Analysis of a Hybrid (Composite-Metal) Spur Gear Subjected to
Stall Torque Using the Finite Element Method.
by
Brenton L Ewing
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
Mechanical Engineering
Approved:
_________________________________________
Ernesto Gutierrez-Miravete, Project Adviser
Rensselaer Polytechnic Institute
Hartford, CT
December 2013
CONTENTS
LIST OF TABLES ............................................................................................................ iv
LIST OF FIGURES ........................................................................................................... v
TABLE OF SYMBOLS .................................................................................................. vii
GLOSSARY ................................................................................................................... viii
KEYWORDS .................................................................................................................... ix
ACKNOWLEDGMENT ................................................................................................... x
ABSTRACT ..................................................................................................................... xi
1. Introduction.................................................................................................................. 1
1.1
Background ........................................................................................................ 1
1.2
Material Properties ............................................................................................. 3
1.3
1.2.1
Tri-axial Braided Composite .................................................................. 3
1.2.2
AISI 9310 Gear Steel ............................................................................. 4
1.2.3
CYCOM PR 520 .................................................................................... 4
Hybrid Gear Specific Geometric Properties ...................................................... 4
2. Theory/Methodology ................................................................................................... 6
2.1
Lewis Bending Stress Equation ......................................................................... 6
3. Analysis ....................................................................................................................... 9
3.1
Mathcad – Lewis Bending Equation .................................................................. 9
3.2
Abaqus FEA– Single Tooth ............................................................................ 10
3.3
Abaqus FEA– Single Tooth Bending with Fixed ID of Gear .......................... 12
3.4
Abaqus FEA – Solid Steel Gear Torsion ......................................................... 14
3.5
Abaqus FEA – Hybrid Gear Torsion ............................................................... 17
3.6
3.5.1
Void Composite ................................................................................... 20
3.5.2
Adhesive............................................................................................... 21
3.5.3
Outer Composite .................................................................................. 22
Abaqus FEA – Metal Gear with Lighting Holes Torsion ................................ 23
ii
4. Results and Discussion .............................................................................................. 25
4.1
Single Tooth Bending Analysis ....................................................................... 25
4.2
Torsion Analysis .............................................................................................. 27
4.2.1
Solid Metal Gear versus Gear with Lightening Holes ......................... 27
4.2.2
Hybrid Gear Components .................................................................... 29
5. Conclusion / Recommendations ................................................................................ 33
6. Appendix A: Mathcad- Lewis Bending Equation ..................................................... 34
7. References.................................................................................................................. 35
7.1
Software ........................................................................................................... 36
iii
LIST OF TABLES
Table 1: Gear Data ............................................................................................................. 4
Table 2: Assumed Gear Properties .................................................................................... 5
Table 3: Results from Lewis Equation .............................................................................. 9
Table 4: Single Tooth FEA Results ................................................................................. 12
Table 5: Single Tooth Whole Gear FEA Results............................................................. 14
Table 6: Solid Metal Gear FEA Results .......................................................................... 16
Table 7: Hybrid Gear Modeling ...................................................................................... 17
Table 8: Hybrid Gear Assembly Legend ......................................................................... 19
Table 9: Void Composite FEA Results ........................................................................... 20
Table 10: Adhesive FEA Results ..................................................................................... 21
Table 11: Outer Composite FEA Results ........................................................................ 22
Table 12: Lightening Hole FEA Results ......................................................................... 24
Table 13: Single Tooth Summary .................................................................................... 25
Table 14: Single Tooth Results Neglecting Stress Concentrations ................................. 26
Table 15: Torsion Comparison- Solid Metal Gear versus Gear with Lightening Holes . 27
Table 16: Hybrid Gear Components, Stress .................................................................... 31
iv
LIST OF FIGURES
Figure 1: Hybrid Gear Assembly Steps [1] ....................................................................... 2
Figure 2: Hybrid Gear Details [1] ...................................................................................... 2
Figure 3: Tri-axial Braided Composite and Single Unit Cell [2] ...................................... 3
Figure 5: Hybrid Gear Assembly....................................................................................... 5
Figure 4: Assumed Dimensions of Hybrid Gear ............................................................... 5
Figure 6: Gear Tooth as Cantilever Beam ......................................................................... 6
Figure 7: Lewis Form Factor Plot [7] ................................................................................ 7
Figure 8: Force applied to Tooth [7].................................................................................. 8
Figure 9: Single Tooth Loading and Boundary Conditions............................................. 10
Figure 10: Single Tooth FEA, fine mesh ......................................................................... 11
Figure 11: Single Tooth FEA, Displacement, fine mesh ................................................. 11
Figure 12: Single Tooth Gear Model with Localized Mesh Refinement ........................ 12
Figure 13: Single Tooth Coarse Mesh Bending Stress .................................................... 13
Figure 14: Single Tooth, Mesh Refinement, Displacement ............................................ 14
Figure 15: Solid Metal Gear, Fine Mesh, Stress .............................................................. 15
Figure 16: Solid Metal Gear, Fine Mesh, Radial Displacement ...................................... 16
Figure 17: Adhesive Connection ..................................................................................... 18
Figure 18: Hybrid Gear Assembly................................................................................... 19
Figure 19: Void Composite Fine Mesh ........................................................................... 20
Figure 20: Adhesive Fine Mesh ....................................................................................... 21
Figure 21: Outer Composite Fine Mesh .......................................................................... 22
Figure 22: Lightening Holes ............................................................................................ 23
Figure 23: Lightening Hole Fine Mesh Stress ................................................................. 24
Figure 24: Lightening Hole Fine Mesh Displacement .................................................... 24
Figure 25: Single Tooth FEA, Stress Comparison .......................................................... 26
Figure 26: Single Tooth FEA, Displacement Comparison .............................................. 26
Figure 27: Torsion FEA, Stress ....................................................................................... 28
Figure 28: Torsion FEA, Radial Displacement, tilted to show detail .............................. 28
Figure 29: Void Composite, Stress Gradients ................................................................. 29
Figure 30: Outer Composite, Stress Gradients ................................................................ 30
v
Figure 31: Adhesive, Stress Gradients............................................................................. 30
Figure 32: Shear Stress versus Shear Strain, Composite [2] ........................................... 32
Figure 33: Tensile Stress versus Strain, Composite [2] ................................................... 32
vi
TABLE OF SYMBOLS
Symbol
Description [units]
E
P
θ
D
F
Y
Modulus of elasticity/ Young’s Modulus [ksi] or [psi]
Diametral pitch [1/in]
Pressure angle [degrees]
Pitch diameter [in]
Face width [in]
Lewis Form Factor [-]
σy
σL
Yield stress [ksi] or [psi]
Wt
ν
Tangential load [lbf]
Poisson ratio [-]
rtooth
Radial distance to gear tooth tip [in]
Lewis Bending Stress [ksi] or [psi]
Tc
c
Critical torque [lbf-in]
Gear tooth tip arc length [in]
fline
h
Wt per unit linear distance [lbf/in]
Inner hub diameter [in]
Lh
Hub hexagonal length [in]
Vl
Void composite outer hexagonal length [in]
Do
Outer composite outer diameter [in]
vii
GLOSSARY
1. Elastic Modulus (Young’s modulus) – slope of the linear portion of a material’s
stress-strain plot.
2. Yield Stress – amount of stress where a material begins to yield (plastically
deform)
3. Spur Gear – gear in which the teeth are parallel to the gear shaft. This type of
gear is used to transfer torsion across parallel shafts.
4. Face Width – thickness of the gear tooth measured parallel to main axis
5. Isotropic – a characteristic of a material which is independent of material
orientation.
6. Involute – curve that follows an imaginary string as it is unwound from a
cylinder. Modern gear tooth profiles are involutes.
7. Diametral Pitch – number of teeth per inch of pitch diameter
8. Pitch Diameter – diameter of the imaginary rolling cylinder that a gear replaces.
9. Pressure Angle – compliment of angle between line of force of gear tooth action
and line connecting gear geometric centers.
10. Poisson Ratio – ratio that relates tangential to axial strain.
11. von Mises Stress – prediction of material failure (yielding) based upon a 3-d
stress state which is converted into an equivalent tensile stress.
viii
KEYWORDS
The following keywords are relevant to this report:
spur gear, composite, torsion, tooth bending, finite element analysis, cohesive
elements, Abaqus, mesh convergence
ix
ACKNOWLEDGMENT
I would like to thank my wife, Hesti, for her tireless dedication. I would also like to
thank Professor Ernesto Gutierrez-Miravete for his support and guidance during this
project.
x
ABSTRACT
Reducing the weight of a component while maintaining strength requirements can
be a design challenge and often one attribute is achieved at the other’s expense. This
study explores an involute spur gear that has been lightened by replacing its center
section with a composite material (see Figure 1 on page 2 for hybrid gear details). A
casualty stall situation where the gear shaft is attempting to turn the gear but the gear
teeth are prevented from moving develops stresses internal to the hybrid gear assembly.
The Lewis bending equation is used to determine how much force is required to
plastically deform the gear teeth. This load is then converted to a maximum torque value
which is then applied to the entire gear.
Two models of a single steel tooth (first model is a single tooth with a fixed base
and the second model is a single tooth on a gear with the gear inside diameter fixed) are
analyzed using the Abaqus Finite Element Analysis (FEA) software to verify results
from the Lewis equation. This loading is then applied to a 3-d model of the hybrid gear
in Abaqus to predict stress levels in the composite parts. Additional considerations
include adhesive failure due to shear stresses at bonded interfaces. For comparison
purposes, a solid metal gear is also analyzed and will be used as a baseline. Finally, a
metal gear is analyzed with lightening holes to simulate a 20 percent reduction in weight
(similar to the weight savings achieved with the composite center section).
The analysis performed on the hybrid spur gear show large (>9) factors of safety for
both the hybrid gear and a metal gear with arbitrary lightening holes.
xi
1. Introduction
1.1 Background
Lightweight components equate to less energy consumption while operating and is
frequently characterized with an increased power to weight ratio. As technology in both
manufacture and material refinement becomes more advanced, components can be
designed to be lighter while still being able to meet or exceed strength and fatigue
requirements. The spur gear is one such component that can benefit from a reduction in
weight.
Spur gears are useful for transmitting torque across parallel shafts. A conventional
spur gear is made from a single material, usually metal, and is placed on a shaft. As this
shaft rotates, the gear teeth mesh with another gear and power is transmitted across their
interface. A hybrid configuration which consists of manufacturing a spur gear from both
metal and composite materials has been proposed [1]. Figure 1 and Figure 2 depict
assembling the gear and show final details. To build the gear, a metal hub is located
centrally. A composite layer (outer composite 1 of 2) is then placed around the hub.
Next, an outer ring of gear teeth is bonded using adhesive. To fill the void created
between the hub and toothed ring, a layer of composite (named void composite) is fitted
and bonded to the outer composite. Finally, the last composite section is bonded to the
assembly (outer composite 2 of 2) and a curing process is specified at elevated pressures
and temperatures. Figure 18 on page 19 shows the assembly in Abaqus. This multimaterial spur gear is approximately 20 percent lighter than a traditional steel gear. This
study evaluates the performance of such a gear using analytical methods. The results are
then compared to an equivalent metal gear with lightening holes.
1
Figure 1: Hybrid Gear Assembly Steps [1]
Figure 2: Hybrid Gear Details [1]
2
1.2 Material Properties
1.2.1
Tri-axial Braided Composite
The baseline composite material assumed in this study is a tri-axial braided carbon
fiber and epoxy laminate. The fibers are TORAYCA T700S carbon fiber and the matrix
is CYCOM PR 520 [2]. This is a relatively expensive and complicated material but the
resulting lamina can be considered quasi-isotropic when several unit cells are included
[2]. Figure 3 is a graphical representation of the material and the size of a unit cell.
Figure 3: Tri-axial Braided Composite and Single Unit Cell [2]
The axial direction (vertical blue arrow in Figure 3) consists of 24k (24 thousand)
flattened tows of carbon fiber. A 24K flattened tow consists of twenty four thousand
carbon fibers in a bundle which is then flattened. The inclined red arrows in Figure 3
show the bias direction (plus and minus 60 degrees off of the axial direction). These
fibers are 12k flattened tows of carbon fiber. For many composite materials, material
data can be looked up in tables, but for this unique material, no such data is readily
available. Testing performed on this material is identified in [2]. For this study, the
composite is considered to be completely isotropic with a modulus of elasticity of 6.4e6
psi and a Poisson ration of 0.3. These values were also used in [1] for modal analysis.
3
1.2.2
AISI 9310 Gear Steel
The gear tooth and hub material are made from AISI 9310 steel and are case
hardened. The material properties are a Young’s modulus of 29e6 psi and a Poisson ratio
of 0.3 [3]
1.2.3
CYCOM PR 520
The bonded interfaces are assumed to be glued using the composite matrix. The
properties for the matrix were obtained from [4]. The resin has the following
characteristics: Young’s modulus: 0.58e6 psi, Poisson ration: 0.398 and a Shear
Modulus of 0.12e6 psi. An arbitrary thickness of 0.001 inches was selected for analysis
purposes.
1.3 Hybrid Gear Specific Geometric Properties
The gear identified in [1] will be analyzed in this project. Table 1 presents the
properties. Note: Computer Aided Drawing (CAD) model obtained [5] for analysis has a
20 degree pressure angle vice the 25 degree angle identified in [1].
Table 1: Gear Data
4
Several gear dimensions were assumed for this analysis because they were not
specifically identified in [1]. Table 2 identifies these dimensions and assumed values;
Figure 4 clarifies where dimensions are located while Figure 5 presents the hybrid
assembly model with part callouts.
Table 2: Assumed Gear Properties
Variable
h
Value
0.5
Description
Inner hub diameter
Units
in
Lh
0.75
Hub hexagonal length
in
Vl
Do
1.5
Void composite outer hexagonal length
in
3.296
Outer composite outer diameter
in
Figure 4: Assumed Dimensions of Hybrid Gear
Figure 5: Hybrid Gear Assembly
Note: front outer composite not shown in isometric for clarity
5
2. Theory/Methodology
2.1 Lewis Bending Stress Equation
The Lewis bending stress equation is one of the oldest (developed in 1892 according
to [6]) and simplest equations to determine stresses in loaded gear teeth. Its simplicity is
derived from the analogy of a gear tooth to a cantilever beam (Figure 6).
Figure 6: Gear Tooth as Cantilever Beam
According to the Lewis equation, the maximum bending stress at the tooth base is given
by equation 2.1.
𝜎
𝑊𝑡 ∗𝑃
𝐹∗𝑌
[2.1]
𝐿=
The variable, Y, above is the Lewis Form Factor. This non-dimensional constant is
based off of the pressure angle, θ, of the gear and its number of teeth, n. Values for Y are
typically found in tables or plots such as in Figure 7.
6
Figure 7: Lewis Form Factor Plot [7]
As a tradeoff to the Lewis equation’s simplicity, there are several key assumptions
and drawbacks involved. These assumptions and subsequent discussion are presented
below:

Radial component of load is neglected

Dynamic effects are not considered

Stress concentrations at tooth fillet are not considered

Highest loading is based on single tooth loaded at tip of tooth

Sufficient contact ratio is obtained (greater than 1.5)
An important drawback to the Lewis equation is that the force transmitted to the
gear due to the mesh is actually at an angle and not tangential as shown in Figure 8. The
radial component of this force would yield a compressive stress in the tooth; this force is
neglected in the Lewis equation.
7
Figure 8: Force applied to Tooth [7]
The Lewis equation also does not include dynamic effects. The effect of cyclic
loading can reduce allowable stress significantly. Since many gears are meant to operate
at a high number of revolutions per minute and for sustained periods of time, the Lewis
equation will not be accurate in these cases.
Finally, the Lewis equation does not accurately predict stress concentrations that
occur at the tooth base fillet. These concentrations are significant and will introduce a
difference when comparing actual stresses and stresses obtained with the Lewis
Equation.
The assumption that the worst case loading occurs when a single tooth is loaded at
its tip would actually not result in the highest stress. According to [6], an accurately
machined gear set with a sufficient contact ratio (greater than 1.5) would have others
gears sharing the load if a tooth was loaded at its tip. A more severe load case would be
when a pair of teeth shares the load equally and that force is applied at the middle of the
gear teeth.
Even with the assumptions and limitations of the Lewis equation, its simplicity
makes it a great starting point to determine stresses in gear teeth. The loads calculated by
the Lewis equation will be used in this study to predict loading internal to the hybrid
gear.
8
3. Analysis
This section includes analysis for several different models and loadings. Section 3.1
presents a hand calculation of tooth bending stress using the Lewis bending equation.
Section 3.2 explores FEA of a single tooth with a fixed base. Next, Section 3.3 depicts a
further refinement of section 3.2. The gear inside diameter (ID) is now fixed while
loading is applied to the tooth. After tooth bending is analyzed and critical loading is
selected, analysis is shifted to torsion. Section 3.4 represents torsion on an idealized
solid metal gear with a fixed ID. Section 3.5 analyzes torsion on the hybrid gear and
presents results from FEA on all composite components as well as mesh convergence
considerations. Finally, section 3.6 uses FEA to predict stresses and displacements on an
idealized metal gear with arbitrary lightening holes which represent a 20 percent
reduction in mass. After each section, results are summarized for convenience. The FEA
portions of this section include modeling considerations and mesh convergence studies.
3.1 Mathcad – Lewis Bending Equation
The following is the Lewis equation analysis which calculates the maximum
tangential load on the tooth as well as some additional values such as different loads to
be applied to the model. Table 3 summarizes results from this section. The Mathcad
calculation is presented in Section 6 on page 34.
Table 3: Results from Lewis Equation
Variable
Value
Description
Tangential load to initiate
tooth bending (plastic deformation)
Wt
1069
Fline
17,558
Line load based on Wt distributed
along top of tooth
lbf/in
Tc
1959
Critical torque - most torque that can be applied
lbf*in
9
Units
lbf
3.2 Abaqus FEA– Single Tooth
The single steel tooth was modeled as a deformable 3-d shell instead of a solid due
to greater shell element performance in bending. This increase in performance is
attributed to a rotational degree of freedom in addition to translation. Solid elements
only contain translation degrees of freedom. The shell thickness is 0.25 inches which is
equal to F, the face width of the gear. The loading was applied to the top of the tooth as a
tangential shell edge load of Fline=17558 lbf/in which equates to a load of Wt=1069 lbf as
calculated by the Lewis equation (see section 3.1, Mathcad – Lewis Bending Equation
for calculation). The bottom edge of the shell has a fixed displacement and rotation
boundary condition. Figure 9 below presents the rendered shell thickness as well as
boundary conditions and loading. The shaded area represents the shell model.
Figure 9: Single Tooth Loading and Boundary Conditions
A convergence study was conducted to get a sense of confidence for the stress
calculated by the Lewis equation. Since the stress concentration at the fillet is not
accurately predicted by the Lewis bending equation, it is expected that FEA results will
be larger than the stresses calculated by the Lewis equation. The model was meshed
using reduced integration 8 node shell elements.
10
Figure 10 shows the bending stress in the single gear tooth with a fine mesh plotted
on the deformed tooth. The center section clearly shows the neutral surface with
compression and tension occurring at the expected locations. The mesh was refined until
the solution became mesh independent. For analysis purposes, the deflection was also
captured through the mesh convergence process.
Figure 10: Single Tooth FEA, fine mesh
Figure 11 shows the displacement of the single tooth under tangential loading. The
distribution is as expected within the model. Results from the single tooth FEA mesh
convergence study are located in Table 4.
Figure 11: Single Tooth FEA, Displacement, fine mesh
11
Table 4: Single Tooth FEA Results
Mesh
Coarse
# Elements
62
S_mises Max [psi]
162,207
% Difference
U_R Max [in]
1.7340E-03
% Difference
Finer
199
166,133
2.36%
1.7340E-03
0.000%
Fine
1221
172,728
3.82%
1.7370E-03
0.173%
7481
179,811
3.94%
1.7370E-03
0.000%
30704
179,955
0.08%
1.7380E-03
0.058%
Super
Fine
Ultra
Fine
3.3 Abaqus FEA– Single Tooth Bending with Fixed ID of Gear
In an attempt to more realistically predict the bending stresses in the gear tooth, an
entire steel gear is modeled with a single gear tooth. The face width is F=0.25 inches and
there is a hole in the center that would represent a 1-inch diameter shaft. The loading is
applied as a shell edge load in shear similar to the loading in Section 3.2. Figure 12
presents the model with rendered shell thickness and localized mesh refinement near the
single gear tooth. The boundary conditions for this model include a coupling constraint
for the gear’s inside diameter (ID) to a reference point at the gear’s geometric center (not
shown). This reference point is then fixed in all directions.
Figure 12: Single Tooth Gear Model with Localized Mesh Refinement
12
The mesh started rather coarse and was refined locally until convergence was
observed. The displacement of the tooth was also selected as a field output for
convergence purposes as well as comparing values to the single tooth model in Section
3.2. The following figures present software outputs for varying meshes.
Figure 13: Single Tooth Coarse Mesh Bending Stress
Figure 13, above, identifies bending stresses in the gear tooth. Even with a very
coarse mesh, the gradients are smooth and are located as expected. The neutral surface is
well defined and located at the tooth center as anticipated. The stresses plotted at the
base of the tooth below the fillet indicate the body portion of the gear does resist the
loading as expected which offers a more accurate bending stress determination. Figure
14 plots displacement of the model. The displacement gradients are reasonable and show
the base of the gear below the fillet has some small deflection referenced to the gear
center, as expected. Table 5 presents a summary from this subsection.
13
Figure 14: Single Tooth, Mesh Refinement, Displacement
Table 5: Single Tooth Whole Gear FEA Results
%
Difference
1206
S_22 Max
[psi]
199,600
3005
191,700
-4.12%
3.28E-03
0.61%
10,081
25,986
50,144
216,000
220,700
222,500
11.25%
2.13%
0.81%
3.29E-03
3.31E-03
3.32E-03
0.12%
0.67%
0.18%
Mesh
# Elements
COARSE
Less
Coarse
Fine
Finer
Finest
U_R Max [in]
%
Difference
3.26E-03
3.4 Abaqus FEA – Solid Steel Gear Torsion
For this section the load case of concern is torsion and how the stress distributes in
the center section of the solid metal gear. A gear is modeled with the teeth removed.
There is a 1-inch diameter hole in the center to accept a shaft. The outer diameter (OD)
of the gear is coupled to a reference point at the geometric center of the gear. This
reference point is then fixed in all directions. The ID of the gear is coupled to a different
14
reference point. A torque of Tc=1959 in*lbf is then applied to this second reference
point. This will simulate the shaft trying to turn the gear with the teeth providing
complete resistance. The model consists of a 3-dimensional shell with a thickness of
F=0.25 inches. The mesh contains 8-node shell elements with reduced integration. The
mesh is initially coarse and is then refined significantly until outputs converge.
Figure 15: Solid Metal Gear, Fine Mesh, Stress
Figure 15 presents the stress distribution for a finely meshed model under a state of
pure torsion. The stress is highest where the torque is applied and tapers down to zero as
a function of the radial distance, as expected.
15
Figure 16: Solid Metal Gear, Fine Mesh, Radial Displacement
Figure 16 shows how the radial displacement is distributed on a coarse mesh model.
The highest displacement occurs at the inside diameter of the idealized gear. The outside
diameter of the gear has a radial displacement of zero which reflects the fixed boundary
condition. The displacement output is a function of radial distance from the gear center
and is as expected. The model is meshed with an increasing number of elements until the
outputs converge. The following figures present a fine mesh. Table 6 summarizes the
results from this section.
Table 6: Solid Metal Gear FEA Results
# Elements
S_mises Max [psi]
592
1682
3806
10,359
29,324
41,939
7180
7507
7845
8052
8226
8249
% Difference
U_R Max [in]
% Difference
4.36%
4.31%
2.57%
2.12%
0.28%
2.0146E-04
2.0095E-04
2.0107E-04
2.0118E-04
2.0130E-04
2.0135E-04
-0.256%
0.062%
0.052%
0.061%
0.024%
16
3.5 Abaqus FEA – Hybrid Gear Torsion
This section places the composite-metallic gear in pure torsion with loading and
boundary conditions similar to that in Section 3.4. The ID of the gear has a moment of
Tc=1959 in*lbf applied and the OD of the gear is fixed. For this analysis each part was
modeled separately and compiled into an assembly. Table 7 presents a summary of
materials and elements used for each part. The metallic parts were meshed with linear
elements to reserve computing resources for composite parts.
Table 7: Hybrid Gear Modeling
Part
Material
Element Type
HUB
AISI 9310 GEAR
STEEL
3-D Solid 8-node (linear) brick with
reduced integration and hourglass control
Void Composite
Composite
3-D Solid 20-node (quadratic) brick with
reduced integration
Outer
Composite
Composite
3-D Solid 20-node (quadratic) brick with
reduced integration
Tooth Ring
AISI 9310 GEAR
STEEL
3-D Solid 8-node (linear) brick with
reduced integration and hourglass control
Adhesive
CYCOM®
PR 520
3-D Linear 8-node cohesive element
To assemble the gear, interactions were identified to allow the different part
surfaces in contact to remain in contact. The hub served as the master surface and was
connected to the void and outer composites using tie constraints with default connection
options. This allows the surfaces to be connected such that during small deformations
the interface will stay together. The void composite was then tied to the tooth ring. To
complete the assembly, the outer composite is tied to the ‘top’ surface of the adhesive
while the void composite and tooth ring were both tied to the ‘bottom’ surface of the
adhesive. Figure 17 shows a graphical representation of the adhesive connections
selected in Abaqus.
17
Figure 17: Adhesive Connection
Figure 18 shows an exploded view of the hybrid gear assembly with an
accompanying legend in Table 8.
18
Figure 18: Hybrid Gear Assembly
Table 8: Hybrid Gear Assembly Legend
Part
Hub
Void Composite
Tooth Ring
Adhesive
Outer Composite
Color
Light Silver
Gray
Blue
Green
Red
Quantity
1
1
1
2
2
The assembly is meshed initially very coarse to get a sense of where the stresses
would be the highest. The assembly was then re-meshed with finer elements until the
results converge. For each part, a table of results is presented as well as screenshots
representing mesh densities.
19
3.5.1
Void Composite
This subsection presents the analysis of the void composite from a coarse mesh to a very
fine mesh. Figure 19 shows a finely meshed void composite with stress gradients plotted
on the undeformed part. Stresses are highest at the inner hexagonal corners while the
outer hexagonal features predict the lowest stress. Table 9 provides a summary of results
obtained.
Figure 19: Void Composite Fine Mesh
Table 9: Void Composite FEA Results
# Global
Elements
4998
10,557
24,262
49,106
61,868
71,268
84,366
# Local
Elements
656
1635
4088
10,320
12,520
16,650
18,810
Maximum von
Mises Stress
1711.425
1870.064
2164.235
2223.565
2154.530
2222.889
2434.615
20
% Difference
8.48%
13.59%
2.67%
-3.20%
3.08%
8.70%
3.5.2
Adhesive
The adhesive layer connects the outer composites to the void composite and tooth
ring. This part is also meshed coarsely initially and then re-meshed until the outputs
converge. Figure 20 shows the stress distribution in the adhesive using a fine mesh. The
largest stresses occur at the outer edge of the gear where the fixed boundary condition is
implemented. The lowest stresses are observed where the adhesive interfaces with the
void composite. A low stress region can be seen where this interface occurs. Table 10
presents mesh convergence data.
Figure 20: Adhesive Fine Mesh
Table 10: Adhesive FEA Results
# Global
Elements
4998
10,557
24,262
49,106
61,868
# Local
Elements
594
1212
2494
4886
6147
Maximum von
Mises Stress
311.148
362.371
376.071
400.120
403.593
21
% Difference
14.14%
3.64%
6.01%
0.86%
3.5.3
Outer Composite
The outer composite is glued to the void composite and tooth ring on both sides of
the gear. Similarly, the part is meshed very coarse and then refined until the change in
stress does not vary significantly. Figure 21 depicts a fine mesh with stress gradients
plotted on the undeformed component. The highest stresses occur at the inner hexagonal
corners and the lowest stresses are at its outer radius. Table 11 summarized the results of
the analysis.
Figure 21: Outer Composite Fine Mesh
Table 11: Outer Composite FEA Results
# Global
Elements
4998
10,557
24,262
49,106
61,868
71,268
84,366
# Local
Elements
1160
2832
7176
14,090
18,110
20,745
26,214
Maximum von
Mises Stress
1797.313
1992.864
2519.917
2330.974
2448.193
2390.645
2457.219
22
% Difference
9.81%
20.92%
-8.11%
4.79%
-2.41%
2.71%
3.6 Abaqus FEA – Metal Gear with Lighting Holes Torsion
The last part of the analysis is to predict the stresses in a solid metal gear with a 20
percent reduction in mass achieved by the removal of material via lightening holes. The
size and quantity of the holes were selected arbitrarily. Loading and boundary conditions
are selected similar to those in Section 3.4. Figure 22 shows the model with holes drilled
to reduce the mass. The model was initially meshed coarsely and then refined until the
solution converged. Stresses and displacement field outputs were obtained. This model
was meshed with 8-node quadratic shells.
Figure 22: Lightening Holes
Figure 23 depicts stress distributions in the metal gear with lightening holes. The
highest stress occurs at the gear ID and the lowest stresses occur at the gear OD. Figure
24 presents displacements for a relatively fine mesh. The largest displacement occurs at
the gear ID where the moment is applied and between the lightening holes and the ID.
The lowest displacements occur between the lightening holes at a constant radial
distance. Table 12 summarizes the outputs from this analysis.
23
Figure 23: Lightening Hole Fine Mesh Stress
Figure 24: Lightening Hole Fine Mesh Displacement
Table 12: Lightening Hole FEA Results
# Elements
1313
8456
18,942
33,926
76,824
S_mises Max [psi]
8810
10,018
10,379
10,524
10,700
% Difference
12.06%
3.48%
1.38%
1.64%
24
UR MAX [in]
3.1126E-04
3.2290E-04
3.2241E-04
3.2410E-04
3.2590E-04
% Difference
3.603%
-.153%
0.523 %
0.550%
4. Results and Discussion
This section expands on results obtained from Section 3 and is divided into two subsections: single tooth analysis and torsion analysis. Comparisons between different
analysis and modeling approaches are presented including commentary on results
obtained.
4.1 Single Tooth Bending Analysis
Results are to be presented for the three methods of predicting the bending stress in
the single gear tooth. The first method used is the Lewis bending Equation (see section
3.1 for full analysis and assumptions). Next, in section 3.2, a model was created of a
detached single tooth. The tooth base is fixed and the load is applied to the top surface.
The last method is the FEA of a gear with a fixed inside diameter. The loading is also
applied to the top surface of the tooth. Table 13 presents data for comparison.
Table 13: Single Tooth Summary
Max Bending
Stress [psi]
Displacement [in]
Lewis Bending
Section 3.1
Single Tooth FEA
Section 3.2
Single Tooth FEA Fixed ID
Section 3.3
130,534
N/A
179,955
1.7380E-03
222,500
3.3160E-03
The values of the bending stress varied from 130,534 [psi] to 222,500 [psi]. This is
due to the simplifications and assumptions embedded in the Lewis Bending Equation
which neglects stress concentrations at the base of the tooth. A side-by-side comparison
of the bending stresses for both FEA models is presented in Figure 25. The bending
stresses occur at the expected locations with clearly defined neutral surfaces. The right
side of Figure 25 shows the stresses developing into the center of the gear. This likely
contributes to the differences in stresses calculated. The model with the fixed ID of the
gear is more realistic than the simple tooth with a fixed base.
25
Figure 25: Single Tooth FEA, Stress Comparison
If one observes the stresses away from the stress concentration at the base of the
tooth, the stresses are closer to the values calculated by the Lewis bending equation, as
shown in Table 14.
Table 14: Single Tooth Results Neglecting Stress Concentrations
Max Bending
Stress [psi]
Lewis Bending
Section 3.1
Single Tooth FEA
Section 3.2
Single Tooth FEA Fixed ID
Section 3.3
130,534
138,605
132,200
Figure 26: Single Tooth FEA, Displacement Comparison
Figure 26 presents a comparison of displacement for the two FEA models. Similar
to the stresses, the gear body below the base of the tooth experiences some measurable
deflection. This accounts for the differences in the values predicted in the two models.
For both models, the displacement gradients are similar and occur where expected. The
tip of the tooth experiences the most deflection while the base does not move too much.
The displacement gradient is a strong function of the radial distance and a weak function
26
of distance from the neutral surface. A numerical comparison of displacement predicted
for both models is not performed different reference points.
4.2 Torsion Analysis
This sub-section provides comparison of both stresses and displacements of the two
FEA models under pure idealized torsion. These models include a solid metal gear and a
gear with lightening holes. For the hybrid gear, results are presented to identify
maximum stresses predicted in the composite parts and adhesive. The factors of safety
presented in section are a worst case scenario which represents the upper limit of elastic
deformation for bending of the gear tooth.
4.2.1
Solid Metal Gear versus Gear with Lightening Holes
The first model is a solid metal gear under idealized, pure torsion. (see section 3.4
for analysis and modeling considerations). Next, in section 3.6, a model was created of a
metal gear with several lightening holes. Table 15 presents data for comparison.
Table 15: Torsion Comparison- Solid Metal Gear versus Gear with Lightening
Holes
Solid Gear
Section 3.4
Gear with Lightening Holes
Section 3.6
Max Bending
Stress [psi]
Maximum Displacement
[in]
8249
10,700
2.30E-04
3.26E-04
Allowable
Stress [psi]
Factor of Safety
130,534
15.82
130,534
12.2
The stresses in the gear with lightening holes are larger that of the solid gear as
expected since there is less material and stress concentrations are introduced. The
displacement is also larger due to the increased deflection locally near the lightening
holes. The allowable stress was calculated by using a 0.02 offset of the elastic modulus,
27
representing the yield stress. A side-by side comparison of the stresses and
displacements are shown in Figure 27 and Figure 28, respectively.
Figure 27: Torsion FEA, Stress
The stresses shown above in Figure 27 are similar in that the highest stress is
located at the ID of the gear and then decreases as a function of radial distance to a
minimum value of zero at the gear OD. The stresses in the gear with lightening holes
show that the geometry more efficiently distributes the stress through the gear in
comparison to the solid gear where the outer 1/3 is a region of relatively low stress.
Figure 28: Torsion FEA, Radial Displacement, tilted to show detail
The radial displacements calculated for both the solid metal gear model and the
model with lightening holes added are shown in Figure 28. The maximum displacement
occurs at the center of the gear on both models which intuitively matches the loading and
boundary conditions established. The lightening holes introduce higher displacements
throughout the gear as expected while the outer third of the solid gear has relatively
28
small displacements. This low stress region corresponds to the lower stress areas in
Figure 27.
4.2.2
Hybrid Gear Components
The stress predictions for both the composite components and adhesive are
presented in this sub section. The analysis for these parts is located in Section 3.5. For
component identification please refer to Figure 18 on page 19.
The void composite is situated at the center of the assembly and is located radially
between the hub and outer tooth ring. Figure 29 shows stress distributions in this part.
The stresses are highest at the inner hexagonal features and decrease as a function of
radial distance. The lowest stresses occur at the outer hexagonal features where an
interface with the outer tooth ring occurs. This is due to the stiffness of the outer metal
ring limiting the displacement.
Figure 29: Void Composite, Stress Gradients
The outer composite is the outer most axial component of the hybrid gear. It is
mounted on the hexagonal hub and attached to the assembly by a layer of adhesive.
Stress gradients are plotted in Figure 30. There are two outer composites per assembly
and they both have the same amount of stress due to the loading of the gear.
29
Figure 30: Outer Composite, Stress Gradients
Similar to the void composite, the highest stresses in the outer composite are located
at the inner hexagonal corners. The stress also decreases as radial distance is increases.
The last part of the hybrid gear to be analyzed is the adhesive layer. This layer has
the same footprint as the outer composite and secures the outer composite to both the
void composite and outer tooth ring. There are two layers of adhesive in the gear
assembly.
Figure 31: Adhesive, Stress Gradients
The adhesive layer analysis predicts highest stress at the outer most portion of the
adhesive layer. This is likely due to the stiffness of the hub-composite interface where
limited relative displacement occurs. This can be seen as the blue region of low stress
that delineates the geometric envelope of the void composite in Figure 31. Radially
beyond the void composite interface, higher relative displacement occurs between the
30
outer tooth ring and outer composite which contributes to the higher stress in the
adhesive. The stress results of the hybrid gear composite components and adhesive are
presented in Table 16. The stress levels for the composite and adhesive are very low and
do not present significant loading with respect to failure. Factors of safety are calculated
for both the composite components and adhesive.
Table 16: Hybrid Gear Components, Stress
Outer
Composite
Void
Composite
Adhesive
Max Bending
Stress [psi]
2457
2434
404
Allowable
Stress [psi]
Factor of Safety
25000
10.18
25000
10.27
3967
9.82
The allowable shear stress for the composite is determined from [2]. Testing was
performed on a composite specimen and the following plots, Figure 32 and Figure 33,
were obtained. The plotted ultimate strength in shear for the composite is 44.8 +/- 1.2
[ksi] [2]. Taking the lower bound of this result and multiplying by 1/3 leads to an
allowable shear stress of 14,533 [psi]. The primary loading of the hybrid gear is torsion
and an equivalent von Mises stress was calculated. For the factor of safety calculation,
the von Mises stress is compared to tensile stress. A conservative value of 75 [ksi] was
selected for the ultimate tensile strength (based on transverse direction properties, see
Figure 33). This value is multiplied by 1/3 to obtain an allowable stress of 25 [ksi].
For the adhesive, the vendor data sheet [3] was used to identify a shear stress at
failure of 8.9 [ksi] and a tensile stress at failure of 11.9 [ksi]. Since the FEA results have
already converted the shear stress to an equivalent von Mises stress, a comparison to
tensile stress was performed to obtain the factor of safety. Again, the ultimate tensile
stress is multiplied by 1/3 to obtain an allowable stress of 3967 [psi] and a factor of
safety was calculated.
31
Figure 32: Shear Stress versus Shear Strain, Composite [2]
Figure 33: Tensile Stress versus Strain, Composite [2]
32
5. Conclusion / Recommendations
The analysis performed in this study show that the hybrid spur gear can withstand
torsional loading large enough to bend a gear tooth. Results also indicate that a gear with
lightening holes can also sustain this load with a substantial factor of safety. It is noted
that the loading applied to the models is an idealized case that does not indicate shafthub attachment. The load case in this study would simulate a gear completely welded to
a rigid shaft. Actual connections would likely be a spline or a keyway machined into the
hub. Further model refinement to include actual shaft connection details would likely
yield more accurate results.
The composite properties in this study were simplified to simulate an isotropic
material due to each lamina containing a 0°/+60°/–60° braided structure (see Figure 3 on
page 3), and the ply stacking sequence which rotates each ply 60° from the previous ply.
This simplification is also validated by the size of the plies which contain more than one
unit cell (see see Figure 3 on page 3). Additional accuracy could be realized if the plies
were modeled individually with complete material properties for compression, tension
and shear in both bias and transverse directions.
From a manufacturing and cost point of view, the gear with lightening holes appears
ideal in comparison to the hybrid gear. The gear could be made of solid metal and a
simple machining process would introduce the holes yielding the mass reduction. This is
in comparison to the assembly and curing process required for the hybrid gear. The
reduction is mass used in this analysis for the lightening holes was not optimized and
was arbitrarily selected to obtain a baseline for comparison. Further modifications to
lightening hole geometry and their impact on stress levels and modal analysis would
probably produce improvements over the baseline.
The original hybrid spur gear was identified for use in rotorcraft applications [1].
This application would generate high revolutions per minute and cyclic loading for an
extended number of cycles. Therefore, the endurance limit of the gear material would be
a substantial consideration for stress analysis.
33
6. Appendix A: Mathcad- Lewis Bending Equation
34
7. References
1. Handschuh, Robert F., Gary D. Roberts, Ryan R. Sinnamon, David B. Stringer,
Brian D. Dykas, and Lee W. Kohlman. "Hybrid Gear Preliminary Results -Application of Composites to Dynamic Mechanical Components." (2012): 1-18.
Web. 24 Sept. 2012. Nasa Report NASA/TM—2012-217630. Glenn Research
Center, Cleveland, Ohio. (2012)
2. Roberts, Gary D., Robert K. Goldberg, Wieslaw K. Binienda, William A. Arnold,
Justin D. Littell, and Lee W. Kohlman. "Characterization of Triaxial Braided
Composite Material Properties for Impact Simulation." (2009): 1-41 Web. 15
Apr. 2013. Nasa Report NASA/TM—2009-215660. Glenn Research Center,
Cleveland, Ohio. (2009)
3. "EFunda: Properties of Alloy Steels Details." EFunda: Properties of Alloy Steels
Details. N.p., n.d. Web. 10 Sept. 2013.
<http://www.efunda.com/Materials/alloys/alloy_steels/show_alloy.cfm?ID=AISI_
9310>.
4. CYCOM® PR 520 RTM RESIN SYSTEM. N.p.: CYTEC, 2012. Print
5. "RushGears.com -- Nobody Makes Custom Gears Faster." RushGears.com -Nobody Makes Custom Gears Faster. N.p., n.d. Web. 21 Aug. 2013.
<http://www.rushgears.com/>. (Gear CAD model)
6. Budynas, Richard G., J. Keith. Nisbett, and Joseph Edward. Shigley. Shigley's
Mechanical Engineering Design. 8th ed. Boston: McGraw-Hill, 2008. Print.
7. "Lewis Factor Equation for Gear Tooth Calculations - Engineers Edge." Lewis
Factor Equation for Gear Tooth Calculations - Engineers Edge. N.p., n.d. Web.
07 Sept. 2013.
8. Gibson, Ronald F. Principles of Composite Material Mechanics. 3rd ed. Boca
Raton, FL: Taylor & Francis, 2012. Print.
35
9. Cook, Robert D. Concepts and Applications of Finite Element Analysis. 4th ed.
New York: Wiley, 2002.
10. Corus Engineering Steels. N.p.: Corus Engineering Steels, n.d. Web. 20 Aug.
2013.
<http://www.tatasteeleurope.com/file_source/StaticFiles/Business%20Units/Engin
eering%20steels/AMS6265.PDF>.
11. "Composite Terminology." Composite Terminology. N.p., n.d. Web. 10 Sept.
2013. <http://www.cstsales.com/terminology.html>.
7.1 Software
The following software was used to perform this study:

Abaqus CAE 6.12, Commercial Version

Mathcad 14

Microsoft Word 2007,2010

Microsoft Excel 2007,2010
36