The tooth surface of a curvilinear-tooth gear was known in past years. Recently, there are several successful generating methods to fabricate the curvilinear-tooth gear.
Liu [1] proposed the manufacture of the cylindrical gear with curvilinear shaped teeth by using a face mill-cutter with a special machine, and stated that the merits of curvilinear-tooth gears include higher bending strength, lower noise, better lubrication effect, and no axial thrust force. Dai et al. [2] proposed the manufacture of a cylindrical gear with curved teeth by using a CNC hobbing machine with an attachment for the hob head, male and female flying cutters. Andrei et al. [3]
developed a special cutting tool to generate the curved face width gears for non-metallic materials. As to theoretically study on the curvilinear-tooth gear, Tseng and Tsay [4] considered an imaginary rack cutter with a curved-tooth to develop the mathematical model of cylindrical gears with curvilinear shaped teeth, and investigated the tooth undercutting of curvilinear-tooth gears.
Owing to easy tool settings, high efficiency and reliable quality, hob cutters have been widely used for manufacturing a variety of gears such as spur, helical, and worm gears. A hob cutting mechanism is a mechanism with multiple degrees of freedom in the process of gear generation. Chang et al. [5] proposed a general gear mathematical
model simulating the generation process of a 6-axis CNC hobbing machine when the hob’s swivel axis is fixed. Fang and Tsay [6] applied an oversized hob cutter to generate the worm gear and investigated the bearing contacts of the ZN-type worm gear set. The geometry of a hob and generating simulation of cylindrical gears are proposed by Kim [7]. However, the tooth surfaces of spur, and helical gear, cut by hobbing with a feed motion, are one parameter enveloping. The necessary and sufficient conditions of the envelope for a two-parameter family of surfaces were proposed by Litvin and Seol [8]. They applied the developed theory to study the generation of helical gears by a ground involute worm.
As is well known, gears with tooth undercutting may decrease the load capacity of a gear pair. Kin [9] applied the contact-line envelope and the pressure-angle limit concepts to prevent the tooth undercutting of worm and worm gear surfaces. Fong and Tsay [10] utilized surface unit normal vectors to investigate the tooth undercutting of spiral bevel gears. Bair and Tsay [11] evaluated the undercutting line of a ZK-type dual-lead worm gear set by searching the zero unit-normal vectors on the tooth surfaces.
An important contribution to the avoidance of tooth undercutting was made by Litvin [12-14] who proposed a detailed investigation on the condition of tooth nonundercutting by considering the relative velocities and the differential equation of meshing. The approach proposed by Litvin to predict tooth undercutting has been applied to study the undercutting phenomenon of various types of gears. The tooth undercutting analysis for noncircular gears generated by shaper cutters was investigated by Chang and Tsay [15]. Liu and Tsay [16] applied Litvin’s method to study the tooth undercutting of beveloid gears. Chen and Tsay [17] presented a mathematical model for modified circular arc helical gears generated by an imaginary
circular arc rack cutter and discussed the tooth undercutting of the proposed helical gears.
Determination of principal curvatures and directions for conjugate surfaces is an important issue in tooth surface design. It can be applied to evaluate the contact deformation, contact stress [18], and minimum oil film thickness of lubricant of the meshing gears [19]. Litvin and Gutman [20] proposed a local synthesis method based on the equations relating the principal curvatures of the two mating surfaces and additional conditions for providing the contact path coincided with a geodesic line on the gear tooth surface. Colbourne [21] constructed the Mohr’s circle to represent the curvature at any point of a helicoids, assuming the shapes and the curvatures are known in the transverse section. Litvin et al. [22] proposed a general approach for determination of the principal curvatures and directions of two surfaces being in continuous tangency along a line at every instant. Kang and Yan [23] derived the equations for evaluating the curvature of variable pitch lead screw transmission mechanism based on the theory of gearing and curvature theory. They further applied the relative normal curvatures to discuss tooth undercutting on the surfaces of variable pitch lead screw. Yan and Cheng [24] presented the general equations of curvature analysis for spatial cam-follower mechanisms with various type of motions.
The shape and level of kinematic errors (KE) induced by gear axial misalignments are the important and efficient factors to predict the noise and vibration of a mating gear pair. If the KE is a discontinuous function, the jumps of gear angular velocity will be induced. Tsay [25] applied tooth contact analysis (TCA) techniques to simulate the meshing conditions for involute helical gears and proposed the compensation method to reduce the KEs induced by horizontal axial-misalignments.
Liu and Tsay [26] investigated the KE, bearing contact and contact ellipse for
beveloid gears with intersected, crossed and parallel axes. Tseng and Tsay [27] studied the contact characteristics of cylindrical gears with curvilinear shaped teeth generated by a rack cutter with a curved-tooth.
To reduce the jumps of gear angular velocity, Litvin et al., [28-32] developed a method for localization of bearing contacts for various gear pairs, and proposed a surface synthesis method by utilizing a predesigned parabolic kinematic-error function to absorb the transmission errors of an approximately linear function of the designed gear pair induced by gear axial-misalignments. Some researchers minimized KEs of the mating gears by using the optimization method. Chen and Tsay [33]
proposed a generating method for helical gears with a pre-designed transmission error and applied the optimization technique to find the adequate gear design parameters to provide the gear pair with a prescribed transmission error. Fong and Tsay [34] studied the sensitivity of the tooth profile of spiral bevel gears due to machine settings, and minimized the KEs of the mating gears. Chang et al. [35] discussed the kinematic optimization for a modified helical gear train.
The contact analysis of the mating gears with a load is much more realistic.
Zhang and Fang [36-37] considered the elastic deformation of tooth surfaces to estimate the transmission errors of helical gears under a load. Umeyama et al. [38]
investigated the loaded transmission errors of helical gears and the relationship between the actual contact ratio and effective contact ratio.
1.3 Motivation
The curvilinear-tooth gear generated by a hob cutter is a new generating method.
The generating method considered the cutting mechanism of a 6-axis CNC hobbing machine with multiple degrees of freedom may result in complex tooth surfaces. The
problems of tooth undercutting and secondary tooth cutting in the new generating process need to overcome. The important work for curvature analysis was proposed by Litvin. He developed a simplified algorithm for computerized determination of principal and normal curvatures of complex gear tooth surfaces. However, the case for which the surfaces as the envelope of two-parameter family of surfaces was not considered.
In this study, a complete mathematical model of the curvilinear-tooth gear cut by a hob cutter is developed firstly, and then the theoretically analyses on tooth undercutting and secondary cutting in the generating process are also investigated. An approach to evaluate principal curvatures and directions for the surfaces of curvilinear-tooth gear is proposed. Besides, the contact characteristics such as contact path, KE, dimension and orientation of contact ellipses of the curvilinear-tooth gear pair under assembly errors are also investigated by utilizing TCA technology. Finally, the finite element model is constructed to investigate the contact and bending stresses of the curvilinear-tooth gears.
1.4 Overview
This dissertation totally includes seven chapters. Chapter 1 is the introduction to the contents of the thesis that contains the feature of curvilinear-tooth gear, reviews of related literatures, research background and the motivation of this thesis.
Chapter 2 derives the mathematical model for ZN worm-type hob cutter surfaces.
According to the cutting mechanism of a CNC hobbing machine, the kinematic relationship between the hob cutter and work piece can be obtained. The mathematical model of the curvilinear-tooth gear hobbing simulation for a 6-axis CNC hobbing machine can be developed based on the proposed cutting mechanism,
generation concept with multiple degrees of freedom, and theory of mechanisms.
Using computer graphics, a three-dimensional tooth surface of curvilinear-tooth gears can be plotted. In addition to developing a mathematical model for cylindrical gears with curvilinear shaped teeth cut by a hob cutter, this chapter investigates the relationship between tooth surface deviations and nominal radius of circular arc tooth traces.
In Chapter 3, tooth undercutting of the curvilinear-tooth gear surfaces are investigated by considering the singularity of the generated tooth surfaces proposed by Litvin [12]. The kinematic method to calculate the differentiated equations of meshing is developed for analyzing tooth undercutting. Numerical examples are presented to demonstrate the tooth undercutting. Due to the geometry character of the hob cutter, The secondary cutting of the gear tooth surface by the hob cutter occurs when the surface of the curvilinear-tooth gears with a small nominal radius of circular arc tooth trace are generated by the hob cutter with a larger outside diameter. To find the relationship between the outside diameter of the hob cutter and the nominal radius of circular arc tooth trace without secondary generating, a computer algorithm for investigation of the secondary cutting is also developed.
Chapter 4 presents investigations on the curvature analysis. An algorithm for computerized determination of principal curvatures and directions of the curvilinear-tooth gear with two-parameter family of surfaces is proposed. Rodrigues’
equation and two differential meshing equations are considered to establish the curvature relationship between the generating surface and the generated surface.
Some illustrative numerical examples are presented to investigate the principal curvatures and directions of the curvilinear-tooth gear surface.
In Chapter 5, the tooth contact analysis technique is applied to find the contact
characteristic of the curvilinear-tooth gear pair such as bearing contacts, kinematic errors, and contact ellipses. Tooth contact simulation model including horizontal axial misalignment, vertical axial misalignment, and center distance variation of the gear pair is established. In this thesis, two methods are used to evaluate the contact ellipses.
Several numerical examples are presented to demonstrate the influence of the assembly errors and gear design parameters on the kinematic errors and contact ellipses of the mating gear pair.
Chapter 6 investigates the contact stress and the bending stress of the proposed curvilinear-tooth gear pair by using the commercial software, ABAQUS/Standard.
Firstly, an automatic mesh-generation program is developed to discretize the three-dimensional tooth model. Then, the finite element models are set up by constructing the finite element meshes, setting the material properties, defining the contact surface, and applying the boundary conditions for loading the gear drive with the desired torque. The input file for ABAQUS computation is generated automatically by a computer program. Some numerical examples are presented to demonstrate the tooth stress with different gear design parameters.
Chapter 7 concludes this study by summarizing the major findings of the accomplished work, and also discusses potential subjects for future study.
CHAPTER 2
Mathematical Model of Cylindrical Gears with Circular Arc Tooth Traces
2.1 Introduction
The hobbing is an economical method for gear manufacturing due to its versatility and high efficiency. Hobbing method can be used to cut various types of gears such as spur, helical, and worm gears. A hob cutter with a given pitch can generate the tooth surface of all involute spur and helical gears with the same normal pitch and pressure angle, including all numbers of teeth and helix angles. Different gear tooth profiles can be generated on the same CNC machine by changing the profile of the hob.
The hobbing process is complicated because of the generating motion with multi-degree of freedom. The tooth surface equations such as spur, helical, and worm can be derived as the envelop to the one-parameter family of hob surfaces. It is not easy to manufacture curvilinear-tooth gears by using an ordinary hobbing machine because the motion of the hob cutter is determined with two independent parameters.
In addition to a rotational motion about the hob’s spindle, the hob rotates with an angular velocity ω about the hob’s swivel axis, and the hob’s swivel translates with a velocity along the worktable axis as the curvilinear-tooth gear is generated. It is noted that the linear velocity is correlated with angular velocity
v
v ω .
In this chapter, we derived tooth surface equation of the curvilinear-tooth gear is derived based on the cutting mechanism of the 6-axis CNC hobbing machine, generation concept with multiple degrees of freedom, and theory of mechanisms.
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