However, the lumped mass modeling makes the model inconsistent. If the differences in the inertial properties between the shell 3-D model and the lumped mass distribution are small, the inconsistency will be negligible. The hybrid model is implemented in WISH-FLEX BEAM and is named WISH-FLEX BEAM+3-D FEM in the results. This section describes how to couple the
fluid models with the 3-D FE model via eigenvectors. There are three topics, which are approximated equation of motion in generalized learn more coordinate system, recalculation of eigenvectors on the panel model using linear interpolation, and external forces. The use of the 3-D FE model is very straightforward for overcoming the disadvantages of the beam theory. Moreover, it is rather simple compared to the sophisticated beam theory conjunction with 2-D analysis of cross-section and consideration for structural discontinuity. However, large degrees of freedom (DOF) should be reduced by modal superposition method in time-domain simulations. There are two assumptions for DOF reduction by modal superposition method. Firstly, motion on the body surface easily converges with a few lower modes because modal stiffness rapidly increases in higher modes except for local modes. It is negligible, the fluid disturbance, due to motions of higher modes. Secondly, responses of higher modes are quasi-static. selleck chemical According to the first assumption, the displacement
vector field in Cartesian coordinate system can be expressed as equation(32) u→(t)=∑j=16⁎mξj(t)A→j≈∑j=16+nξj(t)A→j=[A→1A→2⋯A→6+n]ξ1~6+n(t)where nn is typically smaller than 20. According to the second assumption, the original form of equation of motion can be expressed as equation(33) [MD00MQ]ξ¨1~6+n(t)ξ¨7+n~(t)+[KDKDQKQDKQ]ξ1~6+n(t)ξ7+n~(t)=f1~(t) The mass matrix consists of only diagonal terms of 1 except the rigid body part of 6×6. The rigid body part is defined at the mass center projected on the free surface of the calm water. By applying the two assumptions to Eq. (33) for DOF reduction, it reduces to equation(34) MDξ¨1~6+n(t)+KDξ1~6+n(t)=f1~6+n(t) Eq. (34) will be solved to obtain modal responses in
the coupled-analysis. The response includes both dynamic and quasi-static components. The linear restoring matrix consists of structural stiffness of natural PAK6 mode and fluid restoring. Gravity restoring is also included in the fluid restoring. It is expressed as equation(35) KD=CS+CRCSi,j=ωi2(i=jandi,j>6),CSi,j=0 In addition, quasi-static responses of higher modes can be obtained by solving the decoupled equation as equation(36) KQξ7+n~(t)=[(A→7+n~)T]f7+n~(t)−KQDξ1~6+n(t) In this study, Eq. (36) will not be solved. However, contributions of all modes to sectional force can be considered by direct integration of all external and inertial forces. It will be discussed in Section 3.5. The 3-D FE model is coupled with the 3-D Rankine panel method via eigenvectors.