Synthesis of Six-link, Slider-crank and Four-link Mechanisms for Function, Path and Motion Generation Using Homotopy with m-homogenization

1994 ◽  
Vol 116 (4) ◽  
pp. 1122-1131 ◽  
Author(s):  
A. K. Dhingra ◽  
J. C. Cheng ◽  
D. Kohli
Keyword(s):  
Group Theory ◽  
Matrix Method ◽  
Path Generation ◽  
Link Function ◽  
Homotopy Methods ◽  
Motion Generation ◽  
Function Generator ◽  
Numerical Examples ◽  
The Matrix ◽  
Function Generators

This paper presents solutions to the function, motion and path generation problems of Watt’s and Stephenson six-link, slider-crank and four-link mechanisms using homotopy methods with m-homogenization. It is shown that using the matrix method for synthesis, applying m-homogeneous group theory, and by defining auxiliary equations in addition to the synthesis equations, the number of homotopy paths to be tracked is drastically reduced. To synthesize a Watt’s six-link function generator for 6 through 11 precision positions, the number of homotopy paths to be tracked to obtain all possible solutions range from 640 to 55,050,240. For Stephenson-II and -III mechanisms these numbers vary from 640 to 412,876,800. It is shown that slider-crank path generation problems with 6, 7 and 8 prescribed positions require 320, 3840 and 17,920 paths to be tracked, respectively, whereas for four-link path generators with 6 through 8 specified positions, these numbers range from 640 to 71, 680. The number of homotopy paths to be tracked to body guidance problems of slider-crank and four-link mechanisms is exactly the same as the maximum number of possible solutions given by Burmester-Ball theories. Numerical examples dealing with the synthesis of slider-crank path generators for 8 precision positions, and six-link Watt and Stephenson-III function generators for 9 prescribed positions are also presented.

Complete Solutions to Synthesis of Six-Link, Slider-Crank and Four-Link Mechanisms for Function, Path and Motion Generation Using Homotopy With M-Homogenization

1992 ◽  
Author(s):  
Anoop K. Dhingra ◽  
Jyun-Cheng Cheng ◽  
Dilip Kohli
Keyword(s):  
Group Theory ◽  
Matrix Method ◽  
Numerical Results ◽  
Path Generation ◽  
Link Function ◽  
Homotopy Methods ◽  
Motion Generation ◽  
The Matrix ◽  
Function Generators ◽  
Complete Solutions

Abstract This paper presents complete solutions to the function, motion and path generation problems of Watt’s and Stephenson six-link, slider-crank and four-link mechanisms using homotopy methods with m-homogenization. It is shown that using the matrix method for synthesis, applying m-homogeneous group theory, and by defining compatibility equations in addition to the synthesis equations, the number of homotopy paths to be tracked can be drastically reduced. For Watt’s six-link function generators with 6 thru 11 precision positions, the number of homotopy paths to be tracked in obtaining all possible solutions range from 640 to 55,050,240. For Stephenson-II and -III mechanisms these numbers vary from 640 to 412,876,800. For 6, 7 and 8 point slider-crank path generation problems, the number of paths to be tracked are 320, 3840 and 17,920, respectively, whereas for four-link path generators with 6 thru 8 positions these numbers range from 640 to 71,680. It is also shown that for body guidance problems of slider-crank and four-link mechanisms, the number of homotopy paths to be tracked is exactly same as the maximum number of possible solutions given by the Burmester-Ball theories. Numerical results of synthesis of slider-crank path generators for 8 precision positions and six-link Watt and Stephenson-III function generators for 9 prescribed positions are also presented.

Multiply Separated Synthesis of Six-Link Mechanisms Using Parallel Homotopy With M-Homogenization

1998 ◽  
Author(s):  
A. K. Dhingra ◽  
M. Zhang
Keyword(s):  
Group Theory ◽  
Matrix Method ◽  
Parallel Implementation ◽  
Homotopy Methods ◽  
Numerical Work ◽  
Function Generation ◽  
Connection Machine ◽  
The Matrix ◽  
Generation Problem ◽  
Complete Solutions

Abstract This paper presents complete solutions to the function generation problem of six-link Watt and Stephenson mechanisms, with multiply separated precision positions (PP), using homotopy methods with m-homogenization. It is seen that using the matrix method for synthesis, applying m-homogeneous group theory and by defining auxiliary equations in addition to the synthesis equations, the number of homotopy paths to be tracked in obtaining all possible solutions to the synthesis problem can be drastically reduced. Numerical work dealing with the synthesis of Watt and Stephenson mechanisms for 6 and 9 multiply separated precision points is presented. For both mechanisms, it is seen that complete solutions for 6 and 9 precision points can be obtained by tracking 640 and 286,720 paths, respectively. A parallel implementation of homotopy methods on the Connection Machine on which several thousand homotopy paths can be tracked concurrently is also discussed.

scholarly journals Numerical Solution of Duffing Equation by Using an Improved Taylor Matrix Method

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Berna Bülbül ◽  
Mehmet Sezer
Keyword(s):  
Matrix Method ◽  
Numerical Approach ◽  
Duffing Equation ◽  
Matrix Equations ◽  
Algebraic Equations ◽  
Solution Function ◽  
Numerical Examples ◽  
Collocation Points ◽  
Nonlinear Algebraic Equations ◽  
The Matrix

We have suggested a numerical approach, which is based on an improved Taylor matrix method, for solving Duffing differential equations. The method is based on the approximation by the truncated Taylor series about center zero. Duffing equation and conditions are transformed into the matrix equations, which corresponds to a system of nonlinear algebraic equations with the unknown coefficients, via collocation points. Combining these matrix equations and then solving the system yield the unknown coefficients of the solution function. Numerical examples are included to demonstrate the validity and the applicability of the technique. The results show the efficiency and the accuracy of the present work. Also, the method can be easily applied to engineering and science problems.

Extension of Freudenstein’s Equation to Geared Linkages

1971 ◽  
Vol 93 (1) ◽  
pp. 201-210 ◽  
Author(s):  
A. V. Mohan Rao ◽  
G. N. Sandor
Keyword(s):  
Matrix Theory ◽  
Generalized Equations ◽  
Algebraic Equations ◽  
Principle Of Superposition ◽  
Function Generator ◽  
Closed Form Solutions ◽  
Input And Output ◽  
The Matrix ◽  
Function Generators ◽  
Scalar Expression

Freudenstein’s equation for planar four-bar function generators correlates input and output crank positions implicitly in a scalar expression, with coefficients that are functions of link proportions. Applying this approach to planar geared function generator linkages leads to nonlinear systems of algebraic equations. By the principle of superposition taken from the matrix theory of linear systems and by Sylvester’s dyalitic elimination, closed form solutions are obtained. When the geared linkages are changed into the planar four-bar by setting certain link lengths equal to zero, the generalized equations derived here specialize to Freudenstein’s well-known equation. Results of computer programs for synthesis and analysis based on this theory are tabulated.

scholarly journals The Synthesis of Planar Four-Bar Linkage for Mixed Motion and Function Generation

Sensors ◽  
2021 ◽  
Vol 21 (10) ◽  
pp. 3504
Author(s):  
Bin Wang ◽  
Xianchen Du ◽  
Jianzhong Ding ◽  
Yang Dong ◽  
Chunjie Wang ◽  
...  
Keyword(s):  
Complex Plane ◽  
Path Generation ◽  
Motion Generation ◽  
Numerical Examples ◽  
Function Generation ◽  
Long Time ◽  
Synthesis Procedure ◽  
And Function ◽  
Four Bar Linkage ◽  
Numerical Synthesis

The synthesis of four-bar linkage has been extensively researched, but for a long time, the problem of motion generation, path generation, and function generation have been studied separately, and their integration has not drawn much attention. This paper presents a numerical synthesis procedure for four-bar linkage that combines motion generation and function generation. The procedure is divided into two categories which are named as dependent combination and independent combination. Five feasible cases for dependent combination and two feasible cases for independent combination are analyzed. For each of feasible combinations, fully constrained vector loop equations of four-bar linkage are formulated in a complex plane. We present numerical examples to illustrate the synthesis procedure and determine the defect-free four-bar linkages.

scholarly journals A New Iterative Algorithm for Solving a Class of Matrix Nearness Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-6
Author(s):  
Xuefeng Duan ◽  
Chunmei Li
Keyword(s):  
Iterative Algorithm ◽  
Projection Algorithm ◽  
Frobenius Norm ◽  
Matrix Equations ◽  
Von Neumann ◽  
Numerical Examples ◽  
Matrix Nearness Problem ◽  
Least Frobenius Norm Solution ◽  
Alternating Projection ◽  
The Matrix

Based on the alternating projection algorithm, which was proposed by Von Neumann to treat the problem of finding the projection of a given point onto the intersection of two closed subspaces, we propose a new iterative algorithm to solve the matrix nearness problem associated with the matrix equations AXB=E, CXD=F, which arises frequently in experimental design. If we choose the initial iterative matrix X0=0, the least Frobenius norm solution of these matrix equations is obtained. Numerical examples show that the new algorithm is feasible and effective.

Five Position Synthesis of a Slider-Crank Function Generator

2011 ◽  
Author(s):  
Mark M. Plecnik ◽  
J. Michael McCarthy
Keyword(s):  
Full Range ◽  
Linear Actuator ◽  
Tolerance Zone ◽  
Function Generator ◽  
Front End ◽  
Input And Output ◽  
Function Generators ◽  
Synthesis Procedure ◽  
Position Synthesis ◽  
Branching Solutions

In this paper, we present a synthesis procedure for the coupler link of a planar slider-crank linkage in order to coordinate input by a linear actuator with the rotation of an output crank. This problem can be formulated in a manner similar to the synthesis of a five position RR coupler link. It is well-known that the resulting equations can produce branching solutions that are not useful. This is addressed by introducing tolerances for the input and output values of the specified task function. The proposed synthesis procedure is then executed on two examples. In the first example, a survey of solutions for tolerance zones of increasing size is conducted. In this example we find that a tolerance zone of 5% of the desired full range results in a number of useful task functions and usable slider-crank function generators. To demonstrate the use of these results, we present an example design for the actuator of the shovel of a front-end loader.

Determination of the Force Constants of Non-linear XY2 Molecules in the Gas-Phase by the GF Matrix Method

2004 ◽  
Vol 59 (9) ◽  
pp. 621-622 ◽  
Author(s):  
Fatih Ucun ◽  
Vesile Gūçlü
Keyword(s):  
Gas Phase ◽  
Condensed Phase ◽  
Matrix Method ◽  
Solid Phase ◽  
Force Constants ◽  
Matrix Solution ◽  
The Matrix ◽  
Newton Raphson ◽  
Raphson Method

The force constants of the internal coordinates of nonlinear XY2 molecules in the gas-phase were calculated by using the GF matrix method. The matrix solution was carried out by means a computer program built relative to the Newton-Raphson method and the calculations were listed in a table. The force constants of some molecules in the liquidand solid- phase were also found and compared with these ones, and it was seen that the force constants for more condensed phase are lower as in an agreement with having its lower frequency.

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