Shigley 9E SI Chap15

Shigley’s Mechanical Engineering Design
9th Edition in SI units
Richard G. Budynas and J. Keith Nisbett
Chapter 15
Bevel and Worm Gears
Prepared by
Kuei-Yuan Chan
Associate Professor of Mechanical Engineering
National Cheng Kung University
Copyright © 2011 by The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
15 Bevel and Worm Gears
Bevel Gearing—General
Bevel-Gear Stresses and Strengths
AGMA Equation Factors
Straight-Bevel Gear Analysis
Design of a Straight-Bevel Gear Mesh
Worm Gearing—AGMA Equation
Worm-Gear Analysis
Designing a Worm-Gear Mesh
Buckingham Wear Load
Bevel Gearing-General
• Bevel gears may be classified as follows
Straight bevel gears
Spiral bevel gears
Zerol bevel gears
Hypoid gears
Spiroid gears
• Straight bevel gears are usually used for pitch-line
velocities up to 1000 ft/min (5 m/s) when the noise
level is not an important consideration.
• Spiral bevel gears are recommended for higher
speeds and where the noise level is an important
• The Zerol bevel gear is a patented gear having curved
teeth but with a zero spiral angle.
• Hypoid gears are similar to bevel gears but with the
shafts offset.
• For larger offsets, the pinion begins to resemble a
tapered worm and the set is then called spiroid gearing.
Bevel-Gear Stress and Strength
• Fundamental Contact Stress Equation
• Permissible Contact Stress Number (Strength) Equation
• Bending Stress
• Permissible Bending Stress Equation
AGMA Equation Factors
• Overload Factor Ko (KA)
• Dynamic Factor Kv
AGMA Equation Factors (Cont.)
• Size Factor for Pitting Resistance Cs (Zx)
• Size Factor for Bending Ks (Yx)
• Load-Distribution Factor Km (KHβ)
• Crowning Factor for Pitting Cxc (Zxc)
• Pitting Resistance Geometry Factor I (ZI)
AGMA Equation Factors (Cont.)
Bending Strength Geometry Factor J (YJ)
Stress-Cycle Factor for Pitting Resistance CL (ZNT)
Stress-Cycle Factor for Bending Strength KL (YNT)
Hardness-Ratio Factor CH (ZW)
Temperature Factor KT (Kθ)
Reliability Factors CR (ZZ) and KR (YZ)
AGMA Allowable Stress Numbers
Straight-Bevel Gear Analysis
Design of a Straight-Bevel Gear Mesh
• A useful decision set for straight-bevel gear design is
Design factor
Tooth system
Tooth count
Pitch and face width
Quality number
Gear material, core and case hardness
Pinion material, core and case hardness
A priori decisions
Design variables
Worm Gearing-AMGA Equation
• Crossed helical gears, and worm gears too, usually have a 90◦ shaft
angle, though this need not be so. The relation between the shaft
and helix angles is
Where ∑ is the shaft angle
• The pitch worm diameter d falls in the range
where C is the center-to-center distance.
• AGMA reports the coefficient of friction f as
• The heat loss rate Hloss from the worm-gear case in ft · lbf/min is
where e is efficiency, and Hin is the input horsepower from the worm
Worm Gearing-AMGA Equation(Cont.)
• AGMA relates the allowable tangential force on the worm-gear tooth
(Wt ) all to other parameters by
Worm-Gear Analysis
• Compared to other gearing systems worm-gear
meshes have a much lower mechanical efficiency.
When the worm drives the gearset, the mechanical
efficiency eW is given by
With the gear driving the gearset, the mechanical efficiency eG is given
• To ensure that the worm gear will drive the worm,
where values of fstat can be found in ANSI/AGMA 6034-B92.
• The magnitude of the gear transmitted force WtG can be related to the
output horsepower H0, the application factor Ka , the efficiency e, and
design factor nd by
• The largest lead angle λmax associated with normal pressure angle φn.
Designing a Worm-Gear Mesh
• A usable decision set for a worm-gear mesh includes
Function: power, speed, mG, Ka
Design factor: nd
Tooth system
Materials and processes
Number of threads on the worm: NW
Axial pitch of worm: px
Pitch diameter of the worm: dW
Face width of gear: FG
Lateral area of case: A
A priori decisions
Design variables
Buckingham Wear Load
• Buckingham showed that the allowable geartooth loading for wear
can be estimated from
where Kw = worm-gear load factor
dG = gear-pitch diameter
Fe = worm-gear effective face width

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