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
Major Subject: Mechanical Engineering
Approved:
_________________________________________
Ernesto Gutierrez-Miravete, Project Adviser
Rensselaer Polytechnic Institute
Hartford, CT
October 2013
(For Graduation December 2013)
ii
CONTENTS
LIST OF TABLES iv
LIST OF FIGURES
VARIABLES
iv
vi
ACKNOWLEDGMENT
ABSTRACT
1.
vii
vii
Introduction
1
1.1
Background 1
1.2
Material Properties 2
1.2.1
Tri-axial Braided Composite 3
1.2.2
AISI 9310 Gear Steel 3
1.2.3
CYCOM PR 520
1.3
2.
2.1
3.
3
Hybrid Gear Geometric Properties
Theory/Methodology
5
Lewis Bending Stress Equation
Analysis
3
5
8
3.1
Mathcad – Lewis Bending Equation
3.2
Abaqus – Single Tooth FEA
3.3
Abaqus – Single Tooth Bending with Fixed ID of Gear 16
3.4
Abaqus – Solid Metal Gear Torsion
19
3.5
Abaqus – Hybrid Gear Torsion
23
3.5.1
Void Composite
3.5.2
Adhesive
3.5.3
Outer Composite
3.6
9
11
25
27
29
Abaqus – Metal Gear with Lighting Holes Torsion
4.
Results and Discussion 35
5.
Conclusion
35
6.
Appendices
35
iii
31
7.
References
35
LIST OF TABLES
Table 1: Assumed Gear Properties ................................................................................................. 4
Table 2: Assumptions for Lewis Equation....................................................................................... 7
Table 3: Results from Lewis Equation ........................................................................................... 10
Table 4: Single Tooth FEA Results ................................................................................................. 15
Table 5: Single Tooth Whole Gear FEA Results............................................................................. 18
Table 6: Solid Metal Gear FEA Results .......................................................................................... 23
Table 7: Hybrid Gear Modeling..................................................................................................... 23
Table 8: Hybrid Gear Assembly Legend ........................................................................................ 25
Table 9: Void Composite FEA Results ........................................................................................... 27
Table 10: Adhesive FEA Results .................................................................................................... 29
Table 11: Outer Composite FEA Results ....................................................................................... 31
Table 12: Lightening Hole FEA Results.......................................................................................... 35
LIST OF FIGURES
Figure 1: Hybrid Gear Assembly Steps [Reference 1] ..................................................................... 2
Figure 2: Hybrid Gear Details [Reference 1] ................................................................................... 2
Figure 3: Tri-axial Braided Composite and Single Unit Cell [Reference 2]...................................... 3
Figure 4: Gear Data [Reference 1] .................................................................................................. 4
Figure 6: Hybrid Gear Assembly ..................................................................................................... 5
Figure 5: Assumed Dimensions of Hybrid Gear .............................................................................. 5
Figure 7: Gear Tooth as Cantilever Beam ....................................................................................... 6
Figure 8: Lewis Form Factor Plot [Reference 10] ........................................................................... 7
Figure 9: Force applied to Tooth [Reference 5].............................................................................. 8
Figure 10: Single Tooth Loading and Boundary Conditions .......................................................... 11
Figure 11: Single Tooth FEA, coarse mesh .................................................................................... 12
Figure 12: Single Tooth FEA, Displacement, coarse mesh ............................................................ 12
iv
Figure 13: Single Tooth FEA, finer mesh ....................................................................................... 13
Figure 14: Single Tooth FEA, Displacement, finer mesh ............................................................... 13
Figure 15: Single Tooth FEA, fine mesh ........................................................................................ 14
Figure 16: Single Tooth FEA, Displacement, fine mesh ................................................................ 14
Figure 17: Single Tooth FEA, ultra fine mesh................................................................................ 15
Figure 18: Single Tooth FEA, Displacement, ultra fine mesh........................................................ 15
Figure 19: Single Tooth Gear Model with Localized Mesh Refinement ....................................... 16
Figure 20: Single Tooth Coarse Mesh Bending Stress .................................................................. 17
Figure 21: Single Tooth, Displacement, Coarse Mesh .................................................................. 17
Figure 22: Single Tooth, Mesh Refinement, Bending Stress......................................................... 18
Figure 23: Single Tooth, Mesh Refinement, Displacement .......................................................... 18
Figure 24: Solid Metal Gear, Coarse Mesh, Stress........................................................................ 20
Figure 25: Solid Metal Gear, Coarse Mesh, Radial Displacement ................................................ 20
Figure 26: Solid Metal Gear, Fine Mesh, Stress ............................................................................ 21
Figure 27: Solid Metal Gear, Fine Mesh, Radial Displacement..................................................... 21
Figure 28: Solid Metal Gear, Ultra Fine Mesh, Stress ................................................................... 22
Figure 29: Solid Metal Gear, Ultra Fine Mesh, Radial Displacement............................................ 22
Figure 30: Adhesive Connection ................................................................................................... 24
Figure 31: Hybrid Gear Assembly ................................................................................................. 25
Figure 32: Void Composite Coarse Mesh...................................................................................... 26
Figure 33: Void Composite Fine Mesh .......................................................................................... 26
Figure 34: Void Composite Ultra Fine Mesh ................................................................................. 27
Figure 35: Adhesive Coarse Mesh ................................................................................................ 28
Figure 36: Adhesive Fine Mesh ..................................................................................................... 28
Figure 37: Adhesive Ultra Fine Mesh ............................................................................................ 29
Figure 38: Outer Composite Coarse Mesh ................................................................................... 30
Figure 39: Outer Composite Fine Mesh ........................................................................................ 30
Figure 40: Outer Composite Ultra Fine Mesh ............................................................................... 31
Figure 41: Lightening Holes .......................................................................................................... 32
v
Figure 42: Lightening Hole Coarse Mesh Stress ........................................................................... 32
Figure 43: Lightening Hole Coarse Mesh Displacement ............................................................... 33
Figure 44: Lightening Hole Fine Mesh Stress................................................................................ 33
Figure 45: Lightening Hole Fine Mesh Displacement ................................................................... 34
Figure 46: Lightening Hole Ultra Fine Mesh Stress....................................................................... 34
Figure 47: Lightening Hole Ultra Fine Mesh Displacement .......................................................... 35
VARIABLES
E
Modulus of elasticity/ Young’s Modulus [psi]
P
Diametral pitch [1/in]
θ
Pressure angle [degrees]
D
Pitch diameter [in]
F
Face width [in]
Y
Lewis Form Factor [-]
σy
Yield stress [ksi]
σL
Lewis Bending Stress [ksi]
Wt
Tangential load [lbf]
ν
Poisson ratio [-]
rtooth
radial distance to gear tooth tip [in]
Tc
Critical torque [lbf-in]
c
Gear tooth tip arc length [in]
fline
Wt per unit linear distance [lbf/in]
h
Inner hub diameter [in]
Lh
Hub hexagonal length [in]
Vl
Void composite outer hexagonal length [in]
Do
Outer composite outer diameter [in]
vi
ACKNOWLEDGMENT
I would like to thank my wife, Hesti, for her tireless dedication. I would also like to thank
Professors Ernesto Gutierrez-Miraverte and David Hufner for their support during this project.
ABSTRACT
Reducing the weight of a component while maintaining strength requirements is often a
difficult task and often one attribute is achieved at the others expense. This study explores an
involute spur gear with its center section replaced by a composite material (see Figure 1 for
hybrid gear details). The goal is to explore how a load large enough to cause tooth bending will
generate stresses in the composite. The Lewis bending equation is used to determine this stall
load and subsequent torque. A model of a single tooth is analyzed in 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. Additional
considerations include adhesive failure due to shear stresses at bonded interfaces Finally the
gear is analyzed with lightening holes to simulate a 20 percent reduction in weight (similar to
weight savings with the composite center section).
vii
Introduction
Background
Weight savings in industry is a considerable goal. 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. Lighter components often
equate to less energy consumption while operating and is often characterized as an increased
power to weight ratio.
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 which transmits power across their interface. A hybrid
configuration which consists of manufacturing a spur gear from both metal and composite
materials is originated and presented in [Reference 1]. Figure 1 and Figure 2 depict assembling
the gear and show final details. To assemble the gear, a metal hub is located centrally. This
center section is bonded to a composite material (outer composite 1 of 2). An outer ring of gear
teeth is then positioned on the fixture with a middle layer of composite inserted between the
hub and toothed ring (void composite). Finally the last composite section is bonded to the
assembly (outer composite 2 of 2) and a cure process is specified at elevated pressures and
temperatures. This multi-material spur gear is approximately 20 percent lighter than a
traditional gear.
1
Figure 1: Hybrid Gear Assembly Steps [Reference 1]
Figure 2: Hybrid Gear Details [Reference 1]
Material Properties
2
Tri-axial Braided Composite
The baseline composite material used 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
[Reference 2]. This is a relatively expensive and complicated material but the resulting lamina
can be considered quasi-isotropic when several unit cells are included [Reference 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 [Reference 2]
The axial direction (blue arrow in Figure 3) consists of 24k flattened tows of carbon fiber. A 24K
flattened tow consists of twenty four thousand carbon fibers in a bundle which is then
flattened. The 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. Reference 2 depicts results from testing performed on this material. 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 Reference 1 for modal
analysis.
AISI 9310 Gear Steel
The gear tooth and hub material are made from AISI 9310 metal and are case hardened. The
material properties are a Young’s modulus of 29e6 psi and a Poisson ratio of 0.3.
CYCOM PR 520
The bonded interfaces are assumed to be glued using the composite matrix. The properties for
the matrix were obtained from Reference 11. The resin has the following characteristics:
Young’s modulus: 0.58e6 psi, Poisson ration: 0.398 and a Shear Modulus of 0.12e6 psi.
Hybrid Gear Geometric Properties
3
Reference 1 identifies properties of the gear to be analyzed. Figure 4 presents this data for
reference. Note: Computer Aided Drawing (CAD) model obtained for analysis has a 20 degree
pressure angle vice the 25 degree angle identified in Reference 1. The data from Figure 4 was
the basis for creating the whole hybrid gear model for analysis.
Figure 4: Gear Data [Reference 1]
Several gear dimensions were assumed for this analysis. Table 1 identifies the dimension and
assumed value, Figure 5 clarifies where dimensions are located while Figure 6 presents the
hybrid assembly model with part callouts.
Table 1: Assumed Gear Properties
Variable
Value
Description
Units
h
0.5
Inner hub diameter
in
Lh
0.75
Hub hexagonal length
in
Vl
1.5
Void composite outer hexagonal length
in
Do
3.296
Outer composite outer diameter
in
4
Figure 5: Assumed Dimensions of Hybrid Gear
Figure 6: Hybrid Gear Assembly
Note: front outer composite not shown in isometric for clarity
Theory/Methodology
Lewis Bending Stress Equation
The Lewis bending stress equation is one of the oldest (developed in 1892 according to
reference 5) and simplest equations to determine stresses in loaded gear teeth. Its simplicity is
derived through a comparison of a gear tooth to a cantilever beam (Figure 7).
5
Figure 7: Gear Tooth as Cantilever Beam
The derivation of the Lewis equation can be located in [Reference 5]. The resulting equation is
shown below:
𝜎
𝑊 𝑡 ∗𝑃
𝐿=
𝐹∗𝑌
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 8.
6
Figure 8: Lewis Form Factor Plot [Reference 10]
As a tradeoff to the Lewis equation’s simplicity, there are several key assumptions and
drawbacks involved. A list of assumptions is included in Table 2.
Table 2: Assumptions for Lewis Equation
1.
Radial component of load is neglected
2.
Dynamic effects are not considered
3.
Stress concentrations at tooth fillet are not considered
4.
Highest loading is based on single tooth loaded at tip of tooth
5.
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 9. The radial component
of this force would yield a compressive stress in the tooth; this force is neglected in the Lewis
equation.
7
Figure 9: Force applied to Tooth [Reference 5]
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 higher
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 reference 5, 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 vice the tooth tip.
Even with the assumptions and limitations of the Lewis equation, its simplicity yields itself to a
great starting point to determine stresses in gear teeth. For this reason, the loads calculated by
the Lewis equation will be used in this study to predict loading internal to the hybrid gear.
Analysis
This section includes analysis for several different models and loadings. Section 0 presents a
hand calculation of tooth bending stress using the Lewis Bending equation. Section 0 explores
FEA of a single tooth with a fixed base. Next, Section 0 depicts a further refinement of section
0. 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 0
represents torsion on an idealized solid metal gear with a fixed ID. Section 0 analyzes torsion on
the hybrid gear and presents results from FEA on all composite components as well as mesh
8
convergence considerations. Finally, section 0 uses FEA to predict stresses and displacements
on an idealized metal gear with arbitrary lightening holes which represent a 20 percent
reduction in mass.
Mathcad – Lewis Bending Equation
The following is the Lewis equation analysis which calculates the 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.
9
Table 3: Results from Lewis Equation
Variable
Value
Description
Units
Wt
1069
Tangential load to initiate
tooth bending (plastic deformation)
lbf
Fline
17558
Line load based on Wt distributed
along top of tooth
lbf/in
Tc
1959
Critical torque - most torque that can be applied
lbf*in
10
Abaqus – Single Tooth FEA
The single tooth was modeled as a deformable 3-d shell due to shell element performance in
bending. 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 10 below presents the shell model,
rendered shell thickness as well as boundary conditions and loading.
Figure 10: Single Tooth Loading and Boundary Conditions
A Convergence study was conducted to get a sense of confidence for the load calculated by the
Lewis equation. Since the stress concentration at the fillet is not accurately predicted by the
Lewis 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.
11
Figure 11: Single Tooth FEA, coarse mesh
Figure 11 shows the bending stress in the single gear tooth with a coarse mesh. The center
section clears shows the neutral surface with compression and tension occurring at the
expected locations. The mesh is refined much further until the solution converges. For analysis
purposes, the deflection was also captured through the mesh convergence process.
Figure 12: Single Tooth FEA, Displacement, coarse mesh
Figure 12 shows the displacement of the single tooth under tangential loading. The distribution
is as expected within the model. The following figures show finer meshes for both bending
12
stress and displacements. Results from the single tooth FEA mesh convergence study are
located in
Table 4.
Figure 13: Single Tooth FEA, finer mesh
Figure 14: Single Tooth FEA, Displacement, finer mesh
13
Figure 15: Single Tooth FEA, fine mesh
Figure 16: Single Tooth FEA, Displacement, fine mesh
14
Figure 17: Single Tooth FEA, ultra fine mesh
Figure 18: Single Tooth FEA, Displacement, ultra fine mesh
Table 4: Single Tooth FEA Results
15
Mesh
# Elements
S_mises Max [psi]
% Difference
U_R Max [in]
% Difference
Coarse
62
162207
Finer
199
166133
2.36%
1.7340E-03
0.000%
Fine
1221
172728
3.82%
1.7370E-03
0.173%
Super Fine
7481
179811
3.94%
1.7370E-03
0.000%
Ultra Fine
30704
179955
0.08%
1.7380E-03
0.058%
1.7340E-03
Abaqus – Single Tooth Bending with Fixed ID of Gear
In an attempt to more realistically predict the bending stress in the gear tooth, an entire 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 Section 0. Figure 19 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 ID to a reference point at the gear’s geometric
center. This reference point is then fixed in all directions.
Figure 19: Single Tooth Gear Model with Localized Mesh Refinement
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 0. The following figures present
software outputs for varying meshes. Table 5 summarizes outputs for this section.
16
Figure 20: Single Tooth Coarse Mesh Bending Stress
Figure 20, 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 in the base of the tooth below
the fillet indicate the center portion of the gear does play a non trivial part in tooth bending.
Figure 21 plots displacement of the model. The displacement gradients are reasonable and
show the base of the gear below the fillet has some small deflection. The remaining figures in
this section represent mesh refinements.
Figure 21: Single Tooth, Displacement, Coarse Mesh
17
Figure 22: Single Tooth, Mesh Refinement, Bending Stress
Figure 23: Single Tooth, Mesh Refinement, Displacement
Table 5: Single Tooth Whole Gear FEA Results
18
MESH
S_22 Max [psi]
% Difference
U_R Max [in]
% Difference
COARSE
199600
Less Coarse
191700
-4.12%
3.2840E-03
0.609%
Fine
216000
11.25%
3.2880E-03
0.122%
Finer
220700
2.13%
3.3100E-03
0.665%
Finest
222500
0.81%
3.3160E-03
0.181%
3.2640E-03
Abaqus – Solid Metal 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 inside diameter of the gear is coupled to a different 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 3dimensional 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.
19
Figure 24: Solid Metal Gear, Coarse Mesh, Stress
Figure 24 presents the stress distribution for a coarsely 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.
Figure 25: Solid Metal Gear, Coarse Mesh, Radial Displacement
Figure 25 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.
20
The model is meshed with an increasing number of elements until the outputs converge. The
following figures present this mesh refinement. Table 6 summarizes the results from this
section.
Figure 26: Solid Metal Gear, Fine Mesh, Stress
Figure 27: Solid Metal Gear, Fine Mesh, Radial Displacement
21
Figure 28: Solid Metal Gear, Ultra Fine Mesh, Stress
Figure 29: Solid Metal Gear, Ultra Fine Mesh, Radial Displacement
22
Table 6: Solid Metal Gear FEA Results
# Elements
S_mises Max [psi]
% Difference
U_R Max [in]
% Difference
592
7180
1682
7507
4.36%
2.0095E-04
-0.256%
3806
7845
4.31%
2.0107E-04
0.062%
10359
8052
2.57%
2.0118E-04
0.052%
29324
8226
2.12%
2.0130E-04
0.061%
41939
8249
0.28%
2.0135E-04
0.024%
2.0146E-04
Abaqus – Hybrid Gear Torsion
This section places the composite-metallic gear in pure torsion with loading and boundary
conditions similar to that in Section 0. 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. The adhesive was modeled as a 0.001 inch thick layer. Table 7
presents a summary of modeling information 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 parts to behave as a single
entity. The hub served as the master surface and was connected to the void and outer
23
composites using tie constraints with default connection options. This allows the surfaces to be
connected such that during deformation 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 30Error! Reference source not found. shows a
graphical representation of the adhesive connections. Figure 31 shows an exploded view of the
hybrid gear assembly with an accompanying legend in Table 8.
Figure 30: Adhesive Connection
24
Figure 31: Hybrid Gear Assembly
Table 8: Hybrid Gear Assembly Legend
Part
Color
Quantity
Hub
Light Silver
1
Void Composite
Gray
1
Tooth Ring
Blue
1
Adhesive
Green
2
Outer Composite
Red
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.
Void Composite
25
This subsection presents results on the void composite from a coarse mesh to a very fine mesh.
Figure 32 shows a coarsely meshed void composite with stress gradients plotted on the
undeformed part. Stresses are highest at the inner hexagonal features while the outer
hexagonal features predict the lowest stress. Table 9 provides a summary of results obtained.
Figure 32: Void Composite Coarse Mesh
Figure 33: Void Composite Fine Mesh
26
Figure 34: Void Composite Ultra Fine Mesh
Table 9: Void Composite FEA Results
# Global
Elements
# Local
Elements
Maximum von
Mises Stress
%
Difference
4998
656
1711.425
10557
1635
1870.064
8.48%
24262
4088
2164.235
13.59%
49106
10320
2223.565
2.67%
61868
12520
2154.530
-3.20%
71268
16650
2222.889
3.08%
84366
18810
2434.615
8.70%
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 35
shows the stress distribution in the adhesive. 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.
27
Figure 35: Adhesive Coarse Mesh
Figure 36: Adhesive Fine Mesh
28
Figure 37: Adhesive Ultra Fine Mesh
Table 10: Adhesive FEA Results
# Global
Elements
# Local
Elements
Maximum von
% Difference
Mises Stress
4998
594
311.148
10557
1212
362.371
14.14%
24262
2494
376.071
3.64%
49106
4886
400.120
6.01%
61868
6147
403.593
0.86%
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 38 shows a coarse mesh with stress gradients. The highest stresses
occur at the inner hexagonal corners and the lowest stresses are at its outer radius.
29
Figure 38: Outer Composite Coarse Mesh
Figure 39: Outer Composite Fine Mesh
30
Figure 40: Outer Composite Ultra Fine Mesh
Table 11: Outer Composite FEA Results
# Global
Elements
# Local
Elements
Maximum von
% Difference
Mises Stress
4998
1160
1797.313
10557
2832
1992.864
9.81%
24262
7176
2519.917
20.92%
49106
14090
2330.974
-8.11%
61868
18110
2448.193
4.79%
71268
20745
2390.645
-2.41%
84366
26214
2457.219
2.71%
Abaqus – 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. Similar loading and boundary conditions are
selected similar to those in Section 0. Figure 41 shows the model with holes drilled to reduce
the mass. The model was initially meshed coarsely and then refined until the solution
31
converged. Stresses and displacement field outputs were obtained. This model was meshed
with 8-node quadratic shells.
Figure 41: Lightening Holes
Figure 42: Lightening Hole Coarse Mesh Stress
32
Figure 43: Lightening Hole Coarse Mesh Displacement
Figure 42 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 43 presents
displacements for a coarse mesh model. 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.
Figure 44: Lightening Hole Fine Mesh Stress
33
Figure 45: Lightening Hole Fine Mesh Displacement
Figure 46: Lightening Hole Ultra Fine Mesh Stress
34
Figure 47: Lightening Hole Ultra Fine Mesh Displacement
Table 12: Lightening Hole FEA Results
% Difference
UR R3 MAX [in]
%
Difference
# Elements
S_mises Max [psi]
527
8567
3008
9528
10.09%
3.1760E-04
4.817%
12159
10310
7.58%
3.2300E-04
1.672%
49252
10620
2.92%
3.2500E-04
0.615%
76824
10700
0.75%
3.2590E-04
0.276%
3.0230E-04
Results and Discussion
Conclusion
Appendices
References
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35
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36
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37