The ice-breaking properties of amphibian aircushion vehicles (AACV), which have recently been discovered, [1] make it necessary to solve a number of new applied problems [2]. One of the promising methods of ice breaking with the aid of AACV is the resonance method [2] which is applied at speeds corresponding to maximum wave resistance. In this connection determination of the wave resistance to AACV under ice conditions becomes very important. In the absence of ice this problem has been theoretically solved for the movement of a vehicle in deep and shallow water [3, 4], in a channel [5], and with acceleration [6]. The present paper is concerned with the stationary problem of the wave resistance to AACV in broken ice.1. Let there be a given system of the AACV surface pressures moving at a constant velocity u over an infinite water field covered by broken ice. In accordance with the principle of inverse motion, we assume that a load q(x, y) is applied to the free liquid surface covered by broken ice and moving with velocity -u as x --~ oo.The coordinate system that is stationary relative to the vehicle is located as follows: the plane xOy coincides with the unperturbed ice-water interface, the x axis points in the direction of the vehicle's motion, and the z axis points vertically upward. The water is assumed to be an ideal incompressible liquid with density p2. Broken ice is represented in the form of floating disconnected masses. Interaction forces between separate ice floes are ignored, and their dimensions are considered sufficiently small compared with the wavelength so that ice-floe bending does not occur [7]. Full-scale tests [2] show this approach to be quite justifiable in solving problems on the propulsive properties of AACV in ice broken by the resonance method.Use is made of the assumption that the field covered by broken ice is continuous [7], and the surface density coinciding with the floating-particle mass per unit area is given by the continuous function m(x,y) = plh --p~where p0 is the ice physical density; s(x, y) is a dimensionless function of ice-floe tightness [7] (0 ~ s ~ 1); and h(x, y) is the ice thickness. To simplify the problem, the quantities h and s are further considered constant.In the adopted coordinate system, the velocity potential ~p(x,y, z) of fluid perturbed motion must satisfy the Laplace equation A T = 0 and the linearized boundary conditions z =0: 02~~ 0~o g Oqa plh 03~o _ 10q 0qo 0.(1.2) Oz 2 tt-~z + ~-g; + p20zOz 2 p2u 0-7' z = -H : 0---~ =Here tt > 0 is the coefficient of scattering forces [3, 8]; H = H1 -a; H1 is the water-body depth; and a = hp~ is the ice immersion depth at static equilibrium. For great depths, when H1 >> h, it can be assumed that H ~ H1.According to [3, 9], the wave resistance to AACV is numerically equal to the horizontal projection of the resultant of pressure forces onto the surface R // Ow(z, = q(x,y) -~xY)dxdy, (1.3)
The uniformly accelerated motion of an amphibian air-cushion vehicle on the surface of a basin covered by finely small ice floes is considered.
Steady-state rectilinear motion of a load on an ice sheet modeled by a viscoelastic plate is considered. The viscoelastic properties of ice are described using the linear Maxwell and Kelvin-Voigt models and a generalized Maxwell-Kelvin model. Calculated vertical displacements and strains of the ice plate are compared with available experimental data.Key words: incompressible liquid, viscoelastic plate, steady-state motion of load, ice sheet. 1.The hydrodynamic problem of a load moving on a continuous ice is modeled by a system of surface pressures which moves above a floating ice plate [1,2].We consider an infinite region covered with continuous ice on which a given system of surface pressures q moves at a velocity u. The coordinate system attached to the load is arranged as follows: the plane xOy coincides with the ice-water unperturbed interface, the x direction coincides with the direction of motion of the load, and the z axis is directed vertically upward. It is assumed that water is an ideal incompressible liquid of density ρ 2 and the motion of the liquid is potential. The ice sheet is modeled by a viscoelastic homogeneous isotropic plate which is initially not stressed. The linear viscoelastic material simulating ice is described using the Maxwell and Kelvin-Voigt models and a generalized Maxwell-Kelvin model [3]. In the case of pure shear, the behavior of a Maxwell body can be described by a mechanical model consisting of a spring and a viscous damper connected in series, and the behavior of a Kelvin-Voigt body by a model consisting of a spring and a viscous damper connected in parallel. In the case of a one-dimensional state, the generalized Maxwell-Kelvin model considered in the present paper represents Maxwell and Kelvin units connected in series. This model takes into account instantaneous elastic response, viscous flow and delayed elastic response.By analogy with [1,4], using the generalized model of a viscoelastic Maxwell-Kelvin material to describe the deformation of an ice sheet under the action of a moving load, it is possible to derive the following equation of small oscillations of a floating viscoelastic plate:are the shear elastic moduli of ice corresponding to Maxwell and Kelvin materials, E M and E K are Young moduli for Maxwell and Kelvin bodies, respectively, ν is the Poisson ratio, τ M and τ K are stress and strain relaxation times of ice, respectively, h(x, y) is the ice thickness, ρ 1 (x, y) is the ice density, w(x, y) vertical displacement of ice, and Φ = Φ(x, y, z) is a function of the liquid velocity potential which satisfies the Laplace equation ΔΦ = 0. Below, it is assumed that ρ 1 and h are constants. As the calculated 1
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.