Sessions 2014-2015

Vibrations of stretched beams with clamped ends are analyzed. The numerical solution, asymptotic solution as the first approximation of Vishik and Lyusternik method and solution with the help of dynamic edge effect method for wavenumbers 1 and 5 are obtained. The obtained results are used to examine the applicability of the first approximation of Vishik and Lyusternik method and dynamic edge effect method.

The dynamics of omni-wheeled robot is analyzed for three- and four-wheeled vehicles. In assumprion that the motion of each wheel in the longitudinal direction has no slip the system with nonholonomic constraints is derived. Maggie's equations are used to develop the mathematical model of vehicles. The numerical solution of equations is obtained for different cases of control torques. The direct dynamics problem is solved for some trajectories of motions.

Flows of different fluids through single-walled carbon nanotubes (SWCNTs) with boundary walls having the perfect and defective graphene structures have been investigated by means of molecular dynamics (MD) simulations. Ithas been shown that the boundary wall structure has a very strong influence on not only an average fluid flow rate but also on shapes of trajectories of individual fluid atoms (molecules) and fluid centres of mass. The fluid flows through SWCNTs surrounded by different liquid environments have been also simulated and an influence of these environments on the average fluid flow rates have been studied. Ithas been revealed a strong dependence of the average fluid flow rate on a molecular polarity of fluids flowing through SWCNTs and those surrounding the tubes. Ithas been shown that, for multi-walled carbon nanotubes (MWCNTs), aneffect of liquid environment on the fluid flow can be significantly suppressed.

Fractures of support-motor apparatus are widespread injuries. People fairly often need surgery for fixation of the bone fragments. In this work, mathematical models of osteosynthesis of fractures of the femoral neck and long bones are considered. In the first case, the fixation was done by special screws. Secondly, elastic fixing is used. It is proposed to carry out the analysis of stress-strain state of construction on the basis of the theory of bending of beams.

Buckling under action of external lateral pressure of the cylindrical shell stiffened by identical rings with rectangular cross-section sections and a non-stiffened shell of a neutral surface having the same sizes and made of the same material is considered. It is supposed, that the stiffened and non-stiffened shell lose stability at identical critical pressure. To get approximate formulas for the critical pressure a combination of asymptotic method is used. First we seek solutions as a sum of slowly varying functions and edge effect integrals. Thus the initial singularly perturbed system of differential equations is reduced to an approximate system of the smaller order. Assuming that the rings may be considered as circular beams we obtain the solution of the approximate eigenvalue problem describing buckling of ringstiffened shell by means of homogenization procedure. Using the simple asymptotic formulas for critical pressure the approximate relations for calculation of optimal stiffened shell parameters corresponding to the minimal value of its weight are received in closed form. It is shown that at increase in the ratio of ring width to ring thickness the ratio of weights of stiffened shell to weights of non-stiffened shell decreases. The examples of calculations of optimal parameters are presented. Results of the paper may be used at designing thin-walled structures.

...

Firstly, the 2D problem of propagation of diffusing pollutant on the water surface is analyzed. Such model may be used, for example, to study the lifetime of the toxic pollutant spot on the water surface. For iso-tropic medium the mathematical model consists of the boundary value problem for the diffusion equation, the analytical solution of which may be obtained by means of Fourier method with consequent expansion of the arbi-trary function in Bessel functions. The found analytical solution is compared with the numerical solutions of the boundary value problem obtained with Mathematica and MathLab software packages. The dependence in time of the pollution spot size is studied and the effect of geometrical and physical parameters on the pollution spot radius is discussed. Also the 3D problem of toxic pollutant propagation set on the flat bottom is examined. The size of the domain, where the concentration of the toxic pollutant is higher than the maximum permissible concentration, and the dynamics of this domain are studied.

The critical load and the buckling modes of a transversely isotropic circular cylindric shell under axial compression are found. It is assumed that the shell curvilinear edge is free or weakly supported. In these cases the buckling mode is localized near this edge and the critical load is lower than in the case of the clamped edges. Previously, this problem for isotropic shell is solved based on the 2D Kirchhoff - Love (KL) model. Here it is assumed that the transversely shear modulus is small, hence, the solution is based on the Timoshenko - Reissner (TR) model. The non-dimensional critical load depends on two basic non-dimensional parameters - the wave parameter q and the shear parameter g. If the buckling mode occupies the entire shell surface then this mode is axisymmetric buckling (q = 0). If g = 0, then according to the KL model we get the classical value of critical load. With the increase of g (or with the decrease of the shear modulus), the critical load decreases up to the point g approximately 1. At the point g greater than 1 or equals it, the material loses its stability.

A mathematical model of key mechanisms of the immune response is described. The model gives a consolidated view of proliferation and differentiation processes in nonhomogenous T- and B-cell populations. Original ideas were introduced, allowing to describe complex immune processes by simple PDE models. A specific immunological problems relating to autoimmune disease pathogenesis were successfully solved using the model.

... interaction of mechanical structures with fluid flow is showed.

Small free vibrations of a rotating cylindrical shell which is in a contact with rigid cylindrical rollers are considered. Assumptions of semimomentless shell theory are used. Vibrations modes in the circumferential direction are represented as Fourier series. Increase in number of members of a Fourier series lead to complication of algorithm, but allowed to find additional frequencies, which weren't found in the previous works where the number of number of members was supposed equal to number of rollers. It is especially important that for thin shells among these additional frequencies there were the lowest frequencies representing greatest interest for applications. The algorithm based on analytical solution for the evaluation of frequencies and vibration modes is developed. Calculations of frequencies and vibration modes by the Finite Elements Method are carried out. Analytical results are compared with results of numerical calculations.

The deformation of the orthotropic spherical layer under normal pressure applied on the outer and inner surfaces is analyzed. The layer is assumed to be slightly orthotropic, i.e. it is supposed that the tangential elastic moduli slightly differ from each other, what permits to apply asymptotic methods. For the shell, which is much softer in the transverse direction than in the tangential plane, in the zeroth approximation one gets singularly perturbed boundary value problem. Solution of the problem in zero approximation allows to obtain the asymptotic formula for the change of the relative layer thickness under normal pressure. For the cases of the thick and thin layers the last formula may be simplified. Also the effect of Poisson ratio and the layer thickness on the deformation is studied in the paper. The asymptotic results well agree with the exact solution. The developed formulas are used in analysis of the scleral shell under intraocular pressure and may also be used in solution of the inverse problem, i.e. in evaluation of the elastic moduli of human eye shells.