Analysis of Time–Varying Mesh Stiffness for the Planetary Gear System with Analytical and Finite Element Methods

الموضوعات :

Ali Shahabi ¹ (Department of Mechanical Engineering, University of Sistan and Baluchestan, Zahedan, Iran Nedayedanesh Institute of Higher Education of Hormozgan, Bandar Abbas, Iran)
Amir Hosein Kazemian ² (Department of Mechanical Engineering, University of Sistan and Baluchestan, Zahedan, Iran)

تاريخ الإرسال : 18 السبت , ذو الحجة, 1441 تاريخ التأكيد : 16 الثلاثاء , ربيع الثاني, 1442 تاريخ الإصدار : 28 الثلاثاء , رجب, 1443

الکلمات المفتاحية: Natural frequency, Pressure angle, Vibration mode, Meshing gears,

ملخص المقالة :

In dynamic model of planetary gears, one of the key design parameters and one of the main sources of vibration is time–varying mesh stiffness of meshing gears. According to previous researches, the finite element method and analytical method are two techniques to estimate the mesh stiffness of meshing gears. In this work, in an innovation the periodically time–varying mesh stiffness of meshing gears is examined by both of finite element and analytical methods. The planetary gear set is modeled as a set of lumped masses and springs. Each element such as sun gear, carrier, ring gear and planets possesses three degrees of freedom and is considered as rigid body. The influence of effective parameters on the mesh stiffness of meshing gears and also numerical results of natural frequencies and vibration modes of the system are obtained. Based on the results, the influence of the higher pressure angles on the mesh stiffness of meshing gears is perceptible. By using the proposed mesh stiffness of meshing gears, for the system with numbers of odd and even equally and unequally spaced planets, natural frequencies and vibration modes are validated with a high accuracy.

المصادر:

• Li, S., Wu, Q., and Zhang, Z., Bifurcation and Chaos Analysis of Multistage Planetary Gear Train, Nonlinear Dynamics, 75, No. 1-2,  2014, pp. 217-233.

 

• Chen, E., and Walton, D., The Optimum Design of KHV Planetary Gears With Small Tooth Differences, International Journal of Machine Tools and Manufacture, Vol. 30, No. 1, 1990, pp. 99-109.


• Lin, J., and Parker, R., Planetary Gear Parametric Instability Caused by Mesh Stiffness Variation, Journal of Sound and vibration, Vol. 249, No. 1, 2002, pp. 129-145.


• Sun, T., and Hu, H., Nonlinear Dynamics of a Planetary Gear System with Multiple Clearances, Mechanism and Machine Theory, Vol. 38, No. 12, 2003, pp. 1371-1390.


• Saxena, A., Parey, A., and Chouksey, M., Effect of Shaft Misalignment and Friction Force on Time Varying Mesh Stiffness of Spur Gear Pair, Engineering Failure Analysis, Vol. 49, 2015, pp. 79-91.


• Hao, Z., Xianghe, Y., Qingkai, H., and Hai, H., Dynamic Behaviors of Geared Rotor System in Integrally Centrifugal Compressor, Journal of Vibration Engineering & Technologies, Vol. 7, 2019, pp. 241-249.


• Parker, R. G., and Lin, J., Mesh Phasing Relationships in Planetary and Epicyclic Gears, Journal of Mechanical Design, Vol. 126, No. 2, 2004, pp. 365-370.


• Inalpolat, M., and Kahraman, A., A Dynamic Model to Predict Modulation Sidebands of a Planetary Gear Set Having Manufacturing Errors, Journal of Sound and Vibration, Vol. 329, 2010, pp. 371-393.


• Wei, S., Han, Q. K., Dong, X. J., Peng, Z. K., and Chu, F. L., Dynamic Response of a Single-Mesh Gear System with Periodic Mesh Stiffness and Backlash Nonlinearity Under Uncertainty, Nonlinear Dynamics, Vol. 89, No. 1, 2017, pp. 49-60.


• Ambarisha, V. K., and Parker, R. G., Nonlinear Dynamics of Planetary Gears Using Analytical and Finite element Models, Journal of Sound and Vibration, Vol. 302, No. 3, 2007, pp. 577-595.


• Li, C. H., Chiou, H. S., Hung, C., Chang, Y. Y., and Yen, C. C., Integration of Finite Element Analysis and Optimum Design on Gear Systems, Finite Elements in Analysis and Design, Vol. 38, No. 3, 2002, pp. 179-192.


• Chen, Z. G., Shao, Y. M., and Lim, T. C., Non-Linear Dynamic Simulation of Gear Response Under the Idling Condition, International Journal of Automotive Technology, Vol. 13, No. 4, 2012, pp. 541-552.


• Kahraman, A., Natural Modes of Planetary Gear Trains, Journal of Sound Vibration, Vol. 173, 1994, pp. 125-130.


• Kahraman, A., Load Sharing Characteristics of Planetary Transmissions, Mechanism and Machine Theory, 29, No. 8, 1994, pp. 1151-1165.


• Liu, L., Niu, J., and Li, X., Dynamic Analysis of Gear System Under Rractional-Order PID Control with the Feedback of Meshing Error Change Rate, Acta Mechanica, Vol. 229, 2018, 3833-3851


• Masoumi, A., Pellicano, F., Samani, F. S., and Barbieri, M., Symmetry Breaking and Chaos-Induced Imbalance in Planetary Gears, Nonlinear Dynamics, Vol. 80, 2015, pp. 561-582.


• Sainsot, P., Velex, P., and Duverger, O., Contribution of Gear Body to Tooth Deflections-a New Bidimensional Analytical Formula, Journal of Mechanical Design, Vol. 126, No. 4, 2004, pp. 748-752.


• Shen, Y., Yang, S., and Liu, X., Nonlinear Dynamics of a Spur Gear Pair with Time-Varying Stiffness and Backlash Based on Incremental Harmonic Balance Method, International Journal of Mechanical Sciences, Vol. 48, No. 11, 2006, pp. 1256-1263.


• Shi, J. F., Gou, X. F., and Zhu, L. Y., Modeling and Analysis of a Spur Gear Pair Considering Multi-State Mesh with Time-Varying Parameters and Backlash, Mechanism and Machine Theory, Vol. 134, 2019, pp. 582-603.


• Wang, J., and Howard, I., The Torsional Stiffness of Involute Spur Gears, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, Vol. 218, No. 1, 2004, pp. 131-142.


• Zhou, X., Shao, Y., Lei, Y., and Zuo, M., Time-Varying Meshing Stiffness Calculation and Vibration Analysis for a 16DOF Dynamic Model with Linear Crack Growth in a Pinion, Journal of Vibration and Acoustics, Vol. 134, No. 1, 2012.


• Tian, X., Zuo, M. J., and Fyfe, K. R., Analysis of the Vibration Response of a Gearbox with Gear Tooth Faults. In ASME 2004 International Mechanical Engineering Congress and Exposition, American Society of Mechanical Engineers Digital Collection, 2004, pp. 785-793.


• Jin, G., Ren, W., Zhu, R., and Lu, F., Influence of Backlash on Load Sharing and Dynamic Load Characteristics of Twice Split Torque Transmission System, Journal of Vibration Engineering & Technologies, Vol. 7, 1999, pp. 565-577.


• Yang, D., and Sun, S., A Rotary Model for Spur Gear Dynamics, Journal of Mechanical Design, Vol. 107, No. 4, 1985, pp. 529-535.


• Lin, J., and Parker, R. G., Analytical Characterization of the Unique Properties of Planetary Gear Free Vibration, Journal of Vibration and Acoustics, Vol. 121, 1999, pp. 316-321.