The Displacement Analysis of the Generalized Tracta Coupling

1970 ◽  
Vol 37 (3) ◽  
pp. 713-719 ◽  
Author(s):  
D. M. Wallace ◽  
F. Freudenstein
Keyword(s):  
Constant Velocity ◽  
General Mechanism ◽  
Universal Joint ◽  
Displacement Analysis ◽  
Spatial Mechanism ◽  
Input And Output ◽  
Previous Solution ◽  
The Subject ◽  
Algebraic Solutions ◽  
Special Case

The displacement analysis of spatial linkages has been the subject of a number of recent investigations, using a variety of mathematical approaches. Algebraic solutions have been developed principally, in cases in which the number of links, n, is less than or equal to 4. When n > 4, the complexity of the displacement analysis appears to increase by one or more orders of magnitude. In this paper we describe a method, which we call the geometric-configuration method, which we have used when n > 4. The method is illustrated with respect to the algebraic displacement analysis of a five-link spatial mechanism, which includes the Tracta joint as a special case. The Tracta joint is a spatial linkage of symmetrical proportions functioning as a constant-velocity universal joint for nonparallel, intersecting shafts (Myard, 1933). It has four turning or revolute pairs (R) and one plane pair (E), which is located symmetrically with respect to the input and output shafts. The generalization of this linkage, which we call the generalized Tracta coupling, is the R-R-E-R-R spatial linkage with general proportions. The displacement analysis of the general mechanism, for which we know of no previous solution, has been derived. An analysis of the effect of tolerances in the Tracta joint has been included.

Displacement Analysis of a Special Case of the 7R, Single-Loop, Spatial Mechanism

1983 ◽  
Vol 105 (1) ◽  
pp. 78-87
Author(s):  
Hiram Albala ◽  
David Pessen
Keyword(s):  
Input Output ◽  
Fourth Degree ◽  
Displacement Analysis ◽  
Single Loop ◽  
Spatial Mechanism ◽  
Rotation Axes ◽  
Successive Rotation ◽  
Special Case ◽  
Output Equation

Based on the displacement equations for the general n-bar, single-loop spatial linkage, obtained elsewhere, the displacement analysis for a special case of the 7R spatial mechanism is carried out. In this mechanism the successive rotation axes are perpendicular to each other, the distances between axes 3-4, 4-5, 5-6, are equal and the offsets along axes 4 and 5 are zero, when input axis is labeled axis 1. In this fashion, there still remain nine free linkage parameters. Input-output equation is of the eighth-degree in the tangent of half the output angle. A particular case of this one, where all the distances between axes are equal and all the offsets along axes are zero, leads to an input-output equation of the fourth-degree in the same quantity, with a maximum of four closures. This mechanism resulted to be a double-rocker.

Kinematic and Static Analyses of Tripod Constant Velocity Joints of the Spherical End Spider Type

2005 ◽  
Vol 127 (6) ◽  
pp. 1137-1144 ◽  
Author(s):  
Katsumi Watanabe ◽  
Tsutomu Kawakatsu ◽  
Shouichi Nakao
Keyword(s):  
Constant Velocity ◽  
Closed Loop ◽  
Joint Angle ◽  
Normal Force ◽  
Thrust Force ◽  
Numerical Procedure ◽  
Spatial Mechanism ◽  
Input And Output ◽  
Motion Characteristics ◽  
Constant Velocity Joints

The closed-loop equations of three cylindrical rollers, the spider of three spherical ends, and the housing of the tripod constant velocity joints are deduced as the spatial mechanism. They are solved for prescribed positions of its input, and output shafts and relative motion characteristics of components are made clear. Moreover, a procedure is established for solving, simultaneously, the set of conditional equations with respect to forces and moments acting on three cylindrical rollers, the spider, and the housing, for any values of friction coefficients between cylindrical rollers and its grooves and spherical ends. The established numerical procedure simulates the normal force acting on the roller groove with a period of π and the housing thrust force with a period of 2π∕3 for given values of the joint angle. These results are inspected by experiments.

Displacement Analysis of the Generalized Clemens Coupling, The R-R-S-R-R Spatial Linkage

1975 ◽  
Vol 97 (2) ◽  
pp. 575-580 ◽  
Author(s):  
D. M. Wallace ◽  
F. Freudenstein
Keyword(s):  
Fourth Order ◽  
Constant Velocity ◽  
Degrees Of Freedom ◽  
Input Output ◽  
Closed Form Solutions ◽  
Input And Output ◽  
Order Polynomial ◽  
Special Case ◽  
Output Equation ◽  
Fourth Order Polynomial

The Clemens Coupling is a constant-velocity, universal-type joint for nonparallel intersecting shafts. This mechanism is a spatial linkage with five links connected by four revolute pairs, R, and one spherical pair (ball-and-socket joint), S, which is located symmetrically with respect to the input and output shafts. The Clemens Coupling is a special case of the R-R-S-R-R spatial linkage with general proportions, which will, therefore, be called the Generalized Clemens Coupling. This paper gives the algebraic derivation of the input-output equation for the general R-R-S-R-R linkage and demonstrates that it is a fourth-order polynomial in the half tangents of the crank angles. The effect of housing-error tolerances on the displacements of the Clemens Coupling has also been considered. The results demonstrate feasibility of closed-form solutions for five-link mechanisms with kinematic pairs having more than two degrees of freedom.

Displacement Analysis of the RPRCRR Six-Link Spatial Mechanism

1971 ◽  
Vol 38 (4) ◽  
pp. 1029-1035 ◽  
Author(s):  
M. S. C. Yuan
Keyword(s):  
Algebraic Equation ◽  
Input Output ◽  
Variable Parameters ◽  
Displacement Analysis ◽  
Output Displacement ◽  
Spatial Mechanism ◽  
Numerical Example ◽  
Input And Output ◽  
Displacement Equation

Using the method of line coordinates, the input-output displacement equation of the RPRCRR six-link spatial mechanism is obtained as an algebraic equation of 16th order. For each set of the input and output angles obtained from the equation, all other variable parameters of the mechanism are also determined. A numerical example is presented.

Displacement Analysis of Spatial 7R Mechanisms Suitable for Constant-Velocity Transmission Between Parallel Shafts

1979 ◽  
Vol 101 (4) ◽  
pp. 604-613 ◽  
Author(s):  
M. J. Gilmartin ◽  
J. Duffy
Keyword(s):  
Constant Velocity ◽  
Velocity Ratio ◽  
Numerical Results ◽  
Theory Method ◽  
Unified Theory ◽  
Displacement Analysis ◽  
Loop Method ◽  
Joint Coupling ◽  
Velocity Transmission ◽  
Special Case

Three types of spatial 7R mechanisms are identified as being suitable for transmitting motion with a constant velocity ratio between two parallel shafts. A displacement analysis of each type is made using a vector loop method in conjunction with the Unified Theory method. Numerical results are presented for an example of each type. It is also shown how the double Hooke joint coupling for parallel shafts is a special case of one of these three types.

Closed-Form Displacement Analysis of Five-Link R-C-R-C-P Spatial Mechanism

1973 ◽  
Vol 2 (4) ◽  
pp. 238-240
Author(s):  
R. V. Dukkipati
Keyword(s):  
Closed Form ◽  
Second Order ◽  
Input Output ◽  
Variable Parameters ◽  
Displacement Analysis ◽  
Output Displacement ◽  
Spatial Mechanism ◽  
Dual Number ◽  
Input And Output ◽  
Order Polynomial

Using (3 x 3) matrices with dual-number elements, closed-form displacement relationships are derived for a spatial five-link R-C-R-C-P mechanism. The input-output closed form displacement relationship is obtained as a second order polynomial in the output displacement. For each set of the input and output displacements obtained from the equation, all other variable parameters of the mechanism are uniquely determined. A numerical illustrative example is presented. The derived input-output relationship can be used to synthesize an R-C-R-C-P function generating mechanism for a maximum of 15 precision conditions.

Displacement Analysis of the RRCCR Five-Link Spatial Mechanism

1970 ◽  
Vol 37 (3) ◽  
pp. 689-696 ◽  
Author(s):  
M. S. C. Yuan
Keyword(s):  
Input Output ◽  
Eighth Order ◽  
Variable Parameters ◽  
Displacement Analysis ◽  
Output Displacement ◽  
Spatial Mechanism ◽  
Input And Output ◽  
Displacement Equation ◽  
Order Polynomial

By the method of line coordinates, the input-output displacement equation of the RRCCR five-link spatial mechanism is obtained as an eighth-order polynomial in the half tangent of the output angle. For each set of the input and output angles obtained from the polynomial, all other variable parameters of the mechanism are uniquely determined, and the accuracy of the numerical values of each set of solutions is verified.

Kinematic Investigation of the Deployable Bennett Loop

2006 ◽  
Vol 129 (6) ◽  
pp. 602-610 ◽  
Author(s):  
J. Eddie Baker
Keyword(s):  
Full Advantage ◽  
Spatial Displacement ◽  
Learning Tool ◽  
Working Mechanism ◽  
Efficient Treatment ◽  
Kinematic Characteristics ◽  
The Subject ◽  
The Many ◽  
Special Case

Despite the many studies devoted to it and its value as a learning tool, the Bennett linkage has never been employed as a working mechanism. It has recently found favor, however, among structural analysts as a possible unit in deployable networks owing to the potential for true spatial displacement without flexure. Although the loop can be analyzed in this application by means of purely geometrical methods, a wealth of kinematic examinations is available for more efficient treatment. The particular form that the chain must adopt as a deployable object and the special case of the linkage demanded by the purpose constitute the subject of the present exposition, which takes full advantage of prior analyses of the chain’s kinematic characteristics.

1. On Cases of Instability in Open Structures

1888 ◽  
Vol 14 ◽  
pp. 106-106
Author(s):  
E. Sang
Keyword(s):  
Linear Structures ◽  
Symmetric Structure ◽  
Linear Resistance ◽  
Open Structures ◽  
Extensive Class ◽  
The Subject ◽  
The Stability ◽  
Special Case

AbstractIn the course of some remarks on the design proposed for the Forth Bridge, the author of this paper had enunciated the remarkable theorem, that any symmetric structure built on a rectangular base, and depending on linear resistance alone, is necessarily unstable. The proof of it, given in the eleventh volume of the Transactions of the Royal Scottish Society of Arts, is derived from considerations affecting the special case; but this theorem is only one of an extensive class, and therefore the subject of instability among linear structures in general is here taken up.In the case of regular or semi-regular arrangements, having the corners of an upper supported from the corners of an under polygon, it is shown that when the figures are of odd numbers the structures are stable, while those with even numbers are unstable ; unless indeed the polygons be placed conformably, in which case the stability extends to both classes.

General Relativity and Cosmology

2020 ◽  
pp. 227-360
Author(s):  
P. J. E. Peebles
Keyword(s):  
Constant Velocity ◽  
Large Scale ◽  
Reference Frames ◽  
Physical Sciences ◽  
Inertial Frames ◽  
Moving Body ◽  
History Of ◽  
The Subject ◽  
Freely Moving ◽  
The Universe

This chapter discusses the development of physical sciences in seemingly chaotic ways, by paths that are at best dimly seen at the time. It refers to the history of ideas as an important part of any science, and particularly worth examining in cosmology, where the subject has evolved over several generations. It also examines the puzzle of inertia, which traces the connection to Albert Einstein's bold idea that the universe is homogeneous in the large-scale average called “cosmological principle.” The chapter cites Newtonian mechanics that defines a set of preferred motions in space, the inertial reference frames, by the condition that a freely moving body has a constant velocity. It talks about Ernst Mach, who argued that inertial frames are determined relative to the motion of the rest of the matter in the universe.

Sign in / Sign up

Export Citation Format

Share Document