structure so that all reaction forces during operation except steering moment goes through the slewing bearing from the thruster to the ship structure. Modeling of whole ship structure is very laborious work. Therefore only part of the ship structure was modeled around the assembly block, which represents local stiffness properties of the ship structure. Boundary conditions of this section of the ship structure were varied and corresponding results were also compared to the experiments. Lowest torsional and bending modes of the ship hull could be studied and included to the model for example by using an equivalent beam model. However, global dynamics of the ship hull was excluded from this study. Slewing Bearing Slewing bearing consist of three cylindrical rolling bearings, two axial and one radial as shown in Fig. 2. Individual rolling elements were modeled with two spring elements as shown in the figure. Load-deflection relationship of the individual rolling element and for the complete bearing was calculated according to the Herzian contact theory. Contact model between a cylinder and a plane was applied for axial rolling elements [1]. For radial cylindrical rolling elements equations derived in reference [2] was applied. These equations are also based on the Herzian contact theory. The load-deflection relationships between rolling elements and bearing races are non-linear. The stiffness of individual rolling element contact was linearized around an operation point as follows: ( ) ( ) a a a a a F F k F δ ∂ ∂ = (1) Fig. 2 Slewing bearing Hydraulic Steering System Steering moment of the azimuthing thruster is produced by four hydraulic motors. The hydraulic motors are driven by two adjustable-displacement hydraulic pumps, forming a closed hydraulic circuit. The lowest rotational mode around steering axis is dominated by stiffness of this hydraulic steering system. Most of the flexibility originates from compressibility of the hydraulic fluid, flexibility of the pipelines and hoses. Flexibility of the steering gear mechanism (gear ring and pinions) was assumed to be significantly smaller than flexibility of the hydraulic system and these were neglected in the calculation. The hydraulic fluid “spring” was characterized by the value for the bulk modulus B. Mechanical compliance of the pipelines and hoses were taken into account by calculating radial displacement of the inner surface of the pipe due to internal pressure increase. By using this displacement “bulk modulus” for pipeline Bp and hoses Bh were calculated. The effective bulk modulus for the hydraulic system was then obtained by combining these bulk moduli analogously to the total resistance of parallel resistors as follows [3]: 3
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