On Extreme Velocity Ratios in Spherical Four-Link Mechanisms

1976 ◽  
Vol 98 (4) ◽  
pp. 1260-1265 ◽  
Author(s):  
P. R. Pamidi
Keyword(s):  
Algebraic Polynomial ◽  
Numerical Examples ◽  
The Real ◽  
Real Roots ◽  
Spherical Mechanisms ◽  
Link Mechanism

The conditions for the existence of extreme velocity ratios in spherical four-link mechanisms have been investigated by algebraic means. It is shown that, in the general case, all such extreme velocity ratios must be among the real roots of a 10th-degree algebraic polynomial. The results make it possible to obtain the extreme velocity ratios of any given spherical four-link mechanism. The method of the paper has been illustrated by several numerical examples. Special formulations of the general case have also been considered and applications to some practical spherical mechanisms have been discussed. The results can also be used for synthesizing spherical four-link mechanisms with prescribed extreme velocity ratios.

Velocity Fluctuation in Spatial Four-Link Mechanisms

1981 ◽  
Vol 23 (3) ◽  
pp. 143-148
Author(s):  
M. O. M. Osman ◽  
R. V. Dukkipati
Keyword(s):  
Velocity Fluctuation ◽  
Algebraic Solution ◽  
Kinematic Parameters ◽  
Degree Polynomial ◽  
Numerical Examples ◽  
The Real ◽  
Real Roots ◽  
Geometric Conditions ◽  
Link Mechanism

This paper presents an algebraic solution of the velocity fluctuation in a spatial four-link mechanism. The geometric conditions governing the existence of extreme velocity ratios in a spatial four-link mechanism having one revolute pair, one prismatic pair, one spherical pair, and one cylinder pair have been obtained. It is shown that, in the general case, the extreme velocity ratios must be among the real roots of a tenth degree polynomial in kinematic parameters of the mechanism. Numerical examples are presented to illustrate the described procedure.

On Graeffe's Method for Complex Roots of Algebraic Equations

1924 ◽  
Vol 22 (2) ◽  
pp. 83-87 ◽  
Author(s):  
S. Brodetsky ◽  
G. Smeal
Keyword(s):  
Nineteenth Century ◽  
Real Axis ◽  
Practical Method ◽  
Algebraic Equations ◽  
Argand Diagram ◽  
Considerable Difficulty ◽  
The Real ◽  
Real Roots ◽  
Bipolar Coordinates ◽  
Complex Roots

The only really useful practical method for solving numerical algebraic equations of higher orders, possessing complex roots, is that devised by C. H. Graeffe early in the nineteenth century. When an equation with real coefficients has only one or two pairs of complex roots, the Graeffe process leads to the evaluation of these roots without great labour. If, however, the equation has a number of pairs of complex roots there is considerable difficulty in completing the solution: the moduli of the roots are found easily, but the evaluation of the arguments often leads to long and wearisome calculations. The best method that has yet been suggested for overcoming this difficulty is that by C. Runge (Praxis der Gleichungen, Sammlung Schubert). It consists in making a change in the origin of the Argand diagram by shifting it to some other point on the real axis of the original Argand plane. The new moduli and the old moduli of the complex roots can then be used as bipolar coordinates for deducing the complex roots completely: this also checks the real roots.

3041. On the Real Roots of an Equation

1962 ◽  
Vol 46 (358) ◽  
pp. 317
Author(s):  
Roger North
Keyword(s):  
The Real ◽  
Real Roots

scholarly journals A new method of solving numerical equations

1843 ◽  
Vol 4 ◽  
pp. 300-301
Keyword(s):  
New Method ◽  
Original Equation ◽  
Simple Method ◽  
The Real ◽  
Real Roots ◽  
Significant Figure ◽  
Difficult Question

The object of this paper is to develope a new and remarkably simple method of approximating to the real roots of numerical equations, which possesses several important advantages. After describing the nature of the transformations which are subsequently employed, the author proceeds to develope the process he uses for obtaining one of the roots of a numerical equation. Passing over the difficult question of determining the limits of the roots, he supposes the first significant figure (R) of a root to have been ascertained, and transforms the proposed equation into one whose roots are the roots of the original, divided by this figure (or x /R): one root of this equation lying between 1 and 2, the first significant figure ( r ) of the decimal part is obtained, and the equation transformed into another whose roots are those of the former, divided by 1+ this decimal (or 1 + r ). This last equation is again similarly transformed; these transformations being readily effected by the methods first given. Proceeding thus, the root of the original equation is obtained in the form of a continued product. After applying this method to finding a root of an equation of the 4th, and likewise one of the 5th degree, the author applies it to a class of equations to which he considers it peculiarly adapted, namely, those in which several terms are wanting. One of these is of the 16th degree, having only six terms; and another is of the 622nd degree, having only four terms.

scholarly journals Exact solutions of sixth and fifth degree equations

2021 ◽  
Author(s):  
Mohammed El Amine MONIR
Keyword(s):  
Exact Solutions ◽  
Algebraic Polynomial ◽  
Polynomial Equation ◽  
Polynomial Equations ◽  
Degree Polynomial ◽  
The Real ◽  
Absolute Method ◽  
Quintic Equation ◽  
Degree Equation

Abstract The real problematic with algebraic polynomial equations is how to exactly solve any sixth and fifth degree polynomial equations. In this study, we give a new absolute method that presents a new decomposition to exactly solve a sixth degree polynomial equation, while the corresponding fifth degree equation can be easily transformed into a sixth degree equation of this kind (sixth degree equation solvable by this method), then the sextic equation (sixth degree equation) obtained will be solved by applying the principles of this method; moreover, the solutions of the quintic equation (fifth degree equation) will be easily deduced.

scholarly journals Increasing the interval of convergence for a generalized Newton's method of solving nonlinear equations

2016 ◽  
pp. 7-12
Author(s):  
А.Н. Громов
Keyword(s):  
Continuous Function ◽  
Nonlinear Equations ◽  
Newton’S Method ◽  
Newton's Method ◽  
The Real ◽  
Real Roots ◽  
Complex Roots ◽  
Transcendental Equations ◽  
Generalized Newton’S Method

Рассмотрен подход к построению расширения промежутка сходимости ранее предложенного обобщения метода Ньютона для решения нелинейных уравнений одного переменного. Подход основан на использовании свойства ограниченности непрерывной функции, определенной на отрезке. Доказано, что для поиска действительных корней вещественнозначного многочлена с комплексными корнями предложенный подход дает итерации с нелокальной сходимостью. Результат обобщен на случай трансцендентных уравнений. An approach to the construction of an extended interval of convergence for a previously proposed generalization of Newton's method to solve nonlinear equations of one variable. This approach is based on the boundedness of a continuous function defined on a segment. It is proved that, for the search for the real roots of a real-valued polynomial with complex roots, the proposed approach provides iterations with nonlocal convergence. This result is generalized to the case transcendental equations.

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