Elastic wedge impact onto a liquid surface: Wagner's solution and approximate models

The problem of elastic wedge impact onto the free surface of an ideal incompressible liquid of infinite depth is considered. The liquid flow is two-dimensional, symmetric and potential. The side walls of the wedge are modelled as Euler beams, which are either simply supported or connected to the main structure by linear springs. The liquid flow, the deflection of wedge walls and the size of wetted region are determined simultaneously within the Wagner theory of water impact. We are concerned with the impact conditions of strong coupling between the hydrodynamic loads and the structural response. The coupling is well pronounced for elastic wedges with small deadrise angles. This is the case when the fully nonlinear models fail and approximate models based on the Wagner approach are used. In contrast to the existing approximate models, we do not use any further simplifications within the Wagner theory. Calculations of the velocity potential are reduced to analytical evaluation of the added-mass matrix. Hydrodynamic pressures are not evaluated in the present analysis. In order to estimate the maximum bending stresses, both stages when the wedge surface is partially and totally wetted are considered.Three approximate models of water impact, which are frequently used in practical computations, are examined and their predictions are tested against the present numerical solution obtained by the normal mode method within the Wagner theory. It is shown that the decoupled model of elastic wedge impact, which does not account for the beam inertia, provides a useful formula for estimating the maximum bending stress in thick wedge platings.     

Impact of a surfacewave on an elastic hull

In order to model a ship hull’s response to the impact of surface waves, the two-dimensional problem of wave impact on an elastic beam whose ends are connected by springs with a rigid structure uniformly submerged in a fluid is considered. The fluid is assumed to be ideal and incompressible and its flow symmetric; the lateral bending of the beam is described by the Euler equation. The fluid flow and the size of the wetted region are determined simultaneously with the calculation of the the beam deflection within the framework of the Wagner approach which takes into account the reshaping of the free surface of the fluid on interacting with a body. The stresses and strains arising in the beam and at its ends during impact are found. The numerical algorithm developed makes it possible to analyze the elastic effects in fluid impacts on thin-walled structures of finite length. Moreover, as the stiffness of the connecting springs tends to zero, the solution of this problem describes the impact of an elastic beam with free ends on a weakly curved fluid surface.     

Vibration of beams with general boundary conditions due to a moving random load

Summary  The transverse vibrations of elastic homogeneous isotropic beams with general boundary conditions due to a moving random force with constant mean value are analyzed. The boundary conditions considered are: pinned–pinned, fixed–fixed, pinned–fixed, and fixed–free. Based on the Bernoulli beam theory, the problem is described by means of a partial differential equation. Closed-form solutions for the variance and the coefficient of variation of the beam deflection are obtained and compared for three types of force motion: accelerated, decelerated and uniform. The effects of beam damping and speed of the moving force on the dynamic response of beams are studied in detail. Received 3 December 2001; accepted for publication 30 April 2002     

Plane problem of asymmetrical wave impact on an elastic plate

The problem of wave impact on the edge of an elastic horizontal plate is studied within the framework of the Wagner approach using the normal-modes method. The plate is governed by the Euler beam equation with simply supported ends. The liquid is assumed to be ideal and incompressible. The problem is coupled: the elastic and hydrodynamic characteristics of the impact process and the dimension of the contact region should be found simulatenously. An algorithm that permits a detailed study of the impact on an elastic plate is proposed. The phenomenon of unlimited increase of hydrodynamic loads owing to the plate flexibility (blockage) is revealed for fairly long plates. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirisk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 5, pp. 148–158, September–October, 1998.     

Two-dimensional problem of the impact of a vertical wall on a layer of a partially aerated liquid

The two-dimensional unsteady problem of the impact of a vertical wall on a layer of a liquid which is mixed with air near the wall and does not contain air bubbles away from the wall is solved in a linear approximation. The gas-liquid mixture is modeled by a homogeneous, ideal, and weakly compressible medium with a reduced sound velocity dependent on the air concentration in the gas-liquid mixture. Outside the gas-liquid layer, the liquid is considered ideal and incompressible. During the initial stage of the impact, the liquid flow and the hydrodynamic pressure are determined using the linear theory of the potential motion of an inhomogeneous liquid. The dependence of the amplitude of the impact pressure along the wall on the air concentration in the gas-liquid layer and on the thickness of this layer is investigated. For a small relative thickness of the layer, the thin-layer approximation is used. It is shown that the solution of the original problem tends to the approximate solution as the thickness of the layer decreases. It is shown that the presence of the gas-liquid layer leads to wall pressure oscillations. Estimates are obtained for the pressure amplitude and the oscillation period. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 5, pp. 34–46, September–October, 2006.     

Penetration of an elastic circular cylindrical shell into an incompressible liquid

The plane unsteady problem of impact of a thin elastic cylindrical shell on the surface of an ideal incompressible liquid is considered. The initial stage of interaction between the body and the liquid when the stresses in the shell attain peak values is studied. The problem is treated in a linearized formulation and is solved numerically by the normal modes method within the framework of the Wagner approach. The numerical results agree with experimental data for various types of circular cylindrical shells made from mild steel. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 6, pp. 186–197, November–December, 1999.     

Design of a transversely layered cantilever of minimum mass under specified maximum-deflection constraint

The problem of optimal design of a transversely layered cantilever beam of minimum mass composed of a finite set of elastic homogeneous isotropic materials is considered for the case where a constraint is imposed on the maximum deflection of the beam and information on the loads applied to the beam is incomplete. It is shown that, among these loads, there exists the “worst” point load whose magnitude is equal to the resultant of the forces applied to the beam, for which the deflection is maximal for any material distribution along the beam. The necessary optimality conditions are obtained for the mass-minimization problem of the beam under the “worst” loading, a numerical algorithm is developed for the synthesis of the optimal cantilever, and a numerical example is given. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 4, pp. 104–110, July–August, 2007.     

Unsteady Behavior of an Elastic Beam Floating on the Surface of an Infinitely Deep Fluid

The effect of initial disturbances and unsteady external loading on an elastic beam of finite length which floats freely on the surface of an ideal incompressible fluid is studied in a linear treatment. The fluid flow is considered potential. The beam deflection is sought in the form of an expansion in the eigenfunctions of beam vibrations in vacuum with time-dependent amplitudes. The problem reduces to solving an infinite system of integrodifferential equations for unknown amplitudes. The memory functions entering this system are determined by solving the radiation problem. The beam behavior is studied for various loads with and without allowance for the weight of the fluid. The effect of fluid depth on beam deformation was determined by comparing with the previously obtained solutions of the unsteady problem for a beam floating in shallow water. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 1, pp. 85–94, January–February, 2006.     

Solution of the problem of optimal cut in an elastic beam

The Kirchhoff model of an elastic beam with a transverse cut is considered. The nonpenetration condition proposed by A. M. Khludnev is formulated at the edges of the cut. The equilibrium model of a beam with a restriction on the cut is written in the form of a variational inequality. An analytical solution is obtained with the use of the projection operator. The problem of choosing optimal cuts is formulated for the criterion of minimum opening. Conditions for determining the extremum shapes of the beam are obtained and an example of the solution of the problem is given. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 5, pp. 149–157, September–October, 1999.     

Experimental two-phase flow measurement using ultra fast limited-angle-type electron beam X-ray computed tomography

An experimental evaluation of a novel limited-angle-type ultra fast electron beam X-ray computed tomography approach for the visualization and measurement of a gas–liquid two-phase flow is reported here. With this method, a simple linear electron beam scan is used to produce instantaneous radiographic views of a two-phase flow in a pipe segment of a flow loop. Electron beam scanning can be performed very rapidly, thus a frame rate of 5 kHz is achieved. Radiographic projections are recorded by a very fast detector arc made of zink–cadmium–telluride elements. This detector records the X-ray radiation passing through the object with a sampling rate of 1 MHz. The reconstruction of slice images from the recorded detector data is a limited-angle problem since in our scanning geometry the object’s Radon space is only incompletely sampled. It was investigated here, whether this technology is able to produce accurate gas fraction data from bubbly two-phase flow. Experiments were performed both on a Perspex phantom with known geometry and an experimental flow loop operated under vacuum conditions in an electron beam processing box.     

Impact of a solid on a liquid surface: Asymmetric plane case

The vertical impact of a rigid elliptical cylinder on a compressible liquid surface is considered in a plane asymmetric formulation. Solution of the boundary problem reduces to solving an infinite system of linear Volterra integral equations of the second kind. The results are analyzed in terms of the initial asymmetry angle. Translated from Prikladnaya Mekhanika, Vol. 35, No. 9, pp. 75–84, September, 1999.     

Numerical analysis of free vibrations of a beam with oscillators

The problem of free vibrations of a beam with free ends of variable cross section and mass, from which point masses (oscillators) are suspended by bars, is considered. It is shown that parametric resonances can occur in this oscillating system. Numerical examples showing the efficiency of the calculation method proposed are given. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 4, pp. 135–144, July–August, 2006.     

Motion of an inclusion in uniformly and nonuniformly vibrating liquids

An approach to constructing the quantitative nonuniformity characteristics of liquid vibrations is proposed. A new problem of the motion of an inclusion in a vibrating liquid is considered. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 1, pp. 79–85, January–February, 2007.     

Unsteady gas well operating regimes

A model of a gas well containing liquid is considered, the nature of the steady-state regimes is investigated, and the causes of cyclical output are analyzed. In formulation the problem resembles the oil well gushing problem studied in [1]. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 162–164, November–December, 1986.     

Nonlinear problem of a circular cylinder rising vertically to an interface between liquid media

The nonlinear initial-boundary-value problem of a contour approaching an interface between two liquid media is considered. A solution is constructed using a previously developed numerical method that is based on reducing the original problem to a system of integrodifferential equations for singularities simulating liquid and rigid boundaries and a function that describes the interface between the media. Calculation results for the perturbations generated by a circular cylinder approaching a free surface are presented. The dependences of the flows obtained and the hydrodynamic characteristics of the contour on the Froude number are estimated. Omsk Branch of the Sobolev Institute of Mathematics, Siberian Division, Russian Academy of Sciences, Omsk 644099. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 2, pp. 84–89, March–April, 2000.     

On aspects of damping for a vertical beam with a tuned mass damper at the top

In this paper, the wind-induced, horizontal vibrations of a vertical Euler–Bernoulli beam will be considered. At the top of the beam, a tuned mass damper (TMD) has been installed. The horizontal vibrations can be described by an initial-boundary value problem. Perturbation methods will be applied to construct approximations of the solutions of the initial-boundary value problem, and it will be shown that the TMD uniformly damps the oscillation modes of the beam. In the analysis, it will be assumed that damping, wind-force, and gravity effects are small but not negligible.     

Dispersion and blockage effects in the flow over a sill

The problem of a homogeneous heavy liquid flow over a local obstacle is considered in the long-wave approximation. The steady and unsteady waves in the vicinity of the obstacle are described by second-order models of the shallow-water theory and their hyperbolic approximations. The flow in the vicinity of the leading and trailing edges of bluff bodies (sills and steps) is studied. The solution of the problem of the blocked zone upstream of the step is constructed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 1, pp. 45–58, January–February, 2008.     

Motion of a hot wedge in a melting medium

In [1–3] a series of problems of the motion of heat sources at a temperature higher than the melting point of the surrounding medium was considered. The heat source could be a laser beam or a hot body. Here, the case of a thin wedge heated to a temperature higher than the melting point of the surrounding medium and moving at a constant velocity is investigated. The velocity is high enough for the molten layer formed to be thin. The problem is solved by the method of integral relations. The shape of the molten zone, the drag on the wedge and other flow characteristics of the melt are determined. Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 52–57, September–October, 1988.     

Separation impact of a circular disk floating on the surface of an ideal incompressible fluid of infinite depth

The three-dimensional mixed problem of the separation impact of a circular disk floating on the surface of an ideal incompressible unlimited fluid is considered. The position and shape of the contact area between the body and the fluid (and the separation zone) are not known and depend on the relation between the translational and angular velocities acquired by the disk upon impact. Because of this, the problem in question is nonlinear and belongs to the class of free-boundary problems. The problem is solved using the method of Hammerstein-type nonlinear boundary integral equations. This approach allows the fluid flow after impact and the unknown zone of separation of fluid particles to be determined simultaneously. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 4, pp. 76–86, July–August, 2009.     

Linear theory of the propagation of internal wave beams in an arbitrarily stratified liquid

Beams of harmonic internal waves in a liquid with smoothly changing stratification are calculated in the Boussinesq approximation taking into account the effects of diffusion and viscosity. A procedure of local reduction of the beam in a medium with an arbitrary smooth stratification to the case of an exponentially stratified liquid is constructed. The coefficient of energy losses in the case of beam reflection on the critical level is calculated. Parameters of internal boundary flows with split scales of velocity and density that are formed by a wave beam on discontinuities of the buoyancy frequency and its higher derivatives are determined. Institute of Problems of Mechanics, Russian Academy of Sciences, Moscow 117526. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 5, pp. 88–98, September–October, 1998.