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Analytical Mechanics Of Gears

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Analytical Mechanics Of Gears


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This text is the gratification, to some degree at least, of a suppressed desire of some twenty-five years' standing; this has been the development of a reasonably complete mathematical analysis of the mechanics of gearing.

The subtitle of this treatise could well be Final Report of the ASME Special Research Committees on Worm Gears and the Strength of Gears. '' The information and the inspiration from the work of these committees and from personal contact with their individual members has been the major motive force behind this work.

To quote from a note to one of my colleagues: ‘^If one understands the significance of the calculus, then the subject of conjugate gear-tooth action should be mastered with ease. On the other hand, if one has a visual imagination and caii grasp the significance of this conjugate gear- tooth action, then this presentation should be of material aid to help him to master the significance of the calculus. If however, as with myself, he has neither, then continued application to the problem may give him some slight appreciation of both.”

This is not a text on gear design. It should form, however, a sound foundation upon which logical design practices and design data can be erected. These design structures, however, must be erected by the specialized engineering groups that have need of them. In the final analysis, there are so many unknown and uncertain factors that in critical cases our only answer is to “try and see.” An analysis such as this should be of material aid in the interpretation and in the application of the information that is made available by definite tests and by actual service experience.

The trends of today indicate a growing demand for the seemingly opposed requirements of higher speeds and greater loads with more reli- ability and quietness of operation. These demands are met in part by improved materials, better balancing, more nearly perfect machined sur- faces, and more intensive attention to many details of design. This last should also include a rigorous mathematical analysis of both the kine- matic and dynamic conditions of operation.



1. Conjugate Action on Spur Gears 1
2. Conjugate Action on Internal Gears 35
3. Trochoids, Tooth Fillets, and Undercut 48
4. The Involute Curve and Its Properties 68
5. Involutometry of Spur Gears 78
6. Involutometry of Internal Gears 112
7. Conjugate Action on Helical Gears 141
8. Involutometry of Helical Gears 152
9. Involutometry of Spiral Gears 177
10. Helicoid Sections 193
11. Contact on Worm-gear Drives 212
12. Design of Worm-gear Drives 248
13. Hourglass-worm Drives 276
14. Conjugate Tooth Action on Bevel Gears 301
15. The Octoid Form on Bevel Gears 321
16. Spiral Bevel Gears 338
17. Skew Bevel or Hypoid Gears 352
18. Gear Teeth in Action 383
19. Efficiencies of Gears 395
20. Analysis of Dynamic Loads on Spur-gear Teeth 426
21. Dynamic Loads on Gear Teeth 453
22. Beam Strength of Gear Teeth 474
23. Surface-endurance Limits of Materials 602
24. Limiting Loads for Wear on Gears 627