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

Volume 8: 11th International Power Transmission and Gearing Conference; 13th International Conference on Advanced Vehicle and Tire Technologies ◽

10.1115/detc2011-47202 ◽

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.

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

Journal of Mechanical Design ◽

10.1115/1.4023281 ◽

2013 ◽

Vol 135 (3) ◽

Cited By ~ 9

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.

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

Advances in Difference Equations ◽

10.1186/s13662-021-03636-x ◽

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.

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Effects of meshing density of 1D structural members with non-uniform cross-section along the length on the calculation of eigenfrequencies

MATEC Web of Conferences ◽

10.1051/matecconf/202031300004 ◽

2020 ◽

Vol 313 ◽

pp. 00004

Author(s):

Jakub Rubint

Keyword(s):

Finite Element Method ◽

Finite Element ◽

Nonlinear Equations ◽

Cross Section ◽

System Of Nonlinear Equations ◽

Engineering Practice ◽

Structural Members ◽

Mass Matrices ◽

Uniform Cross Section

Density of division in finite element method does not affect only the accuracy of calculation, but also the necessary calculation time. This is directly influenced by the power of the used hardware, the efficiency of the algorithm used to assemble global stiffness and mass matrices and finally, by the method used to find eigenvalues of matrices for determination of eigenfrequencies. In engineering practice, when commercially available software is used, it is necessary to look for the optimum between the accuracy of the calculation and the length of the calculation. This paper deals with solution of eigenfrequencies of 1D elements with nonuniform cross section using Python 3.7.4 with libraries "numpy 1.18.1" for finding eigenvalues of matrices and "scipy 1.4.1" for finding solution for system of nonlinear equations.

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

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.86.352 ◽

2011 ◽

Vol 86 ◽

pp. 352-356 ◽

Cited By ~ 3

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.

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A class of algorithms for the determination of a solution of a system of nonlinear equations

Optimization Techniques - Lecture Notes in Control and Information Sciences ◽

10.1007/bfb0006510 ◽

1978 ◽

pp. 67-75

Author(s):

B. D. Kiekebusch-Müller

Keyword(s):

Nonlinear Equations ◽

System Of Nonlinear Equations

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Solving System of Nonlinear Equations using Genetic Algorithm

Journal of Computer and Mathematical Sciences ◽

10.29055/jcms/1072 ◽

2019 ◽

Vol 10 (4) ◽

pp. 877-886 ◽

Cited By ~ 1

Author(s):

Chhavi Mangla ◽

Musheer Ahmad ◽

Moin Uddin

Keyword(s):

Genetic Algorithm ◽

Nonlinear Equations ◽

System Of Nonlinear Equations

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Human Umbilical Cord Mesenchymal Stem Cells Induce into Chondrocytes in 3D Cultured System

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.749.198 ◽

2013 ◽

Vol 749 ◽

pp. 198-205

Author(s):

Li Yu ◽

Jing Liu ◽

Chao Xu ◽

Er Mei Luo ◽

Ming Qiao Tang

Keyword(s):

Real Time ◽

Alcian Blue ◽

Culture System ◽

Type Ii Collagen ◽

Human Umbilical Cord ◽

Type Ii ◽

Pellet Culture ◽

Hla Dr

Objective: To investigate a better method of inducing hUC-MSCs into chondrocytes in different culture system in vitro. Method: hUC-MSCs were isolated and cultured by tissue block culture, and the cells surface antigens were identified by flow cytometry, hUC-MSCs were cultured with chondrogenic media and stained with Alcian Blue. The production of matrix was estimated from the determination of hydroxyproline content and Alcian Blue method. Expressions of glycosaminoglycan (GAG), type II collagen and Sox-9 were assayed by real-time fluorescence quantitative PCR. Results: The cultured hUC-MSCs phenotype was CD105+/CD29+/CD44+/ CD31-/CD34-/ CD40-/CD45-/HLA-DR-. hUC-MSCs weakly expressed chondrocyte marker, which strongly expressed GAG and type II collagen after chondrogenic induction, and the cells were incubated in pellet culture with higher expression. Real-time PCR results demonstrated that chondrogenic induction cells were expressed GAG, type II collagen and Sox-9, and the cells were incubated in pellet culture with higher expression. Conclusion: hUC-MSCs incubated in pellet culture is more conducive to differentiate into chondrocytes than those cultured in monolayer culture system.

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