Shigley’s Mechanical Engineering Design
9th Edition in SI units
Richard G. Budynas and J. Keith Nisbett
Chapter 14
Spur and Helical 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.
14 Spur and Helical Gears
Chapter
Outline
14-1
The Lewis Bending Equation
14-2
Surface Durability
14-3
AGMA Stress Equations
14-4
AGMA Strength Equations
14-5
Geometry Factors I and J (ZI and YJ)
14-6
The Elastic Coefficient Cp (ZE)
14-7
Dynamic Factor Kv
14-8
Overload Factor Ko
14-9
Surface Condition Factor Cf (ZR)
14-10
Size Factor Ks
14-11
Load-Distribution Factor Km (KH)
14-12
Hardness-Ratio Factor CH (ZW)
14-13
Stress Cycle Life Factors YN and ZN
14-14
Reliability Factor KR (YZ)
14-15
Temperature Factor KT (Yθ)
14-16
Rim-Thickness Factor KB
14-17
Safety Factors SF and SH
14-18
Analysis
14-19
Design of a Gear Mesh
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.
• A cantilever of cross-sectional dimensions F
and t has a length l and a load W t, uniformly
distributed across the face width F. Its bending
stress is
• Assume that the maximum stress in a gear
tooth occurs at point a. By similar triangles
• Letting y = 2x/3p, we have
This completes the development of the original
Lewis equation.
• The factor y is called the Lewis form
factor.
3
Dynamic Effects
• When a pair of gears is driven at moderate or high speed and noise
is generated, it is certain that dynamic effects are present.
• AGMA standards ANSI/AGMA 2001-D04 and 2101-D04 contain this
caution:
“ Dynamic factor Kv has been redefined as the reciprocal of that used in
previous AGMA standards. It is now greater than 1.0. In earlier AGMA
standards it was less than 1.0. ”
• Barth Equation
• The Barth equation is often modified ,for cut or milled teeth.
• Introducing the velocity factor gives
4
Surface Durability
•
The surfaces of gear teeth wear
•
includes pitting, due to repetitions
of high contact stresses; scoring, a
lubrication failure; and abrasion,
due to the presence of foreign
material.
Replacing F by W t/cos φ, d by 2r,
and l by the face width F, the
surface compressive stress
(Hertzian stress) is found from the
equation
•
The Hertz contact stress between
two cylinders is
r1 and r2 are the radii of curvature on
the pinion- and gear-tooth profiles at the
point of contact.
•
where
Using an elastic coefficient Cp
And a velocity factor Kv
ν1, ν2, E1, and E2 are the elastic constants
and d1 and d2 are the diameters of the two
contacting cylinders.
5
where the sign is negative because σC
is a compressive stress.
AGMA Stress Equation
•
The fundamental equations for
bending resistance are
• The fundamental equation for
pitting resistance is
where for U.S. customary units (SI units),
Wt is the tangential transmitted load, lbf
(N)
Ko is the overload factor
Kv is the dynamic factor
Ks is the size factor
Pd is the transverse diameteral pitch
F (b) is the face width of the narrower
member, in (mm)
Km (KH) is the load-distribution factor
KB is the rim-thickness factor
J (YJ ) is the geometry factor for bending
strength (which includes root fillet stressconcentration factor Kf )
(mt ) is the transverse metric module
Cp (ZE ) is an elastic coefficient, √lbf/in2
6
(√N/mm2)
Cf (ZR) is the surface condition factor
dP (dw1) is the pitch diameter of the
pinion, in (mm)
I (ZI ) is the geometry factor for pitting
resistance
AGMA Strength Equation
•
The equation for the allowable bending stress is
where for U.S. customary units (SI units),
St is the allowable bending stress, lbf/in2 (N/mm2)
YN is the stress cycle factor for bending stress
KT (Yθ ) are the temperature factors
KR (YZ ) are the reliability factors
SF is the AGMA factor of safety, a stress ratio
•
The equation for the allowable contact stress σc ,all is
where the upper equation is in U.S. customary units and the lower
equation is in SI units. Also,
Sc is the allowable contact stress, lbf/in2 (N/mm2)
ZN is the stress cycle life factor
CH (ZW) are the hardness ratio factors for pitting resistance
KT (Yθ ) are the temperature factors
KR (YZ ) are the reliability factors
SH is the AGMA factor of safety, a stress 7ratio
Geometry Factor J
• The determination of I and J
depends upon the face-contact
ratio mF . This is defined as
where px is the axial pitch and F
is the face width.
• Bending-Strength Geometry
Factor J (YJ ) :The AGMA
factor J employs a fatigue
stress-concentration factor Kf ;
and a tooth load-sharing ratio
mN . The resulting equation for
J for spur and helical gears is
8
Geometry Factor I
• The factor I is also called the pitting-resistance geometry factor by
AGMA.
• Define speed ratio mG as
The geometry factor I for external spur and helical gears is the
denominator of the second term in the brackets.
• By adding the load-sharing ratio mN , we obtain a factor valid for both
spur and helical gears.
where mN = 1 for spur gears.
9
The Elastic Coefficient
10
Dynamic Factor
• Dynamic factors are used to account
for inaccuracies in the manufacture and
meshing of gear teeth in action.
• To account for these effects, AGMA
has defined a set of quality numbers
defining the tolerances for gears of
various sizes manufactured to a
specified accuracy.
• Quality numbers 3 to 7 will include most commercial-quality gears.
Quality numbers 8 to 12 are of precision quality.
• The dynamic factor based on Qv
where
11
Overloading Factor
• The overload factor Ko is intended to make allowance for all
externally applied loads in excess of the nominal tangential load W t
in a particular application.
12
Surface Condition Factor
• The surface condition factor Cf or ZR is used only in the pitting
resistance equation.
• It depends on
 Surface finish as affected by, but not limited to, cutting, shaving, lapping,
grinding, shotpeening
 Residual stress
 Plastic effects (work hardening)
• Standard surface conditions for gear teeth have not yet been
established. AGMA specifies a value of Cf greater than unity.
13
Size Factor
• The size factor reflects nonuniformity of material properties due to
size.
• Standard size factors for gear teeth have not yet been established
AGMA recommends a size factor greater than unity.
• If Ks in equation is less than 1, use Ks = 1.
14
Load-Distribution Factor
• The load-distribution factor modified the stress equations to reflect
nonuniform distribution of load across the line of contact.
• The load-distribution factor under these conditions is currently
given by the face load distribution factor, Cmf , where
15
Hardness-Ratio Factor
•
The hardness-ratio factor CH is used
only for the gear. The values of CH are
obtained from the equation
•
When surface-hardened pinions with
hardness of 48 Rockwell C scale
(Rockwell C48) or harder are run with
through-hardened gears (180–400
Brinell), a work hardening occurs.
16
Stress Cycle Factors
• The AGMA strengths are based on 107 load cycles applied. The
purpose of the load cycle factors YN and ZN is to modify the gear
strength for lives other than 107 cycles.
17
Reliability Factor
• The reliability factor accounts for the effect of the statistical
distributions of material fatigue failures.
• The gear strengths St and Sc are based on a reliability of 99 percent.
• A least-squares regression fit is
18
Rim-Thickness Factor
• The rim-thickness factor KB, adjusts the estimated bending stress for
the thin-rimmed gear. It is a function of the backup ratio mB
where tR = rim thickness below the tooth, in, and ht = the tooth
height.
• The rim-thickness factor KB is given by
19
Safety Factor
• The ANSI/AGMA standards contain a safety factor SF guarding
against bending fatigue failure and safety factor SH guarding against
pitting failure.
•
•
• The role of the overload factor Ko is to include predictable
excursions of load beyond W t based on experience. A safety factor
is intended to account for unquantifiable elements in addition to Ko.
20
Analysis Example 1
21
Analysis Example 2
22
Design of a Gear Mesh
• A useful decision set for spur and helical gears includes




Function: load, speed, reliability, life, Ko
Unquantifiable risk: design factor nd
Tooth system: φ, ψ, addendum, dedendum, root fillet radius
Gear ratio mG, Np, NG





Quality number Qv
Diametral pitch Pd
Face width F
Pinion material, core hardness, case hardness
Gear material, core hardness, case hardness
23
a priori
decisions
design
decisions