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.

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.

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.

The Darwin procedure in optics of layered media and the matrix theory

1999 ◽  
Vol 55 (4) ◽  
pp. 613-620 ◽  
Author(s):  
P. Dub ◽  
O. Litzman
Keyword(s):  
Difference Equations ◽  
Continued Fractions ◽  
Matrix Theory ◽  
Dynamical Theory ◽  
Algebraic Equations ◽  
Tridiagonal Matrices ◽  
Linear Algebraic Equations ◽  
Close Relationship ◽  
The Matrix ◽  
Nonhomogeneous System

The Darwin dynamical theory of diffraction for two beams yields a nonhomogeneous system of linear algebraic equations with a tridiagonal matrix. It is shown that different formulae of the two-beam Darwin theory can be obtained by a uniform view of the basic properties of tridiagonal matrices, their determinants (continuants) and their close relationship to continued fractions and difference equations. Some remarks concerning the relation of the Darwin theory in the three-beam case to tridiagonal block matrices are also presented.

scholarly journals Matrix Approach To The Direct Computation Method For The Solution of Fredholm Integro-Differential Equations of The Second Kind With Degenerate Kernels

CAUCHY ◽  
2020 ◽  
Vol 6 (3) ◽  
pp. 100-108
Author(s):  
Nathaniel Mahwash Kamoh ◽  
Geoffrey Kumlengand ◽  
Joshua Sunday
Keyword(s):  
Differential Equations ◽  
Computation Method ◽  
Matrix Approach ◽  
Algebraic Equations ◽  
Direct Computation ◽  
Closed Form Solutions ◽  
Linear Algebraic Equations ◽  
The Matrix ◽  
Sum Of Products ◽  
Finite Sum

In this paper, a matrix approach to the direct computation method for solving Fredholm integro-differential equations (FIDEs) of the second kind with degenerate kernels is presented. Our approach consists of reducing the problem to a set of linear algebraic equations by approximating the kernel with a finite sum of products and determining the unknown constants by the matrix approach. The proposed method is simple, efficient and accurate; it approximates the solutions exactly with the closed form solutions. Some problems are considered using maple programme to illustrate the simplicity, efficiency and accuracy of the proposed method.

scholarly journals Partial Component Consensus of Discrete-Time Multiagent Systems

2019 ◽  
Vol 2019 ◽  
pp. 1-5
Author(s):  
Wenjun Hu ◽  
Gang Zhang ◽  
Zhongjun Ma ◽  
Binbin Wu
Keyword(s):  
Multiagent Systems ◽  
Discrete Time ◽  
Matrix Theory ◽  
Pinning Control ◽  
Directed Network ◽  
Strong Function ◽  
Partial Stability ◽  
Partial Component ◽  
The Matrix ◽  
Error System

The multiagent system has the advantages of simple structure, strong function, and cost saving, which has received wide attention from different fields. Consensus is the most basic problem in multiagent systems. In this paper, firstly, the problem of partial component consensus in the first-order linear discrete-time multiagent systems with the directed network topology is discussed. Via designing an appropriate pinning control protocol, the corresponding error system is analyzed by using the matrix theory and the partial stability theory. Secondly, a sufficient condition is given to realize partial component consensus in multiagent systems. Finally, the numerical simulations are given to illustrate the theoretical results.

scholarly journals Numerical Analysis of Viscoelastic Rotating Beam with Variable Fractional Order Model Using Shifted Bernstein–Legendre Polynomial Collocation Algorithm

2021 ◽  
Vol 5 (1) ◽  
pp. 8
Author(s):  
Cundi Han ◽  
Yiming Chen ◽  
Da-Yan Liu ◽  
Driss Boutat
Keyword(s):  
Numerical Analysis ◽  
Fractional Order ◽  
Governing Equation ◽  
Numerical Solutions ◽  
Bernstein Polynomials ◽  
Matrix Product ◽  
Algebraic Equations ◽  
Rotating Beam ◽  
The Matrix ◽  
The Time Domain

This paper applies a numerical method of polynomial function approximation to the numerical analysis of variable fractional order viscoelastic rotating beam. First, the governing equation of the viscoelastic rotating beam is established based on the variable fractional model of the viscoelastic material. Second, shifted Bernstein polynomials and Legendre polynomials are used as basis functions to approximate the governing equation and the original equation is converted to matrix product form. Based on the configuration method, the matrix equation is further transformed into algebraic equations and numerical solutions of the governing equation are obtained directly in the time domain. Finally, the efficiency of the proposed algorithm is proved by analyzing the numerical solutions of the displacement of rotating beam under different loads.

scholarly journals MATRIX THEORY, HILBERT SCHEME AND INTEGRABLE SYSTEM

1998 ◽  
Vol 13 (34) ◽  
pp. 2731-2742 ◽  
Author(s):  
YUTAKA MATSUO
Keyword(s):  
Hilbert Scheme ◽  
Matrix Theory ◽  
Fock Space ◽  
Homology Class ◽  
Orthogonal Basis ◽  
Inner Product ◽  
Virasoro Symmetry ◽  
Hilbert Scheme Of Points ◽  
The Matrix ◽  
Boson Fock Space

We give a reinterpretation of the matrix theory discussed by Moore, Nekrasov and Shatashivili (MNS) in terms of the second quantized operators which describes the homology class of the Hilbert scheme of points on surfaces. It naturally relates the contribution from each pole to the inner product of orthogonal basis of free boson Fock space. These bases can be related to the eigenfunctions of Calogero–Sutherland (CS) equation and the deformation parameter of MNS is identified with coupling of CS system. We discuss the structure of Virasoro symmetry in this model.

scholarly journals A Simple Algorithm for Finding a Non-negative Basic Solution of a System of Linear Algebraic Equations

2021 ◽  
Vol 28 (3) ◽  
pp. 234-237
Author(s):  
Gleb D. Stepanov
Keyword(s):  
Simplex Method ◽  
Gaussian Elimination ◽  
Simple Algorithm ◽  
Basic Solution ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
The Matrix ◽  
Linear Programming Problems ◽  
Two Stages ◽  
Computational Resources

This article describes an algorithm for obtaining a non-negative basic solution of a system of linear algebraic equations. This problem, which undoubtedly has an independent interest, in particular, is the most time-consuming part of the famous simplex method for solving linear programming problems.Unlike the artificial basis Orden’s method used in the classical simplex method, the proposed algorithm does not attract artificial variables and economically consumes computational resources.The algorithm consists of two stages, each of which is based on Gaussian exceptions. The first stage coincides with the main part of the Gaussian complete exclusion method, in which the matrix of the system is reduced to the form with an identity submatrix. The second stage is an iterative cycle, at each of the iterations of which, according to some rules, a resolving element is selected, and then a Gaussian elimination step is performed, preserving the matrix structure obtained at the first stage. The cycle ends either when the absence of non-negative solutions is established, or when one of them is found.Two rules for choosing a resolving element are given. The more primitive of them allows for ambiguity of choice and does not exclude looping (but in very rare cases). Use of the second rule ensures that there is no looping.

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