Abstract
This paper proposes a dynamic model for the first time in order to investigate nonlinear time-varying dynamic behavior of a drivetrain including parallel axis gears (such as spur and helical gears) and intersecting axis gears (such as spiral bevel gears). Flexibilities of shafts and bearings are included in the dynamic model by the use of finite element modeling. Finite element models of shafts are coupled with each other by the mesh models of gear pairs including backlash nonlinearity and fluctuating mesh stiffness. A system of nonlinear algebraic equations is established from the resulting nonlinear differential equations of motion by utilizing multi-harmonic harmonic balance method (HBM) in conjunction with continuous-time Fourier transform (CFT). Since the number of nonlinear equations is large, potential convergence problems are avoided by utilizing continuous-time Fourier transform, in contrast with gear dynamics studies that use discrete Fourier transform (DFT). Solutions obtained by utilizing CFT and DFT are compared, and the advantages of utilizing CFT are shown. Fourier coefficients are calculated by utilizing analytical integration rather than numerical integration for a further improvement in computational time. A new solution method, modal superposition method, is introduced for the first time to study nonlinear dynamics of drivetrains with multiple gear meshes, which is impractical if traditional solution methods are used due to the increased number of nonlinear equations. Using modal superposition method, the number of nonlinear equations becomes proportional to the number of modes employed which is significantly less than the number of degrees of freedom associated with nonlinearities, especially as the number of gear meshes in the drivetrain increases. Consequently, the proposed method decreases the computational effort drastically in the forced response analysis of multi-mesh, multi-stage gear systems and also makes it possible to model gear shafts by using finite element method. The resulting system of nonlinear algebraic equations is solved by utilizing Newton’s method with arc-length continuation. Solutions obtained by HBM are validated by the solutions obtained by direct numerical integration. Several parametric studies are carried out in order to investigate the effects of design parameters on the dynamics of the drivetrain. It is observed that nonlinear modeling of helical gear pairs is necessary if they are coupled with spur or spiral bevel gear pairs.
Similar content being viewed by others
References
Al-shyyab, A., Kahraman, A.: Non-linear dynamic analysis of a multi-mesh gear train using multi-term harmonic balance method: period-one motions. J. Sound Vib. 284, 151–172 (2005)
Al-shyyab, A., Kahraman, A.: Non-linear dynamic analysis of a multi-mesh gear train using multi-term harmonic balance method: sub-harmonic motions. J. Sound Vib. 279, 417–451 (2005)
Yavuz, S.D., Saribay, Z.B., Cigeroglu, E.: Nonlinear time-varying dynamic analysis of a multi-mesh spur gear train. In: Allen, M., Mayes, R., Rixen, D. (eds.) Dynamics of Coupled Structures. Conference Proceedings of the Society for Experimental Mechanics Series, vol. 4. Springer, Cham (2016)
Liu, G., Parker, R.G.: Nonlinear dynamics of idler gear systems. Nonlinear Dyn. 53, 345–367 (2008)
Liu, G., Parker, R.G.: Nonlinear, parametrically excited dynamics of two-stage spur gear trains with mesh stiffness fluctuation. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci. 226, 1939–1957 (2012)
Zhao, M., Ji, J.C.: Nonlinear torsional vibrations of a wind turbine gearbox. Appl. Math. Model. 39, 4928–4950 (2015). https://doi.org/10.1016/j.apm.2015.03.026
Xiang, L., Gao, N., Hu, A.: Dynamic analysis of a planetary gear system with multiple nonlinear parameters. J. Comput. Appl. Math. 327, 325–340 (2018). https://doi.org/10.1016/j.cam.2017.06.021
Kahraman, A.: Dynamic analysis of a multi-mesh helical gear train. J. Mech. Des. 116, 706–712 (1994)
Kubur, M., Kahraman, A., Zini, D.M., Kienzle, K.: Dynamic analysis of a multi-shaft helical gear transmission by finite elements: model and experiment. J. Vib. Acoust. 126, 398–406 (2004)
Cheng, Y., Lim, T.C.: Vibration analysis of hypoid transmissions applying an exact geometry-based gear mesh theory. J. Sound Vib. 240, 519–543 (2001)
Yang, J., Peng, T., Lim, T.C.: An enhanced multi-term harmonic balance solution for nonlinear period-one dynamic motions in right-angle gear pairs. Nonlinear Dyn. 67, 1053–1065 (2011)
Wang, J., Lim, T.C., Li, M.: Dynamics of a hypoid gear pair considering the effects of time-varying mesh parameters and backlash nonlinearity. J. Sound Vib. 308, 302–329 (2007)
Peng, T., Lim, T.C.: Coupled multi-body dynamic and vibration analysis of high-speed hypoid geared rotor system. In: Proceedings of the SAE Noise and Vibration Conference and Exposition., St. Charles, Illinois, USA (2007)
Yang, J.J., Shi, Z.H., Zhang, H., Li, T.X., Nie, S.W., Wei, B.Y.: Dynamic analysis of spiral bevel and hypoid gears with high-order transmission errors. J. Sound Vib. 417, 149–164 (2018). https://doi.org/10.1016/j.jsv.2017.12.022
Hua, X., Lim, T.C., Peng, T., Wali, W.E.: Dynamic analysis of spiral bevel geared rotor systems applying finite elements and enhanced lumped parameters. Int. J. Automot. Technol. 13, 97–107 (2012)
Yavuz, S.D., Saribay, Z.B., Cigeroglu, E.: Nonlinear time-varying dynamic analysis of a spiral bevel geared system. Nonlinear Dyn. 92, 1901–1919 (2018). https://doi.org/10.1007/s11071-018-4170-9
Menq, C.H., Griffin, J.H., Bielak, J.: The forced response of shrouded fan stages. J. Vib. Acoust. Stress. Reliab. Des. 108, 50–55 (1986). https://doi.org/10.1115/1.3269303
Menq, C.-H., Griffin, J.H., Bielak, J.: The influence of microslip on vibratory response, Part II: a comparison with experimental results. J. Sound Vib. 107, 295–307 (1986)
Cigeroglu, E., An, N., Menq, C.-H.: Forced response prediction of constrained and unconstrained structures coupled through frictional contacts. J. Eng. Gas Turbines Power 131, 022505–022505-11 (2009)
Chen, W., Chen, S., Hu, Z., Tang, J., Li, H.: A novel dynamic model for the spiral bevel gear drive with elastic ring squeeze film dampers. Nonlinear Dyn. 98, 1081–1105 (2019). https://doi.org/10.1007/s11071-019-05250-9
Chen, W., Chen, S., Hu, Z., Tang, J., Li, H.: Dynamic analysis of a bevel gear system equipped with finite length squeeze film dampers for passive vibration control. Mech. Mach. Theory 147, 103779 (2020). https://doi.org/10.1016/j.mechmachtheory.2019.103779
Cao, W., Pu, W., Wang, J.: Tribo-dynamic model and fatigue life analysis of spiral bevel gears. Eur. J. Mech. A Solids 74, 124–138 (2019). https://doi.org/10.1016/j.euromechsol.2018.10.013
Lim, T.C., Singh, R.: Vibration transmission through rolling element bearings, part I: bearing stiffness formulation. J. Sound Vib. 139, 179–199 (1990)
Peng, T.: Coupled Multi-Body Dynamic and Vibration Analysis of Hypoid and Bevel Geared Rotor System. Dissertation, University of Cincinnati (2010)
Kuran, B., Özgüven, H.N.: A modal superposition method for non-linear structures. J. Sound Vib. 189, 315–339 (1996)
Cigeroglu, E., An, N., Menq, C.-H.: A microslip friction model with normal load variation induced by normal motion. Nonlinear Dyn. 50, 609–626 (2007)
Ferhatoglu, E., Cigeroglu, E., Özgüven, H.N.: A new modal superposition method for nonlinear vibration analysis of structures using hybrid mode shapes. Mech. Syst. Signal Process. 107, 317–342 (2018). https://doi.org/10.1016/j.ymssp.2018.01.036
Von Groll, G., Ewins, D.J.: The harmonic balance method with arc-length continuation in rotor/stator contact problems. J. Sound Vib. 241, 223–233 (2001)
Kahraman, A., Blankenship, G.W.: Experiments on nonlinear dynamic behavior of an oscillator with clearance and periodically time-varying parameters. J. Appl. Mech. 64, 217–226 (1997)
Özgüven, H.N., Houser, D.R.: Dynamic analysis of high speed gears by using loaded static transmission error. J. Sound Vib. 125, 71–83 (1988)
Kahraman, A., Singh, R.: Non-linear dynamics of a spur gear pair. J. Sound Vib. 142, 49–75 (1990)
Kahraman, A., Blankenship, G.W.: Steady state forced response of a mechanical oscillator with combined parametric excitation and clearance type non-linearity. J. Sound Vib. 185, 743–765 (1995)
Yavuz, S.D., Saribay, Z.B., Cigeroglu, E.: Nonlinear dynamic analysis of a spiral bevel geared system. In: Di Maio, D., Castellini, P. (eds.) Rotating Machinery, Hybrid Test Methods, Vibro-Acoustics & Laser Vibrometry. Conference Proceedings of the Society for Experimental Mechanics Series, vol. 8. Springer, Cham (2017)
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Yavuz, S.D., Saribay, Z.B. & Cigeroglu, E. Nonlinear dynamic analysis of a drivetrain composed of spur, helical and spiral bevel gears. Nonlinear Dyn 100, 3145–3170 (2020). https://doi.org/10.1007/s11071-020-05666-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-020-05666-8