An efficient method for determining the critical modification coefficient for cylindrical worm gearing in which the worm is generated by a curve

2001 ◽  
Vol 215 (12) ◽  
pp. 1745-1755 ◽  
Author(s):  
C-K Lee ◽  
C-K Chen
Keyword(s):  
Efficient Method ◽  
Linear Equations ◽  
Contact Ratio ◽  
Non Linear ◽  
Worm Gearing ◽  
System Equations ◽  
Curvature Information ◽  
Maximum Contact ◽  
Non Linear Equations

This paper proposes an efficient method for the determination of the critical modification coefficient (CMC) for the cylindrical worm gearing in which the worm is generated by a curve. The term CMC is defined as the modification coefficient that causes the first boundary points to be just on the boundary of the area of meshing. Solved from a system of five non-linear equations, the CMC not only avoids gear undercutting, but also produces the maximum contact ratio. In developing the system equations, the equation of non-undercutting is derived directly from the curvature information of the generating curve and the parameters of relative motion, without the need to derive any partial derivatives to the equation of the generating surface. By using an equivalent generation mechanism rather than the real one, an explicit form of equation of meshing is created. The equation of meshing created can simplify the form of the equation of non-undercutting, reduce the number of non-linear equations and unknowns in the system equations, increase the speed of convergence and decrease numerical instability. Based on the proposed method, a computer program is created and applied to analyse the CMCs of four commonly used worm gearings: ZA, ZN, ZE and ZC3.

scholarly journals On Efficient Iterative Numerical Methods for Simultaneous Determination of all Roots of Non-Linear Function

2020 ◽  
Author(s):  
Mudassir Shams ◽  
Nazir Mir ◽  
Naila Rafiq
Keyword(s):  
Simultaneous Determination ◽  
Iterative Methods ◽  
Linear Equations ◽  
Basin Of Attraction ◽  
Numerical Test ◽  
Computational Cost ◽  
Single Root ◽  
Non Linear ◽  
Non Linear Equations

We construct a family of two-step optimal fourth order iterative methods for finding single root of non-linear equations. We generalize these methods to simultaneous iterative methods for determining all the distinct as well as multiple roots of single variable non-linear equations. Convergence analysis is present for both cases to show that the order of convergence is four in case of single root finding method and is twelve for simultaneous determination of all roots of non-linear equation. The computational cost, Basin of attraction, efficiency, log of residual and numerical test examples shows, the newly constructed methods are more efficient as compared to the existing methods in literature.

scholarly journals Interval and Speed of Convergence on Iterative Methods

2017 ◽  
pp. 112-114
Author(s):  
Iswarmani Adhikari
Keyword(s):  
Iterative Method ◽  
Iterative Methods ◽  
Linear Equations ◽  
Initial Guess ◽  
Similar Process ◽  
Speed Of Convergence ◽  
Non Linear ◽  
Non Linear Equations

The iterative method is a tool of solving the non-linear equations to get their approximate roots with some errors of tolerance. Repetition of the similar process is applied successively on such iterations to compute a sequence of increasingly accurate estimates of the roots. In this paper, the construction of an iterative method for solving an equation, its convergence and the determination of interval of convergence for the approximate choice of initial guess and the speed of convergence are highlighted.The Himalayan Physics Vol. 6 & 7, April 2017 (112-114)

scholarly journals A master exceptional field theory

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Guillaume Bossard ◽  
Axel Kleinschmidt ◽  
Ergin Sezgin
Keyword(s):  
Field Theory ◽  
Gauge Invariance ◽  
Sigma Model ◽  
Linear Equations ◽  
Field Theories ◽  
Gauge Invariant ◽  
Non Linear ◽  
Non Linear Equations

Abstract We construct a pseudo-Lagrangian that is invariant under rigid E11 and transforms as a density under E11 generalised diffeomorphisms. The gauge-invariance requires the use of a section condition studied in previous work on E11 exceptional field theory and the inclusion of constrained fields that transform in an indecomposable E11-representation together with the E11 coset fields. We show that, in combination with gauge-invariant and E11-invariant duality equations, this pseudo-Lagrangian reduces to the bosonic sector of non-linear eleven-dimensional supergravity for one choice of solution to the section condi- tion. For another choice, we reobtain the E8 exceptional field theory and conjecture that our pseudo-Lagrangian and duality equations produce all exceptional field theories with maximal supersymmetry in any dimension. We also describe how the theory entails non-linear equations for higher dual fields, including the dual graviton in eleven dimensions. Furthermore, we speculate on the relation to the E10 sigma model.

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