Kinematic Geometry of Curve and Its Application to Geometric Modeling Flat Gear

2017 ◽  
Vol 5 (2) ◽  
pp. 25-31
Author(s):  
Панчук ◽  
K. Panchuk ◽  
Ляшков ◽  
A. Lyashkov ◽  
Варепо ◽  
...  
Keyword(s):  
Geometric Modeling ◽  
Geometric Model ◽  
Plane Curve ◽  
Geometric Theory ◽  
Analytical Description ◽  
Geometric Interpretation ◽  
Transmission Performance ◽  
Proposed Model ◽  
Kinematic Geometry ◽  
Fixed System

The paper presents the results of investigations in the field of kinematic geometry of the spatial curve of a line. The basis of the research is the method of the movable trihedron of the curve. The components of the trihedron motion along the spatial curve are considered and it is shown that its resultant instantaneous motion is screw motion. This result differs from the representation of the motion of a trihedron known in geometry as a rotation described by the Darboux vector. An analytical description of the set of axes of instantaneous helical motions of a trihedron in a moving and fixed system of assigning a spatial curve is given. The possibility of applying the obtained general results to the investigation of a plane curve is shown. The paper proposes a flat tooth gearing model based on the geometric interpretation of the motions of a trihedron of a plane curve and known in the geometric theory of plane mechanisms of the construction of Bobillier. The geometric scheme of this construction is expanded due to the introduction of evolutes simulating instantaneous motions of trihedron of the corresponding construction curves. As a result, a geometric model is obtained, which is more complete in comparison with the known models of flat gearing. It allows to perform both direct and inverse tasks of profiling the teeth of the wheels while simultaneously obtaining the curvature of the desired profiles in the absence of such. The proposed model can be used as the basis for the development of gears with a planar gearing scheme by the condition of achieving the necessary transmission performance due to the geometric shape of the teeth of the wheels.

GEOMETRIC THEORY OF MULTIDIMENSIONAL INTERPOLATION

2020 ◽  
Vol 2020 (1) ◽  
pp. 9-16
Author(s):  
Evgeniy Konopatskiy
Keyword(s):  
Response Function ◽  
Algebraic Curves ◽  
Geometric Theory ◽  
Analytical Description ◽  
Geometric Interpretation ◽  
Affine Geometry ◽  
Mathematical Apparatus ◽  
Multidimensional Interpolation ◽  
Pass Through

The paper presents a geometric theory of multidimensional interpolation based on invariants of affine geometry. The analytical description of geometric interpolants is performed within the framework of the mathematical apparatus BN-calculation using algebraic curves that pass through preset points. A geometric interpretation of the interaction of parameters, factors, and the response function is presented, which makes it possible to generalize the geometric theory of multidimensional interpolation in the direction of increasing the dimension of space. The conceptual principles of forming the tree of the geometric interpolant model as a geometric basis for modeling multi-factor processes and phenomena are described.

scholarly journals The Method of High Accuracy Calculation of Robot Trajectory for the Complex Curves

2020 ◽  
Vol 28 (4) ◽  
pp. 247-252
Author(s):  
Alexander Lozhkin ◽  
Pavol Bozek ◽  
Konstantin Maiorov
Keyword(s):  
Calculation Method ◽  
Geometric Modeling ◽  
Classical Method ◽  
Geometric Model ◽  
New Method ◽  
Approximation Methods ◽  
Linear Transformations ◽  
Design Quality ◽  
Complex Curves ◽  
Internal Properties

AbstractThe geometric model accuracy is crucial for product design. More complex surfaces are represented by the approximation methods. On the contrary, the approximation methods reduce the design quality. A new alternative calculation method is proposed. The new method can calculate both conical sections and more complex curves. The researcher is able to get an analytical solution and not a sequence of points with the destruction of the object semantics. The new method is based on permutation and other symmetries and should have an origin in the internal properties of the space. The classical method consists of finding transformation parameters for symmetrical conic profiles, however a new procedure for parameters of linear transformations determination was acquired by another method. The main steps of the new method are theoretically presented in the paper. Since a double result is obtained in most stages, the new calculation method is easy to verify. Geometric modeling in the AutoCAD environment is shown briefly. The new calculation method can be used for most complex curves and linear transformations. Theoretical and practical researches are required additionally.

scholarly journals MODELING ARCHITECTURAL FORM SURFACE DEPENDENT SECTIONS

2021 ◽  
pp. 53-58
Author(s):  
A. A. Chekalin ◽  
M. K. Reshetnikov ◽  
V. V. Shpilev ◽  
S. V. Borodulina ◽  
S. A. Ryazanov
Keyword(s):  
Geometric Modeling ◽  
Geometric Model ◽  
Residential Building ◽  
Cubic Splines ◽  
Piecewise Smooth ◽  
Additional Parameter ◽  
Mathematical Apparatus ◽  
Surface Design ◽  
Fourth Degree ◽  
Architectural Form

For the design of surfaces in architecture, as a rule, universal techniques developed for other technical industries are used. First of all, these are general kinematic surfaces and interpolation cubic splines for modeling complex piecewise smooth surfaces. The authors propose to use the fourth degree inerodifferential spline developed by them for problems of geometric modeling of architectural forms. For calculations and constructions on a computer, the proposed spline is not much more complicated than traditional cubic splines, since it has one additional parameter - a coefficient. However, this allows you to locally control the shape of a curve or surface during design, that is, to change the shape in individual areas without affecting other areas. The article proposes a method for constructing a geometric model of the kinematic surface of dependent sections with a fourth degree parabola as a generator. When using cubic splines as a guide, the surface is a 3 × 4 non-uniform (heterogeneous) spline. The article shows that the surface on the basis of the proposed mathematical apparatus can be composite piecewise-smooth. A particular case of surface design is considered on the example of creating a model of the surface of the facade of a residential building according to the existing concept. The algorithm can be easily programmed and added as a tool to existing CAD systems.

scholarly journals Geometric Modelling of Octagonal Lamp Poles

2014 ◽  
Vol XL-5 ◽  
pp. 145-150 ◽  
Author(s):  
T. O. Chan ◽  
D. D. Lichti
Keyword(s):  
Point Cloud ◽  
Geometric Model ◽  
Real Data ◽  
Point Clouds ◽  
Circular Cone ◽  
Basic Point ◽  
The Real ◽  
Cone Model ◽  
Proposed Model ◽  
Pole Point

Lamp poles are one of the most abundant highway and community components in modern cities. Their supporting parts are primarily tapered octagonal cones specifically designed for wind resistance. The geometry and the positions of the lamp poles are important information for various applications. For example, they are important to monitoring deformation of aged lamp poles, maintaining an efficient highway GIS system, and also facilitating possible feature-based calibration of mobile LiDAR systems. In this paper, we present a novel geometric model for octagonal lamp poles. The model consists of seven parameters in which a rotation about the z-axis is included, and points are constrained by the trigonometric property of 2D octagons after applying the rotations. For the geometric fitting of the lamp pole point cloud captured by a terrestrial LiDAR, accurate initial parameter values are essential. They can be estimated by first fitting the points to a circular cone model and this is followed by some basic point cloud processing techniques. The model was verified by fitting both simulated and real data. The real data includes several lamp pole point clouds captured by: (1) Faro Focus 3D and (2) Velodyne HDL-32E. The fitting results using the proposed model are promising, and up to 2.9 mm improvement in fitting accuracy was realized for the real lamp pole point clouds compared to using the conventional circular cone model. The overall result suggests that the proposed model is appropriate and rigorous.

BIOMECHANICAL MODEL PREDICTING VALUES OF MUSCLE FORCES IN THE SHOULDER GIRDLE DURING ARM ELEVATION

2010 ◽  
Vol 10 (04) ◽  
pp. 643-666 ◽  
Author(s):  
ERIC BERTHONNAUD ◽  
MELISSA MORROW ◽  
GUILLAUME HERZBERG ◽  
KAI-NAN AN ◽  
JOANNES DIMNET
Keyword(s):  
Geometric Modeling ◽  
Geometric Model ◽  
Contractile Element ◽  
Muscle Forces ◽  
Active Muscle ◽  
Cross Sectional ◽  
Shoulder Complex ◽  
Arm Elevation

A three-dimensional (3D) geometric model for predicting muscle forces in the shoulder complex is proposed. The model was applied throughout the range of arm elevation in the scapular plan. In vitro testing has been performed on 13 cadaveric shoulders. The objectives were to determine homogeneous values of physiological parameters of shoulder muscles and to locate sites of muscular attachment to any bone of the shoulder complex. Muscular fiber lengths, lengths of contractile element (CE), and muscle volumes were measured, corresponding physiological cross-sectional area (PCSA) were calculated, and force/length muscle relations were found. An in vivo biplanar radiography was performed on five volunteers. The photogrammetric reconstruction of bone axes and landmarks were coupled with a geometric modeling of bones and muscle sites of attachment. Muscular paths were drawn and changes in lengths during movement have been estimated. Directions of muscle forces are the same as that of muscular path at the point of attachment to bone. Magnitudes of muscular forces were found from muscle lengths coupled with force/length relations. Passive forces were directly determined contrary to active muscle forces. A resulting active muscle force is calculated from balancing weight and passive forces at each articular center. Active muscle forces were calculated by distributing the resulting force among active muscles based on the muscular PCSA values.

scholarly journals Geometric modeling of surfaces dependent cross sections in the tasks of spinning and laying

2019 ◽  
Vol 110 ◽  
pp. 01057
Author(s):  
Yuri Deniskin ◽  
Pavel Miroshnichenko ◽  
Andrew Smolyaninov
Keyword(s):  
Composite Materials ◽  
Cross Sections ◽  
Geometric Modeling ◽  
Geometric Model ◽  
Solid Modeling ◽  
Uniform Method ◽  
Calculation Methods ◽  
Related Process

The article is devoted to the development of a geometric model of surfaces of dependent sections to solve the problems of winding by continuous fibers in the direction of the force and its related process of automated winding of composite materials. A uniform method for specifying the surfaces of dependent sections with a curvilinear generator and a method for solid modeling of the shell obtained by winding or calculation methods are described.

scholarly journals Modeling and Precise Processing for Spaceborne Transmitter/Missile-Borne Receiver SAR Signals

2019 ◽  
Vol 11 (3) ◽  
pp. 346 ◽  
Author(s):  
Shiyang Tang ◽  
Ping Guo ◽  
Linrang Zhang ◽  
Chunhui Lin
Keyword(s):  
Geometric Model ◽  
Algorithm Design ◽  
Cross Coupling ◽  
Synthetic Aperture ◽  
Hybrid Domain ◽  
Geometric Configurations ◽  
Proposed Model ◽  
Sar Model ◽  
Simulation Results ◽  
Path Constraint

The spaceborne transmitter/missile-borne receiver (ST/MR) synthetic aperture radar (SAR) could provide several unique advantages, such as wide coverage, unrestricted geography, a small detection probability of the missile, and forward-looking imaging. However, it is also accompanied by problems in imaging, including geometric model establishment and focusing algorithm design. In this paper, an ST/MR SAR model is first presented and then the flight-path constraint, characterized by geometric configurations, is derived. Considering the impacts brought about by the maneuvers of the missile, a non-‘Stop-Go’ mathematical model is devised and it can avoid the large errors introduced by the acceleration, which is neglected in the traditional model. Finally, a two-dimensional (2-D) scaling algorithm is developed to focus the ST/MR data. Without introducing any extra operations, it can greatly remove the spatial variations of the range, azimuth, and cross-coupling phases simultaneously and entirely in the 2-D hybrid domain. Simulation results verify the effectiveness of the proposed model and focusing approach.

DESIGN OF A NOVEL STENTED VALVE AND 3D MODELING OF ITS IMPLANTATION IN THE AORTA

2010 ◽  
Vol 22 (02) ◽  
pp. 157-161 ◽  
Author(s):  
Gideon Praveen Kumar ◽  
Lazar Mathew
Keyword(s):  
Aortic Valve ◽  
Left Ventricle ◽  
3D Modeling ◽  
Geometric Modeling ◽  
Geometric Model ◽  
Bioprosthetic Heart Valve ◽  
3D Cad ◽  
Percutaneous Aortic Valve Replacement ◽  
Improved Design ◽  
Valve Model

Objective: To design a novel percutaneous stented valve and model its implantation in the aorta.Background: The dimensions of stented aortic valve components govern its ability to prevent backflow of blood into the left ventricle. Whilst the theoretical parameters for the best stent performance have already been established, an effective valve model and its suitability along with the stent are lacking.Methods: This article discusses the design of a stented valve suitable for percutaneous aortic valve replacement. Steps involved in 3D CAD-based geometric modeling of the stented aortic valve and its implantation in the aorta are presented. Conceptual designing of individual components was used to build the total geometric model.Results: A novel geometric model of percutaneous stented aortic valve was generated. The improved design enhances its performance during and after implantation.Conclusion: The blunt hooks in the stent model prevent its migration in either direction by getting embedded in the aortic endothelium. This novel stent aortic valve may be of great interest to designers of future bioprosthetic heart valve models, as well as to surgeons involved in minimally invasive valve surgeries.

scholarly journals TO THE THEORY OF n-DIMENSIONAL BRACHISTOCHRONE

2020 ◽  
pp. 53-60
Author(s):  
Gladkov S.O. ◽  
◽  
Bogdanova S.B. ◽  
Keyword(s):  
Three Dimensional ◽  
Plane Curve ◽  
Viscous Friction ◽  
Analytical Description ◽  
The Body ◽  
Viscous Drag ◽  
Friction Forces ◽  
Drag Forces ◽  
Moving Body ◽  
Plane Shape

In this paper, a solution to the problem of the motion of a brachistochrone in the ndimensional Euclidean space is firstly presented. The very first formulation of the problem in a two-dimensional case was proposed by J. Bernoulli in 1696. It represented an analytical description of the trajectory for the fastest rolling down under gravitational force only. Thereafter, a number of problems devoted to a brachistochrone were considered with account for gravitational forces, dry and viscous drag forces, and a possible variation in the mass of a moving body. Analytical solution to the formulated problem is presented in details by an example of the body moving along a brachistochrone in three-dimensional Cartesian coordinates. The obtained parametric solution is confirmed by a graphical interpretation of the calculated result. The formulated problem is solved for an ideal case when drag forces are neglected. If dry and viscous friction forces are taken into account, the plane shape of the brachistochrone remains the same,while the analysis of the solution becomes more complicated. When, for example, a side air flow is taken into account, the plane curve is replaced by a three-dimensional brachistochrone.

scholarly journals A Systematic Approach to Support Design for Manufacturability in Injection Molding and Die Casting

1995 ◽  
Author(s):  
Yuh-Shan Yueh ◽  
R. Allen Miller
Keyword(s):  
Injection Molding ◽  
Geometric Modeling ◽  
Geometric Model ◽  
Die Casting ◽  
Support Design ◽  
Strongly Coupled ◽  
Draw Direction ◽  
Design Cycle ◽  
Boundary Information ◽  
Modeling Data

Abstract In designing a part to be produced by injection molding or die casting, designers need to consider manufacturing characteristics of the part such as filling and ejectability from the dies as well as functional issues. The typical design cycle is iterative, laborious and time-consuming. In this paper, we present a procedure for defining parting information (locations where the mold/die come together), and recognizing the links between part design and die/mold construction. Many decisions and design details, such as draft on surfaces parallel to the draw (die opening) direction, gate and runner locations, vent locations, etc., depend on the parting locations and characteristics. Parting information is normally not part of the geometric model of the part design. Parting design, including draw direction and parting location, is addressed through a custom user interface which contains several options related to different levels of geometric modeling data. The resulting specification is stored in a segment structure which provides a flexible parting description and fits within the B-rep hierarchy. The reasoning about the linking of related surfaces is accomplished by splitting and traversing the extracted geometric entities based on parting definition. The entities covered by the same die/mold component are aggregated as a face group which is a set of complete or partial surfaces with the parting definition as the boundary information and with the draw direction as the moving information. In this approach, manufacturing information can be strongly coupled with geometric data to form a complete part model which supports manufacturability assessment and facilitates any necessary shape transformations to achieve a manufacturable part in a straightforward manner so that design iterations can be controlled and development cost can be reduced.

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