[ ]{ } ~ N p f = p (9) where matrix [Nf] = [N1 N2 … Nm] includes fluid element shape functions. Substituting approximation (9) into equations (8) and (7), using Galerkin’s method of weighted residuals and applying Green’s theorem, following equation can be derived: { }[ ] ( ) { }[ ]{ } [ ] { } [ ]{ } [ ] [ ]{ } { }0 1 2 = + − ∇ ∇ ³ ³ ³ dV c dA dV V T V A T T T N N p N n N u N N p f f s s f f f ρ (10) where matrix [Ns] includes structure element shape functions and {n} denotes a unit normal vector pointing into the structure. Equation (10) can also be written shortly by: { }0 = + − s Q p H p Lu ρ (11) In the structure point of view, a nodal force vector equivalent to the fluid pressure can be expressed using the pressure approximation (9) as: { } [ ] { } [ ] { }[ ] { } { } N n N p L p r N n f s s t T A T A T dA pdA = = = ³ ³ (12) If external force of the structure and damping will be neglected, so called pressure formulation for the coupled fluid-structure problem can be expressed as [9]: { }0 = ¿ ¾ ½ ¯ ® » ¼ º « ¬ ª + ¿ ¾ ½ ¯ ® » ¼ º « ¬ ª − p u 0 H K L p u L Q M 0 T ρ (13) where M and K are structural mass and stiffness matrices. Equation (13) is unsymmetric and therefore it is difficult to solve. However, the fluid-structure matrix system can be symmetrized for instance by defining new fluid unknown as presented in reference [13]. Fig. 3 Boundary conditions of the fluid ABAQUS software was used to create and solve FE-model of combined structure and fluid [14]. Fluid was modeled with acoustic tetrahedron elements. Modeling of the water was first studied by a simple structure such as sphere and circular disk. Meshing properties such as element size, size of the water model and element order as well as boundary conditions at exterior surfaces of the water model were varied. Results of these simple finite element studies were compared with analytical solutions and with some experimental results found in the literature [7, 15, 16, 17, 18]. It was found that if the model of the fluid is large enough, boundary conditions at outer surfaces of the fluid has not significant effect on the natural frequencies of the structure. On the basis of these studies, water model for the azimuthing thruster was constructed. The water model contained about 267 000 elements and 55 000 nodes. 6
RkJQdWJsaXNoZXIy MTMzNzEzMQ==