Methods Of Differential Geometry Research Articles

Designing a helical knife for a shredding drum using a sweep surface

The object of this paper is a helical blade in a shredding drum from a sweep surface. Such drums are used in harvesters for crushing plant mass. If the flat blades are installed on the drum, cutting of the plant mass occurs simultaneously along the entire length of the blade. This could cause a pulsating dynamic load. If a flat knife with a straight blade is installed at an angle to the axis of the drum, then the distances from it to the points of the blade will be different, as well as the cutting conditions along the blade. The elliptical shape enables the same distance from the axis of rotation to the points of the blade, but this does not solve the problem. Many short flat knives with a straight blade can be mounted on the drum, placing them in such a way that the time between the individual knives is minimized. However, all these disadvantages can be eliminated by a helical knife with a blade in the form of a helical line. The design of a helical knife from an unfolding helicoid has been considered. In differential geometry, the bending of unfolded surfaces of zero thickness is considered. Bending of the workpiece into a finished product occurs with minimal plastic deformations, the magnitude of which depends on the thickness of the workpiece sheet. The methods of differential geometry of unfolding surfaces were applied to the analytical description of the surface of the helical knife. The parametric equations of the unfolding helicoid were derived according to the given structural parameters of the knife in space and on the plane. That has made it possible to mathematically describe the contour lines that cut the knife from the surface and on its sweep. Formulae for calculating a flat workpiece through the structural parameters of the knife have been derived. Thus, with the specified structural parameters of the knife R=0.25 m, τ=20°, φ=65°, according to the resulting formula, we find the radius of the knife blade on a flat workpiece: R0=4.8 m.

Computing irregular hypar-based quad-mesh patterns for segmented timber shells

Hyperbolic paraboloids, “hypars,” are special types of ruled surfaces. Their geometric properties provide them with loadbearing and stabilizing capacities, as well as distinct esthetic qualities. These attributes become evident in numerous applications in buildings, in many of which concrete or timber is used for the construction of the hypars. Hypars could also be relevant in the context of circular construction and design for disassembly, and the upcycling of construction waste. Due to the geometric simplicity of straight lines, which generate ruled surfaces, hypar-based structures can be designed and built with relatively simple means. They can consist of self-similar or even identical elements, which could facilitate their reuse.Compared to other types of ruled surfaces, such as conoids, hypars have the advantage of being doubly ruled, meaning that structural grids of straight elements can be formed. This paper investigates another interesting property, which is the possibility of creating flat-quad meshes by diagonally connecting the intersection points of the generatrices. This property has been previously described by other scholars, some of which explored its applicability for glass-clad steel grid shells. In this research, we focus on its potential for segmented timber shells that can serve as stand-alone structures, or as modular and reusable building parts, such as façade or roof components. The reusability of such modular units could be achieved by using reversible joints between them.More specifically, our research investigates the design space of construction systems based on such components via computational design and optimization algorithms, such as the memory limited Broyden–Fletcher–Goldfarb–Shanno (LBFGS) algorithm with automatic computation of the gradient, within the Julia programming environment. By applying principles and methods of differential geometry, we study hypars with irregular tilings, enabling the integration of panels with diverse proportions, shapes and sizes, as they can occur in wood production waste. By reducing construction waste, the work aims at reducing the negative environmental impact of the building construction sector. Moreover, irregular tilings could enable a more customized design of acoustic qualities and offer visual variety in segmented hypar based timber structures.The here presented studies show that the proposed optimization method provides a good fit of many tiles to rhombi, particularly when the steepness is not too large. We also show that optimizing towards rectangles provides better results. Overall, the results support the initial assumption that irregular rulings could be a means of adapting to both homogeneous and diverse material stocks.

ОБОСНОВАНИЕ РАЦИОНАЛЬНОЙ КОНСТРУКЦИИ КРИВОШИПНОГО МЕХАНИЗМА КОЛЕСНО-ШАГАЮЩЕГО ДВИЖИТЕЛЯ

The article substantiates the rational design of the crank mechanism of a wheel-step mover. Kinematics analysis of a prototype of the wheel-step mover revealed the presence of oscillations in the vertical coordinate of the central axis during movement. To eliminate this drawback, differential geometry methods were used, thanks to which a method for calculating the non-circular profile of support shoes was created. Studies of the dynamics of the crank mechanism of the wheel-step mover showed that even in a steady state of motion, periodically acting inertial forces arise. To reduce their effect, non-circular gears should be used in the mover power drive. The dimensions of the crank mechanism links have been calculated, at which the cross-country ability of the running system, its kinematics and dynamics characteristics are improved, energy costs when laying a track are reduced, as well as the negative impact of the machine on the fertile soil layer. The polar coordinates of the involute points have been calculated. They are needed to construct the tooth profiles of non-circular gears. As the power drive of the crank mechanism, non-circular gears have symmetrical geometric parameters along two axes and provide a constant movement speed. Thus, the wheel-step running system can be used in agriculture, the timber industry, mining, and when working in emergency situations. The mover design being developed is intended for unmanned ground vehicles. This type of unmanned vehicles is at the initial stage of its development, inferior in frequency to air and marine unmanned vehicles. A targeted concentration of production and financial resources is necessary to ensure that domestic industry has a leading position in the production and sale of unmanned ground vehicles.

Deformation mechanism and minimum energy path in Silicon–graphite composites with lattice defects

Edge-on atomic layers in layered solids undergo buckling, which creates a structure referred to as a “ripplocation.” A collective set of multiple ripplocations is referred to as a “ripplocation boundary.” In this study, in order to investigate the impact of lattice defects on ripplocation, we investigate the buckling deformation of graphene with lattice defects under confinement. To this end, we conducted confined uniaxial compression simulations by placing graphene sheets with various lattice defects between silicon atomic layers. Different types of lattice defects affect the mechanical properties of graphene, the results show that Young's modulus of graphene with the (1,0)_5 dislocation pair was the maximum at 494.6 GPa, whereas the Young's modulus of the (1,2)_10 dislocation pair was the minimum at 309.5 GPa. In addition, we employed a differential geometry method to study the out-of-plane deformation of a single-layer graphene system. Further, we used the nudged elastic band method for various types of lattice defects in graphene to uncover the minimum transition pathways, activation energy required to move from the (1, 0) dislocation pair to the (1, 1) pair is 44.916 eV. In particular, the results indicate that graphene with lattice defects exhibit kink deformation after buckling compared to that of perfect graphene. Our study not only explores the deformation of ripplocations and kink boundaries in layered solids but also provides a more comprehensive description influencing lattice defects on the nucleation mechanism and mechanical changes of ripplocations in graphene.

Research on the calculation and evaluation method of transmission efficiency of noncircular planetary gear

The torque fluctuation characteristics of noncircular planetary gear (NPG) systems are difficult to obtain, and there is no clear analysis method and evaluation method for transmission efficiency, which seriously affects their service performance. Given that, this article establishes a torque fluctuation model for NPG systems based on the principle of gear meshing and differential geometry theory and methods, taking into account factors such as actual working conditions and speed loads. And proposed, the comprehensive transmission rate, a new method for evaluating the efficiency of NPG systems. By analyzing the actual working conditions of NPG systems, the influence mechanism of alternating load and speed on torque fluctuation and transmission efficiency was explored. The research results indicate that the alternating load can cause an increase in system impact and vibration. The comprehensive transmission rate of NPG systems decreases with increasing rotational speed and increases with increasing load, and is more sensitive to changes in rotational speed. The trend of comprehensive transmission rate and transmission efficiency of NPG systems is basically consistent, indicating that the proposed method of using comprehensive transmission rate to evaluate transmission efficiency is a practical and feasible analysis method. The research results can provide a new method for dynamic measurement and evaluation of torque and transmission efficiency of non-circular gear transmission systems.

Tubular surface generated by a curve lying on a regular surface and its characterizations

<abstract><p>In this research, we have constructed and studied special tubular surfaces in Euclidean 3-space $ \mathbb{R}^{3} $. We examined the singularities and geometrical properties of these surfaces. We achieved some significant results for these surfaces via Darboux frame. Also, we have proposed a few geometric invariants that illustrate the geometric characteristics of these surfaces, such as tubular Weingarten surfaces, using the traditional methods of differential geometry. Additionally, taking advantage of the singularity theory, we have given the classification of generic singularities of these surfaces. At last, we have presented some computational examples as an instance of use to validate our theoretical findings.</p></abstract>

Applications of Differential Geometry of Curves in Roads Design

Abstract An important role in the roads design is played by plane curves, if the road lies on a plane surface and by space curves, if the road can not be embeded in a plane. This article presents some work done in the two and three dimensional roads design, by using the differential geometry methods. A study of a general case for plane roads is given.

Determination of the form of a spiral spring in a free state

The object of research. This article deals with a spiral spring made of tape material with a rectilinear cross-section.
 Investigated problem. If one end of a metal ruler is pinched, and a certain force is applied to the other end, it will bend under the action of the generated moment. The magnitude of the moment depends on the applied force and the length of the ruler. Therefore, the amount of deflection will increase with the increase of these parameters, and the shape of the ruler may take the spiral form. After the termination of the external load, the ruler will regain its original shape, that is, it will become straight. However, such spiral springs do not exist in practice. For compact sizes, the spring in its free state also has a spiral shape. In this regard, the purpose of the study is to find the elastic axis of a tape spiral spring in a free state, which under the action of a concentrated force takes a given shape.
 The main scientific results. The shape of the spiral spring in the free state is determined, which under the action of the applied force acquires the desired shape, which does not lead to the occurrence of frictional forces between adjacent coils.
 The area of practical use of research results. Spiral springs are widely used. One example is mechanical clock movements and measuring devices, in which the determination of indicators depends on the applied force that "unwinds" or "twists" the spring.
 Innovative technological product. The paper calculates the elastic axis of a spiral spring in a free state according to its given final shape, taking into account the moment that arises as a result of the applied force at the end of the spring. For this, the nonlinear theory of absolutely elastic bending using the methods of differential geometry of curved lines as a function of the own length of the arc of the spring axis is applied.
 The scope of application of the innovative technological product. The results obtained in the study can be used in the manufacture of spiral springs for various purposes, as well as for the accumulation of potential energy in the absence of other sources

Integration of the two-dimensional Heisenberg model by methods of differential geometry

Integration of the two-dimensional Heisenberg model by methods of differential geometry

Integration of the two-dimensional Heisenberg model by methods of differential geometry

Для интегрирования двумерной модели Гейзенберга использованы методы классической дифференциальной геометрии. Уравнения модели после преобразования годографа были записаны в терминах метрического тензора, связанного с криволинейной системой координат, и его производных. Показано, что их общее решение описывает все известные ранее точные решения, кроме плоского вихря. Предсказан и проанализирован новый тип вихревой структуры - "вихревая полоса" в двумерном ферромагнетике. Ее отличительные свойства - конечные размеры области определения, конечность полной энергии и отсутствие ядра вихря при наличии вихревой структуры.

Contact dynamic analysis of tether-net system for space debris capture using incremental potential formulation

An efficient computational method is proposed to combine the discrete differential geometry (DDG) method with the multi-dimensional incremental potential (IP) formulation to simulate the contact dynamics between a shooting net and a defunct satellite. The geometrically nonlinearity of the elastic net is described using the DDG method, which discretizes the flexible net into multiple nodes and connected edges. The internal elastic forces and the associated Jacobians are derived analytically. The multi-dimensional incremental potential formulation is used to achieve the intersection-free condition during the physical collision between deformable tether-net and rigid space debris. The smooth IP barrier functional constructed for contact simulation is differentiable, such that the nonlinear elastodynamic system can be robustly solved through a fully implicit time-marching scheme. Our numerical approach is first applied to simple examples, and demonstrating its effectiveness in dealing with non-penetration contacts of elements with different dimensions, such as ‘edge-to-edge’ and ‘edge-to-surface’. Next, the dynamic interaction between the deploying net and defunct satellite is analysed. A parameter sweep is performed with our robust and efficient simulator to show the configuration evolution of the net shot from different relative locations of the target object. Our numerical results can provide an optimal deployment strategy for tether-net capture systems during the active debris removal missions.

Тензор кривизны n-поверхности и ее сферического образа в En+k

The main task of classical multidimensional differential geometry is to study the properties of various n-surfaces. Often these studies use torsion coefficients that are defined for any n-surface with codimension k > 1 in (n + k)-dimensional Euclidean space. For hypersurfaces, the torsion coefficients are not defined.Another important concept used to study the properties of n-surfaces is the spherical Gaussian mapping. The Gaussian mapping defined on submanifolds of Euclidean and pseudo-Euclidean spaces allows one to study the external properties of a submanifold immersed in a Euclidean or pseudo-Euclidean space. In a number of papers, the properties of the Gaussian mapping are studied, as well as the geometric characteristics of the images of submanifolds under a spherical mapping, which are submanifolds of a hypersphere or a Grassmannian.In this article, we study the local properties of the spherical image of a regular n-surface of arbitrary codimension. The spherical mapping is defined for n-surfaces with codimension greater than one in Euclidean space by means of a regular vector field. Each vector of this field at a point of the submanifold is orthogonal to the tangent space of the submanifold at the chosen point.The article uses the methods of differential and Riemannian geometry, as well as tensor analysis to study n-surfaces with a codimension greater than one. Under some additional conditions, a connection is established between the curvature tensor of a given surface and the curvature tensor of its spherical image. Under the same additional conditions, some geometric characteristics of the points of the spherical image of the original n-surface are studied.

Reference shape of bodies with finite incompatible deformations

The work develops differential-geometric methods for modeling of finite incompatible deformations of hyperelastic solids. Deformation incompatibility can be caused, for example, by inhomogeneous temperature fields and distributed defects. As a result, residual stresses and distortion of geometric shape of the body occur. These factors determine the critical parameters of modern high-precision technologies, in particular, in additive manufacturing technologies. In this regard, the development of methods for their quantitative description is an urgent problem of modern solid mechanics.
 The application of methods of differential geometry is based on the representation of a body as a smooth manifold equipped with a metric and a non-Euclidean connection. This approach allows one to interpret the body as a global stress-free shape and to formulate the physical response and material balance equations with respect to this shape. Within the framework of the geometric method, deformations are characterized by embeddings of non-Euclidean shape into physical space, which is still considered to be Euclidean. Measures of incompatibility are identified with the invariants of the affine connection, namely, torsion, curvature, and nonmetricity, and the connection itself is determined by the type of physical process.

Using geometry and biomechanics to understand aortic pathology and endovascular aortic repair outcomes.

Aortic disorders such as dissections and aneurysms affect the structural integrity and strength of the body's major artery. Endovascular aortic repairs have become the preferred course of treatment for such disorders, where a cylindrical stent graft made of metal and fabric is inserted in the aorta to restrict blood flow away from the regions of disease. After endovascular repairs, unfavorable outcomes as a result of the aorta failing to remodel include endoleak, graft migration, neck dilation, graft kinking, and material fatigue. To better understand the interactions between the aorta and stent graft, we need to mechanistically understand how these different biological and synthetic surfaces couple and interact together. Our approach is to apply methods in differential geometry, finite element analysis, and computational fluid dynamics to 3D models of aortas generated from medical imaging. Our results show that there exists a geometric signature that correlates aortic size and shape with clinical outcomes. Additionally, aortas appear to undergo shape preserving growth during normal development; however, pathologic aortas grow with shape changes. For stent grafts to be effective, they must conform to the shape of the aorta and align against the aortic wall. In order to understand why endovascular aortic repairs fail, we capitalize on the enormous availability of medical imaging data on patients with aortic pathology. We hypothesize that geometric incompatibility between stent graft and aorta impacts the mechanical stability of the seal zone. By considering the geometry and biomechanics of the aorta, we hope to develop a risk-stratification and classification scheme to optimize outcomes by defining better patient selection for endovascular surgery and creating design rules for the next generation of stent grafts.

On Exponential Synchronization Rates for High-Dimensional Kuramoto Models With Identical Oscillators and Digraphs

For a high-dimensional Kuramoto model with identical oscillators under a general digraph that has a directed spanning tree, although the exponential synchronization has been proved under some initial state constraints, exponential synchronization rates have not been described exactly until now. In this article, the supremum of exponential synchronization rates is precisely determined as the smallest real part of the nonzero Laplacian eigenvalues of the digraph. Our obtained result extends the existing results from the special case of strongly connected balanced digraphs to the condition of general digraphs owning directed spanning trees, which is the weakest condition for synchronization from the aspect of network structure. Moreover, our adopted method is completely different from and much more elementary than the previous differential geometry method.

Spherical Formation Tracking Control of Non-Holonomic UAVs with State Constraints and Time Delays

This paper addresses a novel spherical formation tracking control problem of multiple UAVs with time-varying delays in the directed communication network, where the dynamics of each UAV is non-holonomic and in the presence of spatiotemporal flowfields. The state constraints (that is, position and velocity constraints) are derived from our previous differential geometry method and the F–S formulas. The state constraints and time delays in the directed communication network bring many difficulties to controller design. To this end, a virtual-structure-like design is given to achieve a formation with delayed information by using Lyapunov–Krasovskii functionals, and then proposing a barrier Lyapunov function for the satisfaction of state constraints to design a novel spherical formation tracking algorithm. The general assumption of the rate of change of time-varying delays, and a certain initial position and velocity adjustment range are given. Simulation results show the feasibility and effectiveness of the proposed algorithm.

Mathematical modeling of the process of vacuum drying of corn cobs

Purpose. To determine the effect of vacuum on the process of drying corn seeds on the cob using mathematical modeling. Methods. Differential calculus methods were used, in particular, methods for solving differential diffusion equations with time-varying boundary conditions, as well as methods of differential and analytical geometry. The mathematical model was analyzed using graphs constructed by the computer program MathCad 13 from the solutions of differential equations and methods of technical analysis. Results. On the basis of the differential diffusion equation with variable boundary conditions in time, the processes of moisture removal from corn seeds in the cob were described. These processes are described by analytical dependencies that describe the kinetics of drying corn seeds and the entire cob, while taking into account diffusion processes both inside the seed and in the cob core. They are used to construct graphs that describe the dynamics of the moisture field inside a single seed and all the seeds along the cob. They show that the part of the seed that is outside the cob takes the longest to dry. The graphs also show the unevenness of drying along the cob: the ends of the cob dry the fastest, and the middle dries the slowest. Conclusions. The vacuum drying of corn cobs consists of 3 processes that take place in parallel: evaporation of moisture from the end surface of the cob, diffusion of moisture to the outer surface of the cob, and diffusion of moisture from the seed to the cob. The outer part of the seed takes the longest to dry, resulting in a moisture gradient that moves toward the core. Reducing the pressure inside the drying chamber reduces the time for moisture equalization, both over the volume of the seed and along the length of the cob, and increases the drying speed of the cob. To increase the uniformity of drying, the distribution of the heating temperature of the seeds along the length of the cob should correspond to a parabolic relationship with the highest temperature in the middle of the cob length, or it is necessary to divide the cob into parts to reduce the length of the moisture path. Keywords: seeds, moisture diffusion, drying time, moisture evaporation, vacuum, humidity, corn cob, cob, drying kinetics, drying chamber.

КОВЗАННЯ ЧАСТИНКИ ПО РУХОМІЙ ГОРИЗОНТАЛЬНІЙ ПЛОЩИНІ

A lot of engineering problems relate to the interaction of working bodies of machines with particles of technological material and require analytical dependencies of particle movement on a moving rough plane. When a particle hits a moving horizontal rough plane, which remains horizontal during movement, the particle begins to slide along it. The character of the movement of the plane determines the shape of the particle's sliding trajectories. The reciprocating oscillations of the horizontal plane, as well as translational oscillations, when all points of the plane describe circles, are sufficiently researched. However, there is no rotation of the plane in these cases. However, numerous parts of machines and mechanisms carry out just such a movement. If the plane moves translationally, the particle moves along a trajectory, which is similar to the curve described by the plane. During the rotational movement of the plane, the particle moves along a spiral of the correct shape. In the case of a combination of these two movements, at the initial stage the relative motion of the particle is chaotic, but over time it acquires the shape of a spiral regardless of point of particle incidence to the plane. The article deals with the relative motion of a particle on a horizontal rough plane, which carries out complex oscillations, which are the result of moving a point of the plane in a circle with a constant angular velocity relative to its center and simultaneous rotation of the plane around this point with the same angular velocity in the opposite direction. The partial case of oscillations, when the lengths of the crank and slider are equal to zero, is considered. Analytical dependencies of particle motion were found by methods of differential geometry. The obtained regularities make it possible to significantly expand the theory of particle movement on the surface. In addition, they can be applied to the geometric design of mechanisms of the crank-slider type, in which the length of the slider is equal to the length of the crank. Key words: relative motion, horizontal plane, oscillations with rotational motion, particle, differential equations.

Justification and calculation of design and strength parameters of screw loaders

Existing designs of screw working bodies of transport and processing machines cannot always provide the required performance and avoid overloading, which causes deformation and damage to parts of the mechanisms. Thus, it is necessary to establish new efficient screw working bodies that would ensure the execution of the technological process. The purpose of the study was to substantiate and explore the design and methodology for calculating the main structural-power parameters of the developed screw loader, to synthesise screw loaders using the method of morphological analysis to obtain designs of mechanisms with better technical and economic characteristics. The methods of surface theory, analytical and differential geometry, computer graphics and mathematics were used to conduct the research. As a result of theoretical and experimental studies, the designs of screw conveyors with hopper loading and screw conveyors with nozzle loading are presented. In addition, a classification of loaders for screw transport and technological mechanisms is presented. The dependencies for calculating the design of screw conveyor loaders are determined. The synthesis of loaders and mixers by the method of morphological analysis was performed to obtain designs of mechanisms with the best technical and economic characteristics and competitive designs of mechanisms were established. A system of coding and synthesis of structural elements and mechanisms of screw implements with improved technological properties is proposed, and competitive designs of screw implement that perform technological processes efficiently and effectively are selected. A methodology for calculating the main structural and power parameters of a screw loader has been developed, which allows for determining the parameters of a screw loader. The materials of the research can be used in further studies of screw conveyors and, in particular, screw loaders to determine their design, technological, energy and power parameters

Study of the adiabatic passage in tripod atomic systems in terms of the Riemannian geometry of the Bloch sphere

We present an analysis of the stimulated Raman adiabatic passage (STIRAP) processes based on the methods of differential geometry. The present work was inspired by an excellent article by Shore et al (Unanyan et al 1999 Phys. Rev. A 59 2910). We demonstrate how a purely geometric interpretation of the adiabatic passage in quantum tripod systems as a Riemannian parallel transport of the dark state vector along the Bloch sphere allows describing the evolution of the system for a given sequence of Stokes, pump and control laser excitation pulses. In combination with the Dykhne–Davis–Pechukas adiabaticity criterion and the minimax principle for circles on a sphere, this approach allows obtaining the analytical form of the optimal laser pulse sequences for a high fidelity tripod fractional STIRAP. In contrast to the conventional STIRAP in Λ-systems, the Gaussian approximations of the optimal laser pulse sequences allow reaching the infidelity of 10−7 for the adiabaticity parameter of 300 without noticeable oscillatory or other detrimental effects on population transfer accuracy.