I think, therefore I am.

A computer does not think, therefore it is not.

Analysis of vibrations of stretched beams by analytical and asymptotic methods.

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-wheel mobile robots

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.

Influence of liquid environment and bounding wall structure on
fluid flow through carbon nanotubes.

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.

Mathematical models of fracture fixation

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.

Optimal design of a stiffened cylindrical shell

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.

Elasto-plastic pure bending of SD beams.

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Propagation dynamics of diffusive pollutants on the water surface and in the water

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.

Buckling under axial compression of a transversely isotropic
cylindrical shell with the weakly supported curvilinear edge.

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.

Mathematical model of the immune response.

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.

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interaction of mechanical structures with fluid flow is showed.

Low-frequencies free vibrations of a rotating cylindrical shell contacted
with cylindrical rollers.

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

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.