1 Spur Gears.- 1. The Law of Gearing.- The Requirement for a Constant Angular Velocity Ratio.- Rack and Pinion.- Law of Gearing for Two Gears.- Conjugate Profiles and the Basic Rack.- 2. Tooth Profile of an Involute Gear.- Basic Involute Rack.- Standard Pitch Circle.- The Involute Tooth Profile.- The Involute Function.- Pressure Angle of a Gear.- Tooth Thickness.- Specifying a Spur Gear.- 3. Gears in Mesh.- A Pinion Meshed With a Rack.- A Pair of Gears in Mesh.- Imaginary Rack.- Fundamental Circles of a Gear.- Advantages of the Involute Profile.- 4. Contact Ratio Interference and Backlash.- Contact Ratio.- Interference.- Backlash.- 5. Gear Cutting I Spur Gears.- Form Cutting.- Shaping with a Pinion Cutter.- Advantages of Generating Cutting.- Shaping with a Rack Cutter.- Hobbing.- Cutter Tooth Tip Geometry.- Undercutting.- 6. Profile Shift.- Definition of Profile Shift.- Geometric Design of a Spur Gear Pair.- Alternative Names for Profile Shift.- 7. Miscellaneous Circles.- Highest and Lowest Points of Single-Tooth Contact.- Form Diameter.- Undercut Circle.- 8. Measurement of Tooth Thickness.- Gear-Tooth Vernier Caliper.- Span Measurement.- Measurement Over Pins.- 9. Geometry of Non-Involute Gears.- General Theory.- Fillet Shape Cut by a Rack Cutter.- Fillet Shape of an Undercut Gear.- Profile Modification.- Fillet Shape Cut by a Pinion Cutter.- 10. Curvature of Tooth Profiles.- Involute Radius of Curvature.- Euler-Savary Equation.- Gear Tooth Fillet Radius of Curvature.- 11. Tooth Stresses in Spur Gears.- Contact Force Intensity.- Contact Stress.- Fillet Stress.- 12. Internal Gears.- Tooth Profile of an Internal Gear.- Meshing Geometry of an Internal Gear Pair.- Tip Interference.- Axial and Radial Assembly.- Cutting Internal Gears.- Shape of the Fillet.- Undercutting.- Rubbing.- Geometric Design of an Internal Gear Pair.- Measurement of Tooth Thickness.- 2 Helical Gears.- 13. Tooth Surface of a Helical Involute Gear.- The Basic Helical Rack.- Standard Pitch Cylinder of a Helical Gear.- The Helix and the Involute Helicoid.- The Generator Through Point A.- Direction of the Normal to the Tooth Surface at A.- Tooth Thickness.- Span Measurement.- Tooth Profile in the Normal Section.- Specifying a Helical Gear.- 14. Helical Gears in Mesh.- A Pinion Meshed With a Rack.- A Pair of Helical Gears in Mesh.- Contact Ratio.- Backlash.- Position and Orientation of the Contact Line.- 15. Crossed Helical Gears.- Rack and Pinion.- A Crossed Helical Gear Pair.- Path of Contact.- Contact Ratio.- Backlash.- Tooth Contact Force and Bearing Reactions.- 16. Gear Cutting II Helical Gears.- Shaping with a Pinion Cutter.- Shaping with a Rack Cutter.- Hobbing.- Hobbing Machine Gear Train Layout.- Use of a Differential in the Hobbing Machine.- Theoretically Correct Shape for the Hob Thread.- Geometric Design of a Helical Gear Pair.- 17. Tooth Stresses in Helical Gears.- Tooth Contact Force.- Contact Length.- Contact Stress.- Fillet Stress, and the Equivalent Spur Gear.
Of all the many types of machine elements which exist today, gears are among the most commonly used. The basic idea of a wheel with teeth is extremely simple, and dates back several thousand years. It is obvious to any observer that one gear drives another by means of the meshing teeth, and to the person who has never studied gears, it might seem that no further explanation is required. It may therefore come as a surprise to discover the large quantity of geometric theory that exists on the subject of gears, and to find that there is probably no branch of mechanical engineering where theory and practice are more closely linked. Enormous improvements have been made in the performance of gears during the last two hundred years or so, and this has been due principally to the careful attention given to the shape of the teeth. The theoretical shape of the tooth profile used in most modern gears is an involute. When precision gears are cut by modern gear-cutting machines, the accuracy with which the actual teeth conform to their theoretical shape is quite remarkable, and far exceeds the accuracy which is attained in the manufacture of most other types of machine elements. The first part of this book deals with spur gears, which are gears with teeth that are parallel to the gear axis. The second part describes helical gears, whose teeth form helices about the gear axis.
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