In the pursuit of maritime energy conservation and emission reduction, I have explored the integration of renewable energy sources on ships. Solar energy, as a clean and widely available resource, offers significant potential. Among various photovoltaic (PV) installations, flexible supports have gained attention for their adaptability and cost-effectiveness. However, applying these systems on ships introduces unique challenges due to dynamic marine environments. In this study, I design a marine solar system flexible support for deployment on ship decks and investigate its vibration response characteristics under combined wind loads and ship motions. The primary goal is to provide technical insights for the design and operation of such solar systems in maritime applications.
The marine solar system comprises flexible components like cables and PV panels, along with rigid supports such as columns and beams. This solar system must withstand not only wind loads but also inertial forces from ship movements, particularly pitch and heave. To analyze these effects, I employ a fluid-structure interaction (FSI) decoupling calculation method. This approach allows me to simulate the complex interactions between aerodynamic forces and structural dynamics efficiently. By fitting periodic load and inertia functions, I can assess the vibration response without extensive coupled simulations, making the analysis feasible for large-scale solar system designs.
I base the design on the parameters of a typical large tanker, similar to the KVLCC2, with a length of 320 meters and a broad deck area. The solar system flexible support consists of multiple PV panels arranged in rows, supported by steel cables under tension. Each panel measures 2.2 m × 1.1 m × 0.035 m, with an installation angle of 0° to minimize wind resistance and align with maritime conditions. The panels are spaced with a ratio of 0.2, and the entire structure spans 56 meters, elevated 4 meters above the deck. The steel cables, with a diameter of 0.03 meters, are made of material with an elastic modulus of 1.95×1011 Pa and yield strength of 1.86×109 Pa. This configuration ensures that the solar system can generate power while enduring harsh sea states.
To evaluate the solar system’s performance, I consider four operational scenarios: wind load only, wind load with heave, wind load with pitch, and wind load with both pitch and heave. These scenarios reflect typical maritime conditions, including severe sea states up to level 8, with wave heights of 9–14 meters and wind speeds up to 37 m/s. The ship’s motions are assumed periodic, with pitch angles ranging from -10° to 10° and a period of 12.35 seconds, and heave displacements with an amplitude of 10.34 meters. The inertial forces from these motions are calculated and combined with aerodynamic loads derived from computational fluid dynamics (CFD) simulations.
The CFD simulations model the wind loads on PV panels at various inclination angles, from -10° to 10°. I use a computational domain of 24 m × 8 m × 12 m with a uniform inflow velocity of 37 m/s, simulating extreme wind conditions. The SST k-ω turbulence model is applied, and pressure coefficients are extracted to determine lift and drag forces. The results show that wind load distribution changes significantly with panel angle, affecting the forces transmitted to the cables. For instance, at a 10° inclination, the lift force on the upper surface is -75.76 N, while on the lower surface, it is -431.42 N. These forces are fitted into time-dependent functions using polynomial equations.
The equations for ship motion are derived as follows. The pitch angle θ as a function of time t is given by:
$$ \theta(t) = 10 \sin(0.5088t) $$
where θ is in degrees and t in seconds. The pitch inertial force F_{Iy} on a mass m at a distance r from the ship’s center of mass is:
$$ F_{Iy} = -m r \ddot{\theta} $$
with \(\ddot{\theta}\) being the angular acceleration. For heave motion, the displacement H is:
$$ H(t) = 10.34 \sin(0.5088t) $$
and the heave inertial force F_{Iz} is:
$$ F_{Iz} = -m \ddot{H} $$
The wind loads on the cables are expressed as functions of θ. For example, the vertical load force F_y on the upper cable is fitted as:
$$ F_{y,\text{upper}} = -0.0048\theta^5 + 0.0201\theta^4 + 0.7546\theta^3 – 3.1603\theta^2 – 29.3712\theta – 171.6203 $$
Similarly, the displacement L_y of the upper cable in the vertical direction is:
$$ L_{y,\text{upper}} = -0.000257\theta^3 + 0.000073\theta^2 + 0.00188\theta – 0.2415 $$
These equations enable the decoupled FSI analysis by providing input forces and displacements for structural dynamics calculations.
I conduct structural dynamics simulations using a simplified model that includes only the steel cables, as they dominate the flexible behavior of the solar system. The cables are fixed at both ends, and periodic loads from wind and inertia are applied at points corresponding to PV panel attachments. I monitor the amplitude and internal stress of the cables, particularly focusing on Cable 2, which represents the mid-span of the solar system. The material properties ensure that stresses remain below yield strength under design conditions.
The results are summarized in tables below. Table 1 lists key parameters of the ship and solar system components, while Table 2 shows the average displacement and amplitude responses of Cable 2 under different scenarios.
| Parameter | Symbol | Value |
|---|---|---|
| Ship Length | LPP | 320.0 m |
| Ship Beam | B | 58.0 m |
| PV Panel Dimensions | – | 2.2 m × 1.1 m × 0.035 m |
| Panel Mass | mp | 30.672 kg |
| Cable Diameter | Dl | 0.03 m |
| Cable Elastic Modulus | E | 1.95×1011 Pa |
| Cable Yield Strength | σs | 1.86×109 Pa |
| Wind Speed (Design) | U | 37 m/s |
| Pitch Period | Tθ | 12.35 s |
| Heave Amplitude | Hmax | 10.34 m |
This solar system design balances structural integrity with energy generation needs, ensuring reliability in marine environments.
| Scenario | Average Vertical Displacement (10-3 m) | Amplitude (10-3 m) | Maximum Stress (MPa) |
|---|---|---|---|
| Wind Load Only | -8.424 (lower), -8.314 (upper) | 0.003 | 45.2 |
| Wind + Heave | -8.398 (lower), -8.288 (upper) | 2.366 | 47.8 |
| Wind + Pitch | -11.017 (lower), -10.260 (upper) | 22.108 | 100.6 |
| Wind + Pitch + Heave | -11.274 (lower), -10.300 (upper) | 20.276 | 98.4 |
The data indicates that pitch motion has the most significant impact on the solar system, increasing cable amplitudes and stresses substantially. Heave motion contributes less, but combined effects are critical for design considerations.
Further analysis of internal stress reveals that the maximum stress occurs near the fixed boundaries of the cables, with values up to 100.6 MPa under pitch conditions. Although this is below the yield strength, it highlights potential fatigue issues over time. The average stress on Cable 2, located at the center of the solar system, shows cyclic variations that could lead to fatigue failure if not addressed. The stress-time curves for different cable segments are derived from simulations, and they follow periodic patterns aligned with ship motions.
To quantify the vibration response, I use the following equation for dynamic stress σ under combined loading:
$$ \sigma(t) = \sigma_{\text{mean}} + \sigma_{\text{amp}} \sin(\omega t + \phi) $$
where \(\sigma_{\text{mean}}\) is the mean stress, \(\sigma_{\text{amp}}\) is the stress amplitude, ω is the angular frequency of ship motion, and φ is the phase angle. For the solar system cables, \(\sigma_{\text{mean}}\) increases by approximately 122.6% under pitch compared to wind load alone, emphasizing the adverse effects of ship dynamics.
In terms of displacement, the vertical movement of cables is more pronounced than the flow-direction movement. The amplitude response A can be expressed as:
$$ A = \sqrt{A_y^2 + A_z^2} $$
where A_y and A_z are amplitudes in vertical and flow directions, respectively. Under pitch, A_y reaches 22.108×10-3 m for the lower cable, indicating large oscillations that could affect the stability of the solar system.
The design of the solar system flexible support must account for these vibrations. I recommend increasing cable pre-tension or using higher-strength materials to mitigate excessive motion. Additionally, regular inspection of boundary regions and mid-span sections is crucial to prevent fatigue-related failures. The solar system’s performance can be optimized by adjusting parameters such as cable diameter or panel spacing, but this requires further analysis beyond the current scope.
In conclusion, this study demonstrates the importance of considering ship motions in the design of marine solar systems. The flexible support exhibits significant vibration responses under pitch and heave, with pitch being the dominant factor. The decoupled FSI method provides an efficient way to simulate these interactions, offering valuable insights for engineers. Future work could explore adaptive control strategies or enhanced materials to improve the durability of such solar systems in maritime applications. Overall, integrating solar energy on ships through flexible supports is feasible, but careful design is essential to ensure long-term reliability and safety.
The solar system’s adaptability to marine environments makes it a promising solution for reducing fuel consumption and emissions. By continuously refining the design based on dynamic analyses, we can advance the adoption of renewable energy in the shipping industry. This research contributes to the growing body of knowledge on maritime solar systems, paving the way for more sustainable operations at sea.