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Dynamic analysis of a multi-stage gear system considering the coupling between mesh phasing angle and coaxial teeth ratio

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Abstract

The mesh phasing angle is an important influencing factor in a gear dynamic system. However, previous studies of mesh phasing in a multi-stage gear system ignore the coupling effects of coaxial teeth ratio. Therefore, this paper derives the coupling relationship between mesh phasing angle and coaxial teeth ratio in a multi-stage gear system. The dynamic model for a two-stage parallel shaft gear train with time-varying mesh stiffness is established. The phase relationship is validated by rigid-flexible coupling model. Through the derived coupling relationship, it is found that the coaxial teeth ratio reduces the periodic variation range of mesh phasing. When the coaxial teeth ratio is not equal to 1, the effect of mesh phasing on the nonlinear vibration response of the system is investigated, and the suppression effect on the system vibration is found. The influence of mesh phasing on the chaotic motion of the system is analyzed in detail. The effect of mesh phasing on the vibration characteristics is compared for different coaxial teeth ratio. Finally, the change law of system vibration is revealed under the interaction of coaxial teeth ratio and mesh phasing, which provides a reference for the dynamic design of a multi-stage gear system.

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The data can be made available on reasonable request.

Abbreviations

\(J_{i} \left( {i = 1,2,3,4} \right)\) :

Inertia of the i gear

\(z_{i} \left( {i = {1},{2},{3},{4}} \right)\) :

Teeth number of the i gear

\({\text{To}}_{i} \left( {i = {1},{2}} \right)\) :

Input and output torque

\(k_{i} (t)\left( {i = {1},{2}} \right)\) :

Time-varying mesh stiffness of the i stage

\(k_{12} (t),k_{12}^{\prime } (t)\) :

The coupling stiffness of the first and second stages

\(k_{{i{\text{m}}}} \left( {i = {1},{2}} \right)\) :

Mean value of the time-varying mesh stiffness of the i stage

\(k_{h} ,k_{{\text{f}}}\) :

Contact stiffness, fillet-foundation stiffness

\(k_{{\text{b}}} ,k_{{\text{s}}} ,k_{{\text{a}}}\) :

Bending stiffness, shear stiffness and axial compressive stiffness

\(k_{e}\) :

Mesh stiffness

\(c_{i} \left( {i = {1},{2}} \right)\) :

Mesh damping of the i stage

\(b_{i} \left( {i = {1},{2}} \right)\) :

Nonlinear backlash of the i stage

\(u_{i} \left( {i = 1,2} \right)\) :

Dynamic transmission error of the i stage

\(q_{i} \left( {i = 1,2} \right)\) :

Dimensionless dynamic transmission error of the i stage

\(\tau\) :

Dimensionless time

\(b_{n}\) :

System characteristic length

\(\omega_{n}\) :

System characteristic frequency

\(\omega_{ij} \left( {i = 1,2;j = 1,2} \right)\) :

Dimensionless characteristic frequency

\(F_{i} \left( {i = 1,2} \right)\) :

Dimensionless input and output load force

\(e_{i} \left( {i = 1,2} \right)\) :

Dimensionless static transmission error of the i stage

\(\alpha\) :

Pressure angle

\(\beta\) :

Operating pressure angle

\(\theta_{f}\) :

Angle between the tooth center-line and the junction with the root circle

\(u_{fi} \left( {i = 1,2} \right)\) :

Distance along the tooth centerline measured from the tooth root to the loading tooth section

\(S_{f}\) :

Tooth root thickness

\(r_{f}\) :

Root circle radius

\(r_{in}\) :

Hub bore radius

\(E,\upsilon\) :

Material Young’s modulus and Poisson's ratio

\(L\) :

Tooth width

\(G\) :

Shear modulus

\(I_{y1} ,I_{y2}\) :

Moment of inertia of the cross-sectional area \(y_{1} ,y_{2}\)

\(A_{y1} ,A_{y2}\) :

Cross-sectional area \(y_{1} ,y_{2}\)

\(\begin{gathered} L_{i}^{*} ,M_{i}^{*} ,P_{i}^{*} \;{\text{and}}\;Q_{i}^{*} \hfill \\ R_{j}^{*} ,S_{j}^{*} ,T_{j}^{*} ,U_{j}^{*} \;{\text{and}}\;V_{j}^{*} \hfill \\ \end{gathered}\) (i = 1,2,3; j = 2,3):

Coefficients expressing the fillet-foundation deformation a based on \(r_{f}\), \(r_{in}\) and \(\theta_{f}\)

\(\varepsilon\) :

Contact ratio

\(k_{i}^{(0)} (t)\left( {i = 1,2} \right)\) :

Average mesh stiffness of the first and second stages

\(k_{i}^{(s)} (t)\left( {i = 1,2} \right)\) :

Mesh stiffness of the s Fourier coefficient

\(\varphi_{is} \left( {i = 1,2} \right)\) :

The s mesh phasing

\(\Omega_{i} \left( {i = 1,2} \right)\) :

Mesh frequency of the first and second stages

\(T_{1} ,T_{2} ,T_{12}\) :

Mesh period of the first stage, the second stage and the total

\(\Phi_{1} ,\Phi_{2} ,\Phi_{12}\) :

Characteristic quantity of the relationship between the coaxial teeth ratio and the mesh phasing of a two-stage gear system

\(F_{{{\text{m}}1}} ,F_{{{\text{m}}2}}\) :

Mesh force of first stage and second stage

References

  1. Zhang, A., Wei, J., Shi, L., Qin, D., Lim, T.C.: Modeling and dynamic response of parallel shaft gear transmission in non-inertial system. Nonlinear Dyn. 98, 997–1017 (2019). https://doi.org/10.1007/s11071-019-05241-w

    Article  Google Scholar 

  2. Kahraman, A.: Dynamic analysis of a multi-mesh helical gear train. Proc. ASME Des. Eng. Tech. Conf. Part F1680, 365–373 (1992). https://doi.org/10.1115/DETC1992-0046

  3. Lin, J., Parker, R.G.: Mesh stiffness variation instabilities in two-stage gear systems. J. Vib. Acoust. Trans. ASME. 124, 68–76 (2002). https://doi.org/10.1115/1.1424889

    Article  Google Scholar 

  4. Parker, R.G.: Physical explanation for the effectiveness of planet phasing to suppress planetary gear vibration. J. Sound Vib. 236, 561–573 (2000). https://doi.org/10.1006/jsvi.1999.2859

    Article  Google Scholar 

  5. Parker, R.G., Lin, J.: Mesh phasing relationships in planetary and epicyclic gears. J. Mech. Des. Trans. ASME. 126, 365–370 (2004). https://doi.org/10.1115/1.1667892

    Article  Google Scholar 

  6. 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). https://doi.org/10.1016/j.jsv.2004.06.010

    Article  Google Scholar 

  7. 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). https://doi.org/10.1016/j.jsv.2003.11.029

    Article  Google Scholar 

  8. Gill-Jeong, C.: Numerical study on reducing the vibration of spur gear pairs with phasing. J. Sound Vib. 329, 3915–3927 (2010). https://doi.org/10.1016/j.jsv.2010.04.005

    Article  Google Scholar 

  9. Guo, Y., Parker, R.G.: Analytical determination of mesh phase relations in general compound planetary gears. Mech. Mach. Theory. 46, 1869–1887 (2011). https://doi.org/10.1016/j.mechmachtheory.2011.07.010

    Article  Google Scholar 

  10. Wang, S., Huo, M., Zhang, C., Liu, J., Song, Y., Cao, S., Yang, Y.: Effect of mesh phase on wave vibration of spur planetary ring gear. Eur. J. Mech. ASolids. 30, 820–827 (2011). https://doi.org/10.1016/j.euromechsol.2011.06.004

    Article  MATH  Google Scholar 

  11. Kang, M.R., Kahraman, A.: An experimental and theoretical study of the dynamic behavior of double-helical gear sets. J. Sound Vib. 350, 11–29 (2015). https://doi.org/10.1016/j.jsv.2015.04.008

    Article  Google Scholar 

  12. Yavuz, S.D., Saribay, Z.B., Cigeroglu, E.: Nonlinear time-varying dynamic analysis of a multi-mesh spur gear train. Conf. Proc. Soc. Exp. Mech. Ser. 4, 309–321 (2016). https://doi.org/10.1007/978-3-319-29763-7_30

    Article  Google Scholar 

  13. Brecher, C., Schroers, M., Löpenhaus, C.: Experimental analysis of the dynamic noise behavior of a two-stage cylindrical gearbox. Prod. Eng. 11, 695–702 (2017). https://doi.org/10.1007/s11740-017-0775-y

    Article  Google Scholar 

  14. Peng, D., Smith, W.A., Randall, R.B., Peng, Z.: Use of mesh phasing to locate faulty planet gears. Mech. Syst. Signal Process. 116, 12–24 (2019). https://doi.org/10.1016/j.ymssp.2018.06.035

    Article  Google Scholar 

  15. Peng, D., Smith, W.A., Borghesani, P., Randall, R.B., Peng, Z.: Comprehensive planet gear diagnostics: use of transmission error and mesh phasing to distinguish localised fault types and identify faulty gears. Mech. Syst. Signal Process. 127, 531–550 (2019). https://doi.org/10.1016/j.ymssp.2019.03.024

    Article  Google Scholar 

  16. Wang, C., Parker, R.G.: Dynamic modeling and mesh phasing-based spectral analysis of quasi-static deformations of spinning planetary gears with a deformable ring. Mech. Syst. Signal Process. 136, 106497 (2020). https://doi.org/10.1016/j.ymssp.2019.106497

    Article  Google Scholar 

  17. Sanchez-Espiga, J., Fernandez-del-Rincon, A., Iglesias, M., Viadero, F.: Planetary gear transmissions load sharing measurement from tooth root strains: numerical evaluation of mesh phasing influence. Mech. Mach. Theory. 163, 104370 (2021). https://doi.org/10.1016/j.mechmachtheory.2021.104370

    Article  Google Scholar 

  18. Sanchez-Espiga, J., Fernandez-del-Rincon, A., Iglesias, M., Viadero, F.: Influence of errors in planetary transmissions load sharing under different mesh phasing. Mech. Mach. Theory. 153, 104012 (2020). https://doi.org/10.1016/j.mechmachtheory.2020.104012

    Article  Google Scholar 

  19. Wang, C., Dong, B., Parker, R.G.: Impact of planet mesh phasing on the vibration of three-dimensional planetary/epicyclic gears. Mech. Mach. Theory. 164, 104422 (2021). https://doi.org/10.1016/j.mechmachtheory.2021.104422

    Article  Google Scholar 

  20. Wang, C., Parker, R.G.: Nonlinear dynamics of lumped-parameter planetary gears with general mesh phasing. J. Sound Vib. 523, 116682 (2022). https://doi.org/10.1016/j.jsv.2021.116682

    Article  Google Scholar 

  21. Li, Z., Peng, Z.: Nonlinear dynamic response of a multi-degree of freedom gear system dynamic model coupled with tooth surface characters: a case study on coal cutters. Nonlinear Dyn. 84, 271–286 (2016). https://doi.org/10.1007/s11071-015-2475-5

    Article  Google Scholar 

  22. Liu, G., Parker, R.G.: Nonlinear dynamics of idler gear systems. Nonlinear Dyn. 53, 345–367 (2008). https://doi.org/10.1007/s11071-007-9317-z

    Article  MATH  Google Scholar 

  23. Vinayak, H., Singh, R., Padmanabhan, C.: Linear dynamic analysis of multi-mesh transmissions containing external, rigid gears. J. Sound Vib. 185, 1–32 (1995). https://doi.org/10.1006/jsvi.1994.0361

    Article  MATH  Google Scholar 

  24. Erltenel, T., Parker, R.G.: A static and dynamic model for three-dimensional, multi-mesh gear systems. Proc. ASME Int. Des. Eng. Tech. Conf. Comput. Inf. Eng. Conf. DETC2005. 5, 945–956 (2005). https://doi.org/10.1115/detc2005-85673

  25. Wang, X.: Stability research of multistage gear transmission system with crack fault. J. Sound Vib. 434, 63–77 (2018). https://doi.org/10.1016/j.jsv.2018.07.037

    Article  Google Scholar 

  26. Li, W., Sun, J., Yu, J.: Analysis of dynamic characteristics of a multi-stage gear transmission system. JVC J. Vib. Control. 25, 1653–1662 (2019). https://doi.org/10.1177/1077546319830810

    Article  Google Scholar 

  27. He, X., Zhou, X., Xue, Z., Hou, Y., Liu, Q., Wang, R.: Effects of gear eccentricity on time-varying mesh stiffness and dynamic behavior of a two-stage gear system. J. Mech. Sci. Technol. 33, 1019–1032 (2019). https://doi.org/10.1007/s12206-019-0203-7

    Article  Google Scholar 

  28. Lu, W., Zhang, Y., Cheng, H., Zhou, Y., Lv, H.: Research on dynamic behavior of multistage gears-bearings and box coupling system. Meas. J. Int. Meas. Confed. 150, 107096 (2020). https://doi.org/10.1016/j.measurement.2019.107096

    Article  Google Scholar 

  29. Bai, W., Qin, D., Wang, Y., Lim, T.C.: Dynamic characteristic of electromechanical coupling effects in motor-gear system. J. Sound Vib. 423, 50–64 (2018). https://doi.org/10.1016/j.jsv.2018.02.033

    Article  Google Scholar 

  30. Wang, S., Zhu, R.: An improved mesh stiffness calculation model for cracked helical gear pair with spatial crack propagation path. Mech. Syst. Signal Process. 172, 108989 (2022). https://doi.org/10.1016/j.ymssp.2022.108989

    Article  Google Scholar 

  31. Kahraman, A., Blankenship, G.W.: Effect of involute contact ratio on spur gear dynamics. J. Mech. Des. Trans. ASME. 121, 112–118 (1999). https://doi.org/10.1115/1.2829411

    Article  Google Scholar 

  32. Dai, H., Long, X., Chen, F., Xun, C.: An improved analytical model for gear mesh stiffness calculation. Mech. Mach. Theory. 159, 104262 (2021). https://doi.org/10.1016/j.mechmachtheory.2021.104262

    Article  Google Scholar 

  33. Marafona, J.D.M., Marques, P.M.T., Martins, R.C., Seabra, J.H.O.: Mesh stiffness models for cylindrical gears: a detailed review. Mech. Mach. Theory 166, 104472 (2021). https://doi.org/10.1016/j.mechmachtheory.2021.104472

    Article  Google Scholar 

  34. Natali, C., Battarra, M., Dalpiaz, G., Mucchi, E.: A critical review on FE-based methods for mesh stiffness estimation in spur gears. Mech. Mach. Theory. 161, 104319 (2021). https://doi.org/10.1016/j.mechmachtheory.2021.104319

    Article  Google Scholar 

  35. Ma, H., Song, R., Pang, X., Wen, B.: Time-varying mesh stiffness calculation of cracked spur gears. Eng. Fail. Anal. 44, 179–194 (2014). https://doi.org/10.1016/j.engfailanal.2014.05.018

    Article  Google Scholar 

  36. Xie, C., Hua, L., Han, X., Lan, J., Wan, X., Xiong, X.: Analytical formulas for gear body-induced tooth deflections of spur gears considering structure coupling effect. Int. J. Mech. Sci. 148, 174–190 (2018). https://doi.org/10.1016/j.ijmecsci.2018.08.022

    Article  Google Scholar 

  37. Kahraman, A., Blankenship, G.W.: Experiments on nonlinear dynamic behavior of an oscillator with clearance and periodically time-varying parameters. J. Appl. Mech. Trans. ASME. 64, 217–226 (1997). https://doi.org/10.1115/1.2787276

    Article  Google Scholar 

  38. Kahraman, A., Singh, R.: Non-linear dynamics of a spur gear pair. J. Sound Vib. 142, 49–75 (1990)

    Article  Google Scholar 

  39. Pizzolante, F., Battarra, M., Mucchi, E.: The role of gear layout and meshing phase for whine noise reduction in ordinary geartrains. Mech. Mach. Theory. 181, 105209 (2023). https://doi.org/10.1016/j.mechmachtheory.2022.105209

    Article  Google Scholar 

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Acknowledgements

The authors would like to acknowledge the financial support from the NSFC, the research is funded by National Natural Science Foundation of China (Contract No. 51775036), these supports are gracefully acknowledged.

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  1. School of Mechanical Engineering, University of Science and Technology Beijing, 30 Xueyuan Road, Haidian District, Beijing, 100083, China

    Wei Li & Ziyuan Li

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Li, W., Li, Z. Dynamic analysis of a multi-stage gear system considering the coupling between mesh phasing angle and coaxial teeth ratio. Nonlinear Dyn 111, 19855–19878 (2023). https://doi.org/10.1007/s11071-023-08896-8

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