We consider the problem of nonstationary strains of an infinite anisotropic Timoshenko plate on an elastic-inertial foundation. A monoclinic symmetry type of an elastic medium, which is characterized by one plane of symmetry, is adopted as a model of anisotropy. Analytical methods are used to construct new fundamental solutions for the nonstationary normal displacement and deflection angles. The Laplace and Fourier integral transforms are used to find fundamental solutions, which serve as the basis for obtaining the integral relations for studies of nonstationary bending waves in a plate under the influence of sets of lumped and distributed loads. An example of the calculations is presented.

The processes of unsteady contact interaction of liquids described by different mathematical models with solid deformable bodies are considered. Closed mathematical formulations of unsteady contact problems in the case of various models of liquids and linear-elastic bodies are developed. The analytical solution of the nonstationary problem of interaction between an acoustic fluid and a deformable solid body is obtained. The time integral Laplace transform is used to construct the solution. The distributions of displacements and stresses in the solid body, as well as pressure and velocity fields in the fluid during unsteady contact interaction are analyzed.

This paper considers a non-stationary problem of heat conduction using a volumetric heat source acting in an isotropic half-space. The process of heat transfer in a half-space is described by the generalized hyperbolic Maxwell–Cattaneo equation. To construct the volumetric influence function, we represent the volumetric heat source as a concentrated source located at a point of a heat-conducting half-space. The problem of continuous heating of a half-space by a concentrated heat source is considered, and the results of calculations are given. The results of the work can be used to assess the contribution of nonstationary thermal conductivity in the processes of heat transfer in materials and structures exposed to intense heat flows (heating by gases with high enthalpy, laser surface treatment, additive technologies, etc.).

When an aircraft passes through a rainy area at high speed, the coating on the front edge of the fuselage will be continuously eroded by raindrops, causing the coating to wear, crack or even peel off. This paper uses carbon fiber T300 material as the base material, and at the different impact speeds and impact numbers, water cutting equipment was used to simulate the erosion of the coating caused by the continuous impact of water droplets. The damage morphology of samples at different damage stages was observed by digital microscope and Scanning Electron Microscope (SEM), and the damage evolution curve was established to analyze and reveal the damage behavior and damage mechanism of rain erosion. The results show that the degree of damage experienced an increasing trend with the increase of impact numbers and speed, until circular peel damage was formed; no damage occurred during the incubation period, and the curvature of the damage evolution curve increased significantly after the expansion period and eventually showed a stable expansion trend. The mechanical properties of the coating material were the main influencing factors of its rain corrosion resistance. Moreover, the axially symmetric unsteady contact problem of droplets impacting the surface of a solid deformable body was studied. And the contact area was determined based on the iterative algorithm boundary positioning method. A mathematical model and closed mathematical formula describing the unsteady interaction between a droplet and a solid deformable obstacle were proposed.

This paper presents a mathematical formulation of a transient problem for an anisotropic plate on an elastic-inertial foundation. The plate features local fixations of various types, located along its perimeter. The plate's contour can be arbitrary. It is subjected to a load with a time-varying amplitude. An original method for solving the posed problem has been developed and implemented. By applying integral transformations, a fundamental solution for an unlimited anisotropic plate has been constructed. This solution is used to formulate resolving integral representations for investigating the transient dynamics of the plate with local supports. Integral representations for transient displacements, moments, and rotation angles of the plate sections have been obtained using the method of compensating loads. The time-dependent compensating loads are determined from the solution of a system of Volterra integral equations of the first kind. By employing the quadrature method at each time step, the problem of compensating loads is reduced to solving a system of linear algebraic equations. A comparison of the obtained results with solutions derived using the finite element method is conducted. Graphical results of the calculations and an evaluation of the convergence of the proposed method are provided.

An effective numerical and analytical method is proposed for studying the force state inside the contact spots of elastic composite spherical shells resting on two flat surfaces inclined at an angle to each other. The method is based on constructing the influence function for the shell in an axisymmetric formulation, followed by its expansion into a Fourier series using the Legendre and Gegenbauer polynomials. A resolving system of equations for the corresponding coefficients is obtained. The developed methodology will allow for a more correct study of the dynamics of various robotic systems, the main controlled elements of which are elastic spherical shells that carry out complex kinematic movements on various surfaces. These systems, in particular, include a “butterfly robot”, the main controlled element of which is a thin spherical shell, rolling with simultaneous spinning and sliding along two parallel plates, as if on rails.

A mathematical formulation of a transient problem for an anisotropic plate on an elastic inertial foundation is given. An original method for solving it has been developed and implemented. The plate has local fixings of various nature located along its boundary. The plate boundary can have an arbitrary shape. A fundamental solution for an unbounded anisotropic plate is found and used in order to construct resolving integral representations. This solution is obtained using the Fourier integral transform in spatial coordinates domain and the Laplace integral transform in time domain. The corresponding original is constructed using analytic inverse method for the Laplace integral transform. The original of the two-dimensional Fourier transform is found numerically using integration methods for rapidly oscillating functions. A special algorithm allowing to obtain the originals of the two-dimensional Fourier transform numerically with a required accuracy is designed and implemented. Using the fundamental solution and the method of compensating loads the integral representations for transient displacements, moments and angles of rotation of plate sections are obtained. The time-dependent compensating loads are derived from the solution of a system of Volterra equations of the first kind. This system is solved step by step in time. At each time step, the problem is reduced to an equivalent system of algebraic equations. The results are compared with the solution obtained using the finite element method. The convergence of the proposed method was evaluated. Graphical results of calculations are given.

Рассмотрена контактная задача для сферической оболочки, опирающейся на абсолютно твёрдую плоскую поверхность. Для оболочки используется система уравнений равновесия в перемещениях, основанная на гипотезах С.П. Тимошенко, учитывающая влияние деформации сдвига. Для построения системы разрешающих уравнений использован принцип суперпозиции, согласно которому нормальные перемещения оболочки связаны с контактным давлением посредством интегрального соотношения. Это приводит к основному разрешающему интегральному уравнению типа Фредгольма I рода. Ядром этого уравнения является функция влияния для оболочки. Функция влияния простроена с применением разложений в ряды по полиномам Лежандра и Гегенбауэра. Для построения корректной системы разрешающих уравнений задача сведена к системе парных рядов-уравнений. Метод решения системы разрешающих уравнений основан на сведении парных уравнений к регулярному интегральному уравнению Фредгольма II рода. Для этого используются известные разложения разрывных функций в ряды по полиномам Лежандра, а также интегральное представление для полиномов Лежандра. В результате задача сведена к интегральному уравнению Фредгольма II рода относительно вспомогательной функции. Через вспомогательную функцию выражаются искомые коэффициенты ряда для контактного давления. Для определения области контакта дополнительно привлекается условие равновесия оболочки.

This article presents the mathematical formulations of transient heat conduction problems corresponding to the models of classical heat conduction using the Fourier law and generalized heat conduction based on the Cattaneo–Vernotta–Lykov law (Maxwell–Cattaneo model), as well as the generalized Green–Nagdy type II and III models. The Fourier transforms in spatial coordinates and the Laplace transforms in time were used to obtain the fundamental solutions of the equations of the Maxwell–Cattaneo and Green–Nagdy type II and III models of classical and generalized heat conduction. The results were displayed graphically and analyzed. Differences between the considered heat conduction models were shown, and suggestions for their practical application were given.

The nonsteady problem of coupled thermoelasticity is solved for a layer filled by a thermoelastic medium of classical, Maxwell–Cattaneo, or Green–Naghdi type. The superposition principle is employed. The Green’s functions are found for all the media. As an example, an initial and boundary value problem is solved. The results for the different models are different.

The purpose of this work is to study the propagation and diffraction of unsteady waves on a thin spherical shell located in an elastic half-space. Analytical methods are used to construct the solution. The problems of diffraction of elastic waves on various types of inhomogeneities are among the most complex and topical problems of the dynamics of deformable bodies. From the applied point of view, this is explained by the fact that the information about the dynamic stress-strain state in the vicinity of these inhomogeneities is of great interest for various purposes. Moreover, the presence of inhomogeneities (inclusions, cavities, notches, local changes in properties, etc.) is an indispensable condition arising in various fields of modern engineering. Such tasks include: creation of new structures working under dynamic loads, development of new composite materials and their introduction in creation of engineering structures, modern tasks of geophysics and seismology, as well as a number of other tasks of scientific and technical character.

Non-stationary longitudinal vibrations of a moment elastic rod of finite length are investigated. To describe the motion of the rod, the system of equations of the general model of moment elastic thin bodies is used without additional hypotheses. The equations of this model take into account longitudinal movements, changes in the angle of independent microrotation, as well as transverse compression of the rod. The rod material is assumed to be homogeneous and isotropic. The system of equations of motion is supplemented by physical relations that describe the relationship of displacements, changes in angles and transverse compression with forces. In contrast to classical models, in addition to normal forces, additional force factors arise in a moment rod. They are: additional moments, moment cutting forces, moments of moment stresses. Accordingly, in addition to the elastic constants of the material, additional physical parameters of the medium are taken into account, which are necessary when taking into account moment effects in the material. The conditions of generalized hinged support are used as boundary conditions at the ends of the rod. The initial conditions are assumed to be zero. To construct the solution, expansions of the desired functions and the external load into trigonometric Fourier series are used. Substituting these expansions into the original relations leads to a system of equations for the coefficients of time-dependent series. To solve it, the integral Laplace transform a in time is used. As a result, expressions for the required coefficients of expansion series in the image space are found. Each of these expressions is the sum of three products. The factors in these products are the Laplace images of the coefficients of the Fourier expansions for the load and for the influence functions. Influence functions are fundamental solutions (Green's functions) of the problem under study. The original coefficients of the series for the influence functions are found analytically using residues. The final expressions for the coefficients of the expansion series of solutions have the form of convolutions in time. The cores of these integral representations are the original coefficients of the series for the influence functions. As an example, the response of a moment elastic rod to the action of a non-stationary axial load is considered. The results obtained are illustrated graphically. The practical convergence of expansion series is estimated.

In this paper, we will present an approach to constructing of dynamical spatial Green’s function (elementary solutions, dominant function) for a thin infinite elastic plate of constant thickness. The plate material is anisotropic with a single plane of symmetry, geometrically coinciding with plate’s middle plane. The Timoshenko theory was used for describing the plate movement. Transient spatial Green’s functions for normal displacements and angles of orthogonal alteration to middle surface before deformation of material fiber are built in the Cartesian coordinate system. To construct Green’s function, direct and inverse Laplace and Fourier integral transformations are applied. The originals of Laplace Green’s functions were analytically found with the theorem of residues. To construct Fourier originals, a specific method was used based on Fourier series transformation inversion integral connection with Fourier series on a variable interval. Green’s function found for normal displacement made it possible to represent the normal transient function as three-fold convolution of Green function with distant load function. The functions of normal distant displacements were constructed in case of the impact of transient total loads concentrated and distributed across rectangular courts. The numerical method of rectangles was used to calculate the convolution integrals. The influence of the concentrated load speed on transient normal displacements of the anisotropic plate was analyzed. As a verification of constructed transient spatial Green’s functions, the results of numerical solutions were compared with the results found using known transient Green’s functions for isotropic thin elastic rectangular simply supported Timoshenko’s plate which solutions are constructed using Laplace integral transformation in time and its decomposition into Fourier series on coordinates. Besides, its confidence was proved analyzing the nature of waves in anisotropic, orthotropic and isotropic plate, found in the process of numerical calculations. The results are represented as diagrams. Examples of calculations are given.

An axisymmetric initial and boundary value problem regarding the impact of a liquid drop on a solid surface at constant speed is considered, for different contact angles. Mathematically, Navier–Stokes equations are used to describe the motion of each phase (liquid, air). A method of numerical solution is described, and sample calculations are presented.

Данная работа посвящена построению аналитического решения задачи о распространении нестационарных волн в тонкой анизотропной пластине большой протяженности. Подход к решению основан на принципе суперпозиции и методе функций Грина. Его суть заключается в связи искомого решения с нагрузкой при помощи интегрального оператора типа свёртки по пространственным переменным и по времени. Ядром этого оператора является функция Грина для анизотропной пластины. Она представляет собой нормальные перемещения в ответ на воздействие единичной сосредоточенной нагрузки. Для математического описания сосредоточенной нагрузки используется дельта-функция Дирака. Пространственные нестационарные функции Грина для анизотропной пластины Тимошенко построены впервые с помощью аналитических методов. В качестве модели анизотропного материала рассматривается упругая среда с единственной плоскостью симметрии, геометрически совпадающей со срединной плоскостью пластины. Движение пластины рассмотрено в декартовой системе координат. В начальный момент времени пластина находится в невозмущенном состоянии. Для решения использованы интегральные преобразования Лапласа по времени и двумерное интегральное преобразование Фурье по координатам. Оригиналы искомых функции по Лапласу построены при помощи второй теоремы разложения для преобразования Лапласа. Оригиналы по Фурье построены с помощью связи интеграла обращения преобразования Фурье с рядом Фурье на переменном интервале. Полученные функции Грина позволили представить искомый нестационарный прогиб и углы поворота в виде тройных сверток функций Грина с функцией нестационарной нагрузки. Для вычисления интеграла свёртки и построения искомого решения использован метод прямоугольников. Результаты решения представлены графически.

The implementations of the theory of multicomponent dry friction [1-19] for analyze the dynamics of some robotic systems, such as a butterfly robot [16-18, 20] or a humanoid robot is proposed. Since the main controlled element of these systems is a spherical, elastic composite shell, it is required to calculate the distribution of normal contact stresses inside the contact spot. The contact pressure distribution for such elements is constructed using the S. A. Ambartsumyan’s equation for a transversally isotropic spherical shell. This equation is modified by introducing the averaged contact pressure and normal displacements for the shell. The construction of the resolving integral equation for the contact pressure is based on the principle of superposition and the method of Green's functions. For this, the corresponding Green's function is constructed, which is the normal displacement of the shell as a solution to the problem of the effect of concentrated pressure. Green's function as well as the contact pressure, it is sought in the form of series expansions in Legendre polynomials, taking into account additional relations for the reduced contact pressure and normal displacements. Using the Green's function, an integral equation solving the problem is constructed. As a result, the problem is reduced to determining the expansion coefficients in a series of the reduced contact pressure. Restricting ourselves to a finite number of terms in the series of expansions, using the discretization of the contact area and the properties of Legendre polynomials, the problem is reduced to solving a system of algebraic equations for the expansion coefficients for the reduced pressure. After that, from the additional relation, the coefficients of the required expansion of the contact pressure in a series in Legendre polynomials are determined. To describe the conditions of shell contact with the surface, the theory of multicomponent anisotropic dry friction is used, taking into account the combined kinematics of shell motion (simultaneous sliding, rotation and rolling). The coefficients of the dry friction model can be calculated using simple explicit formulas [1-19] based on numerical experiments.