Trapped modes of oscillation of a liquid for surface-piercing bodies in oblique waves

The linear problem of the harmonic oscillations of an ideal incompressible heavy liquid with a free surface in the presence of two and more infinitely long partially submerged cylindrical bodies of arbitrary cross-section is considered. It is proved that there are configurations of the bodies which provide examples of the non-uniqueness of the boundary-value problem in the case of an arbitrary frequency of the oscillations and an arbitrary non-zero angle between the generatrix of the cylinders and the direction of propagation of the surface waves. In the case of these configurations, the homogeneous boundary-value problem has non-trivial solutions with a finite energy integral, which describe trapped modes of oscillation of the liquid.     

The non-linear Saint-Venant problem of the torsion,stretching and bending of a naturally twisted rod

Problems on large stretching, torsional and bending deformations of a naturally twisted rod, loaded with end forces and moments, are considered from the point of view of the non-linear three-dimensional theory of elasticity. Particular solutions of the equations of elastostatics are found, which are two-parameter families of finite deformations and which possess the property that, for these deformations, the initial system of three-dimensional non-linear equations reduces to a system of equations with two independent variables. The use of these equations enables one to reduce certain Saint-Venant problems for a naturally twisted rod to two-dimensional non-linear boundary-value problems for a planar domain in the form of the cross-section of a rod. Different formulations of the two-dimensional boundary-value problem for the cross-section are proposed, which differ in the choice of the unknown functions. A non-linear problem of the torsion and stretching of a circular cylinder with helical anisotropy, which is reduced to ordinary differential equations, is considered as a special case.     

A variational formulation and investigation of boundary-value problems of the non-linear theory of plates using an energy approach

A variational formulation of boundary-value problems of the non-linear dynamic theory of elasticity using the Hamilton functional is presented. The quasi-static boundary-value problem for thin plates is considered. The initial system of equations, in a two-dimenonal formulation, is represented in terms of generalized forces and displacements. The sufficient conditions for the existence and uniqueness of a weak solution are established.     

On fractional powers of a pair of matrices and a plate-beam problem

In this paper we consider the transversal deflections of a dynamically-coupled Von Kármán system consisting of a plate which has a beam attached to its one edge. The problem is considered in the form of a non-linear evolution problem in a product space. We show the existence of a unique local solution by following a fractional powers approach to first construct a “weak” solution in a larger space. Regularity properties for this solution yield a unique local strong solution for the original boundary-value problem. This approach entails the introduction of fractional powers of a pair of matrices.     

Generalized parametric oscillations of mechanical systems

An effective numerical-analytical method of investigating parametrically excited oscillatory Hill-type systems, described by general boundary-value problems, is developed. It is assumed that the coefficients of the equation depend in an arbitrary non-linear way on a parameter, the eigenvalues of which are to be obtained. The approach to solving the generalized periodic boundary-value problem is based on the established differential relation between the eigenvalue and the value of the period (the length of the interval). The computational algorithm possesses the property of accelerated convergence, which enables many extremely subtle and difficult problems of constructing the dependences of the eigenvalues and eigenfunctions (the forms with the oscillations) on the index and parameters of the system, difficult to obtain by traditional approaches, to be successfully investigated. To illustrate the high efficiency of the method, a solution of the problem of the spatial angular oscillations of a dynamically symmetrical artificial satellite moving in a circular orbit is constructed.     

On the three-dimensional undulation instability of a rectangular viscoelastic composite plate in biaxial compression

Within the frame work of the second version of small precritical deformation in the three-dimensional linearized theory of stability of deformable bodies (TDLTSDB), the undulation instability problem for a simply supported rectangular plate made of a viscoelastic composite material is investigated in biaxial compression in the plate plane. The corresponding boundary-value problem is solved by employing the Laplace transformation and the principle of correspondence. For comparison and estimation of the accuracy of results given by the TDLTSDB, the same problem is also solved by using various approximate plate theories. The viscoelasticity properties of the plate material are described by the Rabotnov fractional-exponential operator. The numerical results and their discussion are presented for the case where the plate is made of a multilayered viscoelastic composite material. In particular, the variation range of problem parameters is established for which it is necessary to investigate the undulation instability of the viscoelastic composite plate by using the TDLTSDB. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 45, No. 1, pp. 93–108, January–February, 2009.     

The momentum of waves and their effect on the motion of lumped objects along one-dimensional elastic systems

Taking the example of the small longitudinal oscillations of a rod, it is shown that, in order to answer the question concerning wave momentum and its action on an obstacle, the problem of the wave motion in the medium has to be solved in a non-linear formulation. The variational formulation of problems in the dynamics of one-dimensional elastic systems with moving clampings and loads is improved taking account of non-linear factors. The equations of motion and the natural boundary conditions are obtained. The small longitudinal-transverse oscillations of a string and the motion of a bead sliding along it are considered.     

Vortex damping of sloshing in tanks with baffles

The problem of damping the sloshing in tanks with sharp-edged baffles (thin inserts which partially span a longitudinal or transverse cross-section) is considered. Separation of the boundary layer and the formation of vortices occur at these sharp edges. It is assumed that the domains where there is significant vortex motion of the fluid are localized in small neighbourhoods of the sharp edges of the baffles. The non-linear vortex damping is determined from the distribution of the velocity intensity factors at these sharp edges in the same way as the linear damping, caused by the dissipation of energy in a boundary layer close to a wall, is determined from the fluid velocity distribution on the walls of a cavity. Both of the above-mentioned distributions are calculated by solving the same boundary-value problem on the oscillations of an ideal fluid. The second of the distributions characterizes the singular properties of the solutions of this problem on particular lines. A method based on the variation of the area of the baffles, which simplifies the calculation of the velocity intensity factors is described. The distinctive features arising when the method of finite elements is used are considered. The results of numerical calculations of the damping of sloshing in a cylindrical tank with a ring baffle are compared with experimental data.     

The oscillations of layered elastic media with a wavy boundary

Block element methods are used to investigate the oscillations of an elastic layer with a rough surface. For simplification, the investigation is carried out separately for the rotational and potential components of the boundary-value problem, which enables the solution to be obtained by analysing Helmholtz's equations.     

Stability of a smooth flow in a Laval nozzle

A boundary-value problem for a non-linear second-order equation of mixed type in a cylindrical domain is considered. This problem simulates the development of small disturbances in a transonic flow of a chemical mixture in a Laval nozzle. The existence of a regular solution is proved with the help of a priori estimates for a corresponding linear problem and the contractive mapping theorem. The solution of the linear problem is constructed by the Galerkin method.     

The natural oscillations of distributed inhomogeneous systems described by generalized boundary value problems

The problem of the constructive determination of the natural frequencies and modes of oscillations of distributed systems with substantially varying parameters is investigated. Unlike the classical case, the self-adjoint boundary-value problem allows of an arbitrary non-linear dependence of the coefficients of the equation on a numerical parameter, the eigenvalues of which are required to be obtained. An original numerical-analytic method is developed for a highly accurate construction of the desired solution. The computational efficiency of the algorithm, which possesses the property of accelerated (quadratic) convergence, is illustrated by the calculation of model examples. The approach can be extended to other classes of generalized problems of determining the critical values of the parameters and the forms corresponding to them, in particular, to the problem of the loss of stability of elastic systems with variable stiffnesses and inertial and force characteristics. A highly accurate solution of the classical Prandtl problem of determining the critical force which leads to lateral buckling of a long homogeneous cantilever beam is constructed, taking its weight into account.     

Stabilization of the non-trivial relative equilibria of a gyrostat with an elastic element in a circular orbit

The motion of a gyrostat, regarded as a rigid body, in a circular Kepler orbit in a central Newtonian force field is investigated in a limited formulation. A uniformly rotating statically and dynamically balanced flywheel is situated in the rigid body. A uniform elastic element, which, during the motion of the system, is subjected to small deformations, is rigidly connected to the rigid body-gyrostat body. The problem is discretized without truncating the corresponding infinite series, based on a modal analysis or using a certain specified system of functions, for example, of the assumed forms of the oscillations, which depend on the spatial coordinates and which satisfy appropriate boundary-value problems of the linear theory of elasticity. The elastic element is specified in more detail (a rod, plate, etc.), as well as its mass and stiffness characteristics and the form of the fastening, and the choice of the system of functions is determined. Non-trivial relative equilibria of the system (the state of rest with respect to an orbital system of coordinates when the elastic element is deformed) is sought approximately on the basis of a converging iteration method, described previously. It is shown, using Routh's theorem, that by an appropriate choice of the gyrostatic moment and when certain conditions, imposed on the system parameters are satisfied, one can stabilize these equilibria (ensure that they are stable).     

The antiplane problem of electroelasticity for a cylinder with a tunnel crack excited by an arbitrary system of surface electrodes

The antiplane mixed boundary-value problem of electroelasticity of the oscillations of an infinite piezoceramic cylinder, weakened by a curvilinear tunnel crack, is considered. Using special integral representations of the solution, the boundary-value problem is reduced to a system of singular integro-differential equations of the second kind with discontinuous kernels. The results of a numerical realization of the algorithm, characterizing the amplitude-frequency characteristics of a piecewise-uniform cylinder and the behaviour of the components of the electroelastic field in the region and on the boundary of the cylinder under conditions of the inverse piezoelectric effect, are presented.     

The three-dimensional contact problem with friction for an elastic wedge

Solutions of three-dimensional boundary-value problems of the theory of elasticity are given for a wedge, on one face of which a concentrated shearing force is applied, parallel to its edge, while the other face is stress-free or is in a state of rigid or sliding clamping. The solutions are obtained using the method of integral transformations and the technique of reducing the boundary-value problem of the theory of elasticity to a Hilbert problem, as generalized by Vekua (functional equations with a shift of the argument when there are integral terms). Using these and previously obtained equations, quasi-static contact problems of the motion of a punch with friction at an arbitrary angle to the edge of the wedge are considered. In a similar way the contact area can move to the edge of a tooth in Novikov toothed gears. The method of non-linear boundary integral equations is used to investigate contact problems with an unknown contact area.     

Harmonic and pulse excitations of multiply connected cylindrical bodies

The three-dimensional problem of the theory of elasticity of the harmonic oscillations of cylindrical bodies (a layer with several tunnel cavities on a cylinder of finite length) is considered for uniform mixed boundary conditions on its bases. Using the Φ-solutions constructed, the boundary-value problems are reduced to a system of well-known one-dimensional singular integral equations. The solution of the problem of the pulse excitation of a layer on the surface of a cavity is “assembled” from a packet of corresponding harmonic oscillations using an integral Fourier transformation with respect to time. The results of calculations of the dynamic stress concentration in a layer (a plate) weakened by one and two openings of different configuration are given, as well as the amplitude-frequency characteristics for a cylinder of finite length with a transverse cross section in the form of a square with rounded corners, and data of calculations for a trapeziform pulse, acting on the surface of a circular cavity, are presented.     

Transients in a shock-excited hydroelastic system of cylindrical shells of finite length

Forced oscillations of a system of cylindrical shells filled with a liquid are considered. The initial—boundary-value problem is solved by Godunov's method. Plots of kinematic parameters of the hydroelastic system reflecting the behavior of the wave fields are given.Institute of Geophysics of the Ukrainian Academy of Sciences. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 67, pp. 62–66, 1989.     

The construction of filters in non-linear deterministic systems

The method of residuals (see, e.g. [1–31]) is used to solve the problem of estimation when both object and observations involve noise, and the input determination problem [3–5] is considered. These estimation problems are solved by minimizing a certain functional, and this in turn involves solving a boundary-value problem at each instant of time. Depending on the recurrent method used to solve the relevant family of boundary-value problems, one obtains different representations of optimal non-linear niters for the estimated quantities. The choice of a specific representation depends on the degree to which the object with whose help the filter is being designed is well conditioned. A locally optimal filter of a design similar to that of filters for linear problems is constructed.     

Near-resonace transverse oscillations in an elastic layer. Steady-state solutions

The solutions of the equations of the non-linear evolution of transverse oscillations in a layer of an incompressible elastic medium under conditions close to resonance conditions are investigated qualitatively and using analytical methods. The oscillations are created by a small periodic motion of one of the boundaries in its plane, with a period that is close to the period of the natural oscillations of the layer. It is assumed that the medium can possess slight anisotropy and that the amplitude of the oscillations which arise is small. Previously obtained differential equations are used, which describe the slow evolution of the wave pattern of non-linear transverse waves. Two possible formulations of problems for these equations are considered. In the first formulation, it is determined what the external action must be in order that the non-linear evolution of oscillations or periodic oscillations occurs according to a (previously specified) desired law. In the second formulation it is assumed that the periodic motion of one of the boundaries is given. It is shown that a steady-state solution, that does not vary from period to period, can be represented by a continuous solution and also by a solution which contains discontinuities in the strain and velocity components. The mechanism of the overturn of a non-linear wave during its evolution and the formation of a discontinuity are qualitatively described.     

The motion of a body with a cavity partly filled with a viscous liquid

Many papers are concerned with the dynamics of a rigid body with a cavity filled with liquid (see the bibliography in [1]). The present paper deals with the motion of a rigid body having a cavity partly filled with a viscous incompressible liquid, and having a free surface. The shape of the cavity is arbitrary. The problem is considered in a linear formulation. The oscillations of the body with respect to its center of inertia and the motion of the liquid in the cavity are assumed small. The viscosity of the liquid is considered low. The solution of the problem of the oscillations of a body with a cavity partly filled with an ideal liquid is used as an initial approximation [1 to 6]. The viscosity is taken into consideration by the boundary layer method used before in similar problems [1 and 7 to 10). General equations are derived for the dynamics of a body filled with a liquid, for an arbitrary form of cavity. The coefficients of those integro-differential equations depend only on the solution of the problem of the oscillations of a body with a cavity of the given form filled with an ideal liquid. Since the corresponding problem has been solved for cavities of many forms [1 to 6, 11 and 12] in the case of an ideal liquid, the determination of the characteristic coefficients is reduced to the evaluation of quadratures. Several particular cases of motion are considered.     

Method of particular solutions for linear,two-point boundary-value problems

The methods commonly employed for solving linear, two-point boundary-value problems require the use of two sets of differential equations: the original set and the derived set. This derived set is the adjoint set if the method of adjoint equations is used, the Green's functions set if the method of Green's functions is used, and the homogeneous set if the method of complementary functions is used.With particular regard to high-speed digital computing operations, this paper explores an alternate method, the method of particular solutions, in which only the original, nonhomogeneous set is used. A general theory is presented for a linear differential system ofnth order. The boundary-value problem is solved by combining linearly several particular solutions of the original, nonhomogeneous set. Both the case of an uncontrolled system and the case of a controlled system are considered.This research, supported by the NASA-Manned Spacecraft Center, Grant No. NGR-44-006-089, is a condensed version of the investigations described in Refs. 1 and 2.