12 Budynas-Nisbett:Shigley's Ill.Design of Mechanical 14.Spur and Helical Gears T©The McGraw-Hill Mechanical Engineering Elements Companies,2008 Design,Eighth Edition 714 Mechanical Engineering Design This chapter is devoted primarily to analysis and design of spur and helical gears to resist bending failure of the teeth as well as pitting failure of tooth surfaces.Failure by bend- ing will occur when the significant tooth stress equals or exceeds either the yield strength or the bending endurance strength.A surface failure occurs when the significant contact stress equals or exceeds the surface endurance strength.The first two sections present a little of the history of the analyses from which current methodology developed. The American Gear Manufacturers Association'(AGMA)has for many years been the responsible authority for the dissemination of knowledge pertaining to the design and analysis of gearing.The methods this organization presents are in general use in the United States when strength and wear are primary considerations.In view of this fact it is important that the AGMA approach to the subject be presented here. The general AGMA approach requires a great many charts and graphs-too many for a single chapter in this book.We have omitted many of these here by choosing a single pressure angle and by using only full-depth teeth.This simplification reduces the complexity but does not prevent the development of a basic understanding of the approach.Furthermore,the simplification makes possible a better development of the fundamentals and hence should constitute an ideal introduction to the use of the general AGMA method.2 Sections 14-1 and 14-2 are elementary and serve as an examination of the foundations of the AGMA method.Table 14-1 is largely AGMA nomenclature. 14-1 The Lewis Bending Equation Wilfred Lewis introduced an equation for estimating the bending stress in gear teeth in which the tooth form entered into the formulation.The equation,announced in 1892, still remains the basis for most gear design today To derive the basic Lewis equation,refer to Fig.14-1a,which shows a cantilever of cross-sectional dimensions F and t,having a length and a load W,uniformly dis- tributed across the face width F.The section modulus //c is Ft-/6,and therefore the bending stress is M 6W'I (a) Gear designers denote the components of gear-tooth forces as W,Wr,Wa or W,W, Wa interchangeably.The latter notation leaves room for post-subscripts essential to free- body diagrams.For instance,for gears 2 and 3 in mesh,W is the transmitted force of 500 Montgomery Street.Suite 350,Alexandria.VA 22314-1560. 2The standards ANSI/AGMA 2001-D04 (revised AGMA 2001-C95)and ANSI/AGMA 2101-D04 (metric edition of ANSUAGMA 2001-D04).Fundamental Rating Factors and Calculation Methods for Imvolute Spur and Helical Gear Teeth,are used in this chapter.The use of American National Standards is completely voluntary:their existence does not in any respect preclude people,whether they have approved the standards or not,from manufacturing,marketing.purchasing,or using products.processes.or procedures not conforming to the standards. The American National Standards Institute does not develop standards and will in no circumstances give an interpretation of any American National Standard.Requests for interpretation of these standards should be addressed to the American Gear Manufacturers Association.[Tables or other self-supporting sections may be quoted or extracted in their entirety.Credit line should read:"Extracted from ANSI/AGMA Standard 2001-D04 or 2101-D04 Fundamental Rating Factors and Calculation Methods for Involute Spur and Helical Gear Teeth"with the permission of the publisher.American Gear Manufacturers Association, 500 Montgomery Street.Suite 350.Alexandria,Virginia 22314-1560.]The foregoing is adapted in part from the ANSI foreword to these standards.Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 14. Spur and Helical Gears 712 © The McGraw−Hill Companies, 2008 714 Mechanical Engineering Design 1 500 Montgomery Street, Suite 350, Alexandria, VA 22314-1560. 2 The standards ANSI/AGMA 2001-D04 (revised AGMA 2001-C95) and ANSI/AGMA 2101-D04 (metric edition of ANSI/AGMA 2001-D04), Fundamental Rating Factors and Calculation Methods for Involute Spur and Helical Gear Teeth, are used in this chapter. The use of American National Standards is completely voluntary; their existence does not in any respect preclude people, whether they have approved the standards or not, from manufacturing, marketing, purchasing, or using products, processes, or procedures not conforming to the standards. The American National Standards Institute does not develop standards and will in no circumstances give an interpretation of any American National Standard. Requests for interpretation of these standards should be addressed to the American Gear Manufacturers Association. [Tables or other self-supporting sections may be quoted or extracted in their entirety. Credit line should read: “Extracted from ANSI/AGMA Standard 2001-D04 or 2101-D04 Fundamental Rating Factors and Calculation Methods for Involute Spur and Helical Gear Teeth” with the permission of the publisher, American Gear Manufacturers Association, 500 Montgomery Street, Suite 350, Alexandria, Virginia 22314-1560.] The foregoing is adapted in part from the ANSI foreword to these standards. This chapter is devoted primarily to analysis and design of spur and helical gears to resist bending failure of the teeth as well as pitting failure of tooth surfaces. Failure by bend￾ing will occur when the significant tooth stress equals or exceeds either the yield strength or the bending endurance strength. A surface failure occurs when the significant contact stress equals or exceeds the surface endurance strength. The first two sections present a little of the history of the analyses from which current methodology developed. The American Gear Manufacturers Association1 (AGMA) has for many years been the responsible authority for the dissemination of knowledge pertaining to the design and analysis of gearing. The methods this organization presents are in general use in the United States when strength and wear are primary considerations. In view of this fact it is important that the AGMA approach to the subject be presented here. The general AGMA approach requires a great many charts and graphs—too many for a single chapter in this book. We have omitted many of these here by choosing a single pressure angle and by using only full-depth teeth. This simplification reduces the complexity but does not prevent the development of a basic understanding of the approach. Furthermore, the simplification makes possible a better development of the fundamentals and hence should constitute an ideal introduction to the use of the general AGMA method.2 Sections 14–1 and 14–2 are elementary and serve as an examination of the foundations of the AGMA method. Table 14–1 is largely AGMA nomenclature. 14–1 The Lewis Bending Equation Wilfred Lewis introduced an equation for estimating the bending stress in gear teeth in which the tooth form entered into the formulation. The equation, announced in 1892, still remains the basis for most gear design today. To derive the basic Lewis equation, refer to Fig. 14–1a, which shows a cantilever of cross-sectional dimensions F and t, having a length l and a load Wt , uniformly dis￾tributed across the face width F. The section modulus I/c is Ft 2/6, and therefore the bending stress is σ = M I/c = 6Wt l Ft 2 (a) Gear designers denote the components of gear-tooth forces as Wt , Wr, Wa or Wt , Wr, Wa interchangeably. The latter notation leaves room for post-subscripts essential to free￾body diagrams. For instance, for gears 2 and 3 in mesh, Wt 23 is the transmitted force of