Zhang-Hua Fong

Computer-aided manufacturing of spiral bevel and hypoid gears with minimum surface-deviation

The linear regression method to minimize the deviations of a real cut gear-tooth-surface is investigated in this paper. Based on the Gleason hypoid gear generator, a mathematical model of the tooth surface is proposed. Applying the proposed mathematical model, the sensitivities of tooth surface due to the variations of machine-tool setting are also investigated. The corrective machine-tool settings, calculated by using the sensitivity matrix and the linear regression method, are used to minimize the tooth-surface deviations. The minimization problem was solved by using the singular value decomposition (SVD) method. The result of this paper can improve the conventional development process and be also applied to different manufacturing machines and methods. Two examples are presented to demonstrate the proposed methodology.

Computer-aided manufacturing of spiral bevel and hypoid gears by applying optimization techniques

Abstract A mathematical model of an ideal spiral bevel and hypoid gear-tooth surfaces based on the Gleason hypoid gear generator mechanism is proposed. Using the proposed mathematical model, the tooth surface sensitivity matrix to the variations in machine–tool settings is investigated. Surface deviations of a real cut pinion and gear with respect to the theoretical tooth surfaces are also investigated. An optimization procedure for finding corrective machine–tool settings is then proposed to minimize surface deviations of real cut pinion and gear-tooth surfaces. The results reveal that surface deviations of real cut gear-tooth surfaces with respect to the ideal ones can be reduced to only a few microns. Therefore, the proposed method for obtaining corrective machine–tool settings can improve the conventional development process and can also be applied to different manufacturing machines and methods for spiral bevel and hypoid gear generation. An example is presented to demonstrate the application of the proposed optimization model.

MATHEMATICAL MODEL OF SPIRAL BEVEL AND HYPOID GEARS MANUFACTURED BY THE MODIFIED ROLL METHOD

Based on grinding mechanism and machine-tool settings of the Gleason modified roll hypoid grinder, a mathematical model for the tooth geometry of spiral bevel and hypoid gears is developed. Since all the machine-tool settings and machine constants are involved in the proposed mathematical model, excellent correlation between the mathematical model and actual manufacturing machines can be obtained. An example is given to illustrate the application of the proposed mathematical model. Surface deviation between a real cut spiral bevel gear surface and the model surface is investigated. Bearing contacts and kinematic errors in the spiral bevel gear set are also studied.

Kinematical Optimization of Spiral Bevel Gears

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.

The Undercutting of Circular-Cut Spiral Bevel Gears

Undercutting is a serious problem in designing spiral bevel gears with small numbers of teeth. Conditions of undercutting for spiral bevel gears vary with the manufacturing methods. Based on the theory of gearing [1], the tooth geometry of the Gleason type circular-cut spiral bevel gear is mathematically modeled. The sufficient and necessary conditions for the existence and regularity of the generated gear tooth surfaces are investigated. The conditions of undercutting for a circular-cut spiral bevel gear are defined by the sufficient conditions of the regular gear tooth surface. The derived undercutting equations can be applicable for checking the undercutting conditions of spiral bevel gears manufactured by the Gleason Duplex Method, Helical Duplex Method, Fixed Setting Method, and Modified Roll Method. An example is included to illustrate the application of the proposed undercut checking equations.

Evaluating the inter-lobe clearance of twin-screw compressor by the iso-clearance contour diagram (ICCD)

A mathematical procedure is proposed to calculate the inter-lobe clearance between two mating screw rotors, and then represent the clearance field by iso-clearance contour diagram (ICCD). The theory of gearing and the tooth contact analysis (TCA) have been applied to solve the geometrical and kinematic relations of mating rotors. The contact line and the approximate blowhole can be obtained by TCA. However, it is insufficient to describe the inter-lobe leakage. By inspecting the ICCD, we can find all possible inter-lobe leakage paths, not just the contact line and the blowhole. The benefit of utilizing ICCD is that it can also be modeled by a single mathematical model such as the cubic spline interpolation instead of the mathematical model of the multi-segment tooth profiles. Hence, this method avoids the problem of discontinuity and divergence in optimal programming.

Novel Third-Order Correction for a Helical Gear Shaping Cutter Made by a Lengthwise-Reciprocating Grinding Process

Although the Isoform® lengthwise-reciprocating grinding process is considered as one of the most accurate methods for generating the tooth profile geometry of a helical gear shaping cutter, the tooth profile accuracy produced by the Isoform® with a straight cone grinding wheel is not accurate enough for high precision requirement. That is why the shaper cutter is used as a rough cutting tool for most cases. A third-order profile correction to the cone grinding wheel is proposed to increase the accuracy of the work gear profile. A novel topography is developed to schematically show the work gear tooth profile accuracy cut by a resharpened shaping cutter. The profile errors corresponding to the varied resharpening depth are shown in the topography with information of true involute form diameter and semitopping depth. The usable resharpening depth of the shaping cutter can be determined by this topography. The numerical result indicates that third-order correction reduces the profile error of the major cutter enveloping gear to submicro and extends the resharpening depth.