ȍ M I I I ȍ I I M M a z a a d d y x = ¿ ¾ ½ ¯ ® » ¼ º « ¬ ª − + ¿ ¾ ½ ¯ ® » ¼ º « ¬ ª = ¿ ¾ ½ ¯ ® β α β α 0 0 0 0 (4) where Mx and My are moments around two orthogonal axes perpendicular to the spin axis, α and β are angular deviations around corresponding x and y axes, Mz is moment around spin axis, Id is diametral moment of inertia, Ia is moment of inertia around spin axis and Ω is angular velocity around spin axis. The second term in the equation (4) representing gyroscopic torques is called gyroscopic matrix. Considering a continuum model, gyroscopic matrix can also be derived for beam element. Equation of motion including the gyroscopic matrix cannot be diagonalized. However, the equation of motion can be written in state space form by defining new state vector q as follows [6]: 0 = + = ¿ ¾ ½ ¯ ® » ¼ º « ¬ ª− + ¿ ¾ ½ ¯ ® » ¼ º « ¬ ª Aˆ q Bˆ q u u 0 K M 0 u u M C 0 M (5) where M and K are mass and stiffness matrices of the system and damping matrix C includes also terms caused by gyroscopic torques according to equation (4). This first order eigenvalue problem can be solved leading to complex eigenvalues and eigenvectors. The rotor shaft was modeled by beam elements including also gyroscopic matrices. Propeller as well as rotor core was modeled as disk elements. Bearings were modeled as spring elements. Cross coupling terms were neglected. The model did not contain model of thruster body structure nor the ship structure. The model was constructed and solved by using a code created on Matlab environment. Fluid-Structure Interaction As the structure which is in contact with fluid vibrates, the fluid surrounding the structure must vibrate as well. This causes increasing of the kinetic energy and degreasing of natural frequencies of the total system. The coupled problem of fluidstructure interaction of thruster and surrounding water was solved by modeling also water domain with finite elements. The fluid was assumed to be compressible and inviscid medium. In addition small displacements were assumed. At an accelerating fluid-structure interface momentum and continuity considerations require that [8]: nu n p ρ =− ∂ ∂ (6) where p is the dynamic fluid pressure, n is direction of outward normal to the boundary, ρ is fluid density and ün is the normal component of fluid particle acceleration. At rigid wall right side of equation (6) equals zero. At free surface the surface waves were in this case neglected and the boundary condition used was p = 0. Boundary conditions are illustrated in Fig. 3. Generalization of equation (6) to three dimension is given as: { } 2 2 t d p ∂ ∂ ∇ =− ρ (7) Definition of bulk modulus in three dimension is p=-B ∇•d. Three-dimensional wave equation can be constructed by taking divergence of equation (7) and differentiating definition of bulk modulus twice with respect to time, leading to: p c p B p 2 2 1 ∇ = = ρ (8) where ∇2 is Laplacian operator and c = (B/ ρ)0.5 is the wave speed in the fluid. Approximation of the dynamic pressure using the finite element method is then given by [9, 10, 11, 12]: 5
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