Complete Solution of the Nine-Point Path Synthesis Problem for Four-Bar Linkages

1992 ◽  
Vol 114 (1) ◽  
pp. 153-159 ◽  
Author(s):  
C. W. Wampler ◽  
A. P. Morgan ◽  
A. J. Sommese
Keyword(s):  
Unsolved Problem ◽  
Complete Solution ◽  
Synthesis Problem ◽  
Computer Algorithm ◽  
All Solutions ◽  
Path Synthesis

The problem of finding all four-bar linkages whose coupler curve passes through nine prescribed points has been a longstanding unsolved problem in kinematics. Using a combination of classical elimination, multihomogeneous variables, and numerical polynomial continuation, we show that there are generically 1442 nondegenerate solution along with their Roberts cognates, for a total of 4326 distinct solutions. Moreover, a computer algorithm that computes all solutions for any given nine points has been developed.

scholarly journals Comparison of some evolutionary algorithms for optimization of the path synthesis problem

2018 ◽  
Author(s):  
Jakub Krzysztof Grabski ◽  
Tomasz Walczak ◽  
Jacek Buśkiewicz ◽  
Martyna Michałowska
Keyword(s):  
Evolutionary Algorithms ◽  
Synthesis Problem ◽  
Path Synthesis

Synthesis of Defect Free Six Bar Linkages for Body Guidance Through up to Six Finitely Separated Positions

1994 ◽  
Author(s):  
Timothy Tylaska ◽  
Kazem Kazerounian
Keyword(s):  
Design Problem ◽  
Design Methodology ◽  
Complete Solution ◽  
Higher Order ◽  
Synthesis Problem ◽  
Design Constraint ◽  
Solution Sets ◽  
Constraint Equations ◽  
Design Considerations ◽  
Highly Nonlinear

Abstract In the synthesis of watt I six bar linkage, for finitely separated design positions, or in higher order design, constraint equations become highly nonlinear and transcendental. This paper presents a method to decouple the synthesis problem to the synthesis of two path generator 4-bar linkages. Based on this decoupled system, an explicit design methodology is developed, enabling a three, four, five or six body guidance position Watt I linkage to be designed while the designer has choice of some body pivots and ground pivots. Numerical procedures for higher number of positions are also discussed. The methodology allows the designer to obtain an entire set of solutions to a particular design problem. As a spin off from this work, a methodology is also presented to obtain complete solution sets of four bar path generators capable of passing through up to seven precision points, with a procedure that can be eventually extended to eight and nine path points. Design considerations such as branching and transmission angles are also considered.

scholarly journals Finding Only Finite Roots to Large Kinematic Synthesis Systems

2017 ◽  
Vol 9 (2) ◽  
Author(s):  
Mark M. Plecnik ◽  
Ronald S. Fearing
Keyword(s):  
Polynomial Systems ◽  
Synthesis Problem ◽  
New Method ◽  
Homotopy Continuation ◽  
Kinematic Synthesis ◽  
Root Finding ◽  
Large Polynomial Systems ◽  
Path Synthesis

In this work, a new method is introduced for solving large polynomial systems for the kinematic synthesis of linkages. The method is designed for solving systems with degrees beyond 100,000, which often are found to possess quantities of finite roots that are orders of magnitude smaller. Current root-finding methods for large polynomial systems discover both finite and infinite roots, although only finite roots have meaning for engineering purposes. Our method demonstrates how all infinite roots can be precluded in order to obtain substantial computational savings. Infinite roots are avoided by generating random linkage dimensions to construct startpoints and start systems for homotopy continuation paths. The method is benchmarked with a four-bar path synthesis problem.

A novel optimization-based method to find multiple solutions for path synthesis of planar four-bar and slider-crank mechanisms

2021 ◽  
pp. 095440622098336
Author(s):  
Soheil Zarkandi
Keyword(s):  
Feasible Solution ◽  
Heuristic Algorithms ◽  
Classical Method ◽  
Search Algorithm ◽  
Error Function ◽  
Continuation Method ◽  
Gravitational Search Algorithm ◽  
Planar Mechanisms ◽  
All Solutions ◽  
Path Synthesis

The classical method to find possible solutions for path synthesis problem of planar mechanisms is continuation method. However, this method has some disadvantages such as producing unwanted extraneous and degenerate solutions and also producing mechanisms having defects. Moreover, many of the solutions are cognate of each other which can be obtained geometrically. Thus, finding the most feasible solution among all solutions is a cumbersome and time-consuming task. The main purpose of this paper is to explore applicability of heuristic algorithms to find multiple cognate- and defect-free solutions for path synthesis problems of planar four-bar and slider-crank mechanisms. To this aim, a new modified error function and an optimization-based algorithm is presented. The gravitational search algorithm (GSA) is utilized to minimize the modified error function. Efficiency of the method is proved through five case studies for path synthesis of four-bar and slider-crank mechanisms with and without prescribed timing.

Two dimensional theory of stiffened plates

1950 ◽  
Vol 2 (1) ◽  
pp. 1-8 ◽  
Author(s):  
Chang o’chou
Keyword(s):  
Differential Equation ◽  
Isotropic Plate ◽  
Complete Solution ◽  
Stiffened Plates ◽  
Two Dimensional ◽  
Stiffened Plate ◽  
Dimensional Theory ◽  
All Solutions

SummaryThe two-dimensional stiffened plate problem is completely governed by the fundamental differential equation(22)where k1 and k2 are two non-dimensional structural constants depending on the relative proportions of the reinforcing members.The complete solution is(25)This solution renders all solutions for the isotropic plate available for the stiffened plate also. The choice between the several types of solution available is governed by the same considerations as for the isotropic or unstiffened plate.

Fourier series method for path generation of RCCC mechanism

2011 ◽  
Vol 226 (3) ◽  
pp. 816-827 ◽  
Author(s):  
J W Sun ◽  
D Q Mu ◽  
J K Chu
Keyword(s):  
Fourier Series ◽  
Frequency Domain ◽  
Synthesis Problem ◽  
Mathematical Representation ◽  
Path Generation ◽  
Series Method ◽  
Internal Relation ◽  
Fourier Series Method ◽  
Mathematical Equations ◽  
Path Synthesis

The mathematical representation of the coupler curves of RCCC mechanism is established. The coupler curve of RCCC mechanism is analysed using the Fourier series theory. The internal relation between the coupler curve and the dimensional type of the RCCC mechanism in the frequency domain is discovered. Subsequently, a database comprising millions of basic dimensional types, taking the crank–rocker linkages as an example, is established. Based on the information of prescribed coupler and the database, the best suitable basic dimensional type can be determined by identification. The mathematical equations for calculating actual dimensions of mechanism and positional dimensions of installation are deduced. Thus, the path synthesis problem of RCCC mechanism can be solved by the mathematic equations and the basic dimensional type. The calculation illustration is given later in this article.

A Study on Finding Finite Roots for Kinematic Synthesis

2017 ◽  
Author(s):  
Mark M. Plecnik ◽  
Ronald S. Fearing
Keyword(s):  
Random Process ◽  
Synthesis Problem ◽  
Homotopy Continuation ◽  
Synthesis System ◽  
Expected Number ◽  
Kinematic Synthesis ◽  
Scaling Down ◽  
Path Synthesis ◽  
Number Of Iterations

This study presents new results on a method to solve large kinematic synthesis systems termed Finite Root Generation. The method reduces the number of startpoints used in homotopy continuation to find all the roots of a kinematic synthesis system. For a single execution, many start systems are generated with corresponding startpoints using a random process such that start-points only track to finite roots. Current methods are burdened by computations of roots to infinity. New results include a characterization of scaling for different problem sizes, a technique for scaling down problems using cognate symmetries, and an application for the design of a spined pinch gripper mechanism. We show that the expected number of iterations to perform increases approximately linearly with the quantity of finite roots for a given synthesis problem. An implementation that effectively scales the four-bar path synthesis problem by six using its cognate structure found 100% of roots in an average of 16,546 iterations over ten executions. This marks a roughly six-fold improvement over the basic implementation of the algorithm.

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