【正文】 into account hinge rigid modes. These modes are calculated through a numerical analysis of the structure based on the finite element method, rather than being explicitly defined to meet orthogonality conditions. All the modules and connectors are considered to be flexible. The translational and rotational stiffness of the connectors is also considered. This method is validated by a special numerical case, where the hydroelastic response for very high connector stiffness values is shown to be the equivalent to that of a continuous structure. Using the results of this test model, the hydroelastic responses of more general structures are studied, including their displacement and bending moments. Moreover, the effect of connector and module stiffness on the hydroelastic response is studied to provide insight into the optimal design of such structures. 2. Equations of motion for freely floating flexible structures Using the finite element method, the equation of motion for an arbitrary structural system can be represented as ? ? ? ? ? ?? ? ? ?,.... PUKUCUM ??????????????? （ 1） where [M], [C] and [K] are the global mass, damping and stiffness matrices, respectively。 and {P} is the vector of structural distributed forces. All of these entities are assembled from the corresponding single element matrices [Me], [Ce], [Ke], {Ue}, and {Pe} using standard FEM procedures. The connectors are modeled by translational and rotational springs, and can be incorporated into the motion equations using standard FEM procedures. Neglecting all external forces and damping yields the free vibration equation of the system: ? ?? ?UM +? ?? ?KU =0 (2) Assuming that Eq. (2) has a harmonic solution with frequency o, this then leads to the following eigenvalue problem: ? ? ? ?? ?? ? ? ?02 ??? DKMW （ 3） Provided that [M] and [K] are symmetric and [M] is positive definite, and that [K] is positive definite (for a system without any free motions) or semidefinite (for a system allowing some special free motions), all the eigenvalues of Eq. (3) will be nonnegative and real. The eigenvalues 2r? (r=1,2,3， ...6n) represent the squared natural frequencies of the system: 0≤ 21? ≤ 22? ≤...≤. 26n? (4) where 2r? ≥0 when [K] is positive definite, and 2r? ≥0 when [K] is semidefinite. Each eigenvalue is associated with a real eigenvector {Dr}, which represents the rth natural mode: ? ? ? ? ? ? ? ? ? ?? ? ,..., rn21 Trjrrr DDDDD ，? (5) where ??riD is the eigenvector of the ith node which contains 6 degree of freedoms, and i runs over the n nodes of the structural FE model system. ??rd , a submatrix of rD , consists of the rth natural mode ponents of all the nodes associated with one particular element. The rth modal shape ??ru at any point in that element can be expressed as rU =??TI ??N ??L ??rd =? ?Trr WVU r, (6) where [L] is a banded, localtoglobal coordinate transform matrix posed of diagonal submatrices [l], each of which is a simple cosine matrix between two coordinates. [N] is the displacement interpolation function of the structural element. For freely floating, hingeconnected, multimodule structures, Eq. (3) has zerovalued roots corresponding to the 6 modes of global rigid motion and the hinge modes describing relative motion between each module. According to traditional seakeeping theory, the rigid modes of the global system can be described by three translational ponents (uG, vG, wG) and three rotational ponents (yxG, yyG, yzG) about the center of mass in the global coordinate system coincident with equilibrium. Thus, the first six rigid modes (with zero frequency) at any point j on the freely floating body can be expressed by ? ? ? ?TjD 0,0,0,0,0,11 ? , ? ? ? ?0,0,0,0,1,02 ?jD ，