Comprehensive experimental study on the key performance of a semi-submersible aquaculture cage

Summary

The increasing demand for deep-sea aquaculture has spurred advancements in mariculture equipment. This study proposes an innovative semi-submersible aquaculture cage, for which a high-precision physical model was fabricated in accordance with Froude scaling laws. Comprehensive water tank tests were conducted, including towing tests, horizontal restoring stiffness tests, free decay tests, mooring force measurements, and hydrodynamic performance tests, to obtain the comprehensive performance and key parameters of the cage. More importantly, the influence of the netting on the cage’s performance was experimentally investigated, and results demonstrate that the system with netting significantly influences motion performance and load distribution, providing crucial design implications for enhancing the stability and operational safety of semi-submersible aquaculture installations in real marine environments.

Graphical abstract

Highlights

• •An innovative semi-submersible aquaculture cage was proposed
• •High-precision scaled-down model for comprehensive hydrodynamic parametric studies
• •Water tank experiments were conducted to obtain key hydrodynamic parameters
• •The effect of the netting on the system was studied

Applied sciences; Aquaculture

Introduction

The burgeoning global population has driven a surge in demand for protein. Finite terrestrial and nearshore resources are increasingly inadequate to meet human requirements.^1,2,3,4 Consequently, expanding marine aquaculture space and enhancing seafood production have emerged as crucial solutions to this challenge.^5,6,7 In addition, excessive stocking densities in coastal waters have led to increasingly apparent problems, including eutrophication, disease transmission, and sediment pollution. These issues constrain both the available space for aquaculture operations and the diversity of cultured species.^8,9 Hence, developing deep-sea aquaculture represents an imperative option to reconcile the human demand for both the quantity and quality of protein with the imperatives of ecological conservation.¹⁰ Accordingly, advancing offshore mariculture technologies and infrastructure has become a pivotal research focus in the development of deep-sea aquaculture.¹¹

Globally, numerous nations have proposed technical solutions for offshore mariculture equipment, progressively advancing the expansion of aquaculture into deeper waters. For instance, as early as 2004, Huon Aquaculture (Australia) designed and deployed the “Sampson,” an advanced aquaculture service barge.¹² However, such vessels face challenges including high technical complexity and elevated operational costs. In 2017, Norway launched the first semi-submersible offshore salmon farming platform in the world, Ocean Farm 1. It is designed to withstand typhoons at a category of 12 and can boast an annual production capacity of 15,000 tons within the designed aquaculture volume of $250,000{\mathrm{m}}^{\mathrm{3}}$.¹³ In 2018, China developed a fully submersible deep-sea aquaculture net cage “Deep Blue No.1,” which adapts its posture to diverse environmental conditions for extreme weather resilience, enabling the large-scale farming of cold-water fish species in temperate seas.¹⁴ In 2020, China commissioned the semi-submersible truss-girder intelligent deep-sea cage, Dehai No.1. Featuring a hybrid structure of floating bodies and trusses, it has withstood operational tests under typhoon conditions at a category of 17. Furthermore, countries such as Chile, Canada, Australia, New Zealand, and Brazil are actively advancing the development of offshore mariculture technologies.

Among various types of aquaculture equipment, semi-submersible platforms have garnered significant attention owing to their superior seakeeping performance, suitability for deployment and maintenance in abyssal environments, scalability, and high technological maturity.^15,16,17 To elucidate the fundamental hydrodynamic mechanisms governing these platforms under complex wind-wave-current excitations, researchers typically employ numerical simulations and physical model testing. These methodologies enable in-depth analysis of hydrodynamic responses and coefficients, thereby facilitating structural optimization to enhance platform reliability, while reducing design and manufacturing costs.^18,19,20

Numerical modeling represents a prevalent methodology in ocean engineering for simulating complex marine environmental challenges, offering significant advantages in mitigating engineering risks and costs.²¹ Liu et al.²² employed the diffraction theory and the Morison equation to numerically investigate hydrodynamic responses of semi-submersible platforms under regular waves, examining the effects of draft, wave period, and linear elastic coefficients. However, a maximum relative error of $17.2\%$ was observed between numerical and experimental data. In another study, Cheng et al.²³ utilized the Fhsim software to analyze mooring tensions and platform displacements in Norwegian fish farms during partial failures of mooring lines, proposing an autonomous method to identify broken lines and predict maximum residual tensions, with numerical-experimental deviations of approximately $7\%$. Tan et al.²⁴ designed a novel large-scale recreational fishery marine ranch system. Through preliminary calculations and simulations, the platform was demonstrated to meet the design objectives, which aimed to promote the integrated development of marine aquaculture and the tertiary industry, while addressing seawater pollution issues and supporting sustainable marine fisheries. Ma et al.²⁵ numerically analyzed the hydrodynamic characteristics of a multibody articulated floating aquaculture platform under regular waves and the effect of articulation point rotation on the platform stiffness using a boundary element method and integrated mass model, while simulation error of its mooring force was $18.53\%$. In conclusion, current research still faces significant challenges in accurately simulating complex models or operational conditions. These difficulties originate from several fundamental issues: an inadequate understanding of netting scale effects, insufficient insight into the deformation mechanisms of flexible structures, unclarified hydrodynamic response characteristics of netting in complex marine environments, and inaccuracies in the nonlinear modeling of moored cages. Hence, a more in-depth analysis of experimental data remains essential for optimizing mathematical models.

Scaled-down model experiments provide a conventional, intuitive, and accurate approach for validating the hydrodynamic response and mooring performance under controlled environmental conditions in a laboratory setting. The data acquired from the experiments, when correlated with numerical simulations, can support optimized aquaculture system design.^26,27 Combining numerical simulation based on potential flow theory with physical model tests, Jin et al.¹³ investigated the “Ocean Farm 1” platform under various wave conditions to assess its hydrodynamic responses. Tang et al.²⁸ summarized the hydrodynamic changes of an aquaculture cage when the mooring cables failed under irregular waves, and experimental results showed that the mooring force of the remaining cables reached up to twice the original value when one of the upstream mooring cables broke. The findings also point to an over-prediction of mooring line tension by the numerical model. Jiang et al.²⁹ experimentally investigated the platform motion response and the mooring force of an aquaculture integrated with a wind energy converter, and the results showed that the mooring system using a suspension chain line had a small motion response. However, the analysis neither encompassed a comprehensive set of parameters nor did it include a comparative study of scenarios with and without the netting. Y. Shen et al.³⁰ developed a new numerical tool to analyze the survival conditions of marine pastures and validated it through physical experiments. Despite achieving acceptable accuracy, many numerical methods based on model tests continue to face challenges like conservative results and a restricted coverage of working conditions. Thus, obtaining reliable solutions from numerical models remains essential on more comprehensive experimentation and in-depth analysis of the resultant data.

In this study, an innovative semi-submersible aquaculture cage (SSAC) was designed. Given the complexities of the highly nonlinear hydrodynamic and mooring behaviors in the SSAC, which are challenging to characterize and develop a numerical model, this study carried out the following work: a high-fidelity scaled-down model was successfully developed by integrating numerical modeling with experimental methods. Then, a wave tank experiment was designed to measure key parameters, including towing resistance, horizontal restoring stiffness, damping, mooring forces, and motion responses. In particular, the experimental program incorporated conditions both with and without netting, which remain a critical challenging to accurately simulate with existing numerical tools. The experimental data not only provide critical guidance for the design and construction of full-scale aquaculture installations but also establish an essential database for developing and validating high-fidelity numerical models in this field.

The paper is organized as follows. The section experimental model describes fabrication methods for the high-precision scaled-down prototype. The section water tank experiment setup delineates the water tank experiments. The ensuing discussion in results and discussion section encompasses a thorough examination of the outcomes of the physical experiments.

Experimental model

Scaled-down prototype

Figures 1A and 1B show the full-scale SSAC platform involved in this research that has been in commercial use, as well as the corresponding scaled-down prototype. The full-scale SSAC platform measures $100\mathrm{m}$ in length, $36\mathrm{m}$ in width, and has a working draft of $15\mathrm{m}$. Its specific dimensions are detailed in Table 1. To enhance resistance to wind and waves and prevent drifting, a four-point mooring system is employed, which consists of eight mooring chains arranged symmetrically. The scaled-down model was arranged in accordance with the full-scale prototype, as shown in Figure 3.

Figure 1.

SSAC

(A) The upwelling state.

(B) Semi-submersible state of the full-scale prototype.

(C) The scaled-down model fabricated in this paper.

Table 1.

Parameters of the full-scale SSAC and the corresponding scaled-down prototype

Parameter	Full-scale value	Scaled-down value $(\mathrm{m})$
Length	$100.00\mathrm{m}$	$2.22\mathrm{m}$
Width	$36.00\mathrm{m}$	$0.80\mathrm{m}$
Height	$27.00\mathrm{m}$	$0.62\mathrm{m}$
Designed draft	$15.00\mathrm{m}$	$0.33\mathrm{m}$
Mooring line	$350.00\mathrm{m}$ in length, $105.00\mathrm{m}\mathrm{m}$ in diameter	$7.78\mathrm{m}$ in length, $2.0\mathrm{m}\mathrm{m}$ in diameter

Figure 3.

Schematic of experimental setup in water tank under wave action

(A) Side view of the experiment setting.

(B) Top view of the experiment setting.

The netting requires careful consideration. The netting is installed on both the outer and inner frames, as shown in Figure 1C. The side netting of the scaled model has an interlayer spacing of approximately 67 mm and a netting size of about 508 $\times$ 300 mm, the front and rear netting has an interlayer spacing of approximately 67 mm and a netting size of about 300 $\times$ 300 mm, and the bottom netting has an interlayer spacing of approximately 78 mm and a netting size of about 667 $\times$ 489 mm. The netting is constructed from the ultra-high-molecular-weight polyethylene material that has a density of 950 $\mathrm{k}\mathrm{g}/m^3$. This study primarily adheres to the Froude scaling law. If following up the original designed size of the full-scale netting (net opening size of 8 cm, twine diameter of 5 mm), it would be extremely difficult to procure a model-scale net (net opening size of 1.77 mm, twine diameter of 0.11 mm) with geometrically similar mesh openings and twine diameters. However, it is common to replace with different types according to the fish species and their specific growth stages in practical aquaculture cage. Thus, the use of a non-strictly scaled netting (net opening size of 2.5 mm, twine diameter of 0.17 mm) in the model remains operationally representative and does not undermine the validity of the experimental findings. The netting used in this paper, although not scaled from the original full-scale model version, represents an appropriate size that can be adopted during SSAC operation. Therefore, for the full-scale model corresponding to the scaled-down model used in physical experiments, the test results are precise. However, it should be noted that, netting with different mesh sizes and twine diameters may influence the added mass and damping coefficients, which could further affect other relevant data such as response amplitude operators (RAOs), mooring force, and natural frequencies.

With the exception of the iron mooring chain connectors, the entire model was fabricated from cedar wood—a timber characterized by its rot resistance, low water absorption, and a density of approximately $450\mathrm{k}\mathrm{g}/m^3$. The model was then waterproofed using epoxy resin adhesive, paint, and body filler.

Parameter calibrating

To meet the Froude scaling rule for model testing, it is necessary to ensure that the weight, center of gravity, and moment of inertia of the scaled-down prototype are consistent with those of the full-scale platform. Since the scaled-down prototype is fabricated using wood, it is necessary to add ballast blocks of a certain mass inside the columns for adjustment and to measure and adjust the center of gravity and moment of inertia until the parameters are consistent with those of the full-scale aquaculture cage.

When measuring and calculating parameters such as the center of gravity and moment of inertia of the prototype, an inertia test rig is used for measurement, as shown in Figure 2. In the fabricated scaled-down model, 6 ballast tanks, each with dimensions of 191.11$\times$ 26.67$\times$ 300 $\mathrm{m}\mathrm{m}$, were reserved to adjust the center of gravity and moment of inertia, as shown in Figure 2A. A numerical model was then constructed based on the physical model to address the mass allocation problem of the ballast system, from which an optimal solution was derived. Following this optimized distribution, primary lead ballast blocks (Figure 2F) were installed. Should the results fail to meet the required standards, the ballast (iron sand/plates) was manually adjusted by adding or removing material based on real-time calculations. This iterative calibration process was repeated until the deviation in the system’s moment of inertia was reduced to within the acceptable margin of error. Finally, to prevent ballast shifting, the entire ballast tank was filled with foam material and sealed with epoxy resin. The prototype is placed on the test rig, where the longitudinal section of the prototype is perpendicular to the base plane of the inertia test rig and the center of gravity of the prototype is on the same vertical plane as the center point of the knife edge. A certain external force $F$ is applied to the inertia test rig to yield a slight deflection in the prototype and the inertia test rig. Based on the principle of torque equilibrium, the height of the center of gravity of the prototype $Z_C$ from the baseline is calculated as follows:

$$Z_C=\frac{M_D}{M_C}(H-Z_D)+H-\frac{Fd}{M_D\tan \theta },$$ (Equation 1)

where $Z_D$ is the height of the center of gravity of the inertia test rig from the baseline, $M_D$ is the weight of the inertial test rig, $M_c$ is the weight of the SSAC model, $H$ is the vertical distance between the baseline of the inertial test rig and the cutting edge, $d$ is the horizontal distance between the point of application of constant force $F$ and the cutting edge, and $\theta$ is the deflection angle of the inertial frame under the action of constant force $F$.

Figure 2.

Moment of inertia calibration experimentMoment of inertia Calibration Experiment.

(A) Schematic diagram of ballast tanks’ location and dimensions.

(B and C) Ballast system employed.

(D and E) Experimental setup for adjusting rotational inertia.

(F) Fully filled ballast tank.

The moment of inertia is calculated using the principle of small-angle approximation. The differential equation for the small-angle approximation of the inertia frame around the SSAC is:

$$\overline{\phi }+\frac{M_D\alpha }{I_0}\phi =0,$$ (Equation 2)

where $I_0$ is the inertia of the inertial measurement frame relative to the knife edge.

Suppose the time taken for the prototype and the inertia test rig to rotate $n$ cycles around the knife edge is measured as $t$. In that case, the moment of inertia of the entire oscillation system relative to the knife edge is:

$$I_1=g(M_{D}a+M_{C}b){\left(\frac{t}{2n\pi }\right)}^2,$$ (Equation 3)

where $a$ is the vertical distance from the center of gravity of the inertia test rig to the knife edge and $b$ is the vertical distance from the center of gravity of the prototype to the knife edge, which is on the same vertical line as the center of gravity of the inertia frame.

According to the parallel axis theorem, the moment of inertia of the model rotating around its center of gravity axis can be obtained as:

$$I_C=I_1-I_0-M_C{b}^2.$$ (Equation 4)

Under the optimal ballast distribution scheme, the center of gravity and moment of inertia of the scaled-down prototype were obtained, as shown in Table 2.The relative error between the actual calculated values and reference values for all test parameters was controlled within $1.30\%$, with the relative error for the total mass being $0.63\%$ and the relative errors for the moments of inertia $I_{xx}$ and $I_{yy}$ being $1.30\%$ and $1.20\%$, respectively. These data confirm the effectiveness of the ballast scheme in controlling mass distribution and, importantly, provide a reliable physical foundation for subsequent high-fidelity fluid dynamics experiments. The precision level fully meets the stringent requirements of marine engineering scaled experiments, providing a solid experimental basis for full-scale performance prediction.

Table 2.

Results of mass characteristics of SSAC scaled-down prototype under optimal ballast scheme

Parameters	Theoretical value	Measurement value	Relative error height
Total mass $(\mathrm{k}\mathrm{g})$	89.613	89.051	$0.63\%$
Height of center of gravity above baseline $(\mathrm{m})$	0.164	0.162	$1.14\%$
Moment of inertia $I_{xx}$$({\mathrm{k}\mathrm{g}\mathrm{m}}^2)$	7.352	7.446	$1.30\%$
Moment of inertia $I_{yy}$$({\mathrm{k}\mathrm{g}\mathrm{m}}^2)$	51.691	51.075	$1.20\%$

Water tank experiment setup

Water tank

The SSAC model experiments were performed in a water tank (Figure 3) at the South China University of Technology (SCUT). The experimental facility measures $120.0\mathrm{m}$ in length, $8.0\mathrm{m}$ in width, and $4.0\mathrm{m}$ in depth, featuring a sloped energy-dissipating grid installed at the downstream end to attenuate wave reflection effects. The water tank is equipped with several advanced systems: (1) a hinged-flap wave maker capable of generating both regular and irregular wave conditions, (2) a carriage towing system, and (3) high-precision wave probe for spatial-fixed wave elevation measurements.

Towing test setup

The measurement of towing force enables the quantitative assessment of fluid resistance acting on a physical model across varying flow velocities (or Reynolds numbers), facilitating analysis of the relationship between drag, velocity, object geometry, and other influencing factors. The full-scale model in this study was initially constructed and assembled onshore before being transported via tugboat to the deployment site for installation. The towing experiments were specifically designed to evaluate the required design parameters of the tugboat, ensuring stable operation under combined wave and current conditions. For the full-scale model, the towing tests were conducted at a design speed of 4 knots with a significant wave height of $2\mathrm{m}$. Applying the appropriate Froude scaling laws, the corresponding parameters for the scaled-down model were derived as a towing speed of $0.3067\mathrm{m}/\mathrm{s}$ and a significant wave height of $0.044\mathrm{m}$, at which towing force measurements were performed.

The mooringless system model was initially positioned at the upstream end of the water tank. The device was securely mounted to the towing carriage, and a tension-compression load cell was connected to measure towing forces as shown in Figure 4. Following wave generation by the wavemaker system, a stabilization period was observed to ensure consistent wave conditions. The carriage was then driven at a constant velocity from the wave-absorbing (downstream) end toward the wavemaker (upstream) end. During the towing, the tension-compression load cell recorded real-time drag forces acting on the scaled-down model, while a GoPro camera captured the hydrodynamic response of the system. The primary measurement objective was the hydrostatic towing force required to maintain the model’s motion at the prescribed speed under wave loading. The following relationship exists between the tension-compression load cell indication $F_c$ and the hydrostatic towing force $F_r$ due to the existence of an angle $\alpha$:

$$F_r=F_c\cos \alpha .$$ (Equation 5)

Figure 4.

Schematic diagram of towing test setup

(A) Side view of the experiment setting.

(B) Top view of the experiment setting.

Horizontal restoring stiffness test setup

The coupled dynamics of a moored SSAC in waves is governed by the following equation of motion in six degrees of freedom (DOFs)³¹:

$$[\mathbf{M}+\mathbf{A}(\omega )]\ddot{\mathit{\xi }}(t)+\mathbf{B}(\omega )\dot{\mathit{\xi }}(t)+\mathbf{C}\mathit{\xi }(t)+{\mathbf{F}}_{\text{moor}}(\mathit{\xi },\dot{\mathit{\xi }})={\mathbf{F}}_{\text{wave}}(t)+{\mathbf{F}}_{\text{ext}}(t),$$ (Equation 6)

where $\mathit{\xi }(t)={[x,y,z,\varphi ,\theta ,\psi ]}^{\text{T}}$ is the 6-DOF displacement vector, $\mathbf{M}$ is the structural mass/inertia matrix, $\mathbf{A}(\omega )$ is the frequency-dependent added mass matrix, $\mathbf{B}(\omega )$ is the potential damping matrix, $\mathbf{C}$ is the total system restoring stiffness matrix, ${\mathbf{F}}_{\text{moor}}(\mathit{\xi },\dot{\mathit{\xi }})$ is the nonlinear restoring force from the mooring system, ${\mathbf{F}}_{\text{wave}}(t)$ is the wave excitation force (including first- and second-order components), and ${\mathbf{F}}_{\text{ext}}(t)$ includes other external forces (current, wind, etc.).

The total system restoring stiffness matrix $\mathbf{C}$ of the system comprises two components: hydrostatic stiffness ${\mathbf{C}}_{\mathbf{h}}$ and mooring stiffness.³² This study primarily focuses on the horizontal stiffness, which characterizes the core capacity of the system to resist quasi-static environmental loads and maintain station-keeping, and also determines its horizontal natural frequency. In the $x$ and $y$ directions, the ${\mathbf{C}}_{\mathbf{h}}$ of the moored SSAC is $\mathbf{0}$; consequently, the stiffness matrix is predominantly constituted by the horizontal restoring stiffness ${\mathbf{K}}_{\mathbf{h}}$ (horizontal mooring stiffness).

Under low-frequency environmental loads $F_{\text{slow}}$ (primarily second-order wave drift forces), the system’s response is approximately quasi-static.³³ The horizontal restoring stiffness $K_h$ directly governs the mean offset position under sustained environmental forcing, with the steady-state displacement $X_{\text{static}}$ given by

$$X_{\text{static}}\approx \frac{F_{\text{slow}}}{K_h}.$$ (Equation 7)

Under dynamic loading, the horizontal restoring stiffness $K_h$ determines the natural frequency of the system in the horizontal direction, ${\omega }_n$, through the following relationship:

$${\omega }_n=\sqrt{\frac{K_h}{M+A}}.$$ (Equation 8)

When the frequency of the slow-drift forces approaches the natural frequency of the system ${\omega }_n$ (as dictated by $K_h$), severe slow-drift resonance can be triggered.³⁴ Within the wave-frequency range, the horizontal restoring stiffness $K_h$, together with the inertia terms, fundamentally sets the horizontal natural frequency of the system ${\omega }_n$. Furthermore, in the complete 6-DOF equations of motion, off-diagonal coupling terms in the restoring stiffness matrix may involve contributions from $K_h$, potentially exciting pitch or roll motions. Therefore, accurate measurement of horizontal restoring stiffness of the system is crucial for preventing drift instability, suppressing resonant responses, and mitigating the risk of mooring system failure.³⁵

A measurement system employing pulleys and calibrated weights was designed to measure the horizontal restoring stiffness of the model in both the x and y directions, the experimental setup followed the schematic diagram shown in Figure 5. At the beginning of the test, the scaled-down model was positioned in the water tank and connected to the mooring system, ensuring calm water conditions and an equilibrium state. The initial position of the lightweight steel wire rope was recorded when it was in a state of tautness. A calibrated weight was then attached to the free end of the wire rope and gradually lowered to apply a constant tensile force. Under this force, the model shifted until reaching a new equilibrium position, which was recorded to determine the displacement distance. The horizontal restoring stiffness of the scaled-down model was subsequently calculated. After completing each test, the model was reset to its equilibrium position, and the next test commenced only after water conditions stabilized. The stiffness $k$ of the SSAC with mooring system is determined via Hooke’s law,³⁶ establishing a relationship between the experimentally observed displacement $x_k$, the mass of the applied calibrated weight $m_k$, and the restoring force, as follows:

$$k=\frac{m_k\cdot g}{x_k},$$ (Equation 9)

where, $g$ is the gravitational acceleration (9.81 m/s²). Horizontal restoring stiffness serves as a primary metric for quantifying a system’s intrinsic capacity to restore its equilibrium following external disturbances, thereby providing a fundamental characterization of its restorative properties.³⁵

Figure 5.

The principle of the horizontal restoring stiffness test

(A) The horizontal restoring stiffness test in the x direction.

(B) The horizontal restoring stiffness test in the y direction.

Calm water free decay test setup

Free decay test serves as a fundamental methodology for assessing the dynamic performance of marine ranch structures, primarily employed to evaluate their dynamic response characteristics. This approach enables the measurement of the natural frequency of the SSAC in the absence of external excitations (e.g., waves, wind), thereby preventing resonance risks. By analyzing the amplitude decay rate in the free decay curve, the structural damping properties can be quantified, providing critical insights into motion stability.

Four test conditions were conducted: without netting and without mooring, with netting and without mooring, without netting and with mooring, and with both netting and mooring. The IMUs (inertial measurement units) were installed and calibrated at the top of both fore and aft columns of the prototype, as shown in Figure 6. Upon achieving quiescent water conditions, an initial inclination of 5–10° (difficult to control precisely but can be measured using an IMU) was applied to the model to initiate free decay oscillations. The tests were conducted sequentially: first in the roll direction and then, after the water surface calmed once more, in the pitch direction.³⁶ The IMUs, sampling at 10 Hz, recorded the time series of roll and pitch motions, while the damping coefficients and natural frequencies were derived through the Froude method. The time-series motion of the SSAC is recorded through a free decay test. The adjacent amplitude peaks $A_1$ and $A_2$ are measured and then the logarithmic decrement $\delta$ is computed as follows:

$$\delta =\ln \left(\frac{A_1}{A_2}\right).$$ (Equation 10)

Figure 6.

Experimental setup of scaled-down model and sensors

The damping ratio $\zeta$ is givenby:

$$\zeta =\frac{\delta }{2\pi }.$$ (Equation 11)

The damping coefficient $b$ is defined as follows^32,36:

$$b=2\zeta \sqrt{m_{\text{eff}}C}.$$ (Equation 12)

Mooring force and hydrodynamic response test setup

The measurement of mooring force and hydrodynamic responses is crucial for preventing anchor chain failure, optimizing mooring design, assessing motion stability, avoiding resonance, and extending service life. These measurements are crucial for ensuring structural safety, stability, and optimal aquaculture performance. The SSAC model was positioned $70.22\mathrm{m}$ from the wavemaker at the center of the water tank as shown in Figure 3. The mooring system was symmetrically arranged about the center of the model in the x direction, with fore and aft mooring lines deployed $7.88\mathrm{m}$ from the model. Waves were generated along the −x direction of the SSAC model.

The experiment utilized DYMH-103 tension-compression load cells with a capacity of 80 kg and a sensitivity of 1.0–1.5 mV/V, as shown in Table 3. These sensors were properly conditioned, including calibration of the sampling frequency and electromagnetic shielding. Ten tension-compression load cells were used to measure mooring forces: five were installed at the upper end of the mooring chains (near the water surface) and five were installed at the lower end (near the water tank bottle). Considering the higher loads on the weather-side mooring chains under wave excitation, two were installed on each of the four windward mooring chains. The specific arrangement of the tension-compression load cells is shown in Figure 3. The carriage, positioned above the model, was equipped with wave probes, sensor processing units, and a laptop.

Table 3.

Parameters of the DYMH-103 tension-compression load cell

Parameter	Specification
Range	80 kg
Rated output	1.5 mVV−1$\pm$ 10%
Zero output	$\pm$ 2% F.S.
Nonlinearity	0.3% F.S.
Hysteresis	0.1% F.S.
Repeatability	0.1% F.S.
Creep (30 min)	0.2% F.S.
Temperature effect on span	0.1% F.S. per 10°C
Temperature effect on zero	0.1% F.S. per 10°C
Response frequency	10 kHz
Material	Stainless steel
Impedance	350$\mathrm{\Omega }$
Insulation resistance	$\ge$5,000$\text{M}\mathrm{\Omega }$/100VDC
Excitation voltage	0–10 V
Compensated temperature range	−20°C to 80°C
Safe overload	150%
Ultimate overload	200%
Cable specification	$\varphi$3 $\times$ 2 m
Cable tensile strength	10 kg

F.S., Full Scale

The full-scale prototype is situated in the South China Sea, a region prone to typhoons.³⁷ Consequently, verifying the survival of the structure under extreme conditions became the paramount design objective. Therefore, based on the wave characteristics of the deployment site—featuring large wave heights and long periods—regular wave tests were conducted with wave heights set at 0.2 m and 0.25 m. For each wave height, experiments were carried out using five distinct wave periods: 1.0 s, 1.5 s, 1.75 s, 2.0 s, and 2.5 s. Tests began after the water surface became calm. During the tests, IMU data, wave probe data, and incident wave data were recorded. The roll and pitch RAOs of the model were obtained by processing these data according to the definition of RAO. The tests compared mooring forces and hydrodynamic responses between netted and net-free models, providing essential references for studying the influence of netting on the model.

Results and discussion

Towing test results

This study investigates the influence of varying wave periods (T = 1 $\mathrm{s}$, 1.5 $\mathrm{s}$, 2.0 $\mathrm{s}$) on the towing force of an SSAC model under a constant wave height of 0.044 $\left(\mathrm{m}\right)$, as illustrated in Figure 7. Experimental results demonstrate that the towing force ranges from 30 to $50\mathrm{N}$ at T = $1.0\mathrm{s}$, increases to 20 to $55\mathrm{N}$ at T = $1.5\mathrm{s}$, and further expands to 15 to $80\mathrm{N}$ at T = $2.0\mathrm{s}$. The findings reveal a significant positive correlation between wave period and both the magnitude and variability range of towing forces. Specifically, due to experimental constraints, the current dataset exhibits certain limitations in quantitative analysis. Nevertheless, its primary contributions lie in (1) validating the order-of-magnitude reliability of numerical simulations and (2) providing theoretical guidance for towing equipment selection in engineering practice. Subsequent research should employ more precise experimental designs to verify these preliminary findings.

Figure 7.

The towing forces under a wave height of 0.2 m

(A) Wave excitation period of 1.0 s.

(B) Wave excitation period of 1.5 s.

(C) Wave excitation period of 2.0 s.

Horizontal restoring stiffness

To compare the horizontal restoring stiffness of the SSAC system with and without netting, horizontal restoring stiffness tests were first conducted in the x direction (i.e., the wave propagation direction). In this test, calibrated weights of $2.00\mathrm{k}\mathrm{g}$, $4.00\mathrm{k}\mathrm{g}$, $5.00\mathrm{k}\mathrm{g}$, $6.00\mathrm{k}\mathrm{g}$, and $7.00\mathrm{k}\mathrm{g}$ were applied incrementally. Multiple sets of repeated tests were performed, with each set consisting of three valid experimental measurements. The average value of these three measurements was adopted as the final result. The x direction horizontal restoring stiffness test results are presented in Table 4.

Table 4.

Horizontal restoring stiffness in x direction for the SSAC scaled-down model

Weight (kg)	Displacement (cm)	$k$(N/m)	Mean $k$(N/m)
Without netting
2.00	40.60	49.26	50.29
	40.50	49.38
	38.30	52.22
4.00	65.30	61.26	61.30
	64.20	62.31
	66.30	60.33
5.00	74.30	67.29	65.96
	75.40	67.31
	77.80	64.27
6.00	87.00	68.97	70.38
	84.30	71.17
	84.50	71.01
7.00	91.30	76.67	76.26
	93.30	75.03
	90.80	77.09
With netting
2.00	46.60	42.92	41.70
	50.00	40.00
	47.40	42.19
3.00	77.80	38.56	39.90
	73.10	41.04
	75.00	40.00
4.00	93.20	42.92	42.48
	94.20	42.46
	95.10	42.06
5.00	96.80	51.56	50.32
	101.10	49.46
	117.30	59.67

The experiments show that the horizontal restoring stiffness of the scaled-down SSAC model without netting in the x direction exhibits a nonlinear increasing trend with increasing weight (Table 4). Specifically, when the applied mass increases from $2\mathrm{k}\mathrm{g}$ to $7.00\mathrm{k}\mathrm{g}$, the stiffness coefficient $(k)$ rises from $50.29\mathrm{N}/\mathrm{m}$ to $76.26\mathrm{N}/\mathrm{m}$, representing a $51.6\%$ increase, indicating significant weight-dependent stiffness variation. However, the stiffness growth rate gradually decreases with higher weight (e.g., a $21.9\%$ increase in k between 2.00 and $4.00\mathrm{k}\mathrm{g}$, compared to only $15.6\%$ between 5.00 and $7.00\mathrm{k}\mathrm{g}$). This trend suggests material approach toward elastic limits and increasing geometric nonlinearity effects.

A comparative analysis of models with and without netting reveals that the netting significantly reduces system stiffness. Under a $2.00\mathrm{k}\mathrm{g}$ weight, the netted model exhibits an averaged $k$ value of $41.70\mathrm{N}/\mathrm{m}$, $17.1\%$ lower than the case without netting. This reduction is attributed to weight redistribution and delayed structural recovery induced by the netting.

Comparison with the linear spring model shows good agreement at low loads (2.00–$4.00\mathrm{k}\mathrm{g}$), where the experimental values of $k$ deviate by $<10\%$ from theoretical predictions, consistent with small-deformation assumptions. However, at higher loads $(>5\mathrm{k}\mathrm{g})$, discrepancies increase (e.g., a $12.3\%$ lower stiffness at $7.00\mathrm{k}\mathrm{g}$ compared to linear extrapolation).

Upon completion of the aforementioned tests, another experiment was conducted to evaluate the horizontal restoring stiffness (y direction) of the SSAC with and without netting. Given the heightened sensitivity of the structure to motion in this direction, the test employed different counterweight configurations: weights of 1.00, 1.2367, 1.3557, and $1.4747\mathrm{k}\mathrm{g}$ were selected for the netting-free condition, while 0.80, 1.00, 1.30, and $1.40\mathrm{k}\mathrm{g}$ weights were used for the netting condition. Multiple repeated tests were performed, with three valid experimental results selected from each group for averaging.

As presented in Table 5, the horizontal restoring stiffness $(k)$ of the SSAC without netting in the y direction exhibited a nonlinear trend of initial decline followed by stabilization under increasing load. When the counterweight increased from 1.1 to $1.2367\mathrm{k}\mathrm{g}$, the value of $k$ decreased from 22.19 to $17.26\mathrm{N}/\mathrm{m}$ (a $22.2\%$ reduction), potentially attributable to the relaxation of initial pre-tension or localized contact nonlinearity. Within the 1.3557 to $1.4747\mathrm{k}\mathrm{g}$ load range, the value of $k$ stabilized between 17.50 to $17.96\mathrm{N}/\mathrm{m}$.

Table 5.

Horizontal restoring stiffness in y direction for the SSAC scaled-down model

Weight (kg)	Displacement (cm)	$k$ (N/m)	Mean $k$ (N/m)
Without netting
1.0000	50.00	20.00	22.19
	44.50	22.47
	41.50	24.10
1.2367	70.50	17.54	17.26
	71.50	17.30
	73.00	16.94
1.3557	75.50	17.96	17.96
	74.50	18.20
	76.50	17.72
1.4747	82.50	17.88	17.50
	86.50	17.05
	84.00	17.56
With netting
0.80	43.00	18.60	19.63
	41.00	19.51
	38.50	20.78
1.00	60.00	16.67	16.59
	63.00	15.87
	58.00	17.24
1.30	76.00	17.10	17.29
	75.50	17.22
	74.10	17.54
1.40	80.50	17.39	17.39
	82.50	16.96
	77.40	18.09

Comparative analysis of netting configurations revealed that both conditions demonstrated relatively high initial stiffness, likely due to similar pre-tension relaxation or contact nonlinear effects. At higher weight cases, the values of $k$ $(17.39\mathrm{N}/\mathrm{m})$ of the SSAC with netting approached that of the netting-free condition ($17.50\mathrm{N}/\mathrm{m}$ at $1.4747\mathrm{k}\mathrm{g}$). Notably, the values of $k$ (17–$22\mathrm{N}/\mathrm{m}$) in the y direction were significantly lower than x direction measurements (50–$76\mathrm{N}/\mathrm{m}$), demonstrating strong anisotropic characteristics in the SSAC structure.

Calm water free decay

The free decay tests of the SSAC scaled-down model were conducted with multiple sets of measurements, recording the time-series curves of the model during roll and pitch motions, as illustrated in Figure 8. The exponential characteristics of the free decay under mooring conditions for the scaled-down model SSAC without netting were notably evident. Two sets of valid experimental data were selected for in-depth analysis. Using the Froude method, the mean values of the natural period and frequency were calculated, and the damping coefficients corresponding to these conditions were subsequently derived.

Figure 8.

With mooring condition

(A) Calm water decay of roll motion with netting.

(B) Calm water decay of pitch motion with netting.

(C) Calm water decay of roll motion without netting.

(D) Calm water decay of pitch motion without netting.

Effect of netting

Under no mooring conditions (Table 6), the addition of netting significantly increased the roll damping coefficient from 0.181 N$\cdot$ s/m to 0.522 N$\cdot$ s/m, representing a 188% rise. In moored conditions, the netting likewise elevated roll damping from 0.239 N$\cdot$ s/m to 0.504 N$\cdot$ s/m, a 111% increase. In contrast, the netting’s effect on pitch damping was less pronounced, with increases of 109% in no mooring and only 2.6% in moored configurations. The influence on natural frequencies was minimal, with all changes remaining below 3.5%. These results indicate the netting serves as the dominant contributor to roll damping in the model.

Table 6.

Free decay results

	Roll		Pitch
	Case 1	Case 2	Case 1	Case 2
Without netting/without mooring
Natural period (s)	1.476	1.587	1.611	1.763
Natural frequency (rad/s)	4.257	3.959	3.900	3.564
Mean value of natural frequency (rad/s)	4.108		3.732
Damping coefficient	0.200	0.162	0.110	0.204
With netting/without mooring
Natural period (s)	1.481	1.488	1.713	1.703
Natural frequency (rad/s)	4.241	4.223	3.667	3.689
Mean value of natural frequency (rad/s)	4.232		3.678
Damping coefficient	0.652	0.391	0.259	0.399
Without netting/without mooring
Natural period (s)	1.613	1.560	1.773	1.8928
Natural frequency (rad/s)	3.897	4.028	3.543	3.320
Mean value of natural frequency (rad/s)	3.962		3.431
Damping coefficient	0.325	0.153	0.259	0.431
With netting and mooring
Natural period (s)	1.500	1.534	1.835	1.814
Natural frequency (rad/s)	4.189	4.096	3.425	3.464
Mean value of natural frequency (rad/s)	4.142		3.445
Damping coefficient	0.523	0.485	0.320	0.388

Effect of mooring

In the absence of netting, the mooring system produced a substantial increase in pitch damping, from 0.157 N$\cdot$ s/m to 0.345 N$\cdot$ s/m, corresponding to a 120% enhancement. With netting installed, mooring raised pitch damping from 0.329 N$\cdot$ s/m to 0.354 N$\cdot$ s/m. Regardless of netting presence, the mooring system notably reduced the natural frequencies of the platform—by an average of 3.5% in roll and 7.5% in pitch—consistent with its function of providing additional restoring stiffness through elastic restraint. This demonstrates that the mooring system primarily governs pitch damping.

Combined effect of netting and mooring

When both netting and mooring are applied, the system experiences only minor shifts in roll and pitch natural frequencies. However, substantial increases in both pitch and roll damping are observed, indicating that the combined configuration improves the overall hydrodynamic stability of the system.

Mooring force and hydrodynamic response

Mooring force

Tests were conducted using two wave heights of $0.20\mathrm{m}$ and $0.25\mathrm{m}$, with all test periods ranging from 1.0 to $2.5\mathrm{s}$. Comparative experiments were performed for both with and without netting configurations. The experimental data (Figures 9 and 10) reveal that the initial mooring forces recorded by each sensor reflect the pretension state of the anchor chains. Notably, sensor no. 1 registered a negative initial value $(-1.8\mathrm{N})$, which may be attributed to calibration deviation. However, this sensor maintained stable readings throughout the tests and subsequent troubleshooting confirmed its data reliability, rendering its impact on overall results negligible.

Figure 9.

Mooring force of the model without netting under the wave height of $0.2\mathrm{m}$

(A) Wave excitation period of $1.0\mathrm{s}$.

(B) Wave excitation period of $1.5\mathrm{s}$.

(C) Wave excitation period of $1.75\mathrm{s}$.

(D) Wave excitation period of $2.0\mathrm{s}$.

(E) Wave excitation period of $2.5\mathrm{s}$.

(F) Trends in peak-to-peak amplitude (PPA) of mooring forces.

Figure 10.

Mooring force of the model with netting under the wave height of 0.20 m

(A) Wave excitation period of 1.0 s.

(B) Wave excitation period of 1.5 s.

(C) Wave excitation period of 1.75 s.

(D) Wave excitation period of 2.0 s.

(E) Wave excitation period of 2.5 s.

(F) Trends in PPA of mooring forces.

Under wave excitation with a wave height of $0.20\mathrm{m}$, the mooring system without netting exhibited pronounced dynamic response characteristics. Except for the $1.0\mathrm{s}$ wave period case, sensor nos. 4, 6, and 8 displayed typical sinusoidal responses, indicating that these fairlead-positioned chains on the weather side imposed periodic restraining forces on the model. Sensor no. 2 (also on the weather side) showed minimal response, directly correlated with the tension state of its adjacent chains (nos. 7 and 8)—when chain nos. 7 and 8 were taut, chain no. 2 tended to slacken. Sensor no. 6 recorded the maximum dynamic load, while force fluctuations in other sensors remained below $10\mathrm{N}$, demonstrating that the inner-weather-side anchor chain bore the primary wave-induced loads.

In the test with an excitation wave height of $0.25\mathrm{m}$, similar to the results from the $0.20\mathrm{m}$ wave height group, the mooring system without netting did not exhibit distinct sinusoidal responses under $1.0\mathrm{s}$ wave excitation. However, sensor nos. 4, 6, and 8 demonstrated clear sinusoidal responses in other cases. Under a wave period of $1.5\mathrm{s}$, the tension fluctuation recorded by sensor no. 6 reached approximately $15\mathrm{N}$, representing the maximum measured value in this test series.

For the test of the mooring system with netting under $0.20\mathrm{m}$ wave height, significant data fluctuations were observed in sensor nos. 2, 3, 4, 6, and 10. Unlike the previously mentioned sensor nos. 2, 4, 6, and 8 that exhibited sinusoidal responses, sensor no. 8 at the weather side fairlead showed no significant fluctuations, with sensor malfunction having been ruled out. Both sensors nos. 3 and 4 were positioned on the same outer anchor chain on the weather side, with sensor no. 3 demonstrating consistently greater fluctuations than sensor no. 4. The measured values from sensor no. 6 were substantially higher than those from sensor no. 2 on the opposite weather side; sensor no. 10 displayed notable tension characteristics, although its response pattern more closely resembled a half-sinusoidal wave (showing only positive values). Summarizing the above factors and combining them with the response of the system during the experiments, in this set of experiments, the model showed an attitude shift toward the side of sensor no. 2 and the anchor chain where sensor no. 8 was located was in a relaxed state. The fluctuation values of the tension data of the sensors are less than 10 N, except for the fluctuation of the tension indication of sensor no. 3, which will be more than $10\mathrm{N}$ but less than $15\mathrm{N}$.

The experimental results for the mooring system with netting under the wave height of $0.25\mathrm{m}$ are presented in Figure 11. Similar to the $0.20\mathrm{m}$ wave height tests, significant data fluctuations were observed in sensors nos. 2, 3, 4, 6, and 10, although not all sensor responses exhibited sinusoidal characteristics. Consistent with the system response observed during testing, the model again demonstrated attitude deviation toward the sensor no. 2 side, with the mooring line containing sensor no. 8 remaining slack. The net-equipped box model returned to its neutral position, resulting in the disappearance of tension fluctuations in sensor nos. 3, 4, 6, and 8, which exhibited sinusoidal responses, indicating their respective mooring lines actively restrained model motion by applying tensile forces during wave excitation. These responding sensors shared the common characteristic of being located near the water surface at the hinged connections on the weather side. Sensor no. 2 at the weather side fairlead showed no significant data fluctuations in these tests. All sensors recorded tension variations below 10 N under $0.25\mathrm{m}$ wave height with wave periods ranging from 1.0 to $2.5\mathrm{s}$.

Figure 11.

Mooring force of the model with netting under the wave height of 0.25 m

(A) Wave excitation period of 1.0 s.

(B) Wave excitation period of 1.5 s.

(C) Wave excitation period of 1.75 s.

(D) Wave excitation period of 2.0 s.

(E) Wave excitation period of 2.5 s.

(F) Trends in PPA of mooring forces.

The comparative experimental results lead to the following key conclusions. (1) Multi-point mooring forces demonstrate significant sensitivity to model attitude. The variation in the number of active load-bearing sensors between net-equipped and netless configurations, caused by minor attitude deviations, suggests that mooring system load distribution can be optimized through precise attitude control. (2) Sensor no. 6 peaked at $15\mathrm{N}$ ($1.5-\mathrm{s}$ period) without netting, while all sensors fluctuated $<10\mathrm{N}$ with netting; presumably netting structure dissipated wave energy and suppressed extreme loads. (3) Under short-period wave conditions $(1.0\mathrm{s})$, both configurations exhibited non-sinusoidal responses, indicating that high-frequency waves induce chaotic motions and prevent the mooring system from reaching a steady state. (4) The half-sinusoidal response (positive-only values) observed at sensor no. 10 suggests intermittent tensioning of the mooring chain, demonstrating unidirectional impact loads that may correlate with vortex-induced transient forces generated by the net structure. (5) The fatigue resistance for the weather side outer chains should be enhanced (particularly at the sensor no. 3 location) and ballast configuration should be optimized to suppress attitude deviations.

Hydrodynamic analysis

Table 7 and Figure 12A present the RAO results of the SSAC model without netting under mooring conditions. Under the wave height of 0.20 $\mathrm{m}$, the maximum amplitude of the roll motion of the SSAC model without netting reaches 8.70 $\mathrm{deg}/\mathrm{m}$ at an incident wave period of 1.5 $\mathrm{s}$. The maximum amplitude of the pitch motion of the SSAC model without netting attains 41.14 $\mathrm{deg}/\mathrm{m}$ at an incident wave period of $2.0\mathrm{s}$. Under a wave height of $0.25\mathrm{m}$, as shown in Table 7, the maximum amplitude of the roll motion of the SSAC model without netting reaches $12.51\mathrm{deg}/\mathrm{m}$ at an incident wave period of $1.5\mathrm{s}$, representing a peak response. The maximum amplitude of the pitch motion of the SSAC model without netting attains $35.56\mathrm{deg}/\mathrm{m}$ at an incident wave period of $2.0\mathrm{s}$.

Table 7.

RAOs for the SSAC model without netting

Wave height (m)	Period (s)	Roll RAO (degree/m)	Pitch RAO (degree/m)
0.20	1.0	4.74	5.57
0.20	1.5	8.70	28.50
0.20	2.0	2.39	41.14
0.20	2.5	3.11	36.54
0.25	1.0	4.55	5.77
0.25	1.5	12.51	35.56

Figure 12.

Response of the model under a wave height of $0.20\mathrm{m}$

(A) The model without netting.

(B) The model with netting.

Table 8 and Figure 12B present the RAO results of the SSAC model with netting under mooring conditions. Under a wave height of 0.20 $\mathrm{m}$, the maximum amplitude of the roll of the SSAC model with netting reaches 6.36 $\mathrm{deg}/\mathrm{m}$ when the incident wave period is 2.5 $\mathrm{s}$, representing a maximum response; simultaneously, the maximum amplitude of the pitch of the SSAC model with netting attains $37.38\mathrm{deg}/\mathrm{m}$ at the same incident wave period of $2.0\mathrm{s}$. Under a wave height of $0.25\mathrm{m}$, as shown in Table 8, the roll of the SSAC model without netting is minimal, remaining in the range of 1.75 to $3.61\mathrm{deg}/\mathrm{m}$. The maximum amplitude of the pitch of the SSAC model without netting attains $35.02\mathrm{deg}/\mathrm{m}$ at an incident wave period of $2.0\mathrm{s}$.

Table 8.

RAOs for SSAC model with netting

Wave height (m)	Period (s)	Roll RAO (degree/m)	Pitch RAO (degree/m)
Under a wave height of 0.20 m
0.20	1.00	4.37	6.38
0.20	1.50	3.86	25.57
0.20	1.75	2.49	36.15
0.20	2.00	4.69	37.38
0.20	2.50	6.36	35.62
Under a wave height of 0.25 m
0.25	1.00	3.60	5.36
0.25	1.50	3.63	24.03
0.25	1.75	2.21	34.58
0.25	2.00	1.75	35.02
0.25	2.50	3.47	27.28

The RAOs of the SSAC were comparatively analyzed under two configurations: with and without netting structures. The results demonstrate significant variations in both roll and pitch responses induced by the presence of netting structures. Take $0.20\mathrm{m}$ wave height excitation as an example (Table 8). For roll motion, the netting system reduced the roll RAO by an average of 31.7% at wave periods of $1.0\mathrm{s}$ and $1.5\mathrm{s}$, demonstrating its motion-damping effect in certain frequency ranges. However, due to significant increases at higher periods (96.2% at $2.0\mathrm{s}$ and 104.5% at $2.5\mathrm{s}$), the overall influence of the netting led to a 38.2% average increase in roll RAO across all tested wave conditions. Regarding pitch motion, the netting configuration exhibited a consistent damping effect across most frequencies, with RAO reductions ranging from 2.5% to 10.3%, except for a 14.5% increase at $1.0\mathrm{s}$ period. This phenomenon may be attributed to the combined effects of altered resonance period due to the added mass of the system under net panel influence and increased damping coefficient.

The IMU employed in this study is limited to measuring 3 DOFs, lacking data capture for the remaining three linear motions. This constraint may reduce the accuracy of the constructed numerical model for the SSAC, thereby impeding a more in-depth analysis of its motion response and mooring forces. This constitutes a recognized limitation of the present work, which can be addressed and refined by implementing comprehensive 6-DOF measurements in future research.

In conclusion, this study presents a novel SSAC and fabricates a scaled-down model with mass property errors controlled within 1.3%. A comprehensive experimental investigation was conducted via water tank tests to evaluate the hydrodynamic characteristics and performance of the scaled model, elucidating the impact of the netting. The netting functioned as a tunable component, modifying parameters related to the system’s frequency response.

Hydrostatic stiffness was measured using a custom-designed system. The results reveal a nonlinear increase in stiffness with imposed mass and show that the netting reduced stiffness in the x direction by an average of 23.8%. Moreover, the netting increased the roll natural frequency from 3.962 rad/s to 4.142 rad/s and raised the damping coefficient by approximately 111%.

Although mooring forces are highly sensitive to model attitude and wave conditions, the experimental results demonstrate that the netting effectively suppresses loads by dissipating wave energy, as evidenced by a reduction in the maximum mooring force from 15 N (without netting) to 10 N (with netting). The netting mitigates both roll and pitch motions, with RAO average reductions of 10.5% and 13.5%, respectively.

The experimental results provide direct guidance for full-scale model design, with the aim of avoiding resonance risks, enhancing stability, and achieving cost-effective mooring solutions. The dataset facilitates the development of high-fidelity numerical models to optimize the design of large-scale aquaculture platforms. This optimization will ensure structural safety while reducing construction and maintenance costs, ultimately contributing to the development of efficient, reliable, and economical equipment for large-scale offshore aquaculture and supporting the technological advancement of the industry.

Limitations of the study

This study still has the following issues that need to be addressed in the future work. (1) Constrained by experimental conditions, only 3-DOF motions were measured. Future studies should utilize optical motion capture to measure the 6-DOF motions, enabling a more thorough and in-depth analysis of its performance. (2) Comprehensive tests of the SSAC scale model under more regular wave, irregular wave, and fully coupled wind-wave-current conditions are essential to further elucidate its hydrodynamic behavior. (3) Experimental data should be used to accurately model the SSAC, supporting reliable prediction and informing precise design.

Resource availability

Lead contact

Requests for further information and resources should be directed to the lead contact, Liguo Wang (wanglg7@mail.sysu.edu.c).

Materials availability

No new materials were generated by this study.

Data and code availability

• •Data: The data that support the findings of this study are available from the lead contact, upon reasonable request.
• •Code: This article does not report original code.
• •Other Items: Any additional information required to reanalyze the data reported in this article is available from the lead contact upon request.

Acknowledgments

This research is supported by the 2024 Marine Economy Development Project of Guangdong Province (project no. GDNRC[2024]21).

Author contributions

W.K., writing – original manuscript, physical prototype fabrication, and experiments; Z.W., W.D., and X.Y., concept design, methodology, writing – reviewing and editing, and supervision; S.Y. and C.Z., writing – original manuscript, physical model fabrication, and experiments; R.Y., writing – original manuscript and physical model design; M.C. and K.W., concept design, funding acquisition, methodology, writing – reviewing and editing, and supervision; L.W., concept design, funding acquisition, methodology, physical model design, experiments, writing – original manuscript and reviewing and editing, and supervision; W.K. and Z.W. are co-first authors of the article.

Declaration of interests

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

STAR★Methods

Key resources table

REAGENT or RESOURCE	SOURCE	IDENTIFIER
Software and algorithms
MATLAB 2021a	MathWorks	https://ww2.mathworks.cn/products/matlab.html
Microsoft Visio Drawing	Microsoft	https://www.microsoft.com/zh-cn/microsoft-365/visio/
Other
DYMH-103 Tension-Compression Load Cell	CALT	https://caltsensor.com/product/inline-load-cells-dymh-103/
WT9011DCL IMU (Inertial Measurement Unit)	Witmotion	https://wit-motion.cn/product.html
ULD200A Wave probe	Nanjing Yilide Scientific Instrumeng Co.,Ltd	https://www.unilly-tak.com/product/27.html
USB-6343 DAQ(Data Acquisition)	NI	https://www.ni.com/zh-cn/shop/model/usb-6343.html
Water tank	South China Universiy of Technology	https://www2.scut.edu.cn/ships/sytj/list.htm

Experimental model and study participant details

This study did not use experimental models typical in the life sciences (e.g., animals, human participants, cell lines, or microbes). The research focused on physical model testing of a semi-submersible aquaculture cage structure.

Method details

Scaled-down model fabrication

A 1:45 scaled-down physical model of the semi-submersible aquaculture cage (SSAC) was fabricated according to Froude scaling laws. The full-scale prototype measures 100 m in length, 36 m in width, with a working draft of 15 m. The model was constructed from cedar wood (density ≈ 450 kg/m³) and waterproofed using epoxy resin adhesive, paint, and body filler. Six ballast tanks (191.11 × 26.67 × 300 mm each) were reserved inside columns for mass distribution adjustment. Lead ballast blocks and iron sand were used to calibrate weight, center of gravity, and moment of inertia to match full-scale parameters. The final mass properties showed relative errors below 1.30% compared to theoretical values. The model included netting made from ultra-high molecular weight polyethylene (UHMWPE, density 950 kg/m³). Due to procurement limitations, the model-scale netting (mesh opening 2.5 mm, twine diameter 0.17 mm) was not geometrically scaled from the full-scale netting (8 cm opening, 5 mm twine) but remains operationally representative for aquaculture applications. Netting was installed on outer and inner frames, with specific dimensions: side netting interlayer spacing ≈67 mm, netting size ≈508 × 300 mm; front/rear netting interlayer spacing ≈67 mm, size ≈300 × 300 mm; bottom netting interlayer spacing ≈78 mm, size ≈667 × 489 mm.

Inertia and center of gravity measurement

An inertia test rig was used to measure center of gravity height and moment of inertia. The center of gravity height was determined using the torque equilibrium principle by applying a constant force to the inertia rig and measuring the resulting deflection angle. The moment of inertia was measured using the small-angle approximation method. The oscillation period for multiple cycles was recorded, and the moment of inertia relative to the knife edge was calculated. The model’s moment of inertia about its own center of gravity was then obtained using the parallel axis theorem.

Water tank experimental setup

Experiments were conducted in a wave tank (120 m length × 8 m width × 4 m depth) at South China University of Technology, equipped with a hinged-flap wave maker, carriage towing system, and wave probes. The model was positioned 70.22 m from the wavemaker at tank center. A four-point mooring system with eight chains was used, symmetrically arranged in the x-direction with fore and aft lines deployed 7.88 m from the model.

Towing test

Towing tests were performed at a scaled speed of 0.3067 m/s (corresponding to full-scale 4 knots) under regular waves with height 0.044 m and periods of 1.0, 1.5, and 2.0 s. The model was mounted to the towing carriage, and a tension-compression load cell was used to measure towing forces.

Horizontal restoring stiffness test

A pulley and calibrated weight system was used to measure horizontal restoring stiffness in x- and y-directions. The model was positioned in calm water at equilibrium, and incremental weights were applied. Displacement at each new equilibrium position was recorded, and stiffness was calculated accordingly. Tests were conducted with and without netting, using weights of 2.00–7.00 kg for x-direction and 0.80–1.4747 kg for y-direction. Each condition was repeated three times and averaged.

Free decay test

Free decay tests were conducted in calm water to determine natural frequencies and damping coefficients. Four configurations were tested: without netting/without mooring, with netting/without mooring, without netting/with mooring, and with netting/with mooring. IMUs (inertial measurement units) sampling at 10 Hz were mounted on fore and aft columns. An initial inclination of 5–10° was applied, and free decay oscillations in roll and pitch were recorded. Damping coefficients were derived from the decay curves.

Mooring force and hydrodynamic response test

Ten DYMH-103 tension-compression load cells (range 80 kg) were installed on mooring chains: five at upper ends and five at lower ends. Two load cells were installed on each of the four windward mooring chains. Regular wave tests were conducted with wave heights of 0.20 m and 0.25 m, and wave periods of 1.0, 1.5, 1.75, 2.0, and 2.5 s. IMU data, wave probe data, and incident wave data were recorded simultaneously. Roll and pitch Response Amplitude Operators (RAOs) were calculated by normalizing motion data with wave amplitude. Tests were performed for both netted and net-free configurations to assess netting influence.

Quantification and statistical analysis

All experimental data were processed and analyzed using MATLAB (MathWorks). Towing force data were filtered to remove high-frequency noise. Horizontal restoring stiffness values were calculated as means of three repeated measurements for each weight condition. Free decay data were analyzed using the Froude method to extract natural frequencies and damping coefficients. For mooring force analysis, peak-to-peak amplitudes (PPA) were calculated for each load cell time series to quantify force fluctuations. RAOs were computed by dividing the amplitude of motion response (roll or pitch) by the incident wave amplitude at each wave frequency. Values are presented as means where applicable. Due to the physical nature of the experiments and limited sample sizes (single model with multiple test runs), inferential statistical tests were not applied. Experimental uncertainties were minimized through sensor calibration, repeated measurements, and controlled laboratory conditions.

Additional resources

No new websites, forums, or clinical trial resources were generated from this study.

Published: March 11, 2026