Application of Finite Elements to the Analysis of Gear Tooth Stresses

1973 ◽  
Vol 95 (4) ◽  
pp. 1139-1148 ◽  
Author(s):  
L. Wilcox ◽  
W. Coleman
Keyword(s):  
Finite Elements ◽  
Gear Tooth ◽  
Tooth Surface ◽  
Two Dimensional ◽  
Loading Conditions ◽  
The Finite Element Method ◽  
Tooth Shape ◽  
Hypoid Gears ◽  
Wide Range ◽  
New Formula

This paper explains the application of the analytical method of finite elements to the analysis of gear tooth stresses. The details necessary for simulating a two-dimensional tooth shape with finite elements are outlined and a technique for determining stress values at the tooth surface in the root fillet is given. The remainder of the paper uses the finite element method to analyze tensile fillet stress in generated tooth shapes incorporating either symmetric or asymmetric profiles. Special attention has been given to the asymmetric profiles (used in Hypoid gears) since no reliable formulas are available. Stress data thus obtained are used to develop a new simplified stress formula that gives tensile fillet stress as a function of geometric tooth shape and general loading conditions. The accuracy and wide range of applicability of the new stress formula is shown by comparing the new formula to other previously accepted formulas for a variety of tooth shapes and loading conditions.

Simulation of Hypoid Gear Lapping

2008 ◽  
Vol 130 (11) ◽  
Author(s):  
Qimi Jiang ◽  
Claude Gosselin ◽  
Jack Masseth
Keyword(s):  
Gear Tooth ◽  
Tooth Surface ◽  
Wear Coefficient ◽  
Modeling Tools ◽  
Hypoid Gears ◽  
Coordinate Measurement ◽  
Sliding Motion ◽  
Light Load ◽  
Tooth Surfaces ◽  
Initial Results

In the lapping process of hypoid gears, a gear set is run at varying operating positions and under a light load in order to lap the complete tooth surface. Because of the rolling and sliding motion inherent to hypoid gears, the lapping compound acts as an abrasive and refines the tooth surface to achieve smoothness in rolling action and produce high quality gear sets. In this paper, the lapping process is reproduced using advanced modeling tools such as gear tooth vectorial simulation for the tooth surfaces and reverse engineering to analyze the tooth contact pattern of existing gear sets. Test gear sets are measured using a coordinate measurement machine prior to a special lapping cycle where the position of the gear sets on the lapper does not change, and then are remeasured after lapping in order to establish how much and where material was removed. A wear constant named “wear coefficient” specific to the lapping compound is then calculated. Based on the obtained wear coefficient value, an algorithm for simulating the lapping process is presented. Gear sets lapped on the production line at AAM are used for simulation case studies. Initial results show significant scattering of tooth distortion from tooth to tooth and from gear set to gear set, which makes the simulation process difficult. However, it is possible to predict a confidence range within which actual lapping should fall, thereby opening the door to the optimization of the lapping process.

scholarly journals General polytopal H(div)-conformal finite elements and their discretisation spaces.

2020 ◽  
Author(s):  
Elise Le Meledo ◽  
Philipp Öffner ◽  
Remi Abgrall
Keyword(s):  
Finite Elements ◽  
Degrees Of Freedom ◽  
General Setting ◽  
Basis Functions ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Wide Range ◽  
Virtual Element ◽  
Set Up

We present a class of discretisation spaces and H(div) - conformal elements that can be built on any polytope. Bridging the flexibility of the Virtual Element spaces towards the element's shape with the divergence properties of the Raviart - Thomas elements on the boundaries, the designed frameworks offer a wide range of H(div) - conformal discretisations. As those elements are set up through degrees of freedom, their definitions are easily amenable to the properties the approximated quantities are wished to fulfil. Furthermore, we show that one straightforward restriction of this general setting share its properties with the classical Raviart - Thomas elements at each interface, for any order and any polytopial shape. Then, to close the introduction of those new elements by an example, we investigate the shape of the basis functions corresponding to particular elements in the two dimensional case.

Research on Gear Cutter Tooth Profile for Semi-Topping: A Case of a Helical Gear

1992 ◽  
Author(s):  
Y. Ariga ◽  
Shiyeyoshi Nagata
Keyword(s):  
Design Method ◽  
Gear Tooth ◽  
Helical Gear ◽  
Tooth Profile ◽  
Tooth Surface ◽  
Gear Cutting ◽  
Gear Cutter ◽  
Conventional Design ◽  
Wide Range ◽  
Tooth Number

Abstract Gear tooth tips are frequently chamfered to prevent nicks or scuffing on the tooth surface. Some of the hob cutters and pinion cutters can be chamfered but many types of cutters should be used for a particular range of tooth numbers since the amount chamfering largely varies depending on the tooth number. However, intensive efforts in the design have made it possible to produce cutters with little variation of chamfering amount for a wide range of tooth numbers. The error in the amount of chamfering by a single cutter designed by the above method can be maintained within ±10 % for gears with tooth numbers ranging from 16 to 94. It was found that three cutters of the conventional design are required for keeping the error within the same range for cutting gears within a given range of tooth numbers. The paper describes the tooth design method of the hob cutter with little variation of chamfering amount along changes in number of teeth to be machined and demonstrates that chamfering errors are maintained within practically allowable ranges for profile shift cutting or helical gear cutting with the use of this cutter.

Experimental Validation of a Computerized Tool for Face Hobbed Gear Contact and Tensile Stress Analysis

2007 ◽  
Author(s):  
Andrea Piazza ◽  
Martino Vimercati
Keyword(s):  
Experimental Validation ◽  
Gear Tooth ◽  
Tooth Surface ◽  
Spiral Bevel Gear ◽  
Hypoid Gears ◽  
Heavy Loads ◽  
Gear Contact ◽  
Element Technique ◽  
The Mathematical Model ◽  
Good Agreement

While face milled gears have been widely analyzed, about face hobbed ones only very few studies have been developed and presented. Goal of this paper is to propose the validation of an accurate tool, which was presented by the authors in previous works, aimed to the computerized design of face hobbed gears. Firstly, the mathematical model able to compute detailed gear tooth surface representation on both spiral and hypoid gears will be briefly recalled; then, the so obtained 3D tooth geometry is employed as input for an advanced contact solver that, using a hybrid method combining finite element technique with semianalytical solutions, is able to efficiently carry out both contact analysis under light or heavy loads and stress tensile calculation. The validation analyses will be carried on published aerospace face hobbed spiral bevel gear data comparing measurements of root and fillet stresses. Good agreement with experimental results both in the time scale and in magnitude will be revealed.

Effect of Tooth Contact and Gear Dimensions on Transmission Errors of Loaded Hypoid Gears

1991 ◽  
Vol 113 (2) ◽  
pp. 182-187 ◽  
Author(s):  
M. Sugimoto ◽  
N. Maruyama ◽  
A. Nakayama ◽  
N. Hitomi
Keyword(s):  
Gear Tooth ◽  
Transmission Error ◽  
Tooth Surface ◽  
Alignment Error ◽  
Tooth Contact ◽  
Hypoid Gears ◽  
Module Design ◽  
Transmission Errors ◽  
Loaded State ◽  
Final Drive

The effect of the tooth contact and alignment error of the hypoid gear assembly on transmission error was investigated with a new measuring apparatus which can measure the transmission errors of loaded hypoid gears assembled into a final drive unit. Measurements indicate that transmission error predictions made with the TCA and LTCA — analytical tools developed by Gleason Works for a no-load and loaded state, respectively — have sufficient accuracy when actual data on the gear tooth surface and alignment error of the gear assembly are used in the calculations. A systematic examination has also been made of the effects of tooth contact and gear assembly alignment error on transmission error on the basis of the LTCA calculations. It was found that the transmission errors relative to the applied load varied not only according to the tooth contact but also the no-load transmission error of the gears. This relationship was also examined by taking into account the effects of the gear dimensions. It was confirmed through calculation and experiment that a small module design was effective in reducing transmission error.

An Ease-Off Based Method for Loaded Tooth Contact Analysis of Hypoid Gears Having Local and Global Surface Deviations

2010 ◽  
Vol 132 (7) ◽  
Author(s):  
M. Kolivand ◽  
A. Kahraman
Keyword(s):  
Gear Tooth ◽  
Contact Analysis ◽  
Tooth Surface ◽  
Hypoid Gear ◽  
Tooth Contact Analysis ◽  
Tooth Contact ◽  
Hypoid Gears ◽  
Surface Deviations ◽  
Tooth Surfaces ◽  
Loaded Tooth Contact Analysis

Actual hypoid gear tooth surfaces do deviate from the theoretical ones either globally due to manufacturing errors or locally due to reasons such as tooth surface wear. A practical methodology based on ease-off topography is proposed here for loaded tooth contact analysis of hypoid gears having both local and global deviations. This methodology defines the theoretical pinion and gear tooth surfaces from the machine settings and cutter parameters, and constructs the surfaces of the theoretical ease-off and roll angle to compute for the unloaded contact analysis. This theoretical ease-off topography is modified based on tooth surface deviations and is used to perform a loaded tooth contact analysis according to a semi-analytical method proposed earlier. At the end, two examples, a face-milled hypoid gear set having local deviations and a face-hobbed one having global deviations, are analyzed to demonstrate the effectiveness of the proposed methodology in quantifying the effect of such deviations on the load distribution and the loaded motion transmission error.

Generation of Hypoid Gears on CNC Hypoid Generator

2011 ◽  
Vol 133 (12) ◽  
Author(s):  
Vilmos V. Simon
Keyword(s):  
Gear Tooth ◽  
Velocity Ratio ◽  
Transmission Error ◽  
Tooth Surface ◽  
Polynomial Functions ◽  
Gear Pair ◽  
Hypoid Gear ◽  
Kinematic Scheme ◽  
Hypoid Gears ◽  
Fifth Order

In this study, polynomial functions of orders up to five are applied to induce variations in the cradle radial setting and the velocity ratio in the kinematic scheme of the machine tool for the generation of the pinion tooth surfaces corresponding to reduced transmission error amplitudes of a hypoid gear pair. The new CNC hypoid generators have made it possible to perform this nonlinear correction motions for the cutting of the face-milled hypoid gears. An algorithm is developed for the execution of motions on the CNC hypoid generator for the generation of face-milled hypoid gear tooth surface, based on the machine tool setting variation on the cradle-type hypoid generator induced by the optimal polynomial functions up to fifth-order. By using the corresponding computer program, the motion graphs of the CNC hypoid generator are determined for the generation of hypoid gear tooth surface, based on the optimal variation in the velocity ratio in the kinematic scheme and on the variation in the cradle radial setting on a cradle-type generator. The results presented indicate that the variation of the velocity ratio in the kinematic scheme of the hypoid generator induced by a fifth-order polynomial function resulted in a 62% reduction of the maximum transmission error of the gear pair.

Application of EHD Oil Film Theory to Industrial Gear Drives

1976 ◽  
Vol 98 (2) ◽  
pp. 626-631 ◽  
Author(s):  
E. J. Wellauer ◽  
G. A. Holloway
Keyword(s):  
Film Thickness ◽  
Surface Texture ◽  
Rapid Determination ◽  
Gear Tooth ◽  
Tooth Surface ◽  
Composite Surface ◽  
Oil Film ◽  
Wide Range ◽  
Surface Distress ◽  
Gear Drives

The method and assumptions used for the application of EHD theory to the calculation of gear tooth oil film thicknesses for the design and analysis of industrial gear drives is presented. A nomograph, utilizing readily available gear geometry, operational, and lubricant parameters, is illustrated which allows rapid determination of calculated gear tooth oil film thicknesses for a wide range of gear drive conditions. Gear tooth surface distress is related to the specific film thickness, λ, the ratio of calculated oil film thickness to the magnitude of the composite surface texture. The term “surface texture” is introduced for gear contacts to indicate that surface attributes coarser than roughness importantly relate to tooth surface distress, but a sophisticated method for its quantitative assessment has not been developed. Data from several hundred petroleum lubricated laboratory tests and closely followed field applications which include through hardened gears of 1 in. to 15 ft in diameter are used to correlate specific film thickness and gear tooth surface distress. Curves are given to predict the probability of occurrence of such distress over the range of pitch line velocities of 4–35,000 ft/min.

scholarly journals Mathematical modeling of unsteady elastic stress waves in a console with a base (half-plane) under fundamental seismic action

2019 ◽  
Vol 15 (6) ◽  
pp. 477-482
Author(s):  
Vyacheslav K. Musayev
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Finite Elements ◽  
Half Plane ◽  
Stress Waves ◽  
The Body ◽  
Seismic Action ◽  
Two Dimensional ◽  
The Finite Element Method ◽  
First Order

The aim of the work is to consider the problems of numerical modeling of seismic safety of the console with the base in the form of an elastic half-plane under unsteady wave influences. Stress waves of different nature, propagating in the deformed body interact with each other. After three or four times the passage and reflection of stress waves in the body, the process of propagation of disturbances becomes steady, the body is in oscillatory motion. The problem of modeling problems of the transition period is an actual fundamental and applied scientific problem. Methods. The finite element method in displacements is used to solve the two-dimensional plane dynamic problem of elasticity theory with initial and boundary conditions. On the basis of the finite element method in displacements, an algorithm and a set of programs for solving linear plane two-dimensional problems have been developed, which allow solving problems with non-stationary wave effects on complex systems. The algorithmic language “Fortran-90” was used in the development of the complex of programs. The study area is divided by spatial variables into finite elements of the first order. According to the time variable, the study area is also divided into finite elements of the first order. Results. The problem of the influence of a plane longitudinal elastic wave in the form of a Heaviside function on a console with a base (the ratio of width to height is one to ten) is considered. The initial conditions are taken as zero. The system of equations from 16 016 084 unknowns is solved. Contour stresses and stress tensor components are obtained in characteristic areas of the problem. On the basis of the conducted researches it is possible to draw the following conclusions: the console (the ratio of width to height one to ten) is modeled with the elastic basis in the form of an elastic half-plane; the elastic contour stresses on the faces of the console are almost a mirror image of one another, that is, antisymmetric; the console under seismic action works as a rod of variable cross-section, that is, if there are tensile stresses on one face, then compressive stresses on the other; on the contours of the console under seismic action, bending waves mainly prevail.

An Ease-Off Based Method for Loaded Tooth Contact Analysis of Hypoid Gears Having Local and Global Surface Deviations

2009 ◽  
Author(s):  
M. Kolivand ◽  
A. Kahraman
Keyword(s):  
Gear Tooth ◽  
Contact Analysis ◽  
Tooth Surface ◽  
Surface Wear ◽  
Tooth Contact Analysis ◽  
Tooth Contact ◽  
Hypoid Gears ◽  
Manufacturing Errors ◽  
Spiral Bevel ◽  
Loaded Tooth Contact Analysis

Manufacturing errors typically cause real (measured) spiral bevel and hypoid gear surfaces to deviate from the theoretical ones globally. Tooth surface wear patterns accumulated through the life span of the gear set are typically local deviations that are aggravated especially in case of edge contact conditions. An accurate and practical methodology based on ease-off topography is proposed in this study to perform loaded tooth contact analysis of spiral bevel and hypoid gears having both types of local and global deviations. It starts with definition of the theoretical pinion and gear tooth surfaces from the machine settings and cutter parameters, and constructs the theoretical ease-off and roll angle surfaces to compute unloaded contact analysis. Manufacturing errors and localized surface wear deviations are considered to update the theoretical ease-off to form a new ease-off surface that is used to perform a loaded tooth contact analysis according to the semi-analytical method proposed earlier. At the end, a numerical example with locally deviated surfaces is analyzed to demonstrate the effectiveness of the proposed methodology as well as quantifying the effect of such deviations on load distribution and the loaded motion transmission error.

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