本文使用有限元素法 (Finite Element Method)分析多軸人字齒輪轉子軸承系統之動態行為。系統的轉軸以Timoshenko樑模擬；齒輪假設為剛體，並考慮其質量偏心與陀螺效應；齒輪採用兩個螺旋角度相反的斜齒輪左右接合而成的人字齒輪；齒輪對的嚙合以線性彈簧及阻尼器沿壓力線連接來模擬；軸承則以線性彈簧與阻尼器來模擬。本研究探討齒輪的螺旋角度及嚙合勁度係數、軸承勁度係數、不同的軸長及不同的齒輪位置對系統穩態響應的影響。隨著齒輪的螺旋角度越大，系統軸向響應上升。當齒輪的軸承勁度及嚙合勁度上升，系統穩態響應下降。改變軸的長度，對系統的側向響應影響較大，軸向響應影響不大，且改變中間軸的長度對系統響應的影響較大。齒輪的位置越靠近軸承，自然頻率上升，穩態響應下降。

In this paper, dynamic behavior of double-helical geared multi-shaft rotor-bearing system is analyzed by the finite element method. The rotating shafts are modeled as Timoshenko beams that include the effects of shear strain and rotational inertia. The disks are assumed to be rigid and its gyroscopic effects are taken into account. The gears are double-helical gears, each of them is composed of two helical gears joined left and right. The gear mesh stiffness of the gear pair is simulated by connecting a linear spring and a damper along the pressure line. Bearings are modeled as linear spring-damper. In this study, the effects of gear helix angle, gear mesh stiffness, bearing stiffness, shaft length, and gear positions on the steady-state response of the system were investigated. The analysis results show that the axial response of the system increases with the increase of the gear helix angle. When the gear mesh stiffness and bearing stiffness increase, the lateral response of the system decreases. As the shaft length decreases, the resonance frequency of the system increases and the steady-state response decreases, and the effect of changing the length of the intermediate shaft is the most obvious. The closer the position of the gear pair to the bearing, the higher the resonance frequency of the system and the lower the steady-state response.

[1] R. L. Ruhl and J. Booker, "A finite element model for distributed parameter turborotor systems," ASME, Journal of Engineering for Industry, Vol. 94, pp. 126-132,1972.
[2] H. Nelson and J. McVaugh, "The dynamics of rotor-bearing systems using finite elements," ASME, Journal of Engineering for Industry, Vol. 98, pp. 593-600, 1976.
[3] H. Nelson, "A finite rotating shaft element using Timoshenko beam theory," ASME, Journal of Mechanical Design, Vol. 102, pp. 793-803, 1980.
[4] D. Clive and I. Shames, Solid Mechanics: A Variational Approach, New York: McGraw-Hill, 1973.
[5] R. Eshleman and R. Eubanks, "On the critical speeds of a continuous rotor," ASME, Journal of Engineering for Industry, Vol. 91, pp. 1180-1188, 1969.
[6] L. Mitchell, "Torsional-Lateral Coupling in a Geared, High Speed Rotor System," ASME Paper 75-DET-75, 1975.
[7] H. Iida, A. Tamura, K. Kikuchi, and H. Agata, "Coupled torsional-flexural vibration of a shaft in a geared system of rotors: 1st report," Bulletin of JSME, Vol. 23, No. 186, pp. 2111-2117, 1980.
[8] H. Iida, A. Tamura, and H. Yamamoto, "Dynamic characteristics of a gear train system with softly supported shafts," Bulletin of JSME, Vol. 29, No. 252, pp. 1811-1816, 1986.
[9] A. Kahraman, H. N. Ozguven, D. R. Houser, and J. J. Zakrajsek, "Dynamic analysis of geared rotors by finite elements," ASME, Journal of Mechanical Design, Vol. 114, pp. 507-514, 1992.
[10] J.S. Rao, J.R. Chang, and T.N. Shiau, "Coupled bending-torsion vibration of geared rotors," ASME, Design Engineering Technical Conference, DE-Vol. 84-2, Vol. 3, Part B, pp. 977-989, 1995.
[11] S.-T. Choi and S.-Y. Mau, "Dynamic analysis of geared rotor-bearing systems by the transfer matrix method," ASME, Journal of Mechanical Design, Vol. 123, pp. 562-568, 2001.
[12] Y. C. Chen, Dynamic Analysis of a Geared Rotor-Bearing System, Doctoral Dissertation, National Cheng Kung University, 2014.
[13] A. Kahraman, "Effect of axial vibrations on the dynamics of a helical gear pair," ASME, Journal of Vibration and Acoustics, Vol. 115, pp. 33-39, 1993.
[14] A. Kahraman, "Dynamic analysis of a multi-mesh helical gear train," ASME, Journal of Mechanical Design, Vol. 116, pp. 706-712, 1994.
[15] M. Kubur, A. Kahraman, D. Zini, and K. Kienzle, "Dynamic analysis of a multi-shaft helical gear transmission by finite elements: model and experiment," ASME, Journal of Mechanical Design, Vol. 126, pp. 398-406, 2004.
[16] S. Draca, Finite Element Model of a Double-Stage Helical Gear Reduction, Doctoral Dissertation, University of Windsor, 2006.
[17] K. Feng, S. Matsumura, and H. Houjoh, "Dynamic behavior of helical gears with effects of shaft and bearing flexibilities," Applied Mechanics and Materials, Vol. 86, pp. 26-29, 2011.
[18] Y. Zhang, Q. Wang, H. Ma, J. Huang, and C. Zhao, "Dynamic analysis of three-dimensional helical geared rotor system with geometric eccentricity," KSME, Journal of Mechanical Science and Technology, Vol. 27, pp. 3231-3242, 2013.
[19] M. Kang and A. Kahraman, "An experimental and theoretical study of the dynamic behavior of double-helical gear sets," Journal of Sound and Vibration, Vol. 350, pp. 11-29, 2015.
[20] F. C. Yang, Q. L. Huang, Y. Wang, and J. G. Wang, "Research on dynamics of double-mesh helical gear set," Applied Mechanics and Materials, Vol. 215, pp. 1021-1025, 2012.
[21] S. Chen, J. Tang, Y. Li, and Z. Hu, "Rotordynamics analysis of a double-helical gear transmission system," Meccanica, Vol. 51, No. 1, pp. 251-268, 2016.
[22] Q. Chang, L. Hou, Z. Sun, W. Wang, and Y. You, "Nonlinear dynamic modeling of double helical gear system," Jordan Journal of Mechanical and Industrial Engineering, Vol. 8, No. 5, pp. 289-296, 2014.
[23] S. Chen and J. Tang, "Effects of staggering and pitch error on the dynamic response of a double-helical gear set," Journal of Vibration and Control, Vol. 23, No. 11, pp. 1844-1856, 2017.
[24] J. Dong, S. Wang, H. Lin, and Y. Wang, "Dynamic modeling of double-helical gear with Timoshenko beam theory and experiment verification," Advance in Mechanical Engineering, Vol. 8, Issue 5, pp. 1-14, 2016.
[25] K.N. Hsu, Dynamic Analysis of a Double Helical Geared Rotor-Bearing System, Thesis, National Cheng Kung University, 2016.