Mathematical Model for Face-Hobbed Straight Bevel Gears

2012 ◽  
Vol 134 (9) ◽  
Author(s):  
Yi-Pei Shih
Keyword(s):  
Mathematical Model ◽  
Bevel Gear ◽  
Tooth Contact Analysis ◽  
Rolling Circle ◽  
High Productivity ◽  
Bevel Gears ◽  
Straight Line ◽  
Base Circle ◽  
Straight Line Mechanism ◽  
Straight Lines

Face hobbing, a continuous indexing and double-flank cutting process, has become the leading method for manufacturing spiral bevel gears and hypoid gears because of its ability to support high productivity and precision. The method is unsuitable for cutting straight bevel gears, however, because it generates extended epicycloidal flanks. Instead, this paper proposes a method for fabricating straight bevel gears using a virtual hypocycloidal straight-line mechanism in which setting the radius of the rolling circle to equal half the radius of the base circle yields straight lines. This property can then be exploited to cut straight flanks on bevel gears. The mathematical model of a straight bevel gear is developed based on a universal face-hobbing bevel gear generator comprising three parts: a cutter head, an imaginary generating gear, and the motion of the imaginary generating gear relative to the work gear. The proposed model is validated numerically using the generation of face-hobbed straight bevel gears without cutter tilt. The contact conditions of the designed gear pairs are confirmed using the ease-off topographic method and tooth contact analysis (TCA), whose results can then be used as a foundation for further flank modification.

A Lengthwise Modification for Face-Hobbed Straight Bevel Gears

2013 ◽  
Author(s):  
Yi-Pei Shih
Keyword(s):  
Tool Path ◽  
Point Contact ◽  
Contact Condition ◽  
Tooth Contact Analysis ◽  
High Productivity ◽  
Accuracy Requirement ◽  
Bevel Gears ◽  
Tooth Surfaces ◽  
Gear Contact ◽  
Straight Line Mechanism

Face hobbing has been successfully applying in manufacturing straight bevel gears using a virtual hypocycloidal straight-line mechanism. This method is a continuous indexing and double-flank cutting process, and is recognized its high productivity and precision. In order to improve gear contact condition, three types of flank modifications are frequently used in gear industry: profile crowning, lengthwise crowning, and longitudinal twist. In the design of the spiral bevel and hypoid gear, under satisfying a specified accuracy requirement, three types of modifications are blended properly during gear design to absorb assembly and manufacture errors. Circular cutter blades are normally adopted to accomplish the first type modification. The second can be achieved by a cutter radius change or a cutter tilt with adjusted pressure angles. The last can be achieved by a cutter tilt or modified tool path (for example, helical motion and modified roll). This paper proposes a lengthwise crowning method for face-hobbed straight bevel gear (SBG) using a hypocycloidal mechanism. This modification is applied to the pinion only. A numerical example drive with point-contact tooth surfaces is adopted to validate the proposed mathematical model. Finally, two evaluations, ease-off topography and tooth contact analysis (TCA), are made to investigate the contact condition of this numerical case.

Kinematical Optimization of Spiral Bevel Gears

1992 ◽  
Author(s):  
Zhang-Hua Fong ◽  
Chung-Biau Tsay
Keyword(s):  
Mathematical Model ◽  
Bevel Gear ◽  
Tooth Surface ◽  
Spiral Bevel Gear ◽  
Tooth Contact ◽  
Spiral Bevel Gears ◽  
Bevel Gears ◽  
Contact Patterns ◽  
Spiral Bevel ◽  
Kinematic Errors

Abstract Kinematical optimization and sensitivity analysis of circular-cut spiral bevel gears are investigated in this paper. Based on the Gleason spiral bevel gear generator and EPG test machine, a mathematical model is proposed to simulate the tooth contact conditions of the spiral bevel gear set. All the machine settings and assembly data are simulated by simplified parameters. The tooth contact patterns and kinematic errors are obtained by the proposed mathematical model and the tooth contact analysis techniques. Loaded tooth contact patterns are obtained by the differential geometry and the Hertz contact formulas. Tooth surface sensitivity due to the variation of machine settings is studied. The corrective machine settings can be calculated by the sensitive matrix and the linear regression method. An optimization algorithm is also developed to minimize the kinematic errors and the discontinuity of tooth meshing. According to the proposed studies, an improved procedure for development of spiral bevel gears is suggested. The results of this paper can be applied to determine the sensitivity and precision requirements in manufacturing, and improve the running quality of the spiral bevel gears. Two examples are presented to demonstrate the applications of the optimization model.

Reverse on the Spiral Bevel Gear’s Teeth Profile Line

2011 ◽  
Vol 308-310 ◽  
pp. 1596-1599
Author(s):  
Chun Xiang Wang ◽  
Jing Qiang Zhang ◽  
Zhi Jun Liu
Keyword(s):  
Gear Tooth ◽  
Bevel Gear ◽  
Measuring System ◽  
Gear Teeth ◽  
Bevel Gears ◽  
Profile Line ◽  
Simple Processing ◽  
Base Circle ◽  
Spiral Bevel ◽  
Reverse Method

For the purposes of spiral bevel gears of cycloid gear is more and more extensive, the research on the reverse are rare phenomenon, this paper puts forward a reverse method: It’s based on three coordinates measuring system for prolate epicycloids’ bevel gear teeth profile. This method according to the forming principle of the spiral bevel gear tooth profile, with the back cone expanding the measured data of a group of good quality after simple processing, according to the involutes’ parameter equation for theoretical analysis, reverse out the base circle radius and modulus, make the bevel gear tooth profiles to reverse.

Mathematical Model of a Vertical Six-Axis Cartesian Computer Numerical Control Machine for Producing Face-Milled and Face-Hobbed Bevel Gears

2019 ◽  
Vol 142 (4) ◽  
Author(s):  
Ruei-Hung Hsu ◽  
Yi-Pei Shih ◽  
Zhang-Hua Fong ◽  
Chin-Lung Huang ◽  
Szu-Hung Chen ◽  
...  
Keyword(s):  
Mathematical Model ◽  
Correction Method ◽  
Computer Numerical Control ◽  
Bevel Gear ◽  
Numerical Control ◽  
Numerical Illustration ◽  
Bevel Gears ◽  
Machine Stiffness ◽  
The Individual ◽  
Compact Design

Abstract Prior to the development of sophisticated computer numerical control (CNC), both face milling (FM) and face hobbing (FH), the two most popular technologies for bevel gear production, required cradle-type machines with diverse and complicated mechanisms. In the last two decades, however, the gear industry has replaced these traditional machines with six-axis CNC bevel gear cutting machines that have superior efficiency and accuracy. One such machine is a vertical six-axis machine with a vertical spindle arrangement, which offers two industrially proven advantages: compact design and maximum machine stiffness. The technical details of this machine, however, remain undisclosed; so, this paper proposes a mathematical model that uses inverse kinematics to derive the vertical machine's nonlinear six-axis coordinates from those of a traditional machine. The model also reduces manufacturing errors by applying an effective flank correction method based on a sensitivity analysis of how slight variations in the individual machine setting coefficients affect tooth geometry. We prove the model's efficacy by first using the proposed equations to derive the nonlinear coordinates for pinion and gear production and then conducting several cutting experiments on the gear and its correction. Although the numerical illustration used for this verification is based only on FM bevel gears produced by an SGDH cutting system, the model is, in fact, applicable in the production of both FM and FH bevel gears.

Kinematical Optimization of Spiral Bevel Gears

1992 ◽  
Vol 114 (3) ◽  
pp. 498-506 ◽  
Author(s):  
Zhang-Hua Fong ◽  
Chung-Biau Tsay
Keyword(s):  
Mathematical Model ◽  
Bevel Gear ◽  
Tooth Surface ◽  
Spiral Bevel Gear ◽  
Tooth Contact ◽  
Spiral Bevel Gears ◽  
Bevel Gears ◽  
Contact Patterns ◽  
Spiral Bevel ◽  
Kinematic Errors

Kinematical optimization and sensitivity analysis of circular-cut spiral bevel gears are investigated in this paper. Based on the Gleason spiral bevel gear generator and EPG test machine, a mathematical model is proposed to simulate the tooth contact conditions of the spiral bevel gear set. All the machine settings and assembly data are simulated by simplified parameters. The tooth contact patterns and kinematic errors are obtained by the proposed mathematical model and the tooth contact analysis techniques. Loaded tooth contact patterns are obtained by the differential geometry and the Hertz contact formulas. Tooth surface sensitivity due to the variation of machine settings is studied. The corrective machine settings can be calculated by the sensitive matrix and the linear regression method. An optimization algorithm is also developed to minimize the kinematic errors and the discontinuity of tooth meshing. According to the proposed studies, an improved procedure for development of spiral bevel gears is suggested. The results of this paper can be applied to determine the sensitivity and precision requirements in manufacturing, and improve the running quality of the spiral bevel gears. Two examples are presented to demonstrate the applications of the optimization model.

Mathematical Model and 3D Modeling of Involute Spiral Bevel Gears for Nutation Drive

2013 ◽  
Vol 694-697 ◽  
pp. 503-506 ◽  
Author(s):  
Zheng Lin ◽  
Li Gang Yao
Keyword(s):  
Mathematical Model ◽  
3D Modeling ◽  
Bevel Gear ◽  
Transition Curve ◽  
Tooth Surface ◽  
Spiral Bevel Gear ◽  
Spiral Bevel Gears ◽  
Bevel Gears ◽  
Spiral Bevel ◽  
The Mathematical Model

The mathematical model and 3D modeling of involute spiral bevel gears for nutation drive are considered. The basic tooth profile of involute is composed of involute curve and dedendum transition curve, and the equations have been established. The mathematical model of crown gear with involute profile is obtained, and then the mathematical models of the involute spiral bevel gears are developed. The tooth surface modeling of involute spiral bevel gear is proposed, and the 3D modeling of the involute spiral bevel gear for nutation drive is illustrated.

Actual Tooth Contact Analysis of Straight Bevel Gears

2015 ◽  
Vol 137 (9) ◽  
Author(s):  
M. Kolivand ◽  
H. Ligata ◽  
G. Steyer ◽  
D. K. Benedict ◽  
J. Chen
Keyword(s):  
Transmission Error ◽  
Contact Analysis ◽  
Bevel Gear ◽  
Tooth Contact Analysis ◽  
Third Order ◽  
Tooth Contact ◽  
New Approach ◽  
Bevel Gears ◽  
The Difference ◽  
Finishing Processes

Theoretically, spherical involutes are used as one of the base topographies for straight bevel gears. Actual bevel gears, however, have deviations from their intended topographies due to manufacturing errors, heat treatment deviations, and finishing processes. Measuring the physical parts with coordinate measuring machines (CMMs), this study proposes a new approach to capture such deviations. The measured deviations from spherical involute are expressed in form of a third-order two-dimensional (2D) polynomial function and added to the base topography to duplicate the geometry of the actual part; tooth thickness deviation is also accounted for and corrected through changing the theoretical tooth thickness. The resultant surfaces are then used to construct ease-off and surface of roll angle topographies and to perform tooth contact analysis (TCA) and calculate motion transmission error (TE). At the end a sample straight bevel gear set is measured and utilizing the proposed approach its predicted TCA is compared to the experimental TCA obtained from roll tester. The results show very good correlation between the predicted and actual TCA of the parts. Utilizing the proposed methodology, the other bevel gear base profile geometries (such as octoids) can also be analyzed. In the proposed approach, the difference between other base geometries and spherical involutes can be treated as deviations from spherical involutes and can be taken into account to perform TCA.

scholarly journals Digitized Modeling and Machining Simulation of Equal Base Bevel Gear

2018 ◽  
Vol 17 ◽  
pp. 01015
Author(s):  
Bin Wang ◽  
Yunpeng He ◽  
Peiyao Feng ◽  
Xun Sun ◽  
Aijun Xu ◽  
...  
Keyword(s):  
Coordinate System ◽  
Gear Tooth ◽  
Nc Machining ◽  
Bevel Gear ◽  
Tooth Surface ◽  
Automatic Programming ◽  
Machining Simulation ◽  
Bevel Gears ◽  
Base Circle ◽  
Standard Cutter

In order to achieve the NC machining to the equal base circle bevel gear with a standard cutter, to plan the machining path efficiently and to predict machining interference, based on the equal base circle bevel gear theory, the machining coordinate system for the bevel gear was established, according to gear tooth surface equations of the bevel gears, the tooth surface discretization function was derived, and through the calculation and get the discrete points, the gear surface was modeled in the software UG, and then the planning of the machining path, automatic programming and simulation processing were done. At last, simulation processing, analysis and comparison to the gear surface were carried on with the software VERICUT, improved that the gear surface modeling and NC machining methods were correct and feasible.

A Study on the Tooth Geometry and Cutting Machine Mechanisms of Spiral Bevel Gears

1991 ◽  
Vol 113 (3) ◽  
pp. 346-351 ◽  
Author(s):  
Z. H. Fong ◽  
Bill Chung-Biau Tsay
Keyword(s):  
Mathematical Model ◽  
Bevel Gear ◽  
Spiral Bevel Gear ◽  
Helical Motion ◽  
Spiral Bevel Gears ◽  
Bevel Gears ◽  
Gear Cutting ◽  
Cutting Machine ◽  
The Matrix ◽  
Spiral Bevel

The tooth geometry and cutting machine mechanisms of spiral bevel gears are investigated. Based on the kinematics of titled head cutter, machine cradle, sliding base and work head, the matrix presentation of spiral bevel gear’s tooth geometry are developed. The relations between the parameters of the proposed mathematical model and the machine settings of existing spiral bevel gear cutting machines are also investigated. The tilt of head cutter axis, motion of generation, helical motion of sliding base, and nongenerating cutting of spiral bevel gears are taken into consideration. An example is given to illustrate the application of the proposed mathematical model.

A Mathematical Model for the Tooth Geometry of Circular-Cut Spiral Bevel Gears

1991 ◽  
Vol 113 (2) ◽  
pp. 174-181 ◽  
Author(s):  
Z. H. Fong ◽  
Chung-Biau Tsay
Keyword(s):  
Mathematical Model ◽  
Cnc Machining ◽  
Tooth Contact Analysis ◽  
Spiral Bevel Gears ◽  
Bevel Gears ◽  
Finite Element Stress Analysis ◽  
Tool Setting ◽  
Spiral Bevel ◽  
The Mathematical Model ◽  
Made In

A complete tooth geometry of the circular-cut spiral bevel gears has been mathematically modeled. The mathematical model has been divided into several independent modules, each representing an individual kinematic relation or tool-setting, with examples included. A comparison with the spiraloid model has also been made in this paper. The mathematical model can be applied to simulate and calculate the tooth profiles for the Duplex Method, Helical Duplex Method, Formate Method, and Modified Roll Method for circular-cut spiral bevel gears. It can also be applied to the computer numerical controlled (CNC) machining, computer-aided finite element stress analysis, and tooth contact analysis (TCA) for the spiral bevel gear.

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