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We classify the maximal algebraic subgroups of Bir(C x ℙ¹), when C is a smooth CM projective curve of positive genus. [ABSTRACT FROM AUTHOR]

In this study, the involute of a curve is investigated in Minkowski space-time by using extended Darboux frame field and curves of the AW (k) type are studied in Minkowski space-time. [ABSTRACT FROM AUTHOR]

In this paper, we first give necessary conditions in which we can decide whether a given curve is biharmonic or 1-type harmonic and differential equations characterizing the regular curves. Then we research the Frenet formulas of involute of a unit speed curve by making use of the relations between the involute of a curve and the curve itself. In addition we apply these formulas to define the essential conditions by which one can determine whether the involute of a unit speed curve is biharmonic or 1-type harmonic and then we write differential equations characterizing the involute curve by means of Frenet apparatus of the unit speed curve. Finally we examined the helix as an example to illustrate how the given theorems work. [ABSTRACT FROM AUTHOR]

In industrial production, the roller trace design is still based on the trial-and-error method that is more like an art than science. In this paper, we establish the mathematical model of the involute curve roller trace and adopted the forming clearance compensation in the attaching-mandrel process. The backward pass roller trace is optimized to avoid the roller interference due to blank springback. The spinning simulation model of seven forming passes is set up and verified by the experiments with superalloy GH3030. The wall thickness, the strain distribution, and the tool forces are analyzed. The results show that the forming clearance compensation can greatly shorten the forming time and enhance the production efficiency and saving energy. The metal accumulates at the edge of the blank, and the maximum thinning zone appeared near the edge and is prone to crack. In the straightening pass, the tool forces of both the roller and the mandrel are larger than those in the other passes. [ABSTRACT FROM AUTHOR]

In this paper, we investigate special Smarandache curves with regard to Sabban frame of involute curve. We create Sabban frame belonging to spherical indicatrix of involute curve. Smarandache curves are explained by Sabban vectors belonging to spherical indicatrix. Then, we calculate geodesic curvatures of this Smarandache curves. The results found for each curve is given depending on evolute curve. The example related to the subject is given and their figures are drawn with MAPPLE program. [ABSTRACT FROM AUTHOR]

We have already defined and worked on the second order involute curve of a unit speed curve in IL³. In this paper, we consider the second order involute of a spacelike curve with timelike binormal in IL³ . There are three kinds of casual caharacteristics of the second order involute curve. All Frenet apparatus of their are examined in terms of Frenet apparatus of the curve α. [ABSTRACT FROM AUTHOR]

Due to the application in practice and in the process of training students in engineering specialties, it is very important to analyze the possibilities of profiling real involute curves of spure gears. The report presents specific ones opportunities to build a real involute curves of a spure gear through one of the most used CAD systems. The results of the application of the AutoCAD .NET API developed by the authors for the simulation of the profile of cylindrical spur gears with external teeth are presented. The report is presented a comparative analysis between involute profiles built from existing CAD systems and profiles built through the developed application by the authors. The results of the research identified many possibilities for using the developed application for the educational process of students and for use in real specific production of gears obtained through the technologies of direct cutting of the contours of the gears by laser, plasma and water jet cutting. [ABSTRACT FROM AUTHOR]

In this study, the relation between Frenet vectors of the natural lift curve α of the curve α and Frenet vectors of a β the involute of α is given in IR3 and IR31. [ABSTRACT FROM AUTHOR]

In this paper, we have expressed the transformation matrix between the Frenet frames of the spacelike curve with a spacelike binormal and its timelike involute curve as depend on lorentzian spacelike angle between the unit binormal vector and the Darboux vector of the spacelike evolute curve. Furthermore, some new characterizations with relation to the involute-evolute curve couple have been found. [ABSTRACT FROM AUTHOR]

In this paper we consider nine special ruled surfaces associated to an involute of a curve α and its Bertrand mate α** with k1 6= 0. They are called as involute Frenet ruled and Bertrandian Frenet ruled surfaces, because of their generators which are the Frenet vector fields of curve α. First we give the striction curves of all Frenet ruled surfaces. Then the striction curves of involute and Bertrandian Frenet ruled surfaces are given in terms of the Frenet apparatus of the curve α. Some results are given on the striction curves of involute and Bertrand Frenet ruled surfaces based on the tangent vector fields in E³. [ABSTRACT FROM AUTHOR]

Evolute curves were studied by some researchers in 4 dimensional Euclidean space. However the special characters of the curve are not considered which is a research gap in this technique. In this study a kind of generalized involute and evolute curve-couple is considered in 4 dimensional Euclidean space. The necessary and suficient condition for the curve possessing generalized involute as well as evolute is obtained. [ABSTRACT FROM AUTHOR]

We prove a connected sum formula for involutive Heegaard Floer homology, and use it to study the involutive correction terms of connected sums. In particular, we give an example of a three-manifold with d(Y)≠ d(Y)≠ d (Y). We also construct a homomorphism from the three-dimensional homology cobordism group to an algebraically defined Abelian group, consisting of certain complexes (equipped with a homotopy involution) modulo a notion of local equivalence. [ABSTRACT FROM AUTHOR]

The orthogonal trajectories of the first tangents of the curve are called the involutes of x. The hyperspheres which have higher order contact with a curve x are known osculating hyperspheres of x. The centers of osculating hyperspheres form a curve which is called generalized evolute of the given curve x in n-dimensional Euclidean space En. In the present study, we give a characterization of involute curves of order k (resp. evolute curves) of the given curve x in n-dimensional Euclidean space En. Further, we obtain some results on these type of curves in E³ and E4, respectively. [ABSTRACT FROM AUTHOR]

External gear pumps (EGPs) have gained popularity among applications in many fields like fluid power transmissions and systems, automotive, aerospace thanks to their advantage of simplicity, robustness and low cost. Several studies were performed to analyze and innovate the gear profiles of EGPs to achieve better performance, in terms of flow smoothness power to weight ratio. Asymmetric gears represent one of the possible choices. This paper is aimed at developing a methodology of designing asymmetric involute gear, and formulate analytical expression for the instantaneous flowrate and flow non-uniformity given by asymmetric, non-standard involute gear pumps. These analytical expressions are then used within a multi-objective numerical optimization algorithm aimed at minimizing both the flow non-uniformity and the pump size to achieve a specific displacement. The results illustrate the highly constrained nature of the optimization problem, and the relevant impact of certain parameters of the tooth profile. It is shown how gears obtained with the proposed procedure can have significantly higher performance of standard gears, and in particular how the parameters affecting tooth profile asymmetry can further improve the EGP flow irregularity and size. [ABSTRACT FROM AUTHOR]

Purpose This paper aims to present simulation results of a harmonic drive (HD) with involute flexspline (FS) profiles based on two-dimensional (2-D) finite element analysis (FEA).Design/methodology/approach First, the mathematical model of the FS with involute tooth profile was developed using a straight-edge rack cutter based on the theory of gearing. Then the engaging circular spline (CS) with conjugate tooth profile of FS was derived based on the enveloping theory and theory of gearing. Additionally, a mesh generation program was developed to discretize the FS based on the mathematical model. An elliptical wave generator (WG) was inserted into the FS, and a torque was applied to drive the FS meshing with the CS. The WG and the CS were both assumed to be rigid in the finite element model.Findings Finally, a 2-D FEA was conducted to explore the stress distribution on the FS, the engagement movement of the FS, the torsional stiffness and the engaged area of teeth of the HD under various conditions. Moreover, this research also studied the effect of changing pressure angle of the involute FS on the performance of the HD.Research limitations/implications The simulation model and methodology presented in this paper paved the way for further investigation and optimization of the HD with involute tooth profile FS and conjugate CS.Originality/value The simulation model of HD is established on conjugate shape based on the theory of gearing and an automatic mesh generation program is developed to generate the finite element model. The characteristics of the HD can thus be simulated according to the developed model. [ABSTRACT FROM AUTHOR]

This paper concerns iterations of two classical geometric constructions, the evolutes and involutes of plane curves, and their discretizations: evolutes and involutes of plane polygons. In the continuous case, our main result is that the iterated involutes of closed locally convex curves with rotation number one (possibly, with cusps) converge to their curvature centers (Steiner points), and their limit shapes are hypocycloids, generically, astroids. As a consequence, among such curves only the hypocycloids are homothetic to their evolutes. The bulk of the paper concerns two kinds of discretizations of these constructions: the curves are replaced by polygons, and the evolutes are formed by the circumcenters of the triples of consecutive vertices ( $$\mathcal {P}$$ -evolutes), or by the incenters of the triples of consecutive sides ( $$\mathcal {A}$$ -evolutes). For equiangular polygons, the theory is parallel to the continuous case: we define discrete hypocycloids (equiangular polygons whose sides are tangent to hypocycloids) and a discrete Steiner point. The space of polygons is a vector bundle over the space of the side directions; our main result here is that both kinds of evolutes define vector bundle morphisms. In the case of $$\mathcal {P}$$ -evolutes, the induced map of the base is 4-periodic, and the dynamics reduces to the linear maps on the fibers. We prove that the spectra of these linear maps are symmetric with respect to the origin. The asymptotic dynamics of linear maps is determined by their eigenvalues with the maximum modulus, and we show that all types of behavior can occur: in particular, hyperbolic, when this eigenvalue is real, and elliptic, when it is complex. We also study $$\mathcal {P}$$ - and $$\mathcal {A}$$ -involutes and prove that the side directions of iterated $$\mathcal {A}$$ -involutes of polygons with odd number of sides behave ergodically; this generalizes well-known results concerning iterations of the construction of the pedal triangle. In addition to the theoretical study, we performed numerous computer experiments; some of the observations remain unexplained. [ABSTRACT FROM AUTHOR]

In this study we worked on the involute of involute curve of curve α. We called them the second order involute of curve α in E3. All Frenet apparatus of the second order involute of curve α are examined in terms of Frenet apparatus of the curve α. Further we show that; Frenet vector fields of the second order involute curve α2 can be written based on the principal normal vector field of curve α. Besides, we illustrate examples of our results. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to describe ruled surfaces generated by a Frenet trihedron of closed dual involute for a given dual curve. We identify relations between the pitch, the angle of the pitch, and the drall of these surfaces. Some new results related to the developability of these surfaces are also obtained. Finally, we illustrate these surfaces by presenting one example. [ABSTRACT FROM AUTHOR]

In this study we worked on the involute of involute curve of curve α. We called them the second order involute of curve α in E3. All Frenet apparatus of the second order involute of curve α are examined in terms of Frenet apparatus of the curve α. Further we show that; Frenet vector fields of the second order involute curve α2 can be written based on the principal normal vector field of curve α. Besides, we illustrate examples of our results. [ABSTRACT FROM AUTHOR]

With their advantages of low-cost, high-reliability and simplicity, external gear machines (EGMs) are among the most common pumps or motors in high pressure applications. As all positive displacement machines, EGMs are characterized by a substantial flow non-uniformity, which is given by the gear meshing and results in vibrations and noises. Several analytical methods are available but require the knowledge of the tooth profile, involute or cycloidal, and are not directly applicable to unconventional gears. This paper presents a numerical approach for the analysis of the theoretical flow through spur gear EGMs which does not require a-priori knowledge of the gear profile. Starting from a CAD input, the proposed approach automatically determines control volumes for every angular position of the gears to describe the displacing action realized by the unit. After describing the method, the paper shows its application to different geometric morphologies for the gear profiles to calculate the theoretical flow rate. The verification of the numerical model is demonstrated by comparing its results with those provided by the analytical methods available in literature for the case of involute gears. [ABSTRACT FROM AUTHOR]

Skew-polynomial rings, or Ore extensions, constitute an important class in noncommutative ring theory. These structures are currently studied from different points of view such as ideal theory, order theory, Galois theory, homological algebras, etc. Computationally, Ore extensions appear in the context of uncoupling and solving systems of linear differential and difference equations in closed form. In this short note we let K denote a division ring, α : K → K a ring endomorphism and S : K → an a-derivation. We determine the involutions in the Ore extension K[x; α, δ]. [ABSTRACT FROM AUTHOR]

The involute profile is the most commonly used tooth profile for cylindrical gears. However, its counterpart for bevel gears is not commonly used because its generation is not compatible with the use of easy-to-manufacture cutting tools and market-oriented generation methods. Nowadays, with additive manufacturing technology growing across industries to manufacture parts and full-scale products, the spherical involute profile deserves to be given the possibility to become the reference profile for bevel gears generated by additive manufacturing, forging or plastic molding. In this work, both the spherical involute and the octoidal form systems for bevel gears are mathematically defined. The geometry of their respective crown-gears for generation is derived and the computerized modeling of the geometries of the generated bevel gears presented. The comparison between the obtained gear tooth geometries is performed from the point of view of the obtained geometric deviations, their behavior to the change of the shaft angle and the evolution of contact and bending stresses for different values of torque applied to the pinion. The results showed that the spherical involute profile can handle better with the changes in the shaft angle, although the octoidal profile presents lower contact and bending stresses. [ABSTRACT FROM AUTHOR]

We are given a semiprime ringRwith involution *. We show that the following conditions are equivalent. Condition 1: For each elementxofR, [x*,x](=x*x − xx*) = 0. Condition 2: There is a fixed natural numberN′ such that [x*,xN′] = 0, all elementsxofR. Condition 3: There is a fixed natural numberNsuch that for each elementxofR,, wheredxis the inner derivation induced byx, andis the Nth power ofdxunder composition of maps. [ABSTRACT FROM AUTHOR]

The structure of locally finite groups with an involution whose centralizer has finite rank was described by Kuzucuolu and Shumyatsky. In this paper, we investigate the structure of locally finite groups with an element of order 3 whose centralizer has finite rank. Moreover, for anyp, we classify simple locally finite groups having an automorphism of orderpwhose set of fixed points has finite rank. [ABSTRACT FROM AUTHOR]

In this paper, we consider special ruled surfaces associated to the involute curve α* and Bertrand mate α** of a curve α. They are called as Involutive Darboux Frenet ruled surface and Bertrandian Darboux Frenet ruled surface, cause of their generators are the Darboux vector fields of involute curve α* and Bertrand mate α**. We give the parametric equations of striction curves along these Frenet ruled surfaces in terms of the Frenet apparatus of curve α. Further some results are given based on the tangent vector fields of these striction curves along Involutive and Bertrand Darboux ruled surfaces in E3. [ABSTRACT FROM AUTHOR]

A method for fitting a topography measured on an involute spur gear tooth flank to its nominal shape is proposed, so that the exact position along the length of the tooth from which the measured surface originates can be determined, and so that the nominal shape of the surface can be subtracted prior to roughness parameters calculation. [ABSTRACT FROM AUTHOR]

Two-dimensional steepest descent curves (SDC) for a quasiconvex family are considered; the problem of their extensions (with constraints) outside of a convex body K is studied. It is shown that possible extensions are constrained to lie inside of suitable bounding regions depending on K. These regions are bounded by arcs of involutes of ∂K and satisfy many inclusions properties. The involutes of the boundary of an arbitrary plane convex body are defined and written by their support function. Extensions SDC of minimal length are constructed. Self-contracting sets (with opposite orientation) are considered: necessary and/or sufficient conditions for them to be subsets of SDC are proved. [ABSTRACT FROM AUTHOR]

This paper describes methods of teeth shaving and gridding at helical-bevel gears of involute-bevel gearings. Furthermore, the opportunity of doing these operations on machines for teeth processing in common bevel gears is shown. The possibility of these operations’ performance on the equipment that is used for common helical gears teeth processing is showed. [ABSTRACT FROM AUTHOR]

Rigid-body (screw) motions in three-dimensional Euclidean space can be represented by involution (resp. anti-involution) mappings obtained by dual-quaternions which are self-inverse and homomorphic (resp. anti-homomrphic) linear mappings. In this paper, we will represent four dual-quaternion matrices with their geometrical meanings; two of them correspond to involution mappings, while the other two correspond to anti-involution mappings. [ABSTRACT FROM AUTHOR]

In this paper we consider twelve special ruled surfaces associated to the curve α, the involute curve α* and Bertrand mate α** , with k1 ≠ 0. They are called as Frenet ruled, involutive Frenet ruled and Bertrandian Frenet ruled surfaces, cause of their generators are the Frenet vector fields of curve α. First we give all the parametrizations of all Frenet ruled surfaces in terms of the Frenet apparatus of curve α. We examined distribution parameters and Singularity of involutive and Bertrand Frenet ruled surfaces based on the Frenet apparatus of curve α in E3. [ABSTRACT FROM AUTHOR]

Power skiving is an efficient means of producing high accuracy gears; particularly internal gears. However, the literature lacks a systematic mathematical model for the design and analysis of such tools. Accordingly, the present study proposes a comprehensive yet straightforward methodology for the design of resharpening power skiving tools based on conjugate surface theory and full-field angle analysis. The proposed methodology has three important features. First, mathematical models of the virtual-conjugate-surface and rake surfaces are derived, and thus the cutting edges can be obtained. Second, the power skiving is modeled by transformation matrices, and the normal/tangent vectors of the relative surface can be explicitly obtained. As a result, the working rake and clearance angles can be investigated in accordance with recommended ISO-standards. Finally, a method is prescribed for determining the working rake angles, clearance angles and wedge angles of the cutter based on the ISO-defined reference planes. Utilizing this method, a simple technique is proposed for redesigning the power skiving tool so as to avoid negative working clearance angles. The results of the illustrative example are shown that the proposed approach provides a comprehensive, simple and versatile technique for modeling a wide range of power skiving tool design features. [ABSTRACT FROM AUTHOR]

In this paper we consider twelve special ruled surfaces associated to the curve α, the involute curve α* and Bertrand mate α** , with k1 ≠ 0. They are called as Frenet ruled, involutive Frenet ruled and Bertrandian Frenet ruled surfaces, cause of their generators are the Frenet vector fields of curve α. First we give all the parametrizations of all Frenet ruled surfaces in terms of the Frenet apparatus of curve α. We examined distribution parameters and Singularity of involutive and Bertrand Frenet ruled surfaces based on the Frenet apparatus of curve α in E3. [ABSTRACT FROM AUTHOR]

This paper describes helical-bevel gearing (HBG). The helical-bevel gearing commonly includes a cylindrical and a bevel gear. The tooth profiles can be either spur or helical. The involute HBG provides an opportunity to design a broad variety of gear trains, unobtainable by the use of conventional types of gears. The article defines methods of their generation, a contact shape and points out advantages of various arrangements of gears, diagrams of an external helical-bevel gearing skew axis, hypoid gearing, intersecting axes, parallel axes. The expediency of the use of such gear in a machine drive is justified. [ABSTRACT FROM AUTHOR]

This article describes the possibilities controlling technological factors and increasing the reliability of the technological process of tooth processing through integrated management accuracy processing based on the functional dependencies of systematic technological factors and deviations of the parameters of gears, where the argument of functional dependencies is the deviation of a single parameter of the gears deviation of the radius of the involute profiles. [ABSTRACT FROM AUTHOR]

The aim of this technical brief is to provide a new viewpoint of friction factor for contact stress calculations of gears. The idea of friction factor has been coined, for the calculation of contact stresses along the tooth contact for different helical gear pairs. Friction factors were developed by evaluating contact stresses with and without friction for different gear pairs. In this paper, three-dimensional (3D) finite element method (FEM) and Lagrange multiplier algorithm have been used to evaluate the contact stresses. Initially, a spur gear finite element (FE) model was validated with the theoretical analysis under frictionless condition, which is based on Hertz's contact theory. Then, similar FE models were constructed for 5?deg, 15?deg, 25?deg, and 35?deg helical gear pairs. The contact stresses of these models were evaluated for different coefficients of friction. These results were employed for the development of friction factor. [ABSTRACT FROM AUTHOR]

Let FG be the group ring of a group G over a field F of characteristic different from 2, and let FG have an involution induced from one on G . Assuming that G has no elements of order 2 and no dihedral group involved, we determine the conditions under which the set of skew elements of FG is bounded Lie Engel. Furthermore, we make the determination with no restrictions upon G when the involution on FG is classical. [ABSTRACT FROM AUTHOR]

We have already defined the evolutes and the involutes of fronts without inflection points. For regular curves or fronts, we can not define the evolutes at inflection points. On the other hand, the involutes can be defined at inflection points. In this case, the involute is not a front but a frontal at inflection points. We define evolutes of frontals under conditions. T he definition is a generalisation of both evolutes of regular curves and of fronts. By using relationship between evolutes and involutes of frontals, we give an existence condition of the evolute with inflection points. We also give properties of evolutes and involutes of frontals. [ABSTRACT FROM AUTHOR]

It is proved that there exist periodic groups containing elements of even order and only trivial normal 2 -subgroups, in which every pair of involutions generates a 2 -group. This gives a negative answer to Question 11 .11 a in the Kourovka Notebook. Furthermore, we point out examples of finite simple groups that are recognizable by spectrum in the class of finite groups but not recognizable in the class of all groups. [ABSTRACT FROM AUTHOR]

In this paper, we investigate some characterizations of involute - evolute curves in dual space. Then the relationships between dual Frenet frame and Darboux vectors of these curves are found. [ABSTRACT FROM AUTHOR]

This study investigates an approach for parametric modeling and dynamic contact analysis of involute/noninvolute beveloid gears. Firstly, the mathematical models of involute/noninvolute beveloid gear pairs are derived based on the theory of gearing and the generation mechanism. Then the parametric modeling programs of involute/noninvolute beveloid gears are developed to automatically generate exact model via a Matlab code. Subsequently, a numerical example of intersecting axes beveloid gears is presented to evaluate the dynamic stress distribution and dynamic transmission error. Finally, the dynamic contact characteristics of involute and noninvolute beveloid gears are calculated by three-dimensional dynamic contact finite element method, respectively. The results show that the noninvolute beveloid gear pairs can relieve the high dynamic stress and contact shock problem of intersecting axes beveloid gear pairs. [ABSTRACT FROM AUTHOR]