Analysis and study of maximum allowable turning speed for civil aircraft

The maximum allowable turning speed of an aircraft is a critical factor affecting the safety of civil aviation. Evaluating the maximum permissible taxiing turn speed for civil aircraft is an essential aspect of aircraft design. To ensure that the aircraft does not experience unbalanced forces leading to a lateral rollover due to excessive taxiing speed, an engineering estimation method is employed to study the maximum permissible taxiing turn speed for civil aircraft. This method begins with certain assumptions and calculation scenarios, followed by the derivation of a calculation model based on the analysis of aircraft forces. Lastly, a case study of a specific aircraft’s maximum taxiing turn speed is conducted in conjunction with practical engineering considerations. The study reveals that as crosswind forces on civil aircraft increase, the maximum turning speed decreases. Moreover, for lighter aircraft with a forward center of gravity, the risk of lateral rollovers is higher. These findings underscore the engineering applicability of the estimation method.


Introduction
Typically, civil aircraft employ a tricycle landing gear layout.When these aircraft taxi and make turns on the runway, the nose landing gear's wheels are steered.If the aircraft taxis at excessive speeds, it can lead to an unbalanced force moment, potentially causing the aircraft to tip over laterally.To prevent such lateral rollover incidents, it is essential to study the minimum speed at which an aircraft may experience such rollovers during taxiing and turning.This research aims to determine the maximum permissible taxiing turn speed for civil aircraft.Therefore, investigating the ground taxiing and turning characteristics of aircraft, especially focusing on the maximum permissible taxiing turn speed for civil aircraft on the runway, has become an integral component of modern civil aircraft design and development, carrying significant engineering significance and academic value.
Currently, research on aircraft ground movement characteristics, both domestically and internationally, primarily focuses on linear motions such as landing deceleration, runway shock absorption, and yaw oscillations [1][2][3][4][5][6][7] .There is relatively less research concerning civil aircraft's taxiing, turning, or lateral movements on the runway.Wang and Zhu [8] established a mathematical model for the overall aircraft motion and front-wheel steering motion relatively early.Liu and Xu [9] conducted simulation research on the steering characteristics of large civil aircraft for turning.Zhu [10] studied the calculation methods for taxiing turn radius and speed.Xu et al. [11] delved into the limits of aircraft ground taxiing turns.Qian [12] performed simulation research on aircraft front-wheel steering characteristics.From this, it can be observed that there is currently a lack of research on the maximum permissible taxiing turn speed for civil aircraft.This paper will utilize an engineering estimation method to analyze and study the maximum permissible taxiing turn speed for civil aircraft, along with providing an engineering case as a reference.

Calculation scenarios and assumptions
For the sake of convenience in conducting the study, the simplification is applied to consider the front two wheels of the aircraft's landing gear as a single point that passes through the lateral rollover axis, while the same simplification is applied to the four main landing gear wheels.It is assumed that the aircraft will undergo a lateral rollover in a direction opposite to the direction of the aircraft's taxiing and turning, along a line passing through the centers of the front and main landing gear wheels.At this moment, the main landing gear on the side corresponding to the direction of the aircraft's taxiing and turning is in contact with the ground, but the forces between this landing gear and the ground are considered to be zero.
This paper primarily considers the influences of aircraft weight, centrifugal force, engine thrust, and crosswind, as depicted in figure 1.If the restorative moment generated by the aircraft's weight about the rollover axis is greater than the combined moments of centrifugal force, engine thrust, and crosswind, then lateral rollover will not occur.However, if the aircraft's speed is excessive, the moment causing lateral rollover increases due to the significant centrifugal force during the aircraft's turn, resulting in the aircraft tipping outward.We assume that the engine thrust is directed along the aircraft's longitudinal axis.To simplify the calculations, the compression and deformation of the landing gear tires during motion are not considered.The distance 'h' between the aircraft's reference plane (Z=0) and the ground is determined by taking the absolute value of the Z-coordinate of the main landing gear wheel centers when the landing gear is extended, plus the radius of the main landing gear tires.When the aircraft is taxiing at low speeds and making turns by using the manual steering wheel, it is essential to analyze the relationship between the front wheel's steering angle and the maximum allowable turn speed to ensure that the aircraft does not experience lateral rollover.The analysis scenario is as follows: a) Crosswind: 10 knots, 35 knots, and 50 knots.b) Engine thrust: twin-engine ground idle (zero altitudes, zero airspeeds, and 15°C standard temperature).c) Landing gear parameters: coordinates of the points where the main landing gear's ground reaction forces act (X, Y, Z), coordinates of the points where the front landing gear's ground reaction forces act (X, Y, Z). d) Runway conditions: dry runway.e) Front wheel steering angle: 0 to 75 degrees.f) Aircraft weight and center of gravity: the smaller the aircraft's weight is and the further forward the center of gravity (CG), the more susceptible the aircraft is to lateral rollovers.Therefore, calculations are performed for two typical aircraft mass distribution states: operational empty weight (OEW) and maximum design taxi weight (MTW).
To facilitate the calculation of the crosswind moments, the aircraft model is simplified as a cylinder (simulating the fuselage) and a thin plate (simulating the tail fin) based on its actual dimensions, as shown in Figure 2. The diameter D of the cylinder's side is taken as the upper diameter of the fuselage, and the length L of the cylinder represents the total length of the aircraft.Therefore, the characteristic area A1 of the fuselage is calculated as D*L, and the characteristic area A2 of the tail fin is determined in the same manner.Since the lift generated by the wings provides a beneficial restorative moment during aircraft taxiing turns, making the aircraft less prone to lateral rollovers, for conservative calculations, the influence of wing lift due to crosswind is temporarily not considered.Crosswinds acting on the cylinder and the thin plate are separately accounted for, with the point of application of forces being the geometric center of their respective projections on the vertical wind direction.

Calculation model
Under crosswind conditions, the force analysis during aircraft turning is depicted in Figure 1.Factors like tire frictional resistance are not involved in the lateral rollover analysis and are not labeled in the figure .When considering aircraft taxiing turns under crosswind conditions, the influence of aircraft weight, centrifugal force, engine thrust, and crosswind are taken into account.According to theoretical mechanics, the equilibrium equation for lateral rollover moments during the turn can be expressed as: By combining Equations ( 1), ( 2), (3), (4), and (5), the relationship between the maximum allowable taxiing turn speed of the aircraft and the turn radius at the center of gravity can be derived as follows: Based on geometric relationships, the additional relationships between Lgravi, α, θ, β, and R are as follows:     Therefore, the analysis and calculation model is as follows: Whereas, the equation symbols and their interpretations are provided in Table 1.Table 1.Equation symbols and definitions.

Symbols Interpretations θ
The angle between the crosswind direction and the direction perpendicular to the rollover axis, measured in degrees.

H1
The lever arm of the wind force acting on the fuselage (simplified as a cylinder) concerning the rollover axis, which is the distance between the centroid and the ground, measured in meters.

H2
The lever arm of the crosswind force acting on the vertical tail (simplified as a flat plate) concerning the rollover axis, which is the distance between the centroid of the crosswind force point and the ground, measured in meters.

V
The crosswind speed, measured in meters per second (m/s).ρ Air density, 1.225 kg/m³.A A1 and A2 represent the characteristic areas of the fuselage and the vertical tail, measured in square meters (m²).

CD1
The drag coefficient for the cylinder, CD1, is 0.82, which can be obtained from Appendix VI 'Drag Coefficients of Typical Objects' in the English-Chinese Dictionary of Wind Engineering and Industrial Aerodynamics.

CD2
The drag coefficient for the thin plate, CD2, is 0.15, which can be obtained from Appendix VI 'Drag Coefficients of Typical Objects' in the English-Chinese Dictionary of Wind Engineering and Industrial Aerodynamics.
The angle between the direction of centrifugal force and the direction perpendicular to the rollover axis, measured in degrees.

Let
The lever arm of the thrust about the rollover axis, which is the distance between the center of the main mounting surface of the engine and the ground, measured in meters.m Aircraft weight, measured in kilograms (kg).

R
The turning radius at the center of gravity of the aircraft, measured in meters (m).

Vm
The maximum taxiing speed limit for the aircraft, measured in meters per second (m/s).α The front wheel steering angle of the aircraft, measured in degrees (°).xcg The x-coordinate of the aircraft's center of gravity position, measured in meters (m).xcf The x-coordinate of the front landing gear center position, measured in meters (m).

Lnf
The nominal front main wheelbase, measured in meters (m).

H
The lever arm of the centrifugal force about the rollover axis, which is the distance between the center of gravity and the ground, measured in meters (m).

Lgravi
The lever arm of gravity about the rollover axis, which is the distance between the center of gravity and the rollover axis, measured in meters (m).

Fcentr
The centrifugal force acting on the aircraft, measured in Newtons (N).

Fcross
The crosswind force acting on the aircraft, measured in Newtons (N).

Fength
The engine thrust acting on the aircraft, measured in Newtons (N).

Mcross
The moment formed by the crosswind force acting on the aircraft about the rollover axis, measured in Newtonmeters (N•m).

Mength
The moment formed by the engine thrust acting on the aircraft about the rollover axis, measured in Newton-meters (N•m).

Mcentr
The moment formed by the centrifugal force acting on the aircraft about the rollover axis, measured in Newtonmeters (N•m).

Mgravi
The moment formed by the gravity acting on the aircraft about the rollover axis, measured in Newton-meters (N•m).

Lhalfm
Half of the length of the main wheelbase, measured in meters (m).g Gravitational acceleration, 9.80665 m/s².

Case study
Following the above method, the study was conducted on the maximum allowable taxiing speed for a certain aircraft.According to the calculation and analysis results, the critical condition for rollover occurs when the aircraft is turning into a headwind, taking the example of a leftward turn when the left side of the aircraft is facing the wind.It was found that the left main wheel's supporting force first reaches zero.The relationship between the maximum allowable taxiing speed during rollover and the front wheel steering angle for different front wheel steering angles is shown in figure 3, and figure 4 represents the relationship between the maximum allowable taxiing speed and the front wheel steering angle for different engine installation scenarios.The relationship between maximum permissible taxiing turn speed and nose wheel steering angle during lateral rollover for an aircraft (engine type 2).From Figure 3 and Figure 4, it can be observed that as the crosswind increases, the maximum allowable taxiing speed for rollover decreases for different weight conditions.The lowest maximum allowable taxiing speed during rollover corresponds to a crosswind speed of 50 knots.Smaller aircraft weights and a forward center of gravity make the aircraft more susceptible to rollover.The data reveals that for the aircraft under both the empty weight and maximum taxi weight (forward and aft center of gravity limits) conditions, the difference in the maximum allowable taxiing speed for rollover is minimal, with the empty weight condition having a slightly lower maximum allowable taxiing speed.
Taking into account the impact of weight, center of gravity, and crosswind on the maximum allowable taxiing speed for a certain aircraft, when the aircraft is at empty weight and equipped with both the full economy class layout and the three-cabin layout, it can be compared to the turning speed envelope of Boeing and Airbus aircraft of a similar size when subjected to a 50-knot crosswind.The comparison reveals that the taxiing speed envelope for this aircraft is similar to similar-sized aircraft, whether equipped with Engine Type 1 or Engine Type 2. The comparison results are shown in Figure 5 and Figure 6, with data for Type A, Type B, and Type C aircraft sourced from the flight manuals of commonly used Boeing and Airbus aircraft models.

Conclusions 1)
A method for engineering estimation of the maximum allowable taxiing speed for civil aircraft in turns has been proposed, based on the analysis of forces acting on the aircraft during taxiing turns.The calculation model for this engineering estimation method has been provided, along with definitions of the calculation conditions and assumptions.Finally, a case study was conducted to analyze the maximum allowable taxiing speed for a specific aircraft in real engineering scenarios.The research results indicate that this engineering estimation method holds certain practical engineering value.
2) The research using this method reveals that as crosswind intensity increases, the maximum allowable taxiing speed for taxiing turns decreases, and the minimum taxiing speed occurs at a crosswind speed of 50 knots for different weight states.3) The research using this method demonstrates that lighter aircraft with a forward center of gravity are more prone to side tipping during taxiing turns.

Figure 1 .
Figure 1.Schematic diagram of aircraft turning under crosswind conditions.We assume that the engine thrust is directed along the aircraft's longitudinal axis.To simplify the calculations, the compression and deformation of the landing gear tires during motion are not considered.The distance 'h' between the aircraft's reference plane (Z=0) and the ground is determined by taking the absolute value of the Z-coordinate of the main landing gear wheel centers when the landing gear is extended, plus the radius of the main landing gear tires.When the aircraft is taxiing at low speeds and making turns by using the manual steering wheel, it is essential to analyze the relationship between the front wheel's steering angle and the maximum allowable turn speed to ensure that the aircraft does not experience lateral rollover.The analysis scenario is as follows: a) Crosswind: 10 knots, 35 knots, and 50 knots.b) Engine thrust: twin-engine ground idle (zero altitudes, zero airspeeds, and 15°C standard temperature).c) Landing gear parameters: coordinates of the points where the main landing gear's ground reaction forces act (X, Y, Z), coordinates of the points where the front landing gear's ground reaction forces act (X, Y, Z).

Figure 2 .
Figure 2. Schematic diagram of aircraft crosswind force model.

Figure 3 .
Figure 3.The relationship between maximum permissible taxiing turn speed and nose wheel steering angle during lateral rollover for an aircraft (engine type 1).

Figure 4 .
Figure 4.The relationship between maximum permissible taxiing turn speed and nose wheel steeringangle during lateral rollover for an aircraft (engine type 2).From Figure3and Figure4, it can be observed that as the crosswind increases, the maximum allowable taxiing speed for rollover decreases for different weight conditions.The lowest maximum allowable taxiing speed during rollover corresponds to a crosswind speed of 50 knots.Smaller aircraft weights and a forward center of gravity make the aircraft more susceptible to rollover.The data reveals that for the aircraft under both the empty weight and maximum taxi weight (forward and aft center of gravity limits) conditions, the difference in the maximum allowable taxiing speed for rollover is minimal, with the empty weight condition having a slightly lower maximum allowable taxiing speed.Taking into account the impact of weight, center of gravity, and crosswind on the maximum allowable taxiing speed for a certain aircraft, when the aircraft is at empty weight and equipped with