The Image Curve of the Coupler of a Special Spherical Four Bar Linkage

1988 ◽  
Vol 110 (3) ◽  
pp. 276-280
Author(s):  
J. M. McCarthy
Keyword(s):  
Rational Functions ◽  
Global Properties ◽  
Kinematic Mapping ◽  
Curvature And Torsion ◽  
Four Bar Linkage ◽  
Spherical Motion

This paper uses a kinematic mapping of spherical motion to derive an image curve which represents the coupler motion of a doubly folding spherical four bar linkage. The image curve of this linkage, the so called “kite” linkage, can be parameterized by rational functions. This parameterization is presented as well as formulas which allow the computation of its curvature and torsion at any point. These formulas provide a link between the global properties of the coupler motion as represented by the image curve itself and its instantaneous properties given by the curvature and torsion functions.

The Image of the Coupler Motion of a Special Spherical Four Bar Linkage

1987 ◽  
Author(s):  
J. M. McCarthy
Keyword(s):  
Rational Functions ◽  
Global Properties ◽  
Kinematic Mapping ◽  
Curvature And Torsion ◽  
Four Bar Linkage ◽  
Spherical Motion

Abstract This paper uses a kinematic mapping of spherical motion to derive an image curve which represents the coupler motion of a doubly folding spherical four bar linkage. The image curve of this linkage, the so called “kite” linkage, can be parameterized by rational functions. This parameterization is presented as well as formulas which allow the computation of its curvature and torsion at any point. These formulas provide a link between the global properties of the coupler motion as represented by the image curve itself and its instantaneous properties given by the curvature and torsion functions.

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.

Kinematic Mapping Based Evaluation of Assembly Modes for Planar Four-Bar Synthesis

2005 ◽  
Author(s):  
Hans-Peter Schro¨cker ◽  
Manfred L. Husty ◽  
J. Michael McCarthy
Keyword(s):  
Image Space ◽  
Synthesis Problem ◽  
New Method ◽  
Kinematic Mapping ◽  
Position Synthesis ◽  
Four Bar Linkage

This paper presents a new method to determine if two task positions used to design a four-bar linkage lie on separate circuits of a coupler curve, known as a “branch defect.” The approach uses the image space of a kinematic mapping to provide a geometric environment for both the synthesis and analysis of four-bar linkages. In contrast to current methods of solution rectification, this approach guides the modification of the specified task positions, which means it can be used for the complete five position synthesis problem.

The Synthesis of Spherical Motion Generators in the Presence of an Incomplete Set of Attitudes

2014 ◽  
Vol 6 (3) ◽  
Author(s):  
Khalid Al-Widyan ◽  
Jorge Angeles
Keyword(s):  
Rigid Body ◽  
Engineering Design ◽  
Theoretical Framework ◽  
General Methodology ◽  
Four Bar Linkage ◽  
Robust Engineering Design ◽  
Guidance Problem ◽  
Spherical Motion ◽  
Spherical Rigid Body ◽  
Selection Of

Proposed in this paper is a general methodology applicable to the synthesis of spherical motion generators in the presence of an incomplete set of finitely separated attitudes. The spherical rigid-body guidance problem in the realm of four-bar linkage synthesis can be solved exactly for up to five prescribed attitudes of the coupler link, and hence, any number of attitudes smaller than five is considered incomplete in this paper. The attitudes completing the set are determined to produce a linkage whose performance is robust against variations in the unprescribed attitudes. Robustness is needed in this context to overcome the presence of uncertainty due to the selection of the unspecified attitudes, that many a time are specified implicitly by the designer upon choosing, for example, the location of the fixed joints of the dyads. A theoretical framework for model-based robust engineering design is thus, recalled, and a methodology for the robust synthesis of spherical four-bar linkages is laid down. An example is included here to concretize the concepts and illustrate the application of the proposed methodology.

The Instantaneous Kinematics of Line Trajectories in Terms of a Kinematic Mapping of Spatial Rigid Motion

1987 ◽  
Vol 109 (1) ◽  
pp. 95-100 ◽  
Author(s):  
J. M. McCarthy
Keyword(s):  
Rigid Motion ◽  
Distribution Parameter ◽  
Spatial Motion ◽  
Euler Parameters ◽  
Kinematic Theory ◽  
Unit Hypersphere ◽  
Kinematic Mapping ◽  
Curvature And Torsion ◽  
Curve Form

The dual Euler parameters of a rigid spatial motion are used to define a curve on a dual unit hypersphere. The dual velocity, curvature, and torsion of this curve form a set of instantaneous parameters related to the instantaneous invariants of the motion. In this paper these new parameters are used to reformulate the kinematic theory of line trajectories. The distribution parameter, Disteli formulas, and the inflection congruence are examined in detail.

scholarly journals A Unified Approach to Exact and Approximate Motion Synthesis of Spherical Four-Bar Linkages Via Kinematic Mapping

2017 ◽  
Vol 10 (1) ◽  
Author(s):  
Xiangyun Li ◽  
Ping Zhao ◽  
Anurag Purwar ◽  
Q. J. Ge
Keyword(s):  
Null Space ◽  
Solution Space ◽  
Quadratic Equation ◽  
Image Space ◽  
Geometric Constraints ◽  
Design Parameters ◽  
Space Representation ◽  
Motion Synthesis ◽  
Four Bar Linkage ◽  
Spherical Motion

This paper studies the problem of spherical four-bar motion synthesis from the viewpoint of acquiring circular geometric constraints from a set of prescribed spherical poses. The proposed approach extends our planar four-bar linkage synthesis work to spherical case. Using the image space representation of spherical poses, a quadratic equation with ten linear homogeneous coefficients, which corresponds to a constraint manifold in the image space, can be obtained to represent a spherical RR dyad. Therefore, our approach to synthesizing a spherical four-bar linkage decomposes into two steps. First, find a pencil of general manifolds that best fit the task image points in the least-squares sense, which can be solved using singular value decomposition (SVD), and the singular vectors associated with the smallest singular values are used to form the null-space solution of the pencil of general manifolds; second, additional constraint equations on the resulting solution space are imposed to identify the general manifolds that are qualified to become the constraint manifolds, which can represent the spherical circular constraints and thus their corresponding spherical dyads. After the inverse computation that converts the coefficients of the constraint manifolds to the design parameters of spherical RR dyad, spherical four-bar linkages that best navigate through the set of task poses can be constructed by the obtained dyads. The result is a fast and efficient algorithm that extracts the geometric constraints associated with a spherical motion task, and leads naturally to a unified treatment for both exact and approximate spherical motion synthesis.

Differential Kinematics of Spherical and Spatial Motions Using Kinematic Mapping

1986 ◽  
Vol 53 (1) ◽  
pp. 15-22 ◽  
Author(s):  
J. M. McCarthy ◽  
B. Ravani
Keyword(s):  
Geodesic Curvature ◽  
Intrinsic Properties ◽  
Mechanism Synthesis ◽  
Basic Framework ◽  
Kinematic Mapping ◽  
Curvature And Torsion ◽  
Spherical Mechanism ◽  
Spatial Kinematics

This paper develops the basic framework for studying differential kinematics of spherical and spatial motions using a mapping of spatial kinematics. Relationships are derived relating the intrinsic properties of the image curves corresponding to a mapping of spherical and spatial kinematics and the instantaneous invariants of the corresponding spherical and spatial motions. In addition, in the case of spherical motions, the equations for the inflection cone and cubic cone of stationary geodesic curvature, important in spherical mechanism synthesis, are derived in terms of the curvature and torsion of corresponding image curves. Similar relationships defining the polhodes of spherical motions and their curvature at the reference instant are recast as well. A simple example involving a special spherical four-bar motion is also presented.

EXPERIMENTAL DYNAMICS OF A HANGING CABLE CARRYING TWO CONCENTRATED MASSES

1995 ◽  
Vol 05 (01) ◽  
pp. 145-157 ◽  
Author(s):  
FRANCESCO BENEDETTINI ◽  
FRANCIS C. MOON
Keyword(s):  
Chaotic Behavior ◽  
Three Dimensional ◽  
Vertical Motion ◽  
Chaotic Regime ◽  
Chaotic Motions ◽  
Experimental Dynamics ◽  
Global Properties ◽  
The Fourier Transform ◽  
Four Bar Linkage ◽  
Scalar Time Series

The dynamics of a massless hanging cable with two heavy masses is examined experimentally. The experiment is also a model for a three-dimensional four-bar linkage or two coupled spherical pendulums. The system is subjected to a periodic in-phase and out-of-phase vertical motion of the hanging points. The regions of instability of the planar motion in which the system exhibits prechaotic and chaotic behavior are described by means of the Fourier transform, probability density function, and autocorrelation function. Charts of behavior in the frequency-amplitude excitation parameter plane show regions of periodic, quasiperiodic, and chaotic motions. In the chaotic regime a reconstruction from a scalar time series of the global properties of the attractor is done by means of the delay map technique. An analytical model is developed to compare analytical and numerical results. The structure of the experimental global attractor suggests a Shil’nikov model for the transition to chaotic behavior.

scholarly journals On torus knots and unknots

2016 ◽  
Vol 25 (06) ◽  
pp. 1650036 ◽  
Author(s):  
Chiara Oberti ◽  
Renzo L. Ricca
Keyword(s):  
Total Curvature ◽  
Topological Information ◽  
Torus Knots ◽  
Space Curves ◽  
Global Properties ◽  
Inflection Points ◽  
Total Squared Curvature ◽  
Curvature And Torsion ◽  
Total Torsion ◽  
Comprehensive Study

A comprehensive study of geometric and topological properties of torus knots and unknots is presented. Torus knots/unknots are particularly symmetric, closed, space curves, that wrap the surface of a mathematical torus a number of times in the longitudinal and meridian direction. By using a standard parametrization, new results on local and global properties are found. In particular, we demonstrate the existence of inflection points for a given critical aspect ratio, determine the location and prescribe the regularization condition to remove the local singularity associated with torsion. Since to first approximation total length grows linearly with the number of coils, its nondimensional counterpart is proportional to the topological crossing number of the knot type. We analyze several global geometric quantities, such as total curvature, writhing number, total torsion, and geometric ‘energies’ given by total squared curvature and torsion, in relation to knot complexity measured by the winding number. We conclude with a brief presentation of research topics, where geometric and topological information on torus knots/unknots finds useful application.

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