Numerical Method of Determining the Curvature Interference Limit Curve for Modified Hourglass Worm Pairs

2011 ◽  
Vol 86 ◽  
pp. 352-356 ◽  
Author(s):  
Ya Ping Zhao ◽  
Tian Chao Wu
Keyword(s):  
Numerical Method ◽  
Nonlinear Equations ◽  
Double Point ◽  
Jacobi Matrix ◽  
Worm Gear ◽  
Secant Method ◽  
System Of Nonlinear Equations ◽  
Limit Curve ◽  
Numerical Example ◽  
Curvature Interference

The double-point downhill secant method (the DPDS method) is proposed to solve the nonlinear equations to determine the curvature interference limit points for modified hourglass worm drives. Thereupon, the whole curvature interference limit line can be obtained by interpolation. Based on this, the undercutting feature of the corrected worm gear can be investigated. The DPDS method has two main merits in principle. The first is the avoidance of the computation of the Jacobi matrix of the system of nonlinear equations. The second is that the sensitivity to the guess value can be decreased evidently owing to adopt the technique of the norm reduction. The effectiveness of the DPDS method is inspected and verified by a numerical example.

scholarly journals Numerical Solution of Nonlinear Equations in Maple

2021 ◽  
Vol 8 (4) ◽  
Author(s):  
Qani Yalda
Keyword(s):  
Numerical Methods ◽  
Numerical Solution ◽  
Numerical Method ◽  
Nonlinear Equations ◽  
Secant Method ◽  
Bisection Method ◽  
Real Roots ◽  
Newton Raphson ◽  
Root Analysis ◽  
Raphson Method

The main purpose of this paper is to obtain the real roots of an expression using the Numerical method, bisection method, Newton's method and secant method. Root analysis is calculated using specific, precise starting points and numerical methods and is represented by Maple. In this research, we used Maple software to analyze the roots of nonlinear equations by special methods, and by showing geometric diagrams, we examined the relevant examples. In this process, the Newton-Raphson method, the algorithm for root access, is fully illustrated by Maple. Also, the secant method and the bisection method were demonstrated by Maple by solving examples and drawing graphs related to each method.

Determination of the Most Dangerous Meshing Point for Modified Hourglass Worm Drives

2011 ◽  
Author(s):  
Yaping Zhao ◽  
Tianchao Wu
Keyword(s):  
Nonlinear Equations ◽  
Slight Modification ◽  
Second Step ◽  
System Of Nonlinear Equations ◽  
Type Ii ◽  
Severe Condition ◽  
Worm Gearing ◽  
Curvature Interference ◽  
Worm Drive

A kind of modified hourglass worm drives, which is frequently called the type II worm gearing for short, has various favorable meshing features. Nevertheless, its sole shortcoming is the undercutting of the worm wheel. In the condition of adopting slight modification, this problem can be overcome due to the removal of a part of one sub-conjugate area containing the curvature interference limit line. In order to measure the effect of the avoidance of undercutting, a strategy to determine the meshing point in the most severe condition is proposed for a type II worm drive. The presented strategy can be divided into two steps. The first step is to establish a system of nonlinear equations in five variables in accordance with the theory of gearing. The second step is to solve the procured nonlinear equations by numerical iterative method to ascertain the meshing point required. A numerical example is presented to verify the validity and feasibility of the proposed scheme.

Determination of the Most Dangerous Meshing Point for Modified-Hourglass Worm Drives

2013 ◽  
Vol 135 (3) ◽  
Author(s):  
Yaping Zhao ◽  
Yimin Zhang
Keyword(s):  
Nonlinear Equations ◽  
Slight Modification ◽  
System Of Nonlinear Equations ◽  
Type Ii ◽  
Severe Condition ◽  
Numerical Iteration ◽  
Worm Gearing ◽  
Curvature Interference ◽  
Worm Drive

A type of modified-hourglass worm gear drive, frequently called type-II worm gearing for short, has various favorable meshing features. Its sole shortcoming is the undercutting of the worm wheel. By adopting a slight modification, this problem can be overcome due to the removal of a part of one subconjugate area containing the curvature interference limit line. To measure how effectively the undercutting is avoided, a strategy to determine the meshing point in the most severe condition is proposed for a type-II worm drive. The strategy presented consists of two steps. The first step is to establish a system of nonlinear equations in five variables in accordance with the theory of gearing. The second step is to solve the system of nonlinear equations by a numerical iteration method to ascertain the meshing point required. A numerical example is presented to verify the validity and feasibility of the proposed scheme.

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

scholarly journals Application of a Generalized Secant Method to Nonlinear Equations with Complex Roots

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 169
Author(s):  
Avram Sidi
Keyword(s):  
Nonlinear Equations ◽  
Local Convergence ◽  
Real Root ◽  
Numerical Procedure ◽  
Secant Method ◽  
Convergence Theory ◽  
Simple Complex ◽  
Real Roots ◽  
Complex Roots ◽  
Appl Math

The secant method is a very effective numerical procedure used for solving nonlinear equations of the form f(x)=0. In a recent work (A. Sidi, Generalization of the secant method for nonlinear equations. Appl. Math. E-Notes, 8:115–123, 2008), we presented a generalization of the secant method that uses only one evaluation of f(x) per iteration, and we provided a local convergence theory for it that concerns real roots. For each integer k, this method generates a sequence {xn} of approximations to a real root of f(x), where, for n≥k, xn+1=xn−f(xn)/pn,k′(xn), pn,k(x) being the polynomial of degree k that interpolates f(x) at xn,xn−1,…,xn−k, the order sk of this method satisfying 1<sk<2. Clearly, when k=1, this method reduces to the secant method with s1=(1+5)/2. In addition, s1<s2<s3<⋯, such that limk→∞sk=2. In this note, we study the application of this method to simple complex roots of a function f(z). We show that the local convergence theory developed for real roots can be extended almost as is to complex roots, provided suitable assumptions and justifications are made. We illustrate the theory with two numerical examples.

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