Геометрические места точек, равноотстоящих от двух заданных геометрических фигур. Часть 3

The movement of material particles along the inner surface of the cone takes place in cyclones, the designs of which can have both cylindrical and conical parts. Aerodynamic processes occurring in a cyclone are complex in nature, therefore they cannot be accurately modeled on the basis of theoretical approaches. A number of simplifications were introduced during the research: air resistance is not taken into account, since the particle is fed into the cone together with the air, although later their directions of movement do not coincide (the particle damps the speed and falls down, and the air along the central part along the axis of the cone rises up and goes out); the influence of particles on each other, their size, etc. The purpose of the article is to study the motion of a material particle entering the inner surface of a vertical cone with a given initial velocity. If a material particle is directed with an initial velocity to the inner wall of the cone perpendicular to its generator, then its further motion will include both rotation around the axis of the cone and descent down under the action of its own weight. To find the trajectory of motion, a material point was taken as the vertex of the accompanying Frenet trihedron, which has three mutually perpendicular orthogonal planes. The second accompanying Darboux trihedron has a common orthogonal plane tangent to the trajectory with the Frenet trihedron. The balance of the acting forces in the projections onto the orthogonal planes of the Darboux trihedron was considered. This made it possible to determine the projections of the curvature of the curve onto the corresponding orthogonal planes of the Darboux trihedron. The differential geometry apparatus made it possible to find them through the first and second quadratic forms of the surface, which allows avoiding cumbersome transformations. Differential equations of motion of a material particle along the inner surface of a vertical cone were compiled. The equations were solved using the MatLab system. The equation of motion of a particle along the inner surface of the cone was obtained. Analyzing the trajectory of the particle, we can conclude that it is significantly different from the trajectory of motion along the inner surface of the cylinder. The graphs of changes in velocity also show the difference between the motion of a particle along a cone and the same motion along a cylinder. If, upon entering the surface of a cylinder, the particle damps its velocity to a certain limit, and then it begins to increase again, then during movement along a cone the velocity of the particle has a certain periodic character and approaches zero over time. In the absence of friction and air resistance, a material particle, after entering the inner surface of the cone at a certain angle to the generator (except zero), performs an oscillatory motion, alternately rising and falling along a trajectory in the form of a loop, moving for any length of time. Depending on the initial conditions, the particle can describe a finite number of branches of the loop, an infinite number of branches, move along a straight-line generator of the cone, or along an intermediate trajectory between a straight line and a loop. In the presence of friction, the particle will descend to the top of the cone, with possible local rises, the magnitude of which will depend on the initial velocity and the angle of inclination of the generating cone. The velocity in such a motion will damp out, while also having an oscillatory character.

In this paper are studied surfaces which are loci of points (LOP) equally spaced from a point and a conical surface under a variety of the point and conical surface’ mutual arrangement. Mathematical models of such surfaces are studied, and mathematical analysis of their properties is performed, as well as 3D models of considered surfaces are constructed. Possible cases of mutual arrangement for the point and the conical surface: • the point is at the conical surface’s vertex; • the point is on the conical surface; • the point is inside the conical surface: –– on the axis, –– not on the axis; • the point is outside the conical surface. The point is on the vertex of the conical surface Γ — the obtained conical surface Ω has the same vertex, whose generatrixes are perpendicular to the generatrixes of the surface Γ. The point is on the conical surface Γ — LOP equally spaced from the surface Γ and the point O separates into a straight-line l and a surface Φ of 4th order. The line l is located in the axial plane passing through the point O and is perpendicular to the generatrix of the conical surface Γ. Obtained surface Φ has a symmetry plane passing through the axis of the conical surface Γ and the point O. Many sections of the obtained surface Φ are Pascal snails. The point is inside the conical surface on the axis. Obtained surface α is a rotation surface, and the axis z is its axis of rotation. All the sections of the surface by planes perpendicular to the axis z are circles. Point is outside the conical surface. A very interesting surface Ω has been obtained, with the following properties: the surface Ω has a support plane, which is tangent to the surface Ω on a hyperbole; the surface Ω has 2 symmetry planes; there are a circle, parabola and Pascal’s snail among the surface Ω sections. In this paper have been considered analogues between surfaces of LOP equally spaced from the cylindrical surface and the point, and from the conical surface and the point.

Loci of points (LOP) equally spaced from two given geometrical figures are considered. Has been proposed a method, giving the possibility to systematize the loci, and the key to their study. The following options have been considered. A locus equidistant from N point and l straight line. N belongs to l. We have a plane that is perpendicular to l and passing through N. N does not belong to l – parabolic cylinder. A locus equidistant from F point and a plane. In the general case, we have a paraboloid of revolution. The F point belongs to the given plane. We get a straight line perpendicular to the plane and passing through the F point. A locus equidistant from a point and a sphere. The point coincides with the sphere center. We get the sphere with a radius of 0.5 R. The point lies on the sphere. We get the straight line passing through the sphere center and the point. The point does not coincide with the sphere center, but is inside the sphere. We get the ellipsoid. The point is outside the sphere. We have parted hyperboloid of rotation. A locus equidistant from a point and a cylindrical surface. The point lies on the cylindrical surface’s axis. We get the surface of revolution which generatix is a parabola. The point lies on the generatrix of the cylindrical surface of rotation. We get a straight line, perpendicular to that generatrix and passing through the cylinder axis. The point does not lie on the axis, but is located inside the cylindrical surface. We get the surface with a horizontal sketch line – the ellipse, and a front sketch lines – two different parabolas. The point is outside the cylindrical surface. A locus consists of two surfaces. The one with the positive Gaussian curvature, and the other – with the negative one.

The quadrics here considered are the nine real quadrics: parabolic cylinder, elliptic cylinder, hyperbolic cylinder, cone, hyperboloid of one sheet, hyperbolic paraboloid, elliptic paraboloid, ellipsoid and hyperboloid of two sheets. Let Q be one of these quadrics. We consider a polynomial vector field X=(P,Q,R) in R3 whose flow leaves Q invariant. If m1 = degree P, m2 = degree Q and m3 = degree R, we say that m=(m1,m2,m3) is the degree of X. In function of these degrees we find a bound for the maximum number of invariant conics of X that results from the intersection of invariant planes of X with Q. The conics obtained can be degenerate or not. Since the first six quadrics mentioned are ruled surfaces, the degenerate conics obtained are formed by a point, a double straight line, two parallel straight lines, or two intersecting straight lines; thus for the vector fields defined on these quadrics we get a bound for the maximum number of invariant straight lines contained in invariant planes of X. In the same way, if the conic is non-degenerate, it can be a parabola, an ellipse or a hyperbola and we provide a bound for the maximum number of invariant non-degenerate conics of the vector field X depending on each quadric Q and of the degrees m1, m2 and m3 of X.

Purpose. To conduct theoretical studies of the movement of a particle of technological material along the inner surface of a rotary cone with a vertical axis, which carries out rotational movement and translational oscillations in order to find ways to improve grain and seed separation machines. Methods. The methods of linear algebra and differential geometry, theoretical and analytical mechanics, and computer graphics systems were used to conduct the research. Methods of numerical integration of differential equations were used to process research results, thanks to the Simulink package of the MatLab system. Results. The greatest influence on the nature of the movement of the particle of the technological material is exerted by the force of the surface reaction, because as the oscillation parameters increase, the particle may detach from the surface of the cone. During circular translational oscillations of the cone in the horizontal plane, a particle of the technological material of the material in relative motion can describe periodic trajectories between the upper and lower parallels, or slide in a circle (parallels of the cone). The second case is possible provided that the proper initial conditions of motion, which are determined analytically, are ensured. During the rotational movement of the cone, the stabilization of the movement of the particle of the technological material is impossible: it either accelerates, rising along a spiral trajectory upwards, or falls down with a large angle of inclination of the cone and insufficient angular speed of rotation. When the rotational movement of the cone and its harmonic oscillations along the axis of rotation are combined, a particle of the technological material can be detached from the surface. Conclusions 1. The peculiarities of the relative movement of a particle of technological material along the inner rough surface of a vertical cone, which carries out circular translational oscillations in the horizontal plane, rotates around its own axis or combines rotational movement around the axis and harmonic oscillations along the axis, are considered. 2. Constructed trajectories of the movement of a particle of material at different angles of inclination of the generating cone, angular speed of rotation and radius of oscillation. It was established that at large angles of inclination of the generating cone, the particle can move down, describing a spiral curve, while the decrease in pressure and speed are subject to a linear law. At an angle β of the inclination of the generating cone, a smaller angle of friction, and at small angular velocities of oscillations, the particle may not move at all. 3. Research showed that during oscillation, the pattern of particle sliding is the same for different angles β of the inclination of the generators: the particle describes a spiral trajectory, rising up the cone. Stabilization of motion does not occur: the particle either rises up or falls down due to insufficient angular velocity ω of rotation or large the elevation of the particle. 4. The presence of an oscillating component causes a variable surface reaction force. When the frequency of cone oscillations increases to w = 30 s-1, the particle detaches from the surface. Research has also shown that an increase in the amplitude of oscillations also leads to the separation of the particle from the surface. 5. The results of this study can be used in the design of separating machines used at grain processing enterprises. The prospect of these researches is to carry out theoretical calculations and build graphical dependencies to determine the trajectory of movement of process material particles with different masses, initial kinematic and dynamic characteristics. Keywords: a cone with a vertical axis, rotational motion, translational oscillations of a cone, relative motion of a particle, differential equations of motion.

Effects of magnetic field, binary particle loading and rotational conic surface on phase change process in a PCM filled cylinder

In this paper have been investigated the loci equidistant from sphere and plane, and properties of obtained surfaces have been studied. Four options for possible mutual arrangement of plane and sphere have been considered: the plane passes through the center of the sphere; the plane intersects the sphere; the plane is tangent to the sphere; the plane passes outside the sphere.
 In all options of the mutual arrangement of the sphere and the plane, the loci are two surfaces - two coaxial confocal paraboloids of revolution. The general properties of the obtained paraboloids of revolution have been studied: foci and vertices positions, axes of rotation, the distance from the sphere center to the vertices of the paraboloids, the distance between the vertices of the paraboloids, and the position of the directorial planes have been defined.
 Have been derived equations for the surfaces of the loci equidistant from the sphere and the plane: various paraboloids of revolution.
 The loci in each of the four options for the possible mutual arrangement of the plane and the sphere are as follows. 1. The original plane passes through the sphere center – two coaxial confocal multidirectional paraboloids of revolution symmetric relative to the original plane. 2. The initial plane intersects the sphere – two coaxial confocal multidirectional but not symmetrical paraboloids of revolution, since the circle of intersection of the plane and the sphere does not coincide with the diameter of the sphere great circle. 3. The plane is tangent to the sphere – a paraboloid of revolution and a straight line (more precisely, a second order zero-quadric – a cylindrical surface with zero radius) passing through the tangency point of the plane and the sphere and the sphere center. 4. The plane passes outside the sphere – the equidistant loci will be two coaxial confocal unidirectional paraboloids of revolution.

The axial torque until failure of the ligamentous occipito-atlanto-axial complex (C0-C1-C2) subjected to axial angular rotation (theta) was characterized using a biaxial MTS system. A special fixture and gearbox that permitted right axial rotation of the specimen until failure without imposing any additional constraints were designed to obtain the data. The average values for the axial rotation and torque at the point of maximum resistance were, respectively, 68.1 degrees and 13.6 N-m. The specimens offered minimal resistance (approximately 0.5 N-m), up to an average axial rotation of 21 degrees across the complex. The torque-angular rotation (T-theta) curve can be divided into four regions: regions of least and steadily increasing resistances, a transition zone that connects these two regions, and the increasing resistance region to the point of maximum resistance. The regions of least and steadily increasing resistances may be represented by two straight lines with average slopes of 0.028 and 0.383 N-m/degree, respectively. Post-test dissection of the specimens disclosed the following. The point of maximum resistance corresponded roughly to the value of axial rotation at which complete bilateral rotary dislocation of the C1-C2 facets occurred. The types of injuries observed were related to the magnitude of axial rotation imposed on a specimen during testing. Soft-tissue injuries alone (like stretch/rupture of the capsular ligaments, subluxation of the C1-C2 facets, etc.) were confined to specimens rotated up to or close to the point of maximum resistance. The specimens that were subjected to rotations up to the point of maximum resistance of the curve spontaneously reduced completely on removal from the testing apparatus. Spontaneous reduction was not possible for specimens tested slightly beyond their points of maximum resistance.(ABSTRACT TRUNCATED AT 250 WORDS)

This paper is devoted to the topical issue of studying the shapes of geometric surfaces in terms of descriptive geometry and their computer modeling. Definitions of algebraic and geometric surfaces are given. The analysis of features and parameters of surfaces for their classification into the main groups that are most common in practice is performed. The main methods of setting surfaces used in practice, depending on the shape of the surface and the task are given. Analytical, kinematic and frame methods of setting surfaces have been studied. The analytical method is presented on the basis of a conical surface of rotation; the canonical equation of the surface is analyzed, its correspondence for the upper and lower cone cavities is shown. The conical surface is graphically displayed in the AutoCAD graphics system. The peculiarities of the kinematic method of surface formation, which is presented on the example of the surface of rotation, made in the graphics system AutoCAD, and the surface of a straight hyperboloid, the drawing of which is made in the system COMPASS-3D. Examples of practical use of simulated surfaces are given. Enough attention is paid to the study of frame surfaces. A complex drawing of the surface of a cylinder given by a linear framework is given. The three-line frame of the ship's surface, which is formed by waterlines, frames and buttocks, is presented in detail. Simulated frame surface of an axial gas turbine blade is presented in the AutoCAD system. Algorithms and methods of geometric modeling of surfaces in graphic systems AutoCAD and KOMPAS-3D are given. Geometric models of some surfaces have been developed: straight helicoid, single-cavity hyperboloid, hyperbolic paraboloid, surface with return edge, etc. Examples of practical implementation of simulated surfaces are given.

Equine movement symmetry is changed when turning, which may induce alterations in thoracolumbosacral kinematics; however, this has not previously been investigated. Our objectives were to document thoracolumbar movement in subjectively sound horses comparing straight lines with circles on both reins and to relate these observations to the objectively determined symmetry/asymmetry of hindlimb gait. Fourteen non-lame horses were assessed prospectively in a non-random, cross-sectional survey. The horses were trotted in straight lines and lunged on both reins and inertial sensor data collected at landmarks: withers, T13 and T18, L3, tubera sacrale, and left and right tubera coxae. Data were processed using published methods; angular motion range of motion (ROM; flexion-extension, axial rotation, lateral bending) and translational ROM (dorsoventral and lateral) and symmetry within each stride were assessed.The dorsoventral movement of the back exhibited a sinusoidal pattern with two oscillations per stride. Circles induced greater asymmetry in dorsoventral movement within each stride (mean ± standard deviation, up to 9 ± 6%) compared with straight lines (up to 6 ± 6%). The greatest amplitude of dorsoventral movement (119 ± 14 mm in straight lines vs. 126 ± 20 mm in circles) occurred at T13. Circles induced greater flexion-extension ROM (>1.3°; P = 0.002), lateral bending (>16°; P < 0.001), and lateral motion (>16 mm; P = 0.002) compared with straight lines. Circles induced a movement pattern similar to an inside hindlimb lameness, which was significantly associated with the circle-induced greater asymmetry of dorsoventral movement of the thoracolumbar region (P = 0.03). Moving in a circle induces measurable changes in thoracolumbar movement compared with moving in straight lines, associated with alterations in the hindlimb gait.

On the points of inflection in general spatial motion

Revisiting causality using stochastics on atmospheric temperature and CO ₂ concentration

When the location of gear axes and the constant ratio of angular velocities are given, a pair of tooth profiles having a straight path of contact that coincides with the common contact normal at each point of contact is expressed by algebraic equations. The common perpendicular to the gear axes and the axis of relative rotation which is defined in the static space by the given ratio of angular velocities determine the static coordinate system, in which the design point P0 and the path of contact are determined as follows. In the case of cylindrical or bevel gears, P0 is chosen arbitrarily on the axis of relative rotation and the path of contact is chosen as a straight line through P0. In the case of other kinds of gears, P0 is chosen arbitrarily on the horizontal plane which includes the axis of relative rotation and is perpendicular to the common perpendicular, and the path of contact is chosen as a straight line through P0 which lies on the normal plane that is perpendicular to the relative velocity at P0. When the path of contact and its common contact normals are transformed to the coordinate systems rotating with each gear, a pair of tooth profiles without variation of bearing loads is obtained.

I. In order to observe all the markings on the planet Venus, it is necessary to wait for the times when they are turned towards the Sun and to us, and nearer to the Earth. To map them, a place must be chosen which is correctly situated both as regards the observer and the axis of rotation. For either the spectator should be situated outside the globe of the Earth and placed on the axis of the Ecliptic; or located in the plane of the Ecliptic upon the Earth’s globe where we are. II. In either position the lines described by the markings should be considered as they rotate round the axis of revolution, so that the appearance presented to the eye of the spectator can be understood whether in the form of ellipses, circles or straight lines. III. By applying this theory to the observations of the Venusian markings made in 1726, it was established that the plane through the axis of rotation and the Sun’s centre cut the Ecliptic at the beginning of March 1726 about the 20th degree of Leo and Aquarius. IV. When the markings were thus identified, and also their axis of rotation (the inclination of which, 15 or 20 degrees above the plane of the Ecliptic, was deduced from this), armillary machines can be constructed, both of the three-dimensional kind and planispheres, and maps to show the situation of the markings on the globe of Venus and their rotation and revolution day by day. V. Two different types of sheet maps are described, such as those used by geographers and those who map the seas. VI. For the sake of clearer understanding it, it is better to start with the parallel maps. VII. Seven notable markings near the equator of Venus and two near the poles should be given the names of ‘maria’, just as on the Moon. VIII. Names are assigned to each. IX. Another arrangement is proposed of a map made in the form of circles, as geographers use a planisphere to represent the Earth’s globe. X. A solid globe with the same markings reveals all the phases more clearly. XI. The method of constructing this and fitting it on to the armillary machine to imitate the Sun’s light accurately to show the markings is explained.

Aims of research. Studying the possibility of forming Monge carved surfaces, defined by the method of their formation, creating an algorithm and program in the AutoLISP language to demonstrate the formation of surfaces in the AutoCAD environment in a dynamic mode. Methods. Monge carved surfaces are formed by a flat curve, located in the tangent plane to the fixed guide of the developable surface, when the plane and the curve roll along the guide surface without sliding. The described method of formation of these surfaces allows to perform their formation by the kinematic method in the AutoCAD environment using AutoLISP software. The article describes the construction of the Monge surfaces using cylindrical and conical surfaces as guides. A straight line and a sine wave are used as the forming lines. Results. An algorithm and a program in the AutoLISP language were created to form sets of compartments of several Monge surfaces and to visualize the formation of these surfaces in a dynamic mode by sequentially displaying the compartments on the monitor screen. The mini-film about formation of Monge surface by rolling a plane with a straight line along a circular cone is created. In the mini-film the drawings received by transformation of drawings of the AutoCAD environment are used.