Regular wave impact onto an elastic plate

Abstract

A computational analysis of an elastic plate dropped against regular long waves is presented. The problem is considered within the linear potential-flow theory. The liquid flow is two-dimensional and the plate is modelled by an Euler beam. The analysis is based on the normal-mode method with hydroelastic behavior of the plate being of main interest. Different impact conditions are considered to study the dependence of the total energy of the plate-liquid system on impact geometry and plate properties. The contributions of kinetic and potential energies to the total energy are analyzed. It is shown that the kinetic part of the system energy is small at the instant of time when bending stresses in the beam approach their maximum values. Estimations of both the total energy and the maximum of bending stresses are presented. Most of the calculations are performed for the conditions of experiments carried out in MARINTEK. A range of the problem parameters is also considered, to reveal peculiarities of the unsteady interaction between a falling elastic plate and surface waves.

Maximum stress of stiff elastic plate in uniform flow and due to jet impact

“…The infinite systems (28) and (30), as well as the corresponding systems for the averaged deflection with χ(t) changed toχ(t), are truncated and integrated by the Runge-Kutta method of fourth order with corresponding initial conditions. Calculations are performed with N mod = 3, 5, 10, 15 elastic modes in (21). The step of integration ∆t is equal to 1/10 of the non-dimensional period of the highest retained mode with number N mod .…”

“…In reality, the jet front is not parallel to the plate 23 at impact instant ( figure 3(a)), compressibility 24 ( figure 3(b)) and aeration 12,13 ( figure 3(c)) of the fluid in the impact region matter, as well as the presence of the air 10,11,22 in between the plate and the approaching jet front ( figure 3(d)). These effects make the hydrodynamic loading on the plate to be gradual in time and can be described by using the concept of retardation time, T r .…”

“…During this early stage the plate deflection and the flow caused by impact are described in the non-dimensional variables by equations 19,20,21 (tilde is dropped below)…”

“…The mixed boundary value problem (26) is solved in Appendix A by the method of separating variables for γ < 1 and by the theory of analytical functions for γ > 1 with analytic treatment of the flow velocity singularity at the point x = 0, y = 1, where the boundary condition changes its type. Substituting (21) and (25) in the beam equation (12), multiplying both sides of the equation by ψ m (y), m > 1, integrating both sides of the equation in y from y = 0 to y = 1, and using the orthogonality condition (24), we arrive at the infinite system of ordinary differential equations with respect to the principal coordinates of the plate deflection,…”

“…which provides the orthogonality condition (24) where n = m. Here we assume that different modes ψ n (y) correspond to different values of the spectral parameter λ n . The modes ψ n (y) are defined as solutions of the boundary problem (22), (23) up to a constant factor.…”