scholarly journals Geometric and Meshing Properties of Conjugate Curves for Gear Transmission

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Dong Liang ◽  
Bingkui Chen ◽  
Yane Gao ◽  
Shuai Peng ◽  
Siling Qin
Keyword(s):  
Differential Geometry ◽  
Contact Angle ◽  
Power Transmission ◽  
Characteristic Point ◽  
Geometric Analysis ◽  
Gear Transmission ◽  
Axial Position ◽  
Gear Design ◽  
Conjugate Surfaces ◽  
The Given

Conjugate curves have been put forward previously by authors for gear transmission. Compared with traditional conjugate surfaces, the conjugate curves have more flexibility and diversity in aspects of gear design and generation. To further extend its application in power transmission, the geometric and meshing properties of conjugate curves are discussed in this paper. Firstly, general principle descriptions of conjugate curves for arbitrary axial position are introduced. Secondly, geometric analysis of conjugate curves is carried out based on differential geometry including tangent and normal in arbitrary contact direction, characteristic point, and curvature relationships. Then, meshing properties of conjugate curves are further revealed. According to a given plane or spatial curve, the uniqueness of conjugated curve under different contact angle conditions is discussed. Meshing commonality of conjugate curves is also demonstrated in terms of a class of spiral curves contacting in the given direction for various gear axes. Finally, a conclusive summary of this study is given.

Edges of Regression and Limit Normal Point of Conjugate Surfaces1

1998 ◽  
Vol 122 (4) ◽  
pp. 419-425 ◽  
Author(s):  
Ningxin Chen
Keyword(s):  
Differential Geometry ◽  
Contact Boundary ◽  
Boundary Line ◽  
Tangent Point ◽  
Common Tangent ◽  
Numerical Examples ◽  
Envelope Surface ◽  
The Common ◽  
Conjugate Surfaces ◽  
Definition Of

The presented paper utilizes the basic theory of the envelope surface in differential geometry to investigate the undercutting line, the contact boundary line and the limit normal point of conjugate surfaces in gearing. It is proved that (1) the edges of regression of the envelope surfaces are the undercutting line and the contact boundary line in theory of gearing respectively, and (2) the limit normal point is the common tangent point of the two edges of regression of the conjugate surfaces. New equations for the undercutting line, the contact boundary line and the limit normal point of the conjugate surfaces are developed based on the definition of the edges of regression. Numerical examples are taken for illustration of the above-mentioned concepts and equations. [S1050-0472(00)00104-5]

Geometric Analysis and Synthesis of the Metamorphic Robotic Hand

2006 ◽  
Vol 129 (11) ◽  
pp. 1191-1197 ◽  
Author(s):  
Jian S. Dai ◽  
Delun Wang
Keyword(s):  
Differential Geometry ◽  
Geometric Analysis ◽  
Robotic Hand ◽  
Traditional Structure ◽  
Curve Approximation ◽  
Analysis And Synthesis ◽  
Finger Motion ◽  
The Relationship ◽  
Robotic Fingers

This paper presents a novel robotic hand with a metamorphic palm which changes the traditional structure of a robotic hand. Based on this new hand structure, the paper investigates motion of the robotic fingers with respect to the palm by presenting finger operation planes and by revealing the relationship between finger motion and palm motion. The study presents the normals of the finger operation planes as a function of the input angles of the palm and uses these normals to relate finger motion to palm motion. This leads to the coaxial condition of the finger-palm relationship that is then converted to the coplanar condition of normals of all finger operation planes. The condition is then used to generate a coupler trajectory, and the iterative trajectory fitting and curve approximation are used for synthesis of palm links, leading to differential geometry based synthesis of angular lengths of the metamorphic palm.

Analytic program derivation in type theory

1998 ◽  
Author(s):  
Petri Mäenpää
Keyword(s):  
Mathematical Method ◽  
Problem Solving ◽  
Type Theory ◽  
Mathematical Problem ◽  
Geometric Analysis ◽  
Algebraic Method ◽  
Point Of View ◽  
Program Derivation ◽  
Mathematical Problems ◽  
The Given

This work proposes a new method of deriving programs from their specifications in constructive type theory: the method of analysis-synthesis. It is new as a mathematical method only in the area of programming methodology, as it is modelled upon the most successful and widespread method in the history of exact sciences. The method of analysis-synthesis, also known as the method of analysis, was devised by Ancient Greek mathematicians for solving geometric construction problems with ruler and compass. Its most important subsequent elaboration is Descartes’s algebraic method of analysis, which pervades all exact sciences today. The present work expands this method further into one that aims at systematizing program derivation in a heuristically useful way, analogously to the way Descartes’s method systematized the solution of geometric and arithmetical problems. To illustrate the method, we derive the Boyer-Moore algorithm for finding an element that has a majority of occurrences in a given list. It turns out that solving programming problems need not be too different from solving mathematical problems in general. This point of view has been emphasized in particular by Martin-Löf (1982) and Dijkstra (1986). The idea of a logic of problem solving originates in Kolmogorov (1932). We aim to refine the analogy between programming and mathematical problem solving by investigating the mathematical method of analysis in the context of programming. The central idea of the analytic method, in modern terms, is to analyze the functional dependencies between the constituents of a geometric configuration. The aim is to determine how the sought constituents depend on the given ones. A Greek analysis starts by drawing a diagram with the sought constructions drawn on the given ones, in the relation required by the problem specification. Then the sought constituents of the configuration are determined in terms of the given ones. Analysis was the Greeks’ method of discovering solutions to problems. Their method of justification was synthesis, which cast analysis into standard deductive form. First it constructed the sought objects from the given ones, and then demonstrated that they relate as required to the given ones. In his Geometry, Descartes developed Greek geometric analysis-synthesis into the modern algebraic method of analysis.

scholarly journals On the Affine Image of a Rational Surface of Revolution

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2061
Author(s):  
Juan G. Alcázar
Keyword(s):  
Differential Geometry ◽  
Affine Transformation ◽  
Rational Surface ◽  
Affine Differential Geometry ◽  
Affine Mapping ◽  
Surfaces Of Revolution ◽  
Surface Of Revolution ◽  
Affine Image ◽  
Fixed Line ◽  
The Given

We study the properties of the image of a rational surface of revolution under a nonsingular affine mapping. We prove that this image has a notable property, namely that all the affine normal lines, a concept that appears in the context of affine differential geometry, created by Blaschke in the first decades of the 20th century, intersect a fixed line. Given a rational surface with this property, which can be algorithmically checked, we provide an algorithmic method to find a surface of revolution, if it exists, whose image under an affine mapping is the given surface; the algorithm also finds the affine transformation mapping one surface onto the other. Finally, we also prove that the only rational affine surfaces of rotation, a generalization of surfaces of revolution that arises in the context of affine differential geometry, and which includes surfaces of revolution as a subtype, affinely transforming into a surface of revolution are the surfaces of revolution, and that in that case the affine mapping must be a similarity.

Differential geometry of autonomous wheel-legged robots

2019 ◽  
Vol 37 (2) ◽  
pp. 615-637
Author(s):  
Vahide Bulut
Keyword(s):  
Differential Geometry ◽  
Trajectory Planning ◽  
Design Methodology ◽  
Geometric Analysis ◽  
Legged Robots ◽  
Rough Terrain ◽  
Geometric Properties ◽  
Content Type ◽  
Robot Trajectory ◽  
Novel Approach

Purpose The purpose of this study is to obtain the differential geometric analysis of autonomous wheel-legged robots and their trajectories on the terrain. Design/methodology/approach The author uses a wheel using the osculating sphere of the curve on rough terrain. Additionally, the author expresses a triple osculating sphere wheel by taking advantage of differential geometry. Moreover, the author examined the consecutive wheel center-curves to obtain the optimum posture of a micro-hydraulic toolkit (MHT) robot. Findings The author examined the terrain path, which is crucial for trajectory planning in terms of the geometric perspective. The author designed the triple MHT wheel using the osculating sphere of the MHT robot trajectory by taking advantage of local differential geometric properties of this curve on the terrain. The consecutive wheel center-curves were expressed and studied based on differential geometry. Originality/value The author provides a novel approach for the optimum posture of an MHT robot using consecutive wheel-center curves and provides an original perspective to MHT robot and its trajectory by using differential geometry.

Geometry design and mathematical model of a new kind of gear transmission with circular arc tooth profiles based on curve contact analysis

2014 ◽  
Vol 228 (17) ◽  
pp. 3200-3208 ◽  
Author(s):  
Bingkui Chen ◽  
Dong Liang ◽  
Yane Gao
Keyword(s):  
Mathematical Model ◽  
High Performance ◽  
Plane Curve ◽  
Gear Transmission ◽  
Circular Arc ◽  
Smooth Curves ◽  
Gear Drive ◽  
Geometry Design ◽  
Meshing Equation ◽  
The Given

A new meshing relationship for gear drive to characterize the conjugation geometry is studied in this paper based on conjugate curves. Conjugate curves are described as two smooth curves always keep continuous and tangent contact with each other in given contact direction under motion law. The general principle of curve meshing is developed for the given spatial or plane curve. The meshing equation along arbitrary direction of contact angle is derived. The properties of geometric and motion of the contact of conjugate curves are discussed. According to the equidistance-enveloping method, tubular meshing surfaces are proposed to build up circular arc tooth profiles, which inherit all properties of conjugate curves. The geometry design and mathematical model of the gears are established. Three types of contact pattern of tooth profiles are generated: convex-to-convex, convex-to-plane and convex-to-concave. A calculation example for convex-to-concave tooth profiles of gears is provided. Theoretical and numerical results demonstrate the feasibility and correctness of proposed conjugate curves theory and it lays the foundation for the design of high performance gear transmission.

Optimizing involute gear design for maximum bending strength and equivalent pitting resistance

2007 ◽  
Vol 221 (4) ◽  
pp. 479-488 ◽  
Author(s):  
V Spitas ◽  
C Spitas
Keyword(s):  
Bending Strength ◽  
Power Transmission ◽  
Optimization Procedure ◽  
Contact Ratio ◽  
Element Analysis ◽  
Gear Design ◽  
Involute Gear ◽  
Involute Gears ◽  
High Power Transmission ◽  
Complex Algorithm

Standard involute gear designs dominate high-power transmission applications because they combine sufficient bending strength with high pitting resistance, while retaining an adequate contact ratio. In this paper, a non-standard, optimal alternative involute gear design has been presented, which has the same pitting resistance as the standard involute gears but exhibits maximum resistance to bending. The optimization procedure is based on the complex algorithm, where the root stress, as calculated through tabulated boundary element analysis values, is the objective function and the active constraints include all of the kinematical, manufacturing and geometrical conditions, which must be satisfied by the optimal design, including the pitting resistance. The results indicate that optimal designs can achieve up to 8.5 per cent reduction of the fillet stress. Two-dimensional photoelasticity was used to verify the optimization results.

scholarly journals Glaze Icing on Superhydrophobic Coating Prepared by Nanoparticles Filling Combined with Etching Method for Insulators

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Chao Guo ◽  
Ruijin Liao ◽  
Yuan Yuan ◽  
Zhiping Zuo ◽  
Aoyun Zhuang
Keyword(s):  
Contact Angle ◽  
Photoelectron Spectroscopy ◽  
Superhydrophobic Surface ◽  
Power Transmission ◽  
Contact Angle Hysteresis ◽  
Transmission System ◽  
Sliding Angle ◽  
Power Transmission System ◽  
Etching Method ◽  
Xps Characterization

Icing on insulators may cause flashover or even blackout accidents in the power transmission system. However, there are few anti-icing techniques for insulators which consume energy or manpower. Considering the water repelling property, the superhydrophobic surface is introduced for anti-icing of insulators. Among the icing forms, the glaze icing owns the highest density, strongest adhesion, and greatest risk to the power transmission system but lacks researches on superhydrophobic surface. In this paper, superhydrophobic surfaces with contact angle of 166.4°, contact angle hysteresis of 0.9°, and sliding angle of less than 1° are prepared by nanoparticle filling combined with etching method. The coated glass slide and glass insulator showed excellent anti-icing performance in the glaze icing test at −5°C. The superhydrophobicity and anti-icing property of the coatings benefit from the low surface energy and hierarchical rough structure containing micron scale pits and nanoscale coralloid bulges supported by scanning electron microscopy (SEM), atomic force microscopy (AFM), and X-ray photoelectron spectroscopy (XPS) characterization.

scholarly journals Inductive power transmission through printed sprial coils for seizure application

2021 ◽  
Author(s):  
Amin Kalbasi
Keyword(s):  
Power Transmission ◽  
Realistic Model ◽  
Outer Diameter ◽  
Matlab Simulation ◽  
Compact Layer ◽  
Spiral Coil ◽  
Spiral Coils ◽  
Inductive Power Transmission ◽  
The Given ◽  
Inductive Power

This thesis proposes a realistic model for transcutaneous inductive power link for seizure applications using PSCs (Printed Spiral Coils). The benefit of this model is smaller size implanted coil compared to its counterparts while maintaining high loaded system efficiency. The introduced Printed Spiral Coil (PSC) geometric parameters are achieved using MATLAB that searches for the highest efficiency of the inductive coil within the given constraints. The output from the MATLAB simulation is used to created optimum design in AMDSpro tool and is verified. The outer diameter of the implanted coil is introduced to be d₀₂ = 6mm while the simulated efficiency is calculated as η [subscript] sim = 46.67% operating at f [subscript] sim = 2.52MHz for the relative distance of D = 10mm filled with layers of modeled human skull (Outer Compact Layer, Spongiosum, and Inner Compact Layer). The coupling coefficient of the spiral was calculated to be k = 0.69. The implanted PSC is associated with load capacitance and resistance of R [subscript] L = 4.5Ω and C [subscript] L = 95nf.

scholarly journals A Distortion Theorem for Analytic Maps of Annuli

1965 ◽  
Vol 17 ◽  
pp. 185-198
Author(s):  
C. E. Castonguay ◽  
H. G. Helfenstein
Keyword(s):  
Differential Geometry ◽  
Extremal Property ◽  
Riemannian Metrics ◽  
Distortion Theorem ◽  
Point Of View ◽  
Covering Surface ◽  
Open Riemann Surface ◽  
Euclidean Metric ◽  
Analytic Maps ◽  
The Given

Every abstract open Riemann surface can be made "concrete" (in the terminology of (1)) by considering it as a covering surface (in general branched) of the complex plane by means of a suitable projection map p. Since this covering map is not unique, it seems natural to single out some such maps by an extremal property. The use of Riemannian metrics compatible with the conformai structure on the given surface for the study of $1 is well known ; from the point of view of differential geometry it suggests an investigation of the distortion caused by p between such a metric ds^ and the Euclidean metric of .

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