A Comparison of the AGMA Gear Design Stresses, the Lewis Bending
Stress, and the Stresses Calculated by the Finite Element Method for
Spur Gears
by
Andrew Wright
An Engineering Project Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
Master of Engineering
Major Subject: Mechanical Engineering
Approved:
_________________________________________
Ernesto Gutierrez-Miravete, Thesis Adviser
Rensselaer Polytechnic Institute
Hartford, CT
June, 2013
(For Graduation December 2013)
i
CONTENTS
LIST OF TABLES ............................................................................................................ iv
LIST OF FIGURES ........................................................................................................... v
ACKNOWLEDGMENT .................................................................................................. vi
ABSTRACT .................................................................................................................... vii
SYMBOLS AND VARIABLES .................................................................................... viii
GLOSSARY ...................................................................................................................... x
IMPORTANT KEYWORDS ........................................................................................... xi
1. Introduction and Scope ................................................................................................ 1
1.1
Scope and Background ....................................................................................... 1
2. Theory and Methodology ............................................................................................ 4
2.1
Derivation of the Lewis Bending Equation ........................................................ 4
3. Analysis ....................................................................................................................... 6
3.1
3.2
Microsoft Excel Analysis ................................................................................... 6
3.1.1
Determination of the Lewis Bending Stress .......................................... 6
3.1.2
Determination of the AGMA Bending Stress ........................................ 6
3.1.3
Determination of the AGMA Contact Stress ......................................... 6
ABAQUS Finite Element Analysis .................................................................... 7
3.2.1
Lewis Bending Stress Model ................................................................. 7
3.2.2
AGMA Design Stresses Model ............................................................ 14
4. Results and Discussion .............................................................................................. 15
4.1
Comparison of Data ......................................................................................... 15
4.1.1
4.2
Lewis Bending Model vs Excel Analysis ............................................ 15
Sources of Error and Divergence ..................................................................... 16
5. Conclusion ................................................................................................................. 17
6. Appendices ................................................................................................................ 18
ii
6.1
Microsoft Excel Analysis – Lewis Bending Equation ..................................... 18
6.2
Microsoft Excel Analysis – AGMA Design Equations ................................... 19
6.3
Lewis Bending ABAQUS Mesh Convergence Data ....................................... 28
6.4
AGMA ABAQUS Mesh Convergence Data.................................................... 29
7. References.................................................................................................................. 30
iii
LIST OF TABLES
Table 1 – Symbols and Variables ..................................................................................... ix
Table 2 – Mesh Convergence Study Details.................................................................... 12
Table 3 – Lewis Bending Stress by Distance through Tooth as Calculated by Abaqus .. 13
Table 4 – Comparison of Lewis Bending Stresses .......................................................... 15
iv
LIST OF FIGURES
Figure 1 – Gear Tooth Material Resisting Bending Load ................................................. 4
Figure 2 - Cut and Partitioned Pinion Tooth before Meshing ........................................... 7
Figure 3 - Boundary Conditions on the Pinion Gear Tooth............................................... 8
Figure 4 – Load Position, Magnitude, and Coordinate System ......................................... 9
Figure 5 – Coupling Constraint used to Apply Wt ............................................................ 9
Figure 6 - Example Graph of Stress vs Distance along Path through Gear Tooth .......... 12
Figure 7 – The Percentage Change in Standard Deviation versus Iteration .................... 12
Figure 8 – Node Path Used to Determine Bending Stresses ........................................... 13
Figure 9 - Visualization of Completed Pinion Tooth, with Stress Distribution .............. 15
v
ACKNOWLEDGMENT
Type the text of your acknowledgment here.
vi
ABSTRACT
The Lewis bending stress, the AGMA bending stress, and the AGMA pitting
contact stress are calculated for a gear mesh consisting of two spur gears. Gear stresses
and factors of safety are calculated both by hand in Microsoft Excel as well as by using
the finite element method. The “Rush Gears” website has been used to generate gear
CAD files for use in the ABAQUS finite element analysis software package (see
References page). With the gear CAD files imported, ABAQUS was used to mesh,
constrain, and calculate gear stresses. The stresses calculated by these two methods will
be compared in order to determine the effectiveness of the finite element method to
design gears.
vii
SYMBOLS AND VARIABLES
Symbol/Variable
Cf
CH
dg
dp
E
F
HB
I
J
KB
Km
Ko
KR
Ks
Kv
mG
ν
φ
PdG
PdP
Qv
σcG
σcP
σG
σL
σP
satG
satP
ScG
ScP
SFG
SFP
SHG
SHP
SY
Description
Units
Surface condition factor
-
Hardness ratio factor
-
Pitch diameter of the gear
Pitch diameter of the pinion
in
in
Modulus of elasticity
Face width
Brinell hardness of the gears
psi
in
HB
Pitting resistance geometry factor
Bending strength geometry factor
Rim thickness factor
Load distribution factor
Overload factor
-
Reliability factor
Size factor
Dynamic factor
-
Speed ratio
Poisson’s ratio of the gears
Pressure angle
Diametral pitch of the gear
rad
in-1
Diametral pitch of the pinion
in-1
AGMA quality factor
AGMA contact stress, gear
psi
AGMA contact stress, pinion
AGMA bending stress on the gear
Lewis bending stress
psi
psi
psi
AGMA bending stress on the pinion
Allowable bending stress number, gear
psi
psi
Allowable bending stress number, pinion
psi
Contact fatigue strength, gear
psi
Contact fatigue strength, pinion
psi
Bending fatigue failure safety factor, gear
Bending fatigue failure safety factor, pinion
psi
psi
Wear factor of safety, gear
Wear factor of safety, pinion
Yield strength of the gears
psi
viii
Equation
Used
SUT
Tg
Tp
TSG
TSP
Wt
YG
YN
YP
ZNG
ZNP
Ultimate strength of the gears
psi
Operational torque transmitted to the gear
lbf-in
Operational torque transmitted to the pinion
Stall torque of the power source, acting on the
gear
Estimated stall torque of the power source,
acting on the pinion
Tangential transmitted gear load
Lewis form factor of the gear
lbf-in
lbf-in
Stress cycle factor
-
Lewis form factor of the pinion
Pitting resistance stress-cycle factor, gear
-
Pitting resistance stress-cycle factor, pinion
-
Table 1 – Symbols and Variables
ix
lbf-in
lbf
-
GLOSSARY
x
IMPORTANT KEYWORDS
xi
1. Introduction and Scope
1.1 Scope and Background
There are two primary modes of failure for spur gears in contact with each other:
failure by bending and failure by contact stress at the gear tooth surface (Budynas,
2008). The contact stress, or pitting stress, between two contacting gears is a function of
the Hertzian contact equation, and is proportional to the square root of the applied tooth
load (AGMA 2001-D04). The bending stress is calculated by assuming the gear tooth is
a cantilevered beam, with a cross section of face width by tooth thickness. The gear
bending stress is directly proportional to the tooth load. In general, bending failure will
occur when the stress on the tooth is greater than or equal to the yield strength of the
gear tooth material. Pitting failure will occur when the contact stress between the
meshing gears is greater than or equal to the surface endurance strength.
The objective of this project is to establish the effectiveness of the Abaqus finite
element software to calculate bending and contact stresses of spur gears. Gear stresses
are calculated in both Microsoft Excel as well as Abaqus software packages. Two
Abaqus models will be created: one to model the Lewis Bending equation, and another
to model the AGMA gear design equations. The Abaqus analyses will be static (as
opposed to dynamic) in order to simplify the analysis. For the AGMA Abaqus model,
two gears will be modeled: a pinion (smaller) gear and a driven gear. These gears are
cut from solid gears to single teeth in order to simplify the analysis as well as greatly
reduce computation time. In order to match the equations detailed in Ref. 1, various
other assumptions must be made. These assumptions include full-depth teeth, spur
involute gears operating on parallel axes, undamaged gear teeth, elastic isotropic
materials, and gear contact ratios between 1.0 and 2.0. For a list of all assumptions built
into the AGMA equations, see section 1.2 of Ref. 1.
The Lewis Bending equation is one of the oldest and yet most important design
equations to consider when sizing gears (especially spur gears). The equation was
formulated by Wilfred Lewis in 1892, and was the first of its kind to take into account
specific geometric aspects of the tooth profile to determine tooth stresses (Ref. 3). It
remains one of the primary ways to size gears for bending loads, and is by far the easiest
1
way to get reasonable results. Lewis derived his equation by making a few assumptions.
Firstly, he assumed that each gear tooth could be treated separately from the gear mesh.
Next, he applied the transmitted load (Wt in the table of variables) to the tip of the tooth.
This is ideally the most conservative place to apply the load, however it doesn’t quite
match reality. In the instant that a pair of gear teeth comes into contact in a gear mesh,
an adjacent tooth pair is still in contact. Therefore, when contact is created at the tip of a
pair of teeth the load is shared by multiple contact points. It is therefore conservative to
apply the full transmitted load to the tip of the gear tooth. In reality, the full load should
be applied somewhere in the middle of the tooth (say, at the pitch circle). This is the
point of contact on the gear teeth when only one pair of teeth is contacting (Ref. 3).
Lewis assumed that the largest stresses in the gear tooth would be bending, and therefore
modeled the tooth as a cantilevered beam (see Fig. 1 below). Based on this assumption,
the largest stress is located in the root of the tooth at the base, since this location is
furthest away from the neutral axis of bending. Section 2.1 contains a derivation of the
Lewis Bending equation, and appendix 6.1 contains the Lewis Bending Excel analysis.

What do the AGMA equations take into account that the Lewis Bending equation
does not? (which equation is more conservative, and where is the predicted
failure point?)

Lewis bending equation is theoretical, AGMA is empirical – experimentation +
trial / error

Using Rush Gear CAD files to import geometry into Abaqus
2
3
2. Theory and Methodology
2.1 Derivation of the Lewis Bending Equation
(retrieved from http://en.wikipedia.org/wiki/File:Tooth_as_beam.png)
Figure 1 – Gear Tooth Material Resisting Bending Load
4
5
3. Analysis
3.1 Microsoft Excel Analysis
3.1.1
Determination of the Lewis Bending Stress
See section 1.1 for a brief history of the Lewis Bending equation, section 2.1 for
a derivation, and appendix 6.1 for the Lewis Bending analysis in Excel. The values of
the face width, “F”, diametral pitch, “P”, and Lewis Form factor, “Y”, can be found in
appendix 6.2. The transmitted load and yield strength used are consistent between the
Excel and Abaqus analyses.
3.1.2
Determination of the AGMA Bending Stress
3.1.3
Determination of the AGMA Contact Stress
6
3.2 ABAQUS Finite Element Analysis
This section of the report discusses in depth the two Abaqus finite element models
used to calculate the different gear stresses used in the report. For both models, the
organization of the section will follow the different stages of the model formulation:
part(s) creation, material selection, application of boundary conditions, application of the
load, and meshing the part(s). Mesh convergence studies were carried out for both
models in order to ensure that the final mesh density resulted in sufficiently accurate
results. Refer to Appendix 6.2 and 6.3 for the raw data used to carry out the mesh
convergence for both models.
3.2.1
Lewis Bending Stress Model
The first Abaqus model described in this report was created in order to simulate
bending of a single gear tooth, and to compare results with the Lewis Bending equation
as calculated in Excel (see Appendix 6.2). This was a static analysis, with simplified
geometry. The development of this model is described below, with content organized by
the different “windows” of the Abaqus software (Part, Material, Load, etc.).
The pinion (smaller) gear was chosen (instead of the larger gear) for this
analysis. The CAD file was imported as a *step file, and acquired from the Rush Gears
website (Ref. 10). The geometry required some fixing in order to result in a meshable
part. The “geometry edit” tools were used to remove features like redundant edges,
small faces, and invalid features.
Once the
geometry was cleaned up, all but one tooth
were
removed
using
datum
planes
and
extruding commands. Using only one tooth for
this analysis allows for more accurate results
since a higher density of mesh becomes
available for the same memory requirement.
The last step taken in the Parts window was to
create a few partitions in the gear tooth to make
Figure 2 - Cut and Partitioned Pinion Tooth
the mesh elements more symmetric. Figure 2
7
before Meshing
shows the final geometry of the pinion gear tooth, before meshing.
The material chosen for this gear is AISI 4140 steel, to match the corresponding
analysis performed in Excel (INSERT REF FOR MATERIAL).
The material is
assumed to be isotropic and elastic. The modulus of elasticity, E, for the material is
30E6 psi, and the poisson’s ratio is 0.3. The yield strength of the gear, SY, is 61,000 psi,
and the ultimate strength, SUT, is 95,000 psi. The Brinell hardness of this steel is 197.
See the SYMBOLS AND VARIABLES section for the explanation of all variables used
in this report.
Fixed
boundary
conditions (BC), with all
degrees of freedom (DOF)
restricted, were imposed on
three faces of the pinion
tooth: on the bottom face
and on each side (see Figure
3).
There
are
three
important assumptions of the
Figure 3 - Boundary Conditions on the Pinion Gear Tooth
Lewis Bending equation, as
stated in Ref. 3: the gear tooth is treated as a cantilevered beam, it is assumed that other
teeth in the mesh do not share the load, and the max stress will be bending and occur in
the root of the tooth.
The fixed BC on the bottom face emulates a cantilevered
condition. The BC on each side face of the tooth restrict lateral material movement,
forcing the tooth to be rigid and thus resulting in the max stress forming in the root of
the tooth.
8
The next step was to apply
the 800 lbf transmitted load (Wt),
to match the applied load in the
Excel analysis. In order to apply
the load, a continuum distributing
coupling with a reference point
and
custom
datum
coordinate
system are used (see Figure 4).
This coupling applies the load to
the
reference
point
then
redistributes it to the entire surface
(shown in pink in Figure 6).
Figure 4 – Load Position, Magnitude, and Coordinate
Shigley states that the Lewis
Bending equation assumes that the
System
load is applied completely at the
top of the tooth, evenly distributed across the face width, F (Ref. 3). While the use of
the coupling doesn’t exactly match the intent of the equation, I found that the stress
distribution that results from it more closely matches the actual state of stress in the gear
tooth. The stress distribution created using this method is symmetric, with the largest
values in the root on either side of the tooth.
Figure 5 – Coupling Constraint used to Apply Wt
9
Once the gear tooth was simplified, partitioned, bounded, and had an applied
load, the next step was to mesh the geometry. Originally I used Tet elements for the
mesh, however the partitions I created after I cut the tooth enabled the use of Hex
elements. This elements tend to results in more accurate stresses (INSERT REF.?). The
exact element type is a standard, quad, non-reduced integration element (C3D20). I
chose quadratic elements vice linear elements because of the increased number of nodes
and therefore higher accuracy. Figure 10 shows the resulting gear tooth mesh. It was
very symmetric, with zero percent mesh warnings and errors.
A mesh convergence study is an important tool that should be used in any finite
element model to determine when the mesh density is sufficient enough to provide
accurate results. Starting with a course mesh, the model is compiled and submitted and a
result is found. The mesh is then made denser, the model resubmitted, and a new result
documented. This process is iteratively repeated until the result (or some statistical
variable that makes use of the resulting data) shows a minimal percentage change
between iterations. I performed a mesh convergence study with eight iterations and
finished with a seed size (mesh density) in Abaqus of 0.0175. According to Kawalec
and Wiktor (Ref. 5), a successful mesh convergence study will result with stresses
changing by less than 0.4% at the last iteration. Even though the error percentage
calculated in my study was slightly larger (0.57%), I decided to keep the 0.0175 mesh
density. The computer that ran the analysis did not have enough memory to run the
model at a higher number of elements, and a decrease in error of 0.1% will not largely
impact my results. For each iteration I created a node path through the thickness of the
tooth, and then created a data set of the stress along this path (see Table 3). Appendix
6.3 contains this stress versus distance data for all eight iterations, as calculated by
Abaqus.
Table 2 shows information for each iteration of the mesh convergence study: the
number of elements, mesh size, max stress, standard deviation, and percentage change in
standard deviation of the data relative to the last iteration. The percentage change in
standard deviation versus iteration from this Table is shown visually in Figure 7. As you
can see, with each successive iteration the distribution of stress through the tooth face
reaches a steady state. Table 3 shows the stress versus distance data for iteration 8, the
10
final mesh density. The max stress in the root on one side of the tooth was 35,540 psi,
and on the other side it was 38,139 psi. In section 4.1, Comparison of Data, this result
will be compared to the results calculated in Excel.
11
# of
Mesh
Max Stress
%
Iteration Elements
Size
[psi]
STDEV
change
1
240
0.065
35796
10389
N/A
2
460
0.05
35972
10117
-2.694
3
1334
0.035
35782
10947
7.582
4
3120
0.025
37863
10338
-5.889
5
3872
0.0225
37805
10562
2.119
6
3900
0.02
35796
10389
-1.661
7
6642
0.0185
38001
10563
1.647
8
7410
0.0175
38139
10503
-0.570
Table 2 – Mesh Convergence Study Details
Figure 6 - Example Graph of Stress vs Distance along Path through Gear Tooth
FEA Lewis Bending Analysis
Mesh Convergence Study
% Difference
10.000
5.000
0.000
1
2
3
4
5
6
-5.000
-10.000
Iteration No.
Figure 7 – The Percentage Change in Standard Deviation versus Iteration
12
7
Distance Through
Tooth (in)
0.000
35,540
0.025
11,185
0.050
5274
0.075
5811
0.094
8782
0.113
13,046
0.131
20,043
0.150
38,139
Bending Stress (psi)
Table 3 – Lewis Bending Stress by Distance through Tooth as Calculated by Abaqus
Figure 8 – Node Path Used to Determine Bending Stresses
13
3.2.2
AGMA Design Stresses Model
14
4. Results and Discussion
4.1 Comparison of Data
4.1.1
Lewis Bending Model vs Excel Analysis
Pinion
σLP =
24854 Excel analysis
σLP2 =
35540 Abaqus analysis
Table 4 – Comparison of Lewis Bending Stresses
Table 4 shows the final results for the Lewis Bending stress as calculated by both
Microsoft Excel and Abaqus. As stated in section 3.2.1, the bending stress in Abaqus
was determined by finding the stresses along the node path shown in Fig. 7. The
location of the node path was chosen based on proximity to the tooth root and the
strength of the mesh in the area. It was important to compare the Excel bending stress
with the Abaqus stress in the root of the tooth, since that is where the Lewis Bending
equation assumes the maximum stress will occur.
The bending stress as calculated by Abaqus is 30.7% higher than the stress
calculated in Excel. There are various reasons why this error exists between the two
solutions. A few sources of error are discussed below in section 4.2.
Figure 9 - Visualization of Completed Pinion Tooth, with Stress
Distribution
15
4.2 Sources of Error and Divergence
o Mesh elements used?
o Lewis bending equation doesn’t take into account some things that Abaqus does
(stress concentrations?)
o Abaqus load distributed to entire face of tooth
16
5. Conclusion
17
6. Appendices
6.1 Microsoft Excel Analysis – Lewis Bending Equation
4.7 Lewis Bending Analysis
The gears are analyzed for stall torque using the Lewis Bending equation. The transmitted load for section 4.7 is assumed to act at
the top of the tooth, evenly distributed along the face width.
Wt [lbf] =
800
Tangential transmitted gear load. See section 4.1 and 4.2 for inputs.
Pinion
σL [psi] =
σLP =
Gear
σLG =
24854
Factors of safety are calculated on material yield
strength.
FSL = Sy / σL
Pinion
FSL [ ] =
19010
FSLP =
Gear
2.5
FSLG =
18
3.2
Lewis Bending stress.
6.2 Microsoft Excel Analysis – AGMA Design Equations
4.0 CALCULATION / DISCUSSION:
The following analysis calculates the AGMA design stresses for the meshing spur gears.
4.1 Input Loads
Operational Torques:
mG = dg / dp
dg and dp are the pitch diameters of the gear and the pinion, respectively.
mG [ ] =
3.0
Tg [lbf-in] =
1800
Speed Ratio (Ref. 4, Eq. 14-22)
Torque required at the gear to drive the system. This value is the same for the Excel analysis
and the ABAQUS analysis.
Tp = Tg / mG
Tp [lbf-in] =
600
Torque required at the pinion to drive the system.
4.2 Pinion, Idler, and Gear Dimensions
Pinion
F [in] =
d [in] =
Pd [in] =
dp =
PdP =
Qv [ ] =
φ [rad] =
Gear
1.25
1.5
12
1.25
4.5
12
dg =
PdG =
5
5
NG =
0.349
54
200
YG =
0.404
N[]=
HB [ ] =
NP =
0.349
18
200
Y[]=
YP =
0.309
Face width
Pitch diameter
Diametral pitch
AGMA quality
factor
Pressure angle (20 degrees)
Number of teeth
Brinell hardness of the gears (Ref. 3)
Lewis Form Factor (Ref. 4, Table 14-2)
The AGMA quality factor is also known as the transmission accuracy grade number, and is a measure of how accurate the gearing is (see Annex A, Ref. 1). Qv ranges
from 5 to 11, therefore a quality factor of 5 is a conservative estimate.
19
4.3 Material Properties and Other Input Variables
AISI 4140 steel is picked as the gear material. The following material information was retrieved from
[INSERT REF].
E [psi] =
ν[]=
Sy [psi] =
SUT [psi] =
HB [ ] =
30000000
0.3
61000
95000
197
Modulus of Elasticity of the gears (see Ref. )
Poisson's Ratio of the gears
Yield Strength of the gears (Ref. )
Ultimate Strength of the gears (Ref. )
Brinell Hardness of the gears (Ref. )
4.4 Calculation of Pitch Line Velocity
The pitch line velocity is used with the Dynamic factor, Kv, below. Increasing the pitch line velocity of the gear mesh can increase the
max stress of the gears. For this analysis, it is assumed that the gears are static to simplify the analysis. The constants affected by the
pitch line velocity become unity when the velocity is zero.
v = ωr
v [ft/min] =
0
Pitch Line Velocity of the geartrain
4.5 AGMA Bending Stress Analysis
Section 4.5 calculates the AGMA gear bending stress, the bending stress number (similar to a material strength), and a factor of safety
for bending. The calculated bending stress is based on the assumption that the gear tooth is a cantilevered plate, fixed at the base of the
tooth. This bending stress creates fatigue in the gear teeth during operation of the gear mesh. In essence, the AGMA design equations
calculate the maximum input load that the gears can withstand over the life of the gears without creating cracking. If the bending
stresses do cause cracking in the gears, they usually form at the root fillet because this is where the largest stress is. For gears with
small, thin rims, the location of the max stress can change. For this project it is assumed that the rim is of sufficient size to avoid this
situation.
Overload Factor, Ko
Ko [ ] =
1
In most practical purposes the Overlaod Factor is greater than 1 to account for momentary peak torques experienced by most
mechanically driven systems. However, in an attempt to get accurate finite element results this value was kept at 1. A constant
tangential load will be applied in the model, with no transient peaks.
20
Dynamic Factor, Kv
Kv = ((A+V0.5)/A)B
A = 50 + 56(1 - B)
B = 0.25(12 - Qv)0.66
B[]=
0.90
A[]=
55.4
Kv [ ] =
1.00
See Eq. 14-28 in Ref.
4
See Eq. 14-28 in Ref.
4
See Eq. 14-27 in Ref.
4
Size Factor, Ks
The size factor is impacted by many factors, including tooth size, diameter, face width, hardenability, and stress pattern (see Ref. 4,
section 14-10). AGMA suggests either using the equation below or simply assuming unity for this factor. I use the equation listed
because a size factor greater than 1 is conservative.
Ks = 1.192*(F*Y0.5/Pd)0.0535
See section 4.2 above for input variables.
Pinion
Ks [ ] =
KsP =
Gear
1.02
KsG =
1.03
Load Distribution Factor, Km
The load distribution factor is a ratio of the peak load to the average load applied across the entire face of the gear (Ref. 1, Annex D).
When computed analytically, this factor can be very complex. The AGMA gathered empirical data through in service gears and testing
to create the equations and variables below that are used to calculate Km.
Note: The load-distribution factor is equal to the "face load distribution factor", Cmf, under the conditions listed in section 14-11 of
Ref. 4. The gears used herein obey these assumptions, therefore Cmf is used for this factor.
Ref. 4, Eq. 14-30
21
Pinion
Gear
Cmc [ ] =
CmcP =
1
CmcG =
1
Cpf [ ] =
CpfP =
0.06
CpfG =
0.03
Cpm [ ] =
A[]=
CpmP =
Ap =
1
0.1270
CpmG =
AG =
1
0.1270
B[]=
C[]=
Cma [ ] =
Ce [ ] =
Cmf [ ] =
Km [ ] =
Bp =
Cp =
CmaP =
CeP =
CmfP =
KmP =
0.0158
-0.0001
0.15
1
1.21
1.21
BG =
CG =
CmaG =
CeG =
CmfG =
KmG =
0.0158
-0.0001
0.15
1
1.17
1.17
Eq. 14-31 from Ref. 4. Equals 1 for
uncrowned teeth
See Eq. 14-32 in Ref. 4 for 1 < F <= 17
See Eq. 14-33 in
Ref. 4
See Table 14-9 from Ref. 4. Commercial,
enclosed units
See Eq. 14-34 from Ref. 4
See Eq. 14-35 from Ref. 4
Note: Per Ref. 4, for values of F/(10d) < 0.05, F/(10d) = 0.05 is used when computing Cpf above.
Bending Strength Geometry Factor, J
The bending strength geometry factor, J, is impacted by the shape of the teeth in contact. Figure 14-6 in Ref. 4 is used to estimate this
factor based on the number of pinion and gear teeth. This factor assumes spur gears, a 20 degree pressure angle, and full-depth teeth.
Pinion
J[]=
JP =
Gear
0.32
JG =
0.40
See Fig. 14-6 in Ref. 4.
Note: Assumed that load is applied in highest point of single-tooth contact.
Stress Cycle Factor, YN
The Stress Cycle Factor alters the design stress based on the number of design stress cycles. The overall "Service Factor" used by
AGMA combines the Overload Factor, the Reliability Factor, and the Stress Cycle Factor. AGMA 2001-D04 suggests that if designers
are comfortable with the other factors in the Service Factor, unity can be used for the Stress Cycle Factor. Since this is a theoretical
problem, and the FEA software will not take into account total stress cycles, the Stress Cycle Factor has been set at 1.00.
Pinion
YN [ ] =
YNP =
Gear
1.00
YNG =
1.00
Temperature Factor, KT
KT [ ] =
1
This factor is unity unless the working temperature of the gear mesh is higher than 250 degrees
Fahrenheit (see section 14-15 of Ref. 4).
22
Reliability Factor, KR
The reliability factor takes into account normal statistical material failures that occur in material testing. Table 11 shows some common
reliability factors that were calculated from data collected by the US Navy. Unity is picked because this represents a factor in the
middle of the range.
KR [ ] =
1
See Table 11 of Ref. 1.
Rim Thickness Factor, KB
The Rim Thickness Factor is an adjustment factor that takes into account
gears with smaller "rims", the material in between the bore and the base
of the gear teeth. The factor is given in terms of the "backup ratio", the
ratio of the rim thickness of the gear to the whole depth (see Fig. X to the
right from Ref. 1). For backup ratios of greater than 1.2, the Rim
Thickness Factor becomes 1.0. For the sake of making the FEA program
simpler and getting more accurate results between the hand analysis and
the finite element, I will make the rim thickness large enough for the
backup ratio to be greater than 1.2.
KB [ ] =
Pinion
KBP =
Gear
KBG =
1.00
1.00
See Fig. B.1 from Annex B of
Ref. 1.
Gear Bending Stress, σ
Now that all the factors have been calculated, we can determine the gear bending stress. The stress is calculated below using the
equation found in Ref. 4.
Wt = 2Tp/d
Wt [lbf] =
800
σ = WtKoKvKs(Pd/F)(KmKB/J)
Wt is the transmitted tangential load going into the pinion gear.
See Ref. 4, Eq. 14-15
Pinion
σ [psi] =
σP =
Gear
σG =
29674
23
23251
4.5.1 Calculation of the AGMA Bending Fatigue Failure Safety Factor
Allowable Bending Stress Number, sat
The allowable bending stress number, sat, is
similar to a yield strength except that it goes a
step further and takes into account material
composition, cleanliness, the presence of
residual stresses, heat treatments, and materials
processing (see Ref. 1). The AGMA standard
contains tables and charts for various common
engineering gear materials with their associated
bending stress numbers. For AISI 4140, the
material of the gears, Fig. 10 is used. Grade 2,
the larger stress number, is assumed. This
grade is chosen because we will be comparing
the gears against gear models that have ideal
cleanliness and material properties.
sat = 108.6HB + 15890
Pinion
sat [psi]4 =
satP =
Gear
37284
satG =
37284
See Fig. above (from Fig. 10 of
Ref. 1)
Bending Fatigue Failure Safety Factor, SF
In engineering practice, a factor of safety is a design factor that takes into account uncertainty in the calculation of the solution. In
general, it is the ratio of the material strength of a component divided by the stress on that component. Depending on the application,
the risks involved (whether that be cost, time, or safety), and statistical randomness of the inputs, the engineer may decide to design the
component to different factors. For the purposes of the project, I chose to set a factor of safety of 1.5 as a requirement. Not only is this
a standard factor of safety for operational loading, but because AGMA has developed so many factors that make the calculated stress
more accurate we are getting results with less variance.
The AGMA standards use a factor of safety in their gear design process, which can be found in Ref. 4 (see Eq. 14-41). For this project,
the equation simplifies to sat divided by the bending stress since YN, KT, and KR are unity.
SF = [satYN/(KTKR)]/σ
Pinion
SF [ ] =
SFP =
Gear
1.3
SFG =
1.6
24
See Ref. 4, Eq. 1441
4.6 AGMA Pitting Analysis
The second important failure mode for gears is pitting, which is a surface fatigue failure that results from progressive contact stress in
the meshing gears (Ref. 4, section 14-2). Because pitting is a fatigue phenomenon, it may take many cycles to become serious enough
to result in failure of the gear system. AGMA 2001-D04 defines two types of pitting: initial and progressive. In initial pitting, small
defects are formed on the surface of the teeth in areas of high stress. These pits will, over time, correct themselves as the surrounding
high spots get smoothed out by contact with the meshing gear. For this reason, the presence of initial pitting is not a failure criteria for
gear systems. Progressiv pitting, on the other hand, does not correct itself and can occur when the stresses, lifetime cycles, or other
factors are high enough. The AGMA pitting stress equation is designed to calculate the load for which the meshing gears never
experience progressive pitting in their usage lifetime (see Ref. 1, section 4.2). This equation is based on the Hertzian contact stress
equation, modified to account for the effect of gear teeth sliding. The Elastic Coefficient, Cp, is a term that combines the elastic
material constants of the meshing gears. The equation is shown below (see Eq. 14-13, Ref. 4):
Elastic Coefficient, Cp
Cp = [1/(π((1 - νp2)/EP + (1 - νG2/EG)]1/2
Pinion
CPP =
Gear
2291
CPG =
CP [psi1/2] =
2291
See Ref. 4 Eq. 1413.
Surface Condition Factor, Cf
The surface condition factor, Cf, is a factor that takes into account surface finish effects such as cutting, lapping, grinding, or work
hardening. AGMA suggests that if the meshing gears have detrimental surface finishes caused by one of these processes the surface
condition factor should be greater than unity. For our purposes, we will make this factor unity since the finite element model will have
idealized gear surface finishes.
Cf [ ] =
1
See Ref. 4, Section 14-9
Pitting-Resistance Geometry Factor, I
As defined by AGMA, the pitting resistance geometry factor, I, evaluates how the radii of curvature of the contacting tooth profiles of
the gear mesh effects the Hertzian contact stress. Shigley's Mechanical Engineering Design, Ref. 4, provides a useful equation for
calculating I, shown below. Because the pitting resistance geometry factor only depends on the pressure angle, the load-sharing ratio,
and the speed ratio, it is the same for all gears in the mesh.
mN [ ] =
1
Load-Sharing Ratio. Equals unity for spur gears. See Ref. 4, Eq. 14-23
I = [(cosφ*sinφ)/2*mN] * (mG/(mG+1))
I[]=
0.121
See Ref. 4 Eq. 14-23
25
Gear Contact Stress, σc
The numerical value of the contact stress comes from the equation below, from Ref. 4:
σc = Cp(WtKoKvKs(Km / dP*F)(Cf / I))1/2
See Ref. 4, Eq. 14-16. Note: for the gear, dp is actually di in the equation for the
contact stress.
Pinion
σc [psi] =
σcP =
Gear
σcG =
151546
86590
Gear contact stresses
4.6.1 Calculation of the AGMA Wear Safety Factor
Similar to the value that was calculated for the bending stress, a factor of safety is calculated for the pitting stress for the pinion and
gear. Various factors must be calculated first before solving for the factor of safety.
Contact Fatigue Strength, Sc
The contact fatigue strength is calculated in a similar manner as the allowable bending stress number (or bending strength), as shown in
section 4.5.1. Since the gears are made out of AISI 4140, the contact stress number for nitrided through-hardened steel gears from
Table 3 of Ref. 1 can be used directly. Grade 2 is assumed once again in an attempt to get results that match nicely with the results of
the finite element analysis.
Pinion
Sc [psi] =
ScP =
Gear
ScG =
163000
163000
See Table 3 from Ref. 1.
Pitting Resistance Stress-Cycle Factor, ZN
Similar to how the bending stress cycle factor, YN, was handled, the pitting resistance stress cycle factor ZN will be set at unity for this
analysis. This factor alters the pitting strength that AGMA provides based on the number of lifetime stress cycles the gears will
encounter. There is no reason to change this factor from unity since there is no way to set the number of cycles in a finite element
analysis.
ZN [ ] =
Pinion
ZNP =
Gear
1.00
ZNG =
26
1.00
Hardness Ratio Factor, CH
The hardness ratio factor, CH takes into account the fact that the smaller meshing gear will see more stress cycles in the lifteime of the
gears as a result of it's smaller pitch circle. Because both gears in the mesh have the same hardness (making the ratio 1.0), however,
this factor cancels to unity.
HBP/HBG [ ] =
1.0
A' = 0 for HBP/HBG < 1.2
Hardness ratio between the pinion and the gear
Section 14-12 from Ref. 4.
CH = 1.0 + A'*(mG - 1.0)
CH [ ] =
1.0
See Ref. 4, Eq. 14-36. This factor is only used for the gear.
Wear Factor of Safety, SH
SH = [ScZNCH / (KTKR)] / σc
Pinion
SH [ ] =
SHP =
Gear
1.1
SHG =
27
1.9
See Ref. 5.2, Eq. 1442
6.3 Lewis Bending ABAQUS Mesh Convergence Data
Iteration 1
0
0.007505
0.015009
0.022514
0.030019
0.037523
0.045028
0.052533
0.060037
0.067542
0.075047
0.082552
0.11257
0.120075
0.12758
0.135084
0.142589
0.150094
34858.4
25571.7
18201
12746.3
9207.72
7585.35
5895.34
4877.98
4532.59
4859.15
5857.29
6827.63
12446.5
14327.2
17602.3
22272.2
28336.8
35796.1
Iteration 4
0
0.007505
0.015009
0.022514
0.030019
0.037523
0.045028
0.052533
0.060037
0.067542
0.075047
0.082551
0.090056
0.097561
0.105065
0.11257
0.120075
0.150093
36895.9
26400.2
18312.4
12632.4
9585.79
7512.37
5989
5040.63
4705.5
4945.32
5759.78
6871.66
8149.43
9593.35
11082.9
12899.1
15148
37862.5
Iteration 2
0
0.007504
0.015009
0.022513
0.030018
0.037522
0.045027
0.052531
0.060036
0.06754
0.075044
0.082549
0.090053
0.097558
0.105062
0.112567
0.120071
0.127575
0.142584
0.150089
35311.9
27392.3
20739.2
15352.4
11232
8379.7
6354.17
5065.21
4510.88
4691.16
5605.31
6593.34
7840.42
9347.05
11113.2
13137.9
15086.4
18343.7
28786.5
35972.1
Iteration 5
0
0.007505
0.015009
0.022514
0.030019
0.037523
0.045028
0.052533
0.060037
0.075047
0.090056
0.11257
0.120075
0.127579
0.135084
0.142589
0.150093
35548
25615.6
17968.4
12603.2
9677.79
7608.42
6014.46
5003.01
4714.17
5728.19
8163.64
12826.8
14958.9
17372.4
21477.4
28288.4
37805.4
28
Iteration 3
0
0.007505
0.015009
0.022514
0.030018
0.037523
0.045028
0.052532
0.060037
0.067541
0.075046
0.082551
0.090055
0.112569
0.142587
0.150092
35782.1
26555.9
19130.2
13504.4
9680.22
7656.52
6014.95
4996.13
4599.05
4823.79
5669.93
6714.4
7902.6
12330.3
28333.3
33884.1
Iteration 6
0
0.007505
0.015009
0.022514
0.030019
0.037523
0.045028
0.052533
0.060037
0.067542
0.075047
0.082552
0.11257
0.120075
0.12758
0.135084
0.142589
0.150094
34858.4
25571.7
18201
12746.3
9207.72
7585.35
5895.34
4877.98
4532.59
4859.15
5857.29
6827.63
12446.5
14327.2
17602.3
22272.2
28336.8
35796.1
6.4 AGMA ABAQUS Mesh Convergence Data
29
7. References
1.
AGMA 2001-D04, “Fundamental Rating Factors and Calculation Methods for
Involute Spur and Helical Gear Teeth.”
2.
AGMA 908-B89, “Geometry Factors for Determining the Pitting Resistance and
Bending Strength of Spur, Helical, and Herringbone Gear Teeth.”
3.
Budynas, R. G., & Nisbett, J. K. (2008). Shigley's Mechanical Engineering
Design. New York: McGraw-Hill.
4.
Cavdar, K., Karpat, F., & Babalik, F. C. (2005). Computer Aided Analysis of
Bending Strength of Involute Spur Gears with Asymmetric Profile. Journal of
Mechanical Design, 127(3), 477.
5.
Kawalec, A., & Wiktor, J. (2001). Analysis of strength of tooth root with notch
after finishing of involute gears. Archive of Mechanical Engineering, 48(3), 217248.
6.
Kawalec, A., Wiktor, J., & Ceglarek, D. (2006). Comparative Analysis of ToothRoot Strength Using ISO and AGMA Standards in Spur and Helical Gears With
FEM-based Verification. Journal of Mechanical Design, 128(5), 1141.
7.
Li C.-H., Chiou H.-S., Hung C., Chang Y.-Y., & Yen C.-C. (2002). Integration of
finite element analysis and optimum design on gear systems. Finite Elements in
Analysis and Design, 38(3), 179–192.
8.
Li, S. (2002). Gear Contact Model and Loaded Tooth Contact Analysis of a
Three-Dimensional, Thin-Rimmed Gear. Journal of Mechanical Design, 124(3),
511.
9.
Sfakiotakis V.G., Vaitsis J.P., & Anifantis N.K. Numerical simulation of
conjugate spur gear action. Computers and Structures, 79(12), 1153–1160.
doi:10.1016/S0045-7949(01)00014-1
10.
Zhang-Hua F., Ta-Wei C., & Chieh-Wen T. (2002). Mathematical model for
parametric tooth profile of spur gear using line of action. Mathematical and
Computer Modelling, 36(4), 603–614.
11.
Rush Gears Part Search and CAD. Retrieved July 9, 2013,
http://www.rushgears.com/Tech_Tools/PartSearch8/partSearch.php.
30
from