The generalized Wiener–Hopf equations for the elastic wave motion in angular regions

In this work, we introduce a general method to deduce spectral functional equations in elasticity and thus, the generalized Wiener–Hopf equations (GWHEs), for the wave motion in angular regions filled by arbitrary linear homogeneous media and illuminated by sources localized at infinity. The work extends the methodology used in electromagnetic applications and proposes for the first time a complete theory to get the GWHEs in elasticity. In particular, we introduce a vector differential equation of first-order characterized by a matrix that depends on the medium filling the angular region. The functional equations are easily obtained by a projection of the reciprocal vectors of this matrix on the elastic field present on the faces of the angular region. The application of the boundary conditions to the functional equations yields GWHEs for practical problems. This paper extends and applies the general theory to the challenging canonical problem of elastic scattering in angular regions.

Double Tracking Control for the Directed Complex Dynamic Network via the Observer of Outgoing Links

From the large system perspective, the directed complex dynamic network is considered as being composed of the nodes subsystem (NS) and the links subsystem (LS), which are coupled with together. Different from the previous studies which propose the dynamic model of LS with the matrix differential equations, this paper describes the dynamic behavior of LS with the outgoing links vector at every node, by which the dynamic model of LS can be represented as the vector differential equation to form the outgoing links subsystem (OLS). Since the vectors possess the flexible mathematical operational properties than matrices, this paper proposes the more convenient mathematic method to investigate the double tracking control problems of NS and OLS. Under the state of OLS can be unavailable, the asymptotical state observer of OLS is designed in this paper, by which the tracking controllers of NS and OLS are synthesized to ensure achieving the double tracking goals. Finally, the example simulations for supporting the theoretical results are also provided.

Position Vectors of Curves Generalizing General Helices and Slant Helices in Euclidean 3-Space

Inthispaper,wegiveanewcharacterizationofak-slanthelixwhichisageneral- ization of general helix and slant helix. Thereafter, we construct a vector differential equation of the third order to determine the parametric representation of a k-slant helix according to standard frame in Euclidean 3-space. Finally, we apply this method to find the position vector of some examples of 2-slant helix by means of intrinsic equations.

Optimal control of a rocket projectile according to the "minimum force" criterion

The article solves the problem of optimal control of a rocket projectile by its delivery from a given initial position to a given final position, taking into account the air resistance force. The motion of the projectile is described by the vector differential equation of I.V. Meshchersky. The control quality criterion is taken in the form of "minimum force", the minimization of which ensures minimal overloads for the projectile. Three types of the norm of the control force vector are considered. For each of them, an optimal control is obtained that solves the task. The analysis of the results of the numerical experiment is carried out, confirming the general theoretical provisions.

Harmonic waves of a given azimuthal number in a micropolar cylindrical waveguide

Рассматривается система двух связанных векторных дифференциальных уравнений линейной теории микрополярной упругости, сформулированная в терминах перемещений и микровращений в случае гармонической зависимости перемещений и микровращений от времени. Вводятся потенциалы перемещений и микровращений. Выполнено расщепление связанных векторных дифференциальных уравнений микрополярной теории упругости для потенциалов на несвязанные винтовые уравнения, опираясь на пропорциональность (с разными масштабными факторами) вихревых составляющих перемещений и микровращений только одному вихревому винтовому полю. Найдено представление векторов перемещений и микровращений с помощью четырех винтовых векторов. Оно обеспечивает выполнимость связанных векторных дифференциальных уравнений линейной теории микрополярной упругости. Проблема нахождения вихревых составляющих перемещений и микровращений приведена к решению четырех несвязанных между собой векторных винтовых дифференциальных уравнений. Получено представление перемещений и микровращений с помощью двух несвязанных метагармонических векторов. Выполнено разделение пространственных переменных в уравнениях Гельмгольца в цилиндрической системе координат. Определены решения скалярного и векторного уравнений Гельмгольца в бесконечной цилиндрической области, содержащие ряд произвольных постоянных. В явном виде найдены представления векторов перемещений и микровращений в длинном линейном микрополярном цилиндре, содержащие восемь произвольных постоянных. Такого рода решения определяют формы гармонических волн перемещений и микровращений, распространяющихся вдоль оси длинного кругового цилиндра. Полученные представления для гармонических волн перемещений и микровращений имеют смысл только для волн, характеризующихся заданным азимутальным числом. The coupled system of vector differential equations of the linear theory of micropolar elasticity presented in terms of displacements and micro-rotations in the case of a harmonic dependence of physical fields on time is considered in the three different variants of which the two are due to W. Nowacki and H. Neuber. A new scheme of splitting the coupled vector differential equation of the linear theory of micropolar elasticity into uncoupled ones is proposed. The scheme is based on proportionality of the vortex parts of the displacements and micro-rotations to the single vector, which satisfies the screw equation. The problem of determination of the vortex parts of the displacements and micro-rotations fields is reduced to solution of four uncoupled screw differential equations. A new representation of displacement and micro-rotation vectors is obtained by using two uncoupled metaharmonic vectors. The separation of spatial variables in the Helmholtz metaharmonic equations in a cylindrical coordinate net is described. Solutions of the scalar and vector Helmholtz equations in an infinite cylindrical domain containing a series of arbitrary constants are obtained. Representation of displacement and micro-rotation vectors in a long micropolar cylinder containing eight arbitrary constants are explicitly found. The corresponding solutions are proved to determine the modes of harmonic waves of displacements and micro-rotations propagating along the axis of a long circular cylinder. The obtained modes of the harmonic displacements and micro-rotations waves are valid only for those characterized by a given azimuthal number.

On modelling harmonic waves in linear micropolar elastic media by four screw fields

В статье рассматриваются дифференциальные уравнения для потенциалов, обеспечивающие выполнение связанных векторных дифференциальных уравнений линейной теории микрополярной упругости в случае гармонической зависимости поля перемещений и микровращений от времени. Предложена альтернативная схема расщепления связанных векторных дифференциальных уравнений микрополярной теории упругости для потенциалов на несвязанные уравнения первого порядка. Она основана на пропорциональности с разными масштабными факторами вихревых составляющих перемещений и микровращений одному вихревому винтовому полю. Найдено представление векторов перемещений и микровращений с помощью четырех винтовых векторов, обеспечивающее выполнимость связанных векторных дифференциальных уравнений линейной теории микрополярной упругости. В результате проблема нахождения вихревых составляющих перемещений и микровращений сводится к решению четырех несвязанных между собой векторных винтовых дифференциальных уравнений первого порядка с частными производными. Полученные результаты могут быть использованы в прикладных задачах механики, связанных с распространением гармонических (монохроматических) волн перемещений и микровращений вдоль длинных волноводов. The paper is devoted to study of the coupled vector differential equations of the linear theory of micropolar elasticity formulated in terms of displacements and microrotations in the case of a harmonic dependence of the physical fields on time. An alternative analysis aimed at splitting the coupled vector differential equation of the linear theory of micropolar elasticity into uncoupled equations is given. It is based on a notion of proportionality of the vortex parts of the displacements and microrotations to the single vector, which satisfies the screw equation well known from the mathematical physics. As a result, the problem of finding the vortex parts of the displacements and microrotations fields is reduced to solution of four uncoupled screw differential equations of the first order with partial derivatives. Obtained results are to be used in applied problems of the micropolar elasticity and in particular in studies of harmonic wave propagation along waveguides

Representation of Waves of Displacements and Micro-rotations by Systems of the Screw Vector Fields

The present study concerns the coupled vector differential equations of the linear theory of micropolar elasticity formulated in terms of displacements and micro-rotations in the case of a harmonic dependence of the physical fields on time. The system is known from many previous discussions on the micropolar elasticity. A new analysis aimed at uncoupling the coupled vector differential equation of the linear theory of micropolar elasticity is carried out. A notion of proportionality of the vortex parts of the displacements and microrotations to a single vector, which satisfies the screw equation, is employed. Finally the problem of finding the vortex parts of the displacements and micro-rotations fields is reduced to solution of four uncoupled screw differential equations. Corresponding representation formulae are given. Obtained results can be applied to problems of the linear micropolar elasticity concerning harmonic waves propagation along cylindrical waveguides.

Parametric Equations for Space Curves Whose Spherical Images Are Slant Helices

The curve whose tangent and binormal indicatrices are slant helices is called a slant-slant helix. In this paper, we give a new characterization of a slant-slant helix and determine a vector differential equation of the third order satisfied by the derivative of principal normal vector fields of a regular curve. In terms of solution, we determine the parametric representation of the slant-slant helix from the intrinsic equations. Finally, we present some examples of slant-slant helices by means of intrinsic equations.