Dynamic analysis of an aircraft towing slip-out system considering vertical wheel constraints

Abstract

To analyse the vertical dynamic characteristics of the aircraft towing system under different constraints on the nose landing gear wheels of the aircraft during the towing slip-out mode, a dynamic model of the towing system considering the constraints between the clamping mechanism and the aircraft nose landing gear wheels was established based on the general towing system dynamic model. On this basis, an analysis was conducted to determine whether considering the aircraft wheel constraints affects the vertical vibration acceleration of the towing vehicle and the nose landing gear in low-speed (10 km/h) and high-speed (40 km/h) operating conditions. With the consideration of constraints at both ends of the aircraft wheels, the vertical acceleration of the towing vehicle's centre of mass increased by 153% and 172% at low speed and high speed, respectively, compared to not considering the aircraft wheel constraints. Additionally, with the consideration of constraints at both ends of the aircraft wheels, the vertical acceleration of the nose landing gear's centre of mass decreased to 20% and 57% at low speed and high speed, respectively, compared to not considering the aircraft wheel constraints. An analysis of the vertical vibration acceleration of the towing vehicle under different wheel constraint conditions found that the Root Mean Square (RMS) value of the vertical vibration acceleration of the towing vehicle's centre of mass was minimized when the clamping angles of the clamping mechanism to the nose landing gear wheels were 63° and 64°, respectively. Under this clamping angle, the influence of the clamping forces at both ends of the clamping mechanism on the vertical vibration acceleration of the towing vehicle was minimal. The research results provide valuable reference for the direct constraints between the clamping mechanism and the nose landing gear wheels in the aircraft towing slip-out system.

1. Introduction

The current airport towing mode involves the aircraft being pushed from its parking position by a tow vehicle, after which the aircraft's engines operate at low speed to drive it to the takeoff runway. In this mode, the aircraft consumes a significant amount of fuel, generates airport noise and contributes to atmospheric pollution. Operating at low speeds also poses the risk of the aircraft ingesting foreign objects from the ground, which can compromise the safety of the aircraft. The new generation of towing slip-out methods employs towbarless aircraft towing vehicles to directly handle the aircraft's towing from the parking position to the takeoff point. This approach effectively eliminates the issues associated with the current method. In order to enhance airport operational efficiency, aircraft are towed over longer distances at speeds of up to 40 km/h. This towing process involves various manoeuvres, including acceleration, turning and deceleration, making it more complex compared to the current aircraft pushback method. Consequently, it places higher demands on the dynamic characteristics and stability of the towing system. The clamping mechanism, serving as a central component connecting the tow vehicle and the aircraft, has a significant impact on the vibration characteristics of the system under this new towing slip-out mode. Therefore, analysing the influence of the clamping mechanism on the dynamic characteristics of the towing system is of great significance for understanding the operation of the towing system in relation to the new towing slip-out mode.

In recent years, researchers have developed numerous models for the study of the dynamics of towing systems. Xie et al. [1] have proposed that an aircraft towed by a towbarless towing vehicle constitutes a complex multi-body dynamic system. They utilized ADAMS software to create a virtual prototype of the towbarless towing vehicle-aircraft system and conducted dynamic studies considering various factors affecting the towing system. Zhu and Li [2] established a comprehensive dynamic model of the entire towing system using ADAMS software. They combined this model with speed data from towing operations to conduct dynamic simulations. Their research yielded changes in evaluation parameters related to braking stability within the system.

Yu et al. [3] designed a double-layer closed-loop controller for the trajectory tracking of a carrier-to-carrier-based aircraft system on deck under incomplete constraints and various physical conditions, effectively controlling the coordinated motion of carrier-to-tractor; Qi et al. [4] compared the yaw stability of TLTV (towbarless towing vehicle) by using sliding mode control and adaptive fuzzy PID control to compensate for the required torque based on a time-varying nonlinear dynamic model [5]. Considering the inertia force of a ship's multi-degree-of-freedom coupling motion on tractor, a two-degree-of-freedom time-varying nonlinear rodless tractor system dynamics model is established; Zhang et al. [6] investigated the impact patterns generated on airport surfaces during the aircraft towing process using a towbarless aircraft towing vehicle. Their findings contributed to the structural strength design of the clamping mechanism. Wang et al. [7] based their analysis on mechanical models of tires and landing gear. They studied the influence of the towing vehicle's clamping position on system vibrations. They suggested that the clamping mechanism should be positioned as far from the towing vehicle's centre of gravity as possible when clamping the nose landing gear. Mu et al. [8] developed a dynamic model of a towbarless towing system. They simulated and analysed the loads on major components of the nose landing gear under conditions such as steering, braking and passing through trenches. Their model included constraints between the towing vehicle's clamping mechanism and the aircraft's nose landing gear, using a spring model to represent the constraints. Zhang et al. [9] conducted research on hydraulic buffering in towbarless aircraft towing vehicles during braking conditions. The ADAMS simulation model they created used ADAMS Impact function to calculate contact forces. The aforementioned towing system models considered the constraints between the towing vehicle's clamping mechanism and the aircraft's nose landing gear wheels. However, these constraints were often represented in a simplified manner and may not fully capture the complexities of real-world scenarios. Additionally, the research conducted by Gao et al. [10, 11] highlighted the importance of matching the stiffness-damping characteristics of clamping mechanism components, which have a significant impact on aircraft vibration. Therefore, it is crucial to comprehensively account for the constraints between the clamping mechanism and the aircraft's nose landing gear wheels when developing actual motion models for towing systems.

In the field of vehicle vibration analysis. Yu et al. [12] conducted research to analyse the impact of motor vibrations on the vertical dynamics of high-speed train power cars. They developed a coupled dynamic model between the motor and the train, and analysed its time-domain and frequency-domain responses under different conditions. Yu et al. [13] established a vertical coupled dynamic model of the passenger-carriage-track system using Newton's second law. They conducted time-domain and frequency-domain analyses and compared current models with coupled models. Zhu et al. [14, 15] developed a 4-degree-of-freedom (4-DOF) half-car model in the pitch plane using Newton's second law. They performed time-domain analysis to compare the optimized suspension with independent suspension and baseline suspension. To enhance the ride comfort of the vehicle while maintaining good anti-roll performance, the vibration equations of the mechanical-aerodynamic coupled system for the 4-DOF half-car model with a Dynamic Control of Pitch and Inclination System (DCPIS) were derived. These equations include the dynamic equations of the system. Kaldas et al. [16] constructed a 13-degree-of-freedom vertical dynamics model for the entire vehicle and conducted an analysis of factors such as ride comfort under random road profiles. Afterward, Kaldas et al. [17] aimed to develop a semi-active suspension system controller capable of supporting longitudinal dynamic manoeuvres. They established a 16-degree-of-freedom mathematical model for the vehicle and conducted laboratory tests to validate its vertical dynamics. Múčka [18] assessed the relationship between measured vibrations in passenger cars and the Power Spectral Density (PSD) of road roughness. The aforementioned research content and outcomes play a crucial reference role in the dynamic analysis of traction systems and the establishment of road roughness settings. Zhu et al. [19] constructed a 7-degree-of-freedom dynamic model for a towbarless aircraft towing system with an air suspension. They analysed the impact of different vehicle suspensions on the dynamics of the towing system through time-domain analysis.

Wang et al. [20] utilized Newton's second law to formulate the dynamic equations of the towing system. They studied the time-domain and frequency-domain vibration characteristics of the system and compared the smoothness of the system during high-speed and low-speed towing. In addition to the overall dynamics of the towing system, the dynamic characteristics at the nose landing gear are a focus of research. Wang et al. [21] addressed the issue of nose landing gear vibration caused by towbarless towing vehicles during push and tow operations. They simulated time-domain and frequency-domain responses at the landing gear strut under pavement excitation and analysed the effects of various factors on nose landing gear vibration.

Huang et al. [22] considered a 6-degree-of-freedom aircraft body, flexible main landing gear struts, dynamic wheels, shock-absorbing strut elasticity and wheel dynamics in their dynamic model of towing taxiing. They compared wheel resistance under different driving conditions through Adams/Simulink co-simulation. Li et al. [23] studied the dynamic characteristics and dynamic response behaviour of the nose landing gear under full-load and high-speed towing conditions in the context of the new aircraft towing slip-out operation. In the aforementioned studies related to the dynamics of the towing system, there has been limited consideration of the vertical constraints imposed by the aircraft's nose landing gear wheels on the towing vehicle's clamping mechanism. These studies mainly focused on the support provided by the clamping mechanism to the lower end of the aircraft's nose landing gear wheels, while neglecting the constraints exerted by the upper clamping portion on the landing gear wheels. This omission was primarily due to the lower towing speeds and shorter towing distances associated with existing towing methods, resulting in less severe vibrations. However, as research on the new towing slip-out method advances, both the towing speed and distance will significantly increase. Therefore, investigating the dynamic responses of aircraft in the new towing slip-out system and establishing a complete model of constraints on the aircraft's nose landing gear wheels are of vital importance for ensuring the stable operation of the system under this mode.

In conclusion, this paper has taken into consideration the vertical displacement constraints of the aircraft's nose landing gear wheels and the stiffness and damping characteristics of the clamping mechanism's rods in contact with the wheels. By analysing the actual behaviour of the clamping mechanism, it has accounted for the upper constraints imposed by the clamping mechanism on the aircraft's nose landing gear wheels. Utilizing Newton's second law, a 6-degree-of-freedom dynamic model of the aircraft-towbarless towing vehicle system under the new towing slip-out mode has been established. The paper has also analysed the impact of wheel constraint characteristics on the vibration characteristics of the system under this new towing slip-out mode. The study has investigated the dynamic characteristics of the system under various operating conditions in the new towing slip-out mode.

2. Model establishment

2.1. The model of the aircraft taxiing system

The aircraft towing system consists of numerous components connected together through motion pairs and force elements, making it a typical multi-body dynamic system. In this paper, the B737-800 is chosen as the object of towbarless aircraft towing, with the towing vehicle considered as a rigid body. Structurally, it is symmetrical about the centreline and can be simplified as a half-vehicle model consisting of a chassis and nose and rear tyres. Similarly, the aircraft is also simplified for analysis.

A 6-degree-of-freedom model was established, as shown in Fig. 1. Based on Newton's second law, equations of motion for each component can be formulated as follows:

1. The TLTV body bounce equation:

   $$\begin{array}{rcl}{m}_1{\ddot{z}}_1& =& -k_{1f}(z_1-l_f{\theta }_1-q_1)-k_{1r}(z_1+l_r{\theta }_1-q_2)\\ & & +k_1(z_2-z_1-l_1{\theta }_1)-c_{1f}(\dot{z}1-l_f{\dot{\theta }}_1-{\dot{q}}_1)\\ & & -c_{1r}({\dot{z}}_1+l_r{\dot{\theta }}_1-{\dot{q}}_2)+c_1({\dot{z}}_2-{\dot{z}}_1-l_1{\dot{\theta }}_1)\end{array}$$

   (1)
2. The TLTV body pitch motion equation:

   $$\begin{array}{rcl}{J}_1{\ddot{\theta }}_1& =& k_{lf}{l}_1(z_1-l_1{\theta }_1-q_1)-k_{1r}{l}_r(z_1+l_r{\theta }_1-q_2)\\ & & +k_1{l}_1(z_2-z_1-l_1{\theta }_1)+c_{lf}{l}_1({\dot{z}}_1-l_1{\dot{\theta }}_1-{\dot{q}}_1)\\ & & -k_{1r}{l}_r({\dot{z}}_1+l_r{\dot{\theta }}_1-{\dot{q}}_2)+k_1{l}_1({\dot{z}}_2-{\dot{z}}_1-l_1{\dot{\theta }}_1)\end{array}$$

   (2)
3. The nosewheel bounce motion equation:

   $$\begin{array}{rcl}{m}_2{\ddot{z}}_2& =& -k_1(z_2-z_1-l_1{\theta }_1)+k_{2f}(z_3-z_2-l_3{\theta }_2)\\ & & -c_1({\dot{z}}_2-{\dot{z}}_1-l_1{\dot{\theta }}_1)+c_{2f}({\dot{z}}_3-{\dot{z}}_2-l_3{\dot{\theta }}_2)\end{array}$$

   (3)
4. The airport bounce motion equation:

   $$\begin{array}{rcl}{m}_3{\ddot{z}}_3& =& -k_{2f}(z_3-z_2-l_3{\theta }_2)-k_{2r}(z_3-z_4+l_4{\theta }_2)\\ & & -c_{2f}({\dot{z}}_3-{\dot{z}}_2-l_3{\dot{\theta }}_2)-c_{2r}({\dot{z}}_3-{\dot{z}}_4+l_4{\dot{\theta }}_2)\end{array}$$

   (4)
5. The airport bounce motion equation:

   $$\begin{array}{rcl}{J}_2{\ddot{\theta }}_2& =& k_{2f}{l}_3(z_3-l_3{\theta }_2-z_2)-k_{2r}{l}_4(z_3-z_4+l_4{\theta }_2)\\ & & +k_{2f}{l}_3({\dot{z}}_3-l_3{\dot{\theta }}_2-{\dot{z}}_2)-k_{2r}{l}_4({\dot{z}}_3-z_4+l_4{\theta }_2)\end{array}$$

   (5)
6. The rearwheel bounce motion equation:

   $$\begin{array}{rcl}{m}_4{\ddot{z}}_4& =& -k_4(z_4-q_3)+k_{2r}(z_3+l_4{\theta }_2-z_4)\\ & & -c_4({\dot{z}}_4-{\dot{q}}_3)+c_{2r}({\dot{z}}_3+l_4{\dot{\theta }}_2-{\dot{z}}_4)\end{array}$$

   (6)

2.2. Wheel constraint model

The vertical motion of the aircraft's nose landing gear wheels is constrained by the clamping mechanism of the aircraft towing vehicle. Figure 2 shows the clamping force model of the aircraft's nose landing gear wheels on the towing vehicle's clamping mechanism.

In Fig. 2, the clamping rod on the towing vehicle's clamping mechanism restricts the aircraft's nose landing gear wheels on the clamping mechanism. The angles between the clamping rod and the plane of contact with the wheel are denoted as ${\theta }_3$ and ${\theta }_4$⁠, respectively. In the figure, $k_t$ and $c_t$ represent the stiffness and damping when the clamping rod is in contact with the wheel, respectively.

If, during the operation of the towing system, the wheel experiences vertical displacement of $X$⁠, then the radial displacements of the clamping rod along the contact point to the wheel's centre are as follows:

Based on the relationship between radial force and radial deformation of the wheel, we can determine that the radial forces acting along the contact point to the wheel's centre are $F_1$ and $F_2$⁠. Analysing the forces at the contact point, we can deduce that the clamping rod's vertical restraining force on the wheel is given by:

According to Ref. [24], the nose landing gear bears 15% of the aircraft's weight. Based on Equations (3), (4) and (5), the vertical force on the wheel during the operation of the towing system is given by:

Based on force equilibrium:

After considering the clamping rod's constraint on the vertical displacement of the wheel, we also need to account for the effect of the stiffness and damping when the clamping rod contacts the wheel. The nosewheel bounce motion equation, based on Equation (3), is:

The matrix form of the dynamic equations of the traction system can be derived by organizing the aforementioned dynamic Equations (1) to (6) together with Equation (12):

where $\mathit{Z}={(z_1,{\theta }_1,z_2,z_3,{\theta }_1,z_4)}^{\mathrm{T}}$ is the motion response matrix of the TLATS system. $\mathit{X}={(q_1,q_2,q_3)}^{\mathrm{T}}$ is random airport road excitation matrix. $\mathit{M}$⁠, $\mathit{K}$ and $\mathit{C}$ are the mass stiffness and damping matrices of the TLATS system respectively. ${\mathit{B}}_1$ and ${\mathit{B}}_2$ are the coefficient matrices of displacement and velocity excitation of the road.

According to the previous research on the aircraft and TLTV, the parameters of the TLATS model are listed in Table 1.

3. Results analysis

3.1. Random airport road excitation

To analyse the dynamic behaviour of the traction system under wheel constraints, simulations were conducted on Grade A road surfaces at traction speeds of 10 km/h (low-speed aircraft towing) and 40 km/h (high-speed aircraft towing) [24]. Random excitations were applied to the rear wheels of the aircraft towing vehicle and the aircraft's rear landing gear wheels. These excitations were represented in terms of equivalent time delays with respect to the nose wheels, where the time delay depends on the distance between the rear wheels of the towing vehicle and the aircraft's rear landing gear wheels, as well as the forward velocity.

Power spectral density is commonly used to describe statistical characteristics of road surface roughness. According to ISO 8680 standards, road surface power spectral density is used to describe the road profile, and its expression is as follows:

In the above equation, $n$ is the spatial frequency $(m^{-1})$⁠. $n_0=0.1m^{-1}$ is the reference spatial frequency. $f$ is the temporal frequency $(s^{-1})$⁠. $G_q(n_0)$ is the power spectral density of the road surface under the reference spatial frequency; A road surface: $G_q(n_0)=1.6\times {10}^{-6}{m}^3$⁠.

Using the harmonic superposition method, the road surface roughness on the Grade A road surface at speeds of 10 km/h and 40 km/h can be obtained as shown in Figs. 3(a) and (b).

3.2. Traction vehicle vibration response

As shown in Figs. 4 and 5, these are time-domain and frequency-domain analysis plots of the vertical acceleration of the traction vehicle's centre of gravity under different constraints of the aircraft's nose landing gear wheels. Figs. 4(a) and (b) respectively depict the time-domain and frequency-domain analysis of the vertical acceleration of the traction vehicle's centre of gravity when the traction system is engaged in low-speed aircraft towing. Figs. 5(a) and (b) respectively depict the time-domain and frequency-domain analysis of the vertical acceleration of the traction vehicle's centre of gravity when the traction system is engaged in high-speed aircraft towing.

Table 2 represents the Root Mean Square (RMS) values of the vertical acceleration of the traction vehicle's centre of gravity under different operational states as shown in Figs. 4 and 5.

From Figs. 4 and 5 and Table 2, the following observations can be made: 1) Regardless of whether the wheel holding mechanism for the aircraft's nose landing gear wheels is considered or not, an increase in the operating speed of the traction system results in a significant increase in the vertical vibration acceleration of the traction vehicle; 2) The vertical vibration acceleration of the traction vehicle varies significantly under different constraints of the aircraft's nose landing gear wheels. This difference is also evident in both low-speed and high-speed operating states of the traction system. In the low-speed state, the vertical vibration acceleration, with and without considering the constraints of the nose landing gear wheels, is 1.13 m/s² and 0.74 m/s² respectively. In the high-speed state, it is 3.58 m/s² and 2.08 m/s², respectively; 3) In the low-speed operating state of the traction system, when considering the constraints at the upper end of the aircraft wheels, the peak frequency of the vertical vibration acceleration of the traction vehicle is significantly higher (around 3.7 Hz) compared to the case where wheel constraints are not considered. For other frequencies, the results are essentially the same. In the high-speed operating state, the frequency variation pattern of the vertical vibration acceleration of the traction vehicle under different wheel constraints is similar to the low-speed case, with a slightly increased peak frequency.

The consideration of the upper constraint of the aircraft's nose landing gear wheel leads to a significant increase in the vertical vibration acceleration RMS value of the towing vehicle. This increase is substantial, reaching 153% and 172% during low-speed and high-speed operations, respectively. The primary reason behind this lies in the greater influence exerted on the towing vehicle by the upper constraint of the aircraft's nose landing gear and the fuselage. In the dynamic model where the upper constraint of the wheel is not considered, when the vertical upward movement displacement of the current landing gear wheel exceeds the vertical upward movement displacement at the towing vehicle's wheel mechanism, the towing vehicle and the aircraft's nose landing gear wheel would enter a disengaged state. In this state, the towing vehicle would not be affected by the vertical motion of the aircraft's nose landing gear and the fuselage in the vertical direction. Instead, the aircraft's vertical motion would be entirely reflected in the fuselage, creating some disparities from the actual scenario. In reality, considering the upper constraint of the wheel is crucial to establish a more comprehensive dynamic model of the towing system. With the inclusion of the upper constraint of the wheel, the towing vehicle's wheel mechanism and the aircraft's nose landing gear wheel remain in constant contact and constraint. When the aircraft's vertical movement displacement exceeds the vertical upward movement displacement at the towing vehicle's wheel mechanism, the restrained upper wheel causes the vertical vibration of the nose landing gear to transmit through the wheel mechanism to the towing vehicle, thereby resulting in an amplified vertical vibration of the towing vehicle.

3.3. Nose landing gear vertical vibration response

Figs. 6 and 7 present time-domain and frequency-domain analysis plots of the aircraft's nose landing gear under different constraints of the nose landing gear wheels and different towing slip speeds.

Table 3 represents the RMS values of the vertical acceleration of the traction vehicle's centre of gravity under different operational states, as shown in Figs. 6 and 7.

From Figs. 4 and 5 and Table 3, the following observations can be made: 1) Regardless of whether the wheel holding mechanism for the aircraft's nose landing gear wheels is considered or not, an increase in the operating speed of the traction system results in a significant increase in the vertical vibration acceleration of the aircraft's nose landing gear; 2) The vertical vibration acceleration of the nose landing gear varies significantly under different constraints of the aircraft's nose landing gear wheels. This difference is also evident in both low-speed and high-speed operating states of the traction system. In the low-speed state, the vertical vibration acceleration, with and without considering the constraints of the nose landing gear wheels, is 0.43 m/s² and 2.21 m/s², respectively. In the high-speed state, it is 2.65 m/s² and 4.63 m/s², respectively; 3) In both low-speed and high-speed operating states of the traction system, the vertical vibration acceleration in frequency domain exhibits similar trends under different constraints of the nose landing gear wheels. The vibration amplitude is greater without considering the upper wheel constraints in the frequency range of 1 Hz–6 Hz, but the results are reversed in the range of 6 Hz–10 Hz. Additionally, in the high-speed operating state, there is a higher-frequency (3.6 Hz) vibration with greater amplitude than the lower-frequency (8.4 Hz) vibration, which should be a focus in vibration analysis of the traction system.

Considering the clamping constraint of the wheel mechanism on the aircraft's nose landing gear, the influence of the towing vehicle on the aircraft's nose landing gear in the vertical upward direction is increased. This is specifically reflected in a decrease in the vertical vibration acceleration, reducing to 20% and 57% respectively in low-speed and high-speed states. The reasons behind these results are contrary to the reasons for the increased vertical vibration acceleration of the towing vehicle. In the dynamic model considering the clamping constraint of the wheel mechanism on the upper end of the aircraft's nose landing gear, when the vertical upward movement displacement of the current landing gear wheel exceeds the vertical upward movement displacement at the towing vehicle's wheel mechanism, its vertical displacement will be restricted by the clamping mechanism. Consequently, the magnitude of the vertical displacement change decreases. Compared to the scenario without considering the constraint on the nose landing gear wheel, there will be a noticeable decrease in the vertical vibration acceleration RMS at the aircraft's nose landing gear due to the reduced amplitude of vertical displacement.

4. Analysis of constraints at both ends of the aircraft wheels

Due to the increase in towing distance and speed, the vibration characteristics of the towing vehicle have a significant impact on the stability of the towing system and the comfort of the driver. Therefore, the dynamic characteristics of the towing vehicle need to be closely monitored during this process. As a crucial component connecting the aircraft and the towing vehicle, the method of connection has a substantial influence on the vibration behaviour of the towing system, especially when the towing system is operating at high speeds and over long distances. To further clarify the impact of different constraints on the aircraft's nose landing gear wheels on the towing system and to provide reference for the selection of constraints for the new towing method, this study investigates the vibration response of the towing system with a focus on the forces applied at both ends of the aircraft's nose landing gear wheels and the contact angle between the clamping rod and the aircraft's nose landing gear wheels.

According to Ref. [24], the forces exerted by the nose and rear clamping rods of the wheel holding mechanism on the aircraft's nose landing gear wheels are as follows:

when the clamping force at both ends of the aircraft wheels is determined, it is possible to reduce the vibration response of the towing vehicle at high speeds by changing the clamping angle between the clamping rods and the aircraft wheels. By changing the clamping angles at both ends of the aircraft wheels, the variation in the vertical vibration acceleration of the towing vehicle under different clamping angles can be observed, as shown in Fig. 8. From Fig. 8, it can be observed that when the clamping angles of the wheel holding mechanism with the aircraft's nose landing gear wheels are 63° and 64°, the RMS value of the towing vehicle's vertical vibration acceleration reaches its minimum at 2.84 m/s². Fig. 9 shows the corresponding vibration time-domain and frequency-domain responses at this minimum vertical vibration acceleration.

The clamping forces at both ends of the aircraft wheels will affect the radial deformation of the aircraft's nose landing gear wheels, thus influencing the vibration characteristics of the aircraft towing system. To further investigate the impact of different constraint forces on the vibration characteristics of the towing vehicle, with the clamping angles of the wheel holding mechanism with the aircraft's nose landing gear wheels set at 63° and 64° the clamping forces at both ends of the aircraft wheels were changed. The resulting variation in the RMS value of the towing vehicle's vertical acceleration is shown in the graph.

In Fig. 10, with the increase in clamping forces at both ends of the aircraft wheels, the RMS value of the towing vehicle's vertical acceleration initially decreases and then increases. In the range of clamping forces varying from 4000 N to 7000 N, the RMS value of the towing vehicle's vertical acceleration varies within the range of 3.83 m/s² to 3.86 m/s². The numerical difference is relatively small when compared to the clamping angles of the wheel holding mechanism with the aircraft's nose landing gear wheels set at 63° and 64°. This indicates that under these clamping angles, the pre-clamping forces exerted by the wheel holding mechanism on the aircraft's nose landing gear wheels have a minimal impact on the vertical vibration acceleration of the towing vehicle. Therefore, in the state where the hydraulic cylinder output force of the towing vehicle's wheel holding mechanism is determined, achieving a reduction in the vertical vibration of the system can be accomplished by selecting an appropriate clamping angle.

5. Conclusions

1. A 6-degree-of-freedom dynamic model of the aircraft towing system was established. Based on this model, utilizing the previously derived relationship between the forces on the aircraft wheels and their deformation, a dynamic equilibrium equation was obtained to consider the vertical vibration displacement constraint of the aircraft's nose landing gear under wheel constraints. This equation was used to constrain the vertical displacement of the aircraft's nose landing gear. The model also takes into account the stiffness and damping of the tyre when the wheel holding mechanism clamps onto the aircraft's nose landing gear wheels. Based on the vertical displacement constraint of the aircraft's nose landing gear and the tyre contact characteristics, a dynamic model of the vertical constraint of the aircraft's nose landing gear wheels was established.

2. The constraints on the aircraft's nose landing gear wheels have a significant impact on the vertical dynamics characteristics of both the aircraft towing vehicle and the aircraft's nose landing gear. When considering the constraints at both ends of the aircraft wheels, the calculated vertical acceleration of the towing vehicle's centre of gravity is 1.13 m/s² and 3.58 m/s², which increases by 153% and 172% respectively compared to not considering the constraints of the aircraft's nose landing gear wheels. When considering the constraints at both ends of the aircraft wheels, the calculated vertical acceleration of the nose landing gear is 0.43 m/s² and 2.56 m/s², which decreases by 20% and 57% respectively compared to not considering the constraints of the aircraft's nose landing gear wheels.

3. The different clamping angles of the wheel holding mechanism's clamping rods with the aircraft's nose landing gear wheels have a significant impact on the vertical vibration of the towing vehicle. The variation in vertical vibration between the aircraft and the towing vehicle under different clamping angles has been obtained. When the clamping angles of the wheel holding mechanism with the aircraft's nose landing gear wheels are set at 63° and 64°, the RMS value of the towing vehicle's cente of gravity's vertical vibration acceleration is minimized. Moreover, at this clamping angle, the impact of changing the forces applied to the towing vehicle's centre of gravity's vertical vibration acceleration by the aircraft's wheel holding mechanism and the aircraft's nose landing gear wheels is minimal. By analysing the clamping angle and constraint characteristics of the bogie mechanism, it is possible to optimize the towing system, thereby enhancing the stability and safety of system operations under the new towing slip-out mode. The research findings in this article can serve as a theoretical reference for future work on the connection methods between tow vehicles and aircraft, as well as the design and manufacturing of tow vehicles.

Acknowledgements

This study was supported by the the Fundamental Research Funds for the Central Universities, China (Grant No. 3122022066), Postgraduate Research and Innovation Fund, China (Grant No. 2022YJS056) and National Natural Science Foundation of China (Grant No. U2033208). The financial support is gratefully acknowledged.

Conflict of interest statement

The authors declare that they have no conflicts of interest declared.

References