Dynamic Analysis of Active Heave Compensation System for Marine Winch under the Impact of Irregular Waves

Abstract

The six-degree-of-freedom (6 DOF) motion of the mother ship, especially the heave motion, brings difficulties to offshore operations and even causes safety accidents. To ensure the reliability of the launch and recovery system when the active heave compensation (AHC) function works, its dynamic characteristics are analyzed. Firstly, the dynamic model of the launch and recovery system with a rigid rope is constructed, and the mathematical simulation of the launch and recovery system without the AHC function under the excitation of irregular waves is completed. Secondly, a flexible rope based on the finite element method is established, and the dynamic responses of the launch and recovery system with AHC function under different conditions are compared with those without AHC function. The simulation results demonstrate that the dynamic responses of the system are related to the load, the underwater penetration, and the sea condition. In detail, the increase of the load leads to an increase in the dynamic response of the system. In essence, the underwater penetration affects the dynamic response of the system by affecting the activity of load acceleration. Therefore, a short rope is preferred in the maritime operations. Moreover, the harsh sea condition usually causes an undesirable effect of AHC function. In summary, the overall work is conductive to the modeling of the launch and recovery system, as well as the development of AHC technology so as to improve the quality of offshore operations in the future.

1. Introduction

The engineering ship equipped with a marine winch is an essential tool for offshore stevedoring and other hydraulic operations [1]. However, when the engineering ship works in the marine environment, the offshore operations will be affected by wave fluctuations. Specifically, the wave fluctuation affects offshore operations by making the engineering ship present a six-degree-of-freedom (6-DOF) motion, which not only reduces the positioning accuracy of the launch and recovery system, but also increases the risk of operations [2]. The 6-DOF motion of the engineering ship can be divided into horizontal motion and vertical motion [3,4]. Of these, the horizontal motion of the engineering ship (i.e., sway, surge, and yaw) is solved by the dynamic positioning system [5]. However, the remaining vertical motion, especially the heave motion, is the main factor that leads to great reduction of positioning accuracy. Generally, a compensation system is used to solve this problem by decoupling the load motion from the wave-induced ship motion.

Figure 1 shows the process of heave compensation. The marine winch is fixed on the deck, while the load is connected to the marine winch by a rope. Under the excitation of waves, the ship fluctuates up and down, while the load fluctuates under the traction of the rope and deviates from its original working position. Moreover, there is an alternating change of tension and slack, which makes the rope break due to frequent impacts [6]. The heave compensation drives the marine winch to rotate forward and backward, realizing the corresponding retraction of the rope. As a result, the load is stabilized at its original working position, and rope tension is also greatly reduced.

Therefore, many scholars have conducted a lot of research on two aspects. On the one hand, relevant scholars have focused on the dynamic characteristic of the launch and recovery system. For example, Posiadal [7] studied the suspension system of a crane ship and found that the telescopic movement of the rope had a significant impact on the swing angle of the load. Moreover, the complete description of the coupled motions of the load and the truck crane were obtained. Schellin [8] established a three-dimensional model of a crane, which included the surge, heave, and pitch of the ship, and the periodic harmonic wave was used as the external excitation to analyze the model. The work indicated that hook load response, strongly coupled with ship motions, was mainly influenced by first-order wave-exciting forces. Low-frequency horizontal ship motions caused by second-order wave (drift) forces did not generally affect hook load response. Gu [9] developed a virtual AHC system for a draw-works on a hoisting rig and started the system simulation under a sinusoidal wave. The results of heave compensation and the seabed landing of a payload were analyzed and used for optimization in terms of cost and performance. Masoud [10] established the dynamic model of the ship-mounted crane’s launch and recovery system and found that the wave-induced motion of the crane ship could produce large pendulations of the cargo being hoisted and caused operations to be suspended. The research also demonstrated that it was possible to reduce these pendulations significantly by controlling the slew and luff angles of the boom. Ellermann [11] established a single-degree-of-freedom dynamic model of an anchored ship and analyzed the dynamic behavior of the system under the excitation of the combination of a periodic component and an additive disturbance modelled as white noise. The response of this system, which showed multiple attractors for purely periodic forcing, revealed some similarities but also fundamental differences of the disturbed forcing. However, the operations described in the above literature were mainly concentrated on the situation on the sea surface, while it is quite different from the actual offshore operation to use regular waves as the external excitation in the research.

On the other hand, other scholars concentrated on the control strategy to reduce the influence caused by 6-DOF motion. For example, Do et al. [12] proposed a heave compensation system for a hydraulic-driven double-link actuator, which estimated the force between the drill string and the AHC unit. The control system involved in this research reduced the effect of the heave motion of the vessel on the response of the riser by regulating the distance from the upper end of the riser to the seabed. Richter et al. [13] proposed a real-time model predictive trajectory planner, which was based on a model of the hydraulically driven compensation winch. The overall compensation performances, as well as the estimation and prediction accuracy, were evaluated using a full-scale AHC test bench. Liu et al. [14] developed a suite of software programs to enable real-time monitoring of the dynamics of logging tools and assessed the efficiency of wireline heave compensation during downhole operations. The new system could provide an improved level of compensation over previous systems. Küchler et al. [15] proposed a prediction algorithm for the vertical motion of the vessel and formulated an inversion-based control strategy for the hydraulic-driven winch. The proposed method was proven to be able to decouple the motion of the rope-suspended payload from the vessel’s motion. Cuellar and Fortaleza [16] proposed a hydro-pneumatic heave compensator with a semi-active control method. As a result, the compensator with a semi-active control had a satisfactory frequency response for each drill string mass and an accumulator volume comparable with the accumulator volume of the active compensators that were used in the offshore industry at that time. It can be concluded from these works that most compensation systems adopted the hydraulic-driven method, while the electric-driven method has the advantages of effectiveness and convenience in controlling the system. In addition, the system model of the trajectory planning algorithm involved in the literature was based on theoretical modeling, which requires a high level of theoretical knowledge.

Contrary to the mentioned dynamic research, this work assumes that the initial state of the load is underwater and considers the hydrodynamic effect as well as the nonlinear restoring force of the rope. Moreover, the data of irregular waves collected in the actual maritime operation are used as the external excitation of the simulation model. Figure 2 shows the flowchart of the organization of the proposed method in this work. This makes the established dynamic model of the launch and recovery system more consistent with that in reality. The overall work makes the research more comprehensive and meaningful to a great extent. In addition, the simulation results of this work can provide the original data of the model for trajectory planning.

2. System Mathematical Model and Simulation without AHC Function

2.1. Model Assumptions

Figure 3 shows the launch and recovery system for a marine winch; the whole system is mainly composed of the marine winch, the rope, the retractable frame, and the load. It is supposed that the ship-borne coordinate system is oxyz, the coordinate of the lifting point is B, the coordinate of the joint between the lifting point and ship is A, the angle between the retractable frame AB and the horizon is $\alpha$, the angle between the rope and the vertical direction is $\beta$, the initial length of the rope is l, and the center of gravity is G. In addition, O, A, and B are in the same plane, which means the load moves only in the Oxy plane.

As shown in Figure 1, the theory of vibration isolation supposes the rope can be represented as a parallel spring–damper system [17,18], which acts as a mechanical low-pass filter with different values of spring-constant K and damping C. For convenience of modeling, the mechanical system discussed in this work has the following characteristics:

(1)

The work mainly studies the pitch and heave motions in 6-DOF motion.

(2)

The model assumes that the rope is a light rod without mass. Furthermore, the elasticity of the rope and retractable frame is ignored, which means both of them are regarded as rigid bodies.

2.2. Model of Wave Motion

The wave motion is a complex phenomenon in nature. The randomness of wind and the complex structure of the wind field, along with other factors, makes the wave motion present a strong randomness. Generally, the wave motion is the superposition of infinite cosine waves with different frequencies, amplitudes, initial phases, and propagation directions. Therefore, it can be expressed by Formula (1) [19].

$$\zeta \left(t\right)={\displaystyle \sum _{i=1}^{\infty }{\zeta }_{ai}\mathit{cos}\left({\omega }_{wi}t+{\epsilon }_i\right)}$$

(1)

where $\zeta \left(t\right)$ is the amplitude of wave motion, ${\zeta }_{ai}$ is the amplitude of the ith wavelet, ${\omega }_{wi}$ is the angular frequency of the ith wavelet, and ${\epsilon }_i$ is the initial phase of the ith wavelet. In addition, ${\epsilon }_i$ distributes in 0–2π uniformly.

From the spectral density of P–M spectrum, when the increment of angular frequency approaches 0, the amplitude of the wavelet can be calculated by Formula (2) [20].

$$\{\begin{array}{l}{\zeta }_{ai}=\sqrt{2S_{\zeta }{\omega }_{wi}\mathit{\Delta }{\omega }_w}\\ S_{\zeta }({\omega }_{wi})=\frac{8.1\times {10}^{-3}{g}^2}{{\mathsf{\omega }}_{wi}^5}·e^{\left(-\frac{3.11}{h_{1/3}^2{\omega }_{wi}^4}\right)}\end{array}$$

(2)

where $S_{\zeta }$ denotes the spectral density, and $h_{1/3}^2$ denotes the effective wave height.

However, the empirical formula is not completely applicable to the actual sea conditions. In the level 2 sea condition, the amplitude of the wave is very small, and the wavelength is still short, while in the level 5 sea condition, the engineering ship fluctuates violently, and maritime operations need to pay attention to risks. Therefore, this work uses the irregular waves collected under level 3 and level 4 sea conditions, so as to replace the empirical formula with the measured data. As shown in Figure 4, the effective wave height of the level 3 sea condition ranges from −0.8 m to +0.7 m, and the effective wave height of the level 4 sea condition ranges from −1.5 m to +1.3 m.

2.3. Model of Launch and Recovery System

As shown in Figure 3, the inertial coordinate system is o^Ix^Iy^Iz^I, and the ship-borne coordinate system is oxyz. According to the principle of homogeneous coordinate transformation [21,22], the coordinate of point B can be expressed as Equation (3).

$$\left[\begin{array}{c}{x}_{\mathrm{B}}\\ y_{\mathrm{B}}\\ z_{\mathrm{B}}\\ 1\end{array}\right]=\left[\begin{array}{cccc}\cos \theta & 0& \sin \theta & x\\ 0& 1& 0& y\\ -\sin \theta & 0& \cos \theta & z\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{c}{L}_{\mathrm{OA}}+L_{AB}\cos \mathsf{\alpha }\\ 0\\ L_{\mathrm{AB}}\sin \mathsf{\alpha }\\ 1\end{array}\right]=\left[\begin{array}{c}x+L_{\mathrm{OA}}\cos \theta +L_{AB}\cos \mathsf{\alpha }\cos \theta +L_{\mathrm{AB}}\sin \mathsf{\alpha }\sin \theta \\ y\\ z-L_{\mathrm{OA}}\sin \theta -L_{AB}\cos \mathsf{\alpha }\sin \theta +L_{\mathrm{AB}}\cos \mathsf{\alpha }\cos \theta \\ 1\end{array}\right]$$

(3)

Accordingly, the coordinate of point G is expressed as Equation (4).

$$\left[\begin{array}{c}{x}_G\\ y_{\mathrm{G}}\\ z_{\mathrm{G}}\\ 1\end{array}\right]=\left[\begin{array}{c}{x}_{\mathrm{B}}\\ y_{\mathrm{B}}\\ z_{\mathrm{B}}\\ 1\end{array}\right]+\left[\begin{array}{c}l\sin \beta \\ 0\\ -l\cos \beta \\ 1\end{array}\right]$$

(4)

The velocity of the load is the derivative of Equation (4), which can be expressed as Equation (5).

$$\{\begin{array}{l}{\dot{x}}_{\mathrm{G}}=\dot{x}+L_{OA}\dot{\theta }\sin \theta -L_{\mathrm{AB}}\dot{\theta }\cos \alpha \sin \theta \\ +L_{\mathrm{AB}}\dot{\theta }\sin \alpha \cos \theta -\dot{l}\sin \beta -l\dot{\beta }\cos \beta \\ {\dot{y}}_{\mathrm{G}}=0\\ {\dot{z}}_{\mathrm{G}}=\dot{z}+L_{OA}\dot{\theta }\cos \theta -L_{\mathrm{AB}}\dot{\theta }\cos \alpha \cos \theta \\ -L_{\mathrm{AB}}\dot{\theta }\sin \alpha \sin \theta -\dot{l}\cos \beta -l\dot{\beta }\sin \beta \end{array}$$

(5)

The kinetic energy and the potential energy of the load can be expressed by Equation (6) and Equation (7), respectively.

$$T=\frac{1}{2}{m}_{\mathrm{G}}({\dot{x}}_{\mathrm{G}}^2+{\dot{y}}_{\mathrm{G}}^2+{\dot{z}}_{\mathrm{G}}^2)$$

(6)

$$U_{\mathrm{G}}=m_{\mathrm{G}}g(z+L_{\mathrm{OA}}\sin \theta -L_{\mathrm{AB}}\cos \mathsf{\alpha }\sin \theta -L_{\mathrm{AB}}\sin \mathsf{\alpha }\cos \theta -l\cos \beta )$$

(7)

The generalized force of the launch and recovery system mainly refers to the resistance of seawater, which can be expressed by Equation (8).

$$Q_{\mathsf{\beta }}=-\frac{1}{2}\rho C_x{S}_x\left|{\dot{x}}_{\mathrm{G}}\right|{\dot{x}}_{\mathrm{G}}-\frac{1}{2}\rho C_{n}ld{l}^2\left|\dot{\beta }\right|\dot{\beta }$$

(8)

where $\rho$ denotes the density of seawater, $C_x$ denotes the resistance coefficient of the load along the X direction, $S_x$ denotes the resistance surface area of the load along the X direction, $C_n$ denotes the resistance coefficient of the rope, $l$ denotes the length of the rope, and $d$ denotes the diameter of the rope.

Assuming that the swing angle $\beta$ is the generalized coordinate of the system, the dynamic equation of the launch and recovery system can be expressed as Equation (9).

$$\begin{array}{l}{m}_{\mathrm{G}}[l^2\ddot{\beta }+2l\dot{l}\dot{\beta }-l\cos \beta (\ddot{x}+L_{OA}\ddot{\theta }\sin \theta +L_{OA}{\dot{\theta }}^2\cos \theta -L_{\mathrm{AB}}\ddot{\theta }\cos \mathsf{\alpha }\sin \theta \\ -L_{AB}{\dot{\theta }}^2\cos \mathsf{\alpha }\cos \theta +L_{AB}\ddot{\theta }\sin \mathsf{\alpha }\cos \theta -L_{AB}{\dot{\theta }}^2\sin \mathsf{\alpha }\sin \theta )+l\sin \beta (\ddot{z}+g+L_{OA}\ddot{\theta }\cos \theta -\\ L_{OA}{\dot{\theta }}^2\sin \theta -L_{AB}\ddot{\theta }\cos \mathsf{\alpha }\cos \theta +L_{AB}{\dot{\theta }}^2\cos \mathsf{\alpha }\sin \theta -L_{AB}\ddot{\theta }\sin \mathsf{\alpha }\sin \theta -L_{AB}{\dot{\theta }}^2\sin \mathsf{\alpha }\cos \theta )]=Q_{\mathsf{\beta }}\end{array}$$

(9)

The load will be affected by the resistance of seawater when the launch and recovery system works, and the rope tension, which is expressed by Equation (10), can be obtained by Newton’s second law.

$$m_{\mathrm{G}}\left[\begin{array}{l}{\ddot{x}}_{\mathrm{G}}\\ {\ddot{y}}_{\mathrm{G}}\\ {\ddot{z}}_{\mathrm{G}}\end{array}\right]=\left[\begin{array}{c}F\sin \beta -\frac{1}{2}\rho C_x{S}_x\left|{\dot{x}}_{\mathrm{G}}\right|{\dot{x}}_{\mathrm{G}}\\ 0\\ F\cos \beta +\rho gV-m_{\mathrm{G}}g-\frac{1}{2}\rho C_z{S}_z\left|{\dot{z}}_{\mathrm{G}}\right|{\dot{z}}_{\mathrm{G}}\end{array}\right]$$

(10)

where $F$ denotes the rope tension, $V$ denotes the volume of the load, $C_z$ denotes the resistance coefficients when the load moves along the Z direction, $S_z$ denotes the resistance areas when the load moves along the Z direction, and ${\ddot{x}}_{\mathrm{G}}$ and ${\ddot{z}}_{\mathrm{G}}$ denote the accelerations of the load when load moves along the X and Z directions, respectively. The critical parameters of the launch and recovery system are listed in

Table 1. Critical parameters of the launch and recovery system.

Parameter	Description	Value	Parameter	Description	Value
L(m)	Length of ship	73.3	Cz	Resistance coefficient along X direction	0.02
B(m)	Width of ship	10.2	LOA(m)	Distance between O and A	35
d(mm)	Diameter of the rope	32	LAB(m)	Distance between A and B	6
mG(kg)	Quality of the load	55,000	V(m3)	Volume of the load	3.375
l(m)	Length of the rope	1000	Sx	Resistance area along X direction	7.07
Cx	Resistance coefficient along X direction	1.53	Sz	Resistance area along Z direction	7.07
$\rho$ (g/cm3)	Density of seawater	1.025	α(°)	Angle between frame and horizon	45
Cn	Damping coefficient of rope	0.4	g(m/s2)	Acceleration of gravity	9.8

.

The simulation model of the launch and recovery system is shown in Figure 5. There are 7 modules in the simulation model, and the response of the system in the direction of heave is of more concern in this article. Thus, the simulation results of the modules named ‘Load displacement_Z’ and ‘Load velocity_Z’ are mainly investigated.

2.4. Simulation of Launch and Recovery System

To analyze the dynamic response of the system more clearly, this work explores the dynamic response of the system under the level 3 sea condition. As shown in Figure 6, the load is 55 kN, and the underwater penetration is 1000 m. Moreover, the dynamic responses, such as load displacement, load velocity, load acceleration, and rope tension, are mainly studied.

Several characteristics of these figures are of interest, and there is a common phenomenon in these diagrams. Obviously, there are the same laws between the two curves in these figures, except for the difference in amplitude. Combined with the simulation result of rope tension, it is clear that the load fluctuates up and down with the movement of the ship under the traction of the rope.

Table 2. Simulation results of launch and recovery system.

Sea Condition	Object	Displacement (m)	Velocity (m·s−1)	Acceleration (m·s−2)	Tension (kN)
Level 3	Ship	−0.48 to +0.50	−1.50 to +1.70	−5.0 to +3.9	/
	Load	−0.39 to +0.39	−1.38 to +1.35	−4.2 to + 2.9	/
	Rope	/	/	/	0 to +82.0

shows the detailed simulation results of the launch and recovery system.

3. Modeling and Simulation of AHC System

According to the model assumptions in Section 2, the elasticity of the rope is ignored when establishing the mathematical model. However, there is a serious interaction between the engineering ship and underwater load, which is reflected through the rope [23]. In addition, the original mathematical model is relatively complex.

In general, the fluctuation of marine equipment in maritime operations is mainly caused by the fluctuation of the engineering ship, the load displacement caused by seawater damping, and the longitudinal displacement caused by deformation of the flexible rope. Since the deformation of the flexible rope does not matter to a great extent, the AHC function involved in this work only considers the influence of wave and seawater damping on the system. Therefore, the AHC function can be realized by converting the influence of these factors on the load displacement into the angle of the drum, that is, the amount of the rope. For the convenience of applying the AHC function into the dynamic model, it is necessary to rebuild the mechanical model of the launch and recovery system. Figure 7 shows the three-dimensional model of the launch and recovery system without flexible rope, which was then imported into Adams software for the simulation of multi-body dynamics.

3.1. Model of Flexible Rope

The dynamic modeling of the flexible rope can refer to the modeling method of bushing force, the flexible body of the finite element discretization method, and the polyline method [24]. However, the flexible body of the finite element discretization method is not suitable for simulating a rope with large length and deformation, and the polyline method simulates the movement of the rope without considering the mechanical characteristics of the rope. The method of bushing force discretizes a continuous rope into several micro-segments with cylindrical form, which reflects the actual characteristics of the rope to a great extent.

By defining 6 components of force and moment, the modeling method of bushing force adds the bushing forces between adjacent cylindrical micro-segments, realizing the flexible connections between cylindrical micro-segments. The mechanical model between two cylindrical segments is shown in Figure 8.

As shown in Figure 8, the flexible force is applied between 2 segments by defining the 6 components of force and moment $\left\{F_x,F_y,F_z,T_x,T_y,T_z\right\}$ [25]. Furthermore, the calculation equation of the bushing force can be expressed as Equation (11).

$$\begin{array}{l}\left[\begin{array}{l}{F}_x\\ F_y\\ F_{\mathrm{z}}\\ T_x\\ T_y\\ T_z\end{array}\right]=\left[\begin{array}{cccccc}{K}_{11}& 0& 0& 0& 0& 0\\ 0& K_{22}& 0& 0& 0& 0\\ 0& 0& K_{33}& 0& 0& 0\\ 0& 0& 0& K_{44}& 0& 0\\ 0& 0& 0& 0& K_{55}& 0\\ 0& 0& 0& 0& 0& K_{66}\end{array}\right]\left[\begin{array}{l}{r}_x\\ r_y\\ r_z\\ {\theta }_x\\ {\theta }_y\\ {\theta }_z\end{array}\right]\\ -\left[\begin{array}{cccccc}{C}_{11}& 0& 0& 0& 0& 0\\ 0& C_{22}& 0& 0& 0& 0\\ 0& 0& C_{33}& 0& 0& 0\\ 0& 0& 0& C_{44}& 0& 0\\ 0& 0& 0& 0& C_{55}& 0\\ 0& 0& 0& 0& 0& C_{66}\end{array}\right]\left[\begin{array}{l}{v}_x\\ v_y\\ v_z\\ {\omega }_x\\ {\omega }_y\\ {\omega }_z\end{array}\right]+\left[\begin{array}{l}{F}_{x0}\\ F_{y0}\\ F_{z0}\\ T_{x0}\\ T_{y0}\\ T_{z0}\end{array}\right]\end{array}$$

(11)

where ${\left[r_x r_y r_z {\theta }_x {\theta }_y {\theta }_z \right]}^{\mathrm{T}}$ denotes the displacement matrix between adjacent segments, and ${\left[v_x v_y v_z {\omega }_x {\omega }_y {\omega }_z \right]}^{\mathrm{T}}$ denotes the velocity matrix between adjacent segments. $K_r$ and $C_r$ are matrixes of stiffness coefficient and damping coefficient, respectively.

It can be inferred from Equation (11) that the value of the bushing force depends on the displacement and velocity of adjacent segments. In addition, to obtain better mechanical performance of the simulated rope, it is necessary to set the stiffness coefficients and damping coefficients, as shown in Equation (12).

$$\{\begin{array}{c}{K}_{r11}=EA/L_r\\ K_{r22}=K_{r33}=G_{r}A/L_r\\ K_{r44}=G_r\pi R^4/2L_r\\ K_{r55}=K_{r66}=E\pi R^4/4L_r\end{array}$$

(12)

where $E$ denotes the elastic modulus of the rope, $G_r$ denotes the shear modulus of the rope, $A$ denotes the cross-sectional area of a single segment, $R$ denotes the radius of a single segment, and $L_r$ denotes the length of a single segment.

The values of the main parameters when designing cylindrical segments are listed in

Table 3. Values of main parameters of cylindrical segments.

Parameter	Description	Value	Parameter	Description	Value
E(N/mm2)	Elastic modulus of the rope	140,000	C11(N·s/mm)	Stiffness coefficient of the segments	1
Gr(GPa)	Shear modulus of the rope	80	C22(N·s/mm)	Stiffness coefficient of the segments	1
A(mm2)	Cross-sectional area of the segments	804.25	C33(N·s/mm)	Stiffness coefficient of the segments	1
R(mm)	Radius of the segments	16	C44(N·s/mm)	Stiffness coefficient of the segments	10
Lr(mm)	Length of the segments	100	C55(N·s/mm)	Stiffness coefficient of the segments	10
/	/	/	C66(N·s/mm)	Stiffness coefficient of the segments	10

.

When the launch and recovery system works, the contact and extrusion will occur between the rope and the drum, as well as the rope and the hoisting sheave. According to reference [26], the contact is related to the stiffness coefficient collision index, damping coefficient, and penetration depth of contact, and the contact is expressed as Equation (13).

$$F_c=\{\begin{array}{ll}0& x_c>0\\ K_c{x}_c^e+\mathrm{STEP}(x_c,0,0,x_{cmax},c_{max}){\dot{x}}_c& x_c\le 0\end{array}$$

(13)

where $K_c$ denotes the stiffness coefficient of contact, $x_c$ denotes the penetration depth of contacted objects, $x_{c\max }$ denotes the maximum penetration depth, $c_{\max }$ denotes the maximum damping coefficient, $e$ denotes the nonlinear collision index, and STEP denotes the step function in Adams.

The stiffness coefficient of contact can be calculated by Equation (14).

$$\{\begin{array}{c}{K}_c=\frac{4}{3}{R}_c^{\frac{1}{2}}{E}^{*}\\ \frac{1}{R_c}=\frac{1}{R_1}+\frac{1}{R_2}\\ \frac{1}{E^{*}}=\frac{1-{\delta }_1^2}{E_1}+\frac{1-{\delta }_2^2}{E_2}\end{array}$$

(14)

where $R_1$ and $R_2$ denote the equivalent radius of the contacted objects, $E_1$ and $E_2$ denote the elastic module of the contacted objects, and ${\delta }_1$ and ${\delta }_2$ denote Poisson’s ratio of the contacted objects.

The damping coefficient of contact is expressed as Equation (15).

$$c=\frac{3K_c(1-{\epsilon }^2)}{4v_c}{x}_c^e$$

(15)

where $\epsilon$ denotes the restitution coefficient, and $v_c$ denotes the collision velocity when cylindrical surfaces contact each other.

The maximum damping coefficient generally takes 0.1% to 1.0% of the stiffness coefficient, and the parameters of contact force are shown in

Table 4. Critical parameters of contact force.

Parameters	Descriptions	Values
Kc(N/m)	Stiffness coefficient of contact	15,000
xc(mm)	Penetration depth of objects	0.1
e	Nonlinear collision index	1.5

.

3.2. Simulation and Analysis of AHC System under Level 3 Sea Condition

Figure 9 shows the mechanical model of the AHC system for a marine winch, in which a continuous rope is established in multi-body dynamics software. The purpose of establishing a dynamic model is to investigate the influence of different variables on the compensation characteristics of the system so as to conduct further research on the control of the AHC system.

The simulation results under the level 3 sea condition are shown in Figure 10. In Figure 10a, the load displacement without compensation ranges from −0.8 m to +0.6 m, while the load displacement with compensation is greatly reduced, which proves that the effect of the AHC system involved in this work is quite good.

It is assumed that the underwater penetration is 1000 m, and the load is taken as the variable; the simulation result of load displacement is shown in Figure 10b. It can be seen that the load displacement when the load is 25 kN ranges from −0.1 m to 0 m, while most values distribute around −0.05 m. The load displacement when the load is 55 kN ranges from −0.2 m to 0 m, while most values distribute around −0.13 m. The load displacement when the load is 85 kN ranges from −0.25 m to 0 m, while most values distribute around −0.17 m. It can be inferred that with the increase of the load, the average value of load displacements increases too. Moreover, the load displacement with a heavy load tends to be more stable than that with a light load.

The simulation result of load velocity is shown in Figure 10c. The velocity when the load is 25 kN ranges from −0.5 m/s to +0.5 m/s, the velocity when the load is 55 kN ranges from −1.2 m/s to +1.25 m/s, and the velocity when the load is 85 kN ranges from −1.4 m/s to +1.2 m/s. However, the overall result shows that as the load increases, the load velocity will be more stable during the process of AHC. This phenomenon can provide support for the change of load displacement with different loads.

The simulation result of load acceleration is shown in Figure 10d. The acceleration when the load is 25 kN ranges from −3.5 m/s² to +10.9 m/s², the acceleration when the load is 55 kN ranges from −7.5 m/s² to +33.7 m/s², and the acceleration when the load is 85 kN ranges from −8.5 m/s² to +29.3 m/s². Obviously, the average accelerations under these cases fluctuate between −5.0 m/s² and +5.0 m/s². Careful observation shows that the load acceleration changes rapidly with a heavy load, while the load acceleration takes a longer time with a light load. This is the reason for such changes in load displacement and load velocity.

In Figure 10e, the rope tension when the load is 25 kN ranges from 0 kN to 30 kN, the rope tension when the load is 55 kN ranges from 0 kN to 105 kN, and the rope tension when the load is 8.5 t ranges from 0 kN to 140 kN. The result shows that the heavier load leads to greater rope tension in the process of AHC. By comparing the effect of AHC function and rope tension, it was found that the rope tension is roughly equal to the load when the compensation effect of the AHC system is ideal.

The next simulation scheme supposes the load is 55 kN and takes the underwater penetration as the variable; the simulation result of load displacement is shown in Figure 11a. It can be seen that the load displacement when the underwater penetration is 1000 m ranges from −0.2 m to 0 m, while most values distribute around −0.13 m. The load displacement when the underwater penetration is 2000 m ranges from −0.225 m to 0 m, while most values distribute around −0.15 m. The load displacement when the underwater penetration is 3000 m ranges from −0.235 m to 0 m, while most values distribute around −0.17 m. It was found that the underwater penetration determines the range of load displacement, and the larger the underwater penetration is, the larger the absolute value of load displacement is. On the whole, the larger underwater penetration means a slower response of load displacement under the AHC function.

In Figure 11b, the velocity when the underwater penetration is 1000 m ranges from −1.25 m/s to +1.25 m/s, the velocity when the underwater penetration is 2000 m ranges from −1.45 m/s to +1.1 m/s, and the velocity when the underwater penetration is 3000 m ranges from −1.1 m/s to +0.95 m/s. From the average value of velocities under different conditions, it can be found that with the increase of underwater penetration, the average velocity of load decreases, which indicates that a shorter rope leads to a faster load velocity when the AHC system works.

The simulation result of load acceleration is shown in Figure 11c. The acceleration when the underwater penetration is 1000 m ranges from −5.5 m/s² to +30.0 m/s², the acceleration when the underwater penetration is 2000 m ranges from −7.5 m/s² to +34.0 m/s², and the acceleration when the underwater penetration is 3000 m ranges from −8.0 m/s² to +14.8 m/s². Similarly, the average accelerations under these cases fluctuate between −5.0 m/s² and +5.0 m/s². An interesting phenomenon is that a great underwater penetration always means an inactive load acceleration. This is the reason that the load located at shallow water shows a rapid response to the traction of the rope.

The last important indicator of this work is the dynamic rope tension. As Figure 11d shows, the rope tension when the underwater penetration is 1000 m ranges from 0 kN to 105 kN, the rope tension when the underwater penetration is 2000 m ranges from 0 kN to 146 kN, and the rope tension when the underwater penetration is 3000 m ranges from 0 kN to 148 kN. Obviously, as the underwater penetration increases, the average rope tension also increases during the process of AHC. However, ever if the compensation effect of the AHC system is ideal, the rope tension is not equal to the load. Through comparison, it can be inferred that the rope tension is not only related to the load, but it is also related to the underwater penetration.

Compare the simulation results with the AHC function with those without the AHC function, it is clear that the load displacement is greatly reduced after compensation, while the load velocity and load acceleration are reduced to a certain extent, as well as the rope tension. All these results prove that the effect of the AHC function is quite ideal. However, the simulation of the launch and recovery system lacking the AHC function under the level 4 sea condition is difficult to carry out. During the process of this type of simulation, it is found that the simulation data are abnormal. After inspection, the reason is that the rope is easy to break away from the hoisting sheave due to the harsh sea condition.

3.3. Simulation and Analysis of AHC System under Level 4 Sea Condition

The simulation results under the level 4 sea condition are shown in Figure 12. In Figure 12a, the load displacement without compensation ranges from −1.3 m to +1.5 m, while the load displacement after compensation is also greatly reduced. However, with the sea condition getting worse, the compensation effect of the AHC system becomes worse too.

It is assumed that the underwater penetration is 1000 m, and the load is taken as the variable; the simulation result of load displacement is shown in Figure 12b. It can be seen that the load displacement when the load is 25 kN ranges from +0.02 m to +0.13 m, while most values distribute around +0.06 m. The load displacement when the load is 55 kN ranges from −0.13 m to +0.02 m, while most values distribute around −0.06 m. The load displacement when the load is 85 kN ranges from −0.32 m to +0.11 m, while most values distribute around −0.12 m. The same conclusion can be reached as the previous simulation; the variation range of the load displacement under the level 3 sea condition is smaller than that of the simulation under the level 4 sea condition.

The simulation result of load velocity is shown in Figure 12c, the velocity when the load is 25 kN ranges from −0.36 m/s to +0.37 m/s, the velocity when the load is 55 kN ranges from −1.67 m/s to +1.79 m/s, and the velocity when the load is 85 kN ranges from −1.99 m/s to +2.10 m/s. By comparing with the simulation under the level 3 sea condition, it is obvious that the load velocity fluctuates in a larger range. From the average load velocity under different conditions, it can be concluded that a heavier load increases the average load velocity, which indicates that the heavier load is more difficult to control.

The simulation result of load acceleration is shown in Figure 12d. The acceleration when the load is 25 kN ranges from −9.60 m/s² to +4.90 m/s², the acceleration when the load is 55 kN ranges from −8.70 m/s² to +14.5 m/s², and the acceleration when the load is 85 kN ranges from −9.20 m/s² to +21.5 m/s². The accelerations tend to increase with the increase of the load. It can be found that with the same settings of load and underwater penetration, the load acceleration also fluctuates in a larger range than the load acceleration under the level 3 sea condition.

In Figure 12e, the rope tension when the load is 25 kN ranges from 0 kN to 31 kN, the rope tension when the load is 55 kN ranges from 0 kN to 92 kN, and the rope tension when the load is 85 kN ranges from 0 kN to 119 kN. Compared with the results under the level 3 sea condition, it can be speculated that the severe sea condition leads to a drastic change of rope tension. Specifically, the rope tension in the level 4 sea condition always increases suddenly, while the rope tension in the level 3 sea condition tends to be a certain value under the compensation effect. Moreover, the rope tension increases with the increase of load.

The simulation results with underwater penetration as variable, are shown in Figure 13. In Figure 13a, the load displacement when the underwater penetration is 1000 m ranges from −0.13 m to 0.02 m, while most values distribute around −0.06 m. The load displacement when the underwater penetration is 2000 m ranges from −0.24 m to 0.04 m, while most values distribute around −0.10 m. The load displacement when the underwater penetration is 3000 m ranges from −0.26 m to 0.11 m, while most values distribute around −0.08 m. Compared with the same type of simulation under the level 3 sea condition, although the load displacement still increases with the increase of load, the amplitude of load displacement has increased. Moreover, the load displacement seems to change faster.

In Figure 13b, the load velocity when the underwater penetration is 1000 m ranges from −1.98 m/s to +2.05 m/s, the load velocity when the underwater penetration is 2000 m ranges from −1.87 m/s to +1.88 m/s, and the load velocity when the underwater penetration is 3000 m ranges from −1.31 m/s to +1.50 m/s. Similarly, the results of load velocity in this case show that a shorter rope brings a sensitive response of the AHC system. However, the amplitude of the load velocity is much bigger than that of the level 3 sea condition. This is the reason for the poor effect of the AHC system under the level 4 sea condition.

The result of load acceleration is shown in Figure 13c. The acceleration when the underwater penetration is 1000 m ranges from −9.80 m/s² to +22.3 m/s², the acceleration when the underwater penetration is 2000 m ranges from −9.5 m/s² to +17.5 m/s², and the acceleration when the underwater penetration is 3000 m ranges from −7.40 m/s² to +10.3 m/s². The same phenomenon as the same type of simulation is that the greater the underwater penetration is, the gentler the change of load acceleration is. However, the amplitude of load acceleration is also increased due to the harsh sea condition.

As Figure 13d shows, the rope tension when the underwater penetration is 1000 m ranges from 0 kN to 92 kN, the rope tension when the underwater penetration is 2000 m ranges from 0 kN to 138 kN, and the rope tension when the underwater penetration is 3000 m ranges from 0 kN to 155 kN. This situation follows the rule of rope tension as summarized in the previous conclusion.

Table 5. Simulation results of AHC system.

Sea Condition	Subcondition	Load Displacement (m)	Load Velocity (m·s−1)	Load Acceleration (m·s−2)	Rope Tension (kN)
Level 3	mG = 25 kN, l = 1000 m	−0.10 to 0	−0.50 to +0.50	−3.50 to +10.90	0 to +30
	mG = 55 kN, l = 1000 m	−0.20 to 0	−1.20 to +1.25	−7.50 to +33.70	0 to +105
	mG = 85 kN, l = 1000 m	−0.25 to 0	−1.40 to +1.20	−8.50 to +29.3	0 to +140
	mG = 55 kN, l = 2000 m	−0.225 to 0	−1.45 to +1.10	−7.80 to +34.0	0 to +146
	mG = 55 kN, l = 3000 m	−0.235 to 0	−1.10 to +0.95	−8.00 to +14.8	0 to +148
Level 4	mG = 25 kN, l = 1000 m	0 to +0.13	−0.36 to +0.37	−9.60 to +4.90	0 to +31
	mG = 55 kN, l = 1000 m	−0.13 to +0.02	−1.98 to +2.05	−9.80 to +22.3	0 to +92
	mG = 85 kN, l = 1000 m	−0.32 to +0.11	−1.99 to +2.10	−9.20 to +21.5	0 to +119
	mG = 55 kN, l = 2000 m	−0.24 to +0.04	−1.87 to +1.88	−9.50 to +17.5	0 to +138
	mG = 55 kN, l = 3000 m	−0.26 to +0.11	−1.31 to +1.50	−7.40 to +10.3	0 to +155

shows the detailed simulation results of the AHC system under different conditions.

4. Conclusions

This article explored the influence of various factors on the AHC system in maritime operations. The work establishes a model of launch and recovery systems by using a homogeneous coordinate transformation and replaces the empirical wave with the irregular wave, which is used as external excitation of the simulation model. Considering the flexibility of the rope and the import of the AHC function, a dynamic simulation model of the AHC system with a flexible rope is simulated under different conditions.

From the simulation results, the following deductions can be obtained.

(1)

There are some differences between the results of simulation without AHC and those with AHC function, while this work still has a reliable reference for its ideal effect of AHC function. During the process of modeling and simulation, the launch and recovery system lacking AHC function is easily affected by harsh sea conditions, and the rope is easy to break away from the hoisting sheave. However, the rope runs normally under the effect of the AHC function.

(2)

On the whole, the increase of the load and underwater penetration will lead to the increase of the dynamic response of the system. Through the simulation results, it is strongly recommended to choose a light load and short rope in actual maritime operations.

(3)

A harsh sea condition will cause an increase of load displacement, load velocity, and load acceleration, as well as rope tension, and brings difficulty to the AHC system. Especially, the results of this work also demonstrate the importance of the AHC function in maritime operations.

The overall work can provide a reference for the selection of marine equipment as well as the reliable marine operation. In addition, under the background that the combination of wave compensation and predictive control has gradually become the mainstream practice in this field, it provides a theoretical basis for the automatic control of the launch and recovery systems.