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Research Papers: Design of Direct Contact Systems

Implementation of a Finite Element Model for Gear Stress Analysis Based on Tie-Surface Constraints and Its Validation Through the Hertz's Theory

[+] Author and Article Information
Ignacio Gonzalez-Perez

Department of Mechanical Engineering,
Polytechnic University of Cartagena,
Cartagena 30202, Spain
e-mail: ignacio.gonzalez@upct.es

Alfonso Fuentes-Aznar

Department of Mechanical Engineering,
Rochester Institute of Technology,
Rochester, NY 14623
e-mail: afeme@rit.edu

1Corresponding author.

Contributed by the Power Transmission and Gearing Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received July 14, 2017; final manuscript received October 10, 2017; published online November 13, 2017. Assoc. Editor: Qi Fan.

J. Mech. Des 140(2), 023301 (Nov 13, 2017) (13 pages) Paper No: MD-17-1480; doi: 10.1115/1.4038301 History: Received July 14, 2017; Revised October 10, 2017

A new finite element model for stress analysis of gear drives is proposed. Tie-surface constraints are applied at each tooth of the gear model to obtain meshes that can be independently defined: a finer mesh at contact surfaces and fillet and a coarser mesh in the remaining part of the tooth. Tie-surface constraints are also applied for the connection of several teeth in the model. The model is validated by application of the Hertz's theory in a spiral bevel gear drive with localized bearing contact and by observation of convergency of contact and bending stresses. Maximum contact pressure, maximum Mises stress, maximum Tresca stress, maximum major principal stress, and loaded transmission errors are evaluated along two cycles of meshing. The effects of the boundary conditions that models with three, five, seven, and all the teeth of the gear drive provide on the above-mentioned variables are discussed. Several numerical examples are presented.

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References

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Handschuh, R. F. , and Litvin, F. L. , 1991, “ A Method for Determining Spiral-Bevel Gear Tooth Geometry for Finite Element Analysis,” National Aeronautics and Space Administration, Washington, DC, AVSCOM Technical Report No. 91-C-020. http://www.dtic.mil/dtic/tr/fulltext/u2/a242332.pdf
Argyris, J. H. , Fuentes, A. , and Litvin, F. L. , 2002, “ Computerized Integrated Approach for Design and Stress Analysis of Spiral Bevel Gears,” Comput. Methods Appl. Mech. Eng., 191(11–12), pp. 1057–1095. [CrossRef]
Gonzalez-Perez, I. , Roda-Casanova, V. , Fuentes, A. , Sanchez-Marin, F. T. , and Iserte, J. L. , 2012, “ A Finite Element Model for Consideration of the Torsional Effect on the Bearing Contact of Gear Drives,” ASME J. Mech. Des., 134(7), p. 071007. [CrossRef]
Li, S. , 2002, “ Gear Contact Model and Loaded Tooth Contact Analysis of a Three-Dimensional, Thin-Rimmed Gear,” ASME J. Mech. Des., 124(3), pp. 511–517. [CrossRef]
Gonzalez-Perez, I. , Fuentes, A. , Roda-Casanova, V. , Sanchez-Marin, F. T. , and Iserte, J. L. , 2013, “ A Finite Element Model for Stress Analysis of Lightweight Spur Gear Drives Based on Thin-Webbed and Thin-Rimmed Gears,” International Conference on Gears, VDI-Society for Product and Process Design, Garching, Germany, Oct. 7–9, pp. 75–86.
Roda-Casanova, V. , Sanchez-Marin, F. T. , Gonzalez-Perez, I. , Iserte, J. L. , and Fuentes, A. , 2013, “ Determination of the ISO Face Load Factor in Spur Gear Drives by the Finite Element Modeling of Gears and Shafts,” Mech. Mach. Theory, 65, pp. 1–13. [CrossRef]
Fuentes, A. , Ruiz-Orzaez, R. , and Gonzalez-Perez, I. , 2016, “ Compensation of Errors of Alignment Caused by Shaft Deflections in Spiral Bevel Gear Drives,” Theory and Practice of Gearing and Transmissions (Mechanisms and Machine Science), Vol. 34, Springer, Cham, Switzerland, pp. 301–319. [CrossRef]
Vijayakar, S. M. , 1987, “ Finite Element Methods for Quasi-Prismatic Bodies With Application to Gears,” Ph.D. thesis, The Ohio State University, Columbus, OH. https://etd.ohiolink.edu/pg_10?114811821050336::NO:10:P10_ETD_SUBID:138872
Gonzalez-Perez, I. , and Fuentes, A. , 2017, “ Implementation of a Finite Element Model for Stress Analysis of Gear Drives Based on Multi-Point Constraints,” Mech. Mach. Theory, 117, pp. 35–47. [CrossRef]
Mao, K. , 2007, “ Gear Tooth Contact Analysis and Its Application in the Reduction of Fatigue Wear,” Wear, 262(11–12), pp. 1281–1288. [CrossRef]
Vijayakar, S. M. , 1991, “ A Combined Surface Integral and Finite Element Solution for a Three-Dimensional Contact Problem,” Int. J. Numer. Methods Eng., 31(3), pp. 525–545. [CrossRef]
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Fuentes, A. , Ruiz-Orzaez, R. , and Gonzalez-Perez, I. , 2014, “ Computerized Design, Simulation of Meshing, and Finite Element Analysis of Two Types of Geometry of Curvilinear Cylindrical Gears,” Comput. Methods Appl. Mech. Eng., 272, pp. 321–339. [CrossRef]
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Figures

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Fig. 1

For mesh definition of tooth body region: (a) designed volume with obtained points from application of gear theory, (b) auxiliary surfaces and intermediate surface S, (c) nodes, and (d) finite elements

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Fig. 2

For mesh definition of contact-fillet region: (a) designed volume with obtained points from application of gear theory and (b) nodes and finite elements

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Fig. 3

For illustration of the tie-surface constraint between the nodes of the slave surface and the master surface for one tooth of the gear model

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Fig. 4

Scheme in a transverse section at the heel of the gear teeth for illustration of: (i) slave and master surfaces to connect several teeth and (ii) rigid surface for application of boundary conditions

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Fig. 5

For illustration of the rigid surface in, (a) a three-tooth finite element model and (b) a all-tooth finite element model, of the pinion

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Fig. 6

Gear drive model and contact patterns corresponding to three cases of design

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Fig. 7

Evolution of stresses underneath the contacting surfaces for case 2 of design at mean contact position 12 obtained from application of Hertz's theory

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Fig. 8

Finite element model (a) for investigation of convergency of contact stresses and contact area for case 2 of design when varying the number of profile elements np (b), and the number of longitudinal elements nl (c)

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Fig. 9

Contact pressure distribution in case 2 of design for (a) ct=0.2 and (b) ct=0.125 with three layers of quadratic elements

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Fig. 10

Variation of maximum contact pressure (a) and contact area (b) in case 2 of design considering models 64×48×16 with different coefficients ct, linear or quadratic elements, and three or six layers of elements

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Fig. 11

Mises and Tresca stresses for several analysis of case 2 of design considering models 64×48×16: (a) Mises distribution for ct=0.2 and three layers of quadratic elements, (b) maximum Mises, and (c) maximum Tresca for several values of coefficient ct, type of element and number of layers

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Fig. 12

For illustration of convergency of maximum major principal stress σ1o in the fillet region of the middle tooth for several values of the number of elements in the fillet direction nf and different types of elements

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Fig. 13

Finite element models based on (a) five pairs of contacting teeth (five-tooth pair model) and (b) five pairs of contacting teeth and all the other teeth of the gear drive (five-tooth pair complete model)

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Fig. 14

Maximum contact pressures po and maximum major principal stresses σ1o along two cycles of meshing at the middle tooth of a three-tooth pair model for illustration of different behavior of element types C3D20, C3D8I, and C3D8

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Fig. 15

Maximum contact pressure po along two cycles of meshing in a: (a) three-tooth pair model, (b) three-tooth pair complete model, (c) five-tooth pair model, and (d) five-tooth pair complete model

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Fig. 16

For illustration of the periodicity grade of the function of total transmission errors (TE) in a: (a) three-tooth pair model, (b) three-tooth pair complete model, (c) five-tooth pair model, (d) five-tooth pair complete model, (e) seven-tooth pair model, and (f) seven-tooth pair complete model

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