Bevel Gear Paper

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1 Bevel Gear Tooth Bending Stress Evaluation Using Finite Element Analysis Prepared by Emre Turkoz, BSME | [email protected] Can Ozcan, MSME | [email protected] AKRO R&D Ltd. Phone: +90 (262) 678-7215 KEMAL NEHROZOGLU CAD. GOSB TEKNOPARK HIGH TECH BINA 3.KAT B5 GEBZE/KOCAELI/TURKEY - 41480

Transcript of Bevel Gear Paper

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Bevel Gear Tooth Bending Stress Evaluation Using Finite Element Analysis

Prepared by

Emre Turkoz, BSME | [email protected]

Can Ozcan, MSME | [email protected]

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

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