Dynamic Modeling of Gears: An Innovative Hybrid FEM–Analytical Approach

Dynamic Modeling of Gears: An Innovative Hybrid FEM–Analytical Approach

Franco Concli Concli Gorla

Free University of Bolzano/Bozen, Faculty of Science and Technology, Italy

Politecnico di Milano, Dept. of Mechanical Engineering, Italy

Available online: 
| Citation

© 2021 IIETA. This article is published by IIETA and is licensed under the CC BY 4.0 license (http://creativecommons.org/licenses/by/4.0/).



Gearboxes are widely used in several applications ranging from the automotive to the industrial and robotic sectors. A planetary gearbox is a special kinematic gear arrangement that, taking advantage of a planet carrier, ensures high reduction ratios together with a very small design. Therefore, they are widely employed for transmissions which require a high power density. There are several fields of applications including, but not limited to, mechatronic, automation and wind power generation. To improve the design of new solutions, for performing monitoring activities on actual gearboxes and for the definition of maintenance schedules, the availability of physical models able to accurately describe the behavior of the system, both in healthy and damaged conditions, would represent a great support. Experimental and numerical studies of the behavior of gearboxes are already available in the literature. Nevertheless, while the experimental approaches are valid only for the specific configuration tested, the numerical techniques show limitations related to the computational effort required. This paper presents an innovative approach for the characterization of the behavior of two different geared transmissions. It is based on a hybrid approach that combines finite elements (FE) with analytical formulations. In detail, the solver computes separately the macro deformation of the bodies (numerical solution based on a coarse grid) and the contacts (solved analytically avoiding the need of mesh refinements). The computational effort is reduced significantly without affecting the accuracy of the results significantly. This approach was used to investigate and understand the vibro-dynamical behavior of a back-to-back test rig (typically used for the characterization of the surface fatigue strength of gears) and of an indus- trial planetary gearbox. The results obtained for the healthy – not damaged – gearboxes were compared with experimental measurements for both configurations in order to validate the hybrid approach. Once the models were validated, the same methodology was eventually used to study the effects of typical gear failures and in specifically surface fatigue (pitting), on the vibrational response. The capability to reproduce the effect of damages with the model of a gearbox represents the first indispensable step of a Structural Health Monitoring strategy. State-of-art and challenges are analyzed and discussed in the paper.


experimental, gears, hybrid FE–analytical approach, SHM, simulation, Transmission3D


[1] Ragheb, A. & Ragheb, M., Wind turbine gearbox technologies, 2010 1st International Nuclear and Renewable Energy Conference, 2010.

[2] Austin, J.L., A Multi-Component Analysis of a Wind Turbine Gearbox using A High Fidelity Finite Element Model, 2013.

[3] Guo, Y., Keller, J., Errichello, R. & Halse, C., Gearbox Reliability Collaborative Analytic Formulation for the Evaluation of Spline Couplings, 2013.

[4] Kahraman, A., An Experimental and Theoretical Investigation of Micropiiting in Wind Turbine Gears and Bearings. United States, 2012. doi:10.2172/1037344

[5] Qu, A., Hong, L., Jian, X., He, M., He, D., Tan, Y. & Zhou, Z., Experimental study of dynamic strain for gear tooth using fiber Bragg gratings and piezoelectric strain sensors. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci., 0(0), pp. 1–12, 2017.

[6] Concli, F., Austempered Ductile Iron (ADI) for gears: Contact and bending fatigue behavior, Procedia Struct. Integr., 8, pp. 14–23, 2018. https://doi.org/10.1016/j. prostr.2017.12.003

[7] Gorla, C., Conrado, E., Rosa, F. & Concli, F., Contact and bending fatigue behaviour of austempered ductile iron gears. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci., 232(6), 2018. https://doi.org/10.1177/0954406217695846

[8] Concli, F., Cortese, L., Vidoni, R., Nalli, F. & Carabin G., A mixed FEM and lumped- parameter dynamic model for evaluating the modal properties of planetary gearboxes. Journal of Mechanical Science and Technology, 32(7), pp. 3047–3056, 2018. https:// doi.org/10.1007/s12206-018-0607-9

[9] Wehrle, E., Concli, F., Cortese, L. & Vidoni, R., Design optimization of planetary gear trains under dynamic constraints and parameter uncertainty. Proceedings of the 8th ECCOMAS Thematic Conference on MULTIBODY DYNAMICS 2017, 2017.

[10] Concli, F., Conrado, E. & Gorla, C., Analysis of power losses in an industrial planetary speed reducer: Measurements and computational fluid dynamics calculations. Proc. Inst. Mech. Eng. Part J J. Eng. Tribol., 228(1), pp. 11–21, 2014. https://doi.org/10.1177/1350650113496980

[11] Concli, F. & Gorla, C., Influence of lubricant temperature, lubricant level and rotational speed on the churning power loss in an industrial planetary speed reducer: computational and experimental study. Int. J. Comput. Methods Exp. Meas., 1(4), pp. 353–366, 2013. https://doi.org/10.2495/cmem-v1-n4-353-366

[12] Concli, F., Low-loss gears precision planetary gearboxes: reduction of the load dependent power losses and efficiency estimation through a hybrid analytical-numerical opti- mization tool [Hochleistungsund Präzisions-Planetengetriebe: Effizienzschätzung und Reduzierung]. Forsch. im Ingenieurwesen/Engineering Res., 81(4), pp. 395–407, 2017. https://doi.org/10.1007/s10010-017-0242-0

[13] Concli, F. & Gorla, C., CFD simulation of power losses and lubricant flows in gearboxes. American Gear Manufacturers Association Fall Technical Meeting 2017, 2017.

[14] Vijayakar, S., A combined surface integral and finite element solution for a three‐dimen- sional contact problem. Int. J. Numer. Methods Eng., 31(3), pp. 525–545, 1991. https:// doi.org/10.1002/nme.1620310308

[15] Prueter, P.E., Parker, R.G. & Cunliffe, F., A study of gear root strains in a multi-stage planetary wind turbine gear train using a three dimensional finite element/contact mechanics model and experiments. Proceedings of the ASME Design Engineering Technical Conference, 8, pp. 621–633, 2011.

[16] Vijayakar, S.M., Busby, H.R. & Houser, D.R., Linearization of multibody frictional contact problems. Comput. Struct., 29(4), pp. 569–576, 1988. https://doi.org/10.1016/0045- 7949(88)90366-5

[17] Parker, R.G., Vijayakar, S.M. & Imajo T., Non-linear dynamic response of a spur gear pair: modelling and experimental comparisons. J. Sound Vib., 237(3), pp. 435–455, 2000. https://doi.org/10.1006/jsvi.2000.3067

[18] Hamand, Y.C. & Kalamkar, V., Analysis of stresses and deflection of sun gear by theoretical and ANSYS method. Modern Mechanical Engineering, 1(2), pp. 56–68, 2011. https://doi.org/10.4236/mme.2011.12008

[19] Haastrup, M., Hansen, M.R., Ebbesen, M.K. & Mouritsen, O.Ø., Modeling and parameter identification of deflections in planetary stage of wind turbine gearbox. Modeling, Identification and Control, 33(1), pp. 1–11, 2012. https://doi.org/10.4173/mic.2012.1.1

[20] Lin, J., Analytical characterization of the unique properties of planetary gear free vibration. Journal of Vibration and Acoustics, 121(3), pp. 316–321, 2018. https://doi. org/10.1115/1.2893982

[21] Drive, O.C., Internal gear strains and load sharing in planetary. Journal of Mechanical Design, 130(7), pp. 1–12, 2007. https://doi.org/10.1115/1.2890110