A transmission node of an offshore underwater acoustic sensor network consists of a buoy, a mooring system and an underwater acoustic modem for communication (shown in Figure 1). To simplify the problem, we assume that the buoy is a cylinder with a bottom diameter of 2m and a height of 2m. The mass of the buoy is 1000kg. The mooring system consists of 4 steel tubes, a sealed cylindrical drum, a heavy ball, a series of welded chain and an anti-dragged anchor. The mass of the anchor is 600kg. For the welded chain, commonly used types and their parameters are listed in the supplemental table. For each steel tube, the length is 1m and the diameter is 50mm, while the mass is 10kg. The angle between seabed and the welded chain at the anchor point must be less than or equal to 16 degrees. Otherwise, the anchor would be dragged, and the node will be lost. The underwater acoustic modem is installed in a sealed cylindrical drum. The length of the drum is 1m and the diameter is 30cm. The total mass of the modem and drum is 100kg. The drum is linked between the 4th steel tube and the welded chain. If the drum keeps vertical, the acoustic modem works best. Otherwise, when the drum tilts, the performance of the modem is hurt. If the tilt angle (the angle between the drum and the vertical line) of the drum exceeds 5 degrees, then the performance becomes so poor that is not acceptable. To control the tilt angle of the drum, sometimes people hang a heavy ball at the bottom of the drum.
The optimal design problem of the mooring system is to select the type and the length of the welded chain, the mass of the heavy ball, which minimize the tilt angle of the drum, the immersion depth and the possible swimming area of the buoy.
#Question 1. We use the type II welded chain with the length of 22.05m. Suppose that the mass of the heavy ball is 1200kg. We allocate the node in a test area, where the seabed can be simplified as flat and the depth can be simplified as 18m, while the density of seawater is ρ=1.025×103kg/m3. Without considering the currents, please calculate the tilt angle of the drum and the tilt angles of the steel tubes, and describe the shape of the welded chain, and estimate the immersion depth and the swimming area of the buoy, when the speeds of wind are 12m/s and 24m/s, respectively.
#Question 2.
Under the assumptions of question 1, please calculate the tilt angle of the drum and the tilt angles of the steel tubes, and describe the shape of the welded chain, and estimate the immersion depth and the swimming area of the buoy when the speed of wind is 36m/s. Please adjust the mass of the heavy ball such that the tilt angle of the drum is less than or equal to 5 degrees, and the angle between the seabed and the welded chain at the anchor point is less than or equal to 16 degrees.
#Question 3.
Due to tide and other factors, the depths of the sea water are ranging between 16m and 20m. The maximum seawater current velocity is 1.5m/s and the maximum wind velocity is 36m/s. Please design the optimized mooring system considering the acting forces of wind and seawater, the different sea water depth due to the tide. Please analyze the tilt angle of the drum and the tilt angles of the steel tubes, and describe the shape of the welded chain, and estimate the immersion dept and the swimming area of the buoy.
#Note
The offshore wind load can be approximated by the formula: F=0.625×Sv2(N) , where S is the normal plane projection area (m2) of an object, and v is the wind speed (m/s). The offshore water load can be approximated by the formula: F=374×Sv2(N), where S is the normal plane projection area (m2) of an object and v is the seawater current velocity (m/s).