Estimation of a Planetary Gear Mesh Stiffness: An Approach Based on Minimising Error Function


  •   K. A. Olanipekun


The mesh stiffness of gear teeth is one of the major sources of excitation in gear systems. Many analytical and finite element methods have been proposed in order to determine the mesh stiffness of gears especially parallel axis spur gears. Most of these methods are not trivial because they involve complicated analyses which incorporate parameters like gear tooth error, gear spalling sizes and shapes, nonlinear contact stiffness and sliding friction before mesh stiffness can be determined. In this work, a method is proposed to determine the sun-planet and ring-planet mesh stiffnesses of a planetary gear system. This approach involves fitting a relationship between the measured natural frequencies from an experimental modal test and natural frequencies predicted using an analytical model of a planetary gear. This method is relatively easier compared to the existing methods which involve complicated analyses. For this study, the average mesh stiffness estimated is 12.5 MN/m.

Keywords: Error function, natural frequency, mesh stiffness, minimisation, planetary gear


N. K. Raghuwanshi and A. Parey, “Mesh stiffness measurement of cracked spur gear by photoelasticity technique,” Meas. J. Int. Meas. Confed., vol. 73, 2015, doi: 10.1016/j.measurement.2015.05.035.

H. Nevzat Özgüven and D. R. Houser, “Mathematical models used in gear dynamics—A review,” Journal of Sound and Vibration, vol. 121, no. 3. 1988, doi: 10.1016/S0022-460X(88)80365-1.

C. G. Cooley and R. G. Parker, “A review of planetary and epicyclic gear dynamics and vibrations research,” Applied Mechanics Reviews, vol. 66, no. 4. 2014, doi: 10.1115/1.4027812.

C. G. Cooley, C. Liu, X. Dai, and R. G. Parker, “Gear tooth mesh stiffness: A comparison of calculation approaches,” Mech. Mach. Theory, vol. 105, 2016, doi: 10.1016/j.mechmachtheory.2016.07.021.

R. G. Parker, S. M. Vijayakar, and T. Imajo, “Non-linear dynamic response of a spur gear pair: modelling and experimental comparisons,” J. Sound Vib., vol. 237, no. 3, 2000, doi: 10.1006/jsvi.2000.3067.

V. K. Ambarisha and R. G. Parker, “Nonlinear dynamics of planetary gears using analytical and finite element models,” J. Sound Vib., vol. 302, no. 3, 2007, doi: 10.1016/j.jsv.2006.11.028.

C. J. Bahk and R. G. Parker, “Analytical solution for the nonlinear dynamics of planetary gears,” J. Comput. Nonlinear Dyn., vol. 6, no. 2, 2011, doi: 10.1115/1.4002392.

C. J. Bahk and R. G. Parker, “Analytical investigation of tooth profile modification effects on planetary gear dynamics,” Mech. Mach. Theory, vol. 70, 2013, doi: 10.1016/j.mechmachtheory.2013.07.018.

T. Kiekbusch, D. Sappok, B. Sauer, and I. Howard, “Calculation of the combined torsional mesh stiffness of spur gears with two- and three-dimensional parametrical FE models,” Stroj. Vestnik/Journal Mech. Eng., vol. 57, no. 11, 2011, doi: 10.5545/sv-jme.2010.248.

I. Howard, S. Jia, and J. Wang, “The dynamic modelling of a spur gear in mesh including friction and a crack,” Mech. Syst. Signal Process., vol. 15, no. 5, 2001, doi: 10.1006/mssp.2001.1414.

L. Chang, G. Liu, and L. Wu, “A robust model for determining the mesh stiffness of cylindrical gears,” Mech. Mach. Theory, vol. 87, 2015, doi: 10.1016/j.mechmachtheory.2014.11.019.

M. B. Sánchez, M. Pleguezuelos, and J. I. Pedrero, “Approximate equations for the meshing stiffness and the load sharing ratio of spur gears including hertzian effects,” Mech. Mach. Theory, vol. 109, 2017, doi: 10.1016/j.mechmachtheory.2016.11.014.

O. D. Mohammed, M. Rantatalo, and J. O. Aidanpää, “Improving mesh stiffness calculation of cracked gears for the purpose of vibration-based fault analysis,” Eng. Fail. Anal., vol. 34, 2013, doi: 10.1016/j.engfailanal.2013.08.008.

H. Ma, Z. Li, M. Feng, R. Feng, and B. Wen, “Time-varying mesh stiffness calculation of spur gears with spalling defect,” Eng. Fail. Anal., vol. 66, 2016, doi: 10.1016/j.engfailanal.2016.04.025.

F. Karpat, O. Dogan, C. Yuce, and S. Ekwaro-Osire, “An improved numerical method for the mesh stiffness calculation of spur gears with asymmetric teeth on dynamic load analysis,” Adv. Mech. Eng., vol. 9, no. 8, 2017, doi: 10.1177/1687814017721856.

Z. Chen and Y. Shao, “Mesh stiffness calculation of a spur gear pair with tooth profile modification and tooth root crack,” Mech. Mach. Theory, vol. 62, 2013, doi: 10.1016/j.mechmachtheory.2012.10.012.

A. Saxena, A. Parey, and M. Chouksey, “Time varying mesh stiffness calculation of spur gear pair considering sliding friction and spalling defects,” Eng. Fail. Anal., vol. 70, 2016, doi: 10.1016/j.engfailanal.2016.09.003.

J. Lin and R. G. Parker, “Analytical characterization of the unique properties of planetary gear free vibration,” J. Vib. Acoust. Trans. ASME, vol. 121, no. 3, 1999, doi: 10.1115/1.2893982.


Download data is not yet available.


How to Cite
Olanipekun, K.A. 2021. Estimation of a Planetary Gear Mesh Stiffness: An Approach Based on Minimising Error Function. European Journal of Engineering and Technology Research. 6, 3 (Apr. 2021), 164–169. DOI: