Transient response analysis concerning the bottom slamming and green water effect has to be calculated in the time domain. In the linear system, the calculation result in the frequency domain can easily be transferred to time sequences using a relevant method, such as Fourier transform. Determination of the transient response required suitable models for excitation due to slamming and green water. Bottom impact slamming is a three-dimensional phenomenon and occurs over a very short time. When a body enters the water, there is an air-gap between the body and water surface, which reduces the slamming impact force. It is well known that the slamming force is proportional to velocity squared and a function of body geometry. The distribution of slamming force in space and time is complicated to evaluate, which makes the problem difficult. Therefore, some empirical formulae for the slamming were suggested and applied.
Ochi and Motter (1971) suggested the empirical non-dimensional slamming pressure factor for Wagner’s (1932) model depending on the section shape, using a seakeeping test and a drop test. The sectional coefficients are determined using the conformal mapping technique and the sectional distribution of the local pressure is assumed to be linear from maximum value at bottom to zero at the effective area level (i.e. one tenth of draught). Stavovy and Chuang (1976) suggested an empirical slamming pressure by regression analysis of measured data from drop tests as a function of deadrise angle. They also assumed the local pressure distribution as linear. .
A time domain mathematical model with convolution integral formulation can consider fluid memory effects on ship response to arbitrary excitation such as transient slamming. Bishop et al. (1978b) devised a linear method to estimate
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slamming responses. The slamming responses for sinusoidal waves were obtained by superimposing on the steady-state responses induced by the waves in the manner discussed by Bishop et al. (1978b). They identified two distinct ways of slamming. The first method, so-called ‘impact slamming’ (Ochi and Motter, 1971, Stavovy and Chuang, 1976), evaluates the forces at the instant when the hull strikes the free surface of the waves. The second method, so-called ‘momentum slamming’ (Leibowitz 1963), describes the effect of pressure variations around the hull surface as it penetrates the moving fluid after the initial entry. They discussed two distinct issues: One is that the hull vibrates due to slamming while the bow is deeply immersed or emerged above the mean water line; another is that the constants of hydrodynamic coefficients for added mass and damping (A, B) are, in general, frequency dependent, but treated as constant. Belik et al. (1987) considered the rate of change of fluid momentum (Leibowitz 1963) and additional flare buoyancy when bow sections plunge into/re-emerge from the water before the sea surface reaches the still water draught (flare slamming). The total transient excitation consists of impact (Ochi and Motter, 1971, Stavovy and Chuang, 1976) and momentum slamming. The results show that the flare slamming effect is important for large flared ships. Aksu et al. (1995) carried out probability analysis using time simulation results. Both hydroelasticity investigations are based on linear strip method, representation of irregular waves by a combination of a large number of regular waves.
Kaplan (1987) presented the analytical/computational determination of the slamming forces arising from flat bottom impacts on the water surface of ships advancing in waves. In this work, the concepts of fluid momentum theory, with a three- dimensional model rather than the conventional two-dimensional strip theory methods, are applied to the impact problems of a flat surface.
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Zhao and Faltinsen (1993) presented a numerical method for studying water entry of a two-dimensional body of arbitrary cross-section using a nonlinear boundary element method with a jet flow approximation. This method was verified by comparison with similarity theoretical solution of water entry wedges derived by Dobrovol’skaya (1969) and Wagner’s (1932) model for small dead rise angle.
Kvålsvold and Faltinsen (1995) presented a theoretical and numerical slamming model for the wet deck of a multi-hulled vessel. The disturbance of the wetted surface as well as the local hydroelastic effects in the slamming area were accounted for. The elastic deflections of the wet deck, modelled as a beam, are expressed in terms of ‘dry’ normal modes. The structural deformation of beam model accounts for the shear deformations and the rotary inertia effects. The results indicated that an important effect arises from the body boundary condition as an angle of attack effect. The maximum bending moment stress was proportional to curvature of the wave crest in the impact region. Both theory and drop test results did not predict the maximum pressure deterministically; however, the bending stresses and deflections agree with each other.
Ramos et al. (2000) compared the empirical formulae for the slamming forces described above and proposed a simple formula for sectional distributions of the slam force. The study showed that Ochi and Motter’s (1971) formula gave a slightly smaller slamming force, whilst other empirical formulae showed good agreement.
Storhaug et al. (2003) measured global vibrations in terms of whipping and springing of a large ocean-going ship, using the DNV structural monitoring system. From the measurements it became apparent that a possible cause of vibration may be stern slamming (bottom slams are rare); hence, small impacts and low damping may cause
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the springing. The predicted wave frequency results are in fairly good agreement with measurement, while, for high frequency, the results do not capture the measured trend.
Buchner (1995) presented the green water phenomena based on model tests with a frigate. He indicated that the rate of change of water height on the deck has an important effect on the maximum deck pressure as well as static load and an inertia load for vertical acceleration. The calculations showed good agreement with the measurements. Buchner’s approach is widely accepted for modelling the effects of green water on the global wave-induced vertical bending moment, as shown by Wang et al. (1998) and Jensen and Mansour (2003).
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