Post on 13-May-2017
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Bevel Gear Tooth Bending Stress Evaluation Using Finite Element Analysis
Prepared by
Emre Turkoz, BSME | emre.turkoz@akroengineering.com
Can Ozcan, MSME | can.ozcan@akroengineering.com
AKRO R&D Ltd.
Phone: +90 (262) 678-7215
KEMAL NEHROZOGLU CAD. GOSB TEKNOPARK
HIGH TECH BINA 3.KAT B5
GEBZE/KOCAELI/TURKEY - 41480
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1. Introduction Bevel gears are used widely at different applications in Industrial Machinery, especially in the
automotive industry. Since bevel gears have such a great range of applications, it’s crucial to be able to
analyze their deformation under an applied load. In this work, our aim is to investigate the behavior of a
bevel gear set under a given moment using Finite Element Analysis.
The results are evaluated both numerically and analytically. For the analytic solution, formulas from
Norton [1] is used. Finite element analysis is used as the numerical method. Autodesk Simulation
Mechanical 2012 is used to perform finite element analysis.
2. Properties of the Bevel Gear Set The bevel gear set consists of two helical gears. In gear terminology, the smaller one is called pinion, and
the larger one is called gear.
The properties of the gear and the pinion are as follows:
3. Description of the Problem The problem solved is the static application of a moment of 600000 Nmm on the pinion, which tries to
rotate but is hindered by the grounded gear. The torque is transferred to the gear through contact faces
on tooth pairs. The moment causes on these pairs a contact force to be generated. Apart from the
contact stress this force forms, roots of the contact teeth also suffer from tooth root bending stress. The
aim of this work is the evaluation of this root bending stress generated during the static application of
the moment. The numerical and the analytical solutions are compared to validate the model used for
the finite element analysis.
4. Analytical Solution of the Problem The methodology for the analytical solution is obtained from Norton [1], as stated before. The formula
for the tooth root bending stress, for the gear or for the pinion, is given below:
# of tooth in Gear: Ng = 29
# of tooth in Pinion: Np = 17
Facewidth: F = 62.354 mm
Gear pitch diameter: dg = 220 mm
Pinion pitch diameter: dp = 130 mm
Pressure angle: Θ = 20o
Spiral angle: ϕ = 35o
Module: m = d/N = 7.58 mm
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: Tooth root bending stress [Mpa]
T : Torque applied or transformed to the gear/pinion [Nmm]
d : Diameter of the gear/pinion [mm]
F : Facewidth [mm]
J : Geometry factor of the gear/pinion
m: Module [mm]
Ka: Application factor
Km: Load distribution factor
Ks: Size factor
Kv: Dynamic factor
Kx: Gear geometry factor (spiral/straight)
The analytical calculation is performed for the gear in our problem. The parameters and the result are
given in the table at right above.
5. Modeling of the Problem Using the Finite Element Analysis
As stated above, Autodesk Simulation Mechanical 2012 is used to perform the finite element analysis.
As the Analysis Type, “Static Stress with Linear Material Models” is chosen, since the model doesn’t
include any nonlinearity.
As Mesh settings, an absolute surface mesh size of 6 mm is imposed. Solid mesh is set to the option “All
tetrahedral”. Contact setting is left as the default setting, bonded. To get more accurate results, mesh of
the contact region is refined. The vertices in the contact zone are selected as refinement points and they
are forced to have the mesh size of 1.25 mm and the radius of 7 mm.
Gear Ratio 0.586207
Tp [Nmm] 600000
d [mm] 220
F [mm] 62.354
J 0.21
m [mm] 7.58
Ka 1
Km 1.6
Ks 1
Kv 1
Kx 1.15
[MPa] 44.82072
Figure 5.1: Top and Right views of the meshed bevel gear set
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The boundary conditions and moment applied should be specified in the Load and Constraint Groups
section. Since the tetrahedron element doesn’t possess rotational degree of freedoms, separate beam
joint elements should be defined, on which rotational degree of freedoms can be imposed. Two surfaces
encapsulating the inner cylindrical area of the gear and the two surfaces encapsulating the cylindrical
surface of the shaft connected to the pinion are selected and joints are added to these surface pairs. The
two joint vertices of the gear are selected and fixed boundary conditions are imposed, whereas the two
joint vertices of the pinion are imposed only one rotational degree of freedom, which is y direction in
our model. The two vertices in the pinion joint are selected and imposed a moment of -300000 Nmm in
y direction, which add up to -600000 Nmm, the desired amount for our model.
For shorter solution times, contact region surfaces can be separated from their corresponding parts by
assigning a new surface attribute to the participating line elements of each part. Then these contact
surfaces should be selected and specified as in surface contact.
6. Numerical Solution of the Problem Using the Finite Element
Analysis The finite element analysis results are pretty much close to the analytically evaluated results. The mean
tooth root bending stress, evaluated by selecting the nodes at the root of the contact teeth, has the
value of 37,016 MPa, which corresponds to the 17% difference with the analytical result.
Figure 6.1: The gear mesh. The contact region has the finest elements
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To calculate the tooth root bending stress, the vertices in the root region are selected and the mean
stress is calculated. The mean, as it can be seen from the figure below, is 37,016 MPa.
Figure 6.2: A closer look at the contact region. Pay attention to the contact pattern where the stresses are high
Figure 6.3. Calculating the root tooth bending stress
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To see the big picture more clearly, stress distribution plot of the root vertices are generated as can be
seen below. It is to be observed that the stress values in general lie between 20-50 MPa. The values are
concentrated between 35 and 40 MPa.
Figure 6.4. Stress distribution of the root vertices.
7. Discussion We see a great similarity between the numerical and the analytical results. From the results evaluated, it
can be said that the FEA Analysis is validated. The difference in between is caused by many factors. The
accuracy of the FEA results may be increased by using more mesh elements, which encapsulate contact
regions more densely. Also a smaller tolerance for the solution of the stiffness matrix can be imposed.
8. References [1] Norton, Robert L., “Machine Design: An Integrated Approach”, Third Edition, 2006, Pearson Prentice
Hall .
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