Open AccessArticle

Winch Traction Dynamics for a Carrier-Based Aircraft Under Trajectory Control on a Small Deck in Complex Sea Conditions

School of Energy and Power Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China

Author to whom correspondence should be addressed.

Aerospace 2024, 11(11), 885; https://doi.org/10.3390/aerospace11110885

Submission received: 21 August 2024 / Revised: 22 October 2024 / Accepted: 23 October 2024 / Published: 27 October 2024

(This article belongs to the Special Issue Advances in Thermal Fluid, Dynamics and Control)

Browse Figures

Figure 1
Virtual prototyping model of the tractor–aircraft system [<a href="#B20-aerospace-11-00885" class="html-bibr">20</a>]. "> Figure 2
Schematic diagram of aircraft winch traction. "> Figure 3
Schematic diagram of the whole system of carrier-based aircraft traction. "> Figure 4
Cardan Angles(The conversion relationships between different coordinate systems). "> Figure 5
The mathematical model of a landing gear-tire system in a z-direction. "> Figure 6
Schematic diagram of wind load. "> Figure 7
Graph of PID control for aircraft speed. "> Figure 8
The generated trajectory diagram (Bessel curve). "> Figure 9
Traveling trajectory diagram of the aircraft with the control (curve) (a) Planar view from z to −z; (b) Three-dimensional view. "> Figure 10
Traveling trajectory diagram of the aircraft without the control (straight line). (a) Planar view from z to −z; (b) Three-dimensional view. "> Figure 11
Curves of rope force changing with time under different pitching amplitudes (θ1 = 5°, 2°, 0.8° and 0.1°, φ1 = 5°). "> Figure 12
Curve of the vertical force of each tire over time (φ1 = 5°, θ1 = 2°). "> Figure 13
Curve of tire force over time in each direction for tire three (φ1 = 5°; θ1 = 2°). "> Figure 14
Curve of the vertical force of tire three over time at different pitching amplitudes (θ1 = 5°, 2°, 0.8° and 0.1°; φ1 = 5°). "> Figure 15
Curve of the force of the front rope over time at different rolling angle frequencies (ω1 = 2π/Tφ1 = 0.93 rad/s, 0.63 rad/s, 0.23 rad/s and 0.1rad/s; φ1 = 5°, θ1 = 2°). "> Figure 16
Curve of the vertical force of tire three changing with time at different rolling angle frequencies (ω1 = 2π/Tφ1 = 0.93 rad/s, 0.63 rad/s, 0.23 rad/s and 0.1 rad/s; φ1 = 5°; θ1 = 2°). "> Figure 17
Curve of the force of the front rope over time at different pitching amplitudes (θ1 = 2°, 0.8°, 0.4° and 0.1°; θ2 = 1°). "> Figure 18
Curve of the vertical force of tire three over time at different pitching amplitudes (θ1 = 2°, 0.8°, 0.4° and 0.1°; θ2 = 1°). "> Figure 19
Curve of the force of the front rope over time at different rolling angle frequencies (ω2 = 2π/Tφ2 = 2 rad/s, 1 rad/s, 0.63 rad/s and 0.23 rad/s). "> Figure 20
Curve of the vertical force of tire three changing with time at different rolling angle frequencies (ω2 = 2π/Tφ2 = 2 rad/s, 1 rad/s, 0.63 rad/s and 0.23 rad/s). "> Figure 21
Curve of the force of the front rope over time at different heaving amplitudes (z1 = 0.19 m, 0.1 m, 0.05 m, and 0.019 m). "> Figure 22
Curve of the vertical force of tire three over time at different heaving amplitudes (z1 = 0.19 m, 0.1 m, 0.05 m, and 0.019 m). "> Figure 23
Curve of the front rope force over time under single-frequency excitation and multi-frequency excitation with trajectory control. "> Figure 24
Curve of the vertical force of tire three over time under single-frequency excitation and multi-frequency excitation with trajectory control. "> Figure 25
Curve of the front rope force over time with trajectory control and without trajectory control under multi-frequency excitation. "> Figure 26
Curve of the vertical force of tire three over time with trajectory control and without trajectory control under multi-frequency excitation. "> Figure 27
Simulation of five-winch traction of aircraft. "> Figure 28
Local amplication view of landing gear in five-winch traction of aircraft. "> Figure 29
Front landing gear model. "> Figure 30
Rear landing gear model. "> Figure 31
Comparison of the front rope forces obtained by ADAMS and MATLAB. "> Figure 32
Comparison of vertical forces for tire three obtained by ADAMS and MATLAB. ">

Versions Notes

Abstract

When the winch traction system of a carrier-based aircraft works under complex sea conditions, the rope and the tire forces are greatly changed compared with under simple sea conditions, and it poses a potential threat to the safety and stability of the aircraft’s traction system. The accurate calculation of the rope and tire forces of a carrier-based aircraft’s winch traction under complex sea conditions is an arduous problem. A novel method of dynamic analysis of the aircraft-winch-ship whole system under complex sea conditions is proposed. A multiple-frequency excitation is adopted to describe the complex sea conditions and the influences of pitching amplitude, and the rolling frequency on the traction dynamics of a carrier-based aircraft along the setting trajectory under complex sea conditions are studied. The advantages and disadvantages of a winch traction system with trajectory control and without trajectory control in complex sea conditions are analyzed. For realizing the trajectory control of the aircraft, the vector difference between the center of mass for the carrier-based aircraft and the position on the predetermined Bessel curve is calculated, so as to obtain the azimuth vector in the aircraft coordinate system. This research is innovative in the modeling of the whole system and the trajectory control of a carrier-based aircraft’s winch traction system under the complicated sea condition of the multi-frequency excitation. ADAMS (Automatic Dynamic Analysis of Mechanical System) is used to verify the correctness of the theoretical calculation for the winch traction. The results show that the complex sea environment has a certain influence on the winch traction safety of the aircraft; in the range of 10–15 s for the traction, the rope force amplitude of complex sea conditions under the multi-frequency excitation is 29.5% larger than that of the single-frequency amplitude, while the vertical force amplitude of the tire is 201.1% larger than that of the single-frequency amplitude. This research has important guiding significance for the selection of rope and tire models for a carrier-borne aircraft’s winch traction in complex sea conditions.

Keywords:

carrier-based aircraft; winch traction; complex sea conditions; rope and tire forces; trajectory control

1. Introduction

As a tool for carrier-based aircraft traction, winch traction has more advantages over other traction modes for traction operation on an aircraft carrier with a relatively narrow space. When the aircraft is parked on the deck, it can be pulled into the warehouse using the ropes; the front rope provides the tension, the rear rope provides the back tension, and the left and right ropes provide the auxiliary tension to keep the aircraft stable. The winch traction process of a carrier-based aircraft is susceptible to the influence of a complex marine environment. Establishing the winch traction model and studying the traction characteristics has important academic value and engineering application value [1,2].

1.1. Literature Review

At present, many scholars have carried out a lot of research on the modeling of the aircraft traction system. Ferhatoglu et al. [3] presented a method for deriving the conservative upper and lower bounds of frequency response by a nonlinear modeling method, and studied the change of frequency response of a friction-damped structure caused by non-unique residual traction. Li et al. [4] constructed a tractor–aircraft system that enabled the pilot to complete the traction and taxiing operations of the aircraft to solve the problem of the lack of information interaction between the aircraft and the tractor. However, this modeling did not consider the effect of the lateral wind load on the traction trajectory and velocity and, therefore, this research had certain limitations. Heirendt et al. [5] proposed a steady-state conceptual model for studying the state thermal characteristics of slider bearings in lubricated aircraft landing gear. Yang et al. [6] introduced a dynamic model of helicopters and tractors based on the independent modeling for motion subjects (IMMS) method to simulate the traction operation of a shipboard helicopter on the deck. Long et al. [7] established the aileron oil traction cable dynamics model and analyzed the unstable region of cable parameter-excited vibration. Fricke et al. [8] proposed an analytical method to link the tire-soil interaction model of representative critical aircraft components with the finite element tire stress simulation to study the landing gear failure of critical aircraft components under dangerous conditions. Zhao et al. [9] developed the dynamic equations of the tractor and the aircraft’s multi-body system using the Lagrange method, and analyzed the ride comfort of the tractor and aircraft by applying the tire model represented by the spring and damper. Linn et.al [2] established a dynamic model of a helicopter-landing gear tire-auxiliary device to carry out a maneuver simulation. Nevertheless, there were some deficiencies in the study, and the influence of ship motions on the tire forces and the probe forces was not taken into account. Zhang et al. [10] introduced a high-fidelity aircraft gliding dynamics model and designed a controller based on deep reinforcement learning to improve mobility performance.

In terms of the traction trajectory planning and the safety of the carrier-based aircraft, Itagaki et al. [11] proposed an improved method of traditional traction curve estimation based on the creep theory. Wang et al. [12] studied technology and presented research progress on scheduling path planning for carrier-based aircrafts on deck. Liu et al. [13] developed the Homogeneous-Planning-Tracking (HPT) method and a backward horizon controller to track the reference trajectory obtained in the planning layer. Liu et al. [14] presented an online tracking method based on the bending horizon control (RHC) theory and the offline optimal control algorithm to accurately track the obtained trajectory. Jiao et al. [15] established an algorithm to analyze the friction characteristics between the runway and the tire which was only based on the aircraft’s wheel speed signal, which can correctly identify the runway condition and improve the braking efficiency. Gao et al. [16] discussed the two-stage comprehensive optimization of the aircraft’s frame configuration and tractor route to achieve an energy-saving operation at the airport. Wang et al. [17] created a centralized optimal control method to solve the path planning problem for achieving efficient and robust taxi coordination planning. Yu et al. [18] researched a co-evolution mechanism for aircraft systems to ensure coordinated trajectory planning among multiple aircrafts, and used a hybrid RRT algorithm to generate trajectories suitable for tractor systems. Zhang et al. [19] proposed a deck path planning algorithm based on the aircraft’s carrier deck, and the effectiveness of the path planning algorithm is verified using a simulation. Wang et al. [20] designed a path-tracking controller based on the fuzzy logic theory, and adopted a compensation tracking method for direction control. The results showed that the tractor accurately follows the reference route on flat ground, as shown in Figure 1 where the tractor pulls the aircraft through the traction rod. Guo et al. [21] proved the importance of hyperelastic material characterization in the development of detailed finite element tire models. Zhan Wen et. al. [22] developed a dynamic model of the carrier-based aircraft’s landing-gear ship and treated the rolling and pitching motions of the ship as single-frequency excitation forces. However, the two motions of a ship under complex sea conditions are actually multi-frequency excitations instead of single-frequency excitations and, therefore, the modeling approach has certain limitations in terms of complexity. Williams [23] built a real-time optimal control strategy for the traction cable control system. The cable winch is applied to control the height of the traction cable body, which can safely avoid a collision with the terrain. Wang et al. [24] established the dynamic model of the aircraft traction system, and studied the influence of tire friction coefficient on the braking of an aircraft traction system. Zhang [25] analyzed the load of the traction aircraft when the tire was flat and normal, respectively.

1.2. Problem Description

Investigators have done lots of work on aircraft traction and trajectory control, and these studies were mainly focused on the local model for the aircraft traction system, the robustness and the trajectory control in the process of traction with simple sea conditions. This simple sea condition is often treated as a single-frequency excitation [22]. However, there are some limitations within whole system modeling and the excitation complexity of sea conditions within this research. In the ocean environment, there may be different frequencies of wave excitation caused by tsunamis, explosions, and other vibration sources. These complex wave excitations of different frequencies bring fierce shocks to the hull and affect the traction dynamics of a carrier-based aircraft, which is undoubtedly a big safety problem for the actual project.

In this paper, a whole system model composed of the helicopter, the landing gear, the tires, and the winch traction is developed considering complex sea conditions. The model developed in this research is suitable for studying winch traction systems on a small deck. As a small deck is more vulnerable to the influence of waves, especially during complex sea conditions, i.e., multi-frequency wave excitation, the safety and stability of aircraft winch traction faces even bigger challenges. The traction dynamics characteristics of the winch on a small deck in complex sea conditions are studied. It is innovative in whole system dynamic modeling and path control under the complicated sea condition of multi-frequency excitation. Considering complex sea conditions as multi-frequency excitations rather than single-frequency excitations means this excitation is closer to the actual situation, and the calculation is more accurate. This research provides more guiding significance to the safe operation of carrier-based aircraft winch-traction systems during adverse sea conditions.

1.3. Structure of This Paper

The structure of this paper is as follows: Section 2 establishes the whole system model including the aircraft, the winch, the landing gear, the tire, and the wind. Section 3 develops the control approach of the aircraft’s speed and traveling trajectory. In Section 4, the dynamics of carrier-based aircrafts under single-frequency and multi-frequency wave excitation are analyzed. In Section 5, a comparison of traction dynamics between complex and simple sea conditions is conducted. Section 6 verifies the theoretical modeling method based on ADAMS software (ADAMS2020). Section 7 presents the conclusions.

2. Modeling of the Traction Dynamics of a Carrier-Based Aircraft Winch

The traction system is characterized by low movement speed and the existence of both holonomic and nonholonomic constraints in the traction system. The universal model of the traction system is established based on these physical characteristics. A schematic diagram of the basic traction system is shown in Figure 2, where the front winch provides tension, rear winch 1 and rear winch 2 provide back tension, and auxiliary winch 1 and auxiliary winch 2 provide auxiliary tension. The aircraft fuselage is hypothesized to be a rigid body, and its mass is concentrated in the center of mass of the aircraft. In the process of winch traction, the thrust generated by the helicopter rotor wing and the influence of the wheel inclination angle on the tire force is neglected.

2.1. Establishment and Transformation of Coordinate Systems

A winch traction system considering complex sea conditions involves the movement of multiple parts of the aircraft, the tires, and the ship, and this requires the establishment of different coordinate systems for analysis. In order to study the relationship between the motion of each part in the whole system, it is necessary to establish the conversion relationship of each coordinate system.

2.1.1. Establishment of Coordinate System

Figure 3 is a schematic diagram of the whole system model of carrier-based aircraft traction, including the winch model, landing gear-tire model, and ship motion model. In Figure 3, m₁ is the tire mass; the details are shown in the red box. The rope force analysis is seen in the blue box; F₂ is the loose edge pull, and F₁ is the tight edge pull. Each coordinate system is defined as follows:

(1): Inertial coordinate system o_ix_iy_iz_i. The origin o_i is located at sea level and is relatively stationary with the earth, the o_ix_i axis coincides with the speed direction of the ship and points forward, the o_iz_i axis is vertically upward, and the o_iy_i axis points to larboard.
(2): Ship coordinate system o_sx_sy_sz_s. The origin o_s is located in the center of rotation of the ship, the o_sx_s axis is consistent with the longitudinal axis of the ship, pointing forward, the o_sy_s axis points to larboard, and the o_sz_s axis is perpendicular to the ship benchmark surface. There is a rolling angle $φ$ , a pitching angle $θ$ , and a heaving displacement z in the inertial coordinate system.
(3): Fuselage coordinate system o_px_py_pz_p. The o_p point is at the center of mass of the fuselage, the o_px_p axis is along the longitudinal axis of the fuselage, the direction points forward, the o_py_p axis is perpendicular to o_px_p in the horizontal plane of the fuselage, the direction points to the left of the fuselage, and the o_pz_p axis is the yaw axis of the fuselage, perpendicular to the horizontal plane of the fuselage, and the direction is vertically upwards.

2.1.2. Transformation of Coordinate System

The orientation of the rigid body can be described by three independent angular coordinates of the Cardan angle (φ₁, φ₂, φ₃), a fixed coordinate system Ox_iy_iz_i, and a dynamic coordinate system Ox_py_pz_p consolidated on the aircraft, which are established through the fixed point O. The consolidated coordinate system is called the fixed coordinate system. As shown in Figure 4, the two coordinate systems established coincide; the coordinate system Ox_py_pz_p first rotates at an angle φ₁ around the x-axis, then rotates φ₂ around the y-axis, and finally rotates φ₃ around the new z-axis. The matrix of the three transformations is as follows:

A (φ_{1}, φ_{2}, φ_{3}) = [\begin{matrix} c_{φ_{2}} c_{φ_{3}} & c_{φ_{1}} s_{φ_{3}} + s_{φ_{1}} s_{φ_{2}} c_{φ_{3}} & s_{φ_{1}} s_{φ_{3}} - c_{φ_{1}} s_{φ_{2}} c_{φ_{3}} \\ c_{φ_{2}} c_{φ_{3}} & c_{φ_{1}} c_{φ_{3}} - s_{φ_{1}} s_{φ_{2}} s_{φ_{3}} & s_{φ_{1}} s_{φ_{3}} + c_{φ_{1}} s_{φ_{2}} c_{φ_{3}} \\ s_{φ_{2}} & - s_{φ_{1}} c_{φ_{2}} & c_{φ_{1}} c_{φ_{2}} \end{matrix}]

(1)

where

A (φ_{1}, φ_{2}, φ_{3})

is an orthogonal matrix, and its inverse matrix is the same as the transpose matrix, also

s_{φ_{1}} = \sin φ_{1}

s_{φ_{2}} = \sin φ_{2}

s_{φ_{3}} = \sin φ_{3}

c_{φ_{1}} = \cos φ_{1}

c_{φ_{2}} = \cos φ_{2}

c_{φ_{3}} = \cos φ_{3}

The relationship between the angular velocity ω and the time derivative of the transformation angle (φ_1, φ_2, φ₃) in the definite coordinate system is

ω = [\begin{matrix} ω_{1} \\ ω_{2} \\ ω_{2} \end{matrix}] = [\begin{matrix} \sin φ_{2} \sin φ_{3} & \sin φ_{3} & 0 \\ - \cos φ_{2} \sin φ_{3} & - \cos φ_{3} & 0 \\ \sin φ_{2} & 0 & 1 \end{matrix}] [\begin{matrix} {\dot{φ}}_{1} \\ {\dot{φ}}_{2} \\ {\dot{φ}}_{3} \end{matrix}]

(2)

where ω₁, ω₂, and ω₃ is the angular velocity of the rigid body around the x, y, and z axes, respectively,

{\dot{φ}}_{1}

{\dot{φ}}_{2}

{\dot{φ}}_{3}

are the first-order derivatives of the Euler angular at the three transformations of the coordinate system,

[\begin{matrix} {\dot{φ}}_{1} \\ {\dot{φ}}_{2} \\ {\dot{φ}}_{3} \end{matrix}] = [\begin{matrix} \cos φ_{3} / \cos φ_{2} & - \sin φ_{3} / \cos φ_{2} & 0 \\ \sin φ_{3} & \cos φ_{3} & 0 \\ - \tan φ_{2} \cos φ_{3} & - \tan φ_{2} \sin φ_{3} & 1 \end{matrix}] [\begin{matrix} ω_{1} \\ ω_{2} \\ ω_{3} \end{matrix}]

(3)

[\begin{matrix} {\ddot{φ}}_{1} \\ {\ddot{φ}}_{2} \\ {\ddot{φ}}_{3} \end{matrix}] = \frac{\partial G^{- 1}}{\partial t} ω + G^{- 1} ω

(4)

where G is not an orthogonal matrix, and its verse matrix is G⁻¹, they can be expressed, respectively, as

G = [\begin{matrix} \cos φ_{3} \cos φ_{2} & \sin φ_{3} & 0 \\ - \cos φ_{2} \sin φ_{3} & \cos φ_{3} & 0 \\ \sin φ_{2} & 0 & 1 \end{matrix}]

(5)

G^{- 1} = [\begin{matrix} \cos φ_{3} / \cos φ_{2} & - \sin φ_{3} / \cos φ_{2} & 0 \\ \sin φ_{3} & \cos φ_{3} & 0 \\ - \tan φ_{2} \cos φ_{3} & \tan φ_{2} \sin φ_{3} & 1 \end{matrix}]

(6)

2.2. The Fuselage Model

The fuselage can be simplified to a rigid body, and the external forces and moments of the fuselage are then calculated. By appropriately defining the aircraft’s mass, the angular velocity matrix, and the moment of inertia matrix, the translational and rotational acceleration of the aircraft can be calculated using the classical Newton–Euler equation. The specific dynamical equations are as follows:

\begin{array}{l} {\ddot{X}}_{p i} = \frac{1}{m} [Σ F_{x i}; Σ F_{y i}; Σ F_{z i}] \\ {\dot{ω}}_{x} = (M_{p x} + (I_{p y} - I_{p z}) * ω_{y} ω_{z}) / I_{p x} \\ {\dot{ω}}_{y} = (M_{p y} + (I_{p z} - I_{p x}) * ω_{x} ω_{z}) / I_{p y} \\ {\dot{ω}}_{x} = (M_{p z} + (I_{p x} - I_{p y}) * ω_{x} ω_{y}) / I_{p z} \end{array}

(7)

where m is the mass of the aircraft; M_px, M_py, and M_pz are the torques of the x, y, and z directions, respectively; ω_x, ω_y, ω_z are the angular velocities of the aircraft around the x, y, and z directions, respectively,

I = [I_{p x}; I_{p y}; I_{p x}]

is moment of inertia matrix;

[Σ F_{x i}; Σ F_{y i}; Σ F_{z i}]

is the resultant external force acting on the aircraft system.

2.3. The Winch Model

The winch system consists mainly of the capstans and ropes which will be separately modeled in this section.

(1): Capstan model

The force analysis of the cable on the capstan is shown in the box of Figure 3; the cable is wound on the traction capstan, and a small section of cable corresponding to the angle of the dα micro-arc on the winch is taken for the force analysis. The capstan is rotated clockwise, and the tension on the cable is F + dF. The output tension is F because the friction of the micro-arc dα reduces the tension by dF. dF is the frontal pressure of the cable against the winch, and μ is the coefficient of friction between the cable and the groove of the capstan. Based on the analysis above, the following equation can be derived:

\begin{array}{l} (F + d F) \cos \frac{d α}{2} = μ d N + F \cos \frac{d α}{2} \\ (F + d F) \sin \frac{d α}{2} + F \sin \frac{d α}{2} = d N \end{array}

(8)

The angle of the micro-arc dα is small, so a relationship exists

\sin \frac{d α}{2} \approx \frac{d α}{2}

\cos \frac{d α}{2} \approx 1

, and the following expressions can be obtained:

\begin{array}{l} d F = μ d F \\ F d α = d N \end{array}

(9)

\frac{d F}{F} = μ d α

(10)

Integrating Equation (11) leads to

F_{1} = F_{2} e^{μ α}

(11)

where α is the glide wrap corner; F₂ is the loose edge pull; F₁ is the tight edge pull.

(2): Rope model

In this study, the rope model is simplified as a spring model under ideal conditions. The formula for calculating the rope force value is as follows

F_{r o p e} = k_{r} \times Δ L + c_{r} \times Δ \dot{L}

(12)

where k_r is the stiffness of the rope, c_r is the rope damping, ΔL is the length of the rope deformation, and

Δ \dot{L}

is the deformation velocity of the rope.

2.4. Landing Gear-Tire Modeling

Telescopic landing gear is used in this research, and it generally consists of the pillar, the buffer, and the machine wheel. According to the structure and working principle of the landing gear, it can be equivalent to the mass-spring-damper model. If the equivalent damping and equivalent stiffness of the landing gear are c₂ and k₂, respectively, the dynamic model of the landing gear can be established, as shown in the red box in Figure 3. The vertical force of the landing gear is [6]

F = k_{2} d + c_{2} \dot{d}

(13)

where d is the strut displacement, and

\dot{d}

is the strut velocity.

The tire is a significant connection between the landing gear and the deck, and it plays a very important role in the stability, braking ability, and safety of the aircraft. When the aircraft moves, the tire is subjected to the corresponding forces, such as the longitudinal force, the lateral force, the vertical force, etc. The vertical force of the tire in the z-direction is similar to the landing gear, so the z-direction force of the landing gear-tire system can be equivalent to the two-mass-spring-damping system, as shown in Figure 5 where 0 represents the z-direction motion of the deck, 1 represents the z-direction motion of the center of the wheel, 2 represents the z-direction motion of the pillar, m₁ is the mass of the wheel, and m₂ is the mass of the aircraft fuselage. The equations of motion for the system can be written as

\{\begin{cases} m_{1} {\ddot{z}}_{1} = k_{2} (z_{2} - z_{1}) + c_{2} ({\dot{z}}_{2} - {\dot{z}}_{1}) - k_{1} (z_{1} - z_{0}) - c_{1} ({\dot{z}}_{1} - {\dot{z}}_{0}) - m_{1} g \\ m_{2} {\ddot{z}}_{2} = - k_{2} (z_{2} - z_{1}) - c_{2} ({\dot{z}}_{2} - {\dot{z}}_{1}) - m_{2} g \end{cases}

(14)

The force acting on the vertical direction of the aircraft is

F = k_{2} (z_{2} - z_{1}) + c_{2} ({\dot{z}}_{2} - {\dot{z}}_{1})

(15)

The longitudinal force (friction force) of the tire is

F_{l} = \{\begin{matrix} k_{t} d_{t} + c_{t} {\dot{d}}_{t} & F_{l} < F_{f r i c} \\ μ_{x} F n & F_{l} \geq F_{f r i c} \end{matrix}

(16)

where k_t and c_t are the tire’s longitudinal stiffness and damping, respectively; µ_x is the longitudinal friction coefficient.

The lateral force of the tire is

F_{c} = \{\begin{matrix} k_{c} d_{c} + c_{c} {\dot{d}}_{c} & F_{c} < F_{f r i c} \\ μ_{y} F n & F_{c} \geq F_{f r i c} \end{matrix}

(17)

where the k_c and c_c are the tire’s lateral stiffness and damping, respectively; µ_y is the lateral friction coefficient.

2.5. Ship Movement Characteristics

The traction process of the carrier-based helicopter is affected by a complex marine environment. The marine environment such as the wave and wind load will have an impact on the ship’s movements with six DoFs, among which the roll (φ), the pitch (θ), and the heave (z) have great effects on the traction process, and these three motions will be focused on. The rolling is along the longitudinal axis direction of the ship, the pitch is along the lateral axis direction of the ship, and the heave is along the vertical direction of the ship, and the sinusoidal function is applied to describe the three movements.

A ship is excited by sea waves and wind loads which are often multi-frequency. In traditional research, these loads are simplified to be a single-frequency excitation. In this research, these complex sea loads are regarded as multi-frequency excitation which is described by Equation (13) [6].

\{\begin{array}{l} φ = φ_{1} \sin (2 π t / T_{φ 1} + η_{φ 1}) + φ_{2} \cos (2 π t / T_{φ 2} + η_{φ 2}) \\ θ = θ_{1} \sin (2 π t / T_{θ 1} + η_{θ 1}) + θ_{2} \cos (2 π t / T_{θ 2} + η_{θ 2}) \\ z = z_{1} \sin (2 π t / T_{z 1} + η_{z 1}) + z_{2} \cos (2 π t / T_{z 2} + η_{z 2}) \end{array}

(18)

where

φ_{1}

θ_{1}

z_{1}

and

φ_{2}

θ_{2}

z_{2}

are the amplitude of the rolling, the pitching, and the heaving motions, respectively;

T_{φ 1}

T_{θ 1}

T_{z 1}

are period 1 of the rolling, the pitching, and heaving motions, respectively;

T_{φ 2}

T_{θ 2}

T_{z 2}

are period 2 of the rolling, the pitching, and heaving motions, respectively;

η_{φ 1}

η_{θ 1}

η_{z 1}

and

η_{φ 2}

η_{θ 2}

η_{z 2}

are the initial phase angles of the rolling, the pitching, and the heaving motions. The schematic diagram of the ship’s motions is shown in the dashed box of Figure 2.

2.6. Wind Load

The carrier-based aircraft traction system is prone to interference by wind load which causes the aircraft to overturn in certain cases. The wind speed is mainly composed of the wind speed generated by the movement of the aircraft, the wind speed generated by the ship’s navigation, and the wind speed on the sea’s surface. Moreover, it is supposed that the wind load mainly acts on the centroid of the aircraft, and the wind load can be simplified as follows [26]

Q = 1676 \times S \times {(V / 100)}^{2}

(19)

where Q is the resultant force of the wind load, its unit is N; 1676 is the product of the air density and power coefficient; S is the area of the helicopter in the plane perpendicular to the wind direction, and its unit is m²; and

V = V_{w i n d} + V_{s h i p}

is the synthetic wind speed, V_ship is the speed of the ship, V_wind is the wind speed, and its unit is kn. The direction of the wind load is the synthesis velocity direction. The specific schematic diagram is shown in Figure 6.

3. Control of Aircraft Speed and Trajectory

The aircraft is pulled into the warehouse under the setting trajectory at a certain speed. In order to effectively control the driving trajectory and speed of the aircraft, the PID (Proportional Integral Derivative)is used and the vector difference between the centroid of the aircraft and the position on the predetermined Bessel curve is calculated for realizing the trajectory control of the aircraft.

3.1. Control of Aircraft Speed

PID is a classical control algorithm with strong adaptability and robustness. The traction system established in this paper is suitable for using PID to control the aircraft’s speed. In the winch traction system of the shipborne aircraft, PID is used to control the speed of collecting rope for the winch so as to control the driving speed of the aircraft. By constantly adjusting the proportion, integral, and differential parameters, a stable speed of the aircraft is maintained under the different working conditions.

The PID parameters and the adjustment process can be briefly described as follows. Proportional (P) is the proportional control part which is proportional to the error; it produces the control output based on the deviation between the current speed and the desired speed. The role of proportional control is to approach the system rapidly to the desired value. Integration (I) is the integral control part under consideration of the accumulation of errors over time, and it is used to eliminate static errors. Differential (D) is the differential control part considering the change rate of the error, and it is used to suppress the sensitivity of the system to input changes. The formula of the PID control algorithm is written as

u (t) = k_{p} e (t) + k_{i} \int_{0}^{t} e (t) d t + k_{d} \frac{d e (t)}{d t}

(20)

where u (t) is the output control amount which is the first-order derivative of the rope force; k_p is the proportional amplification coefficient; k_i is the integral time constant; and k_d is the differential time constant. Their specific parameters are shown in Table 1. The principle of aircraft speed control by PID is shown in Figure 7. The aircraft speed control process is performed as follows: The given target speed is input first, and then the deviation between the actual aircraft speed and the target speed e(t) is calculated. Afterwards, the control quantity u(t) produced by the PID controller is output, and the aircraft is made to travel according to the speed of this control amount. The true speed of the aircraft is reached, and again, the real aircraft speed and the target speed are compared. Then, the deviation between the actual aircraft speed and the target speed is computed again, and the system enters the control cycle. The aircraft controlled by multiple cycles will travel at the given target speed. In this research, e(t) is the deviation between the setting rope collecting speed and the speed of the rope-aircraft traction point. It can be expressed by

e (t) = y_{s} (t) - y (t)

(21)

where y_s(t) is the setting rope collecting speed, and y(t) is the speed of the rope-aircraft traction point.

3.2. Trajectory Control

The travel trajectory of the aircraft is controlled using the Bessel curve in the research. The Bessel curve is an important parametric curve whose parameters mainly include the starting, the ending, and the controlling points. There are position vectors of n reference points in the space, P_i, i = (0, 1, 2, 3,…, n), and the interpolation formula for a point on the Bessel curve of n times is

p (t) = \sum_{i = 0}^{n} P_{i} B_{i, n} (t), t \in [0, 1]

(22)

where t is the independent variable, P_i is the control point, and B_i,n (t) is the n order Bernstein basis function. Its expression is

B_{i, n} (t) = \frac{n!}{i! (n - 1)!} t^{!} {(1 - t)}^{n - 1}, i = 0, 1, 2, 3, \dots, n

(23)

The Bessel curve is mainly determined by its control points which here contain four control points, P₀, P₁, P₂, and P₃, and the relationship between the control points and the curve is written as

P_{3} (t) = {(1 - t)}^{3} P_{0} + 3 t {(1 - t)}^{2} P_{1} + 3 t^{2} (1 - t) P_{2} + t^{3} P_{3}

(24)

The control points P₀, P₁, P₂, and P₃ are chosen as (0, 0), (20, 5), (30, −5), and (60, 0) in this work, respectively, and the generated trajectory diagram is shown in Figure 8; the longitudinal coordinates are the displacement of aircraft in the y direction and the horizontal coordinates are the displacement in the x direction; their units are m.

The shape of the curve is controlled by changing the related parameters. The vector difference between the centroid of the carrier-based aircraft and the position on the predetermined Bessel curve is calculated, so as to obtain the azimuth vector in the aircraft coordinate system for realizing the trajectory control of the aircraft. The implementation plan of the aircraft trajectory control is as follows: At moment t₁, taking the point on the trajectory curve which is nearest and at the front of the aircraft centroid, the orientation vector between the point and the transient centroid of the aircraft is calculated. Then, the azimuth angle of the aircraft is calculated at that moment, and the steering angle of the wheel at that moment is obtained using the transformation equation of the coordinates. The steering angle at the next moment t₂ can be obtained according to this method. The steering angles at all moments are formed into the control command flow which is used to control the steering wheels of the aircraft. In this way, the aircraft can follow the setting trajectory curve. Figure 9 shows the traveling trajectory diagram of the aircraft with the control. Figure 9a is the planar view from z to −z and Figure 9b is the corresponding three-dimensional view. It can be seen from Figure 9 that the trajectory is a curve. Figure 10 presents the trajectory diagram of the aircraft without trajectory control. Figure 10a is the planar view from z to −z and Figure 10b is the corresponding three-dimensional view. It can be shown from Figure 10 that the traveling trajectory is approximately a straight line.

4. Calculation Results and Analysis

The traction process of a carrier-based aircraft is affected by a complicated marine environment which inspires the hull through waves. This excitation from complex sea conditions is often a multi-frequency excitation. Traditional studies treat this excitation as a single-frequency excitation. In this study, the dynamics of traction under multi-frequency excitation will be analyzed, and a comparison of traction dynamics under single and multiple-frequency excitation will be conducted; the influence of the hull roll, the pitch, and the heave on the tire and rope forces during the traction process will also be researched in this section. Based on the dynamic model of the aircraft winch traction system developed, the five-winch traction is chosen to pull the front three-point aircraft. The mass of the aircraft is 13,000 kg, the coordinates of the center of the aircraft are (0, 0, 2.8), and the inertia of the fuselage I_xx is 18,000 kg/m², I_yy is 32,666.7 kg/m², and I_zz is 50,666.7 kg/m². Specific parameters are shown in Table 2 and Table 3, and these parameters are derived from real project data obtained from experimental tests.

4.1. Single-Frequency Excitation

The deck movements under the single-frequency excitation include various types of movement such as the roll, the pitch, and the heave. The amplitude and angular frequency of these movements have effects on the rope force and tire force. The dynamic analysis of the five-winch traction system considering the wind load and the control of trajectory will be carried out in the following section.

4.1.1. Effect of Pitching Amplitude on Rope and Tire Forces

The pitching amplitude of the hull reflects the intensity of the sea conditions. After the calculation, the tire and the rope forces of the aircraft traction system under the single-frequency excitation are obtained, as shown in Figure 11, Figure 12, Figure 13 and Figure 14. The first 10 s in each figure is the self-equilibrium stage of the aircraft. In order to study the rules of the rope force changing with time under the different amplitudes, the pitching amplitudes of 5°, 2°, 0.8°, and 0.1° are chosen for calculation to obtain Figure 11, where the front rope force increases suddenly to 40,412 N at the end of the self-equilibrium phase. This is because the initial pulling force required for the aircraft is large; then, the front rope force amplitude increases with the pitching amplitude. Figure 12 is the curve of the vertical force of each tire over time. It can be seen from Figure 12 that the vertical force of tires two and three changes periodically over time after self-equilibrium (10 s later), and the vertical force of tire one is almost stable at a constant value of 43,472 N after the initial self-equilibrium. In order to study the safety of the tire during the traction process, tire three with the largest force value of all the tires is mainly analyzed. Figure 13 shows the curve of the force over time in each direction of tire three. It can be seen from Figure 13 that the vertical force and lateral force of tire three fluctuates greatly over time, while the longitudinal force is stable at −2623 N, basically without fluctuation; the vertical force of the three forces is relatively larger, and this is caused by the dead weight of the aircraft. Figure 14 shows the curve of the vertical force of tire three over time at different amplitudes. It can be shown from Figure 14 that the single-frequency excitation amplitude has little effect on the vertical force of tire three.

4.1.2. Effect of the Rolling Angular Frequency on Rope and Tire Forces

To further investigate the effect of the rolling angle frequency on the dynamics of the traction system, the rolling angular frequencies ω₁ = 0.93 rad/s, 0.63 rad/s, 0.23 rad/s, and 0.1 rad/s under the constant amplitude of the rolling and pitching are chosen to obtain the time-varying curves of the rope and tire forces. Figure 15 is the curve of the force of the front rope over time at different rolling angle frequencies. It can be seen from Figure 15 that with the gradual increase of the rolling angle frequency, the fluctuation frequency of the front rope tension is becoming higher, and the fluctuation amplitude also shows an increasing trend. Figure 16 is the curve of each tire force changing with time under the different rolling angle frequencies. It can be seen from Figure 16 that the amplitude and fluctuation frequency of the tire force increases with the increase of rolling angle frequency. In addition, between 10 and 20 s, the tire force has small fluctuations which are related to the initial state. The fluctuations gradually tend to disappear after 20 s because of the vibration damping. In conclusion, when the rolling angle frequency of the ship increases, the fluctuation frequency of rope force and tire force increases, and the fluctuation amplitude becomes larger. This poses a potential threat to traction safety; in the initial stage, there is transient vibration caused by the initial state which decreases with time and has little impact on traction safety.

4.2. Multiple-Frequency Excitation

4.2.1. Effect of the Pitching Amplitude on Rope and Tire Forces

The pitching of the hull caused by the complex sea conditions can affect the safety and stability of the traction system. In order to study the effect of the pitching amplitude on the rope force and the tire force, the pitching amplitude θ₁ is selected as 2°, 0.8°, 0.4° and 0.1°, respectively, and θ₂ remains unchanged. The curve of the front rope force over time under the variable amplitudes is shown in Figure 17. It can be seen from Figure 17 that with the increase of the pitching amplitude, the front rope force shows a large fluctuation of frequency between 10 s and 20 s.

In order to study the influence of the pitching amplitude on the tire force, the vertical force of a typical tire three is selected as the research focus, and the influence of the pitching amplitude on the tire force is calculated, as shown in Figure 18. As can be seen from Figure 18, the peak value of the tire vertical force is at 10.5 s, and the value increases with the pitching amplitude, but after 12 s, the tire force gradually decreases, and the influence of the pitching amplitude on the vertical force gradually decreases.

4.2.2. Effect of the Rolling Angular Frequency on Rope and Tire Forces

For studying the effect of the different rolling angular frequency on the aircraft traction system under the multi-frequency wave excitation, the rolling angular frequencies ω₂ of 2 rad/s, 1 rad/s, 0.63 rad/s, and 0.23 rad/s are chosen for calculation. Figure 19 shows the curve of the force of the front rope over time at the different rolling angle frequencies. As can be seen from Figure 19, the fluctuation frequency of the front rope force increases with the increase of the excitation frequency. Figure 20 shows the curve of the vertical force of tire three changing with time at different rolling angle frequencies. As can be seen from the figure, the vertical force at each amplitude is almost equal before 12 s, and after 12 s with the increase of the frequency, the amplitude of the vertical force of tire three has no certain rules.

4.2.3. Effect of Heaving Amplitude on Traction Characteristics

In order to research the influence of heaving amplitude on the dynamics of an aircraft traction system, the heaving amplitude z₁ is 0.19 m, 0.1 m, 0.05 m, and 0.019 m, respectively, for calculations. The curves of rope force and tire force with different heaving amplitudes under the multi-frequency excitation are drawn, as shown in Figure 21 and Figure 22. Figure 21 shows the curve of the front rope force over time at different heaving amplitudes. As can be seen from Figure 21, between 11 s and 15 s, the rope force at each amplitude has a large oscillation which is caused by the transient effect in the starting stage of the aircraft. The changing curves of the rope force at each amplitude after 15 s basically coincide, and the influence of the heaving amplitude is minimal. Figure 22 is the curve of the vertical force of tire three over time at different heaving amplitudes. It can be seen from Figure 22 that the oscillation amplitude of tire force is related to the heaving amplitude of the hull.

5. Comparison of Traction Dynamics Between Complex and Simple Sea Conditions

It is of great significance for the safety and stability of the system to study the difference in traction dynamics under complex and simple sea conditions. The differences in dynamics between the multiple and single-frequency excitations, with trajectory control and without trajectory control, will be studied in this section.

5.1. Comparison of Multi-Frequency Excitation and Single-Frequency Excitation under Trajectory Control

The time-varying curves of the front rope force and tire force under the single-frequency and multi-frequency excitations with trajectory control are obtained, as shown in Figure 23 (rope force) and Figure 24 (tire force). As can be seen from Figure 23, during 10~15 s, the rope force amplitude of multi-frequency excitation is 29.5% larger than that of the single frequency, and the oscillation frequency is larger. In the stable stage after 15 s, the two amplitudes are not much different and there is only a certain phase difference. Figure 24 shows the curve of the vertical force of tire three over time under single-frequency excitation and multi-frequency excitation with trajectory control. It can be seen from Figure 24 that the vertical force amplitude of tire three under multi-frequency excitation is 201.1% larger than that of single-frequency excitation during 10–15 s. The two amplitudes are roughly the same after the stabilization. It is deduced that the multi-frequency excitation has a great influence on the initial rope force and tire force.

5.2. Comparison of Trajectory Control and No Trajectory Control Under Multi-Frequency Excitation

The controlled trajectory of the aircraft is a Bessel curve, and the trajectory is a straight line when there is no trajectory traction. Figure 25 shows the curve of the front rope force over time with trajectory control and without trajectory control under multi-frequency excitation. It can be seen from Figure 25 that in the initial phase (10–15 s, region I), the rope force amplitude under trajectory control is 26.2% larger than that of no trajectory control, but only 7.0% larger at a steady state (20–35 s, region II). This is because the aircraft trajectory under the control is the curve, the centripetal force of the rope is larger, and the starting stage is even greater compared with the straight line. Figure 26 shows the curve of the vertical force of tire three over time with trajectory control and without trajectory control under multi-frequency excitation. At the starting time (10 s), the amplitude of tire force under no trajectory control is 27.1% larger than that of trajectory control. It can also be seen that the transient oscillation time of the tire is greatly reduced.

6. Verification of the Calculation Results

For verifying the correctness of the theoretical calculation for the winch traction, the general software ADAMS (ADAMS2020) is used to model the winch traction system with the same parameters as theoretical modeling, and the calculation results are compared with the theoretical calculation results. Figure 27 shows the dynamic model of the five-winch traction system of the aircraft based on ADAMS. The aircraft is treated as a mass point for simplifying the model. Winch 1 provides the rope force as the front winch, winches 3 and 4 provide the back tension as the rear winches, and winch 2 and 5 are auxiliary winches to prevent abnormal conditions such as the side slip and off the ground. Figure 28 shows the local amplification view of the landing gear in five-winch traction of the aircraft; the front and rear landing gears are telescopic, as shown in Figure 29 and Figure 30.

The rope force of the front winch (winch 1) and the vertical force of tire three are obtained and compared with the theoretical calculation results obtained by MATLAB. Figure 31 is the comparison of the front rope force obtained by ADAMS and MATLAB; Figure 32 shows the comparison of the vertical forces for the right rear wheel (tire three) obtained by ADAMS and MATLAB. As can be seen from Figure 31 and Figure 32, the maximum difference between the result of the general software calculation and the result of the theoretical calculation is only 12.1%, which is within the allowable range and meets the engineering requirements. The general software calculation results agree well with the theoretical calculation results, and this proves the correctness and reliability of the whole system analysis method for the carrier-based aircraft winch traction.

7. Conclusions

The theoretical analysis method of the dynamics for the aircraft-winch system is proposed in this paper. The conversion relationship among the fuselage, the ship, and the inertial coordinate systems is developed; the five-winch traction and the telescopic landing gear are adopted in the research. Considering the pitching, rolling, and heaving motions of the ship, the wave excitation in complex sea conditions is described as a multiple-frequency harmonic function to study the dynamic characteristics of the whole system. The theoretical analysis method is validated by the general software ADAMS. The main research conclusions are as follows:

(1): The influences of the pitching amplitude and the rolling frequency on the traction dynamics of a carrier-based aircraft driven by setting trajectory under complex sea conditions are studied. In multi-frequency excitation under complex sea conditions, the pitching amplitude has a greater impact on the tire force compared with that of single frequency, especially in the starting time of aircraft traction. In 10~15 s, the rope force amplitude of the multi-frequency excitation is 29.5% larger than that of the single-frequency amplitude, and the frequency of the rope force oscillation is larger, while the vertical force amplitude of the tire is 201.1% larger than that of the single-frequency amplitude. With the increase of the rolling angular frequency, the amplitude increase of the rope force in the stability period (after 15 s) under multiple-frequency excitation is smaller than that of single-frequency excitation. For the winch traction system working in complex sea conditions, the strength of tires and ropes should be checked in advance with sufficient safety factors, and attention should be paid to the change in dynamic force in real time during the traction process to ensure the safety and stability of traction.
(2): The advantages and disadvantages of a winch traction system with trajectory control and without trajectory control in complex sea conditions are analyzed. Under the complex sea conditions of the multi-frequency wave excitation, the amplitude of the front rope force under trajectory control is greater than that of no trajectory control, up to 26.2%. The front rope force during the aircraft turn (under trajectory control) is larger than that of traveling along a straight line (without trajectory control), and this is because the centripetal force during the aircraft turn is larger. The tire force at the initial moment (10 s) without trajectory control is 27.1% greater than that of the trajectory control. Therefore, in complex sea conditions, comprehensive consideration should be taken and appropriate measures should be taken to ensure the safety and stability of the traction system.
(3): The general software ADAMS is adopted to model the winch traction system for verifying the correctness of the theoretical calculation for the winch traction. The general software calculation results agree well with the theoretical calculation results, and this proves the correctness and reliability of the whole system analysis method of winch traction for carrier-based aircrafts.

Author Contributions

Software, Y.L.; Validation, S.Y.; Writing—original draft, G.N.; Writing—review & editing, Y.Z. All authors have read and agreed to the published version of the manuscript.

Funding

This project is supported by the National Natural Science Foundation of China (Grant No. 52275118).

Data Availability Statement

Data is contain within the article.

Conflicts of Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Nomenclature

Variables	Annotation
k₁, c₁	Stiffness and damping of tire in a vertical direction
k₂, c₂	Stiffness and damping of strut (landing gear) in a vertical direction
d, $\dot{d}$	Displacement and velocity of the strut
m₁, m₂	Mass of wheel and aircraft fuselage
µ_x, µ_y	Longitudinal and lateral friction coefficient
F_fric	Tire static friction
L_yt	Force acting on the vertical direction of the aircraft
F_l	Longitudinal force (friction force) of tire
F_c	Lateral force of tire
${\ddot{X}}_{p i}$	Acceleration of aircraft centroid
ω_x, ω_y, ω_z	Angular velocities of aircraft around x, y, and z directions,
${\dot{ω}}_{x}$ , ${\dot{ω}}_{y}$ , ${\dot{ω}}_{z}$	Angular accelerations of aircraft around x, y, and z directions,
I_px, I_py, I_pz	Moment of inertia matrix of aircraft in x, y, and z directions,
α	Glide wrap corner
µ	Friction coefficient between capstan and rope
F₁, F₂	Pull force of tight end and loose end
F_rope	Rope force
ω₀, θ₀, z₀	Amplitude of rolling, pitching, and heaving motions
T_ω, T_θ, T _z	Period of rolling, pitching, and heaving motions
η_ω, η_θ, η_z	Initial phase angles of rolling, pitching, and heaving motions
Q	Resultant force of the wind load
S	Area of the helicopter in plane perpendicular to wind direction
V	Synthetic wind speed

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Figure 1. Virtual prototyping model of the tractor–aircraft system [20].

Figure 2. Schematic diagram of aircraft winch traction.

Figure 3. Schematic diagram of the whole system of carrier-based aircraft traction.

Figure 4. Cardan Angles(The conversion relationships between different coordinate systems).

Figure 5. The mathematical model of a landing gear-tire system in a z-direction.

Figure 6. Schematic diagram of wind load.

Figure 7. Graph of PID control for aircraft speed.

Figure 8. The generated trajectory diagram (Bessel curve).

Figure 9. Traveling trajectory diagram of the aircraft with the control (curve) (a) Planar view from z to −z; (b) Three-dimensional view.

Figure 10. Traveling trajectory diagram of the aircraft without the control (straight line). (a) Planar view from z to −z; (b) Three-dimensional view.

Figure 11. Curves of rope force changing with time under different pitching amplitudes (θ₁ = 5°, 2°, 0.8° and 0.1°, φ₁ = 5°).

Figure 12. Curve of the vertical force of each tire over time (φ₁ = 5°, θ₁ = 2°).

Figure 13. Curve of tire force over time in each direction for tire three (φ₁ = 5°; θ₁ = 2°).

Figure 14. Curve of the vertical force of tire three over time at different pitching amplitudes (θ₁ = 5°, 2°, 0.8° and 0.1°; φ₁ = 5°).

Figure 15. Curve of the force of the front rope over time at different rolling angle frequencies (ω₁ = 2π/T_φ1 = 0.93 rad/s, 0.63 rad/s, 0.23 rad/s and 0.1rad/s; φ₁ = 5°, θ₁ = 2°).

Figure 16. Curve of the vertical force of tire three changing with time at different rolling angle frequencies (ω₁ = 2π/T_φ1 = 0.93 rad/s, 0.63 rad/s, 0.23 rad/s and 0.1 rad/s; φ₁ = 5°; θ₁ = 2°).

Figure 17. Curve of the force of the front rope over time at different pitching amplitudes (θ₁ = 2°, 0.8°, 0.4° and 0.1°; θ₂ = 1°).

Figure 18. Curve of the vertical force of tire three over time at different pitching amplitudes (θ₁ = 2°, 0.8°, 0.4° and 0.1°; θ₂ = 1°).

Figure 19. Curve of the force of the front rope over time at different rolling angle frequencies (ω₂ = 2π/T_φ2 = 2 rad/s, 1 rad/s, 0.63 rad/s and 0.23 rad/s).

Figure 20. Curve of the vertical force of tire three changing with time at different rolling angle frequencies (ω₂ = 2π/T_φ2 = 2 rad/s, 1 rad/s, 0.63 rad/s and 0.23 rad/s).

Figure 21. Curve of the force of the front rope over time at different heaving amplitudes (z₁ = 0.19 m, 0.1 m, 0.05 m, and 0.019 m).

Figure 22. Curve of the vertical force of tire three over time at different heaving amplitudes (z₁ = 0.19 m, 0.1 m, 0.05 m, and 0.019 m).

Figure 23. Curve of the front rope force over time under single-frequency excitation and multi-frequency excitation with trajectory control.

Figure 24. Curve of the vertical force of tire three over time under single-frequency excitation and multi-frequency excitation with trajectory control.

Figure 25. Curve of the front rope force over time with trajectory control and without trajectory control under multi-frequency excitation.

Figure 26. Curve of the vertical force of tire three over time with trajectory control and without trajectory control under multi-frequency excitation.

Figure 27. Simulation of five-winch traction of aircraft.

Figure 28. Local amplication view of landing gear in five-winch traction of aircraft.

Figure 29. Front landing gear model.

Figure 30. Rear landing gear model.

Figure 31. Comparison of the front rope forces obtained by ADAMS and MATLAB.

Figure 32. Comparison of vertical forces for tire three obtained by ADAMS and MATLAB.

Table 1. Parameters of PID.

Three-Winch		Five-Winch
Proportional (P)	0.15	Proportional (P)	0.1
Integral (I)	0.102	Integral (I)	0.2
Derivative (D)	0	Derivative (D)	0

Table 2. The parameters of aircraft modeling.

Parameters	Values
Maximum take-off mass of carrier-based aircraft (m₂/kg)	13,000
Mass of each tire for the carrier aircraft (m₁/kg)	150
Vertical distance from the aircraft centroid to the deck/m	2.8
Horizontal distance from the aircraft centroid to the main landing gear shaft/m	2
The inertia of rotation around the X axis (Ixx/kg·m²)	18,000
The inertia of rotation around the Y axis (Iyy/kg·m²)	32,666.7
The inertia of rotation around the Z axis (Izz/kg·m²)	50,666.7
Horizontal distance between the centerlines of the two main landing gears/m	4
Horizontal distance between front gear shaft and main gear shaft/m	6.6

Table 3. The parameters of aircraft landing gear.

Parameters	Values
Stiffness of main landing gear (k₂/N/mm)	884.4
Damping of main landing gear (c₂ N·s/mm)	15.3
Stiffness of front landing gear (k₂/N/mm)	884.4
Damping of front landing gear (c₂/N·s/mm)	15.3

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MDPI and ACS Style

Nan, G.; Yang, S.; Li, Y.; Zhou, Y. Winch Traction Dynamics for a Carrier-Based Aircraft Under Trajectory Control on a Small Deck in Complex Sea Conditions. Aerospace 2024, 11, 885. https://doi.org/10.3390/aerospace11110885

AMA Style

Nan G, Yang S, Li Y, Zhou Y. Winch Traction Dynamics for a Carrier-Based Aircraft Under Trajectory Control on a Small Deck in Complex Sea Conditions. Aerospace. 2024; 11(11):885. https://doi.org/10.3390/aerospace11110885

Chicago/Turabian Style

Nan, Guofang, Sirui Yang, Yao Li, and Yihui Zhou. 2024. "Winch Traction Dynamics for a Carrier-Based Aircraft Under Trajectory Control on a Small Deck in Complex Sea Conditions" Aerospace 11, no. 11: 885. https://doi.org/10.3390/aerospace11110885

APA Style

Nan, G., Yang, S., Li, Y., & Zhou, Y. (2024). Winch Traction Dynamics for a Carrier-Based Aircraft Under Trajectory Control on a Small Deck in Complex Sea Conditions. Aerospace, 11(11), 885. https://doi.org/10.3390/aerospace11110885

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