Mathematical Model for Parametric Tooth Surface of Cylindrical Gears Based on Kinematic Synthesis

2007 ◽  
Author(s):  
Carlos Garci´a-Masia´ ◽  
Juan D. Morillas-A´lvarez
Keyword(s):  
Mathematical Model ◽  
Tooth Surface ◽  
Asymmetry Ratio ◽  
Contact Ratio ◽  
Motion Generation ◽  
Kinematic Synthesis ◽  
Transmission Errors ◽  
Cylindrical Gears ◽  
Tooth Surfaces ◽  
And Function

A generalized approach for parametrizing conjugate tooth surfaces in cylindrical gears is presented in this work. Developed are the polynomials expressions to define the tooth surfaces of pinion and gear based on kinematics synthesis for planar gears. The polynomials expressions incorporate the motion generation (points or positions of precision) and function of transmission errors. It is interesting to note that if the desired pressure angle for the tooth profile is constant, the output polynomial of profile becomes a conventional involute. Polynomials expressions are given for the profile modifications necessary to compensate for any specified or anticipated errors of assembly and/or manufacturing. In addition property of rack as the limits of zone active, transverse contact ratio and contact asymmetry ratio are analysed.

Application and Investigation of Modified Helical Gears With Several Types of Geometry

2007 ◽  
Author(s):  
Ignacio Gonzalez-Perez ◽  
Alfonso Fuentes ◽  
Faydor L. Litvin ◽  
Kenichi Hayasaka ◽  
Kenji Yukishima
Keyword(s):  
Gear Tooth ◽  
Tooth Surface ◽  
Contact Ratio ◽  
Tooth Contact Analysis ◽  
Helical Gears ◽  
Transmission Errors ◽  
Advantages And Disadvantages ◽  
Bearing Contact ◽  
Tooth Surfaces ◽  
Gear Drives

Involute helical gears with modified geometry for transformation of rotation between parallel axes are considered. Three types of topology of geometry are considered: (1) crowning of pinion tooth surface is provided only partially by application of a grinding disk; (2) double crowning of pinion tooth surface is obtained applying a grinding disk; (3) concave-convex pinion and gear tooth surfaces are provided (similar to Novikov-Wildhaber gears). Localization of bearing contact is provided for all three types of topology. Computerized TCA (Tooth Contact Analysis) is performed for all three types of topology to obtain: (i) path of contact on pinion and gear tooth surfaces; (ii) negative function of transmission errors for misaligned gear drives (that allows the contact ratio to be increased). Stress analysis is performed for the whole cycle of meshing. Finite element models of pinion and gear with several pairs of teeth are applied. A relative motion is imposed to the pinion model that allows friction between contact surfaces to be considered. Numerical examples have confirmed the advantages and disadvantages of the applied approaches for generation and design.

Effects of Gear Dimensions and Tooth Surface Modifications on the Loaded Transmission Error of a Helical Gear Pair

1998 ◽  
Vol 120 (1) ◽  
pp. 119-125 ◽  
Author(s):  
M. Umeyama ◽  
M. Kato ◽  
K. Inoue
Keyword(s):  
Transmission Error ◽  
Tooth Surface ◽  
Contact Ratio ◽  
Rotational Angle ◽  
Actual Contact ◽  
Transmission Errors ◽  
Effective Contact ◽  
Tooth Surfaces ◽  
Helical Gear Pair ◽  
Tooth Pair

Analysis of the loaded transmission error proved that the actual contact ratio and the effective contact ratios are the valid indices. In order to calculate the loaded transmission error, deformations of a pair of teeth are estimated using Hertzian formulas for the approach deformation and approximate formulas based on the FEM for the bending deflection. The actual contact ratio εr is defined using the rotational angle during which a tooth pair is actually in contact with each other. εr increases with the increase of applied loads. The effective contact ratio, εn, is determined geometrically by gear dimensions and modified tooth surfaces based on the path of contact. Adopting these ratios, the following characteristics are derived. 1) The loaded transmission error correlates to εr when it is smaller than εn and correlates to εn when εr exceeds εn. 2) Loaded transmission errors have their minimums and maximums at the same values of εr. 3) No load transmission error is the largest among the maximums. 4) Gear pairs with higher values of εn show lower loaded transmission error.

scholarly journals Meshing principle analysis of 3-DOF involute spherical gear

2020 ◽  
Vol 213 ◽  
pp. 02029
Author(s):  
Baichao Wang ◽  
Xue Zhang ◽  
Litong Zhang ◽  
Xianting Lu
Keyword(s):  
Mathematical Model ◽  
Tooth Profile ◽  
Bevel Gear ◽  
Tooth Surface ◽  
Gear Pair ◽  
Tooth Surfaces ◽  
Three Degree Of Freedom ◽  
Relationship Of ◽  
The Mathematical Model ◽  
Spherical Gear

In this paper, a mathematical model of meshing motion of three degree of freedom involute spherical gear pair is constructed. The mathematical model can realize continuous meshing transmission between gear pairs without transmission principle error. Based on the meshing principle and motion analysis of the gear, the tooth profile of the spherical gear is designed by combining the two tooth surfaces of the involute ring gear and the hemispherical bevel gear. According to the conjugate motion relationship of spherical gear pair, a mathematical model of arc tooth surface of hemispherical bevel gear is established, and the mathematical description of the tooth profile of spherical gear is completed by combining the equation of ring tooth surface. It provides the basis and Reference for the meshing design of ball gear.

New developments in the design and generation of gear drives

2001 ◽  
Vol 215 (7) ◽  
pp. 747-757 ◽  
Author(s):  
F. L. Litvin ◽  
A Fuentes ◽  
A Demenego ◽  
D Vecchiato ◽  
Q Fan
Keyword(s):  
Point Contact ◽  
Tooth Surface ◽  
Helical Gears ◽  
Transmission Errors ◽  
Bevel Gears ◽  
Bearing Contact ◽  
Analysis And Synthesis ◽  
Tooth Surfaces ◽  
Design Generation ◽  
Gear Drives

Design, generation and simulation of the meshing and contact of gear drives with favourable bearing contact and reduced noise are considered. The proposed approach is based on replacement of the instantaneous line of contact of tooth surfaces by point contact and on application of a predesigned parabolic function of transmission errors that is able to absorb linear discontinuous functions of transmission errors caused by misalignment. Basic algorithms for analysis and synthesis of gear drives are presented. The developed theory is applied for design and generation of the following gear drives with modified geometry: (a) spur and helical gears, (b) a new version of Novikov-Wildhaber (N-W) helical gears, (c) asymmetric face gear drives with a spur pinion, (d) formate-cut spiral bevel gears. Generation of the tooth surface of a worm gear is presented as the formation of a two-branch envelope. The discussed topics are illustrated with examples.

On a General Method of Spatial Kinematic Synthesis by Means of a Stretch-Rotation Tensor

1969 ◽  
Vol 91 (1) ◽  
pp. 115-121 ◽  
Author(s):  
G. N. Sandor ◽  
K. E. Bisshopp
Keyword(s):  
Mathematical Model ◽  
Kinematic Chain ◽  
Higher Order ◽  
Motion Generation ◽  
Rotation Tensor ◽  
Kinematic Synthesis ◽  
Rotation Operator ◽  
Key Concepts ◽  
The Mathematical Model ◽  
General Method

One of the key concepts in a general method of spatial kinematic synthesis is a stretch-rotation operator applied to members of a general spatial kinematic chain. The latter consists of one or more interconnected loops of successively ball-jointed bar-slideball members. Each member is represented by a vector free to stretch-rotate with the motion of the chain. In the mathematical model of the general chain, displacement is simulated by means of stretch-rotation tensors operating on each member vector. Appropriate mathematical constraints render the general chain and its mathematical model equivalent to a particular mechanism. With this approach and by taking derivatives, first, second, and higher-order loop equations can be developed which form the basis for a general method of spatial kinematic synthesis, applicable to path, function and motion generation (body guidance) with first, second, and higher-order as well as for combined “point-order” approximations.

Normal Vector Based Tooth Surface Generation of Helical Noncircular Gear

2012 ◽  
Vol 538-541 ◽  
pp. 2739-2744
Author(s):  
Jing Lin
Keyword(s):  
Mathematical Model ◽  
Normal Direction ◽  
Surface Generation ◽  
Tooth Surface ◽  
Normal Vector ◽  
Tooth Surfaces ◽  
General Mathematical Model

a general mathematical model for describing tooth surfaces of a helical noncircular gear is established. Normal vector of tooth surface is introduced to deal with the model. First the normal direction angles and modulus are solved and then tooth surface could be generated directly, logically and systematically. Some examples are given to illustrate the application of the proposed mathematical model.

Fourth-Order Kinematic Synthesis for Face-Milling Spiral Bevel Gears With Modified Radial Motion (MRM) Correction

2005 ◽  
Vol 128 (2) ◽  
pp. 457-467 ◽  
Author(s):  
Pei-Yu Wang ◽  
Zhang-Hua Fong
Keyword(s):  
Fourth Order ◽  
Face Milling ◽  
Bevel Gear ◽  
Radial Motion ◽  
Numerical Control ◽  
Tooth Surface ◽  
Kinematic Synthesis ◽  
Bevel Gears ◽  
Tooth Surfaces ◽  
Spiral Bevel

The use of a fourth-order motion curve is proposed by Stadtfeld and Gaiser to reduce the running noise of a bevel gear set recently. However, the methodology of synthesizing the tooth surfaces was not clearly shown in the literature. In this work, we proposed a methodology to synthesize the mating tooth surfaces of a face-milling spiral bevel gear set transmitting rotations with a predetermined fourth-order motion curve and contact path. A modified radial motion (MRM) correction in the machine plane of a computer numerical control (CNC) hypoid generator is introduced to modify the pinion tooth surface. With MRM correction, an arbitrary predetermined contact path on the pinion tooth surface with predetermined fourth-order motion curve can be achieved. Parameters of MRM correction are calculated according to the predetermined contact path and motion curve. As shown by the numerical examples, the contact path and the motion curve were obtained as expected by applying the MRM correction. The results of this work can be applied to the pinion, which is generated side-by-side (for example, fixed setting method, formate method, and Helixform method) and can be used as a basis for further study on the motion curve optimizations.

Mathematical Model of Klingelnberg Cyclo-Palloid Hypoid Gear

2013 ◽  
Vol 341-342 ◽  
pp. 572-576 ◽  
Author(s):  
Jin Fu Du ◽  
Zong De Fang ◽  
Min Xu ◽  
Xing Long Zhao ◽  
Yu Min Feng
Keyword(s):  
Mathematical Model ◽  
Contact Analysis ◽  
Tooth Surface ◽  
Hypoid Gear ◽  
Tooth Contact Analysis ◽  
Tooth Contact ◽  
Hypoid Gears ◽  
Tooth Surfaces ◽  
Normal Vectors

The geometry of the tooth surface is important for tooth contact analysis, load tooth contact analysis and the ease-off of gear pairs. This paper presents a mathematical model for the determination of the tooth geometry of Klingelnberg face-hobbed hypoid gears. The formulation for the generation of gear and pinion tooth surfaces and the equations for the tooth surface coordinates are provided in the paper. The surface coordinates and normal vectors are calculated and tooth surfaces and 3D tooth geometries of gear and pinion are obtained. This method may also applied to other face-hobbing gears.

Concave modifications of tooth surfaces of beveloid gears with crossed axes

2018 ◽  
Vol 233 (4) ◽  
pp. 1411-1425 ◽  
Author(s):  
Siyuan Liu ◽  
Chaosheng Song ◽  
Caichao Zhu ◽  
Qi Fan
Keyword(s):  
Film Thickness ◽  
Contact Ratio ◽  
Tooth Contact Analysis ◽  
Transmission Errors ◽  
Maximum Value ◽  
Contact Patterns ◽  
Beveloid Gears ◽  
Minimum Film Thickness ◽  
Tooth Surfaces ◽  
Crossed Axes

The mathematical models of the beveloid gear surfaces with different scenarios of combinations of profile concave modification and lead crowning are derived. Four schemes of modifications were proposed for beveloid gears with crossed axes. Tooth contact analysis is developed to study the influences of different schemes of concave modifications on the mesh behaviors including film thickness, transmission errors, contact ratio, root stresses, and contact patterns. Comparison of the contact characteristics of a beveloid gear drive with and without concave modifications is conducted. The results show that all the concave modification schemes can increase the area of contact patterns and decrease the maximum value of contact stresses, while the minimum film thickness can be increased. For the scheme i.e. the pinion with profile crowning modification and gear with profile concave modification, the contact ratio increases firstly then decreases to a relative lower value. Also, the root stresses are increased obviously. For the scheme for pinion without modification and gear with lead concave modification and the scheme for both pinion and gear with lead concave modification, the transmission errors are decreased slightly. The scheme for the pinion with combined crowning modification and gear with combined concave modification shows the largest improvement for the mesh behaviors in terms of the transmission errors and contact patterns where an almost contact condition can be found for the crossed beveloid gear pair.

Mathematical Model and Surface Deviation of Cylindrical Gears With Curvilinear Shaped Teeth Cut by a Hob Cutter

2004 ◽  
Vol 127 (5) ◽  
pp. 982-987 ◽  
Author(s):  
Jui-Tang Tseng ◽  
Chung-Biau Tsay
Keyword(s):  
Mathematical Model ◽  
Tooth Surface ◽  
Design Parameters ◽  
Cutting Mechanism ◽  
Cylindrical Gear ◽  
Cylindrical Gears ◽  
Tooth Gear ◽  
Surface Deviations ◽  
Surface Deviation ◽  
Cutter Design

The generating motion of a cylindrical gear with curvilinear shaped teeth cut by a CNC hobbing machine is proposed. On the basis of the cutting mechanism and the gear theory, the surface equation of this type of gear is developed as a function of hob cutter design parameters. Computer graphs of the curvilinear-tooth gear are presented based on the developed gear’s mathematical model, and the tooth surface deviations due to machine-tool settings with nominal radius of circular tooth traces are also investigated.

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