Flight Dynamic Characteristics of Wide-Body Aircraft with Wind Gust and Turbulence
Abstract
:1. Introduction
1.1. Stability and Performance Optimization—A Design Dilemma
1.2. Extended Flight Envelope and Unconventional Aircraft Design
1.3. Atmospheric Influence on Stability
1.4. Flight Dynamic Models
2. Aircraft Non-Linear Dynamics
2.1. Equations of Motion (EOMs)
2.2. COESA Atmospheric Models
2.3. Mathematical Representation of Turbulence Modeling
3. Modeling Methodology
3.1. Schematic of UDF
3.2. Axis System
3.3. Designing of Simulink Canvas
3.4. Validation of Non-Linear Model
- Validation using the Boeing 747-200: A state–space model was built using the values for the Boeing 747-200 taken from [61]. The characteristic equation was formulated using frequencies and the damping ratio calculated for Phugoid and short periods. The frequencies were found to be in agreement with the frequencies captured during UDF simulations for the control input, thus validating the UDF accuracy (see Table 4). Moreover, the trim conditions calculated for the approach and cruise conditions were found to be in agreement with the data published in [61] and provide further authentication of the accuracy of the non-linear model. The time history plots obtained from the present canvas also predicted the same trim conditions under the same operating conditions as those mentioned in [61] and had a maximum error of 5.1%.
- 2.
- Validation using a Civil Airplane Model (RCAM): Further validation of the designed canvas was performed using RCAM (an open-source model from the Group of Aeronautical Research and Technology in Europe (GARTEUR) [62,63]). The non-linear UDF was updated with the RCAM model parameters; simulation results were found to be in agreement with the RCAM results available in [62]; this further validated the functionality and structure canvas for computing non-linear flight dynamics. After successful validation, simulations were carried out for two flight conditions: (i) at sea level and (ii) at 40,000 feet cruise flight conditions; the results are discussed in the subsequent sections.
4. Aircraft Trim Conditions and Control Response
4.1. Sea Level Flight Condition
- (1)
- Without control inputs: The model was updated using the sea level coefficients and was subjected to various initial conditions. The initial conditions acted as a disturbance in the system and the aircraft free response was captured; Figure 6 shows the variation in the longitudinal variables. The aircraft entered phugoid mode, which died out subsequently and the aircraft achieved trim conditions. The trim longitudinal velocity achieved was approximately 225 m/s (documented value of 221 m/s in literature [63]).
- (2)
- With elevator control input: In order to evaluate the free response, a 5° impulse elevator input was initiated (acting as a longitudinal disturbance); the aircraft entered a long-period oscillation that subsequently died out; then, the aircraft returned to its trimmed state (see Figure A7 of Appendix B).
- (3)
- With lateral control inputs: Under the influence of lateral controls (aileron and rudder impulse inputs), the longitudinal states remained unaffected; the lateral states depicted short-period oscillation as a response to lateral control deflection, which quickly died out and regained its original position. The bank angle ϕ recovered its original zero position; however, the yaw angle ψ reached a new trim position (as expected from theoretical knowledge—see Figure A9 and Figure A10 of Appendix B).
4.2. Cruise Flight Condition
5. Aircraft Response to Wind Turbulence and Wind Shear
- (1)
- Wind gust. In order to check the aircraft response to a wind gust, a constant velocity vertical wind gust of 20 ft/s was generated at sea level flight conditions for 50 s using the Simulink wind shear and gust blocks. As expected, the aircraft angle of attack was disturbed for the period of wind gust (see Figure 9a); once the gust was removed, the aircraft regained its trim conditions. The aircraft response to a 3D gust of 10 ft/s in all directions was also simulated; the response of the aircraft to the 3D gust is presented in Figure 9b. All the state variables were disturbed from its trim conditions for the period of gust, and once the gust was removed, the aircraft regained position.
- (2)
- Atmospheric turbulence. In order to analyze the behavior of the aircraft for atmospheric turbulence, the turbulence block was used during the simulations. As wind turbulence is a three-dimensional phenomenon, all the modes are affected. However, the random fluctuations were small in amplitude and continued for the entire duration of the simulation; therefore, all the modes kept fluctuating about their free-response behavior.
- (3)
- Atmospheric conditions at landing. As the landing approach is the most critical phase of flight and sudden changes in wind profile in the proximity of the ground have catastrophic effects on flight safety; therefore, a detailed analysis with a variety of wind gust profiles was conducted to ascertain the free response of the aircraft during landing conditions.
- (i)
- Landing in headwind and tailwind conditions. When considering an aircraft on a standard 3° glide slope that was subjected to 25 ft/s headwinds (see Figure 10), the effective speed of the aircraft increased due to headwind and generated more lift. As a result, the aircraft gained height and was disturbed from its preset glide course. The aircraft experienced an overshoot of approximately 3000 ft from the touchdown point. Similarly, an undershoot of approximately 4000 ft was recorded for a 25 ft/s tailwind condition due to the reduction in the effective forward speed. The undershoot and overshoot were increased to approximately 6000 ft once the wind velocity was increased to 30 ft/s.
- (ii)
- Landing in crosswind conditions. A case of a 25 ft/s 90° crosswind condition was also simulated; as expected, the aircraft started drifting in the direction of the crosswind. Under the simulated conditions, the aircraft became offset from its landing approach by approximately 450 ft (see Figure 11), with the glide slope and touchdown point being negligibly affected. Due to inherent coupling, the roll and yaw rates along with the variation in roll and yaw angle were observed. Once the crosswind conditions were reduced to 10 ft/s, an offset was reduced to 180 ft from the initial course.
- (iii)
- Landing in vertical wind shear conditions. Vertical wind shear simulations were also conducted for a wind velocity of 25 ft/s. During an upward gust condition, the effective angle of attack increased for the aircraft, thus increasing the overall lift of the aircraft. The aircraft momentarily gained height, departing from its preset 3° glide slope before diving again to a glide slope of approximately 10° and overshooting the touchdown point by a couple of hundred feet (see Figure 12). In the case of a downward gust condition of 25 ft/s, the aircraft’s angle of attack decreased, and this put the aircraft into a steep glide slope of approximately 11~12° and undershot the touchdown point by approximately 4000 ft.
- (iv)
- Landing in microburst conditions. A microburst with a maximum velocity of 25 ft/s was also simulated. As the physical spread of the velocity profile was higher, the burst simulations were started at 32,000 ft short of the actual touch-down point. In the first phase, a headwind of 25 ft/s was built and sustained for 10,000 ft and then transitioned into a sustained downwind of 25 ft/s for another 10,000 ft. In the last phase, the downwind was transitioned into a tailwind of 25 ft/s, which continued until touchdown. As expected, initially, the aircraft gained height in the headwind condition due to an increase in the effective speed (departing from its 3° glide course), followed by a sharp dip under the influence of the sustained downwind. The aircraft momentarily leveled off while transitioning from a downwind condition to a tailwind condition; it then glided down at a sharp angle of ~14.5° under the influence of tailwind (as the aircraft’s forward speed was reduced). Aircraft touchdown was approximately 8000 ft short of the actual touchdown point (see Figure 13); additionally, the descent rate was also high. It was the high glide slope angles and higher descent rate that led to aircraft crashes while encountering microbursts short of the runway. This dangerous phenomenon associated with aircraft crashes encountering microbursts during the approach was successfully simulated and analyzed.
6. Conclusions
7. Future Work
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Nomenclature
I | Inertia tensor |
Ix, Iy, Iz | Moment of inertia about axis (rotational inertia that resists change in the rotational velocity of an object on its axis |
Ixz, Iyz, Ixy | Product of inertia (feature of an object that describes an imbalance relative to a defined set of coordinate axes) |
, M, N | Moments about x, y, and z-axes in a body frame, respectively. corresponds to rolling, M corresponds to pitching, and N corresponds to yawing moment of the aircraft |
Velocity components about x, y, and z-axes in a body frame | |
Translational accelerations about x, y, and z-axes in a body frame | |
Angular velocities about x, y, and z-axes in a body frame | |
Rotational accelerations about x, y, and z-axes in a body frame | |
Euler angles of aircraft about x, y, and z-axes of inertial frame | |
Rate of change of Euler angles about x, y, and z-axes of inertial frame | |
Position coordinates of the aircraft with respect to inertial frame | |
Translational velocities of the aircraft with respect to inertial frame | |
Direction cosines for transformation of vectors from body frame to inertial frame |
Appendix A
- Assumption. While formulating EoMs for aircraft (747-200), the following assumptions were made:
- (a)
- The aircraft is a rigid body.
- (b)
- The Earth is taken as an inertial frame—stationary flat surface.
- Formulation of non-linear EoMs: According to Newton’s second law, the motion of a rigid body in an inertial frame is governed by Equation (A1).
- 3.
- Calculation of external forces: To solve Equation (A5), we need to calculate external forces applied on the aircraft. External forces in a body frame are a combination of gravitational force, aerodynamics force, and propulsive forces acting on the aircraft.
- (A)
- Gravitational force: In an inertial frame, weight is always acting downward toward the center of the Earth (z-axis of inertial frame) and can be represented as Equation (A16). Gravitational force into the body frame can be calculated using direction cosines as shown in Equation (A17) using Figure A1.
- (B)
- Propulsion forces: The Boeing 747-200 has four engines installed on its wings, two on each side (ref. Figure A2). It is assumed that the thrust produced by all the engines is in line with the x-axis of the body frame. Therefore, a simple expression can be used to express the propulsion forces (ref. Equation (A18)). These propulsive forces are throttle-dependent forces and require the throttle position as an input condition.
- (C)
- Aerodynamics Forces: There are three forces acting along the axes of the body frame. These are normal, axial, and side forces. Generally, the forces are captured in the wind axes by definition and need to be transformed back to body axes using a transformation matrix (Ref Equation (A19)). The transformation variables are selected as per Figure A3.
- 4.
- Calculation of External Moments: External moments in a body frame are a combination of moments generated by gravitational force, aerodynamics force, and propulsive forces.
- 5.
- Calculation of aerodynamics forces: Aerodynamic forces are a function of a lot of variables, which include translational velocities, angular velocities, the rates of change of velocities, control input, etc. The aerodynamic forces and moments can be expressed as a function of all the motion variables; however, only the significant variables are retained, and the rest are neglected.
- 6.
- State space representation: Equations (A5), (A10), (A12), and (A14) represent the equations of motion and can be easily represented in a state space notation. For ease, the input conditions and are placed in a separate column vector as per standard procedures of state space representation (ref. Equation (A27)).
- 7.
- Simulink modeling: The function formulated in Equation A27 was interpreted in a Simulink environment as a user-defined function. The sketch is shown in Figure A4. and are provided as inputs; the block calculates , which is passed through an integrator to obtain the state space . is routed again to the block as an input. The cycle continues until convergence is achieved.
- 8.
- Wind turbulence modeling: To introduce the effect of wind turbulence and atmospheric gust/shear, Simulink blocks are separated; the Dryden turbulence model, wind shear, and wind gust were incorporated into the canvas. The schematic of the improved canvas is shown in Figure A5. The density variation with height was captured through the COESA atmospheric block. The linear and angular wind velocities were summed up with the velocities obtained from the interpreted MATLAB function and rerouted to the interpreted MATLAB function block for subsequent iteration until the solution is converged.
Appendix B
- (1)
- Without Control Inputs.
- (2)
- With Elevator Control Input.
- (3)
- With Aileron/Rudder Control Inputs.
- (1)
- Landing Under Headwind and Tailwind Conditions.
- (2)
- Landing Under Crosswind Conditions.
- (3)
- Landing Under Vertical Wind Shear Conditions.
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Input/Output | Definition | Remarks |
---|---|---|
U(1) | Aileron deflection for roll control | Input |
U(2) | Elevator deflection for pitch control | Input |
U(3) | Rudder deflection for yaw control | Input |
U(4), U(5) | Throttle controls for engines | Input |
du/dt, dv/dt, dw/dt | Rate of change of linear velocities | Output |
dϕ/dt, dθ/dt, dψ/dt | Rate of change of Euler velocities | Output |
Parameter | Approach | Cruise | Units | ||
---|---|---|---|---|---|
W | 564,000 (255,826) | 636,636 (288,773) | Ib (kg) | ||
S | 5500 (~510.9) | 5500 (~510.9) | ft2 (m2) | ||
c | 27.3 (8.3) | 27.3 (8.3) | ft (m) | ||
cg | 0.25 | 0.25 | |||
Ixx | 13,700,000 (18,574,705) | 18,200,000(24,675,886) | Slug-ft2 (kg-m2) | ||
Iyy | 30,500,000 (41,352,447) | 33,100,000 (44,877,574) | Slug-ft2 (kg-m2) | ||
Izz | 43,100,000 (58,435,753) | 49,700,000 (67,384,152) | Slug-ft2 (kg-m2) | ||
Ixz | 830,000 (1,125,328) | 970,000 (1,315,143) | Slug-ft2 (kg-m2) | ||
Longitudinal Coefficients | Lateral Coefficients | ||||
Coeff | Approach | Cruise | Coeff | Approach | Cruise |
CL1 | 1.76 | 0.52 | Clb | −0.281 | −0.095 |
CD1 | 0.263 | 0.045 | Clp | −0.502 | −0.32 |
CM1 | 0 | 0 | Clr | 0.195 | 0.2 |
CD0 | 0.0751 | 0.0305 | Cyb | −1.08 | −0.9 |
Cdu | 0 | 0.22 | Cyp | 0 | 0 |
Cda | 1.13 | 0.5 | Cyr | 0 | 0 |
CL0 | 0.92 | 0.29 | Cnb | 0.184 | 0.21 |
CLu | −0.22 | −0.23 | Cnp | −0.222 | 0.02 |
CLa | 5.67 | 5.5 | Cnr | −0.36 | −0.33 |
CLa. | 6.7 | 8 | Clda | 0.053 | 0.014 |
CLq | 5.65 | 7.8 | Clδr | 0 | 0.005 |
Cmu | 0.071 | −0.09 | Cyδa | 0 | 0 |
Cma | −1.45 | −1.6 | Cyδr | 0.179 | 0.06 |
Cma. | −3.3 | −9 | Cnδa | 0.0083 | −0.0028 |
Cmq | −21.4 | −25.5 | Cnδr | −0.113 | −0.095 |
CDδe | 0 | 0 | |||
CLδe | 0.36 | 0.3 | |||
Cmδe | −1.4 | −1.2 |
Parameter | Units | Approach | Cruise |
---|---|---|---|
H | ft (m) | 0 | 40,000 (12,192) |
M | - | 0.198 | 0.9 |
TAS | ft/s (m/s) | 221.0 (67.4) | 870.91 (265.24) |
ρ ρ | Slug/ft3 (kg/m3) | 0.0023769 (4.6 × 10−6) | 0.0005873 (1.14 × 10−6) |
Q | Ib/ft2 (N/m-) | 58.0 (2777) | 222.72 (10,664) |
α | Deg | 8.5 | 2.4 |
Parameter | Literature Value [61] | Present Work | Error |
---|---|---|---|
Approach Velocity | 221 ft/s | 225 ft/s | 1.8% |
Cruise Velocity | 871 ft/s | 882 ft/s | 1.2% |
Time Period (Phugoid) | 45.9 s | 43.5 s | 4.3% |
Time Period (SP) | 9.13 s | 9.6 s | 5.1% |
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Mehmood, K.; Ali Shah, S.I.; Ali Shams, T.; Mumtaz Qadri, M.N.; Khan, T.A.; Kukulka, D. Flight Dynamic Characteristics of Wide-Body Aircraft with Wind Gust and Turbulence. Fluids 2023, 8, 320. https://doi.org/10.3390/fluids8120320
Mehmood K, Ali Shah SI, Ali Shams T, Mumtaz Qadri MN, Khan TA, Kukulka D. Flight Dynamic Characteristics of Wide-Body Aircraft with Wind Gust and Turbulence. Fluids. 2023; 8(12):320. https://doi.org/10.3390/fluids8120320
Chicago/Turabian StyleMehmood, Kashif, Syed Irtiza Ali Shah, Taimur Ali Shams, Muhammad Nafees Mumtaz Qadri, Tariq Amin Khan, and David Kukulka. 2023. "Flight Dynamic Characteristics of Wide-Body Aircraft with Wind Gust and Turbulence" Fluids 8, no. 12: 320. https://doi.org/10.3390/fluids8120320