A Precise and Efficient Method to Manipulate the Amplitude of Parabolic Function of Transmission Errors

2014 ◽  
Vol 1064 ◽  
pp. 183-190
Author(s):  
Cheng Kang Lee
Keyword(s):  
Nonlinear Equations ◽  
Design Variable ◽  
Three Dimensional ◽  
Spur Gear ◽  
System Of Nonlinear Equations ◽  
Unknown Parameters ◽  
Transmission Errors ◽  
Root Finding ◽  
Parabolic Function ◽  
Root Finding Method

This paper proposes a system of nonlinear equations to manipulate the amplitude of parabolic function of transmission errors. Firstly, the characteristics of parabolic function of transmission errors are defined. Then, a system of nonlinear equations for manipulating the amplitude of parabolic function of transmission errors is created based on both the conditions of contact and the constraint on the amplitude of function of transmission errors. As the number of independent scalar equations in the system minus the number of unknown parameters is one, one extra design parameter can be applied to manipulate the amplitude of parabolic function of transmission errors. The solution to the extra design variable is automatically, precisely, and efficiently determined by the computer program which is created based on the Newton’s root finding method. The time-consuming manual iterations for trying the value of design variable are eliminated. The proposed method can be applied to both two-and three-dimensional gearing problems. At last, a pair of meshing gears composed of a circular-arc spur gear and an involute spur gear is presented to verify the methodology proposed in this paper.

scholarly journals On iterative techniques for estimating all roots of nonlinear equation and its system with application in differential equation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mudassir Shams ◽  
Naila Rafiq ◽  
Nasreen Kausar ◽  
Praveen Agarwal ◽  
Choonkil Park ◽  
...  
Keyword(s):  
Nonlinear Equation ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Basin Of Attraction ◽  
Numerical Test ◽  
Computational Cost ◽  
System Of Nonlinear Equations ◽  
Single Root ◽  
Root Finding

AbstractIn this article, we construct a family of iterative methods for finding a single root of nonlinear equation and then generalize this family of iterative methods for determining all roots of nonlinear equations simultaneously. Further we extend this family of root estimating methods for solving a system of nonlinear equations. Convergence analysis shows that the order of convergence is 3 in case of the single root finding method as well as for the system of nonlinear equations and is 5 for simultaneous determination of all distinct and multiple roots of a nonlinear equation. The computational cost, basin of attraction, efficiency, log of residual and numerical test examples show that the newly constructed methods are more efficient as compared to the existing methods in literature.

scholarly journals SOLVING DUAL FUZZY NONLINEAR EQUATIONS VIA MODIFIED STIRLING’S METHOD

2019 ◽  
Vol 4 (14) ◽  
pp. 84-91
Author(s):  
A. O Umar ◽  
M Mamat ◽  
M.Y Waziri
Keyword(s):  
Fixed Point ◽  
Nonlinear Equations ◽  
Unique Fixed Point ◽  
Numerical Results ◽  
Benchmark Problems ◽  
Parametric Form ◽  
Root Finding ◽  
Root Finding Method ◽  
Finding Method

Stirling’s method is a root-finding method designed to approximate a locally unique fixed point and cannot be used to solve fuzzy nonlinear equations. In this paper, we present a modified Stirling’s method for solving dual fuzzy nonlinear equations. The fuzzy coefficient is presented in parametric form. Numerical results on some benchmark problems indicate that the proposed method is efficient.

Two-Component Generalization of a Generalized the Short Pulse Equation

2019 ◽  
pp. 1760-1765
Author(s):  
Mohammed Allami
Keyword(s):  
Nonlinear Equations ◽  
Short Pulse ◽  
Three Dimensional ◽  
Continuous Group ◽  
System Of Nonlinear Equations ◽  
Similarity Reduction ◽  
Short Pulse Equation ◽  
Point Transformations ◽  
Two Component ◽  
Lie Brackets

     In this article, we introduce a two-component generalization for a new generalization type of the short pulse equation was recently found by Hone and his collaborators. The coupled of nonlinear equations is analyzed from the viewpoint of Lie’s method of a continuous group of point transformations. Our results show the symmetries that the system of nonlinear equations can admit, as well as the admitting of the three-dimensional Lie algebra. Moreover, the Lie brackets for the independent vectors field are presented. Similarity reduction for the system is also discussed.

scholarly journals CONDITIONS FOR THE COINCIDENCE OF THE LS AND AITKEN ESTIMATIONS OF THE HIGHER COEFFICIENT OF THE QUADRATIC REGRESSION MODEL

2019 ◽  
pp. 33-42
Author(s):  
Marta Savkina
Keyword(s):  
Regression Model ◽  
Nonlinear Equations ◽  
Line Segment ◽  
Quadratic Model ◽  
System Of Nonlinear Equations ◽  
Quadratic Regression ◽  
Unknown Parameters ◽  
Quadratic Regression Model ◽  
Equidistant Points

In the paper in the case of heteroscedastic independent deviations a regression model whose function has the form $ f (x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are unknown parameters, is studied. Approximate values (observations) of functions $f (x)$ are registered at equidistant points of a line segment. The theorem proved in the paper states that Aitken estimation of the higher coefficient of the quadratic model in the case of odd the number of observation points coincides with its estimation of LS iff values of the variances satisfy a certain system of nonlinear equations. Under these conditions, the Aitken and LS estimations of $b$ and $c$ will not coincide. The application of the theorem for some cases of a specific quantity of observation points and the same values of the variances at nodes symmetric about the point $\frac{1}{2}$ is considered. In all these cases it is obtained that the LS estimation will be coincide Aitken estimation if the variance in two points accepts arbitrary values, and at all others does certain values that are expressed through the values of variances in these two points.

Finding All Solutions of Systems of Nonlinear Equations Using Spiral Dynamics Inspired Optimization with Clustering

2015 ◽  
Vol 19 (5) ◽  
pp. 697-707 ◽  
Author(s):  
Kuntjoro Adji Sidarto ◽  
◽  
Adhe Kania
Keyword(s):  
Nonlinear Equations ◽  
Systems Of Nonlinear Equations ◽  
System Of Nonlinear Equations ◽  
Advantages And Disadvantages ◽  
Root Finding ◽  
Spiral Dynamics ◽  
All Solutions ◽  
Computational Sciences ◽  
System Equations ◽  
Novel Method

Nowadays the root finding problem for nonlinear system equations is still one of the difficult problems in computational sciences. Many attempts using deterministic and meta-heuristic methods have been done with their advantages and disadvantages, but many of them have fail to converge to all possible roots. In this paper, a novel method of locating and finding all of the real roots from the system of nonlinear equations is proposed mainly using the spiral dynamics inspired optimization by Tamura and Yasuda [1]. The method is improved by the usage of the Sobol sequence of points for generating initial candidates of roots which are uniformly distributed than of pseudo-random generated points. Using clustering technique, the method localizes all potential roots so the optimization is conducted in those points simultaneously. A set of problems as the benchmarks from the literature is given. Having only a single run for each problem, the proposed method has successfully found all possible roots within a bounded domain.

Solving System of Nonlinear Equations using Genetic Algorithm

2019 ◽  
Vol 10 (4) ◽  
pp. 877-886 ◽  
Author(s):  
Chhavi Mangla ◽  
Musheer Ahmad ◽  
Moin Uddin
Keyword(s):  
Genetic Algorithm ◽  
Nonlinear Equations ◽  
System Of Nonlinear Equations

Study on the working state of the five hundred-meter aperture spherical telescope using step-wise assignment method

2014 ◽  
Vol 229 (2) ◽  
pp. 316-324 ◽  
Author(s):  
Yu Liu ◽  
Feng Gao
Keyword(s):  
Nonlinear Equations ◽  
Analytical Solutions ◽  
Driving Mechanism ◽  
Initial Value ◽  
Unknown Parameters ◽  
Support Structure ◽  
Assignment Method ◽  
New Type ◽  
Set Up ◽  
The Mathematical Model

The working state of the five hundred-meter aperture spherical telescope (FAST) is solved using the step-wise assignment method. In this paper, the mathematical model of the cable-net support structure of the FAST is set up by the catenary equation. There are a large number of nonlinear equations and unknown parameters of the model. The nonlinear equations are solved by using the step-wise assignment method. The method is using the analytical solutions of the cable-net equations of one working state as the initial value for the next working state, from which the analytical solutions of the nonlinear equations of the cable-net for each working state of the FAST and the tension and length of each driving cable can be obtained. The suggested algorithm is quite practically well suited to study the working state of the cable-net structures of the FAST. Also, the working state analysis result of the cable-net support structure of a reduced model of the cable-net structure reflector for the FAST is given to verify the reliability of the method. In order to show the validity of the method, comparisons with another algorithm to set the initial value are presented. This method has an important guiding significance to the further study on the control of the new type of flexible cable driving mechanism, especially the FAST.

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