Resonant frequencies for a system of time-harmonic elastic wave

In this work, we consider the propagation of elastic waves outside a (compact) obstacle with smooth boundary. Existence, uniqueness results for the solution are established in a simple way. We study the meromorphic continuation to the whole complex plane of the solution. A proof alternative of the existence of resonant frequencies is given.     

A finite element assessment of natural frequencies of undamped elastic

A curved parametric element, capable of giving the natural unforced and undamped frequencies of thin rotational shells has been developed. The new formulation possesses a number of interesting features including true nodal conformity, genuine curved generator, optional number of degrees of freedom and variable thickness. The nature of the displacements and the generation of the stiffness and the mass matrices are outlined in the text. Finally the accuracy of the present formulation is examined by considering a selection of well known examples.     

The Cauchy problem for a one-dimensional nonlinear elastic peridynamic model

This paper studies the Cauchy problem for a one-dimensional nonlinear peridynamic model describing the dynamic response of an infinitely long elastic bar. The issues of local well-posedness and smoothness of the solutions are discussed. The existence of a global solution is proved first in the sublinear case and then for nonlinearities of degree at most three. The conditions for finite-time blow-up of solutions are established.     

Calculation of the spectral dependence of the Anderson localization criterion in a one-dimensional system with correlated diagonal disorder

We consider the problem of calculating the Anderson criterion for a one-dimensional disordered chain with correlated disorder. We solve this problem by the perturbation method with the inverse correlation length as the small parameter. We show that in a correlated system, the degree of localization not only naturally decreases but its spectral dependence also differs significantly from the spectral dependence in uncorrelated chains. The calculations are based on the method for constructing joint statistics of Green’s functions, which was previously used to analyze uncorrelated one-dimensional systems. We illustrate the theoretical calculations with a numerical experiment.     

On the length equalities for one-dimensional rings

In this article we characterize noetherian local one-dimensional analytically irreducible and residually rational domains (R,m_R) which are non-Gorenstein, the non-negative integer is equal to τ_R-1 and ?(R/(C+xR))=2, where τ_R is the Cohen-Macaulay type of R, C is the conductor of R in the integral closure of R in its quotient field Q(R) and xR is a minimal reduction of m by giving some conditions on the numerical semi-group v(R) of R.     

Precise integration method for natural frequencies and mode shapes of ocean risers with elastic boundary conditions

The effects of the elastic constraints simulating damaged and undamaged boundaries on the natural frequencies and mode shapes of ocean risers with a variable axial tension are investigated using the precise integration method (PIM). The classical high-order variable-coefficient partial-differential governing equation of the free vibration of ocean risers is reduced to a set of first-order ordinary differential equations and efficiently solved by the PIM. The main advantages of the PIM are that the numerical results can be calculated with high accuracy even when total element number n, number of iterations N and Taylor expansion terms r are small. Moreover, the computing time is quite short. Various boundary conditions are modeled as linear elastic constraints using a pair of translational and torsional springs, and four types of boundary damage coefficients are proposed to investigate the effects of a damaged boundary on the natural frequencies. The results for specific boundary conditions show agreement with those reported in the literature, and the calculation errors are very small in comparison with the analytical solution. Overall, the methodology of PIM is applicable for the investigation of the natural frequencies and mode shapes of ocean risers with a variable axial tension and cross-section and various boundary conditions.     

Some inequalities concerning the characteristic frequencies of elastic continuum

Summary We consider the boundary value problem αz″(x)+m(x)y(x)=0, αy″(x)+p(x)z(x)=0, xε[0, 1], y(0)=y(1)=z(0)=0, where the functions m(x) and p(x) are assumed integrable and positive everywhere in [0, 1]. As the main result we obtain the inequalities for n=1, 2, ... where δ_n(m, p) stands for the product of the first n eigenvalues α_i(m, p) of the above system and where δ_n(m) abbreviates δ_n(m, m). Entrata in Redazione il 6 febbraio 1976.     

Mathematical justification of a one-dimensional model for general elastic shallow arches

We present a bending model for a shallow arch, namely the type of curved rod where the curvature is of the order of the diameter of the cross section. The model is deduced in a rigorous mathematical way from classical tridimensional linear elasticity theory via asymptotic techniques, by taking the limit on a suitable re-scaled formulation of that problem as the diameter of the cross section tends to zero. This model is valid for general cases of applied forces and material, and it allows us to calculate displacements, axial stresses, bending moments and shear forces. The equations present a more general form than in the classical Bernoulli–Navier bending theory for straight slender rods, so that flexures and extensions are proved to be coupled in the most general case. © 1998 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Stability of the one-dimensional motions of a viscous gas with a linear dependence of the velocity on the coordinates

The stability of a new solution of the equations of one-dimensional gas dynamics is investigated. This solution is a generalization of the solutions of Sedov /1, 2/ to the case of a viscous, thermally conducting ideal gas with an exponential dependence of the coefficient of viscosity and thermal conductivity on temperature. The linearized equations for small perturbations (the effects of thermal conductivity are not allowed for in the equations for the perturbations), which contain functions of time and the radial coordinate in the coefficients, can be solved by separation of the variables. The conditions under which instability arises are determined from an analysis of the time parts of the solutions. The stability of the solutions /1/ has been considered in /3–5/.     

A justification of the one-dimensional linear model of elastic beam

In this paper we prove that the one-dimensional model of elastic beam is an approximation to the three-dimensional linear theory of elasticity.     

The natural vibration frequencies of multilayer orthotropic shells of revolution

We propose a method of computing the natural vibrations of thin elastic multilayer shells of revolution. The generalized mathematical model makes it possible to take account of the orthotropy of the material of the component layers, the angle of orientation of the elastic constants relative to the coordinate lines, and attached solid bodies. The solution is constructed for various boundary conditions on the basis of the linear theory of shells using the Ritz method. We give the results of numerical studies. Four figures. Bibliography: 5 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 27, 1997, pp. 114–120.     

On a one-dimensional version of the dynamical Marguerre-Vlasov system

A one-dimensional version of the so-called Marguerre-Vlasov system of equations describing the vibrations of shallow shells is considered. The system depends on a parameter 0 in a singular way and undergoes the effect of damping mechanisms. We show that the system converges to a nonlinear beam equation while the energy decays exponentially uniformly (on 0) as time goes to infinity.     

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.     

Natural frequencies of a multi-degree-of-freedom vibration system

A chain structured mass-spring vibration system with N degrees of freedom is considered where massesM _j = m_jm = m/j and spring stiffnesses k_j = c_jk = (N + 1 - j)k, j = 1, …, N, have been applied. The conjecture of Mikota [1] on the natural frequencies is discussed showing how difficult it is to find an explicit solution of the related eigenvalue problem. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Limit theorems for one-dimensional nonhomogeneous stochastic diffusion equations under irregular dependence of the coefficients on a parameter

The limiting behavior of solutions of stochastic Ito differential equations is investigated with coefficients which could be -type sequences or possess a degeneracy of some other form. Application of the results obtained to a study of the limiting behavior of a solution of the Cauchy problem for parabolic differential equations in partial derivatives is considered.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 4, pp. 435–443, April, 1990.