Generalized Cardan Motion

1969 ◽  
Vol 91 (1) ◽  
pp. 135-141 ◽  
Author(s):  
D. Tesar ◽  
H. Anderson
Keyword(s):  
Design Procedures ◽  
Center Point ◽  
Point Curve ◽  
Moving Plane ◽  
Position Problem ◽  
Graphical Design

Cardan motion has been treated in the literature principally in terms of infinitesimally separated positions. Here Cardan motion is synthesized for all combinations of up to five infinitesimally and finitely separated positions of the moving plane. For each of the five distinct combinations which exist for four positions, the circle point curve is shown to degenerate into a circle and a line and the center point curve into a line at infinity and a hyperbola. Elementary analytical and graphical design procedures are provided and supported by examples. The five position problem is shown to be constrained only by the well-known Scott-Russell mechanism or a double-slider mechanism.

The Opposite Pole Quadrilateral as a Compatibility Linkage for Parameterizing the Center Point Curve

1993 ◽  
Vol 115 (2) ◽  
pp. 332-336 ◽  
Author(s):  
J. M. McCarthy
Keyword(s):  
Complex Number ◽  
Kinematic Analysis ◽  
Special Form ◽  
Body Form ◽  
Kinematic Theory ◽  
Opposite Pole ◽  
Center Point ◽  
Point Curve ◽  
Position Problem ◽  
Four Bar Linkage

In planar four-position kinematics, the centers of circles containing four positions of a point in a moving rigid body form the center point curve. This curve can be parameterized by analyzing a “compatibility linkage” obtained from a complex number formulation of the four-position problem. In this paper, we present another derivation of the center point curve using a special form of dual quaternions and the fact that it is identical to the pole curve. The defining properties of the pole curve lead to a parameterization by kinematic analysis of the opposite pole quadrilateral as a four-bar linkage. Thus the opposite pole quadrilateral becomes the compatibility linkage. This derivation generalizes to provide parameterizations for the center point cone of spherical kinematics and the central axis congruence of spatial kinematic theory.

Geometric Characteristics of the Center-Point Curve Based on the Kinematics of the Compatibility Linkage

1992 ◽  
Author(s):  
J. A. Schaaf ◽  
J. A. Lammers
Keyword(s):  
Complex Number ◽  
Change Point ◽  
Double Point ◽  
Classification Scheme ◽  
Center Point ◽  
Degenerate Form ◽  
Point Curve ◽  
Position Synthesis ◽  
Point Form

Abstract In this paper we develop a method of characterizing the center-point curves for planar four-position synthesis. We predict the five characteristic shapes of the center-point curve using the kinematic classification of the compatibility linkage obtained from a complex number formulation for planar four-position synthesis. This classification scheme is more extensive than the conventional Grashof and non-Grashof classifications in that the separate classes of change point compatibility linkages are also included. A non-Grashof compatibility linkage generates a unicursal form of the center-point curve; a Grashof compatibility linkage generates a bicursal form; a single change point compatibility linkage generates a double point form; and a double or triple change point compatibility linkage generates a circular-degenerate or a hyperbolic-degenerate form.

A Parameterization of the Central Axis Congruence Associated with Four Positions of a Rigid Body in Space

1993 ◽  
Vol 115 (3) ◽  
pp. 547-551 ◽  
Author(s):  
J. M. McCarthy
Keyword(s):  
Rigid Body ◽  
The Body ◽  
Central Axis ◽  
Spatial Mechanisms ◽  
Center Point ◽  
Point Curve ◽  
Screw Surface ◽  
Computer Based

Given four positions of a rigid body in space, there is a congruence of lines that can be used as the central axes of cylindric cranks to guide the body through the four positions. This “central axis congruence” is a generalization of the center point curve of planar kinematics. It is known that this congruence is identical to the screw congruence which arises in the study of complementary screw quadrilateral. It is less well-known that the screw congruence is the “screw surface” of the 4C linkage formed by the complementary screw quadrilateral, and it is this relationship that we use to obtain a parameterization for the screw congruence and in turn, the central axis congruence. This parameterization should facilitate the use of this congruence in computer based design of spatial mechanisms.

Identifying Sets of Four and Five Positions That Generate Distinctive Center-Point Curves

2009 ◽  
Author(s):  
David H. Myszka ◽  
Andrew P. Murray
Keyword(s):  
Closed Form ◽  
The Other ◽  
Specific Design ◽  
Opposite Pole ◽  
Center Point ◽  
Point Curve ◽  
Position Synthesis ◽  
Two Sides ◽  
Cubic Function

The fixed pivots of a planar 4R linkage that can achieve four design positions are constrained to a center-point curve. The curve is a circular cubic function and plots can take one of five different forms. The center-point curve can be generated with a compatibility linkage obtained from an opposite pole quadrilateral of the four design positions. This paper presents a method to identify design positions that generate distinctive shapes of the center-point curves. The form of the center-point curve is dependent on whether the shape of the opposite pole quadrilateral is an open or closed form of a rhombus, kite, parallelogram, or when the sum of two sides equals the other two. Interesting cases of three and five position synthesis are also explored. Four and five position cases are generated that have center points at infinity allowing a PR dyad with line of slide in any direction to achieve the design positions. Further, a center-point curve for five specific design positions is revealed.

Slider-Cranks as Compatibility Linkages for Parametrizing Center-Point Curves

2010 ◽  
Vol 2 (2) ◽  
Author(s):  
David H. Myszka ◽  
Andrew P. Murray
Keyword(s):  
Direct Method ◽  
Center Point ◽  
Crank Angle ◽  
Point Curve ◽  
A Direct Method

In synthesizing a planar 4R linkage that can achieve four positions, the fixed pivots are constrained to lie on a center-point curve. It is widely known that the curve can be parametrized by a 4R compatibility linkage. In this paper, a slider-crank is presented as a suitable compatibility linkage to generate the center-point curve. Furthermore, the center-point curve can be parametrized by the crank angle of a slider-crank linkage. It is observed that the center-point curve is dependent on the classification of the slider-crank. Lastly, a direct method to calculate the focus of the center-point curve is revealed.

Slider Cranks as Compatibility Linkages for Parameterizing Center Point Curves

2009 ◽  
Author(s):  
David H. Myszka ◽  
Andrew P. Murray
Keyword(s):  
Direct Method ◽  
Center Point ◽  
Crank Angle ◽  
Point Curve ◽  
A Direct Method

In synthesizing a planar 4R linkage that can achieve four positions, the fixed pivots are constrained to lie on a center-point curve. It is widely known that the curve can be parameterized by a 4R compatibility linkage. In this paper, a slider crank is presented as a suitable compatibility linkage to generate the centerpoint curve. Further, the center-point curve can be parametrized by the crank angle of a slider crank linkage. It is observed that the center-point curve is dependent on the classification of the slider crank. Lastly, a direct method to calculate the focus of the center-point curve is revealed.

Planar Circle-Point Equations for Finitely Separated and Double-Point Positions

1998 ◽  
Vol 123 (1) ◽  
pp. 157-160
Author(s):  
Hyoung Jun Kim ◽  
Raj S. Sodhi
Keyword(s):  
Rigid Body ◽  
Double Point ◽  
Rigid Body Motion ◽  
Body Motion ◽  
New Method ◽  
Planar Mechanisms ◽  
Point Position ◽  
Point Curve ◽  
New Form ◽  
Position Problem

The rigid body motion is studied for a combination of finitely and infinitesimally separated positions in planar kinematics. A general new method is developed for determining the locations of points in a rigid body moving through finitely and infinitesimally separated positions. These points would satisfy the constraints of the crank links for planar mechanisms. A new form of the circle-point curve equations is derived for the double-point position problem and also for the finitely separated position problem in planar kinematics.

Graphical Synthesis of Fourbar Mechanisms by Three-Position, Instant-Center Specification

2012 ◽  
Author(s):  
Thomas J. Thompson
Keyword(s):  
Computational Method ◽  
Front Suspension ◽  
Point Pair ◽  
Prosthetic Knee ◽  
Center Point ◽  
Practical Synthesis ◽  
Instantaneous Center ◽  
Triangle Theory ◽  
Position Problem ◽  
Position Synthesis

In four-bar mechanism synthesis, solutions to both the three-position and four-position synthesis problems are well-known. However, certain practical synthesis problems also require consideration of the instantaneous center of velocity for one of the precision positions. Examples are the double-wishbone front suspension of an automobile (camber in jounce and rebound, along with roll center), and four-bar prosthetic knee (standing stability, flexion length, and sitting cosmetic advantage). Because specifying the location of the instant center constrains the solution by one free choice per dyad, it reduces the number of free choices available in a three-position problem from two to one. Thus, center point and circle point solutions to the three-position, instant center specified synthesis (TPICS) problem are located along point-pair solution curves similar to the Burmester curves in four-position synthesis. The purpose of this paper is to present a direct, graphical method for finding pivot locations in three-position, instant-center synthesis of four-bar mechanisms. The method uses pole triangle theory to determine pivot locations along center point and circle point curves. A summary of a previously-presented computational method is included. As an example, both the graphical and the computational method are used to generate TPICS center-point curves for an automotive front suspension.

A Unified Analysis and Implementation of Four Multiply Separated Position Synthesis of Four-Bar Linkages

1992 ◽  
Author(s):  
P. Srikrishna ◽  
Kenneth J. Waldron
Keyword(s):  
Rigid Body ◽  
Design Procedure ◽  
Unified Approach ◽  
Center Point ◽  
Point Curve ◽  
Position Synthesis

Abstract The objective of this paper is to derive analytically the circle-point and center-point curve equations for the synthesis of four-bar linkages for rigid body guidance through four multiply separated design positions. A unified approach is evolved to deal with the different combinations of four finitely and infinitesimally separated design position, namely the PP-P-P, PP-PP and PPP-P cases. The design procedure incorporates the rectification procedures developed by Waldron (1977) to eliminate the branch and order problems and is implemented in the interactive synthesis package RECSYN.

Assessing Position Order in Rigid Body Guidance: An Intuitive Approach to Fixed Pivot Selection

2008 ◽  
Vol 131 (1) ◽  
Author(s):  
David H. Myszka ◽  
Andrew P. Murray ◽  
James P. Schmiedeler
Keyword(s):  
Rigid Body ◽  
Single Point ◽  
New Method ◽  
Entire Plane ◽  
Fixed Frame ◽  
Center Point ◽  
Point Curve ◽  
Pivot Selection ◽  
Line Passing ◽  
Pass Through

Several established methods determine if an RR dyad will pass through a set of finitely separated positions in order. The new method presented herein utilizes only the displacement poles in the fixed frame to assess whether a selected fixed pivot location will yield an ordered dyad solution. A line passing through the selected fixed pivot is rotated one-half revolution about the fixed pivot, in a manner similar to a propeller with infinitely long blades, to sweep the entire plane. Order is established by tracking the sequence of displacement poles intersected. With four or five positions, fixed pivot locations corresponding to dyads having any specified order are readily found. Five-position problems can be directly evaluated to determine if any ordered solutions exist. Additionally, degenerate four-position cases for which the set of fixed pivots corresponding to ordered dyads that collapse to a single point on the center point curve can be identified.

Sign in / Sign up

Export Citation Format

Share Document