Plane linear problem of the immersion of an elastic plate in an ideal incompressible fluid

Abstract

Introduction. A plane unsteady-state problem of the immersion of an elastic plate of finite length in an ideal incompressible weightless fluid is considered. At the initiM moment (t ~ = 0), the fluid occupies the lower half-plane (yl ~< 0); the segments of its boundary -L < x ~ < L and y~ = 0 correspond to the elastic plate, and the segments z ~ > L, x ~ < -L, and y~ = 0 to the free boundary of the fluid (Fig. 1). The dimensional variables are primed. The plate is hinged to a structure which is being immersed in the fluid with constant velocity V. The impact phenomena, which are connected with the beginning of the motion, determine the initial plate deflection w'(x', 0) and the velocity of its points (Ow'/Ot)(x', 0), which are assumed to be known and are denoted by W~o(X ') and w~(x'), respectively. At the initial stage of immersion, when the structure is displaced to a much smaller extent than the length of the plate, one needs to determine the plate deflection and the distribution of bending stresses in it.The problem is considered within the framework of a linear approximation. The fluid flow is assumed to be plane and potential. The plate is modeled by an Euler beam, and the bending stresses in the transverse direction are assumed to be negligible.The impact by a shallow wave on a plate of finite size is divided into two stages [1]. At the first (impact) stage, the plate is wetted only partially, and the hydrodynamic loads on the plate are great and depend on the rate of expansion of the region of contact between the plate and the fluid. Generally, for shallow waves, this stage is short, and the stresses in the plate do not reach maximum values. At the second stage (immersion), the plate is wetted completely and continues to be immersed in the fluid. Here, the hydrodynamic loads on the plate are already insignificant and cannot be classified as impact. The plate vibrates mainly owing to the potential energy of elastic strains and to the kinetic energy accumulated in the plate during the impact stage. At both stages, the fluid boundary can be replaced by a plane boundary if the wave is quite shallow, and the depth of immersion of the plate is small compared with its size. The last remark explains the problem formulation for the second stage considered in the present work as a problem of immersion of a floating plate in an ideal weightless fluid for which the initial deflection and the velocity distribution are given.

Regular wave impact onto an elastic plate

“…The beam deflection is sought as a superposition of eigenfunctions of the spectral problem for a totally wetted elastic plate. Details of the analysis of "wet" modes and formulae for the elements of the matrix of hydrodynamic coefficients S nm (see (25)) can be found in [14]. It was found that the first "dry" and "wet" modes are practically identical, for the second modes the difference is less than 8%, for the third modes less than 7% and for the fourth modes less than 6%.…”

“…This problem is equivalent to that of a floating elastic plate, which starts to enter the liquid at t = t 3 with a constant velocity and both the initial deflection of the plate and the initial velocity of its elements being prescribed. A theoretical analysis of this stage was given by Faltinsen [7]; see also [14]. In the last paper the problem is formulated with respect to the hydrodynamic pressure p(x, y, t) and the beam deflection w(x, t).…”

“…That is why we do not expect any difficulties with its numerical study, even if the choice of scales for the penetration stage are not optimal. More details about the orders of the unknown quantities during the penetration stage can be found in [2] and [14].…”

Skimming impacts and rebounds on shallow liquid layers

“…At slower speeds the flow under a fixed surf board or an inclined sluice gate is considered by Binder & Vanden-Broeck (2005, 2007, in a regime in which surface tension has a significant effect on the free-surface evolution. Interactions between the liquid layer and body have been incorporated in a number of previous impact studies, including in the form of an elastic response in the body (Korobkin & Khabakhpasheva 1999) and a trapped gas cavity in the region of impact (Korobkin 1996). The latter study is notable in that a system of ordinary differential equations governing the boundaries of the contact patch between fluid and solid and the response of the body to the impact is again derived.…”

Impact of a surfacewave on an elastic hull

“…The study continues [1][2][3][4][5] in which the case of fluid impact on a hinged beam was considered. However, the possibility of beam edge displacements leads to the appearance of new eigenvalues and, hence, of new normal modes, which makes it much more difficult to find the corresponding elements of the apparent-mass matrix analytically.…”