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ENT345 Mechanical
Components Design Sem 1-
2015/2016
Dr. Haftirman
School of Mechatronic Engineering
1
LECTURE NOTES
ENT345
MECHANICAL COMPONENTS DESIGN
Lecture 6, 7
29/10/2015
SPUR AND HELICAL GEARS
Dr. HAFTIRMAN
MECHANICAL ENGINEEERING PROGRAM
SCHOOL OF MECHATRONIC ENGINEERING
UniMAP
COPYRIGHT©RESERVED 2015
AGMA
The American Gear Manufacturers
Association (AGMA) has for many years
been the responsible authority for the
dissemination of knowledge pertaining to
the design analysis of gearing.
ENT345 Mechanical
Componets Design Sem 1-
2015/2016
Dr. School of Mechatronic Engineering 2
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
formula.
The equation, announced in 1892, still
remains the basis for most gear design
today.
ENT345 Mechanical
Componets Design Sem 1-
2015/2016
Dr. School of Mechatronic Engineering 3
The Lewis Bending Equation
To derive the basic Lewis equation refer to Figure,
which shows a cantilever of cross-sectional
dimensions F and t, having a length l and a load
Wt, uniformly distributed across the face width F.
The section modulus: I/c = Ft2/6
The bending stress (σ ).
ENT345 Mechanical
Componets Design Sem 1-
2015/2016
Dr. School of Mechatronic Engineering 4
ENT345 Mechanical
Componets Design Sem 1-
2015/2016
Dr. School of Mechatronic Engineering 5
The Lewis bending equations
2
23 6
6)12(
2
/ Ft
lWFt
t
Ft
c
I
cI
M t
l
tx
t
l
x
t
4
2/
2/
2
62
34
3/2
14
6
4
1
4
1
6/
16222
xx
l
tF
W
ltF
W
Ft
lW ttt
ENT345 Mechanical
Componets Design Sem 1-
2015/2016
Dr. School of Mechatronic Engineering 6
The Lewis bending equations
Fpy
pW
p
xy
xpF
pW
t
t
3
2
3
2
3
2
,
xPY
FY
PW
yYp
P
t
y is The Lewis form factor
Y means that only the bending of the tooth is considered and
that the compression due to the radial component of the force is neglected.
Wr
Wt
Values of the Lewis form factor
Y
ENT345 Mechanical
Componets Design Sem 1-
2015/2016
Dr. School of Mechatronic Engineering 7
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.
If a pair of gears failed at 500 lbf tangential load at
zero velocity and at 250 lbf at velocity V1, then a
velocity factor, designated Kv, of 2 was specified
for the gears at velocity V1.
ENT345 Mechanical
Componets Design Sem 1-
2015/2016
Dr. School of Mechatronic Engineering 8
ENT345 Mechanical
Componets Design Sem 1-
2015/2016
Dr. School of Mechatronic Engineering 9
Dynamic effects
Kv= the velocity factor
V = the pitch-line velocity in
ft/min
SI units
)(78
78
)(50
50
)(1200
1200
)),(600
600
profilegroundorshavedV
K
profileshapedorhobbedV
K
profilemildorcutV
K
profilecastironcastV
K
v
v
v
v
)(56.5
56.5
)(56.3
56.3
)(1.6
1.6
)),(05.3
05.3
profilegroundorshavedV
K
profileshapedorhobbedV
K
profilemildorcutV
K
profilecastironcastV
K
v
v
v
v
FY
PWK t
vFmY
WK t
v
The metric versions
Example 14–1
Shigley’s Mechanical Engineering Design
Example 14–1
Shigley’s Mechanical Engineering Design
X
X
Example 14-1
A stock spur gear is available having a module of 4 mm, a
44 mm face width, 18 teeth, and a pressure angle of 20°
with full-depth teeth. The material is AISI 1020 steel in as-
rolled condition. Use a design factor of nd = 3 to rate the
power output of the gear corresponding to a speed of n
= 25 rev/s and moderate applications.
Solution
The term moderate applications seems to imply that the
gear can be rated by using the yield strength as a criterion
of failure. AISI 1020 steel in as-rolled, from Table A-18,
Sut = 380 MPa and Sy = 210 MPa.
ENT345 Mechanical
Components Design Sem 1-
2014/2015
Dr. Haftirman
School of Mechatronic Engineering
12
Example 14-1
Solution
A design factor of 3 means that the allowable
bending stress is 𝑆𝑦
𝑛𝑑=
210
3.5= 60 𝑀𝑃𝑎
The pitch diameter of d = Nm = 18 (4) = 72 mm.
The pitch-line velocity is
V = πdn= π(0.072) 25 =5.65487m/s
The velocity factor (Eq 14-6b):
𝐾𝑣 =6.1+𝑉
6.1=
6.1+5.65487
6.1= 1.92703
ENT345 Mechanical
Components Design Sem 1-
2014/2015
Dr. Haftirman
School of Mechatronic Engineering
13
Table 14-2, Y=0.309 for 18 teeth.
The tangential component of load Wt
𝑊𝑡 =𝐹𝑌𝜎𝑎𝑙𝑙
𝐾𝑣𝑃⇒ 𝜎𝑎𝑙𝑙 =
𝑆𝑦
𝑛𝑑=
210
3.5= 60 𝑀𝑃𝑎
𝑃 =𝑁
𝑑⇒
𝑁
𝑑=
1
𝑚
𝑊𝑡 =𝐹𝑌𝜎𝑎𝑙𝑙
𝐾𝑣𝑃=
𝑚𝐹𝑌𝜎𝑎𝑙𝑙
𝐾𝑣
=4𝑚𝑚 44𝑚𝑚 0.309 60 𝑁/𝑚𝑚2
(1.92703)= 1693.30𝑁
The power that can be transmitted is
Hp=Wt V= (1693.30 N)(5.65487 m/s)= 9575.391W
ENT345 Mechanical
Components Design Sem 1-
2014/2015
Dr. Haftirman
School of Mechatronic Engineering
14
√
Example 14–2
Shigley’s Mechanical Engineering Design
𝑚 = 4 𝑚𝑚 ⇒ 𝑃 =25.4
𝑚=
25.4
4= 6.35
𝑚 = 3 𝑚𝑚 ⇒ 𝑃 =25.4
𝑚=
25.4
3= 8.47
Example 14–2
Shigley’s Mechanical Engineering Design
Example 14–2
Shigley’s Mechanical Engineering Design
Example 14–2
Shigley’s Mechanical Engineering Design
Example 14–2
Shigley’s Mechanical Engineering Design
Example 14–2
Shigley’s Mechanical Engineering Design
Fatigue Stress-Concentration Factor
A photoelastic investigation gives an estimate of fatigue stress-
concentration factor as
Shigley’s Mechanical Engineering Design
Surface durability
Wear is the failure of the surfaces of gear
teeth.
Pitting is a surface fatigue failure due to
many repetitions of high contact stresses.
Scoring is a lubrication failure, and
abrasion, which is wear due to the presence
of foreign material.
ENT345 Mechanical
Componets Design Sem 1-
2015/2016
Dr. School of Mechatronic Engineering 22
Surface durability
To obtain an expression for the surface-contact
stress, we shall employ the Hertz theory. The
contact stress between two cylinders may
computed from the equation;
pmax =largest surface pressure.
F= force pressing the two cylinders together.
l = length of cylinders.
ENT345 Mechanical
Componets Design Sem 1-
2015/2016
Dr. School of Mechatronic Engineering 23
Cylindrical Contact Stress
Two right circular cylinders with length l and
diameters d1 and d2
Area of contact is a narrow rectangle of width
2b and length l
Pressure distribution is elliptical
Half-width b
Maximum pressure
Shigley’s Mechanical Engineering Design
Fig. 3−38
Surface durability
Half-width b is obtained from
The surface compressive stress (Hertzian stress)
ENT345 Mechanical
Componets Design Sem 1-
2015/2016
Dr. School of Mechatronic Engineering 25
Surface durability
The radii of curvature of the tooth profiles at the
pitch point are
An elastic coefficient Cp
ENT345 Mechanical
Componets Design Sem 1-
2015/2016
Dr. School of Mechatronic Engineering 26
Example 14–3
Shigley’s Mechanical Engineering Design
Example 14–3
Shigley’s Mechanical Engineering Design
SPUR GEAR BENDING
Based on ANSI/AGMA 2001-D04 (US. Customary units)
Shigley’s Mechanical Engineering Design
Fig. 14–17
SPUR GEAR WEAR
Based on ANSI/AGMA 2001-D04 (US. Customary units)
Shigley’s Mechanical Engineering Design
Fig. 14–18
AGMA equations
Two fundamental stress equations are used in the
AGMA methodology, one for bending stress and
another for pitting resistance (contact stress).
In AGMSA terminology, these are called stress
numbers, as contrasted with actual applied stress
(σ).
ENT345 Mechanical
Componets Design Sem 1-
2015/2016
Dr. School of Mechatronic Engineering 31
AGMA Bending Stress
equations
ENT345 Mechanical
Componets Design Sem 1-
2015/2016
Dr. School of Mechatronic Engineering 32
AGMA Contact Stress
equations
ENT345 Mechanical
Componets Design Sem 1-
2015/2016
Dr. School of Mechatronic Engineering 33
AGMA Strengths
AGMA uses allowable stress numbers rather than strengths.
We will refer to them as strengths for consistency within the
textbook.
The gear strength values are only for use with the AGMA stress
values, and should not be compared with other true material
strengths.
Representative values of typically available bending strengths are
given in Table 14–3 for steel gears and Table 14–4 for iron and
bronze gears.
Figs. 14–2, 14–3, and 14–4 are used as indicated in the tables.
Tables assume repeatedly applied loads at 107 cycles and 0.99
reliability.
Shigley’s Mechanical Engineering Design
Bending Strengths for Steel Gears
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Bending Strengths for Iron and Bronze Gears
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Bending Strengths for Through-hardened Steel Gears
Shigley’s Mechanical Engineering Design Fig. 14–2
Bending Strengths for Nitrided Through-hardened Steel Gears
Shigley’s Mechanical Engineering Design Fig. 14–3
Bending Strengths for Nitriding Steel Gears
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Fig. 14–4
Allowable Bending Stress
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Allowable Contact Stress
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Nominal Temperature Used in Nitriding and Hardness Obtained
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Table 14–5
Contact Strength for Steel Gears
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Contact Strength for Iron and Bronze Gears
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Contact Strength for Through-hardened Steel Gears
Shigley’s Mechanical Engineering Design Fig. 14–5
Geometry Factor J (YJ in metric)
Accounts for shape of tooth in bending stress equation
Includes
◦ A modification of the Lewis form factor Y
◦ Fatigue stress-concentration factor Kf
◦ Tooth load-sharing ratio mN
AGMA equation for geometry factor is
Values for Y and Z are found in the AGMA standards.
For most common case of spur gear with 20º pressure angle, J can be read directly from Fig. 14–6.
For helical gears with 20º normal pressure angle, use Figs. 14–7 and 14–8.
Shigley’s Mechanical Engineering Design
Spur-Gear Geometry Factor J
Shigley’s Mechanical Engineering Design Fig. 14–6
Helical-Gear Geometry Factor J
Get J' from Fig. 14–7, which assumes the mating gear has 75 teeth
Get multiplier from Fig. 14–8 for mating gear with other than 75
teeth
Obtain J by applying multiplier to J'
Shigley’s Mechanical Engineering Design Fig. 14–7
Modifying Factor for J
Shigley’s Mechanical Engineering Design
Fig. 14–8
Surface Strength Geometry Factor I (ZI in metric)
Called pitting resistance geometry factor by AGMA
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Elastic Coefficient CP (ZE)
Obtained from Eq. (14–13) or from Table 14–8.
Shigley’s Mechanical Engineering Design
Elastic Coefficient
Shigley’s Mechanical Engineering Design
Dynamic Factor Kv
Accounts for increased forces with increased speed
Affected by manufacturing quality of gears
A set of quality numbers Qv define tolerances for gears
manufactured to a specified accuracy.
Quality numbers 3 to 7 include most commercial-quality gears.
Quality numbers 8 to 12 are of precision quality.
The AGMA transmission accuracy-level number Av is basically the
same as the quality number.
Shigley’s Mechanical Engineering Design
Dynamic Factor Kv
Dynamic Factor equation
Or can obtain value directly from Fig. 14–9
Maximum recommended velocity for a given quality number,
Shigley’s Mechanical Engineering Design
Dynamic Factor Kv
Shigley’s Mechanical Engineering Design Fig. 14–9
Overload Factor KO
To account for likelihood of increase in nominal tangential load
due to particular application.
Recommended values,
Shigley’s Mechanical Engineering Design
Surface Condition Factor Cf (ZR)
To account for detrimental surface finish
No values currently given by AGMA
Use value of 1 for normal commercial gears
Shigley’s Mechanical Engineering Design
Size Factor Ks
Accounts for fatigue size effect, and non-uniformity of material
properties for large sizes
AGMA has not established size factors
Use 1 for normal gear sizes
Could apply fatigue size factor method from Ch. 6, where this size
factor is the reciprocal of the Marin size factor kb. Applying
known geometry information for the gear tooth,
Shigley’s Mechanical Engineering Design
Load-Distribution Factor Km (KH)
Accounts for non-uniform distribution of load across the line of
contact
Depends on mounting and face width
Load-distribution factor is currently only defined for
◦ Face width to pinion pitch diameter ratio F/dp ≤ 2
◦ Gears mounted between bearings
◦ Face widths up to 40 in
◦ Contact across the full width of the narrowest member
Shigley’s Mechanical Engineering Design
Load-Distribution Factor Km (KH)
Face load-distribution factor
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Load-Distribution Factor Km (KH)
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Load-Distribution Factor Km (KH)
Shigley’s Mechanical Engineering Design Fig. 14–10
Load-Distribution Factor Km (KH)
Cma can be obtained from Eq. (14–34) with Table 14–9
Or can read Cma directly from Fig. 14–11
Shigley’s Mechanical Engineering Design
Load-Distribution Factor Km (KH)
Shigley’s Mechanical Engineering Design
Fig. 14–11
Hardness-Ratio Factor CH (ZW)
Since the pinion is subjected to more cycles than the gear, it is
often hardened more than the gear.
The hardness-ratio factor accounts for the difference in hardness of
the pinion and gear.
CH is only applied to the gear. That is, CH = 1 for the pinion.
For the gear,
Eq. (14–36) in graph form is given in Fig. 14–12.
Shigley’s Mechanical Engineering Design
Hardness-Ratio Factor CH
Shigley’s Mechanical Engineering Design Fig. 14–12
Hardness-Ratio Factor
If the pinion is surface-hardened to 48 Rockwell C or greater, the
softer gear can experience work-hardening during operation. In
this case,
Shigley’s Mechanical Engineering Design
Fig. 14–13
Stress-Cycle Factors YN and ZN
AGMA strengths are for 107 cycles
Stress-cycle factors account for other design cycles
Fig. 14–14 gives YN for bending
Fig. 14–15 gives ZN for contact stress
Shigley’s Mechanical Engineering Design
Stress-Cycle Factor YN
Shigley’s Mechanical Engineering Design
Fig. 14–14
Stress-Cycle Factor ZN
Shigley’s Mechanical Engineering Design
Fig. 14–15
Reliability Factor KR (YZ)
Accounts for statistical distributions of material fatigue failures
Does not account for load variation
Use Table 14–10
Since reliability is highly nonlinear, if interpolation between table
values is needed, use the least-squares regression fit,
Shigley’s Mechanical Engineering Design Table 14–10
Temperature Factor KT (Yq)
AGMA has not established values for this factor.
For temperatures up to 250ºF (120ºC), KT = 1 is acceptable.
Shigley’s Mechanical Engineering Design
Rim-Thickness Factor KB
Accounts for bending of rim on a gear that is not solid
Shigley’s Mechanical Engineering Design
Fig. 14–16
Safety Factors SF and SH
Included as design factors in the strength equations
Can be solved for and used as factor of safety
Or, can set equal to unity, and solve for traditional factor of safety
as n = all/
Shigley’s Mechanical Engineering Design
Comparison of Factors of Safety
Bending stress is linear with transmitted load.
Contact stress is not linear with transmitted load
To compare the factors of safety between the different failure
modes, to determine which is critical,
◦ Compare SF with SH2 for linear or helical contact
◦ Compare SF with SH3 for spherical contact
Shigley’s Mechanical Engineering Design
Summary for Bending of Gear Teeth
Shigley’s Mechanical Engineering Design
Fig. 14–17
Summary for Surface Wear of Gear Teeth
Shigley’s Mechanical Engineering Design
Fig. 14–18
Example 14–4
Shigley’s Mechanical Engineering Design
Example 14–4
Shigley’s Mechanical Engineering Design
Example 14–4
Shigley’s Mechanical Engineering Design
Example 14–4
Shigley’s Mechanical Engineering Design
Example 14–4
Shigley’s Mechanical Engineering Design
Example 14–4
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Example 14–4
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Example 14–4
Shigley’s Mechanical Engineering Design
Example 14–4
Shigley’s Mechanical Engineering Design
Example 14–4
Shigley’s Mechanical Engineering Design
Example 14–4
Shigley’s Mechanical Engineering Design
Example 14–4
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Example 14–5
Shigley’s Mechanical Engineering Design
Example 14–5
Shigley’s Mechanical Engineering Design
Example 14–5
Shigley’s Mechanical Engineering Design
Example 14–5
Shigley’s Mechanical Engineering Design
Example 14–5
Shigley’s Mechanical Engineering Design
Example 14–5
Shigley’s Mechanical Engineering Design
Example 14–5
Shigley’s Mechanical Engineering Design
Example 14–5
Shigley’s Mechanical Engineering Design
Comparing Pinion with Gear
Comparing the pinion with the gear can provide insight.
Equating factors of safety from bending equations for pinion and
gear, and cancelling all terms that are equivalent for the two, and
solving for the gear strength, we get
Substituting in equations for the stress-cycle factor YN,
Normally, mG > 1, and JG > JP, so Eq. (14–44) indicates the gear
can be less strong than the pinion for the same safety factor.
Shigley’s Mechanical Engineering Design
Comparing Pinion and Gear
Repeating the same process for contact stress equations,
Neglecting CH which is near unity,
Shigley’s Mechanical Engineering Design
Example 14–6
Shigley’s Mechanical Engineering Design
Example 14–7
Shigley’s Mechanical Engineering Design
Example 14–8
Shigley’s Mechanical Engineering Design
Example 14–8
Shigley’s Mechanical Engineering Design
Example 14–8
Shigley’s Mechanical Engineering Design
Example 14–8
Shigley’s Mechanical Engineering Design
Example 14–8
Shigley’s Mechanical Engineering Design
Example 14–8
Shigley’s Mechanical Engineering Design
Example 14–8
Shigley’s Mechanical Engineering Design
Example 14–8
Shigley’s Mechanical Engineering Design
Example 14–8
Shigley’s Mechanical Engineering Design
Example 14–8
Shigley’s Mechanical Engineering Design
Assignment 1
Page 773-774
Problem:
14-15
14-16
14-17
14-18
Shigley’s Mechanical Engineering Design
ENT345 Mechanical
Components Design Sem 1-
2015/2016
Dr. Haftirman
School of Mechatronic Engineering
113
Thank you
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