Invariant algorithms of spatial constructions elements forming and cutting

This report clearly demonstrates the capabilities of an advanced research area of applied mathematics, i.e., computational geometry to be applied for shaping dimensional structures. Vector-matrix models are provided to cover the composite membrane piecewisesmooth structures composed of surface elements with zero Gaussian curvature. General algorithms are presented for finding cutting lines for cylindrically and conicallyshaped elements, into which the curves contained in the developable surfaces transform. A dome-like peak-shaped structure comprising components of both cylindrical and conical shape is given as an example to present equations describing the cutting lines explicitly, which makes it possible to implement a high-precision technique for producing such a structure

Applying Gaussian distributions on SO(3) for modeling the texture and predicting the properties of texturized polycrystalline materials

This paper considers a quantitative description of the crystallographic texture. When the crystallographic texture is modeled, the generalized Gaussian distribution is applied on Riemannian manifolds. A simple model of the orientation distribution function (ODF) is obtained, which corresponds to the Gaussian distribution of the axial crystallographic texture. It is demonstrated that an equally probable distribution of crystallographic axes and an ideal axial texture are realized as special cases of the ODF model. An anisotropy evaluation of elastic modulus is carried out. The indicatrix transformation of Young's modulus is demonstrated. © 2016 Author(s)

Mathematical modelling of the spatial network of bone implants obtained by 3D-prototyping

In this paper, the mathematical model suitable for bone implants 3D-prototyping is proposed. The composite material with the spatial configuration of reinforcement with matrix of hydroxyapatite and titanium alloys fibers is considered. An octahedral cell is chosen as an elementary volume. The distribution of reinforcing fibers is described by textural parameters. Textural parameters are integrated characteristics that summarize information on the direction of reinforcing fibers and their volume fractions. Textural parameters, properties of matrix and reinforcing fibers allow calculating effective physical and mechanical properties of the composite material. The impact of height and width of the octahedral reinforcement cells on textural parameters of the composite material is investigated in this work. The impact of radius of fibers is also analyzed. It is shown that the composite becomes quasi-isotropic under certain geometrical parameters of cell. © 2016 Author(s)

Texture parameter variation region for orthotropic polycrystals with cubic symmetry of the crystal lattice

The variation region of texture parameters (which are integral characteristics of the preferable orientation of crystallographic axes) allows solutions to be found for managing anisotropic properties, illustrating all possible textured states of an orthotropic polycrystalline material with a crystal lattice of cubic symmetry. Each point in this region is matched by certain anisotropy of both elastic and plastic properties. The region of texture parameter variation is defined both analytically and using a numerical experiment of statistic simulation. Analytically, the solution is found via determining the effective eigenvalues of the elasticity operator for a textured anisotropic cubic polycrystal. The algorithm to be followed for visualizing the region-forming elements implies determining the lines of intersection of planes with a conical surface. The numerical solution is based on the determination of texture parameters, i.e, on the starting assumption that the variation region is bounded and lies in the first octant. The task of constructing the variation region is solved via finding the texture parameters using the Monte-Carlo method according to the density of distribution of crystallographic axes in space. When modeling the variation region, octets are used, which are symmetrical reflections of randomly taken orientations in all the octants of space. The constructed regions have the required symmetry. The numerically obtained cloud of textured states and the analytically constructed variation region have geometric centers coinciding at the point corresponding to the non-textured state. At various stages of thermal and mechanical treatment of metallic materials, texture evolution can be represented geometrically as a texture state trajectory that is seen to be within the determined texture parameter variation region. © 2018 Author(s)

Geometric representation of polycrystalline material texture by axis-angle parametrization

Texture is the preferential orientation of crystallographic axes in polycrystal. For its mathematical modeling, the orientation distribution function of the crystallographic axes is used. Traditionally, the orientation distribution function is written with the help of directional cosine matrices, Miller indices or Euler-Krylov angles. Recently, texture has increasingly often been described using quaternions, Rodrigues parameters and the vector space of axis-angle parameters. Axis-angle parameters allow us to describe all possible rotations of the SO(3) group, which corresponds to all possible orientations of crystallographic axes in polycrystalline materials. The SO(3) group is a set of rotations to all possible angles around all possible axes given by all vectors of the unit sphere. The set of such rotations corresponds to points set on a ball of radius π in three-dimensional Euclidean space. The description of the crystallographic texture using axis-angle parameters made it possible to visualize the distribution of crystallographic axes and obtain a new geometric representation of the texture. © 2018 Author(s)

Quaternion model of programmed control over motion of a Chaplygin ball

This paper deals with the problem of program control of the motion of a dynamically asymmetric balanced ball on the plane using three flywheel motors, provided that the ball rolls without slipping. The center of mass of the mechanical system coincides with the geometric center of the ball. Control laws are found to ensure the motion of the ball along the basic trajectories (line and circle), as well as along an arbitrarily given piecewise smooth trajectory on the plane. In this paper, we propose a quaternion model of ball motion. The model does not require using the traditional trigonometric functions. Kinematic equations are written in the form of linear differential equations eliminating the disadvantages associated with the use of Euler angles. The solution of the problem is carried out using the quaternion function of time, which is determined by the type of trajectory and the law of motion of the point of contact of the ball with the plane. An example of ball motion control is given and a visualization of the ball-flywheel system motion in a computer algebra package is presented. © 2019 Udmurt State University. All rights reserved

Smooth movement of a rigid body in orientational space along the shortest path through the uniform lattice of the points on SO(3)

Many tasks of motion control and navigation, robotics and computer graphics are related to the description of a rigid body rotation in three-dimensional space. We give a constructive solution for the smooth movement of a rigid body to solve such problems. The smooth movement in orientational space is along the shortest path. Spherical solid body motion is associated with the movement of the point on the hypersphere in four-dimensional space along the arcs of large radius through the vertices of regular four-dimensional polytope. Smooth motion is provided by the choice of a special nonlinear function of quaternion interpolation. For an analytical presentation of the law of continuous movement, we use the original algebraic representation of the Heaviside function. The Heaviside function is represented using linear, quadratic and irrational functions. The animations in the computer program MathCad illustrate smooth motion of a rigid body through the nodes of a homogeneous lattice on the group SO(3). The algorithm allows one to change in a wide range the time intervals displacements between nodes, as well as the laws of motion on these intervals

Mathematical model of building envelope element insolation

In this paper, the problem of estimation of building envelope oriented element insolation by direct sunlight is solved using mathematical model of Earth's rotation. Quaternions are used as mathematical tool for description of rotation. The model allows to obtain automatically estimation of building element insolation by direct sunlight for given latitude and given time interval (month, week, day, etc.). The model takes into account schedule of change of day and night, change of direct sunlight angle on a given element of the surface caused by sun motion over horizon (changes of its height and azimuth during time of day). Distinctive feature of the model is simplicity of program-algorithmic implementation due to using description of rotation by means of quaternions. © Published under licence by IOP Publishing Ltd

Междисциплинарный подход к реализации проектного обучения

The problem of transition from the fragmented knowledge to learning outcome of the all-engineering module can be solved by introduction of interdisciplinary projects in bachelor curricula.Задача перехода от фрагментированных знаний к результатам обучения по общеинженерному модулю может быть решена введением в учебные планы бакалавриата междисциплинарных проектов