Faydor L. Litvin

Gear Geometry and Applied Theory: Spiral Bevel Gears

INTRODUCTION Spiral bevel gears have found broad application in helicopter and truck transmissions and reducers for transformation of rotation and torque between intersected axes. Design and stress analysis of such gear drives has been a topic of research by many scientists including the authors of this book [Krenzer, 1981; Handschuh & Litvin, 1991; Stadtfeld, 1993, 1995; Zhang et al ., 1995; Gosselin et al ., 1996; Litvin et al ., 1998a, 2002a; Argyris et al ., 2002; Fuentes et al ., 2002]. Reduction of noise and stabilization of bearing contact of misaligned spiral bevel gear drives are still very challenging topics of research although manufacturing companies [Gleason Works (USA), Klingelnberg–Oerlikon (Germany–Switzerland)] have developed skilled methods and outstanding equipment for manufacture of such gear drives. The conditions of meshing and contact of spiral bevel gears depend substantially on the machine-tool settings applied. Such settings are not standardized but have to be determined for each case of design, depending on the parameters of the gears and generating tools, to guarantee the required quality of the gear drives. This chapter covers an integrated approach for the design and stress analysis of spiral bevel gears that has been developed by the authors of the book and their associates. The approach provides the solution to the following problems: Determination of machine-tool settings for generation of low-noise stable bearing contact spiral bevel gear drives. Computerized analysis of meshing and contact of gear tooth surfaces. […]

Gear Geometry and Applied Theory: Planetary Gear Trains

INTRODUCTION Planetary gear trains were the subject of intensive research directed at determination of dynamic response of the trains, vibration, load distribution, efficiency, enhanced design, and other important topics [Lynwander, 1983; Ishida & Hidaka, 1992; Kudrjavtzev et al ., 1993; Kahraman, 1994; Saada & Velex, 1995; Chatterjee & Tsai, 1996; Hori & Hayashi, 1996a, 1996b; Velex & Flamand, 1996; Lin & Parker, 1999; Chen & Tseng, 2000; Kahraman & Vijajakar, 2001; Litvin et al ., 2002e]. This chapter covers gear ratio, conditions of assembly, relations of tooth numbers, efficiency of a planetary train, proposed modification of geometry of tooth surfaces, determination of transmission errors, etc. Special attention is given to the regulation of backlash for improvement of load distribution. GEAR RATIO A planetary gear mechanism has at least one gear whose axis is movable in the process of meshing. Planetary Mechanisms of Figs. 23.2.1 (a) and (b) Figures 23.2.1(a) and (b) represent two simple planetary gear mechanisms formed by two gears 1 and 2 that are in external or internal meshing, respectively, and a carrier c on which the gear with the movable axis is mounted. Gear 1 is fixed and planet gear 2 performs a planar motion of two components: (i) transfer rotation with the carrier, and (ii) relative rotation about the carrier.

Gear Geometry and Applied Theory: Involute Helical Gears with Crossed Axes

INTRODUCTION Involute helical gears are widely applied in the industry for transformation of rotation between parallel and crossed axes. Figure 16.1.1 shows an involute helical gear drive with crossed axes in 3D-space. A gear drive formed by a helical gear and a worm gear is a particular case of a gear drive with crossed axes (Figure 16.1.2). Gear tooth surfaces are in line contact for involute helical gear drives with parallel axes and in point contact for involute helical gear drives with crossed axes. The theory of involute gears and research in this area have been presented by Litvin [1968], Colbourne [1987], Townsend [1991], and Litvin et al . [1999, 2001a, 2001c, 2001d] and the theory of shaving and honing technological processes are discussed in the works of Townsend [1991] and Litvin et al . [2001a]. Despite the broad investigation that has been accomplished in this area, the quality of misaligned involute helical gear drives is still a concern of manufacturers and designers. The main defects of such misaligned gear drives are (i) appearance of edge contact, (ii) high levels of vibration, and (iii) the shift of the bearing contact far from the central location. To overcome the defects mentioned above, some corrections of gear geometry have been applied in the past: (i) correction of the lead angle of the pinion (requires regrinding), and (ii) crowning in the areas of the tip of the profile and the edge of the teeth (based on the experience of manufacturers).

Gear Geometry and Applied Theory: Face-Gear Drives

INTRODUCTION A conventional face-gear drive is formed by an involute spur pinion and a conjugated face-gear (Fig. 18.1.1). Such a gear drive may be applied for transformation of rotation between intersected and crossed axes. An important example of application of a face-gear drive with intersected axes is in the helicopter transmission (Fig. 18.1.2). The manufacturing of face-gears by a shaper was invented by the Fellow Corporation. The basic idea of generation is based on simulation of meshing of the generating shaper with the face-gear being generated as the meshing of the pinion of the drive with the face-gear. In the process of generation, the surfaces of the teeth of the shaper and the face-gear are in line contact at every instant. However, when the shaper is exactly identical to the pinion of the face-gear drive, the generated face-gear drive becomes sensitive to misalignment. This causes an undesirable shift of the bearing contact and even separation of the surfaces. Therefore, it is necessary to provide an instantaneous point contact between the tooth surfaces of the pinion and the face-gear instead of a line contact. Then, the bearing contact will be localized and the face-gear drive will be less sensitive to misalignment. Point contact between the pinion and face-gear tooth surfaces is provided by application of a shaper of number of teeth N s > N p where N p is the number of teeth of the pinion of the drive (see Section 18.4).

Gear Geometry and Applied Theory: Worm-Gear Drives with Cylindrical Worms

INTRODUCTION There are two types of worm-gear drives: (i) those with cylindrical worms (Fig. 19.1.1) (single-enveloping worm-gear drives), and (ii) those with hourglass worms (see Chapter 20) (double-enveloping worm-gear drives). The terms “single-enveloping” and “double-enveloping” are confusing because in both cases the surface of the worm-gear tooth is the envelope to the one-parameter family of worm thread surfaces that are generated in the coordinate system rigidly connected to the worm-gear. The thread surface of a cylindrical worm is a helicoid. (We recall that a helicoid is the surface that is generated by a given curve while it performs a screw motion.) This chapter covers (i) the generation and geometry of cylindrical worms, and (ii) the basic design problems (relations between design parameters). Depending on the method for generation, we differentiate henceforth the following types of cylindrical worms (see German Standards DIN 3975): (i) ZA worms, with surface A . The worm surface is a ruled surface that is generated by a straight line while it performs a screw motion with respect to the worm axis. The generating line intersects the worm axis and therefore the axial section of the worm surface is a straight line that is just the generating line. The cross section of the ZA worm is an Archimedes spiral (see Section 19.4). (ii) ZN worms, with surface N . The worm surface is also a ruled surface. However, the generating line lies in a plane that passes through the perpendicular to the worm axis and forms angle λ p with the worm axis (see Section 19.5). Here, λ p is the lead angle on the pitch cylinder on the worm. The cross section of the worm is an extended involute (see Section 19.5) […]

Gear Geometry and Applied Theory: Gear Geometry and Applied Theory

1. Coordinate Transformation. 2. Relative Velocity. 3. Centrodes. Axodes. Operating Pitch Surfaces. 4. Planar Curves. 5. Surfaces. 6. Conjugate Surfaces and Curves. 7. Curvatures of Surfaces and Curves. 8. Mating Surfaces: Curvature Relations, Contact Ellipse. 9. Computerized Simulation of Meshing and Contact. 10. External Involute Gears. 11. Internal Involute Gears. 12. Noncircular Gears. 13. Cycloidal Gearing. 14. Involute Helical Gears with Parallel Axes. 15. Helical Involute Gears with Crossed Axes. 16. Double-Circular Arc Helical Gears. 17. Face-Gear Drives. 18. Worm-Gear Drives with Cylindrical Worms. 19. Double-Enveloping Worm-Gear Drives. 20. Hypoid Gears. 21. Generation of Helicoids. 22. Design of Flyblades. 23. Generation of Surfaces by Application of Computer Numerically Controlled Machines. 24. Overwire (Ball) Measurements. 25. Coordinate Measurements and Minimization of Deviations. References.

Gear Geometry and Applied Theory: Index

1. Coordinate Transformation. 2. Relative Velocity. 3. Centrodes. Axodes. Operating Pitch Surfaces. 4. Planar Curves. 5. Surfaces. 6. Conjugate Surfaces and Curves. 7. Curvatures of Surfaces and Curves. 8. Mating Surfaces: Curvature Relations, Contact Ellipse. 9. Computerized Simulation of Meshing and Contact. 10. External Involute Gears. 11. Internal Involute Gears. 12. Noncircular Gears. 13. Cycloidal Gearing. 14. Involute Helical Gears with Parallel Axes. 15. Helical Involute Gears with Crossed Axes. 16. Double-Circular Arc Helical Gears. 17. Face-Gear Drives. 18. Worm-Gear Drives with Cylindrical Worms. 19. Double-Enveloping Worm-Gear Drives. 20. Hypoid Gears. 21. Generation of Helicoids. 22. Design of Flyblades. 23. Generation of Surfaces by Application of Computer Numerically Controlled Machines. 24. Overwire (Ball) Measurements. 25. Coordinate Measurements and Minimization of Deviations. References.

Gear Geometry and Applied Theory: Contents

1. Coordinate Transformation. 2. Relative Velocity. 3. Centrodes. Axodes. Operating Pitch Surfaces. 4. Planar Curves. 5. Surfaces. 6. Conjugate Surfaces and Curves. 7. Curvatures of Surfaces and Curves. 8. Mating Surfaces: Curvature Relations, Contact Ellipse. 9. Computerized Simulation of Meshing and Contact. 10. External Involute Gears. 11. Internal Involute Gears. 12. Noncircular Gears. 13. Cycloidal Gearing. 14. Involute Helical Gears with Parallel Axes. 15. Helical Involute Gears with Crossed Axes. 16. Double-Circular Arc Helical Gears. 17. Face-Gear Drives. 18. Worm-Gear Drives with Cylindrical Worms. 19. Double-Enveloping Worm-Gear Drives. 20. Hypoid Gears. 21. Generation of Helicoids. 22. Design of Flyblades. 23. Generation of Surfaces by Application of Computer Numerically Controlled Machines. 24. Overwire (Ball) Measurements. 25. Coordinate Measurements and Minimization of Deviations. References.