolic drag polar,

For maximum , differentiate with respect to and set the result to zero:

This gives:

Therefore,

At this condition, parasite drag equals induced drag:

The maximum ratio is:

For the parabolic drag polar

the maximum lift-to-drag ratio occurs when:

• Parasite and induced drag coefficients are equal:
  
• The corresponding lift coefficient is:
  
• The total drag coefficient is:
  
• The maximum lift-to-drag ratio is:
  
• The glide angle is minimum for an unpowered aircraft because:
  
• Drag and thrust required are minimum in steady level flight.

Graphically, maximum is found where a line drawn from the origin is tangent to the versus drag-polar curve.

In steady level flight,

The total drag is:

Therefore, thrust required is:

where

The thrust-required curve is U-shaped because:

• At low speed, a high lift coefficient is required, causing large induced drag proportional to .
• At high speed, parasite drag dominates and increases as .
• At an intermediate speed, total drag and thrust required are minimum.

The minimum-thrust condition corresponds to maximum , where induced drag equals parasite drag.

Power required is the rate at which work must be done to overcome drag:

Using

we obtain:

Thus,

The power-required curve has the following characteristics:

• At low speed, induced power dominates and varies as .
• At high speed, parasite power dominates and varies as .
• The curve has a minimum at an intermediate speed.
• Minimum power required does not occur at the same speed as minimum thrust required.

For a propeller-driven aircraft, the minimum-power condition is particularly important because it is associated approximately with maximum endurance.

The power required is:

For minimum power:

Therefore,

Since parasite power is and induced power is , this condition gives:

In coefficient form, minimum power occurs when:

Minimum thrust required, however, occurs when:

Therefore, minimum power occurs at a higher lift coefficient and lower speed than minimum thrust. Their speed relationship is:

Thus, the minimum-power speed is approximately of the minimum-thrust speed.

Turbojet aircraft:

• Thrust available is often approximated as nearly constant with speed over the subsonic operating range.
• Consequently, the thrust-available curve is approximately horizontal on a thrust-versus-speed graph.
• Actual thrust may change because of ram effects, engine characteristics, and altitude.

Propeller-driven aircraft:

• Engine shaft power may be approximately constant over a useful speed range.
• Propulsive power is related to thrust by:
  
• Hence,
  
• Thrust available generally decreases as speed increases when power available is treated as constant.

Thus, jets are commonly analyzed using approximately constant thrust available, whereas propeller aircraft are commonly analyzed using approximately constant power available.

Power available is related to thrust available by:

Jet aircraft:

• If thrust available is approximately constant, power available increases approximately linearly with speed.
• Therefore, the curve is nearly a straight line through the origin in an idealized model.

Propeller-driven aircraft:

• Shaft power may remain approximately constant with speed.
• Useful propulsive power is:
  
  where is propeller efficiency.
• If is assumed constant, power available is approximately horizontal.
• In reality, propeller efficiency changes with airspeed, so the curve is not perfectly constant.

The difference between power available and power required is called excess power, which determines climb capability.

The intersections of the thrust-available and thrust-required curves represent equilibrium flight speeds at which:

At these points, excess thrust is zero, and the aircraft can maintain steady level flight without accelerating.

Important implications are:

• The lower-speed intersection represents the minimum steady level-flight speed permitted by available thrust, provided it is above stall speed.
• The higher-speed intersection represents the maximum steady level-flight speed.
• Between the intersections:
  
  so excess thrust is available for acceleration or climb.
• Outside the intersections:
  
  so steady level flight cannot be maintained.

The maximum vertical separation between the two curves gives maximum excess thrust, which is related to the maximum climb angle for a jet aircraft.

The thrust-to-weight ratio is defined as:

where is available thrust and is aircraft weight.

Its importance includes:

• A higher provides greater acceleration.
• It improves takeoff performance and reduces takeoff distance.
• It increases climb angle and climb rate by providing more excess thrust or excess power.
• It can increase maximum level-flight speed.
• For steady level flight:
  
• The minimum thrust-to-weight ratio needed to sustain level flight is therefore:
  

Although a large improves performance, it can increase engine mass, fuel consumption, cost, and structural requirements.

The wing loading of an aircraft is its weight per unit wing planform area:

Its effects include:

• Stall speed:
  
  Hence, greater wing loading increases stall speed.
• Takeoff and landing: Higher wing loading generally increases takeoff and landing speeds and runway distance.
• Maneuverability: Higher wing loading requires greater speed or lift coefficient to generate a specified load factor and may increase turn radius at a given speed.
• Gust response: Aircraft with high wing loading are generally less sensitive to atmospheric gusts.
• Drag and cruise: Wing loading affects the lift coefficient required at a given speed and therefore changes induced drag and the speeds for minimum drag and minimum power.

Thus, wing loading is a major aircraft-sizing parameter that involves trade-offs among low-speed performance, cruise performance, and structural design.

In steady level flight, lift equals weight:

The lift equation is:

The lowest speed occurs when the wing reaches its maximum usable lift coefficient . Therefore:

Solving for minimum velocity:

Equivalently, using wing loading:

This is normally identified with the stall speed for the specified configuration and load factor. It increases with aircraft weight and wing loading but decreases with air density, wing area, and maximum lift coefficient.

Effect of weight:

Characteristic speeds obtained from the lift equation vary as . Thus,

An increase in weight shifts the thrust-required curve toward higher speeds and increases minimum drag because:

Effect of altitude:

As altitude increases, air density decreases. The true airspeed required to generate a specified lift therefore increases as:

Consequently:

• True stall speed increases with altitude.
• True minimum-drag speed increases with altitude.
• The thrust-required curve shifts toward higher true airspeeds.
• For the ideal parabolic drag model, minimum drag at a given weight is independent of density.
• Equivalent or indicated characteristic speeds remain approximately unchanged for fixed weight and configuration.

Available engine thrust or power usually decreases with altitude, reducing excess performance.

Minimum-drag speed :

• Occurs at maximum .
• Parasite drag equals induced drag.
• It is the speed for minimum thrust required.
• For an unpowered aircraft, it gives the best glide angle and maximum still-air glide range.
• For a jet aircraft, it is approximately associated with maximum endurance under ideal assumptions.

Minimum-power speed :

• Occurs when induced drag is three times parasite drag.
• It is lower than minimum-drag speed.
• The relationship is:
  
• For a propeller-driven aircraft, it is approximately associated with maximum endurance.
• For a glider, it corresponds to minimum sink rate.

Therefore, maximum range and maximum endurance generally occur at different aerodynamic operating conditions.

The thrust required equals drag:

where

For minimum thrust:

Therefore:

The corresponding speed is:

Substituting for and gives:

At this speed, induced drag equals parasite drag. Since minimum thrust equals minimum drag:

Hence:

The minimum steady level-flight speed is:

Substituting the given values:

Therefore:

This is approximately:

Physical meaning:

• At this speed, the aircraft must operate at to balance its weight.
• Below this speed, the required lift coefficient would exceed .
• The wing would stall unless weight, load factor, or configuration changed.
• In practice, operational flight speeds are maintained above this theoretical minimum to provide a safety margin.