Sergei B. Filippov

Star‐like Fullerene Containing Poly(Vinylpyrrolydone) Derivatives: Chloroform Solution Properties

Abstract Star‐like fullerene C60 derivatives with different branch number, synthesized by the reaction of fullerene with poly(vinylpyrrolydone) (PVP) macromolecules bearing the terminal amino‐groups, were investigated in solution by the viscometry, dielectric, and electrooptical Kerr‐effect methods in comparison with the ordinary linear PVPs of the same molecular mass. It was shown that covalent linkage of branches to fullerene through amino‐groups leads to appearance of a polar and optically anisotropic nanoparticle (amine‐substituted C60) in the center of the coil of star‐like polymers that radically changes dielectric and electrooptical properties of the initial polymer. Effect of fullerene on the dimension of polymer coil had been detected also.

Asymptotic methods in mechanics of solids

Asymptotic Estimates.- Asymptotic Estimates for Integrals.- Regular Perturbation of ODEs.- Singularly Perturbed Linear ODEs.- Linear ODEs with Turning Points.- Asymptotic Integration of Nonlinear ODEs.- Bibliography.- Index.

Numerical and Asymptotic Modeling of Annular Plate Vibrations

Free axisymmetric flexural vibrations of an annular elastic thin plate are studied. Numerical solutions of eigenvalue problem for various boundary conditions are obtained. The plate can be used as a model of the supporting frame of a shell. In this connection the boundary conditions corresponding the attaching of the plate to a cylindrical shell are also considered. The plate is called narrow if the ratio of its width to the radius of the inner edge is small. For the vibrations analysis of a narrow plate new asymptotic methods are elaborated. Comparison asymptotic and numerical results shows, that the error of the approximate formulae quickly decreases with reduction of the plate width.

Asymptotic Integration of Nonlinear Differential Equations

There are several types of asymptotic expansions for the solutions of nonlinear differential equations. Regularly perturbed nonlinear equations were considered in Chap. 3.

Asymptotic Estimates for Integrals

Mechanical problems can be described by differential equations, the solutions of which often cannot be expressed by elementary functions, but have an integral representation.

Singularly Perturbed Linear Ordinary Differential Equations

In this chapter, we study systems of linear differential equations with variable coefficients.

Singularly Perturbed Linear Ordinary Differential Equations with Turning Points

In this chapter, we consider systems of linear ordinary differential equations with variable coefficients and a small parameter \(\mu \) in the derivative terms.

Regular Perturbation of Ordinary Differential Equations

In this chapter we find asymptotic solutions of regularly perturbed equations and systems of equations, to which problems in mechanics are reduced. We consider Cauchy problems, problems for periodic solutions and boundary value problems.

Finite Axisymmetric Deformation of an Inflatable Anisotropic Toroidal Membrane

Finite axisymmetric deformation of a thin toroidal shell under action of internal pressure is studied. The shell is reinforced by two systems of threads located along parallels and meridians and is considered as anisotropic membrane. The nonlinear theory of membranes is used. To find membrane deformations and displacements the system of ordinary differential equations of the fourth order is delivered. The method of asymptotic integration in the case when the meridian radius is much smaller than the parallel one is elaborated. Asymptotic and numerical results are compared.

Free Vibrations of Square Elastic Tubes with a Free End

The localized vibrations of a thin-walled square tube with a free end are studied. By means of asymptotic methods, an expression for the natural frequency is found. The asymptotic results agree well with the numerical results obtained using the Finite Element Method. The dependence of the natural frequency on the tube length is analyzed.