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solution. These papers did not couple the elastic response of the hull with the hydrodynamic problem.

The following sections will present the theory of transient impact, experimental validation, and numerical calculations of coupled wedge impact with elastic hull response. A time-dependent free surface impact boundary value problem (Vorus 1992, 1996) will be used to estimate the impact load acting on a rigid, constant deadrise section. The paper will discuss the significant difficulties associated with solving the fully nonlinear boundary value problem when time dependent spray sheets are present and assumptions based upon similarity flows are no longer valid. In addition, the impact problem for a finite wedge where the wedge surface becomes vertical at a hard chine will be briefly reviewed (Vorus, 1996). Comparisons between theory and experiments will also be presented. The experiments involve the drop testing of nearly prismatic sections of typical planing hulls. The analysis is then extended to include the coupling of a discrete mass attached to the rigid hull by springs and dashpots. This model has direct application to the shock problem associated with high speed planing vessel impact in waves. Numerical studies are presented which show the effect of different parameter values in system mass ratio, stiffness, and damping on maximum acceleration response.

THEORETICAL MODELING OF IMPACT

The characteristics of the flow during impact, which include the hull pressure distribution, jet velocity, and free surface deformation, change dramatically as the jet head passes over severe hull geometric variations. When the jet head reaches a location on the hull's surface where the surface curvature exceeds that which would normally occur in an unrestrained jet, such as at a chine, the pressure drops significantly. See Figure 1 for a schematic defining the “chines dry” and “chines wet” stages of impact. A variation of the model described in the following paragraphs (Vorus, 1992, 1996) has been compared extensively with steady planing pressure distributions (Lai and Troesch, 1995) which include the essential characteristics of impact hydrodynamics. A summary of the impact model is reviewed here briefly for completeness. Details can be found in Vorus (1996).

The theoretical formulation of Vorus (1992, 1996) can be viewed as a solution to the complete two dimensional nonlinear impact initial-boundary value problem in all respects except that the nonlinear boundary conditions are satisfied on the horizontal axis. This is argued to be consistent to lowest order in the flatness limit. Physically, as the cylinder flattens toward coincidence with the horizontal axis, the boundary conditions more and more accurately apply on the axis, implying a limit of geometric linearity. However, with increasing flatness, the transverse flow velocity tends toward infinity over the entire material contour (except with a singular zero at the plane of symmetry for symmetric impact). This implies the limiting condition of uniform hydrodynamic nonlinearity. The theory is therefore mixed: It is geometrically linear in that the boundary conditions are satisfied on the horizontal axis, but it is hydrodynamically nonlinear in that the large transverse perturbation velocity is fully retained in the axis boundary conditions.

Figure 1: Definition of “chines dry” (a) and “chines wet” (b) impact phases

The mathematics problem is defined with the aid of Figure 2. Although depicted on Figure 2 as a semi-infinite wedge, the cylinder contour is of arbitrary shape and can include a hard chine where separation is forced to occur. The impact velocity V(t) is also arbitrary and can include, for example, forced deceleration which also produces contour flow separation, as is demonstrated in Vorus (1996).

Referring to Figure 2, the principal solution unknowns are the zero pressure point offset zc(t), the jet head offset zb(t) and the jet velocity distribution, Vs(z,t), between zc(t) and zb(t). A multi-layered nonlinear iteration is required in computing these unknowns starting from an initial condition corresponding to the self-similar semi-infinite wedge at the initial impact velocity V(0). The solution equations that must be iterated derive from the nonlinear boundary conditions satisfied on the axis segments indicated in Figure 2 b.

The dynamic boundary condition of zero pressure on the jet head free vortex sheet leads to a one-dimensional Burger's equation in terms of the Vs(z,t) and zc(t) unknowns:



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