5.4 Examples
5.4.2 Case Study – Balance of Forces and Wind Interference
Performance data for a number of aircraft are provided in Table 16.
These data include mass, TAS, coefficients of lift and drag, thrust per mass, and the component of thrust along the x-coordinate. Wind data are also provided. The aircraft are to fly a given course, at specified altitudes, atmospheric conditions (i.e., temperature, pressure, and wind), and angles of attack that may vary depending on their maneuvering positions. Ignore magnetic variations in this problem. Using the data presented in Table 16, for the cases 1 to 10 provided, calculate:
(a) angle of att ack in radians, (b) angle of wind with respect to the aircraft heading in degrees and radians, (c) wind magnitude including half of the gust factor, (d) crosswind component of the wind, (e) headwind component
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of the wind, (f) aircraft crab angle, (g) true heading, (h) ground speed, (i) air pressure in inHg and kPa, (j) density, (k) assuming the standard- adiabatic lapse rate (1.98 °C/1,000 ft), calculate the temperature for the given altitudes, (l) ratio of the lift to the drag, assuming wing surface area as the reference area, (m) drag force per unit area, (n) lift force per unit area, (o) drag force given wing surface area, (p) lift force given wing surface area, (q) total thrust along the x-coordinate using its dimensionless value (i.e., thrust per 0.5 v2A, (r) total thrust, (s) required velocity for the aircraft to remain balanced, and (t) compare calculated and given thrusts and conclude if the calculated thrust (maximum available thrust) is sufficient for the aircraft to remain aloft given the specific scenarios.
Table 16 presents the performance data for the list of the aircraft in this case study. To calculate the wind magnitude to be used to find the vertical and horizontal components of the wind, half of the gust factor—Gf (w G)—is to be added to the given wind magnitude (w w Gf /2), where G is the gust. The Angle of Attack (AOA) is the angle between the relative air with respect to the aircraft’s chord line of its airfoil ().
As the aircraft pitch angle increases (positive and negative pitch angles may represent ascending and descending positions, respectively), the component of weight parallel to the motion of the aircraft (mgSin) resists or encourages the forward motion of the aircraft, respectively. The crosswind (wSin) and headwind (wCos) components of the wind are the cosine and sine of the angle between the nose and the wind ( ) multiplied by the wind magnitude (w)—considering the gust factor (Gf )—
where is the true course. The crab angle is the arc(sine) of the ratio of the crosswind component to the TAS (CA arc Sin(wSin /TAS)) using vector algebra (Table 17).
TABLE 16 Aircraft angle of attack, mass, and wind direction.
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Depending on the direction of the wind (easterly or westerly), the aircraft is to crab into the wind; therefore, to calculate the aircraft crab angle, the wind direction is to be added to or deducted from the true course to obtain the crab angle—THdg . Crosswinds from the right or left sides are to be added to or deducted from the true course to result in the true heading.
If the magnetic variation were to be included, the magnetic heading would have been calculated by adding it to (westerly variations) or deducting it from (easterly variations) the true heading (THdg Var MagHdg), and the results would also have been used for landing and takeoff calculations.
The effect of geographic variation is ignored in this exercise. The magnetic variation may also be added to (westerly variations) or subtracted from (easterly variations) the difference between the two angles as well (Var).
Note that this data may then be used to obtain the compass heading. When selecting the runway to take off or land, however, the variation is to be combined with the true wind data to generate the wind magnetic heading, which can then be used to identify the crosswind and headwind components given the runway magnetic heading.
The ground speed (GS) is the speed of the aircraft relative to the ground.
Depending on the wind direction, headwind, or tailwind, the ground speed will be less (headwind) or more (tailwind) than the TAS. Note that GS
TASCos —the headwind/tailwind component is calculated based on the angle between the wind direction and longitudinal axis along the fuselage.
The pressure altitude (PAL) may be assumed the same as the flight level (FL) in this scenario. Given the elevation (H) and the standard atmospheric condition for pressure altitude (101,325 Pa, 1 atm, 29.92 inHg, 14.67 psi), pressure (P) at the given altitude (H) may be calculated in inches of mercury (inHg) and then converted to kPa—PA H 1000(29.92 PinHg) (Table 18).
Knowing this pressure (PinHg), the outside air temperature (T OAT), and the air specific gas constant for dry air (Rd 287.0 J/kgK), the air density may be calculated— P/RT. This density is then used in combination with the speed of the aircraft (TAS) to find the multiplier for the lift and drag coefficients—(0.5 v2)—to obtain the force per unit area or pressure (P F/A) for both drag and lift forces. Note that the wind tunnel experiments and the related data are based on a common reference plane when deducing the lift and drag coefficients. Once the wind tunnel tests are conducted and the lift (FL) and drag (FD) forces are measured based on the wing surface
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area (A) as the reference area, the lift and drag coefficients can then be calculated. For this problem, the lift and drag coefficients are already provided (Table 19). Drag and lift coefficients may also be adopted from scholarly sources or obtained from wind tunnel experiments [133].
The remaining steps are as follows: (a) using the lift coefficient, assuming that the lift force equals the vertical component of weight, calculate the minimum velocity magnitude required to ensure the force balance, ve 2mgCos / ACL ; (b) lift force is essentially the same as the vertical component of the weight and is a function of the lift coefficient,
0.5 2
L e L
F A v C ; (c) given the calculated balancing velocity and using the given drag coefficient, the drag force can be calculated, FD0.5A v C e2 D; (d) thrust is essentially the same as the drag plus the horizontal component of the weight, and therefore it depends on the pitch angle, FT FDmgSin;
(e) thrust per mass is obtained as FT /m; (f) minimum available thrust is provided but also can be calculated using the total power per cruise speed (assume TAS), FTlimit Power/TAS, where is the efficiency of the engine; (g) the horizontal component of the thrust is calculated using the dimensionless horizontal component of the thrust multiplied by the equivalent lift or drag forces per associated lift and drag coefficients (0.5 Av2), but based on the TAS, Fx 0.5FT xTAS A2 ; and (h) eventually, by balancing the forces applied along the aircraft motion (i.e., components of weight, thrust, and drag parallel to the fuselage) (F FT FD Fw), and dividing the result by the total takeoff mass, the acceleration (a F/m, if any, can be estimated—it equals zero if the forces are correctly balanced (Table 20).
This example considers the balance of the forces along the direction of the aircraft motion. The total thrust obtained from these calculations may be compared with the available thrust for the given cases. If the calculated value is under the available thrust for the specific angle of attack, performance parameters, and given atmospheric conditions, the aircraft is able to generate the thrust required to remain aloft.
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TABLE 17 Relative angle of wind, wind components, crab angle, a nd ground speed.
TABLE 18 Outside air properties, pressure altitude, and wing sur face area.
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TABLE 19 Aircraft lift, drag, and thrust ratio coefficients.
TA BLE 20 Aircraft lift, drag, thrust, and balance velocity magnitu de.
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C H A P T E R