The Synthesis of the Pitch Surfaces of Internal and External Skew-Gears and Their Racks

2005 ◽  
Vol 128 (4) ◽  
pp. 794-802 ◽  
Author(s):  
Giorgio Figliolini ◽  
Jorge Angeles
Keyword(s):  
Euclidean Space ◽  
Dual Space ◽  
Spatial Motion ◽  
Screw Axis ◽  
Dual Algebra ◽  
Screw Motion ◽  
Bevel Gears ◽  
The Subject ◽  
Instant Screw Axis ◽  
Spherical Motion

The synthesis of the pitch surfaces of any pair of external and internal skew gears, using dual algebra and the principle of transference, is the subject of this paper. The spatial motion of the Euclidean space is transferred to the dual space in order to obtain a simplified dual spherical motion, thus emulating the motion of bevel gears. The relative screw motion is hence analyzed by determining the position of the instant screw axis and the angular and sliding velocities. Moreover, the hyperboloid pitch surfaces of the driving and driven gears are synthesized, along with the helicoid pitch surface of their rack. Several numerical results are reported.

The Synthesis of the Axodes of Spatial Four-Bar Linkages

2012 ◽  
Author(s):  
Giorgio Figliolini ◽  
Pierluigi Rea ◽  
Jorge Angeles
Keyword(s):  
Motion Analysis ◽  
Unit Sphere ◽  
Dual Space ◽  
Moving Frames ◽  
Spatial Motion ◽  
Screw Axis ◽  
Dual Algebra ◽  
Different Types ◽  
Instant Screw Axis ◽  
Four Bar Linkage

The paper introduces a procedure for the motion analysis of a four-bar linkage by means of dual algebra and the Principle of Transference. This procedure allows the mapping of the motion from the Euclidean to the spherical dual space. In particular, the position analysis of a spatial four-bar linkage is formulated by referring to a spherical four-bar linkage, which moves on the dual unit sphere. Moreover, both fixed and moving axodes of the coupler link are obtained, as traced by the spatial motion of the instant screw axis (ISA) with respect to the fixed and moving frames. These ruled surfaces reproduce the spatial motion of the coupler link upon relatively rolling and sliding around and along the ISA. Finally, the proposed procedure has been implemented in MatLab, in order to analyze the motion of the different types of four-bar linkages, including the Bennett mechanism and the Hooke joint. The motion is illustrated by means of animations of the four-bar linkages and their axodes.

Generation of Noncircular Bevel Gears With Free-Form Tooth Profile and Curvilinear Tooth Lengthwise

2016 ◽  
Vol 138 (6) ◽  
Author(s):  
Fangyan Zheng ◽  
Lin Hua ◽  
Xinghui Han ◽  
Dingfang Chen
Keyword(s):  
Screw Theory ◽  
Mapping Method ◽  
Tooth Profile ◽  
Bevel Gear ◽  
Free Form ◽  
Arc Length ◽  
Screw Axis ◽  
Bevel Gears ◽  
Generating Method ◽  
Instant Screw Axis

Noncircular bevel gear is applied to intersecting axes, realizing given function of transmission ratio. Currently, researches are focused mainly on gear with involute tooth profile and straight tooth lengthwise, while that with free-form tooth profile and curvilinear tooth lengthwise are seldom touched upon. Based on screw theory and equal arc-length mapping method, this paper proposes a generally applicable generating method for noncircular bevel gear with free-form tooth profile and curvilinear tooth lengthwise, covering instant screw axis, conjugate pitch surface, as well as the generator with free-form tooth profile and curvilinear tooth lengthwise. Further, the correctness of the proposed method is verified through illustrations of computerized design.

The Synthesis of the Axodes of RCCC Linkages

2015 ◽  
Vol 8 (2) ◽  
Author(s):  
Giorgio Figliolini ◽  
Pierluigi Rea ◽  
Jorge Angeles
Keyword(s):  
Ruled Surface ◽  
Line Contact ◽  
Output Function ◽  
Input Output ◽  
Screw Axis ◽  
Dual Algebra ◽  
Synthesis Procedure ◽  
Instant Screw Axis ◽  
Four Bar Linkage

As the coupler link of an RCCC linkage moves, its instant screw axis (ISA) sweeps a ruled surface on the fixed link; by the same token, the ISA describes on the coupler link itself a corresponding ruled surface. These two surfaces are the axodes of the linkage, which roll while sliding and maintaining line contact. The axodes not only help to visualize the motion undergone by the coupler link but also can be machined as spatial cams and replace the four-bar linkage, if the need arises. Reported in this paper is a procedure that allows the synthesis of the axodes of an RCCC linkage. The synthesis of this linkage, in turn, is based on dual algebra and the principle of transference, as applied to a spherical four-bar linkage with the same input–output function as the angular variables of the RCCC linkage. Examples of RCCC linkages are included. Moreover, to illustrate the generality of the synthesis procedure, it is also applied to a spherical linkage, namely, the Hooke joint, and to the Bennett linkage.

The Role of the Orthogonal Helicoid in the Generation of the Tooth Flanks of Involute-Gear Pairs With Skew Axes

2015 ◽  
Vol 7 (1) ◽  
Author(s):  
Giorgio Figliolini ◽  
Hellmuth Stachel ◽  
Jorge Angeles
Keyword(s):  
Smooth Surface ◽  
Tangent Plane ◽  
Line Contact ◽  
Planar Surface ◽  
Screw Axis ◽  
Bevel Gears ◽  
Involute Gears ◽  
G2 Continuity ◽  
Instant Screw Axis

Camus' concept of auxiliary surface (AS) is extended to the case of involute gears with skew axes. In the case at hand, we show that the AS is an orthogonal helicoid whose axis (a) lies in the cylindroid and (b) is normal to the instant screw axis of one gear with respect to its meshing counterpart; in general, the helicoid axis is skew with respect to the latter. According to the spatial version of Camus' Theorem, any line or surface attached to the AS, in particular any line L of AS itself, can be chosen to generate a pair of conjugate flanks with line contact. While the pair of conjugate flanks is geometrically feasible, as they always share a line of contact and the tangent plane at each point of this line, they even have the same curvature, G2-continuity, when L coincides with the instant screw axis (ISA). This means that the two surfaces penetrate each other, at the same common line. The outcome is that the surfaces are not realizable as tooth flanks. Nevertheless, this is a fundamental step toward the synthesis of the flanks of involute gears with skew axes. In fact, the above-mentioned interpenetration between the tooth flanks can be avoided by choosing a smooth surface attached to the AS, instead of a line of the AS itself, which can give, in particular, the spatial version of octoidal bevel gears, when a planar surface is chosen.

The Evaluation of Moments of Regions With Piecewise-Linearly Approximated Boundaries via Line Integration

1989 ◽  
Author(s):  
J. Angeles ◽  
M. J. Al-Daccak
Keyword(s):  
Euclidean Space ◽  
Three Dimensional ◽  
Inertia Tensor ◽  
Divergence Theorem ◽  
Dimensional Euclidean Space ◽  
Repeated Application ◽  
The Subject ◽  
First Moment

Abstract The subject of this paper is the computation of the first three moments of bounded regions imbedded in the three-dimensional Euclidean space. The method adopted here is based upon a repeated application of Gauss’s Divergence Theorem to reduce the computation of the said moments — volume, vector first moment and inertia tensor — to line integration. Explicit, readily implementable formulae are developed to evaluate the said moments for arbitrary solids, given their piecewise-linearly approximated boundary. An example is included that illustrates the applicability of the formulae.

Kinematic Analysis of Spherical Four-Bar Linkages via the Inflection Cubic and Thales Ellipse

2015 ◽  
Author(s):  
Giorgio Figliolini ◽  
Jorge Angeles
Keyword(s):  
Special Interest ◽  
Kinematic Analysis ◽  
Vector Fields ◽  
Three Dimensions ◽  
Unique Point ◽  
Planar Case ◽  
Acceleration Vector ◽  
The Subject ◽  
Four Bar Linkage ◽  
Spherical Motion

The subject of this paper is the formulation of a specific algorithm for the kinematic analysis of spherical four-bar linkages via the inflection spherical cubic and spherical Thales ellipse by devoting particular attention to the crossed four-bar linkage (anti-parallelogram). Moreover, both the inflection and the elliptic cones, which represent the equivalent of the Bresse cylinders of the planar case in three-dimensions, are obtained by showing the particular properties of the spherical motion in terms of the curvature of a coupler curve and both the velocity and acceleration vector fields. Of special interest are also the cases in which the three acceleration poles coincide at one unique point or in two plus one, which depends on the intersections of two spherical curves of third and second degree.

Synthesis of the Pitch Surfaces of Non-Circular Skew-Gears

2010 ◽  
Author(s):  
Giorgio Figliolini ◽  
Pierluigi Rea ◽  
Jorge Angeles
Keyword(s):  
Transmission Ratio ◽  
Dual Algebra ◽  
Cylindrical Gears ◽  
Geometrical Characteristics ◽  
Variable Transmission ◽  
The Subject ◽  
Gear Engagement

The subject of this paper is the synthesis of the pitch surfaces of non-circular skew gears, intended to generate any motion program with a periodically varying transmission ratio. This is done by extending an existing algorithm, which was formulated through the application of dual algebra and the Principle of Transference. In particular, the variable transmission ratio of N-lobed elliptical and logarithmical cylindrical gears is expressed and analyzed along with their main characteristics to test the proposed algorithm, which is implemented in Matlab. The code generates the pitch surfaces of N-lobed elliptical and logarithmical skew gears, along with those of indexing skew gears. Finally, significant numerical and graphical results are shown to analyze the geometrical characteristics of the gear engagement. Not unexpectedly, cylindrical and bevel non-circular gears become particular cases thereof.

Generic substitutions

2005 ◽  
Vol 70 (1) ◽  
pp. 61-83 ◽  
Author(s):  
Giovanni Panti
Keyword(s):  
Continuous Function ◽  
Free Algebra ◽  
Classical Logic ◽  
Propositional Logic ◽  
Dual Space ◽  
Extensive Literature ◽  
Cantor Space ◽  
Natural Measure ◽  
The Subject ◽  
Stochastic Properties

AbstractUp to equivalence, a substitution in propositional logic is an endomorphism of its free algebra. On the dual space, this results in a continuous function, and whenever the space carries a natural measure one may ask about the stochastic properties of the action. In classical logic there is a strong dichotomy: while over finitely many propositional variables everything is trivial, the study of the continuous transformations of the Cantor space is the subject of an extensive literature, and is far from being a completed task. In many-valued logic this dichotomy disappears: already in the finite-variable case many interesting phenomena occur, and the present paper aims at displaying some of these.

scholarly journals The Scepticism of Descartes’s Meditations

2011 ◽  
Vol 67 (2) ◽  
pp. 271-279
Author(s):  
James Thomas
Keyword(s):  
Euclidean Space ◽  
Thirteenth Century ◽  
Sensory Experience ◽  
Controlling Factor ◽  
New Science ◽  
The Will ◽  
Aristotelian Science ◽  
The Subject

What I’m suggesting is that the model for Descartes’s defence of Renaissance science would be Aquinas’s own defence of thirteenth-century Aristotelian science, except that the coherence of the will took on the role of the consistency of concepts, as the controlling factor in the analyses of all types of science. As a result, the new science would incorporate the awareness of Platonic ideas and the divisibility of Euclidean space as equally valid input into a dialectical knowledge of sensory experience. You can read the early arguments to doubt the reality of sensory experience and reason as a way of dividing out the experience of the will in affirming or denying an object’s nature, as the subject for subsequent inquiry.

scholarly journals Magnetic curves in a Euclidean space: One example, several approaches

2013 ◽  
Vol 94 (108) ◽  
pp. 141-150 ◽  
Author(s):  
Marian Munteanu
Keyword(s):  
Magnetic Field ◽  
Vector Field ◽  
Euclidean Space ◽  
Energy Level ◽  
Killing Vector ◽  
Short Review ◽  
Killing Vector Field ◽  
Screw Motion ◽  
Fixed Energy

This is a short review of different approaches in the study of magnetic curves for a certain magnetic field and on the fixed energy level. We emphasize them in the case when the magnetic trajectory corresponds to a Killing vector field associated to a screw motion in the Euclidean 3-space.

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