4.1.1. Simulation Method of the Three-Cylinder Buoy
The buoy structure mainly consists of a floating three-cylinder frame, a cabin, and a counterweight. Firstly, considering that the buoy was floating and the cabin was located above the water surface, the waves had no direct or less influence on the cabin. Therefore, it was considered possible to integrate the mass into the floating three-cylinder frame. Next, the counterweight was located at the bottom of the floating three-cylinder frame, which was of a smaller volume and had less wave influence on the entire buoy structure, and the mass was able to be integrated into the floating three-cylinder frame. Thus, to simplify the simulation, the whole buoy structure could be simplified into a floating three-cylinder frame.
The floating three-cylinder frame was a rigid structure. Under the action of waves, the floating three-cylinder frame was mainly affected by gravity, buoyancy, wave force, and the tension of the mooring cable. The movement of water mass points generated the wave force. Because of the velocity of water mass points gradually decreased from water surface to seabed, the wave force of the floating three-cylinder frame needed to be calculated using the finite unit method. The three-cylinder buoy of smaller diameter belonged to the small-scale category compared to the wave length, so the wave force could be calculated using the Morison equation. Thirdly, the hydrodynamic coefficient of the cylinder was directional and related to the relative movement velocity direction of the water particle [
23]. When the floating three-cylinder frame rotates, the wave force of each unit should be calculated by considering the angle with the wave incident direction. Therefore, it was necessary to establish the local coordinate system. Above all, the detailed derivation and numerical simulations of the movement of the floating three-cylinder frame were studied [
24].
- 2.
Translation and rotation of the floating three-cylinder frame
Only considering translation, the forces of all units of the floating three-cylinder frame were accumulated in the global coordinate system. The acceleration of the floating three-cylinder frame was calculated by Newton’s second law, and the displacement per unit time step was analyzed by the fourth-order Runge–Kutta method [
25]. Relative to the origin of the global coordinate system, the center of mass of the rigid structure in the global coordinate system was usually taken as the origin of the moving coordinate for rotation calculation. The movement of the floating three-cylinder frame not only included 3D translational movement, but also 3D rotational movement. A local coordinate system was established on the floating three-cylinder frame by referring to the principle of rigid structure dynamics, and then converting it to the global coordinate using the Blaine angle formula. The relevant formulas and theories could be found in the literature [
24]. The center-of-mass coordinate and rotational inertia of the floating structure were deduced in this paper.
The floating three-cylinder frame is composed of three floating cylinders. The structure was relatively complex, and the calculation was difficult. In this simplified model, the floating three-cylinder frame was regarded as three independent floating cylinders, and the moment of inertia of each floating cylinder was considered separately. In order to ensure the water resistance of the floating three-cylinder frame, the cylinders were filled with the foam glue. Although the mass was very small, it was necessary to recalculate the density of the floating three-cylinder frame. As shown in
Figure 10,
l1 represents the length of the floating cylinder above the water surface, and
l2 the depth of water inflow.
rf indicates the radius of the floating cylinder, and
rT the radius of the counterweight;
lT is the thickness of the counterweight, and
ρT is the density of the counterweight.
ρf is the desired density of the floating cylinder. The formula is as follows:
As shown in
Figure 10, the floating cylinder was regarded as a cylindrical structure with uniform medium, and its centroid was the geometric centroid. If a counterweight was added, the centroid will move down the length of
xc. The size of
xc can be calculated according to the following formulas, and the position of the centroid of the floating cylinder can be obtained.
According to the parallel axis theorem, the rotation inertia of the single floating cylinder in the three-axis was calculated as follows:
Figure 11 shows the density and centroid coordinates of the floating three-cylinder frame, which can be deduced by Formula (4), composed of three cylinders.
Figure 11 is the plan view of the floating three-cylinder frame. The centroid of the triangle was regarded as the origin from which to establish coordinate axis. The horizontal direction is the
x-axis, and the vertical direction is the
y-axis. According to the model parameters, the coordinates of points A, B, and C are (13.3/120.5, 13.3/2), (−13.3/120.5, 0), and (13.3/120.5, −13.3/2), respectively,
Lr is the radius of the circumscribed circle of the plane triangle. According to the established coordinate system, the moment of inertia of the buoy in three axis was calculated by using the parallel axis theorem:
4.1.2. Simulation Method of the Mooring Cable
The mooring cable was a typical flexible structure that was simulated by the lumped mass method. Assuming that the mooring cable was composed of a finite massless spring connected by the lumped mass points, the shape of the mooring cable and the tension between each mass point were further obtained by calculating the offset of the lumped mass point under the influence on dynamic boundary conditions. The force of the lumped mass point of the mooring cable mainly included the gravity, the buoyancy, the tension of the mooring cable, the drag force, and the inertia force. The wave force received by the mooring cable could still be calculated using the Morrison formula. Firstly, the mooring cable mass point was regarded as a cylinder, and the calculation method of wave force was the same as the buoy; then, the particle movement equation was established by using Newton’s second law, as shown in Formula (6).
where
l0 is the original length of the mooring cable, and
l is the length after deformation. The unit was m.
C1 and
C2 represent the elastic coefficients of the mooring cables. d indicates the diameter of the mooring cable.
In the model experiment, it was difficult to tighten the three mooring cables and to ensure the preset water depth of the buoy. Therefore, the mooring cable had a certain degree of relaxation so that the water depth of the buoy could be ensured. This research proposed an efficient and accurate method to simulate the movement of the mooring cable in the non-tightened state. The initial position of the mass point of the mooring cable was designed by the position of tightened state. Then, when judging the tension of the mooring cable, the length l0 of ∆l in Formula (6) was increased. The mooring cable was freely able to move under no wave action, and the balanced state is the position of the mooring cable under the relaxed state. In particular, it was pointed out that the physical quantities in bold in Equation (6) were vectors in three directions of the global coordinate system (x, y, z), while the calculation of the tension T of the mooring cable was the physical quantity along the rope direction. It was necessary to calculate the projection under the overall coordinate system and convert it into T.