Two Precision Position Synthesis of Planar Linkages With Positional Rectification

1996 ◽  
Author(s):  
John A. Mirth
Keyword(s):  
Numerical Algorithms ◽  
All Solutions ◽  
Pass Through ◽  
Planar Linkages ◽  
Position Synthesis ◽  
Exact Positions

Abstract Precision position synthesis is used to generate planar linkages that pass through two exact positions and an additional number of approximate positions. The approximate positions provide a means of rectifying the solution linkages such that all solutions presented are more likely to describe a motion that remains within acceptable positional bounds. The rectification method involves the development of three different numerical algorithms that may be applied to a particular step in the dyad/triad method of precision position synthesis. The three algorithms presented can be applied in a variety of combinations to allow for the synthesis of both simple (four-bar) and complex (multiloop) planar linkages.

Four-Bar Linkage Synthesis Methods for Two Precision Positions Combined With N Quasi-Positions

1995 ◽  
Author(s):  
John A. Mirth
Keyword(s):  
Synthesis Method ◽  
Synthesis Methods ◽  
Bounded Motion ◽  
Transmission Angle ◽  
Pass Through ◽  
Position Synthesis ◽  
Four Bar Linkage ◽  
Exact Positions ◽  
Linkage Synthesis

Abstract Mechanisms seldom need to pass through more than one or two exact positions. The method of quasi-position synthesis combines a number of approximate or “quasi” positions with two exact positions to design four-bar linkages that will produce a specified, bounded motion. Quasi-position synthesis allows for the optimization of some linkage characteristic (such as link lengths or transmission angles) by using the three variables that describe a single quasi-position. Procedures for circuit and transmission angle rectification are also easily incorporated into the quasi-position synthesis method.

The Synthesis of Planar Linkages to Satisfy an Approximate Motion Specification

1993 ◽  
Author(s):  
John A. Mirth
Keyword(s):  
Design Space ◽  
Accurate Position ◽  
Solution Region ◽  
Pass Through ◽  
Position Problem ◽  
The Individual ◽  
Planar Linkages ◽  
Position Synthesis ◽  
Individual Design ◽  
Complete Process

Abstract This paper introduces the method of quasi-precision position synthesis for planar linkages. A quasi-precision position is defined by an approximate region that the mechanism must pass through. The quasi-precision position problem specification generates an increased design region by not requiring the accurate position specification that is characteristic of optimization and precision position methods. The design space is generated by intersecting the individual design regions of different four position sets. Each four position set consists of three exact positions plus one quasi-precision position. The method allows for the use of any number of quasi-precision positions. The result of using quasi-precision positions is an increase in the available design space without violating the basic problem constraints. The increased solution region gives the designer a greater variety of choices while reducing the number of required design iterations. The complete process of quasi-precision position synthesis is presented through the use of an example.

Quasi-Precision Position Synthesis of Four-Bar Linkages

1994 ◽  
Author(s):  
John A. Mirth
Keyword(s):  
Two Dimensional ◽  
Exact Position ◽  
Planar Linkages ◽  
Position Synthesis ◽  
Exact Positions

Abstract Quasi positions describe portions of a mechanism’s motion that do not require an exact position. Quasi positions are combined with three exact positions to define a two-dimensional design plane for the synthesis of planar linkages. Quasi-precision position synthesis is based on precision position techniques. The method preserves the computational advantages of precision position synthesis with the added advantage of being able to specify a much greater number of design positions.

Finding All Solutions to Unconstrained Nonlinear Optimization for Approximate Synthesis of Planar Linkages Using Continuation Method

1999 ◽  
Vol 121 (3) ◽  
pp. 368-374 ◽  
Author(s):  
A.-X. Liu ◽  
T.-L. Yang
Keyword(s):  
Continuation Method ◽  
Optimization Method ◽  
Global Optimum ◽  
Kinematic Synthesis ◽  
All Solutions ◽  
Approximate Synthesis ◽  
Unconstrained Nonlinear Optimization ◽  
Optimum Solution ◽  
Planar Linkages ◽  
Four Bar Linkage

Generally, approximate kinematic synthesis of planar linkage is studied using optimization method. But this method has two defects: i) the suitable initial guesses are hard to determine and ii) the global optimum solution is difficult to find. In this paper, a new method which can find all solutions to approximate kinematic synthesis of planar linkage is proposed. Firstly, we reduce the approximate synthesis problem to finding all solutions to polynomial equations. Polynomial continuation method is then used to find all solutions. Finally, all possible linkages can be obtained. Approximate syntheses of planar four-bar linkage for function generation, rigid-body guidance and path generation are studied in detail and three examples are given to illustrate the advantages of the proposed method.

Approximate Velocities in Mixed Exact-Approximate Position Synthesis of Planar Mechanisms

1999 ◽  
Vol 123 (3) ◽  
pp. 388-394 ◽  
Author(s):  
Jennifer E. Holte ◽  
Thomas R. Chase ◽  
Arthur G. Erdman
Keyword(s):  
Two Dimensional ◽  
Computer Implementation ◽  
Motion Generation ◽  
Planar Mechanisms ◽  
Form Complex ◽  
New Approach ◽  
Realistic Representation ◽  
Velocity Constraints ◽  
Position Synthesis ◽  
Exact Positions

A new approach to the synthesis of planar linkage mechanisms with approximate velocity constraints is proposed. The paper presents the first closed-form complex-number dyad solution to the ground pivot specification problem for two precision positions with velocity specified at one of the positions. The solution is then manipulated in order to add approximate velocity constraints to design methods for two exact positions and an unlimited number of approximate positions. The approximate position and velocity constraints facilitate more realistic representation of design objectives. Solution spaces are presented using two-dimensional ground-pivot maps. Computer implementation of the proposed methodologies would allow designers with little or no knowledge of the synthesis techniques to interactively explore maps of solutions for four-bar motion generation.

The Duality Between Planar Kinematics and Statics

2005 ◽  
Author(s):  
Offer Shai ◽  
Gordon R. Pennock
Keyword(s):  
Mechanical Systems ◽  
Unique Point ◽  
Singular Configurations ◽  
New Concepts ◽  
Basic Concepts ◽  
Instantaneous Center ◽  
Pass Through ◽  
Planar Linkages ◽  
Insight Into ◽  
General Perspective

This paper shows that there is a correlation between basic concepts underlying the kinematics of mechanisms and the statics of trusses. The implication of this correlation, referred to here as duality, is that the science of kinematics can be utilized in a systematic manner to yield insight into the statics of mechanical systems. The paper begins by proving the existence of a unique line (referred to as the equimomental line) where the moments, at each point on this line, caused by two arbitrary co-planar forces are equal. The dual concept in kinematics is the instantaneous center of zero velocity and two theorems are presented based on the duality between equimomental lines and instantaneous centers. The first theorem states that the three equimomental lines defined by three co-planar forces must intersect at a unique point. The second theorem states that the equimomental line for two co-planar forces acting in a trusss with two degrees of indeterminacy must pass through a unique point. The paper presents several practical examples to demonstrate how the duality between kinematics and statics provides a better understanding of planar linkages and trusses. The new concepts are used to identify the singular configurations of linkages and the configurations of determinate trusses where they are not rigid. Finally, the paper takes advantage of some important relationships between linkages and trusses to provide a general perspective of the duality between the kinematics of mechanisms and the statics of trusses.

Geometric Conditions for the Satisfaction of Order in Four Position Synthesis of Four-Bar Linkages

1992 ◽  
Author(s):  
John A. Mirth
Keyword(s):  
Type A ◽  
Computer Programs ◽  
Order Condition ◽  
Order Problem ◽  
Geometric Characteristics ◽  
Geometric Conditions ◽  
Linkage Design ◽  
Correct Sequence ◽  
Pass Through ◽  
Position Synthesis

Abstract A mechanism that cannot pass through all precision positions in the correct sequence is said to violate the order condition. This paper establishes the order requirements for the precision position synthesis of all Grashof and non-Grashof four-bar linkages. These order requirements are based on three geometric characteristics of the four-bar mechanism: the angle of one of the rotating links that is attached to the ground link; the limits that the entire mechanism imposes on this rotating link; and the dyad configurations for the dyad that is opposite the ground pivot of this link. The results improve upon traditional discussions of the order problem by including all four-bar linkages, rather than just those of a specified Grashof type. A table summarizes the results in a form that allows for their implementation into computer programs for linkage design and analysis.

Mixed Exact-Approximate Position Synthesis of Planar Mechanisms

1998 ◽  
Vol 122 (3) ◽  
pp. 278-286 ◽  
Author(s):  
Jennifer E. Holte ◽  
Thomas R. Chase ◽  
Arthur G. Erdman
Keyword(s):  
Three Dimensional ◽  
Solution Space ◽  
Motion Generation ◽  
Planar Mechanisms ◽  
Dimensional Solution ◽  
New Approach ◽  
Fuzzy Constraints ◽  
Position Synthesis ◽  
Synthesis Approach ◽  
Exact Positions

A new approach to the synthesis of planar linkage mechanisms with fuzzy constraints is proposed. Design methods for two exact positions and an unlimited number of approximate positions are presented. The use of approximate specifications allows the designer to represent design objectives more realistically. A precision position synthesis approach is used to generate a three-dimensional solution space of dyads satisfying all exact and approximate constraints. The three-dimensional solution space is reduced to a two-dimensional ground-pivot map. Computer implementation of the proposed methodologies would allow designers with little or no knowledge of the synthesis techniques to interactively explore maps of solutions for four-bar motion generation. [S1050-0472(00)00803-5]

Patterned Bootstrap: A New Method Which Gives Efficiency for Precision Position Synthesis of Planar Linkages

2005 ◽  
Author(s):  
Zhenjun Luo ◽  
Jian S. Dai
Keyword(s):  
New Method ◽  
Continuation Methods ◽  
Interval Methods ◽  
Homotopy Continuation ◽  
Root Finding ◽  
Starting Point ◽  
Homotopy Continuation Methods ◽  
Upper Level ◽  
Planar Linkages ◽  
Position Synthesis

This paper presents a new method, termed as patterned bootstrap (PB), which is suitable for precision position synthesis of planar linkages. The method solves a determined system of equations using a new bootstrapping strategy. In principle, a randomly generated starting point is advanced to a final solution through solving a number of intermediate systems. The structure and the associated parameters of each intermediate system is defined as a pattern. In practice, a PB procedure generally consists of two levels: an upper level which controls the transition of patterns, and a lower level which solves intermediate systems using globally convergent root-finding algorithms. Besides introducing the new method, tunnelling functions have been added to several systems of polynomials derived by formal researchers in order to exclude degenerated solutions. Our numerical experiments demonstrate that many precision position synthesis problems can be solved efficiently without resorting to time-consuming polynomial homotopy continuation methods or interval methods. Finding over 95 percentages of the complete solutions of the 11 precision position function generation problem of a Stephenson-III linkage has been achieved for the first time.

Mixed Exact-Approximate Planar Mechanism Position Synthesis

1977 ◽  
Vol 99 (2) ◽  
pp. 434-439 ◽  
Author(s):  
G. H. Sutherland
Keyword(s):  
Rigid Body ◽  
Digital Computer ◽  
Algebraic Solution ◽  
Numerical Examples ◽  
Planar Mechanism ◽  
Straight Line ◽  
Position Synthesis ◽  
Scientific Calculator ◽  
Exact Positions ◽  
Solution Techniques

Design equations are developed for rigid body guidance through K equals one, two, three, and four exact positions and N – K approximate positions. Both circle and straight-line points are considered. Algebraic solution techniques for use with a pocket scientific calculator and a digital computer are illustrated using numerical examples.

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