A Comparison of the Tooth-Root Stress and Contact Stress of an
Involute Spur Gear Mesh as Calculated by FEM and AGMA Standards
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, Project Adviser
Rensselaer Polytechnic Institute
Hartford, CT
June, 2013
(For Graduation December 2013)
i
CONTENTS
LIST OF TABLES ............................................................................................................ iv
LIST OF FIGURES ........................................................................................................... v
ACKNOWLEDGMENT .................................................................................................. vi
SYMBOLS AND VARIABLES ..................................................................................... vii
ACRONYMS .................................................................................................................... ix
ABSTRACT ...................................................................................................................... 1
1. Introduction and Scope ................................................................................................ 2
1.1
Scope and Background ....................................................................................... 2
2. Theory and Methodology ............................................................................................ 4
2.1
Derivation of the Lewis Bending Equation ........................................................ 4
3. Analysis ....................................................................................................................... 7
3.1
3.2
Microsoft Excel Analysis ................................................................................... 7
3.1.1
Determination of the Lewis Bending Stress .......................................... 7
3.1.2
Determination of the AGMA Bending Stress ........................................ 7
3.1.3
Determination of the AGMA Contact Stress ......................................... 7
Abaqus Finite Element Analysis ........................................................................ 8
3.2.1
2D Lewis Bending Stress Model............................................................ 8
3.2.2
3D AGMA Tooth-Root Bending Stress Models .................................. 13
3.2.3
2D AGMA Tooth-Root Bending Stress Models .................................. 17
3.2.4
3D AGMA Contact Stress Model ........................................................ 20
3.2.5
2D AGMA Contact Stress Model ........................................................ 25
4. Results and Discussion .............................................................................................. 28
4.1
Comparison of Data ......................................................................................... 28
4.2
Sources of Error and Divergence ..................................................................... 29
5. Conclusion ................................................................................................................. 31
ii
6. Appendices ................................................................................................................ 32
6.1
Microsoft Excel Analysis – Lewis Bending Equation ..................................... 32
6.2
Microsoft Excel Analysis – AGMA Design Equations ................................... 33
6.3
Mesh Convergence Studies .............................................................................. 41
6.3.1
2D Lewis Bending Stress Model.......................................................... 42
6.3.2
3D AGMA Tooth-Root Bending Models ............................................ 45
6.3.3
2D AGMA Tooth-Root Bending Models ............................................ 48
7. References.................................................................................................................. 51
iii
LIST OF TABLES
Table 1 – Max Bending Stress for the Pinion and the Gear in the 2D Lewis Bending
Stress Model ............................................................................................................ 11
Table 2 - Results of the 3D AGMA Tooth-Root Bending Stress Models ....................... 16
Table 3 - Results of the 2D AGMA Tooth-Root Bending Stress Models ....................... 18
Table 4 - Results of the 3D AGMA Tooth-Root Bending Stress Models ....................... 24
Table 5 - Results of the 2D AGMA Contact Stress Model ............................................. 27
Table 6 - Comparison of Stresses for the 2D Lewis Bending Analysis .......................... 28
Table 7 – Comparison of Stresses for the 3D Tooth-Root Bending Analysis ................. 28
Table 8 – Comparison of Stresses for the 2D Tooth-Root Bending Analysis ................. 28
Table 9 – Comparison of Stresses for the 3D Contact Analysis...................................... 28
Table 10 – Comparison of Calculated Stresses for the 2D Contact Analysis ................. 28
Table 11 – Mesh Convergence Data for the Pinion in the 2D Lewis Bending Load Case
................................................................................................................................. 42
Table 12 - Mesh Convergence Data for the Gear in the 2D Lewis Bending Load Case . 44
Table 13 - Mesh Convergence Data for the Pinion in the 3D AGMA Bending Load Case
................................................................................................................................. 45
Table 14 - Mesh Convergence Data for the Gear in the 3D AGMA Bending Load Case
................................................................................................................................. 46
Table 15 - Mesh Convergence Data for the Pinion in the 2D AGMA Bending Load Case
................................................................................................................................. 48
Table 16 - Mesh Convergence Data for the Gear in the 2D AGMA Bending Load Case
................................................................................................................................. 49
iv
LIST OF FIGURES
Figure 1 – Gear Tooth Material Resisting Bending Load (modified from Ref. 12) .......... 4
Figure 2 – Gear Tooth Dimensions used in ....................................................................... 5
Figure 3 – Mesh Distribution of the Gear for Lewis Bending Case .................................. 9
Figure 4 – Mesh Distribution of the Pinion for Lewis Bending Case ............................... 9
Figure 5 – S22 (Vertical) Stress Distribution in the Pinion ............................................. 12
Figure 6 - S22 (Vertical) Stress Distribution in the Gear ................................................ 12
Figure 7 – Boundary Condition and Coupling used for Gears ........................................ 14
Figure 8 – Load Coupling Used for Each Gear Face....................................................... 14
Figure 9 – Isometric View of the Mesh Distribution for the Gear .................................. 15
Figure 10 – Isometric View of the Bending Stress Distribution in the Gear ................... 15
Figure 11 – Pinion Mesh Distribution for the 2D AGMA Bending Model..................... 18
Figure 12 – Pinion Vertical Stress (S22) Distribution in the 2D AGMA Bending Model
................................................................................................................................. 19
Figure 13 – Isometric View of the Gear Teeth in Contact, with Coordinate Systems .... 20
Figure 14 – Contact Interaction Surfaces for the 3D AGMA Contact Model ................. 21
Figure 15 – Mesh Distribution in the Pinion Tooth ......................................................... 22
Figure 16 – CPRES Distribution in the Pinion Face ....................................................... 23
Figure 17 – Isometric View of the Von Mises Stress in Both Gears ............................... 24
Figure 18 – Contact Interaction Master and Slave Surfaces Chosen in the 2D AGMA
Contact Model ......................................................................................................... 25
Figure 19 – Mesh Distribution in the Teeth of the Pinion and Gear for the 2D AGMA
Contact Model ......................................................................................................... 26
Figure 20 – Resulting Von Mises Stress Distribution of the Contacting Gears in the 2D
AGMA Contact Model ............................................................................................ 27
Figure 21 – Path Used for the 2D Lewis Bending Model Pinion Mesh Convergence
Study (Gear is Similar) ............................................................................................ 43
Figure 22 – Path Used in the Gear in the Mesh Convergence Study for the 3D AGMA
Bending Load Case (Pinion Similar) ....................................................................... 47
Figure 23 - Path Used for the Pinion in the Mesh Convergence Study for the 2D AGMA
Bending Load Case (Gear Similar).......................................................................... 50
v
ACKNOWLEDGMENT
I had a lot of assistance and support from friends, family, and loved ones while
developing this Masters project. Thanks go out to Nick, Taylor, and Chet for their
technical advice and assistance during the development of the finite element models. In
addition, I’d like to thank my former coworker Mike for his guidance in the development
of the analysis that this project was based on.
Lastly thanks go to my girlfriend,
Lindsey, for her patience and encouragement throughout the process.
vi
SYMBOLS AND VARIABLES
Symbol/Variable
Description
Units
Cf
Surface condition factor
-
CH
dg
dp
E
F
HB
I
J
KB
Km
Ko
KR
Ks
Kv
mG
ν
φ
PdG
PdP
Qv
σcG
σcP
σG
Hardness ratio factor
Pitch diameter of the gear
in
Pitch diameter of the pinion
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
AGMA quality factor
AGMA contact stress, gear
AGMA contact stress, pinion
in-1
psi
psi
AGMA bending stress on the gear
psi
σL
σP
satG
satP
ScG
ScP
SFG
SFP
SHG
SHP
SY
Lewis bending stress
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
Bending fatigue failure safety factor, gear
psi
psi
Bending fatigue failure safety factor, pinion
Wear factor of safety, gear
psi
-
Wear factor of safety, pinion
-
Yield strength of the gears
psi
vii
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
-
viii
lbf-in
lbf
-
ACRONYMS
1. AGMA – The American Gear Manufacturers Association. AGMA publishes
standards on gears, and is one of the trusted authorities on gear design and
analysis. The AGMA standard 2001-D04 is used in this report to calculate the
maximum tooth-root bending stress as well as the max pitting contact stress in
the gears.
2. Pitting – A failure mode in gears caused by high or repeated surface contact
stresses. Pitting is a fatigue phenomenon, and the expression for the contact
stress is based on Hertzian contact.
3. Pressure Line – The pressure line goes through the contact point of two meshing
gear teeth, and represents the direction of contact force between the two gears. It
is perpendicular to the base circles of both meshing gears.
4. Pressure Angle – The angle of the pressure line with respect to the horizontal.
5. Base Circle – The base circle of a gear is an imaginary circle that is
perpendicular to the pressure line between two meshing gears. In addition, it is
used to create the involute profile of a gear tooth profile.
6. Addendum and Dedendum Circles – The circle swept out by the top edges of the
gear teeth is the addendum circle. The dedendum circle is the circle tangent to
the base of the gear teeth, below the root radius.
7. Pitch Diameter – The pitch diameters of two meshing gears will be tangent at the
pressure line. This diameter is larger than the base circle, but smaller than the
addendum of the gear.
8. Involute Profile – An involute profile is the traditional method of creating
“involute” gear teeth. It is formed by connecting points created from lines
perpendicular to the base circle of the gear.
9. Abaqus – The finite element software used in this report to calculate gear
stresses.
10. CAD – Computer Aided Drafting.
11. Module – A ratio of the pitch diameter to the number of teeth in a gear.
ix
12. Diametral Pitch –A measure of the number of teeth per inch along the pitch
diameter. It is the reciprocal of the module.
13. Face Width – The thickness of a gear tooth (distance parallel to the main axis of
the gear).
14. Mesh Convergence Study – In finite element analysis, it is important to ensure
that the mesh density used in the model is resulting in accurate data. One way to
confirm this is to iteratively increase the mesh until the percentage change in
some reference value levels off.
15. Yield Strength – The stress at which a material begins to plastically deform.
x
ABSTRACT
The Lewis bending stress, the American Gear Manufacturers Association
(AGMA) tooth-root bending stress, and the AGMA pitting contact stress are calculated
for a gear mesh consisting of two spur gears. Gear stresses are calculated in Microsoft
Excel using the AGMA standards as well as in Abaqus 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 were compared in order to determine the
effectiveness of the finite element method to design gears. In conclusion, there was a
strong correlation between AGMA and Abaqus for tooth-root bending stresses. The
contact stresses calculated by Abaqus did not match as well with the AGMA standards.
1
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 may be
estimated using 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 compare the gear tooth-root bending stress and
contact stress as calculated by the finite element method and the AGMA standards for a
spur gear mesh. In addition, a comparison is made between the Lewis Bending stress as
calculated by hand and as calculated in Abaqus. The analyses are based on two gears: a
18-tooth, 1.5” pitch diameter pinion spur gear and a 54-tooth, 4.5” pitch diameter spur
gear. Microsoft Excel is used to calculate the AGMA and Lewis Bending stresses, and
Abaqus is used to calculate the finite element stresses. Seven Abaqus models were
created, including both 3D and 2D elements. To calculate the contact stress, one 3D
model and one 2D model were created. Four models calculate bending stresses, with 3D
and 2D versions of the gear and the pinion. The last model is a 2D model created to
calculate the Lewis Bending stress on one tooth of the pinion gear. The Abaqus analyses
will be static (as opposed to dynamic) in order to simplify the analysis. For the 3D
Abaqus contact model, the gears modeled were reduced from solid gears to single teeth
in order to simplify the analysis as well as greatly reduce computation time. This is a
strategy that was used effectively by Patel to model 3D gear stresses (Ref. 14). In the
3D tooth-root bending models, all but one tooth was removed from the gear to reduce
computation time.
2
In order to good correlation between the Abaqus stresses and the AGMA stresses
(Ref. 1), various 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.
3
2. Theory and Methodology
2.1 Derivation of the Lewis Bending Equation
Wt
F
t
Figure 1 – Gear Tooth Material Resisting Bending Load (modified from Ref. 12)
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
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 above). Based on this assumption,
4
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.
The following section derives the Lewis bending stress equations used in the
analyses. It is based on a derivation in “Shigley’s Mechanical Engineering Design”
Chapter 14 (Ref. 3).
As mentioned previously, the Lewis Bending stress is based on
bending of a cantilevered beam. As shown in Fig. 1, the cross sections of the “fixed”
end of the gear tooth are “F” x “t”. The load (Wt) is applied at a height of “L” above the
base. Based on these variables, the moment, “M”, is be Wt*L, and the section modulus,
I/c, is (F*t2) /6. The bending stress is therefore M/(I/c), or (6*Wt*L)/( F*t2).
Figure 2 – Gear Tooth Dimensions used in
Lewis Bending Equation (Ref. 3)
A separate form of the equation which may be more useful for engineers makes
use of the diametral pitch and a factor y called the Lewis Form Factor. We introduce a
variable “x”, as shown in Fig. 2 (see Ref. 3, Figure 14-1(b)). Using the similar triangles
relationship:
(t/2)/x = L/(t/2)
x = t2/(4*L)
(1)
We can now rearrange the bending stress equation calculated in the previous paragraph:
σ = (6*Wt*L)/( F*t2) = (Wt/F)*1/(t2/6*L) = (Wt/F)*1/(t2/4*L)*1/(4/6)
5
Note that the equation for “x” can be substituted into the stress equation above. We also
multiply the numerator and denominator by the circular pitch, “p”:
σ = (Wt*p)/(F*(2/3)*x*p)
Introducing a new factor y = (2*x)/(3*p), the above equation changes to:
σ = Wt/(F*p*y)
(2)
The new factor, “y”, is called the Lewis form factor (Ref. 3). As stated above, it is often
easier for engineers to work with the diametral pitch (instead of the circular pitch) since
it is more commonly presented in reference books. We therefore substitute the diametral
pitch, “P”, into the equation by setting P = π/p. In addition, we set Y = π*y. This
version of the Lewis form factor is most commonly used, and its values are tabulated for
various types and sizes of gears. The final version of the Lewis bending equation is
shown below:
σ = (Wt*P)/(F*Y)
6
(3)
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. In an attempt to get the most accurate results, dimensions
“t” and “L” from the first form of the Lewis bending equation (see section 2.1) were
measured directly from the Abaqus model of the gear tooth profile.
3.1.2
Determination of the AGMA Bending Stress
The Excel analysis of the AGMA bending stress uses the equations found in
AGMA 2001-D04 (Ref. 1). In order to simplify this analysis to the point where a
comparison could be made between the Excel stresses and those in the Abaqus models,
some of the factors found in the AGMA standard needed to be reduced to unity. They
are the overload factor, Ko, the dynamic factor, Kv, the stress cycle factor, YN, the
temperature factor, KT, the reliability factor, KR, and the rim thickness factor, KB. An
explanation of all the factors, as well as rationale for reducing them, can be found in
appendix 6.2.
3.1.3
Determination of the AGMA Contact Stress
AGMA 2001-D04 is also used to calculate the max AGMA contact stress in Excel.
Similar to the AGMA bending stress excel analysis, the surface condition factor, Cf, was
assumed to be unity to simplify the analysis. Many of the factors mentioned in section
3.1.2 are also used to calculate the AGMA contact stress. An explanation of all the
factors, as well as rationale for reducing them, can be found in appendix 6.2.
7
3.2 Abaqus Finite Element Analysis
This section of the report discusses the various Abaqus finite element models used
to calculate the stress distributions in the gears. 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 convergences were carried about for some of the models to
ensure an adequate mesh density. See section 4.1 for a comparison between the Abaqus
results and the AGMA results as calculated by Excel.
3.2.1
2D 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.).
Both the pinion gear and the larger gear were analyzed in this model. A 2D CAD
file was acquired from the Rush Gears website (Ref. 10), and turned into a *step sketch
file in Abaqus. A 2D deformable shell part was then created using this sketch to capture
the tooth profile of the gears. The geometry required some fixing in order to result in a
meshable part, such as removing redundant and invalid edges. Once the geometry was
cleaned up, all but one tooth was removed from the hub of the gear. The other teeth
were removed to decrease computation time and increase the mesh in the areas of
interest. A solid, homogeneous section was applied to both gears. The thickness of the
section was set to 0.5 in, to match the Excel analysis. Figures 3 and 4 show the mesh
distributions of the final geometry of the pinion and gear used in this analysis.
8
Figure 3 – Mesh Distribution of the Gear for Lewis Bending Case
Figure 4 – Mesh Distribution of the Pinion for Lewis Bending Case
9
The material chosen for both gears is AISI 4140 steel, to match the
corresponding analysis performed in Excel (see Ref. 13). 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.
There are three important assumptions of the Lewis Bending equation, as stated
in Ref. 3: the gear tooth is treated as a cantilevered beam, only one tooth in the mesh
resists the load, and the max stress will be bending and occur in the root of the tooth. To
emulate a fixed condition of the gear tooth, a continuum distributing coupling was
created. This coupling originates at a reference point in the center of the gear, and
couples to the circle of nodes on the inner diameter of each gear. A boundary condition
was created at this reference point, constraining both gears in all degrees of freedom (inplane translations and out of plane rotation).
The next step was to apply the transmitted load (Wt) to the tooth, to match the
applied load in the Excel analysis. Shigley states that the Lewis Bending equation
assumes that the load is applied completely at the top of the tooth, evenly distributed
across the face width, F (Ref. 3). In order to accurately reflect this loading condition, a
load was applied using a shear surface traction at the top face of the tooth, directed
tangential to the diameter of the gear. The load was 31,372 lbf/in2 for the pinion, and
24615 lbf/in2 for the gear. These loads correspond to a 800 lbf load divided by the area
of the top face of the tooth. The top face of the gear tooth is larger than that of the
pinion, hence the smaller pressure applied. This method of applying the load results in
more accurate data because it does not create stress concentrations at the application of
the load.
Once the gear tooth was simplified, partitioned, bounded, and had an applied
load, the next step was to mesh the geometry. 8-node biquadratic plane stress quad
element types were used, with “free” quad element shapes. Plane stress elements were
used because the stress distribution in the gear teeth are uniform through the thickness of
10
the gear tooth, and also because it resulted in more realistic 2D stress distributions than
plane strain elements.
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 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. A mesh convergence study was carried out for this model, as well as
various other models in this report. See Appendix 6.3 for data for all the convergence
studies.
The geometry of the tooth profile as well as the location of the calculated stress
has a significant impact on the resulting stress. To stay consistent with the Excel
analysis, stresses in Abaqus were calculated at the surface of the gear tooth at a distance
“L” from the top face of the gear. The “L” distance used for the pinion and the gear can
be seen in Appendix 6.1. The thickness of the gear tooth, dimension “t”, at this height
was used in the Excel analysis. The resulting max bending stresses for both the pinion
and the gear are shown in Table 1 below. Figures 5 and 6 show elevation views of stress
distributions for both the pinion and the gear. See section 4.1 for a comparison of the
data to what was calculated in Excel.
Max Bending Stress (S22) [psi]
Pinion
64366
Gear
52346
Table 1 – Max Bending Stress for the Pinion and the Gear in the 2D Lewis Bending Stress Model
11
Figure 5 – S22 (Vertical) Stress Distribution in the Pinion
Figure 6 - S22 (Vertical) Stress Distribution in the Gear
12
3.2.2
3D AGMA Tooth-Root Bending Stress Models
Both the gear and the pinion were generated from the “Rush Gears” website (see
Ref. 11) as *step files. These files were imported into Abaqus as 3D solid parts. To
reduce computation time, all but one tooth was removed via extrusion for both gears.
Some work needed to be done to the geometry, such as removing invalid edges.
The gears were modeled as AISI 4140 steel (see Ref. 13), with modulus of elasticity
rounded to 30E6 psi and poisson’s ratio 0.3. Solid, homogeneous section properties
were used for both gears.
For each gear, a boundary condition with all degrees of freedom fixed was imposed
at the center point. This point was coupled to the inside diameter face using a continuum
distributing coupling, as shown in Fig. 7.
The load, Wt, was applied to both gears perpendicular to the gear tooth face, at the
pitch diameter of the gear. This was done because when two involute spur gears mesh
together, they contact at the tangent between the two pitch diameters. A continuum
distributing coupling was used to distribute the load from a reference point to the rest of
the gear tooth face for both the pinion and the gear (see Fig. 8), with a custom
rectangular coordinate system originating at the top of the gear tooth. Using a coupling
to apply the load instead of applying it to an edge on the tooth resulted in more accurate
stress results.
Figure 9 shows the final mesh distribution for the gear (the pinion was similar). The
mesh was separated between the hub of the gears and the gear tooth. In both sections,
20-node quadratic hex elements were used. In the gear teeth, “reduced integration” was
turned off, and structured hex elements were used (C3D20 elements). In the hubs,
“reduced integration” was turned on to reduce computation time (C3D20R elements).
Appendix 6.3.2 contains the mesh convergence study performed for this analysis.
13
Figure 7 – Boundary Condition and Coupling used for Gears
Figure 8 – Load Coupling Used for Each Gear Face
14
Figure 9 – Isometric View of the Mesh Distribution for the Gear
Figure 10 – Isometric View of the Bending Stress Distribution in the Gear
15
Max Bending Stress [psi]
Pinion
74072
Gear
53571
Table 2 - Results of the 3D AGMA Tooth-Root Bending Stress Models
The maximum bending stress calculated by Abaqus for the 3D gear and pinion are
shown in Table 2 above. Fig. 10 shows the resulting bending stress (stress in the vertical
direction, perpendicular to the base of the tooth) distribution in the gear (the pinion had a
similar distribution). See section 4.1 for a comparison between these results and the
stresses calculated in Excel.
16
3.2.3
2D AGMA Tooth-Root Bending Stress Models
For the 2D AGMA bending models, 2D sketches (matching the same profiles as the
gears used in the 3D models) were downloaded from the “Rush Gears” website (see Ref.
11). These sketches were used to create 2D planar, deformable, shell parts. Some
geometry edits were required to use the parts, such as removing redundant edges. In
order to reduce computation time, all but one tooth were removed via extrusion. Lastly,
the remaining tooth (for both the pinion and the gear) was partitioned at the location of
the pitch diameter. This created a marker to apply the tangential bending load.
The gears were modeled as AISI 4140 steel (see Ref. 13), the same material used in
the other models.
A solid, homogeneous section was applied to both gears, with
thickness equal to 0.5”.
Similar to the 3D AGMA bending models, two continuum distributing coupling
constraints were used for each part. A reference point was created at the center of both
gears and a coupling was distributed from the point to the nodes at the ID of the gears.
This coupling was used with the boundary conditions of the gears. A second reference
point was created at the pitch diameter partition, on the surface of the gear tooth. A
coupling was then applied at this point and distributed to one surface of the gear tooth.
This coupling is used with the load applied to each part.
For each gear, a boundary condition with all degrees of freedom fixed was imposed
at the center reference point. The coupling applied at the same point distributed the
boundary condition to the rest of the gear.
The load, Wt, was applied to both gears at the second coupling created. Wt is a
horizontal force, tangent to the circumference of the gear hub. The load was applied at
the pitch diameter because when two involute spur gears mesh together this is where
they contact. Using a coupling to apply the load instead of applying it to an edge on the
tooth resulted in more accurate stress results.
Figure 11 shows the final mesh distribution for the pinion (the gear was similar). 8node biquadratic plane stress quad element types were used, with “free” quad element
shapes. Edge seeds were created in the gear tooth to increase the mesh density of the
model at the locations of the max stresses.
convergence study performed for this analysis.
17
Appendix 6.3.3 contains the mesh
Figure 11 – Pinion Mesh Distribution for the 2D AGMA Bending Model
Max Bending Stress (S22) [psi]
Pinion
69318
Gear
53547
Table 3 - Results of the 2D AGMA Tooth-Root Bending Stress Models
The maximum bending stress calculated by Abaqus for the 3D gear and pinion are
shown in Table 3 above. Fig. 12 shows the resulting bending stress (stress in the vertical
direction, perpendicular to the base of the tooth) distribution in the pinion (the gear had a
similar distribution). See section 4.1 for a comparison between these results and the
stresses calculated in Excel.
18
Figure 12 – Pinion Vertical Stress (S22) Distribution in the 2D AGMA Bending Model
19
3.2.4
3D AGMA Contact Stress Model
The same 3D solid gears were used in the 3D AGMA contact stress model as were
used in the 3D tooth-root bending stress models. Instead of using the entire hub with
one tooth modeled, the gears were reduced to single-tooth sectors in the 3D contact
model (see Fig. 11). Minimizing the number of elements in the model was paramount
because contact analyses require much more computation time than simple bending
models.
The gears were modeled as AISI 4140 steel (see Ref. 13), with modulus of elasticity
rounded to 30E6 psi and poisson’s ratio 0.3. Solid, homogeneous section properties
were used for both gears.
Figure 13 – Isometric View of the Gear Teeth in Contact, with Coordinate Systems
For the gear mesh assembly, a “parallel face” position constraint was used on the
two gears to ensure that there wasn’t any misalignment in the mesh. The two gear
instances were translated such that the distance between their centers was equal to the
sum of the pitch radii. Since the pinion has a 1.5” pitch diameter and the gear has a 4.5”
pitch diameter, the center-to-center distance was 3.0”.
Two steps were used: an initial step where boundary conditions and couplings were
initiated, and a static, general load step. In order to resolve the initial gap between the
pinion and the gear, contact controls are used with automatic stabilization. This tool
helps Abaqus to close the gap and converge. A contact interaction was created to detect
the contact stresses between the two gears. The gear faces were chosen as the master,
and the pinion gear faces were chosen as the slave (see Fig. 14). Interaction properties
were used in conjunction with the contact interaction, defining “normal” contact and
20
“tangential” contact. The normal contact is set for “hard” contact, and the tangential
contact uses the “penalty” option with a coefficient of friction set at 0.2.
Figure 14 – Contact Interaction Surfaces for the 3D AGMA Contact Model
For all boundary conditions, cylindrical coordinate systems were used. The systems
originated at the center of each gear, with the r-axis in the radial direction, the theta axis
tangent to the pitch diameter, and the z-axis through the thickness of the gears. Similar
to the 3D tooth-root bending models, boundary conditions were applied to a reference
point at the center of the gears. The reference point was coupled using a kinematic
coupling to the inner diameter face of each gear. For the larger (driven) gear, this
boundary condition constrained all degrees of freedom. For the smaller (driver) pinion,
rotation about its axis was unconstrained since it must be able to rotate. In addition to
these boundary conditions, symmetry boundary conditions were used on the faces that
were cut. Based on how the gears were cut, symmetry about the theta axis is applicable.
Naturally, for the larger gear the exposed faces were constrained in all degrees of
freedom except for translation along the r-axis. The boundary condition on the exposed
21
face of the smaller pinion gear needed to be altered since it has to move in the theta
direction to strike the larger gear. For these boundary conditions, all degrees of freedom
were constrained except translation in the theta direction.
A moment (torque) was applied to the center reference point of the smaller (driver)
pinion. The torque was 600 in-lbf, equal to Tp in the Excel analysis (see Appendix 6.2).
This torque rotates the pinion into the gear, resulting in the contact stress on the surfaces
of the gears and bending stress in the roots of both gears.
20-node quadratic, non-reduced integration mesh elements were used. The element
shape was hex structured. Figure 15 shows the mesh distribution in the pinion tooth (the
gear tooth was similar). An edge seed was created on the face of the tooth to locally
increase the mesh near the contact point of the two gears.
Figure 15 – Mesh Distribution in the Pinion Tooth
22
Figure 17 shows the resulting von mises stress distribution of the deformed gears in
contact. As you can see, the highest stress in the assembly occurs at the contact point.
Figure 16 shows the contact pressure (CPRES) distribution in the pinion gear, with
results detailed in Table 4 below. See section 4.1 for a comparison between these results
and the stresses calculated in Excel.
Figure 16 – CPRES Distribution in the Pinion Face
23
Figure 17 – Isometric View of the Von Mises Stress in Both Gears
Max CPRES (Contact Pressure) [psi]
Pinion
Gear
200437
116796
Table 4 - Results of the 3D AGMA Tooth-Root Bending Stress Models
24
3.2.5
2D AGMA Contact Stress Model
The same 2D shell gear parts were used in the 2D AGMA contact stress model as
were used in the 2D tooth-root bending stress model. Similar to the 2D AGMA bending
stress models, all but one tooth were removed via extrusion. It is especially important to
minimize the number of elements in models involving contact stress because the analysis
requires much more computation time than simple bending models.
The gears were modeled as AISI 4140 steel, the same material used in the other
models. A solid, homogeneous section was used for both gears, with thickness equal to
0.5”.
The two gear instances were translated such that the distance between their centers
was equal to the sum of the pitch radii. Since the pinion has a 1.5” pitch diameter and
the gear has a 4.5” pitch diameter, the center-to-center distance was 3.0”.
Figure 18 – Contact Interaction Master and Slave Surfaces Chosen in the 2D AGMA Contact Model
25
Two steps were used: an initial step where boundary conditions and couplings were
initiated, and a static, general load step. In order to resolve the initial gap between the
pinion and the gear, contact controls are used with automatic stabilization. This tool
helps Abaqus to close the gap and converge. A contact interaction was created to detect
the contact stresses between the two gears. The gear faces were chosen as the master,
and the pinion gear faces were chosen as the slave (see Fig. 18). Interaction properties
were used in conjunction with the contact interaction, defining “normal”
contact and “tangential” contact. The normal contact is set for “hard” contact, and the
tangential contact uses the “penalty” option with a coefficient of friction set at 0.2.
Two boundary conditions were created: one at the center of the gear, and the other
at the center of the pinion. The boundary conditions were the same as those used in the
3D AGMA contact model: the gear was constrained in all degrees of freedom, while the
pinion was constrained from translating but allowed to rotate about its axis. The load
was applied in the same was as in the 3D AGMA contact model.
Figure 19 – Mesh Distribution in the Teeth of the Pinion and Gear for the 2D AGMA Contact Model
26
8-node biquadratic plane strain quadrilateral mesh elements were used. The element
shape was quadrilateral “free”.
The elements in the tooth of the gear were non-
integration reduced, while the elements in the hub were integration reduced to minimize
computation time. Figure 19 shows the mesh distribution in the pinion tooth (the gear
tooth was similar). An edge seed of 30 elements was created on the face of the tooth to
locally increase the mesh near the contact point of the two gears.
Figure 20 – Resulting Von Mises Stress Distribution of the Contacting Gears in the 2D AGMA
Contact Model
Figure 20 shows the resulting Von Mises stress distribution of the gears in contact.
Obviously, the highest stress in the assembly occurs at the contact point. Table 5 below
contains the max CPRES (contact pressure) stresses computed by Abaqus for the pinion
and the gear. See section 4.1 for a comparison between these results and the stresses
calculated in Excel.
Max CPRES (Contact Pressure) [psi]
Pinion
98173
Gear
98269
Table 5 - Results of the 2D AGMA Contact Stress Model
27
4. Results and Discussion
4.1 Comparison of Data
This section details the calculated stresses from Abaqus and Excel for each load
case. In addition, a percentage error between the two results is calculated.
2D Lewis Bending Model vs Excel Analysis
Stresses
Error
Pinion [psi]
Gear [psi]
Pinion
Gear
Abaqus
Excel
Abaqus
Excel
1.2%
3.8%
64366
63573
52346
50418
Table 6 - Comparison of Stresses for the 2D Lewis Bending Analysis
3D AGMA Tooth-Root Bending Stresses vs. Excel Analysis
Stresses
Error
Pinion [psi]
Gear [psi]
Pinion
Gear
Abaqus
Excel (AGMA)
Abaqus
Excel (AGMA)
9.8%
1.4%
74072
67454
53571
54352
Table 7 – Comparison of Stresses for the 3D Tooth-Root Bending Analysis
2D AGMA Tooth-Root Bending Stresses vs. Excel Analysis
Stresses
Error
Pinion [psi]
Gear [psi]
Pinion
Gear
Abaqus
Excel (AGMA) Abaqus
Excel (AGMA)
2.8%
1.5%
69318
67454
53547
54352
Table 8 – Comparison of Stresses for the 2D Tooth-Root Bending Analysis
3D AGMA Contact Stresses vs. Excel Analysis
Stresses
Error
Pinion [psi]
Gear [psi]
Pinion
Gear
Abaqus
Excel (AGMA) Abaqus
Excel (AGMA)
12.2%
11.7%
200437
224890
116796
130461
Table 9 – Comparison of Stresses for the 3D Contact Analysis
2D AGMA Contact Stresses vs. Excel Analysis
Stresses
Error
Pinion [psi]
Gear [psi]
Pinion
Gear
Abaqus
Excel (AGMA) Abaqus
Excel (AGMA)
57.1%
25.9%
98173
224890
98269
130461
Table 10 – Comparison of Calculated Stresses for the 2D Contact Analysis
28
4.2 Sources of Error and Divergence
There are various sources of error both in the Abaqus models as well as the AGMA
equations that could explain the error in the results of this report. The sources of error
are discussed below, divided into sections based on the type of analysis performed.
The contact models, both for 3D and 2D elements, contained the highest number
of potential error sources. To begin with, contact stress analysis in any finite element
software involves non-linear equations. Any time results are calculated in a non-linear
system, small deviations in input parameters lead to larger errors in output as compared
to a linear analysis. In the case of a gear mesh, the majority of these input errors can be
explained by the tooth geometry and the tooth mesh itself. The AGMA contact stresses
are based on the assumption that the tooth profiles are perfect, resulting in tangential
pitch diameters between the two gears. The CAD files used in this report to create the
gear parts were not exact; when they were assembled and constrained at the theoretical
center-to-center distance, an interference existed. The most likely cause of this error is
that the software methods used by Rush Gears (Ref. 11) to create their CAD files is
flawed. The center-to-center distances had to be tweaked such that a good gear mesh
existing, thus changing the location of the contact line. The next source of error comes
from the assumptions of AGMA itself. The AGMA design equations are intended for
use with spinning gears, operating for many cycles. To simplify the analysis, the speed
of the gears was reduced to zero and the cycles were unity. While you can still calculate
AGMA stresses for these conditions (various factors reduce to unity), it’s possible that
the results aren’t as accurate since the empirical data is based on operational gears.
Lastly, error was introduced into the results due to the boundary conditions applied to
the gears. The gears were cut into single sectors to reduce computation time, however in
doing so the exposed faces needed to be constrained. Symmetry conditions were applied
to these faces, however the result isn’t exactly comparable to modeling full gears in
contact.
The tooth-root bending stress models had their own sources of error. Similar to
the contact models, errors in the gear CAD files introduced error in the results. The
loads in these models were applied at the theoretical pitch diameter of the gear, based on
where theoretically perfect meshing gears would contact. Errors in the geometry of the
29
gears used will alter the bending stresses calculated in Abaqus. The second source of
error came from how the load was applied to each gear. Using a coupling to distribute
the force more evenly on the face of the tooth provides smoother stress distributions,
however the most realistic technique.
While the load is still applied at the pitch
diameter, it is distributed to the rest of the tooth face. In reality, the load is transferred to
the tooth over a much smaller area around the contact point between the two gears.
Using a surface traction on a face to apply a load is the most accurate method, since the
load can be applied exactly where you want without the risk of stress concentrations at
the application point. Unfortunately, this method cannot be used when the load is not
applied parallel to an exposed face.
The Lewis bending models resulted in the most accurate data in the report. The
imperfections of the gear CAD files introduced error in these models just the same as the
rest, however it did not impact results as much since the Lewis bending stresses were
calculated away from the root of the gear tooth. As shown by Figs. 5 & 6, the stresses in
the root of the tooth were much higher than the stress calculated by the Lewis bending
equation. This is because the Lewis bending equation does not take into account stress
concentrations created by the geometry of the tooth. This makes the Lewis bending
equation highly dependent on geometry, because the stress must be calculated far
enough away from the root such that there are no stress spikes.
30
5. Conclusion
Comparisons of bending and contact stresses have been made for various load cases
between the Abaqus finite element software and the AGMA standards (as calculated by
Excel). Overall, the bending stresses were very similar when compared between the two
calculated methods. The contact analyses were not as accurate, possibly due to the error
sources discussed in the previous section. For future analyses, there are a few things that
can be changed which would reduce solution errors. Firstly, the gear tooth profiles
should be created in CAD using the theoretical involute profile. It is a fairly simple
process, and can be completed as long as the pitch diameters, number of teeth, and
pressure angle are known for both meshing gears. This process would reduce any
geometrical error in the gear mesh, and result in data more in line with theoretical
predictions. Secondly, for contact analyses, symmetry boundary conditions should be
avoided. If a high-powered computer can be acquired, both meshing gears in full should
be meshed and compiled in the analysis. If computer resources are limited, utilizing
cyclic symmetry could be a useful tool. Cyclic symmetry allows the user to impose
symmetry to a sector and specify how many times that sector repeats. Abaqus contains a
symmetry interaction, however it can only be used around one revolution point. Perhaps
a different finite element software package has a cyclic symmetry option that can be
used with meshing gears.
In conclusion, the finite element method should be used with caution when
analyzing gears. Static, implicit models like those discussed in this report provide good
results when calculating static load cases with uniform boundary conditions. However,
most gear failures are the result of fatigue loading (surface pitting, cracking). These
types of failures often occur after hundreds of thousands of cycles, and are affected by
many variables. The AGMA equations are perhaps a more accurate tool when it comes
to predicting gear failure. They are based on empirical results from decades of data on
gearing, and incorporate many parameters that a finite element software package cannot.
31
6. Appendices
6.1 Microsoft Excel Analysis – Lewis Bending Equation
4.7 Lewis Bending Analysis during Stall Condition
The gears are analyzed for stall torque using the Lewis Bending equation. The transmitted load for is equal to the tangential load caused by
the operational torque used in the AGMA analyses.
Wt [lbf] =
800
Tangential transmitted gear load. See section 4.1 and 4.2 for inputs.
*See Fig. 14-1(b) in Ref. 4.
t [in] =
L [in] =
Pinion
0.15
0.149
Gear
0.169
0.15
F [in] =
0.5
0.5
σ = (12*Wt*L)/(2*F*t^2)
Tooth thickness
Length of tooth
Face width of
tooth
Modified equation for the Lewis Bending stress. This version is used because dimensions t, L,
and F can be measured directly from the tooth profile of the CAD drawing. See Eq. (d) in
section 14-1 of Ref. 4.
Pinion
σL [psi] =
σLP =
FSL = Sy / σL
Gear
σLG =
63573
Factors of safety are calculated on material yield strength.
Pinion
FSL [ ] =
50418
FSLP =
Gear
1.0
FSLG =
32
1.2
Lewis Bending stress.
6.2 Microsoft Excel Analysis – AGMA Design Equations
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
0.5
1.5
12
0.5
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.
4.3 Material Properties and Other Input Variables
AISI 4140 steel is picked as the gear material. The following material information was retrieved from Ref. 13:
E [psi] =
ν[]=
Sy [psi] =
SUT [psi] =
HB [ ] =
30000000
0.3
61000
95000
197
Modulus of Elasticity of the gears
Poisson's Ratio of the gears
Yield Strength of the gears
Ultimate Strength of the gears
Brinell Hardness of the gears
33
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.
Dynamic Factor, Kv
Kv = ((A+V0.5)/A)B
A = 50 + 56(1 - B)
B = 0.25(12 - Qv)0.66
B[]=
A[]=
Kv [ ] =
0.90
55.4
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.
34
Pinion
Ks [ ] =
Gear
KsP =
0.97
KsG =
0.98
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
Pinion
Gear
Cmc [ ] =
CmcP =
1
CmcG =
1
Cpf [ ] =
CpfP =
0.02
CpfG =
0.02
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
0.13
1
1.15
1.15
BG =
CG =
CmaG =
CeG =
CmfG =
KmG =
0.0158
-0.0001
0.13
1
1.15
1.15
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.
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.
35
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).
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 =
The Backup Ratio for the Rim Thickness
Factor (Ref. 1)
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] =
σ = WtKoKvKs(Pd/F)(KmKB/J)
800
Wt is the transmitted tangential load going into the pinion gear.
See Ref. 4, Eq. 14-15
36
Pinion
σP =
σ [psi] =
Gear
σG =
67454
54352
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.
The Allowable Bending Stress Number, sat, for AISI 4140 (see Fig. 10,
Ref. 1)
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
0.6
SFG =
0.7
37
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
Gear Contact Stress, σc
38
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 =
228485
132390
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 =
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.
39
Wear Factor of Safety, SH
SH = [ScZNCH / (KTKR)] / σc
Pinion
SH [ ] =
SHP =
Gear
0.7
SHG =
40
1.2
See Ref. 5.2, Eq. 1442
6.3 Mesh Convergence Studies
The following appendices contain mesh convergence studies for some of the
models in the report. As described in section 3.2.1, a mesh convergence study increases
the fidelity of a finite element model by showing that the results are not changing
dramatically as the mesh density is increased. For each study below, the model was
meshed and data was recorded along a “path” in the model. The location of the path
used for each model was kept constant between iterations to maintain a good reference
point. Each iteration contains 20 equally spaced points along the same path. The max
stress as well as the standard deviation of the stress distribution in the path was
calculated for each iteration. The number of elements was increased at each iteration, in
most cases by symmetrically increasing the seed density in critical areas. Lastly, for
every iteration the percentage change in standard deviation is calculated. Using the
standard deviation instead of the max stress to determine convergence of the mesh
increases the trustworthiness of the data because it is calculated from data along the
entire path of the geometry. 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. This standard is adopted for my report, and most of the appendices below
reached this value. In certain cases convergence was deemed accepted for values above
0.4% in order to reduce computation time and maintain manageable file sizes. Mesh
convergence studies were not performed for the two contact models due to the excessive
computing power and computation time required for such a task.
41
6.3.1
2D Lewis Bending Stress Model
This section contains mesh convergence data for the 2D Lewis Bending Model
(pinion and gear). Tables 11 and 12 contain the raw convergence data, and Fig. 21
shows the path used in each iteration to calculate stresses.
Pinion
Horizontal
Stress (It #1)
Stress (It #2)
Position
[psi]
[psi]
[in]
0
64636
64210
0.007
60518
61029
0.015
53915
54226
0.022
45378
45283
0.030
37280
37119
0.037
29863
29752
0.045
23275
23293
0.052
17091
17113
0.060
11260
11154
0.067
5557
5555
0.075
-38
-98
0.082
-5654
-5649
0.090
-11368
-11393
0.097
-17195
-17233
0.105
-23393
-23481
0.112
-30031
-29962
0.120
-37503
-37237
0.127
-45543
-45477
0.135
-54164
-53903
0.142
-60826
-60979
0.150
-64673
-64366
Iteration
Max Stress
%
# of Elements Seed Density [psi]
STDEV [psi] Change
1
338
8
64673
39208
N/A
2
386
10
64366
39171
-0.09
Table 11 – Mesh Convergence Data for the Pinion in the 2D Lewis Bending Load Case
42
Figure 21 – Path Used for the 2D Lewis Bending Model Pinion Mesh Convergence Study (Gear is
Similar)
43
Gear
Horizontal
Position Stress (It #1) [psi] Stress (It #2) [psi] Stress (It #3) [psi] Stress (It #3) [psi]
[in]
0
52161
52518
52154
52346
0.008
46647
47511
47106
47216
0.017
39779
40967
40572
40593
0.025
33944
34300
33989
34154
0.034
28377
28470
28562
28627
0.042
23140
23298
23239
23216
0.051
18338
18397
18385
18436
0.059
13616
13684
13688
13706
0.068
9046
9081
9063
9086
0.076
4485
4526
4427
4541
0.085
2
11
2
9
0.093
-4559
-4529
-4571
-4515
0.101
-9080
-9083
-9124
-9053
0.110
-13749
-13713
-13731
-13613
0.118
-18417
-18469
-18499
-18439
0.127
-23303
-23384
-23384
-23307
0.135
-28676
-28643
-28643
-28709
0.144
-34677
-34462
-34251
-34313
0.152
-41128
-40986
-40722
-40969
0.161
-47398
-47535
-47248
-47597
0.169
-53151
-52570
-52474
-52209
Iteration
1
2
3
3
# of Elements
610
767
802
860
Seed Density
8
10
12
14
Max Stress [psi]
53151
52570
52474
52346
STDEV [psi]
31271
31418
31259
31317
Table 12 - Mesh Convergence Data for the Gear in the 2D Lewis Bending Load Case
44
% Change
N/A
0.47
0.51
0.19
6.3.2
3D AGMA Tooth-Root Bending Models
This section contains mesh convergence data for the 3D AGMA Tooth-Root
Bending Model (pinion and gear). Tables 13 and 14 contain the raw convergence data,
and Fig. 22 shows the path used in each iteration to calculate stresses.
It should be noted that for this appendix the max stress calculated does NOT
necessarily match the final calculated max bending stress in the model. This is because
the location of the max stress changed slightly in each iteration while the path had to
remain the same. Once the mesh convergence revealed the steady state mesh density,
the max stress used in the report (see section 4.1) was calculated.
Pinion
Horizontal
Stress (It #1) [psi] Stress (It #2) [psi]
Position [in]
0
0.008
0.015
0.023
0.031
0.038
0.046
0.061
0.069
0.077
0.085
0.092
0.100
0.108
0.115
0.131
0.138
0.146
0.154
Iteration
1
2
54918
25784
18116
12437
9730
7436
5501
2406
762
-527
-1859
-3548
-5139
-6809
-8913
-14295
-19713
-26592
-50171
# of Elements
5580
7952
54229
25798
17225
12795
9565
7371
2538
675
-535
-1790
-3692
-5176
-6871
-8847
-11265
-14725
-19184
-26807
-49603
Seed Density
Max Stress [psi] STDEV [psi] % Change
12
54918
21520
N/A
14
54229
21424
-0.45
Table 13 - Mesh Convergence Data for the Pinion in the 3D AGMA Bending Load Case
45
Gear
Horizontal
Position
[in]
0
0.009
0.018
0.027
0.036
0.046
0.055
0.064
0.073
0.082
0.091
0.100
0.109
0.118
0.128
0.137
0.146
0.155
0.164
0.173
0.182
Iteration
1
2
3
3
4
Stress (It #1)
[psi]
Stress (It #2)
[psi]
Stress (It #3)
[psi]
Stress (It #3)
[psi]
Stress (It #4)
[psi]
39513
16859
9209
8939
6208
5372
4366
3109
1648
472
-299
-1013
-2319
-3875
-5223
-6551
-7633
-10519
-11823
-17168
-31819
39404
15269
12219
7751
6389
4963
3897
3067
1722
637
96
-825
-2497
-3930
-5102
-6294
-8054
-9475
-13434
-16129
-31292
38465
14835
10998
7592
6349
5070
4042
3417
2070
387
-121
-726
-2698
-4229
-5234
-6547
-7873
-9558
-12572
-15872
-30673
37918
14389
9436
7809
6100
5013
4051
3164
2061
245
-242
-785
-2646
-3811
-4936
-6224
-7729
-9844
-12256
-16264
-30606
37284
15374
9641
7791
6375
5210
4106
3124
2167
-81
927
-487
-3792
-4460
-5237
-6270
-7676
-9488
-11941
-16491
-29971
# of
Elements
4688
5890
8592
11424
14896
Seed Density
8
10
12
14
16
Max Stress [psi]
39513
39404
38465
37918
37284
STDEV [psi]
13481
13365
13050
12873
12803
% Change
N/A
0.87
2.41
1.38
0.55
Table 14 - Mesh Convergence Data for the Gear in the 3D AGMA Bending Load Case
46
Figure 22 – Path Used in the Gear in the Mesh Convergence Study for the 3D AGMA Bending Load
Case (Pinion Similar)
47
6.3.3
2D AGMA Tooth-Root Bending Models
This section contains mesh convergence data for the 2D AGMA Bending Model
(pinion and gear). Tables 15 and 16 contain the raw convergence data, and Fig. 23
shows the path used in each iteration to calculate stresses.
It should be noted that for this appendix the max stress calculated does NOT
necessarily match the final calculated max bending stress in the model. See Appendix
6.3.2 for explanation.
Pinion
Horizontal
Position [in]
0
0.008
0.015
0.023
0.030
0.038
0.046
0.053
0.061
0.069
0.076
0.084
0.091
0.099
0.107
0.114
0.122
0.130
0.137
0.145
0.152
Iteration
1
2
3
4
Stress (It #1) [psi]
Stress (It #2) [psi]
Stress (It #3)
[psi]
Stress (It #4)
[psi]
64996
38687
24716
19800
15298
11614
8941
6447
4152
1961
-176
-2326
-4539
-6848
-9359
-12077
-15874
-20288
-25501
-37708
-58791
67287
37682
26057
19595
15218
11813
8942
6434
4129
1947
-191
-2325
-4527
-6862
-9409
-12296
-15687
-20166
-26357
-36941
-60199
68134
37278
26235
19574
15227
11811
8942
6432
4119
1939
-192
-2327
-4522
-6858
-9400
-12309
-15741
-20092
-26460
-36597
-60642
68799
37160
26157
19557
15194
11789
8932
6412
4112
1930
-204
-2338
-4536
-6868
-9421
-12320
-15751
-20223
-26478
-36515
-60929
# of Elements
764
854
953
1151
Max Stress [psi]
64996
67287
68134
68799
STDEV [psi]
25573
25978
26092
26194
% Change
N/A
1.56
0.44
0.39
Table 15 - Mesh Convergence Data for the Pinion in the 2D AGMA Bending Load Case
48
Gear
Horizontal
Position
[in]
0
0.009
0.018
0.027
0.036
0.045
0.053
0.062
0.071
0.080
0.089
0.098
0.107
0.116
0.125
0.134
0.142
0.151
0.160
0.169
0.178
Iteration
1
2
3
Stress (It #1)
[psi]
Stress (It #2)
[psi]
Stress (It
#3) [psi]
47467
26427
17513
14159
11104
8765
6655
4792
3065
1412
-225
-1840
-3525
-5273
-7147
-9199
-11554
-14624
-18156
-25866
-41220
50422
25698
18897
14039
11249
8714
6646
4780
3049
1380
-239
-1871
-3551
-5310
-7157
-9350
-11747
-14448
-18476
-25246
-42136
50070
25436
18298
14267
11177
8729
6659
4787
3050
1388
-239
-1863
-3530
-5277
-7144
-9207
-11573
-14494
-18808
-25040
-42235
# of Elements
695
732
935
Max Stress [psi]
47467
50422
50070
STDEV [psi]
18207
18670
18592
% Change
N/A
2.48
0.42
Table 16 - Mesh Convergence Data for the Gear in the 2D AGMA Bending Load Case
49
Figure 23 - Path Used for the Pinion in the Mesh Convergence Study for the 2D AGMA Bending
Load Case (Gear Similar)
50
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.
12.
Design and manufacturing of gears. Retrieved July 12, 2013, from
http://en.wikipedia.org/wiki/Design_and_manufacturing_of_gears.
51
from
13.
Alloy
Steel
AISI
4140.
Retrieved
July
21,
2013,
from
http://www.efunda.com/materials/alloys/alloy_steels/show_alloy.cfm?ID=AISI_4
140&prop=all&Page_Title=AISI%204140.
52