Parametric vibrations of viscoelastic beams

The method of averaging is used for an analysis of the effect which the viscous resistance of a beam material has on the stability range of beam vibrations under the most general stipulations regarding the beam supports.     

The dual nature of the transverse vibrations of an elastic rod

The transverse vibrations of an elastic rod, to one of which displacements are applied while the other end is free, are investigated. It is assumed that the propagation velocity of the perturbations in the rod is finite. The unperturbed part performed rotational motion around the centre line. The angle of rotation is expressed by the angle of curvature of the centre line of the perturbed part of the rod. Two types of elastic vibrations are obtained: (1) the rod vibrates elastically due to displacements applied at the end, and (2) when performing rotational motion elastic vibrations and additional forces occur in the rod due to elasticity [1].     

Flexural vibrations of a moving rod

The problem of the transverse natural vibrations of part of a rod between two coaxially fixed guides, moving with an arbitrary constant velocity, is investigated. The conditions of rigid clamping are taken as the boundary conditions. Additional shear stresses, due to longitudinal tension or compression are taken into account. Relations defining the natural frequencies and forms are constructed in an exact formulation by Fourier method. The dependence of the natural frequencies and forms of the lowest vibration modes on the rate of displacement, unknown in the literature, are constructed, and their features are established. A modelling and animation of unusual wave motions of the rod are presented. The main characteristics for the higher vibration modes are constructed.     

Parametric vibrations of an orthotropic cylindrical shell with an elastic core. 2. Some numerical results

It is shown that the operator of the problem belongs to the class of oscillation operators. The first five regions of dynamic instability of an elastic orthotropic cylindrical shell with an elastic isotropic core are determined in the first approximation. The effect of the core and transverse shear in the shell on the width and location of these regions of dynamic instability of the system is determined. The effect of transverse shear on the natural frequencies of the empty shell is established.     

Parametric vibrations of cylindrical shells with a viscoelastic core

The method of calculating the axisymmetric and nonaxisymmetric parametric vibrations of a cylindrical shell bonded to an elastic core [2] is extended to the case of hollow and solid viscoelastic cores by substituting for the material moduli in the equations of motion of the core integral operators with kernels in the form of an exponential and a sum of exponentials. Expressions are given for the reaction of the viscoelastic core, together with the equation of the boundaries of the spectrum of principal regions of dynamic instability. The effect of relaxation time and the long-term modulus of elasticity of the core on the shape and location of the regions of dynamic instability is analyzed.     

A parametric investigation of the harmonic vibrations of a poroelastic rod

The problems of calculating the forced harmonic vibrations of porous structures saturated with a fluid are considered. The equations of motion are obtained by starting from general relations of the Biot theory of poroelasticity and continuous mechanics after taking into account the anisotropy of the elastic and hydraulic properties of the medium. The influence of the mechanical constants of the material on its dynamical characteristics is investigated using the example of the flexural vibrations of a simple rod-shaped structure.     

Discrete transparent boundary conditions for the equation of rod transverse vibrations

Local perturbations of an infinitely long rod travel to infinity. On the contrary, in the case of a finite length of the rod, the perturbations reach its boundary and are reflected. The boundary conditions constructed here for the implicit difference scheme imitate the Cauchy problem and provide almost no reflection. These boundary conditions are non-local with respect to time, and their practical implementation requires additional calculations at every time step. To minimise them, a special rational approximation, similar to the Hermite - Padé approximation is used. Numerical experiments confirm the high “transparency” of these boundary conditions and determine the conditional stability regions for finite-difference scheme.     

Parametric resonances in the problem of longitudinal impact on a thin rod

The longitudinal impact on a thin elastic rod, which generates a periodic system of longitudinal waves in it, is considered. At definite values of the parameters of the problem in the linear approximation, these waves induce parametric resonances, which are accompanied by an unlimited increase in the amplitude of the transverse vibrations. To obtain finite values of the amplitudes, a quasilinear system is considered in which the effect of the transverse vibrations on the longitudinal vibrations is taken into account. This system was previously solved using the Bubnov–Galerkin method and beats accompanied by energy transfer between the transverse and longitudinal vibrations were obtained. In this work, an approximate analytical solution of the system has been derived that is based on double-scale expansions. A qualitative analysis of this solution has been carried out. An estimate of the maximum transverse bending has been obtained for various methods of loading. Both shortand long-term pulses have been considered. It has been shown that, in the case of a spontaneously applied long-term pulse that is lower than the Euler critical load, intensive transverse vibrations can occur.     

Optimization of the boundary displacement control of vibrations of a rod consisting of two dissimilar parts

We optimize the boundary displacement control that is applied at one end of a rod consisting of two dissimilar parts and brings the rod vibrations from a given initial state to a given terminal state for the case in which the other end of the rod is fixed.     

Optimization of elasticity-induced boundary control for the vibrations of a rod consisting of two different segments

Doklady Mathematics -     

Control of damped vibrations of a rod by means of a concentrated force or moment loading

The optimal local loading regimes of a vibrating rod (one endpoint of the rod is fixed, while the second is stress-free), which ensure the damping of certain modes of elastic vibration over a minimal time interval, for a minimal norm of the control function and also under the simultaneous minimization of the time and the norm of the load function are given.Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 25, pp. 71–75, 1987.     

Analysis of an impact damper of vibrations

An analysis of the system with two degrees of freedom, which – provided that its parameters are properly selected – can operate as an impact damper of vibrations, is presented. The investigations have been carried out in two directions. In the first one, it has been assumed that the mass of the system playing the role of a damper is 10 times lower than the mass of the basic system and that this mass is suspended from a spring characterised by low stiffness. In the second case, the damper mass has been assumed to be four times smaller than the basic mass and it is suspended from a spring whose stiffness is four times lower than the stiffness of the basic system spring. Moreover, it has been shown that the impact damper of vibrations operates more efficiently than a classical dynamic vibration damper.     

On vibrations of an inhomogeneous anisotropic plate

An equation of motion of a flat inhomogeheous anisotropic plate is considered. Formal asymptotic solutions are constructed by applying the space-time ray method. An equation describing the flow of energy is obtained in the form of a continuity equation. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 73–82. Translated by A. S. Golubeva.     

Heat distribution in an infinite rod

We construct an asymptotic expansion of the solution of the Cauchy problem for the one-dimensional heat equation for the case in which the initial function at infinity has power asymptotics.     

Parametric curve with an implicit domain

In this paper we present a new representation of curve, named parametric curve with an implicit domain (PCID), which is a curve that exists in parametric form defined on an implicit domain. PCID provides a bridge between parametric curve and implicit curve because the conversion of parametric form or implicit form to PCID is very convenient and efficient. We propose a framework model for mapping given points to the implicit curve in a homeomorphic manner. The resulting map is continuous and does not overlap. This framework can be used for many applications such as compatible triangulation, image deformation and fisheye views. We also present some examples and experimental results to demonstrate the effectiveness of the framework of our proposed model.     

Axisymmetric vibrations of an orthotropic non-uniform cylindrical shell

The axisymmetric vibrations of a circular cylindrical shell of constant thickness, orthotropic along the principal geometrical directions and non-uniform along the generatrices, are considered. It is shown, using a particular problem as an example, that the non-uniform physical-mechanical characteristics of the shell material may lead to qualitatively new results in problems of shell dynamics.     

Steady-state vibrations of an unbounded linear piezoelectric medium

We consider fundamental (Dirichlet and Neumann-type) boundary value problems in a theory of generalized plane strain for the steady-state vibrations of an infinite piezoelectric medium with transversely isotropic symmetry (6 mm). Using integral equation methods with the appropriate Sommerfeld-type radiation conditions, we prove existence and uniqueness results for the corresponding exterior boundary value problems. Exact solutions are obtained in the form of integral potentials. (Received: September 27, 2005)