Non-linear near-resonance oscillations of an elastic incompressible layer

Plane one-dimensional waves of small amplitude, propagating transverse to an incompressible elastic layer and reflected successively from its boundaries, are considered. The oscillations are caused by small periodic (or close to periodic) external action on one of the layer boundaries, when the period of the external action is close to the period of natural oscillations of the layer. One of the boundaries of the elastic layer is fixed, while the other performs small specified two-dimensional motion in its plane. In such a near-resonance situation, non-linear effects occur which may build up over time. A system of equations is obtained which describes the slow change in the functions characterizing the oscillations of the medium in each period of the external action. It is assumed that all the quantities depend both on real time, any change of which in the approach considered is limited to one period, and on “slow” time, for which one period of real time serves as a small quantity. It is assumed that the evolution of the solution occurs when the slow time changes, while the role of real time is similar to the role of a spatial variable. This system of equations is obtained by the method of averaging over a period of the quantities representing nonlinear terms and the effect of the boundary conditions in the equations. It contains derivatives with respect to the real and slow times and also values of the functions characterizing the solution averaged over a period of the real time. If the averaged values are known, the equations have a hyperbolic form and their solutions can be both continuous and contain weak and strong discontinuities.     

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.     

Torsional oscillations of a circular stamp on an elastic layer with a cylindrical cavity

The problem of torsional oscillations of a stamp that is linked with an elastic stratum which contains a cylindrical cavity is considered. The problem is formulated in the form of conjugate integral equations that are related to the integral Weber transforms. The conjugate equations are reduced to an equivalent Fredholm equation of the second kind.Translated from Dinamicheskie Sistemy, No. 9, pp. 54–59, 1990.     

A path integral approach for solving Euler''s equation describing the oscillations of an infinite bar

We provide a path integral representation of the solution to Euler's equation, describing the transversal oscillations of a rail under action of damping, restoring, and external forces, with respect to the trajectories of a quasi-Markov process naturally associated with Euler's equation for the free, infinite rail.     

Harmonic oscillations of a rigid punch on a porous elastic layer

The plane dynamic contact problem of the harmonic oscillations of a rigid punch on the free surface of an elastic layer of porous isotropic material with linear properties is considered. The Fourier transformation of the problem is reduced to a Fredholm integral equation of the first kind in the contact pressure. The properties of the kernel of the fundamental integral equation are investigated and a numerical method of solving it is constructed. Numerical results are compared with existing results in classical limiting cases.     

A dynamic contact problem for an elastic half-plane in the case of high frequency oscillations

Harmonic high frequency oscillations of a rigid stamp coupled without friction tc an elastic half-plane are considered. The main difficulty in constructing the high-frequency asymptotic forms is that of carrying out the effective factorization of the kernel of the basic integral equation. A function is proposed, which takes into account all properties of the kernel, enables it to be uniformly approximated and is easily factorized. Such a solution of the problem of approximate factorization makes it possible to write, in a simple explicit form, the principal term of the asymptotic expression of the solution. The nature of the distribution of contact stresses under the stamp is studied, as well as the compliance of the foundation and phase shift between the applied force and the displacement of the stamp.     

A free boundary problem for a parabolic system describing an ecological model

In this paper, the one-dimensional quasilinear Verigin problem and the relative diffraction problem are studied. This free boundary problem describes a motion of the interface between two groups of animals. The existence and uniqueness of a classical solution locally in time are obtained. The continuous dependence of the solution on the internal boundary and the smoothness of the free boundary are also given.     

A mathematical model for the angular motion of large space structures with gyroscopic drive for an active compensation of elastic oscillations

Doklady Mathematics -     

On an effective model describing a layered periodic elastic medium with slide contacts on the interfaces

Effective models are derived for layered periodic elastic media'with slide contacts on all interfaces. In the case where each period consists of n layers with different plate velocities, the effective model has n phases. These models are investigated for typical media. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210, 1994, pp. 192–212. Translated by L. A. Molotkov.     

The oscillations of a rod in an inhomogeneous elastic medium

The local length-dependence of the natural frequencies and forms of plane transverse oscillations of a thin inhomogeneous rod in an elastic medium with a variable stiffness and arbitrary elastic-fastening boundary conditions is investigated. It is established that the presence of an external elastic medium, described by the Winkler model, can lead to an anomalous effect – an increase in the natural frequencies of lower oscillation modes as the length of the rod increases continuously. The extremely fine properties of this change as a function of the length, the mode number and the method of fastening are revealed. The oscillations in the case of standard methods of fastening are investigated separately. Simple examples, which illustrate the anomalous dependence of the natural oscillation frequencies of the rod in an extremely inhomogeneous elastic medium with different boundary conditions are calculated.     

Free and nonsteady-state oscillations of elastic fluid-filled systems situated on an elastic base

We develop a method of computing the nonsteady-state and free oscillations of a framed elastic structure situated on an elastic base and containing an ideal compressible fluid. The solution uses the method of integral transforms in conjunction with the method of orthogonal polynomials. In the transform space the problem reduces to systems of linear algebraic equations. The Fourier transform is applied to return to the original space. Examples of the computation are given.Translated fromDinamicheskie Sistemy, No. 6, 1987, pp. 69–72.     

A two-dimensional model for extensional motion of a pre-stressed incompressible elastic layer near cut-off frequencies

A two-dimensional model for extensional motion of a pre-stressedincompressible elastic layer near its cut-off frequencies isderived. Leading-order solutions for displacement and pressureare obtained in terms of the long wave amplitude by direct asymptoticintegration. A governing equation, together with correctionsfor displacement and pressure, is derived from the second-orderproblem. A novel feature of this (two-dimensional) hyperbolicgoverning equation is that, for certain pre-stressed states,time and one of the two (in-plane) spatial variables can changeroles. Although whenever this phenomenon occurs the equationstill remains hyperbolic, it is clearly not wave-like. The second-ordersolution is completed by deriving a refined governing equationfrom the third-order problem. Asymptotic consistency, in thesense that the dispersion relation associated with the two-dimensionalmodel concurs with the appropriate order expansion of the three-dimensionalrelation at each order, is verified. The model has particularapplication to stationary thickness vibration of, or transientresponse to high frequency shock loading in, thin walled bodies.     

Study of oscillations in a viscoelastic cylindrical shell in an elastic medium

An equation was obtained and solved for oscillations in a viscoelastic cylindrical shell located in the ground. A numerical example is examined.Moscow Electronic Engineering Institute. Translated from Mekhanika Polimerov, No. 1, pp. 178–181, January–February, 1974.     

A singular transport model describing cellular division

In this Note, we study a system of partial differential equations with a singular transport term describing blood cellular production. The population of cells considered is capable of simultaneous proliferation and maturation. We prove that uniqueness of solutions depends only on stem cells.     

Calculation of the nonlinear viscoelastic oscillations of an antivibration layer

It is proposed to use a viscoelastic layer to protect equipment against vibration. The principal quadratic theory of hereditary viscoelasticity is used as the physical relation between the forces and displacements. The solutions obtained for the integrodifferential vibration equation make it possible to minimize the displacements and accelerations of the protected equipment.Moscow Institute of Electronic-Machine Building. Translated from Mekhanika Polimerov, No. 2, pp. 321–326, March–April, 1972.     

Internal layer oscillations in FitzHugh-Nagumo equation

A controller-propagator system with a FitzHugh-Nagumo equation can be reduced to a free boundary problem when a layer parameter ε is equal to zero. We shall show the existence of solutions and the occurence of a Hopf bifurcation for this free boundary problem as the controlling parameter τ varies.