Influence of the Upstream Channel of a Ship Lift on the Hydrodynamic Performance of a Fleet Entry Chamber and Design of Traction Scheme

Abstract

This study investigates the hydrodynamic performance of ships entering a ship lift compartment that is under the influence of upstream channel geometry and proposes a mechanical traction scheme to enhance operational safety and efficiency. Utilizing a Reynolds-averaged Navier–Stokes (RANS)-based computational fluid dynamics (CFD) approach with overlapping grid technology, numerical simulations were conducted for both single and grouped ships navigating through varying water depths, speeds, and shore distances. The results revealed significant transverse force oscillations near the floating navigation wall due to unilateral shore effects, posing risks of deviation. The cargo ship experienced drastic resistance fluctuations in shallow-to-very-shallow-water transitions, while tugboats were notably affected by hydrodynamic interactions during group entry. A mechanical traction system with a four-link robotic arm was designed and analyzed kinematically and statically, demonstrating structural feasibility under converted real-ship traction forces (55.1 kN). The key findings emphasize the need for collision avoidance measures in wall sections and validate the proposed traction scheme for safe and efficient ship entry/exit. This research provides critical insights for optimizing ship lift operations in restricted waters.

1. Introduction

Due to narrow and congested ship lift compartments and gate head channels, when a ship enters or leaves compartments, the shore wall effect produced by the ship lift’s gate wall causes the ship to produce a shore suction force and turning head moment. This could result in yawing, collision, and other dangers. Furthermore, shallow water results in the squat effect and the bottoming out of the ship’s body. Improper driver operation during the entry or exit of the ship lift compartment can easily lead to accidents, resulting in substantial economic losses. Therefore, it is necessary to conduct a numerical study on the hydrodynamic changes in large ships navigating in restricted waters, such as within ship lift compartments.

Several studies have focused on the hydrodynamic problems of ships in restricted waters. Most have adopted experimental physical models to examine the resistance performance of ships in restricted areas, including the analysis of the following: the impact of different water depth–draught ratios, h/T; ship types; sailing speed effects on the resistance of the ship and the hull-sinking amount; methods for resistance estimation; and empirical formulas for the resistance of shallow-water ships. Norrbin [1] conducted an experimental study on the shore wall effect generated during the passage of a certain type of ship under different shore wall conditions. He found that the transverse force and the first rocking moment of a ship sailing through a shore wall were significantly affected by the shore wall effect: The closer the ship is to the shore wall, the greater the transverse force and first rocking moment experienced during travel. Duffy [2] conducted an experimental study on ships sailing at different water depths, shore distances, and quay wall heights to analyze the effects of ship maneuverability and hydrodynamic characteristics—such as transverse force and bow rocking moment—under a unilateral quay wall and derived relevant regression formulas. Ch’Ng et al. [3] explored the hydrodynamic effects of a quay wall on ships maneuvering through a constrained environment and reported the hydrodynamic effects on the transverse force and bow rocking moment. Lo et al. [4] used CIP and MAC methods to calculate the rising waves generated during the sailing of a medium-sized ship through a restricted channel. Terziev et al. [5] calculated the hydrodynamic sinking, longitudinal pitching, and drag characteristics of a ship under shallow-water conditions and with respect to different channel cross-sections via the computational fluid dynamics (CFD) method. Wang and Zou [6] utilized a non-stationary RANS solver to numerically investigate the hydrodynamic characteristics of a 12,000 TEU container ship sailing through the Panama Canal locks. Toxopeus and Bhawsinka [7] estimated the hydrodynamic interaction forces experienced by a ship entering the Pierre Vandamme Lock using ReFRESCO computational predictions. Yao et al. [8] conducted a computational study using the first-order Rankine source surface element method and examined the transverse force and bow rocking moment generated by 60 ship types sailing through different shore walls. Zhang [9] used a two-dimensional current mathematical model to simulate ship maneuverability in common shallow-water navigation areas of inland waterways and constructed a mathematical ship-maneuvering-motion model under the shallow-water effect. Yang [10] considered the influence of free surfaces and fluid viscosity on numerical calculations combined with slip grid technology and performed a numerical study on the surrounding flow field and hydrodynamic characteristics of the ship navigation process in restricted waters. Hadi et al. [11] evaluated the hydrodynamic interaction of the 750 DWT Perintis Ship moving through different canal types to determine the relative effects of width and depth cross-section limitations on a ship’s resistance. Ma et al. [12] used the RANS equation, the volume of fluid method, and the body force method to investigate the hydrodynamic characteristics of a model ship (model scale ratio of λ = 31.6) and a full-scale ship advancing through confined waters at low speeds.

Ship lifts are autonomously operated by ship drivers when maneuvering in and out of compartments. However, from a safety and reliability perspective, traditional autonomous operations cannot guarantee safety, and there are problems associated with energy consumption and the wear and tear of the main engine, etc. From a safety and economic perspective, using push-assist methods via tugboats is a feasible approach to help large ships enter and exit a ship compartment. However, the tugboat used to assist the ship could cause interference and result in a course change; ship collisions are also possible, in addition to other accidents resulting in deterioration. Thus, a study of the hydrodynamic interference between the two ships is of important theoretical significance and practical value. In current research on inter-ship interference forces, Lo [13] used CFD to analyze the interaction between the flow field and the ship, including changes in all six degrees of freedom. Based on maritime overtaking and head-on encounter phenomena, they proposed a numerical model that facilitated accurate computation. Hua-fua et al. [14] proposed a method for predicting the hydrodynamic interaction forces of ships during shallow-water encounters and overtaking, using a three-dimensional high-order surface element method based on non-uniform rational B-spline (NURBS) curves. Raven [15] discussed the effect of simulated pool widths on shallow-water ship modeling tests. Wang et al. [16] used a numerical method to study the hydrodynamic interactions between moored and passing ships entering and exiting one-way ship locks. The power grid method and sliding interface technique were used to calculate the relative motion between passing ships and locks. Numerical results were obtained for the transverse and longitudinal positions of ships at different speeds, water depths, and berths. Consequently, the effects of these factors on the hydrodynamic interactions between ships and locks were analyzed. Deng et al. [17] performed numerical calculations of hydrodynamic interference between two ships during the process of assisting a large ship to berth away from a pier using a CFD-modeled tugboat, regressing the results of the calculations in the modeling of the tugboat. Park et al. [18] conducted time-domain numerical simulations to study hydrodynamic interactions between passing and moored ships. Zou and Larsson [19,20] performed a systematic calculation of ship–ship interactions in shallow water. The forces and moments acting on the hull, as well as sinking and longitudinal pitching, were investigated at different relative longitudinal and transverse positions. Yuan [21] investigated the effects of free water on the interactions of ships at different speeds.

In studying traction solutions to assist ship navigation, the most widely used traction aids are floating docks. Lian et al. [22] proposed a set of traction ship programs that can be applied to a floating dock and analyzed the practicality of an auxiliary traction program. Furthermore, they discussed the calculation method of the traction force required for the traction program; the scale of the traction trolley; the scale of the traction track; the design of the operation and control system; the installation of the track on the floating dock; and the strength of the track trolley calibration. Chen and Zhang [23] studied a method for calculating the traction force required to tow a ship into a floating dock under differing wind, wave, and current conditions and determined the parameters of the auxiliary towing device. Li [24] conducted a model test on the process of docking a sailing ship into a floating dock and selected the rail form and trolley size parameters with reference to the results of the model test. Furthermore, they performed a comparative analysis of the formulae for calculating the tractive force required for towing a ship using various existing traction schemes, and the strength of the system needed for towing a ship through the floating dock was improved by employing a finite element method.

Currently, track traction systems are widely used domestically and internationally to assist ships in entering and exiting ship chambers. However, given the unalterable external navigation environment of the Three Gorges ship lift, such a track traction system cannot be installed. Presently, navigation during chamber entry and exit relies entirely on the autonomous operation experience of vessel pilots. To enhance the operational efficiency of the ship lift and compensate for varying skill levels among vessel pilots, this study aims to research ship towing schemes for entering and exiting the chamber under the complex external navigation conditions of the Three Gorges ship lift. This research is conducted in conjunction with hydrodynamic theoretical calculations and analyses of ships entering and exiting the chamber. Thus, this study employed the CFD method based on RANS to simulate a ship sailing straight into the lift compartment, and fluid analysis software was applied to numerically simulate the viscous flow field of the ship. Consequently, the hydrodynamic behavior of the ship at different positions in the channel and the influence of channel shape on the ship’s hydrodynamics during entry and exit from lock-restricted waters were analyzed. In addition, building upon single-ship autonomous entry, the hydrodynamic conditions during group ship entry were explored. Consequently, research on the mechanical traction mode was conducted. Furthermore, a comparison and analysis were performed according to the ship lift’s site environment to propose a traction mode suited for the ship lift and to develop a mechanical traction technology program for ships entering and exiting the compartments.

2. Numerical Model and Method

2.1. Ship Parameters

This study focused on calculating the impact of a channel’s shape on ship resistance when a ship passes through a ship lift. The ship model used in our calculations is a typical ship type; the parameters of the ship type are presented in

Table 1. Main elements of ship model.

Ship Type	Length (m)	Beam (m)	Draught (m)	Depth (m)	Scale Ratio
Cargo ship	105	16.3	2.7	5.2	20:1
Full-swing tugboat	36.6	10.5	3.1	4.5

. Regarding the ship lift, most RO/RO vessels transporting newly manufactured vehicles are large and exhibit low transit efficiency. To enhance the transit efficiency of these vehicle-carrying RO/RO vessels, they should be designated as representative vessel types for conducting research on towing technology. The front ship model used in the calculation is an RO/RO vessel, and the rear ship is a full-swing tugboat. Figure 1 shows the RO/RO vessel’s model, and Figure 2 shows the three-dimensional diagram of the tugboat model.

2.2. Numerical Simulation Method

In this study, the SST k-ω turbulence model was selected. As the ship approached the head of the upper lock and ship compartment channel, the narrowing and shallowing of the channel caused the flow field within the vicinity of the ship to become chaotic and complex. Thus, the near-wall model was used for near-wall treatment. In addition, due to the blocking effect, the water level inside the lock chamber changes significantly when the ship enters the compartment, and neglecting the free surface reduces the accuracy of hydrodynamic and flow calculations for the ship. Thus, the influence of free surfaces must be considered during calculations. Consequently, the volume of fluid (VOF) model was used to capture the free surface of the flow field, and the Reynolds-averaged Navier–Stokes (RANS) equations were solved using the SST k-ω turbulence model. The center difference was used for discretization in space, whereas time discretization employed a first-order implicit scheme.

In contrast to the ship-following coordinate system, the fixed coordinate system had no incoming velocity throughout the computational domain. This study relied on the overlap technique to realize the movement of the ship relative to the channel. Overlapping grids are structured grid methods in which individual grid subregions can overlap, nest, or overlay each other, transferring flow-field information via interpolation between individual subgrids [25].

2.3. Boundary Conditions

2.3.1. Velocity Inlet Boundary Condition

Setting the inlet boundary condition requires defining the velocity components, direction, temperature, and turbulence parameters. The inlet boundary was configured based on a prescribed freestream velocity.

2.3.2. Pressure Outlet Boundary Condition

The pressure outlet boundary condition specifies static pressure and reverse flow characteristics at the domain outlet. To ensure accuracy, the outlet should be positioned sufficiently far from the simulated object, allowing the assumption that gradients of all physical quantities (except pressure) approach zero at the boundary. Additional specifications include total pressure, total temperature, reverse flow conditions, and turbulence parameters.

2.3.3. Wall Boundary Condition

Wall boundary conditions play a critical role in viscous flow simulations. They enforce the no-slip condition, where fluid velocity relative to the solid surface reduces to zero. This study focuses on flow around a ship hull in shallow water. Thus, both the hull surface and domain bottom were defined as no-slip boundaries.

2.3.4. Overset Mesh Boundary Condition

The overset mesh technique was selected for simulating ship transit through lock chambers. Originally proposed by Steger et al., this approach embeds component grids (e.g., moving ship) within a background grid. The overset region undergoes hole-cutting to exclude non-physical domains, while flow data exchange occurs through interpolation at overlapping boundaries.

This technique simplifies meshing for complex geometries and efficiently handles large relative motions without remeshing. In this study, STAR-CCM+’s automatic meshing generated unstructured grids, with overset methodology enabling ship motion simulation.

2.4. Calculation Domain Setting

In this study, the calculation channel was set according to the actual layout of the ship lift, which comprised an upstream channel, floating navigation wall, upper gate head, and a ship lift compartment with four segments. In the design of the calculation domain, as shown in Figure 3, the entire channel’s water depth changed from infinite to medium, shallow, and finally very shallow along the channel’s length. The water depth settings are presented in

Table 2. Channel depths.

	Length (m)	Depth (m)	Depth-to-Draft Ratio
Upstream channel	230	130
Floating Navigation Wall	130	10	3.7
Upper gate	130	4	1.48
Ship lift compartments	119	3.5	1.29

. The scale ratio of the domain to the ship model was set to 20:1.

The ship lift channel has a vertical shore wall and water bottom, where the upper gate head and the width of the ship lift compartments are 18 m; from the right side of the gate, there is an outward projection of a 130 m long floating navigation wall. This study calculated the initial position of the ship’s first plumbline distance from the gate: 1.75Lpp. The channel’s geometric model is shown in Figure 4.

The maximum speed of a ship through the ship lift compartment is 0.5 m/s. Based on the overall design specification of the lock, the initial speed of the ship entering the compartment from the upstream channel was set as 1.1118 m/s. The ship begins to decelerate when the bow touches the floating navigational wall. This deceleration is completed when the ship’s stern fully enters the head of the lock, and the speed is then maintained at a constant level from the lock head until the ship completely enters the ship lift compartment. At this time, the ship model speed was set at 0.1118 m/s to satisfy the maximum speed permitted in the ship lift compartment (Figure 5). Among all simulation conditions, we fix the speed of each stage of the tugboat, similar to the form of the constraint mode of the towing tank tests.

2.5. Mesh and Boundary Conditions

2.5.1. Boundary Condition Settings

The boundary conditions included the velocity inlet, pressure outlet, solid wall conditions, and overlapping grid boundary conditions. The computational domain comprises the background and overlap regions—which can be moved or rotated in the background region—and the two regions are connected via the overlapping grid interface interpolation link (Figure 6).

2.5.2. Grid Generation

In this study, the CFD calculation software STAR-CCM+ was used to automatically divide the nonstructural mesh of the cut-body model, as shown in Figure 7, which shows the entire background channel’s mesh cutting.

Figure 8 shows the grid generation of the overlap region, which provides the translation speed. In the calculations, the ship was translated into the compartment, with the ship fixed on the coordinate system; it was then translated to the overlap region, which simulated the process of the ship moving into the compartment. In this study, considering the influence of the turbulence model, the grid size was divided in a sufficiently fine manner to ensure the grid’s quality and maintain computational costs. Thus, the overlapping grid size was set as 0.02 × 0.02 × 0.02 m, and the number of grids in the overlapping area was approximately 3.25 million. Moreover, due to the limitation of the ship’s water area, the lift compartment and the bottom portion of the ship were encrypted locally and individually. The hull surface’s boundary layer was generated using a prismatic layer mesh generator.

According to the International Towed Tank Conference’s (ITTC) (2014) recommendation for time steps, when using one-equation or two-equation turbulence models, the time step should be less than 0.01 L/U, where L denotes the length of the boat and U denotes the advection velocity. In this study, the time step was set to 0.03 s.

2.6. Grid Dependency Validation

To ensure the reasonableness of the grid division settings, the first-order RE method was used to analyze the convergence of grid-scale settings. According to the rate of change in the grid scale, three sets of grids were created from coarse to fine: G3, G2, and G1. The corresponding resistance values are $F_{G3},F_{G2}$, and $F_{G1}$, respectively. The convergence factor $R_G$ is expressed in Equation (1).

$$R_G={\displaystyle \frac{{\epsilon }_{G_{21}}}{{\epsilon }_{G_{32}}}}={\displaystyle \frac{F_{G2}-F_{G1}}{F_{G3}-F_{G2}}}$$

(1)

The convergence of the calculations was classified into four categories based on the magnitude of RG (as shown in

Table 3. Convergence.

${\mathit{R}}_{\mathit{G}}$	Result
$R_G<-1$	Oscillatory dispersion
$-1<R_G<0$	oscillatory convergence
$0<R_G<1$	monotonous and astringent
$R_G>1$	monotonous and diffuse

):

This study encrypted the meshing scheme according to the scale ratio recommended by ITTC; forecasted the drag resistance results of the restrained mode test of the cargo ship under different grid scales; and analyzed the convergence according to the grid scale’s rate of change of $\sqrt{2}$. Three sets of different meshes—coarse, medium, and fine—were considered. For the more complex wall boundary layer of the flow field, a 5-layer prismatic layer grid was adopted to encrypt the grid near the wall. The total thickness of the prismatic layer grid was 0.018 m, and the elongation rate was 1.05. The main body and side of the surface of the mesh model are generally less than 10‰ of the length of the ship; thus, the surface of the hull mesh was set to the length of the ship’s main body at 7.5‰ of the hull. Furthermore, the Kelvin wave area and the free surface of the liquid were encrypted. The specific grid division is shown in Figure 9.

Table 4. Drag prediction results for different grid schemes.

Grid	Meshes (Million)	Resistance (N)	Error with G1 (%)
G1	10.59	16.89	-
G2	5.40	17.04	0.89
G3	2.54	17.33	1.72

presents the number of grids for three course, medium, and fine grids—from dense to sparse—and the drag forces on the hull of the ship during simulated towing tests are demonstrated for the grid cases.

The difference between the “G2–G1” and “G3–G2” resistance prediction results was calculated, and the convergence rate of the grid was $R_G=$ 0.52, which falls within $0<R_G<1$. Furthermore, the results of the calculation were monotonically convergent; the comprehensive calculation accuracy and calculation resources were considered; the medium-sized grid was chosen as the calculation grid in subsequent calculations.

$$U_G={\displaystyle \frac{1}{2}}(S_U-S_L)$$

With three solutions in conclusion (i), only the leading term can be estimated, which provides one-term estimates for error and order of accuracy:

$${\delta }_{RE}={\displaystyle \frac{{\epsilon }_{k_{21}}}{r_G^{p_G}-1}}=0.161$$

$$p_G={\displaystyle \frac{\ln ({\epsilon }_{k_{32}}/{\epsilon }_{k_{21}})}{\ln (r_G)}}=1.903$$

The uncertainty was then estimated using a correction factor, which provides a quantitative metric for defining the distance of the solution from the asymptotic range and approximate the effect of the higher order terms in the initial error and uncertainty estimates. Here p_Gest is an estimate for the limiting order of accuracy.

$$C_G={\displaystyle \frac{r_G{p}_G-1}{r_G{P}_{Gest}-1}}=0.934$$

Benchmark result S_C can be an estimates form:

$${\mathrm{S}}_{\mathrm{C}}=\mathrm{S}-{\delta }_{\mathrm{RE}}=16.739$$

The uncorrected U_k solution uncertainty estimates given by:

$$U_G=[9.6{(1-C_G)}^2+1.1]\left|{\delta }_{RE}\right|=0.184$$

while 1 − C_G = 0.067 < 0.125.

3. Hydrodynamic Performances of Single Ship

Before analyzing the hydrodynamic performance of a group of ships in a ship lift compartment, the hydrodynamic situation of a cargo ship autonomously sailing into the compartment was studied. In addition, to facilitate the analysis of the calculation results, the change in speed with respect to position is shown in Figure 5. Deceleration was completed before the ship entered the upper gate head, and a uniform speed was maintained when the ship entered the compartment.

3.1. Analysis of the Total Resistance of a Single Ship into the Compartment

As shown in Figure 10, $X<-1.19L_{pp}$ for the beginning stage, and the ship in the upstream channel has a uniform speed of 0.6118 m/s. From stage $X=-1.19L_{pp}$ to $X=0$, the ship traveled to the floating navigation wall section. However, as this stage was part of the deceleration stage, the total resistance of the ship decreased. At $X=0$, the bow of the ship entered the upper gate head, and the ship’s speed was reduced to 0.1118 m/s in order to enter the uniform speed stage. The total resistance tended to level off. At $X=1.19L_{pp}$, when the ship’s bow entered the ship lift compartment, the ship’s resistance began to increase until the ship’s transom entered the compartment. At $X=1.19L_{pp}$ to $X=2.28L_{pp}$, the ship sailed into the upper gate head and the ship lift compartment, and the blockage rate of the upper gate head and the compartment section of the ship lift was extremely high. Here, the ship sailed from shallow water to very shallow water. Due to the influence of the shore wall of the ship lift compartment and the shallow-water effect, the ship’s resistance was considerably greater than in the case of open water, and its resistance changed drastically.

3.2. Analysis of Transverse Force of Single Ship Entering Compartment

Due to the influence of the floating navigation wall, in addition to the total resistance, the influence of the transverse force along the ship’s width on the ship’s hydrodynamic performance when entering the compartment must be considered. As shown in Figure 11, when the ship was not within the floating navigation wall’s range ($X<-1.19L_{pp}$), the transverse force was low. As it entered the deceleration stage ($-1.19L_{pp}<X<0$), due to the influence of the unilateral shore wall, the transverse force fluctuations began to increase, with a peak value of 11.7 N. Here, $0<X<1.19L_{pp}$ was the uniform velocity stage, where the ship entered the upper gate head’s range and completely left the floating navigation wall’s range. Due to the influence of the two sides of the shore wall, the ship’s transverse force was observed earlier within a large range of oscillations. When the ship completely entered the upper gate head and before it entered the ship lift compartment, the ship was subjected to a transverse force, with a smaller range of oscillations.

3.3. Flow Field Analysis

When the ship model traveled to the floating navigation wall section, according to the previous analysis, the total resistance and transverse force at this stage changed drastically. We consider X = −0.1 as an example. At this moment, the total resistance and transverse force of the ship were close to the lowest peak, and the ship model was located in the vicinity of the floating navigation wall. Furthermore, the surface pressure cloud diagram in Figure 12 indicated that, at this stage, the pressure of the ship’s bow increased. The surface pressure of the ship model’s port side (both above and below) was greater than on the starboard side, pushing water into the upper gate head compartment. Consequently, the water level in the upper gate head rose.

At this time, the ship had completed the deceleration stage. Within the area from the floating navigation wall to the head section of the upper gate, the total resistance was maintained at a low value, and the lateral force with respect to $0<X<0.5L_{pp}$ dramatically increased, as shown in Figure 13. For the ship’s surface pressure at $X=0.31L_{pp}$, as the ship’s first half entered the upper head section, the sudden decrease in water depth resulted in a decrease in the section coefficient of the gate. Furthermore, the water level at the ship’s starboard side, which was close to the floating navigation wall, increased. The pressure on the starboard side (including the lower part of the port side corresponding to the starboard side) was considerably greater than that on the port side. At this stage, the ship’s transverse force reached its peak. However, as the ship moved up to the head of the gate, due to the gradual symmetry of the two sides of the channel, the pressure on the left and right sides was the same, and the transverse force tended to be zero.

Figure 14, which presents a comparison of the free surface cloud diagrams near the ship at different $X=0.31L_{pp}$, $X=0.51L_{pp}$, and $X=0.72L_{pp}$ positions, shows that the water level inside the compartment gradually increased as the ship entered the upper gate head of the ship lift. Moreover, due to the ship’s compartment, as the ship approached the ship-bearing compartment, the liquid area between the ship’s bow and the compartment decreased gradually. This caused the water level in the ship lift compartment to rise over time, and the ship’s resistance gradually increased as it entered the upper gate head. As the ship entered the upper lock head, its resistance increased gradually.

When the ship traveled into the upper gate head and compartment sections, due to the symmetry of the shore wall on both sides, the lateral force tended to be zero, maintaining a small range of oscillations. Furthermore, the total resistance exhibited a dramatic change in the range of $1.19L_{pp}<X<2.28L_{pp}$ due to the change in water depth. Combined with the surface pressure diagram of each position, as shown in Figure 15, when the ship began to enter the compartment, the differing water depths of the upper gate head and compartment sections caused the reflux in the compartment to be delayed. Consequently, the ship’s resistance increased dramatically with fluctuations. The ship gradually entered the ship lift compartment due to the small change in the compartment’s water level; however, because of the impact of the shore wall, shallow water, and other factors, the resistance of the ship fluctuated within a relatively small range.

The pressure change analysis indicated that the ship’s pressure resistance in the floating navigation wall section was mainly related to the unilateral shore wall, causing the most violent lateral force oscillations, which were easy to offset in this section. In actual navigation, it is necessary to focus on the lateral force affecting the ship’s maneuvering performance. The fleet must pay extra attention to ship maneuvering challenges when navigating the floating navigation wall section and especially at the intersection of the upper gate head and the floating navigation wall section. Because of the change in water depth, the ship model’s pressure resistance increased. As the ship model traveled from shallow to very shallow water in the upper gate head section and the ship lift compartment, its resistance changed drastically.

Because the ship in the compartment experiences a high water body obstruction ratio and very shallow water depth, the rated speed from the upper gate head to the ship lift compartment causes changes in water depth along the waterway. Consequently, the rear water level rises, and the front water level decreases when the ship experiences bottom-touching phenomena.

4. Analysis of Group Ship

Considering the autonomous entry of cargo ships into ship lift compartments, this study further investigates the hydrodynamic performance of group ship entry to provide a theoretical basis for the group entry and exit of ships, aiming to improve operational efficiency.

In this study, the initial position of the cargo ship was 1.75Lpp away from the origin, and the initial position of the pusher was 2.77Lpp away from the origin. The geometric model of the waterway is shown in Figure 16. The computational domain included the background and overlapping areas, and the distance from the initial position of the cargo ship was set as the pressure outlet. The surfaces of the cargo ship and the pusher’s wheel were set as no-slip walls. Moreover, the single side wall of the floating navigation wall, the upper gate head, the underwater surface of the carrier compartment, and the wall surface were also set as no-slip walls. The remaining boundaries were set as velocity inlets.

4.1. Analysis of Resistance of Cargo Ship

As shown in Figure 17, the cargo ship was less affected by the tugboat when $1.19L_{pp}<X_{\mathrm{c}\arg \mathrm{oship}}<1.65L_{pp}$ (of which $L_{pp}$ is the captain of the cargo ship). When the ship traveled within the upper gate head section, the total resistance of the ship increased (Figure 18), and the single ship entering the ship compartment was subjected to the lateral force of oscillations. However, when $1.19L_{pp}<X_{\mathrm{c}\arg \mathrm{oship}}$, the value of the transverse force was between ±0.5 N. The total resistance during this stage of the floating navigation wall decreased significantly with speed; however, the group of ships was affected by the floating navigation wall, and the total resistance decreased significantly. At $-1.19L_{pp}<X_{\mathrm{c}\arg \mathrm{oship}}$, with respect to the floating navigation wall, the total resistance decreased with a reduction in speed, and the group of ships was affected by the floating navigation wall (the lateral force oscillation was obvious).

4.2. Analysis of Tugboat Resistance

As shown in Figure 19, the tugboat was more obviously affected by the cargo ship at the front: that is, $-0.22L_{pp}<X_{\mathrm{tugboat}}<0.83L_{pp}$ (at this time, $0.76L_{pp}<X_{\mathrm{c}\arg \mathrm{oship}}<1.81L_{pp}$). From the floating navigation wall to the head of the gate, the resistance of the tugboat was affected by the larger ship.

Compared to the cargo ship, the tugboat was smaller and more stable overall due to the lateral force, as shown in Figure 20. Even after the floating navigation wall section, the maximum lateral force was ± 4 N or less. At the $-0.96L_{pp}<X_{\mathrm{tugboat}}<0.19L_{pp}$ position, the tugboat arrived at the floating navigation wall. In this time period, because of the impact of the unilateral quay wall, the lateral force changed; however, compared to the change in the lateral force of the cargo ship, the tugboat by the unilateral quay wall was less affected.

4.3. Analysis of Inter-Ship Interference Force of Grouped Ships

A comparison of the hydrodynamic results of the group of ships and a single ship entering the compartment revealed interesting results. Compared with the tugboat, the cargo ship was almost unaffected. The tugboat in the 0.76L_pp < X_{cargo ship} < 1.81L_pp section—compared to the tugboat with a single ship entering the compartment in the 0.76L_pp < X_{cargo ship} < 1.39L_pp section—first experienced a sharp decrease in total resistance, followed by an increase within 1.39L_pp < X_{cargo ship} < 1.81L_pp.

As shown in Figure 20, at 0.76L_pp < X_{cargo ship} < 1.39L_pp, the cargo ship was located at the upper gate head, and the tugboat traveled in the floating navigation wall section. As the cargo ship gradually approached the carrier compartment, the water level of the stern of the cargo ship gradually increased. We consider X_{cargo ship} = 1.18L_pp as an example. At this stage, the water level of the stern of the cargo ship and the section from the bow to the carrier compartment was rising due to the increased water level at the tugboat’s bow. As the tugboat reached 0.76L_pp < X_{cargo ship} < 1.18L_p, the resistance of the bow of the ship increased. In the case of the free surface at X_{cargo ship} = 1.39L_pp, as shown in Figure 21, the water level at the stern of the tugboat increased as the tugboat entered the upper gate head. After X_{cargo ship} = 1.39L_pp, as the cargo ship entered the carrier compartment, because the cargo ship was traveling from shallow to very shallow water, the water level in the carrier compartment decreased at the bow of the cargo ship. In contrast, the water level at the stern of the cargo ship increased significantly. Consequently, the tugboat experienced an increase in resistance.

According to the hydrodynamic comparison between a single ship and a group of ships in the lift compartment, the stern’s water level increase during the cargo ship’s entry caused the tugboat to be affected more substantially than the cargo ship. In contrast, in the process of entering the compartment, the cargo ship was almost unaffected by the tugboat. When entering the compartments, the blockage rate of the pushers was considerably lower than that of the cargo ship, and the influence of the channel’s shape was lower than that of the cargo ship.

To further study the optimal towing distance of a ship in the restricted channel, numerical calculations were conducted for different towing distances and groups of tugboats, and the resistance values of two ships in the narrow channel—with respect to different towing distances—are shown in

Table 5. Forces on grouped ships at different towing distances.

Pulling Distance (m)	Ship Group
	Cargo Ship Resistance (N)	Tugboat Resistance (N)
0.05	2.654	0.311
0.10	2.639	0.315
0.15	2.879	0.335
0.20	2.829	0.340
0.25	2.947	0.343
0.30	2.947	0.346
0.35	2.632	0.340
0.40	2.811	0.359

. The hydrodynamic conditions were more complex for groups of ships entering and exiting the lift compartment due to the influence of shallow water and shore wall effects. In contrast to open water, the resistance of ships at different distances was slightly different, and the resistance of a group of ships entering the narrow channel was minimized when the distance between the cargo ship and tugboat was 0.1 m. As for the pusher, because the size of the pusher was approximately 1/3 of the cargo ship, it was affected by the cargo ship, and the resistance of the cargo ship increased with an increase in distance.

4.4. Conversion of Traction Force

Based on the CFD calculations in Section 4.1, Section 4.2 and Section 4.3, several results were obtained in terms of the ship model’s force. The traction force required by the mechanical arm requires the force of a real ship; therefore, it is necessary to convert the force of the ship model to that of a real ship. Because the ship travels in the ship lift at a low Froude number, wave-making resistance can be ignored, making the total resistance equivalent to viscous resistance. Therefore, the use of the three-factor conversion method is applied based on actual ship resistance.

The three-factor conversion method assumes that the ratio of frictional resistance to the viscous pressure resistance of a real ship is constant. To determine the total resistance of a real ship, it is necessary to first determine the frictional resistance of a real ship. According to Schlichting’s intermediate velocity theory, due to the existence of velocity reflux, the resistance of a ship traveling with a speed of V in restricted waters is equal to that of a ship traveling with a speed of VC = (V − ∆V) in unrestricted waters, where ∆V is the reflux velocity, and VC is the intermediate velocity. To determine the real ship’s resistance, the return velocity of the ship must first be determined.

Although the velocity vector distribution near the surface of the ship model can be obtained via CFD post-processing, the flow field vector distribution is very irregular; therefore, it is not possible to directly determine the return velocity of the flow field near the ship model using a numerical calculation method. Because the frictional resistance of the ship model is known and there is a clear correlation between the frictional resistance and intermediate velocity, the intermediate velocity of the ship model can be obtained by solving the ITTC equation.

$$\begin{array}{l}{C}_f={\displaystyle \frac{0.075}{{(\lg Re-2)}^2}}\\ R_f={\displaystyle \frac{1}{2}}{C}_f\rho S{v}^2\end{array}$$

(2)

The characteristic length of the Reynolds number (Re) was Lpp = 5.031 m. The wet surface area of the ship’s model was S = 4.395 m², the fluid density was ρ = 998 kg/m³, and the ship’s model speed was v = 0.1118 m/s.

The 13th ITTC proposed eight correction formulas for the calculation of a ship’s return velocity under the blocking effect As the authors of [26] suggested, regardless of the correction formula, the velocity correction ratio is only related to the Froude number, ship coefficient, and channel coefficient. The numerical simulation of the ship model ensures that the ship model is consistent with the Froude number, ship coefficient, and channel coefficient of the real ship. Therefore, it can be assumed that the velocity correction ratio between the ship model and the real ship is the same. After solving for the return velocity of a real ship, the friction resistance of the real ship can be obtained according to the ITTC formula.

Combined with the aforementioned force curve of a ship model entering and exiting the compartment, the ship experienced maximum resistance when it entered the compartment at X = 1.601L_pp. At this moment, the resistance of the ship model was R_m = 8.93 N: That is, the ship model experienced residual resistance. According to Equation (2), the real ship’s friction resistance was obtained as R_f = 1.84 × 10⁴ N.

$$C_r={\displaystyle \frac{R_r}{\frac{1}{2}\rho v^{2}S}}$$

(3)

According to Equation (3), with a residual drag coefficient of C_mr = 0.975, the known frictional resistance and the residual drag coefficient were used to calculate the resistance of the real ship as Rs = 5.51 × 10⁴ N.

5. Design of Mechanical Traction Program for Entering and Leaving the Ship Compartment

Combined with the actual arrangement of the ship lift, Section 5 discusses the traction system for assisting ships in entering and exiting the ship compartment, and a traction scheme that is suitable for assisting entry into and exit from an actual ship lift compartment is proposed. Furthermore, the structural feasibility of the auxiliary mechanical arm is analyzed based on the results of the CFD numerical calculations from both kinematic and static perspectives, as obtained in the previous section. Finally, the effectiveness of the traction system is analyzed.

5.1. Mechanical Traction Design

5.1.1. Design of Quick Connecting Device

The connection between the tugboat and the vessel passing through the ship lift must be carried out with special devices that can effectively adapt to the dry side and stern line characteristics of different ship types, enabling fast docking and disengagement between the tugboat and the ship. This study proposes the use of a robot to connect two ships, with the tugboat pushing the cargo ship into the ship compartment. When a cargo ship fully enters a ship lift compartment, the tugboat assists in braking; it is then separated from the cargo ship and exits the ship compartment. Consequently, the ship compartment is lifted or lowered into place, after which the traction device pulls the cargo ship out of the ship compartment. In this study, we design a quick-connecting robotic arm device.

A four-link arm was used for arm flexibility. Figure 22 shows the structure of the traction arm, which mainly comprises a large arm, wrist, small arm, and end effector. The large arm cylinder is responsible for adjusting the orientation of the large arm, and the small arm cylinder is responsible for controlling the direction of the small arm. Furthermore, the wrist connects the small arm to the electromagnetic suction cup of the end effector and can be adjusted along the Z-direction to meet the traction connection requirements of ships with different parameters.

5.1.2. Guiding System and Braking System

Figure 23 shows the overall layout of the ship lift. Considering the efficiency of ships entering and leaving the ship compartment and the related safety issues, four pairs of guide wheels are placed at the upper and lower heads of the ship lift compartment. In addition, the electric tugboat is equipped with a separate guidance system in the ship lift compartment, which is positioned along both sides and the upstream and downstream navigation walls. The guiding system should include anti-collision and ship mooring functions. Two structural designs are proposed.

First, the guiding device could be uniformly arranged in the upstream and downstream navigation walls along the current direction, and the device is designed as a floating structure capable of moving vertically to accommodate fluctuations in the water level of the shifting waterway.

Second, the guiding device could be symmetrically arranged along the longitudinal centerline of the compartment, following the principle of denser placement at the upstream and downstream ends in the direction of the current and sparser placement in the middle section.

In addition to the electric tugboat’s braking system, for safety reasons, wire rope winch-type braking devices were installed near the upstream and downstream ends of the cabin navigation wall on both sides. These devices move vertically with water level fluctuations in the lift channel, exerting a damping effect. These devices were also installed midway along both sides of the ship compartment to serve as the ship’s emergency cable brakes in case of loss of ship control. In addition, bollards were installed near the floating navigation wall at the upper gate head and the lower gate head near the ship compartment to provide emergency braking during the ship’s entry and exit.

5.2. Modeling of Mechanical Traction Kinematics

The robotic traction arm, as an operational tool moving in space, must be analyzed in terms of the position and direction of its end effector. Mechanical arm kinematics were used to characterize spatial motion aspects. The arm’s kinematics were solved for the position and attitude of the end effector of the arm under the condition that the relative positions of the arm links are known. The inverse kinematics of the robot arm were solved to determine the joint angle values based on the known position and attitude of the arm’s end effector in a Cartesian coordinate system.

For the positive kinematics of the robotic arm, the coordinate system of each joint was first established according to the D-H method, and the position matrix of the end of the robotic traction arm was obtained by multiplying the change matrix of the coordinate system of each linkage. This indicated the functional relationship between the base of the robotic arm and the position of the end effector, thus obtaining the positive kinematic equations of the robotic traction arm. The kinematic inverse of the robotic traction arm was solved using an algebraic method. Based on the positive kinematics of the robotic traction arm, simulations and analyses were performed using MATLAB R2022b to obtain the joint angles required for the robotic arm to accommodate different ship types.

5.2.1. Mechanical Arm Parts Modeling

A four-degree-of-freedom robotic arm was used as the traction connection mechanism between two ships, and the main structure of the robotic arm comprised a base, a large arm, a small arm, a wrist, and the electromagnetic suction cup of the end effector, all of which had rotating joints. To observe the simplified kinematic model, the base was set as a fixed vice.

Figure 24 shows the overall model diagram of the robotic traction arm. The upper part of the robotic traction arm’s base drives the entire robot arm to perform rotary movement in the horizontal plane. The lower part is fixed as the foundation, providing support on the ground or working surface, and the hydraulic motor housed in this section outputs power to drive the upper base, enabling rotation.

The large arm is primarily used to connect the small arm and base through the large-arm cylinder support, facilitating pitching movement. In contrast, the small arm is designed using a small-arm cylinder to drive rotary movement. To tow the ship out of the compartment, the end of the electromagnetic suction cup is connected to the small arm, and the end of the small arm is connected to the internal hydraulic motor. This facilitates the electromagnetic suction cup in performing pitching and swinging motions.

5.2.2. Kinematic Modeling of the Robotic Arm

In the positional description of a robotic arm, the D-H method is typically used to perform calculations [27]. The focus of this method is on the rules for establishing the coordinate system of each joint, based on which four defined parameters are used to describe the positional relationship between the coordinate systems of the two neighboring joint linkages. The kinematic equations of the robotic arm were established via the sequential multiplication of the chi-square transformation matrices of each joint in order to obtain a mapping relationship between the end position of the robotic arm and the base coordinate system [28].

When the D-H method was used to build the linkage coordinate system for the robotic arm, joint i and joint i + 1 were considered in writing the formulas; the establishment of the coordinate system should comply with the following rules [29]:

(1)

The direction of the rotation axis of each joint is the direction of the coordinate axis.

(2)

The direction of the z-axis, which is perpendicular to the neighboring joints, is the direction of the coordinate axis, and it points away from joint i.

(3)

The intersection of the common perpendiculars of the Z-axis of the neighboring joints is the origin of the coordinate system $O_i-X_i{Y}_i{Z}_i$. Furthermore, if the axes of the neighboring joints intersect, the origin of the coordinate system is the point of intersection of the axes.

(4)

The base coordinate system is fixed to the base joint of the robot arm in the same direction.

The coordinate system of each joint linkage of the lightweight underwater robotic arm was established based on the following parameters: linkage length, joint angle, linkage torsion angle, and deflection distance. The chi-square transformation matrix of the neighboring joint linkage coordinate system can thus be obtained by substituting these parameters.

The above four D-H parameters are defined as follows:

(1)

Length of the Connecting Rod: The distance along the axis from one joint to the next, which is positive in the same direction.

(2)

Rod Torsion Angle: The angle through which the shaft is rotated around its axis to align with the next axis, where counterclockwise rotation is defined as positive.

(3)

Prismatic Joint Distance: The distance that the shaft moves along the direction of the axis, where movement along the positive direction of the axis is considered positive.

(4)

Joint Angle: The angle by which the shaft rotates around the axis to be parallel to the axis, where the rotation around the axis in the counterclockwise direction is positive.

According to the definitions above, the expression of the linkage transformation matrix is as follows:

$${}_i{}^{i-1}T=Rot(x_{i-1},{\alpha }_{i-1})Trans(a_{i-1},0)Rot(z_i,{\theta }_i)Trans(0,0,d_i)$$

(4)

According to Equation (4), the generalized equation for the transformation of the joint coordinate system of adjacent connecting rods can be obtained as

$${}_i{}^{i-1}T=\left[\begin{array}{cccc}\cos {\theta }_i& -\sin {\theta }_i& 0& a_{i-1}\\ \sin {\theta }_i\cos {\alpha }_{i-1}& \cos {\theta }_i\cos {\alpha }_{i-1}& -\sin {\alpha }_{i-1}& -d_i\sin {\alpha }_{i-1}\\ \sin {\theta }_i\sin {\alpha }_{i-1}& \cos {\theta }_i\sin {\alpha }_{i-1}& \cos {\alpha }_{i-1}& -d_i\cos {\alpha }_{i-1}\\ 0& 0& 0& 1\end{array}\right]$$

(5)

where ${}_i{}^{i-1}T$ represents the positional relationship between two coordinate systems, and the joint angle ${\theta }_i$ is the only variable parameter in the general transformation equation. If there are i joint coordinates of the robot arm, then the transformation matrix of the end coordinate system and the base coordinate system can be obtained from a general transformation, via Equation (5), of the connecting rod and the parameter values in the D-H parameter table:

$${}_i{}^{0}T={}_1{}^{0}T{}_2{}^{1}T{}_3{}^{2}T\dots {}_i{}^{i-1}T$$

(6)

Therefore, the robotic arm was transformed from a joint coordinate system into a Cartesian coordinate system.

According to the above D-H method, to establish the joint linkage coordinate system of the traction robot arm, the base coordinate system, $O_i-X_i{Y}_i{Z}_i$, was fixed to the bottom slewing joint, and the linkage coordinate system of the traction robot arm was obtained, as shown in Figure 25.

Table 6. Parameters of robotic arm linkage D-H.

i	${\mathit{a}}_{\mathit{i}-1} \left(\mathbf{mm}\right)$	${\mathit{\alpha }}_{\mathit{i}-1} (°)$	${\mathit{d}}_{\mathit{i}} \left(\mathbf{mm}\right)$	${\mathit{\theta }}_{\mathit{i}} (°)$
1	0	0	0	${\theta }_1$
2	$l_1$	90	0	${\theta }_2$
3	$l_2$	0	0	${\theta }_3$
4	$l_3$	0	0	${\theta }_4$(0)
5	$l_4$	−90	0	${\theta }_5$

presents the robotic arm’s connecting rod parameters that were substituted into the coordinate system transformation of adjacent connecting rods in the general equation. Consequently, the solution to the robotic arm’s kinematic equation can be obtained.

5.2.3. Kinematic Equation Solving of Robotic Arm

The inverse kinematic solution involves the determination of the position of the robotic arm and the structural parameters of each linkage; moreover, the angle of the rotary joint of the robotic arm is obtained according to the position matrix of the robotic arm’s end effector. There are three types of solution methods for the inverse kinematics of robotic arms: algebraic, geometric, and iterative methods. When the degree of freedom of the robotic arm is low, the use of the algebraic method is less computationally intensive, and solutions can be obtained quickly; however, as the degree of freedom increases, computation also increases rapidly. Geometric methods are typically used when the robotic arm has a low degree of freedom or when its configuration is special. For a robotic arm with more degrees of freedom, it is more difficult to solve inverse kinematics via algebraic and geometric methods, and the use of iterative methods has better applicability. However, the iterative method suffers from large and complex computations. The structure of the designed traction robot arm was relatively simple; therefore, in this study, an algebraic method was used to solve the inverse kinematics.

The directions of the rotational axes of robotic arm joints 2, 3, and 4 were parallel to one another, and the kinematic model could be simplified by multiplying the three matrices as follows:

$${}_4{}^{1}T={}_2{}^{1}T{}_3{}^{2}T{}_4{}^{3}T=\left[\begin{array}{cccc}{c}_{234}& -s_{234}& 0& l_1+l_2{c}_2+l_3{c}_{23}\\ 0& 0& -1& 0\\ s_{234}& c_{234}& 0& l_2{s}_2+l_3{s}_{23}\\ 0& 0& 0& 1\end{array}\right]$$

(7)

$$\begin{array}{l}{\theta }_1=\arctan (-p_x/p_y)\\ {\theta }_2=\arcsin (-(p_y+l_3{\sin }_{23})/l_3)\\ {\theta }_3=\arccos ((p_x^2+p_y^2-l_2^2-l_3^2)/2l_2{l}_3{\sin }_{23})\\ {\theta }_2+{\theta }_3+{\theta }_4=0\end{array}$$

(8)

5.2.4. Kinematic Simulation of Traction Robotic Arm

The forward and inverse motion algorithms were input into MATLAB. During the compartment process, the traction points were located on both sides of the ship. The traction point is defined by $P_x=l+\Delta x+\mathrm{\Phi }$, where l (l = 1.6 m) denotes the distances of the center of the traction arm chassis from the tugboat to the ship’s head; the electromagnetic suction cup’s diameter is $\mathrm{\Phi }=1.6 \mathrm{m}$; and $\Delta x$ denotes the two ship’s traction distance, obtained from the compartment process’s traction point of $P_x=l+\Delta x+r+\Delta l$. Relative to the movement out of the compartment, we must consider the distance between the traction point’s location and the distance from the bow of the cargo ship $\Delta l$. We selected $\Delta l=3 \mathrm{m}$ according to $\Delta l$ in order to further determine the position of the ship’s width, with a traction point of $P_y=(B-r)/2$ and base radius of $r=2.5 \mathrm{m}$. When towing the ship in and out of the compartments, the traction point $P_z$ value should be considered in accordance with the dry height of the cargo ship and when the tugboat’s wheel is in place. This ensures that the electromagnetic suction cup is located between the waterline and the deck.

Based on the analysis of the towing distances of typical ships in restricted waters in Section 3 and the actual scales of cargo ships and pushers, the optimal towing points for typical ships were determined, as shown in

Table 7. Optimum towing point for ship.

Optimum Towing Point	Enter a Compartment $[\begin{array}{ccc}{\mathit{p}}_{\mathit{x}}& {\mathit{p}}_{\mathit{y}}& {\mathit{p}}_{\mathit{z}}\end{array}]$
Cargo ship	$\left[\begin{array}{ccc}6800& 6150& 550\end{array}\right]$

.

To ensure that the end linkage reached the optimal traction point position, the traction position data from Table 7 was substituted into Equation (7). This matrix equation was substituted into the inverse kinematics equation, and MATLAB was used to solve the equation. According to the robotic arm’s equation, some traction configurations fall outside the rotation angle’s range. As a result, the back end of the angle’s solution cannot be solved; valid solutions corresponding to the equation are shown in

Table 8. Traction angle.

		Traction Angle $[\begin{array}{ccc}{\mathit{\theta }}_1& {\mathit{\theta }}_2& \begin{array}{ccc}{\mathit{\theta }}_3& {\mathit{\theta }}_4& \begin{array}{cc}{\mathit{\theta }}_5& {\mathit{\theta }}_6\end{array}\end{array}\end{array}]$
Cargo ship	Enter a compartment	{24.8434° 22.5058° 91.0888° 68.5830° 65.1566°}

.

According to the traction angles of different ships entering and exiting the compartment, it is evident that the design of the traction arm can satisfy typical ship movements concerning traction connections.

5.3. Traction Mechanical Arm Static Analysis

The previous section on the kinematics of the robotic arm focused on determining the joint angles based on the spatial position of the electromagnetic suction cup at the arm’s end. Considering the safety of the robotic traction arm and according to the hydrodynamic results obtained from the previous sections—converted to the traction force required here—the load inflicted was calculated using the SolidWorks v2024 simulation module, and static analysis was carried out considering the results of the strength verification of the robotic traction arm. We examine the strength of the robotic traction arm in this section.

5.3.1. Theoretical Basis of Statics

The static analysis in this section is based on the fourth strength theory, also known as the maximum distortion energy density theory, which is the primary factor that causes materials to yield to the maximum distortion energy. The maximum distortion energy density theory assumes that, regardless of the stress state of the material, the maximum change in the critical point of a material can reach the limiting value found in simple tensile yielding, and the material will yield.

The relationship between the distortion energy density $v_d$ and yield stress ${\sigma }_s$ is expressed as follows:

$$v_d={\displaystyle \frac{1+\mu }{6E}}({\sigma }_s{}^2)$$

(9)

where $\mu$ is the Poisson ratio of the material, and $E$ is the modulus of elasticity of the material.

The distortion energy density formula in an arbitrary state is expressed as follows:

$$v_d={\displaystyle \frac{1+\mu }{6E}}[{({\sigma }_1-{\sigma }_2)}^2+{({\sigma }_2-{\sigma }_3)}^2+{({\sigma }_3-{\sigma }_1)}^2]$$

(10)

where ${\sigma }_i(i=1,2,3)$ is unit principal stress.

The yield criterion can be derived from the above equation:

$${\sigma }_s=\sqrt{{\displaystyle \frac{1}{2}}\left[{({\sigma }_1-{\sigma }_2)}^2+{({\sigma }_2-{\sigma }_3)}^2+{({\sigma }_3-{\sigma }_1)}^2\right]}$$

(11)

Therefore, it can be deduced that the fourth strength theory condition is as follows:

$$\sqrt{{\displaystyle \frac{1}{2}}\left[{({\sigma }_1-{\sigma }_2)}^2+{({\sigma }_2-{\sigma }_3)}^2+{({\sigma }_3-{\sigma }_1)}^2\right]}\le {\displaystyle \frac{{\sigma }_s}{n_s}}=\left[\sigma \right]$$

(12)

where ${\sigma }_s$ is the yield stress, $[\sigma ]$ is the allowable stress, and $n_s$ is the safety factor.

5.3.2. Force Analysis Under Traction Connection Condition of Mechanical Arm

As the mechanical structure responsible for the traction connection between two ships, the traction arm will bear certain loads when ships enter and exit compartments. Furthermore, the arm will undergo certain deformations under loading; however, the size of the deformation is related to the safety and reliability of the designed arm. The following section analyzes the static mechanics of the arm. The force analysis of the arm is based on the cargo ship’s entry into the compartment.

According to the previous section, the traction force required for a group of ships entering the compartment was $T=5.51\times {10}^{4}N$. The following is a derivation of the forces at the arm joints based on the required traction force: The force analysis of the small arm is shown in Figure 26. The small arm is subjected to the force of the small-arm cylinder F_b, the force of the large arm F₂, the reaction force of the wrist ${F_3}^{\prime }$, and its own gravity G₂.

Using the conventional method, the load $F_2$ is solved as follows:

$${F_3}^{\prime }=\sqrt{{(G_3+G_4)}^2+{({\displaystyle \frac{1}{2}}T\cos {\theta }_1)}^2}$$

(13)

$$F_2\cos ({\theta }_4-{\theta }_3)+F_b\cos ({\theta }_b)={\displaystyle \frac{1}{2}}T\sin {\theta }_1$$

(14)

$$F_b\sin {\theta }_b+F_2\sin ({\theta }_4-{\theta }_3)=G_2+G_3+G_4$$

(15)

where ${\theta }_b$ is the angle between the small-arm cylinder and the ground, and $G_3$ and $G_4$ are the wrist and electromagnetic suction cup weights, respectively.

The force analysis of the large arm is shown in Figure 27, where it can be observed that the large arm is subjected to the force of the small-arm cylinder ${F_b}^{\prime }$, the force of the small-arm bar $F_a$, the force of the base $F_1$, the force of the small-arm cylinder ${F_b}^{\prime }$, and the gravity of the large arm itself $G_1$.

The loads applied are solved as follows:

$$F_1\cos {\theta }_2+{F_2}^{\prime }\cos {\theta }_2+{F_b}^{\prime }\cos {\theta }_b=F_a\cos {\theta }_a$$

(16)

$$F_1\sin {\theta }_2+{F_2}^{\prime }\sin {\theta }_2+{F_b}^{\prime }\sin {\theta }_b+G_1=F_a\sin {\theta }_a$$

(17)

where θ_a is the angle between the boom cylinder and the ground.

The force analysis of the base is shown in Figure 28. As is shown, the base was subjected to the force ${F_a}^{\prime }$ of the boom cylinder, the rising force ${F_1}^{\prime }$ of the boom, its own gravity $G_0$, and the support force. The loads are solved as follows:

From the action force, the reaction force relationship was obtained as $F_a={F_a}^{\prime }$ and $F_1={F_1}^{\prime }$.

The working conditions for each arm of the mechanical traction system during the first four stages of the ship’s compartment entry are listed in

Table 9. Traction mechanical arm of each part of the force situation.

Component		Force (N)
Lower arm	${F_3}^{\prime }$	33,916.08
	$F_b$	136,197.30
	$F_2$	−44,612.19
Maximal arm	$F_1$	70,162.69
	$F_a$	197,377.63
	${F_b}^{\prime }$	136,197.30
	${F_2}^{\prime }$	−44,612.19
Pedestals	${F_a}^{\prime }$	197,377.63
	${F_1}^{\prime }$	70,162.69

.

5.3.3. Finite Element Static Analysis

According to the calculation of the traction force and the derivation of the joint connection points, finite element static analysis was performed to verify the safety and reliability of the traction arm. In this study, static finite element analysis was completed using the simulation module in SolidWorks with respect to the key components of the traction arm. Considering the ship’s entry into the compartment as a working example, we set the material properties of the parts and select the design of the traction arm as “ordinary carbon steel”. The specific parameters are listed in

Table 10. Parameters of plain carbon steel.

Material	Poisson’s Ratio	Density $({\mathbf{kg}/\mathbf{m}}^3)$	Yield Strength $\left(\mathbf{Mpa}\right)$	Modulus of Elasticity $({\mathbf{N}/\mathbf{m}}^2)$
Plain Carbon Steel	0.28	7800	220.59	2.1 × 1011

.

The stress and deformation maps of each component of the traction robot arm under compartment entry conditions are shown in Figure 29 and Figure 30, and the maximum stress and strain values of each component are listed in

Table 11. Maximum stress and deformation values for each component.

Component	Maximum Stress Value (Mpa)	Maximum Deformation Value (mm)
Maximal arm	14.25	0.2454
Lower arm	35.58	0.4710

.

The results of the finite element analysis indicate that the maximum stress values of the traction arm, small arm, and key components remained below the yield limit of the ordinary carbon steel material used. Furthermore, the maximum deformation value was within the allowable range. Therefore, the stiffness and strength of the designed traction arm satisfied the design requirements.

6. Conclusions

In this study, a numerical simulation was performed for the entire process of a typical ship entering a downstream ship lift. Moreover, the influence of the ship lift channel’s shape on the drag performance and flow characteristics during the ship’s entry and exit from the lift compartment was analyzed. The hydrodynamic performances of a group of ships and a single ship entering the compartments were comparatively analyzed. The following conclusions were drawn:

(1)

The hydrodynamic conditions during a ship’s descent into a compartment were more complex due to the influence of the channel’s shape. When the ship was in the floating navigation wall section, due to the influence of the unilateral shore wall, the lateral force oscillation was most violent at this moment, and the ship easily deviated in this section. Therefore, attention must be given to the impact of lateral forces on the ship’s maneuverability during actual navigation, and the installation of collision avoidance devices along the floating navigation wall should also be considered.

(2)

As a typical ship in the ship lift compartment has a large water obstruction ratio and the water depth is very shallow, the ship travels to the ship lift compartment from the head of the upper gate according to a rated speed. Because of the change in the water depth in the waterway, the water level at the rear increases, and the water level at the front reduces.

(3)

According to the comparative analysis of hydrodynamic performance for different entry methods, in the section where the cargo ship enters the upper gate head and the ship lift compartment, the tugboat is significantly affected by the cargo ship due to changes in the water level within the channel. Thus, extra attention needs to be paid to safety in this section during actual operations.

Based on the above results, a mechanical traction scheme was investigated. Furthermore, comparisons and analyses were performed according to the site of the ship lift in order to propose an adapted traction method and formulate a mechanical traction technology scheme for ship entry and exit. In addition, based on the simulation analysis software, dynamic and static mechanical analyses of the ship traction mechanism were performed to analyze its engineering feasibility and safety. Furthermore, a theoretical analysis of the improvement in efficiency was conducted.