Basis property of eigenfunctions of the Frankl problem with a nonlocal evenness condition and with a discontinuity in the solution gradient

In the present paper, we write out the eigenfunctions of the Frankl problem with a nonlocal evenness condition and with a discontinuity of the normal derivative of the solution on the line of change of type of the equation. We show that these eigenfunctions form a Riesz basis in the elliptic part of the domain. In addition, we prove the Riesz basis property on [0, π/2] of the system of cosines occurring in the expressions for the eigenfunctions. Earlier, the Riesz basis property was proved for the eigenfunctions of the Frankl problem with a nonlocal evenness condition and with continuous solution gradient.     

The coupled problem of thermoelastic vibrations of a cylinder with cavities

Using a method based on the machinery of analytic functions of complex variables, we construct the solution of the problem of steady planar strain of an isotropic cylinder with curvilinear cavities caused by uneven heating of the surfaces of the cylinder and the cavities. It is an exact solution of the equations and an approximate solution of the boundary conditions. The results of numerical studies are given for a hollow cylinder undergoing thermoelastic radially symmetric vibrations. Two figures. Bibliography: 4 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 26, 1996, pp. 82–86.     

The uncoupled problem of thermoelastic vibrations of a cylinder

Using the theory of functions of a complex variable, we construct an approximate solution of the problem of steady thermoelastic vibrations of a cylinder with curvilinear cavities, not taking account of the interaction of the strain and temperature fields. The results of numerical computations are given in the case of radially symmetric vibrations of a cylinder with a cavity. Two figures. Bibliography: 5 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 28, 1998, pp. 107–112.     

Basis property of the eigenfunctions of a generalized gasdynamic Frankl problem with a nonlocal evenness condition and with a discontinuity in the solution gradient

We find the eigenfunctions of a generalized Frankl problem with the use of Bessel functions. We prove that these eigenfunctions form a Riesz basis in the space L ₂(D ₊), where D ₊ is the elliptic part of the domain. In addition, we prove the Riesz basis property of a trigonometric function system and the completeness of this system in the space L ₂(0, π/2).     

On the basis property of eigenfunctions of the Frankl problem with a nonlocal parity condition

The Frankl problem without the spectral parameter was considered by Bitsadze and Smirnov. The present paper gives the eigenvalues and eigenfunctions of the Frankl problem with the odd parity condition. We prove the completeness of eigenfunctions. The Frankl problem with a nonlocal parity condition for the Lavrent’ev-Bitsadze equation is studied. The eigenvalues and eigenfunctions are found, and the basis property of the eigenfunctions in the elliptic part of the domain in the space L ₂ is proved.     

Homogeneous solutions of the problem of steady vibrations of a piezoceramic cylinder

Steady vibrations of a hollow piezoceramic cylinder with radial polarization are considered. An analytical and numerical analysis is performed of the homogeneous solutions, and the behaviour of the dispersion curves of the real and complex modes is investigated as a function of the geometrical parameters.     

On the basis property of eigenfunctions of the Frankl problem with a nonlocal parity condition of the second kind

In the present paper, we write out the eigenvalues and the corresponding eigenfunctions of the modified Frankl problem with a nonlocal parity condition of the second kind. We analyze the completeness, the basis property, and the minimality of the eigenfunctions depending on the parameter of the problem.     

Basis property of eigenfunctions of nonselfadjoint operator pencils generated by the equation of nonhomogeneous damped string

We consider a class of nonselfadjoint quadratic operator pencils generated by the equation, which governs the vibrations of a string with nonconstant bounded density subject to viscous damping with a nonconstant damping coefficient. These pencils depend on a complex parameterh, which enters the boundary conditions. Depending on the values ofh, the eigenvalues of the above pencils may describe the resonances in the scattering of elastic waves on an infinite string or the eingenmodes of a finite string. We obtain the 7asymptotic representations for these eigenvalues. Assuming that the proper multiplicity of each eigenvalue is equal to one, we prove that the eigenfunctions of these pencils form Riesz bases in the weightedL ²-space, whose weight function is exactly the density of the string. The general case of multiple eigenvalues will be treated in another paper, based on the results of the present work.     

Analysis of a Signorini problem with friction

This paper deals with the study of a nonlinear problem of frictionalcontact between an elastic body and a rigid foundation. Theelastic constitutive law is assumed to be nonlinear and thecontact is modelled with Signorini's conditions and a versionof Coulomb's law of dry friction. We present two weak formulationsof the problem and establish existence and uniqueness results,using arguments of elliptic variational inequalities and a fixed-pintproperty. Moreover, we prove some equivalence results and studythe behaviour of the solution when the coefficient of frictiontends to zero.     

Expansion in eigenfunctions of the Sturm-Liouville problem on a bundle graph

In this paper we consider the applicability of the Fourier method for partial differential equations on spatial grids (we choose a bundle graph as a model). This leads to an important problem, namely, to the expansion of a given function in eigenfunctions of the corresponding Sturm-Liouville problem on a grid. We study a model problem which describes a symmetric case, when one considers physically identical one-dimensional continuums on the bundle graph. Such problems arise, for example, in the modeling of oscillating processes of an elastic mast with supporting elastic ties.     

Avoidance of friction induced vibrations on seals

In many fields of technical and non–technical applications occur friction induced vibrations. If undesired, which applies to most of these oscillations, different methods exist for their avoidance or at least reduction. A recently investigated method, the systematic treatment of contacting surfaces by different techniques is shown in this work with the example of a pneumatic seal. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Steady-state vibrations of a compound isotropic cylinder under a plane strain

We solve the problem of two-dimensional strain of a compound isotropic cylinder deformed by distributed pulsing forces. We study the influence of the geometric and elastic characteristics of the layers on the magnitude of the stresses and the lowest natural frequency of radial-symmetric vibrations of a three-component cylinder. Two figures Bibliography: 3 titles Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 25, 1995, pp. 103–108.     

Half-linear Sturm-Liouville problem with weights: Asymptotic behavior of eigenfunctions

We consider a nonlinear Sturm-Liouville problem with weights. Under suitable assumptions on weights we prove an asymptotic estimate for the decay of the eigenfunctions. An application to the radial problem on the entire ? ^N is given.     

On the basis property of eigenfunctions of a singular second-order differential operator

We consider the first boundary value problem for a singular differential operator of second order on an interval with transmission conditions at an interior point of the interval. We show that the system of eigenfunctions corresponding to this problem is complete in the space L ²(0, 1) and forms a Riesz basis in that space.