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 ◽  
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.

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.

Study on the Computer Numerical Control Process of Variable Pitch, Groove Depth and Groove Width Screw

2011 ◽  
Vol 201-203 ◽  
pp. 85-88 ◽  
Author(s):  
Min Han ◽  
Shan Li ◽  
Lu Tao Deng
Keyword(s):  
Mathematical Model ◽  
Control Process ◽  
Groove Depth ◽  
Computer Numerical Control ◽  
Cnc Machining ◽  
Numerical Control ◽  
Groove Width ◽  
Variable Pitch ◽  
Design Techniques

The article introduces three design techniques of variable pitch & groove depth & groove width screw. We found parameterized mathematical model of the type of heterotypic screw by analyzing and computing. Then it can realize to CNC machining of variable pitch & groove depth & groove width screw on lathe.

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.

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.

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.

Designing and Manufacturing Spiral Bevel Gears Using 5-Axis Computer Numerical Control (CNC) Milling Machines

2013 ◽  
Vol 135 (2) ◽  
Author(s):  
Joël Teixeira Alves ◽  
Michèle Guingand ◽  
Jean-Pierre de Vaujany
Keyword(s):  
Manufacturing Process ◽  
Computer Aided Design ◽  
Bevel Gear ◽  
Numerical Control ◽  
Spiral Bevel Gear ◽  
Milling Machine ◽  
Spiral Bevel Gears ◽  
Bevel Gears ◽  
Computer Aided ◽  
Spiral Bevel

The design of spiral bevel gears remains complex since tooth geometry and the resulting kinematic performance stem directly from the manufacturing process. Spiral bevel gear cutting up to now has relied on the works of several manufacturers. Recent advances in milling machine technology and computer aided manufacturing (CAM) now make it possible to manufacture good quality spiral bevel gears on a standard 5-axis milling machine. This paper describes the computer aided design (CAD) definition and manufacturing of spiral bevel gear tooth surfaces. Process performance is assessed by comparing the resulting surfaces after machining with the predefined CAD surfaces. This manufacturing process makes it possible to obtain geometry analytically, making design easier than with standard spiral bevel gears.

Free-Form Flank Correction in Helical Gear Grinding Using a Five-Axis Computer Numerical Control Gear Profile Grinding Machine

2012 ◽  
Vol 134 (4) ◽  
Author(s):  
Yi-Pei Shih ◽  
Shi-Duang Chen
Keyword(s):  
Correction Method ◽  
Computer Numerical Control ◽  
Numerical Control ◽  
Free Form ◽  
Profile Grinding ◽  
Wheel Profile ◽  
Grinding Machine ◽  
Tooth Flank ◽  
Five Axis ◽  
Gear Profile

To reduce form grinding errors, this paper proposes a free-form flank topographic correction method based on a five-axis computer numerical control (CNC) gear profile grinding machine. This correction method is applied not only to the five-axis machine settings (during grinding) but also to the wheel profile (during wheel truing). To achieve free-form modification of the wheel profile, the wheel is formulated as B-spline curves using a curve fitting technique and then normal correction functions made up of four-degree polynomials are added into its working curves. Additionally, each axis of the grinding machine is formulated as a six-degree polynomial. Based on a sensitivity analysis of the polynomial coefficients (normal correction functions and CNC machine settings) on the ground tooth flank and the topographic flank errors, the corrections are solved using the least squares method. The ground tooth flank errors can then be efficiently reduced by slightly adjusting the wheel profile and five-axis movement according to the solved corrections. The validity of this flank correction method for helical gears is numerically demonstrated using the five-axis CNC gear profile grinding machine.

A Novel Lengthwise Crowning Method for Face-Hobbed Straight Bevel Gears

2017 ◽  
Vol 139 (6) ◽  
Author(s):  
Yi-Pei Shih
Keyword(s):  
Mathematical Model ◽  
Numerical Control ◽  
High Productivity ◽  
Bevel Gears ◽  
Cutting Machine ◽  
Performance Indexes ◽  
Tooth Surfaces ◽  
Recent Addition ◽  
Machine Setting ◽  
The Many

A recent addition to the many milling processes used in manufacturing to cut straight bevel gears (SBGs) is a new face-hobbing (FH) method that uses a virtual hypocycloid straight-lined mechanism to produce straight-lined teeth. Despite earning much attention because of its high productivity, however, this method is unable to handle lengthwise crowning on tooth surfaces, which results in poor contact performance. This paper therefore proposes a novel lengthwise crowning method, applicable on a modern six-axis computer numerical control (CNC) bevel gear cutting machine, in which the gear blank performs a swinging motion during machining. This swinging motion is enabled by machine setting modifications, which here are derived from a mathematical model of a double (profile and lengthwise) crowned gear. After the model's correctness is confirmed using ease-off and tooth contact analyses, a final investigation examines the effect of two key parameters related to contact performance indexes whose interrelations are graphed to provide a designer reference.

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