Dynamic Analysis and Numerical Simulation of Arresting Hook Engaging Cable in Carried-Based UAV Landing Process

: Carrier-based unmanned aerial vehicles (UAVs) require precise evaluation methods for their landing and arresting safety due to their high autonomy and demanding reliability requirements. In this paper, an efﬁcient and accurate simulation method is presented for studying the arresting hook engaging arresting cable process. The ﬁnite element method and multibody dynamics (FEM-MBD) approach is employed. By establishing a rigid–ﬂexible coupling model encompassing the UAV and arresting gear system, the simulation model for the engagement process is obtained. The model incorporates multiple coordinate systems to effectively capture the relative motion between the rigid and ﬂexible components. The model considers the material properties, arresting gear system characteristics, and UAV state during engagement. Veriﬁcation is conducted by comparing simulation results with experimental data from a referenced arresting hook rebound. Finally, simulations are performed under different touchdown points and roll angles of the UAV to analyze the stress distribution of the hook, center of gravity variations, and the tire touch and rollover cable response. The proposed rigid–ﬂexible coupling arresting dynamics model in this paper enables the effective analysis of the dynamic behavior during the arresting hook engaging arresting cable process.


Introduction
The engagement of the arresting hook with the arresting cable is a critical maneuver that most directly reflects the success of arrestment and serves as the most complex stage in terms of dynamics in the carrier-based aircraft landing process [1,2]. Carrier-based UAVs usually land in a collision [3]. To achieve a successful arrest on a carrier, UAVs must approach with the specified speed and attitude, and land precisely on the desired area of the carrier deck [4]. The arrested landing process of UAVs includes multiple intricate dynamic stages. The arresting hook, located at the tail of the aircraft, makes contact with the deck upon landing and rebounds to a limited height under the influence of the hook's damper. Subsequently, the arresting hook engages with the arresting cable on the deck, transferring the arresting force to the fuselage and bringing the aircraft to a rapid stop.
Due to the complexity of carrier arresting systems and the unique operational environment, it is challenging to accurately measure real-time loads via experiments [5]. During the design phase of a carrier-based aircraft, the analysis of the hook-cable engagement process plays an important role in confirming the safety area for deck landing [6]. The primary research employed analyzes the arresting performance through theoretical analysis and simulation methods [7,8]. Thomlinson [9] conducted research on the motion of the aircraft arresting hook within the plane of symmetry after impacting the deck. In the paper, it was assumed that the carrier-based aircraft had no yaw deviation during the arresting process. Jones [10] obtained a fitting equation and curve for the arresting force based on statistical data from the arresting system. Gao [11], Liu [12] and Peng [13,14] established the mechanical relationships among the aircraft, deck and arresting hook for an arresting hook impacting the deck. Zhu [15] considered the influence of frictional force on the collision between the arresting hook and deck, and established a more accurate model for the arresting hook and deck collision. The previous research provides a better understanding of the interaction between the arresting hook and deck during the collision process. Additionally, the enhanced model proposed has a guiding effect on the design of the arresting hook system. However, it is acknowledged that the model assumptions in these studies limit the comprehensive analysis of the dynamics involved in the arresting process in different scenarios.
In order to conduct a comprehensive analysis of a hook's cable engagement process, it is essential to establish accurate model for both the aircraft and the arresting cable [16,17]. Various methods can be employed to model cable dynamics, including the absolute node coordinate method [18] and finite segment method [19,20]. Deng [21] developed a 2D nonmaterial variable-domain co-rotational element to perform a nonlinear dynamic analysis of arresting gears, and the nonlinear equation of the hydraulic damper sub-system was formulated. The propagation mechanism of longitudinal waves and kink waves was investigated. Software such as LS-Dyna [22] and PAM-Crash [23] have also been widely used to simulate the dynamic characteristics of aircraft landing and arrest. Shen [24] established a full-scale dynamic model of an MK7-type arresting gear system based on the multi-body dynamics method. Zhang [25] adopted arbitrary Lagrangian-Eulerian formulation to efficiently simulate hook/pulley-cable moving contact in arresting cable systems. Zhang [26] developed a dynamic model for the rebound of the arresting hook during collision, using the numerical iteration calculation method to obtain the longitudinal safety envelope of the aircraft during the landing and arresting process. Peng [27] conducted an impact rebound test for the arresting hook and subsequently refined the coefficient of restitution for the deck coating based on the test results. The comprehensive examination of the arresting hook engaging cable process necessitates a meticulous analysis encompassing the model of landing gear shock absorbers and flexible tires, as well as the material properties and mechanical characteristics of the arresting cable and hook.
In this paper, a novel rigid-flexible coupling model of a carrier-based UAV for arresting engagement is established based on the FEM-MBD approach. The dynamic model is verified by comparing simulation results with experimental data from references, and simulations are conducted with different touchdown points and roll angles. The proposed simulation method can accurately capture the process of UAV arresting hook engagement, including the rebound motion of the hook upon deck impact, the bending of the cable during engagement and the stress distribution on the hook. The results obtained from this analysis offer valuable insights into the performance of the engagement process, which can be used to test various design configurations virtually and improve the design of carrier-based UAVs for carrier landing.

Finite Element Discretization and Contact
The finite element method primarily employs eight-node hexahedral elements and four-node shell elements to describe the structural components. The dynamic equation for each finite element is as follows:   The contact force between elements is illustrated in Figure 1. n i represents the slave node, S j represents the master surface, m 1 , m 2 , m 3 , m 4 represents the master surface node, H cont represents contact thickness, δ i represents the penetration between elements, v i represents relative sliding velocity at the contact point with respect to the master surface, F i represents the contact force acting on the slave node n i , and f 1 , f 2 , f 3 and f 4 represent the equivalent force acting on the master node. The contact force between elements is illustrated in Figure 1. i n represents the slave node, j S represents the master surface, 1 2 3 4 , , , m m m m represents the master surface node, cont H represents contact thickness, i δ represents the penetration between elements, i v represents relative sliding velocity at the contact point with respect to the master surface, i F represents the contact force acting on the slave node i n , and 1 f , 2 f , 3 f and 4 f represent the equivalent force acting on the master node. The contact force, i F , between elements can be resolved into the normal component f s and tangential component c f , which are determined via Equation (2).
where s,e f is the normal elastic force, , s v f is the normal viscous force, f c,e is the tangential elastic, u is the coefficient of friction, n is the unit vector in the normal direction of contact, and t is the unit vector in the tangential direction of contact.
The normal elastic force, s,e f , and normal viscous force, , s v f , between elements are given by Equations (3) and (4), respectively.
where i k is the local contact stiffness, ε is the proportionality factor of the contact force, ξ is the contact damping coefficient, The contact force, F i , between elements can be resolved into the normal component f s and tangential component f c , which are determined via Equation (2).
where f s,e is the normal elastic force, f s,v is the normal viscous force, f c,e is the tangential elastic, u is the coefficient of friction, n is the unit vector in the normal direction of contact, and t is the unit vector in the tangential direction of contact.
The normal elastic force, f s,e , and normal viscous force, f s,v , between elements are given by Equations (3) and (4), respectively.
where k i is the local contact stiffness, ε is the proportionality factor of the contact force, c Ni = 2ξ i √ k Ni m i is the internal damping, ξ i is the contact damping coefficient, m i is the mass of the slave node, and v i is the relative velocity in the normal direction at the contact point.
The tangential elastic force, f c,e , and tangential friction force, f c , between elements are as follows.
where k Ni is the normal stiffness, ν is Poisson's ratio, and δ j is the tangential relative displacement at contact nodes.

Model Description
Noticing a common technique to simplify system dynamic equations and reduce computational costs, this paper adopts the approach of connecting multiple rigid bodies through joints. The UAV model consists of the fuselage, landing gear, and arresting hook. The simulation model of the arresting gear system includes wire rope supports, a deck, an arresting cable, a damper and sheaves. The simulation model of the integrated UAV and arresting gear system is shown in Figure 2.

Model Description
Noticing a common technique to simplify system dynamic equations and reduce computational costs, this paper adopts the approach of connecting multiple rigid bodies through joints. The UAV model consists of the fuselage, landing gear, and arresting hook. The simulation model of the arresting gear system includes wire rope supports, a deck, an arresting cable, a damper and sheaves. The simulation model of the integrated UAV and arresting gear system is shown in Figure 2.

Configuration of UAV
The deformation and stress of the UAV structure are not the main concern. Therefore, it is modeled using rigid bodies described by the degree of freedom (DoF) kinematic and dynamic differential equations. The UAV model in this paper consists of the arresting hook, nose landing gear (NLG), main landing gear (MLG) and fuselage as shown in Figure  3a. Aerodynamic force is applied as a 6 DoF load (X, Y, Z, L, M, and N) on the fuselage rigid body, with the point of application being converted into the center of gravity. X, Y, Z, L, M, and N are calculated via Equation (7).  The deformation and stress of the UAV structure are not the main concern. Therefore, it is modeled using rigid bodies described by the degree of freedom (DoF) kinematic and dynamic differential equations. The UAV model in this paper consists of the arresting hook, nose landing gear (NLG), main landing gear (MLG) and fuselage as shown in Figure 3a. Aerodynamic force is applied as a 6 DoF load (X, Y, Z, L, M, and N) on the fuselage rigid body, with the point of application being converted into the center of gravity. X, Y, Z, L, M, and N are calculated via Equation (7).
where ρ is the density of air, u is the UAV velocity, and C x , C y , C z , C l , C m , and C n are aerodynamic coefficients, respectively. c, b, and S re f are the reference chord, reference span and reference wing area of the UAV, respectively. where ρ is the density of air, u is the UAV velocity, and x The influence of the engine's rotational torque is neglected, and the engine thrust is decoupled into a three-axis force acting at a point in the fuselage's rigid body [28]. The thrust during this process remains constant at 14 tons in this paper. The relative positions of aerodynamic force and thrust with respect to the UAV body are depicted in Figure 3b.

Model of Landing Gear
The landing gear is the ground support system of the UAV and plays a crucial role as an energy-absorbing component during the landing process. As shown in Figure 4, the dynamic model of the landing gear comprises the upper strut, lower strut, torque link, wheel axle and tire assemblies. The upper strut is collected in the same rigid body as the fuselage, while the lower strut is collected in the same rigid body as the wheel axle. Three revolute joints are set between the upper strut and upper torque link, the upper strut and lower torque link, and the lower strut and lower torque link, respectively. The influence of the engine's rotational torque is neglected, and the engine thrust is decoupled into a three-axis force acting at a point in the fuselage's rigid body [28]. The thrust during this process remains constant at 14 tons in this paper. The relative positions of aerodynamic force and thrust with respect to the UAV body are depicted in Figure 3b.

Model of Landing Gear
The landing gear is the ground support system of the UAV and plays a crucial role as an energy-absorbing component during the landing process. As shown in Figure 4, the dynamic model of the landing gear comprises the upper strut, lower strut, torque link, wheel axle and tire assemblies. The upper strut is collected in the same rigid body as the fuselage, while the lower strut is collected in the same rigid body as the wheel axle. Three revolute joints are set between the upper strut and upper torque link, the upper strut and lower torque link, and the lower strut and lower torque link, respectively.  The displacement of the shock absorber is determined via the relative motion of the upper and lower struts [23]. The hydraulic force, , can be expressed as (8) where is the air spring force, and is the hydraulic damping force. The air spring force, , can be expressed as where is the initial pressure area of the low-pressure air chamber, is the pressure area of the high-pressure air chamber, and is the initial pressure in the low-pressure air chamber.
is the initial pressure in the low-pressure air chamber. is the atmospheric pressure, is the stroke of the damper, and is the initial stroke of the highpressure chamber.
is the initial volume of the low-pressure chamber, is the initial volume of the high-pressure chamber, and is the polytropic exponent. The hydraulic damping force can be expressed as follows: where ρ is the oil density, is the stroke velocity, A is the effective area of the buffer, A is the main oil cavity oil hole area, and are the flow coefficient of the main oil hole under the forward and reverse stroke, A is the effective area of the back oil hole, and are the effective flow areas of the oil return hole under the forward and reverse stroke, and and are the flow coefficient of the back oil hole under the for- The displacement of the shock absorber is determined via the relative motion of the upper and lower struts [23]. The hydraulic force, F S , can be expressed as where F a is the air spring force, and F u is the hydraulic damping force. The air spring force, F a , can be expressed as where A L a is the initial pressure area of the low-pressure air chamber, A H a is the pressure area of the high-pressure air chamber, and P L a0 is the initial pressure in the low-pressure air chamber. P H a0 is the initial pressure in the low-pressure air chamber. P atm is the atmospheric pressure, S is the stroke of the damper, and S H0 is the initial stroke of the high-pressure chamber. V L a0 is the initial volume of the low-pressure chamber, V H a0 is the initial volume of the high-pressure chamber, and γ is the polytropic exponent.
The hydraulic damping force can be expressed as follows: where ρ h is the oil density, .
S is the stroke velocity, A h is the effective area of the buffer, A d is the main oil cavity oil hole area, C + d and C − d are the flow coefficient of the main oil hole under the forward and reverse stroke, A hL is the effective area of the back oil hole, A + dL and A − dL are the effective flow areas of the oil return hole under the forward and reverse stroke, and C + dL and C − dL are the flow coefficient of the back oil hole under the forward and reverse stroke.
In addition to the load of the shock absorber, the flexibility of the tire also contributes significantly to the impact load during UAV landing. The compression of the tire under the impact load constitutes a substantial proportion of the overall compression stroke of the landing gear's damping system. The internal structure of the tire is illustrated in Figure 5a. The inner layer of the tire is defined as the fabric material and the change of volume surrounded by the wheel rim and the inner fabric layer of the tire conforms to the ideal gas equation. In addition to the load of the shock absorber, the flexibility of the tire also contributes significantly to the impact load during UAV landing. The compression of the tire under the impact load constitutes a substantial proportion of the overall compression stroke of the landing gear's damping system. The internal structure of the tire is illustrated in Fig The tread and the wheel rim share common nodes on the adjacent surface, and the rotational constraints of the tire are defined using a coordinate system, O-XYZ, located at the center of the wheel rim as shown in Figure 5b. The rubber material of the tire is modeled using eight-node hexahedral elements, employing the Mooney-Rivlin material model. The constitutive equation for this model is as follows: , A and B are the Rivlin constants determined through uniaxial tensile testing, v is Poisson's ratio, and I , II and III are the Green-Lagrange strain tensor constants.

Model of Arresting Hook
The dynamic model of the arresting hook [29] is shown in Figure 6. The model of the arresting hook comprises fuselage assembly, the hold down damper, lateral damper, hook shank, hook, joint part and two revolute joints. Fuselage assembly and the fuselage are set to a rigid body. The hold down damper has one end node located at the fuselage assembly and the other end node located at the joint part. By defining revolute joints 1 and 2, the longitudinal and lateral rotations of the arresting hook are, respectively, determined. The tread and the wheel rim share common nodes on the adjacent surface, and the rotational constraints of the tire are defined using a coordinate system, O-XYZ, located at the center of the wheel rim as shown in Figure 5b. The rubber material of the tire is modeled using eight-node hexahedral elements, employing the Mooney-Rivlin material model. The constitutive equation for this model is as follows: where , A and B are the Rivlin constants determined through uniaxial tensile testing, v is Poisson's ratio, and I, II and III are the Green-Lagrange strain tensor constants.

Model of Arresting Hook
The dynamic model of the arresting hook [29] is shown in Figure 6. The model of the arresting hook comprises fuselage assembly, the hold down damper, lateral damper, hook shank, hook, joint part and two revolute joints. Fuselage assembly and the fuselage are set to a rigid body. The hold down damper has one end node located at the fuselage assembly and the other end node located at the joint part. By defining revolute joints 1 and 2, the longitudinal and lateral rotations of the arresting hook are, respectively, determined.
Prior to engaging the arresting cable, the arresting hook collides with the deck and rebounds with a certain velocity. The rebound height of the arresting hook is limited by the hold down damper to ensure the successful engagement of the arresting hook with the cable. The hold down damper of the arresting hook plays a critical role in determining the height of the rebound [25], and it is modeled as a bar element.
The dynamic model of the arresting hook [29] is shown in Figure 6. The model of the arresting hook comprises fuselage assembly, the hold down damper, lateral damper, hook shank, hook, joint part and two revolute joints. Fuselage assembly and the fuselage are set to a rigid body. The hold down damper has one end node located at the fuselage assembly and the other end node located at the joint part. By defining revolute joints 1 and 2, the longitudinal and lateral rotations of the arresting hook are, respectively, determined.

Model of Arresting System
The deck is partitioned into the arresting hook contact area and non-contact area as shown in Figure 7. Additionally, the arresting gear system in this paper is based on the MK7-3 hydraulic arresting gear system [8]. The complete hydraulic arresting gear system model is divided into three components: the pulley system, the hydraulic system and arresting cable system. The pulley system comprises fixed pulleys, moving pulleys, and steering pulleys. The hydraulic system is composed of the damper sheave, main hydraulic cylinder and cable anchor damper system, which are modeled using the shell element and spring damper beam. Prior to engaging the arresting cable, the arresting hook collides with the deck and rebounds with a certain velocity. The rebound height of the arresting hook is limited by the hold down damper to ensure the successful engagement of the arresting hook with the cable. The hold down damper of the arresting hook plays a critical role in determining the height of the rebound [25], and it is modeled as a bar element.

Model of Arresting System
The deck is partitioned into the arresting hook contact area and non-contact area as shown in Figure 7. Additionally, the arresting gear system in this paper is based on the MK7-3 hydraulic arresting gear system [8]. The complete hydraulic arresting gear system model is divided into three components: the pulley system, the hydraulic system and arresting cable system. The pulley system comprises fixed pulleys, moving pulleys, and steering pulleys. The hydraulic system is composed of the damper sheave, main hydraulic cylinder and cable anchor damper system, which are modeled using the shell element and spring damper beam.

Modeling of Rigid Bodies in the Arresting Cable System and Constraints
The arresting hook of a carrier-based UAV engages with the arresting cable, resulting in the cable being pulled out during the arresting process. The cable is threaded through the arresting gear system, forming a block and tackle mechanism. This mechanism is designed to transfer the load from the UAV to the hydraulic machine. Within the hydraulic machine, the kinetic energy of the UAV is converted into heat and subsequently dissipates.

Modeling of Rigid Bodies in the Arresting Cable System and Constraints
The arresting hook of a carrier-based UAV engages with the arresting cable, resulting in the cable being pulled out during the arresting process. The cable is threaded through the Drones 2023, 7, 530 8 of 19 arresting gear system, forming a block and tackle mechanism. This mechanism is designed to transfer the load from the UAV to the hydraulic machine. Within the hydraulic machine, the kinetic energy of the UAV is converted into heat and subsequently dissipates. The arresting gear system is a complex mechanical hydraulic system. In this paper, the arresting gear system is modeled using three arresting cables (one deck pendant and two purchase cables), block and tackles (forty-eight sheaves), and five hydraulic dampers (two damper sheave installations, two cable anchor dampers and one hydraulic cylinder). In this research, the deformation and stress of the pulley, piston and cylinder system are not the main concern. Therefore, they are modeled as rigid bodies. The fixed sheave assembly is merged into the deck, which prevents there from being unnecessary fixed joints between the deck and itself. The other sheaves, piston and cylinder shown in Figure 7 undergo translation and rotation simultaneously.

Arresting Cable
The configuration of the pendant and wire rope support is shown in Figure 8. To improve computational efficiency, the arresting cable is divided into two parts based on the connecting muffle, the pendant and the purchase cable, as shown in Figure 8. The purchase cable is modeled using a nonlinear tension bar. This modeling approach takes into account the nonlinear dynamic characteristics of the cable under tension by accounting for factors such as material properties, cable diameter and applied tension force.
Drones 2023, 7, x FOR PEER REVIEW 9 of 20 The arresting gear system is a complex mechanical hydraulic system. In this paper, the arresting gear system is modeled using three arresting cables (one deck pendant and two purchase cables), block and tackles (forty-eight sheaves), and five hydraulic dampers (two damper sheave installations, two cable anchor dampers and one hydraulic cylinder). In this research, the deformation and stress of the pulley, piston and cylinder system are not the main concern. Therefore, they are modeled as rigid bodies. The fixed sheave assembly is merged into the deck, which prevents there from being unnecessary fixed joints between the deck and itself. The other sheaves, piston and cylinder shown in Figure 7 undergo translation and rotation simultaneously.

Arresting Cable
The configuration of the pendant and wire rope support is shown in Figure 8. To improve computational efficiency, the arresting cable is divided into two parts based on the connecting muffle, the pendant and the purchase cable, as shown in Figure 8. The purchase cable is modeled using a nonlinear tension bar. This modeling approach takes into account the nonlinear dynamic characteristics of the cable under tension by accounting for factors such as material properties, cable diameter and applied tension force. The FEM model of the pendant is modeled using a shell and spring beam element with 6 DoF as shown in Figure 9. The beam element is connected to the shell element at the node Ni and the nodes Ni1-Ni8 of the shell element. These nodes collectively form a rigid body. In the local coordinate system (Ni-SiRiTi), Ni represents the origin point of the coordinate system, Si denotes the direction vector along the beam axis, Ri represents the direction vector along the cross-sectional plane of the beam, and Ti represents the direction vector perpendicular to both Ri and Si. Defining the constitutive characteristics of the beam in the local coordinate system enables an accurate representation of the beam's deformation and response to applied loads.  The FEM model of the pendant is modeled using a shell and spring beam element with 6 DoF as shown in Figure 9. The beam element is connected to the shell element at the node N i and the nodes N i1 -N i8 of the shell element. These nodes collectively form a rigid body. In the local coordinate system (N i -S i R i T i ), N i represents the origin point of the coordinate system, S i denotes the direction vector along the beam axis, R i represents the direction vector along the cross-sectional plane of the beam, and T i represents the direction vector perpendicular to both R i and S i . Defining the constitutive characteristics of the beam in the local coordinate system enables an accurate representation of the beam's deformation and response to applied loads. The arresting gear system is a complex mechanical hydraulic system. In this paper, the arresting gear system is modeled using three arresting cables (one deck pendant and two purchase cables), block and tackles (forty-eight sheaves), and five hydraulic dampers (two damper sheave installations, two cable anchor dampers and one hydraulic cylinder). In this research, the deformation and stress of the pulley, piston and cylinder system are not the main concern. Therefore, they are modeled as rigid bodies. The fixed sheave assembly is merged into the deck, which prevents there from being unnecessary fixed joints between the deck and itself. The other sheaves, piston and cylinder shown in Figure 7 undergo translation and rotation simultaneously.

Arresting Cable
The configuration of the pendant and wire rope support is shown in Figure 8. To improve computational efficiency, the arresting cable is divided into two parts based on the connecting muffle, the pendant and the purchase cable, as shown in Figure 8. The purchase cable is modeled using a nonlinear tension bar. This modeling approach takes into account the nonlinear dynamic characteristics of the cable under tension by accounting for factors such as material properties, cable diameter and applied tension force. The FEM model of the pendant is modeled using a shell and spring beam element with 6 DoF as shown in Figure 9. The beam element is connected to the shell element at the node Ni and the nodes Ni1-Ni8 of the shell element. These nodes collectively form a rigid body. In the local coordinate system (Ni-SiRiTi), Ni represents the origin point of the coordinate system, Si denotes the direction vector along the beam axis, Ri represents the direction vector along the cross-sectional plane of the beam, and Ti represents the direction vector perpendicular to both Ri and Si. Defining the constitutive characteristics of the beam in the local coordinate system enables an accurate representation of the beam's deformation and response to applied loads.

Wire Rope Supports
Wire rope supports serve as a means of elevating the cable above the deck to guarantee the engagement of the arresting cable on the incoming arresting hook. The four wire rope supports are equidistantly placed across the deck and maintain a minimum cross-deck cable height of 0.5 m, measured from the bottom of the cable to the deck at its lowest point (Figure 8). Each wire rope support is directly tied to the deck [30].
As shown in Figure 10, the model of the wire rope support is established based on a four-node shell element. The forward end of the wire rope support spring is secured using a cam mounted in a deck recess and a follower pinned at the end of the wire rope support. The aft end of the wire rope support is also pinned and set between adjustable forward stops as required.

Wire Rope Supports
Wire rope supports serve as a means of elevating the cable above the deck to guarantee the engagement of the arresting cable on the incoming arresting hook. The four wire rope supports are equidistantly placed across the deck and maintain a minimum crossdeck cable height of 0.5 m, measured from the bottom of the cable to the deck at its lowest point (Figure 8). Each wire rope support is directly tied to the deck [30].
As shown in Figure 10, the model of the wire rope support is established based on a four-node shell element. The forward end of the wire rope support spring is secured using a cam mounted in a deck recess and a follower pinned at the end of the wire rope support. The aft end of the wire rope support is also pinned and set between adjustable forward stops as required.

Verification of the Dynamic Model of the Collision and Rebound of the Arresting Hook
All the computations presented in this study are performed using an in-house-developed code solver and a combination of tools of PAM-Crash was utilized. The code solver solves the governing finite element equations based on a central difference explicit integration scheme in time [31]. Simulation is performed to verify the collision rebound of the arresting hook. As shown in Figure 11a, fuselage assembly and main landing gear sleeve are defined as a rigid body. Below the arresting hook is a rotating disc, and the collision point between the hook and disc is located at the edge of the disc. The rotation direction of the disc is shown in Figure 11b. In this simulation, the relative linear velocity between the hook and disc is equal to the velocity of the UAV at the collision of point. The simulation conditions including the sinking velocity, horizontal velocity and arresting hook configuration are consistent with the results of the collision rebound test in reference [27]. The elastic modulus of the hook and rotating disc material is 210 GPa, and Poisson's ratio is 0.3. The collision rebound height and distance are matched to the test in the reference by adjusting the spring and damping parameters of the arresting hook's longitudinal buffer.

Verification of the Dynamic Model of the Collision and Rebound of the Arresting Hook
All the computations presented in this study are performed using an in-house-developed code solver and a combination of tools of PAM-Crash was utilized. The code solver solves the governing finite element equations based on a central difference explicit integration scheme in time [31]. Simulation is performed to verify the collision rebound of the arresting hook. As shown in Figure 11a, fuselage assembly and main landing gear sleeve are defined as a rigid body. Below the arresting hook is a rotating disc, and the collision point between the hook and disc is located at the edge of the disc. The rotation direction of the disc is shown in Figure 11b. In this simulation, the relative linear velocity between the hook and disc is equal to the velocity of the UAV at the collision of point. The simulation conditions including the sinking velocity, horizontal velocity and arresting hook configuration are consistent with the results of the collision rebound test in reference [27]. The elastic modulus of the hook and rotating disc material is 210 GPa, and Poisson's ratio is 0.3. The collision rebound height and distance are matched to the test in the reference by adjusting the spring and damping parameters of the arresting hook's longitudinal buffer.
The simulation results for the collision rebound height of the arresting hook, corresponding to sinking velocities of 3.6 m/s, 4 m/s, and 5 m/s, are presented in Figure 12. With the requirement of an increased sinking velocity, the bounce height on the aircraft hook increases correspondingly. The satisfactory agreement observed between the simulated and experimental data regarding the rebound height and rebound span of the arresting hook demonstrates the accuracy of the dynamic model established in this paper for the collision rebound process. The simulation results for the collision rebound height of the arresting hook, corresponding to sinking velocities of 3.6 m/s, 4 m/s, and 5 m/s, are presented in Figure 12. With the requirement of an increased sinking velocity, the bounce height on the aircraft hook increases correspondingly. The satisfactory agreement observed between the simulated and experimental data regarding the rebound height and rebound span of the arresting hook demonstrates the accuracy of the dynamic model established in this paper for the collision rebound process.  The simulation results for the collision rebound height of the arresting hook, corresponding to sinking velocities of 3.6 m/s, 4 m/s, and 5 m/s, are presented in Figure 12. With the requirement of an increased sinking velocity, the bounce height on the aircraft hook increases correspondingly. The satisfactory agreement observed between the simulated and experimental data regarding the rebound height and rebound span of the arresting hook demonstrates the accuracy of the dynamic model established in this paper for the collision rebound process.  Figure 13a illustrates the initial condition of the UAV, and the mass of the fuselage's rigid body is set as 16,500 kg. The horizontal and sinking velocity of the carrier-based UAV at the moment the hook touches the deck are 60 m/s and 4.5 m/s, respectively. The diagram of the distance between the touchdown point, cable d and the roll angle, ϕ , is shown in Figure 13 b, c. The position of the aerodynamic force and thrust is same as that described in Section 2.2.1. The control of variables facilitates a comparative study of the similarities and differences between different touchdown points and roll angles of an UAV.

Influence of Touchdown Point on the Engagement of Hook with Cable
The hook engages with the arresting cable after the UAV hook touches down on the carrier deck. Subsequently, the UAV starts to slow down under the action of the arresting gear system. In this section, the influence of distance between the touch point and cable  Figure 13a illustrates the initial condition of the UAV, and the mass of the fuselage's rigid body is set as 16,500 kg. The horizontal and sinking velocity of the carrier-based UAV at the moment the hook touches the deck are 60 m/s and 4.5 m/s, respectively. The diagram of the distance between the touchdown point, cable d and the roll angle, ϕ, is shown in Figure 13b,c. The position of the aerodynamic force and thrust is same as that described in Section 2.2.1. The control of variables facilitates a comparative study of the similarities and differences between different touchdown points and roll angles of an UAV.  Figure 13a illustrates the initial condition of the UAV, and the mass of the fuselage's rigid body is set as 16,500 kg. The horizontal and sinking velocity of the carrier-based UAV at the moment the hook touches the deck are 60 m/s and 4.5 m/s, respectively. The diagram of the distance between the touchdown point, cable d and the roll angle, ϕ , is shown in Figure 13 b, c. The position of the aerodynamic force and thrust is same as that described in Section 2.2.1. The control of variables facilitates a comparative study of the similarities and differences between different touchdown points and roll angles of an UAV.

Influence of Touchdown Point on the Engagement of Hook with Cable
The hook engages with the arresting cable after the UAV hook touches down on the carrier deck. Subsequently, the UAV starts to slow down under the action of the arresting gear system. In this section, the influence of distance between the touch point and cable

Influence of Touchdown Point on the Engagement of Hook with Cable
The hook engages with the arresting cable after the UAV hook touches down on the carrier deck. Subsequently, the UAV starts to slow down under the action of the arresting gear system. In this section, the influence of distance between the touch point and cable on the engagement process is studied. The initial condition of the UAV is represented in Figure 13a, and the distance between the touchdown point and cable d is set at 0, 2 m, 4 m 6 m, 8 m and 10 m. The distance between the touchdown points and the cable significantly influences the rebound height of the arresting hook at the moment of engagement. The engagement location of the arresting hook with the arresting cable varies depending on the distance from the touchdown point to the arresting cable. However, the contact force between the arresting hook and cable are nearly equal, at approximately 54 kN, for different touchdown point distances.
The stress-strain situation of the arresting hook is depicted in Figure 14. At d = 1 m, the arresting hook rebounds and engages with the cable in the upward phase of the first rebound. The maximum stress, reaching 187 MPa, occurs at the groove of the arresting hook. At d = 2 m, the arresting hook rebounds and engages with the cable in the downward phase of the first rebound. The maximum stress at this point is 179 MPa. At d = 4 m, the arresting hook engages with the cable during the second rebound phase. The engagement location is in the upper part of the hook, resulting in a maximum stress of 172 MPa. When d = 6 m, the arresting hook engages with the cable during the third rebound phase. The engagement location is at the connection between the hook head and the hook shank, leading to a larger high-stress area. Additionally, the maximum stress in this case is 184 MPa. For d = 8 m, the arresting hook engages with the cable during the deck drag phase. The engagement location is at the connection between the hook head and the hook shank. Additionally, the maximum stress is 188 MPa. When d = 10 m, the arresting hook engages with the cable during deck drag on the deck. However, there is an occurrence of the cable being crushed by the tire, which significantly affects the location of the cable's engagement. The location of maximum stress in this case is at the point where the hook shank and the cable engage, with a maximum stress of 165 MPa. on the engagement process is studied. The initial condition of the UAV is represented in Figure 13a, and the distance between the touchdown point and cable d is set at 0, 2 m, 4 m 6 m, 8 m and 10 m. The distance between the touchdown points and the cable significantly influences the rebound height of the arresting hook at the moment of engagement. The engagement location of the arresting hook with the arresting cable varies depending on the distance from the touchdown point to the arresting cable. However, the contact force between the arresting hook and cable are nearly equal, at approximately 54 kN, for different touchdown point distances.
The stress-strain situation of the arresting hook is depicted in Figure 14. At d = 1 m, the arresting hook rebounds and engages with the cable in the upward phase of the first rebound. The maximum stress, reaching 187 MPa, occurs at the groove of the arresting hook. At d = 2 m, the arresting hook rebounds and engages with the cable in the downward phase of the first rebound. The maximum stress at this point is 179 MPa. At d = 4 m, the arresting hook engages with the cable during the second rebound phase. The engagement location is in the upper part of the hook, resulting in a maximum stress of 172 MPa. When d = 6 m, the arresting hook engages with the cable during the third rebound phase. The engagement location is at the connection between the hook head and the hook shank, leading to a larger high-stress area. Additionally, the maximum stress in this case is 184 MPa. For d = 8 m, the arresting hook engages with the cable during the deck drag phase. The engagement location is at the connection between the hook head and the hook shank. Additionally, the maximum stress is 188 MPa. When d = 10 m, the arresting hook engages with the cable during deck drag on the deck. However, there is an occurrence of the cable being crushed by the tire, which significantly affects the location of the cable's engagement. The location of maximum stress in this case is at the point where the hook shank and the cable engage, with a maximum stress of 165 MPa.  Figure 15 illustrates the process of the arresting cable being run over by the tire during the aircraft's arrest. As the tire rolls over the arresting cable, a significant bending effect occurs at the point of contact, leading to the propagation of bending waves along the cable's length. Notably, at 0.03 s, when the bending wave reaches the middle section of the arresting cable, it forcefully makes contact with the deck, resulting in a rebound effect. During the rebound, the cable momentarily loses tension at 0.04 s and then regains tension at 0.08 s. The simulation model effectively captures the dynamic behavior of the arresting cable after the tire rolls over, providing valuable insights into the engagement process of the cable and hook.  Figure 15 illustrates the process of the arresting cable being run over by the tire during the aircraft's arrest. As the tire rolls over the arresting cable, a significant bending effect occurs at the point of contact, leading to the propagation of bending waves along the cable's length. Notably, at 0.03 s, when the bending wave reaches the middle section of the arresting cable, it forcefully makes contact with the deck, resulting in a rebound effect. During the rebound, the cable momentarily loses tension at 0.04 s and then regains tension at 0.08 s. The simulation model effectively captures the dynamic behavior of the arresting cable after the tire rolls over, providing valuable insights into the engagement process of the cable and hook.  Figure 15 illustrates the process of the arresting cable being run over by the tire during the aircraft's arrest. As the tire rolls over the arresting cable, a significant bending effect occurs at the point of contact, leading to the propagation of bending waves along the cable's length. Notably, at 0.03 s, when the bending wave reaches the middle section of the arresting cable, it forcefully makes contact with the deck, resulting in a rebound effect. During the rebound, the cable momentarily loses tension at 0.04 s and then regains tension at 0.08 s. The simulation model effectively captures the dynamic behavior of the arresting cable after the tire rolls over, providing valuable insights into the engagement process of the cable and hook. At a sinking velocity of 4.5 m/s, the bounce height of the arresting hook remains below the height of the arresting cable. Therefore, under the above conditions, no failures in engaging the cable were observed during UAV arrestment. The optimal distance between the touchdown point and cable for the UAV on the carrier deck is 4 m. At this distance, the arresting hook undergoes a secondary collision with the deck before engaging the cable, resulting in a low bounce height and a high success rate of engagement.

Influence of Roll Angle on the Engagement of Hook with Cable
In this section, the influence of roll angle on the engagement process is studied. The initial condition of UAV is represented in Figure 12. The distance between the touchdown point and cable d is 4m and the roll angle, φ, is set at 0, 2°, 4° and 6°. Figure 16 depicts the change in the height of the UAV's center of gravity after the tire makes contact with the deck. One side of the landing gear touches down earlier, resulting in a higher altitude of the UAV's center of gravity at the moment of touchdown compared to that in the situation without any roll angle. The greater the roll angle, the larger the distance of the center of gravity descent. The stress distribution at the moment of the hook engaging the cable is shown in Figure 17. It can be observed that the roll angle has a minor influence on the contact force during cable engagement, which remains around 54 kN. However, the roll angle affects the contact area of the hook, resulting in an increased high-stress region in the arresting hook. At a sinking velocity of 4.5 m/s, the bounce height of the arresting hook remains below the height of the arresting cable. Therefore, under the above conditions, no failures in engaging the cable were observed during UAV arrestment. The optimal distance between the touchdown point and cable for the UAV on the carrier deck is 4 m. At this distance, the arresting hook undergoes a secondary collision with the deck before engaging the cable, resulting in a low bounce height and a high success rate of engagement.

Influence of Roll Angle on the Engagement of Hook with Cable
In this section, the influence of roll angle on the engagement process is studied. The initial condition of UAV is represented in Figure 12. The distance between the touchdown point and cable d is 4 m and the roll angle, ϕ, is set at 0, 2 • , 4 • and 6 • . Figure 16 depicts the change in the height of the UAV's center of gravity after the tire makes contact with the deck. One side of the landing gear touches down earlier, resulting in a higher altitude of the UAV's center of gravity at the moment of touchdown compared to that in the situation without any roll angle. The greater the roll angle, the larger the distance of the center of gravity descent. At a sinking velocity of 4.5 m/s, the bounce height of the arresting hook remains below the height of the arresting cable. Therefore, under the above conditions, no failures in engaging the cable were observed during UAV arrestment. The optimal distance between the touchdown point and cable for the UAV on the carrier deck is 4 m. At this distance, the arresting hook undergoes a secondary collision with the deck before engaging the cable, resulting in a low bounce height and a high success rate of engagement.

Influence of Roll Angle on the Engagement of Hook with Cable
In this section, the influence of roll angle on the engagement process is studied. The initial condition of UAV is represented in Figure 12. The distance between the touchdown point and cable d is 4m and the roll angle, φ, is set at 0, 2°, 4° and 6°. Figure 16 depicts the change in the height of the UAV's center of gravity after the tire makes contact with the deck. One side of the landing gear touches down earlier, resulting in a higher altitude of the UAV's center of gravity at the moment of touchdown compared to that in the situation without any roll angle. The greater the roll angle, the larger the distance of the center of gravity descent. The stress distribution at the moment of the hook engaging the cable is shown in Figure 17. It can be observed that the roll angle has a minor influence on the contact force during cable engagement, which remains around 54 kN. However, the roll angle affects the contact area of the hook, resulting in an increased high-stress region in the arresting hook. The stress distribution at the moment of the hook engaging the cable is shown in Figure 17. It can be observed that the roll angle has a minor influence on the contact force during cable engagement, which remains around 54 kN. However, the roll angle affects the contact area of the hook, resulting in an increased high-stress region in the arresting hook. It is noteworthy that at a roll angle of 8°, one side of the tire makes contact with the arresting cable, and the engagement location of the arresting hook and the cable is at the connection between the hook and hook shank. This particular process of the tire touching the cable is depicted in Figure 18. Upon tire-cable contact, the arresting cable undergoes bending, leading to induced vertical movements. The arresting cable bends and transmits the force to the wire rope support. At 0.05 s, the bending is effectively transmitted to the wire rope support, resulting in the near elimination of the vertical movements of the arresting cable. Due to the impact from the tire collision, the arresting cable experiences a slight forward movement after 0.05 s until the hook engages with the cable. It is noteworthy that at a roll angle of 8 • , one side of the tire makes contact with the arresting cable, and the engagement location of the arresting hook and the cable is at the connection between the hook and hook shank. This particular process of the tire touching the cable is depicted in Figure 18. Upon tire-cable contact, the arresting cable undergoes bending, leading to induced vertical movements. The arresting cable bends and transmits the force to the wire rope support. At 0.05 s, the bending is effectively transmitted to the wire rope support, resulting in the near elimination of the vertical movements of the arresting cable. Due to the impact from the tire collision, the arresting cable experiences a slight forward movement after 0.05 s until the hook engages with the cable. Furthermore, a scenario involving one side tire rolling over the cable is observed when the distance between the touchdown point and the cable is 8 m, and the UAV experiences a roll angle of 4°. As depicted in Figure 19 at 0.1 s after the tire makes contact with the cable, it is evident that tire rollover exerts a considerably larger interference on the motion of the arresting cable compared to the scenario where the tire touches the cable. Specifically, when the tire of the right landing gear rolls over the arresting cable, it applies a substantial bending force on the cable due to the pressure and friction at the contact point, leading to significant cable flexing and deformation along its length. Subsequently, the bending action propagates towards the wire rope supports on the other side, consequently impacting the cable's tension and dynamics. In this case, the bending is transmitted to the left-side wire rope support at 0.05 s. Simultaneously, the arresting cable undergoes a forward motion in the direction of the tire, affecting the engagement location of the hook and cable. Furthermore, a scenario involving one side tire rolling over the cable is observed when the distance between the touchdown point and the cable is 8 m, and the UAV experiences a roll angle of 4 • . As depicted in Figure 19 at 0.1 s after the tire makes contact with the cable, it is evident that tire rollover exerts a considerably larger interference on the motion of the arresting cable compared to the scenario where the tire touches the cable. Specifically, when the tire of the right landing gear rolls over the arresting cable, it applies a substantial bending force on the cable due to the pressure and friction at the contact point, leading to significant cable flexing and deformation along its length. Subsequently, the bending action propagates towards the wire rope supports on the other side, consequently impacting the cable's tension and dynamics. In this case, the bending is transmitted to the left-side wire rope support at 0.05 s. Simultaneously, the arresting cable undergoes a forward motion in the direction of the tire, affecting the engagement location of the hook and cable. a substantial bending force on the cable due to the pressure and friction at the contact point, leading to significant cable flexing and deformation along its length. Subsequently, the bending action propagates towards the wire rope supports on the other side, consequently impacting the cable's tension and dynamics. In this case, the bending is transmitted to the left-side wire rope support at 0.05 s. Simultaneously, the arresting cable undergoes a forward motion in the direction of the tire, affecting the engagement location of the hook and cable.

Conclusions
Successful engagement requires careful coordination and attention to detail, including aspects such as the configuration of a UAV and arresting cable setup, as well as a determination of the precise landing point for the UAV. In this paper, a FEM-MBD numerical method is used to study the dynamic characteristics of a carrier-based UAV. Simulation was conducted to investigate the engagement process under different touch down points and UAV roll angles. Analysis of the scenarios involving cable and tire contact was performed.
(1) A rigid-flexible coupling model of the hook-cable engagement process is established for a specific carrier-based UAV using the FEM-MBD method. To validate the rationality of the finite element model in solving the hook's rebound dynamics, the results are compared with experimental data from a relevant reference. The comprehensive model incorporates three key elements: the coupling of the carrier deck, aircraft body and the landing gears; a detailed arresting hook and cable model which considers their material properties and contact interactions; wire rope supports and a full-scale arresting gear system model. The established FEM-MBD model offers a framework for examining the dynamic behavior of the hook-cable engagement process. (2) The touchdown position significantly influences the height of the arresting hook at the moment of hook and cable engagement, leading to variations in the maximum stress location on the arresting hook. When the touchdown point is 10 m away from the arresting cable, the cable is rolled over by the tires of the UAV. As a consequence, the arresting cable undergoes bending at the point of contact with the tires, and this bending effect is transmitted to both sides of the cable. The propagation of bending along the cable leads to the cable's contact with the deck and subsequent rebound. At the same time, the arresting cable experiences rapid movement in the direction of the tire roll-over. This movement has an impact on the engagement of the hook and cable, affecting the hook and cable's engagement location. (3) In the case of an UAV carrier landing with a roll angle, the main landing gear on one side makes contact with the deck first, resulting in a more significant variation in the height of the UAV's center of gravity during hook engagement compared to that in situations without any roll angle. As the roll angle increases, the downward displacement of the center of gravity increases. The roll angle has a minimal effect on the contact force at the moment of cable engagement, but it influences the location of Figure 19. Hook engaging cable after one side of the tire rolls over the cable.

Conclusions
Successful engagement requires careful coordination and attention to detail, including aspects such as the configuration of a UAV and arresting cable setup, as well as a determination of the precise landing point for the UAV. In this paper, a FEM-MBD numerical method is used to study the dynamic characteristics of a carrier-based UAV. Simulation was conducted to investigate the engagement process under different touch down points and UAV roll angles. Analysis of the scenarios involving cable and tire contact was performed.
(1) A rigid-flexible coupling model of the hook-cable engagement process is established for a specific carrier-based UAV using the FEM-MBD method. To validate the rationality of the finite element model in solving the hook's rebound dynamics, the results are compared with experimental data from a relevant reference. The comprehensive model incorporates three key elements: the coupling of the carrier deck, aircraft body and the landing gears; a detailed arresting hook and cable model which considers their material properties and contact interactions; wire rope supports and a full-scale arresting gear system model. The established FEM-MBD model offers a framework for examining the dynamic behavior of the hook-cable engagement process. (2) The touchdown position significantly influences the height of the arresting hook at the moment of hook and cable engagement, leading to variations in the maximum stress location on the arresting hook. When the touchdown point is 10 m away from the arresting cable, the cable is rolled over by the tires of the UAV. As a consequence, the arresting cable undergoes bending at the point of contact with the tires, and this bending effect is transmitted to both sides of the cable. The propagation of bending along the cable leads to the cable's contact with the deck and subsequent rebound. At the same time, the arresting cable experiences rapid movement in the direction of the tire roll-over. This movement has an impact on the engagement of the hook and cable, affecting the hook and cable's engagement location. (3) In the case of an UAV carrier landing with a roll angle, the main landing gear on one side makes contact with the deck first, resulting in a more significant variation in the height of the UAV's center of gravity during hook engagement compared to that in situations without any roll angle. As the roll angle increases, the downward displacement of the center of gravity increases. The roll angle has a minimal effect on the contact force at the moment of cable engagement, but it influences the location of cable contact at the engagement moment. A larger roll angle increases the high-stress area of the arresting hook at the engagement moment. (4) The detailed analysis of this tire-cable interaction sheds light on the complex dynamics involved in the engagement process. Under a roll angle of 8 • and distance of 4 m between the touch point and the cable, the tire makes contact with the cable. The cable undergoes bending at the point where the tire touches the cable, and the bending leads to the small-scale vertical movement of the arresting cable. Under the condition of a 4 • roll angle and an 8 m distance between the touchdown point and the cable, the tire rolls over the arresting cable on one side. The bending caused by the tire rolling over is greater than that of the tire touching the cable. As the single tire rolls over the arresting cable, it causes a rapid movement of the cable in the direction of the UAV's motion and transmits force to the wire rope support. Simultaneously, this movement of the arresting cable amplifies the uncertainty of engagement.