Application of the generalized body-fixed coordinate system for the wave-body interaction problem of a small-depth elastic structure in head seas

The body-fixed coordinate system is applied to the wave-body interaction problem of a small-depth elastic structure which has both rigid and elastic body motions in head waves. In the weakly non-linear assumption, the perturbation scheme is used and the expansion is conducted up to second-order to consider several non-linear quantities. To solve the boundary value problem, linearization is carried out based not on inertial coordinate but on body-fixed coordinate which could be accelerated by a motion of a body. At first, the main feature of the application of body-fixed coordinate system for a seakeeping problem is briefly described. After that the transformation of a coordinate system is extended to consider an elastic body motion and several physical variables are re-described in the generalized mode. It has been found that the deformation gradient could be used for the transformation of a coordinate system if several conditions are satisfied. Provided there are only vertical bending in elastic modes and the structure has relatively small depth, these conditions are generally satisfied. To calculate an elastic motion of a body, the generalized mode method is adopted and the mode shape is obtained by solving eigen-value problem of dynamic beam equation. In the boundary condition of the body-fixed coordinate system, the motion effect reflected to free-surface boundary is considered by extrapolating each mode shape to the horizontal direction from a body. At last, simple numerical tests are implemented as a validation process. The second-order hydrodynamic force of a freely floating hemisphere is first calculated in zero forward speed condition. Next, motion and added resistance of a ship with forward speed are considered at different flexibility to confirm the effect of an elastic body motion in body-fixed coordinate system.