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)
CONTENTS
LIST OF TABLES ............................................................................................................ iii
LIST OF FIGURES .......................................................................................................... iv
NOMENCLATURE .......................................................................................................... v
ACKNOWLEDGMENT .................................................................................................. vi
ABSTRACT .................................................................................................................... vii
1. Introduction.................................................................................................................. 1
1.1
Background ........................................................................................................ 1
1.2
Material Properties – Composite and Metal....................................................... 3
1.3
1.2.1
Tri-axial Braided Composite .................................................................. 3
1.2.2
AISI 3190 Gear Steel ............................................................................. 3
1.2.3
E-glass .................................................................................................... 4
Gear Properties ................................................................................................... 4
2. Theory/Methodology ................................................................................................... 5
2.1
Lewis Bending Stress Equation ......................................................................... 5
3. Analysis ....................................................................................................................... 9
3.1
Mathcad – Lewis Bending Equation .................................................................. 9
3.2
Abaqus – Single Tooth FEA .............................................................................. 9
3.3
Abaqus – Hybrid Gear FEA ............................................................................. 10
3.4
Abaqus – Metal Gear with Lighting Holes ...................................................... 10
4. Results and Discussion .............................................................................................. 11
5. Conclusion ................................................................................................................. 12
6. Appendices ................................................................................................................ 13
7. References.................................................................................................................. 14
ii
LIST OF TABLES
Table 1: Assumptions for Lewis Equation ........................................................................ 6
iii
LIST OF FIGURES
Figure 1: Hybrid Gear Assembly Steps [Reference 1] ...................................................... 2
Figure 2: Hybrid Gear Details [Reference 1] .................................................................... 2
Figure 3: Gear Tooth as Cantilever Beam ......................................................................... 5
Figure 4: Lewis Form Factor Plot [Reference 10] ............................................................. 6
Figure 5: Force applied to Tooth [Reference 5] ................................................................ 7
iv
NOMENCLATURE
n
Number of teeth
E
Modulus of elasticity [ksi]
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
Wt
Tangential load [lbf]
ν
Poisson ratio
G12,G13,G23
Planar shear moduli
v
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.
vi
ABSTRACT
Reducing the weight of a component while maintaining strength requirements is
often a difficult task and is typically a compromise on both ends. 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 create stresses in the composite. Lewis equations are used to determine this
stall load. A model of a single tooth is analyzed in Abaqus Finite Element Analysis
(FEA) software to verify results from the Lewis equation. This load is then applied to a
3-d model of the hybrid gear in Abaqus to predict stress levels in the composite.
Additional considerations include shear stresses at bonded interfaces and if a different
composite material would perform equally as well as the baseline composite (tri-axial
braided carbon fiber/epoxy lamina). 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
1. Introduction
1.1 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 equate to less energy consumption while operating
and is often characterized in 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 meshes 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]. Figures 1
and 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. 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. Finally the last composite section is bonded
to the assembly. 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]
2
1.2 Material Properties – Composite and Metal
1.2.1
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
The axial direction (blue arrow in figure 3) consists of 12k flattened tows of carbon
fiber. A 12K flattened tow consists of twelve thousand carbon fibers in a bundle which is
then flattened. The red arrows in figure 3 depict the bias direction (plus and minus 60
degrees off of the axial direction). These fibers are 24k flattened tows of carbon fiber.
The following table is material properties for this composite.
1.2.2
AISI 9310 Gear Steel
Gear Steel Properties and Info Here
3
1.2.3
E-glass
E-Glass Properties and Info Here
1.3 Gear Properties
Gear Properties Here
4
2. Theory/Methodology
2.1 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
4).
Figure 4: 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 figure 4.
5
Figure 5: 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 1.
Table 1: 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 5. The
6
radial component of this force would yield a compressive stress in the tooth; this force is
neglected in the Lewis equation.
Figure 6: 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,
7
the loads calculated by the Lewis equation will be used in this study to predict loading
internal to the hybrid gear.
8
3. Analysis
3.1 Mathcad – Lewis Bending Equation
Insert Mathcad analysis here
3.2 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 17558 lbf/in which equates to a load of 1069 lbf as calculated by the Lewis
equation. The bottom edge of the shell has a fixed displacement and rotation boundary
condition. Figure 7 below presents the shell model, rendered shell thickness as well as
boundary conditions and loading.
Figure 7: 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.
9
3.3 Abaqus – Hybrid Gear FEA
Modeling
Convergence Study
3.4 Abaqus – Metal Gear with Lighting Holes
Modeling
Convergence Study
10
4. Results and Discussion
11
5. Conclusion
12
6. Appendices
6.1 Mathcad Analysis
13
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.
<http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20120005332_2012004461.p
df>.
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.
3. Gibson, Ronald F. Principles of Composite Material Mechanics. 3rd ed. Boca
Raton, FL: Taylor & Francis, 2012. Print.
4. Cook, Robert D. Concepts and Applications of Finite Element Analysis. 4th ed.
New York: Wiley, 2002.
5. Budynas, Richard G., J. Keith. Nisbett, and Joseph Edward. Shigley. Shigley's
Mechanical Engineering Design. 8th ed. Boston: McGraw-Hill, 2008. Print.
6. 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>.
7. "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>.
14
8. "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)
9. "Composite Terminology." Composite Terminology. N.p., n.d. Web. 10 Sept.
2013. <http://www.cstsales.com/terminology.html>.
10. "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.
15