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Twenty-First Symposium on NAVAL HYDRODYNAMICS
tools of less sophisticated methods at present and in the near future.
On the other hand, driven by the need of ship design some simplified approaches emerged at the expense of accuracy, see e.g. Meyerhoff and Schlachter(1980), Yamamoto et al(1978–1979), Jensen and Pedersen(1979,1981), Schlachter(1989) and Xia et al(1995). Almost all of them use the combination of the conventional strip theory and nonlinear modifications of some kind. As a new effort in this direction, Wu and Moan(1996) presented a nonlinear hydroelastic simulation method for the prediction of wave-induced structural responses in ships with large amplitude motion in head or following sea. The total response is decomposed into linear and nonlinear parts. The linear part is evaluated by using appropriate linear potential flow theory. The nonlinear part comes from the convolution of the impulse response function of the ship-fluid system and the nonlinear hydrodynamic force caused by slamming and nonlinear modifications of added mass, damping, restoring and wave forces. Unlike the previous ones, it can be used with high-speed strip theory or three-dimensional flow theory and the frequency dependence of added mass and damping has been taken care of, to some extent. However, it has not yet been verified by experiments.
In this paper, we will apply the method to a high-speed catamaran model in regular head waves. We will compare the calculated structural responses with those from model tests which were conducted at MARINTEK. The main purpose of the model tests is to verify the linear hydroelastic formulation of high-speed strip theory (Wu et al 1993, Hermundstad et al 1995, Hermundstad 1995). However, strong nonlinear effects are observed at some frequencies of regular waves. It offers an opportunity for verifying the proposed formulation which is outlined in the next section. Although the comparison is rather limited, the satisfactory agreement between numerical and experimental results is quite encouraging.
Consider a flexible ship moving in the long-crested head waves on deep water. Let (x,y,z) be a right-handed coordinate system with the positive z-direction vertically upwards. The ship has a forward speed U in the negative x-direction. Only the global structural responses are investigated here and the hydrodynamic forces are understood in this sense. We assume that
the nonlinearity comes from the large ship motions in heave and pitch while the structural deformation remains small,
the incident waves can be described sufficiently by linear wave theory,
the influence of ship motions on the incident wave elevation is not significant,
the hydrodynamic interaction among the multiple hulls of high speed vessels(Froude number Fn>0.4) is negligible.
In addition, the radiation and diffraction velocity potentials are assumed to be zero on the free surface which means there is no free surface memory effect. The memory effect of free surface will be introduced later and therefore this assumption is actually removed, to a great extent. The dynamic vertical force per unit length exerted by the fluid on each hull of the ship at position x may be expressed as(e.g. Faltinsen 1990)
where m(x,t) is the high-frequency added mass of the submerged cross section. f(x,t) consists of the Froude-Krylov force and hydrostatic restoring force per unit length on the instantaneous wetted surface. D/Dt represents the total derivative with respect to time t,
ξ(x,t) is the vertical displacement of the ship hull relative to the wave surface,
The vertical displacement of the ship hull w(x,t) is approximated by the s lowest symmetric dry eigenmodes, wk(x), including heave and pitch which are the dominant part,
Here p(t)=[p1(t),p2(t),…,ps(t)]T is the vector of generalized coordinates. Multiplying Eq.(1) by wk(x) and integrating over the length of hulls, we get the kth generalized, or modal hydrodynamic force