在本論文中將探研齒輪轉子軸承系統的動態分析。首先，考慮轉軸用Timoshenko樑理論來描述轉軸之變形，並藉由使用Lagrangian法發展齒輪轉子軸承系統的有限元素模型來推導出系統的運動方程式，並求得系統的自然頻率與穩態響應。接著將齒輪對的嚙合壓力角、接觸比與嚙合勁度都視為隨時間變化的參數，並探討系統具非線性齒輪嚙合勁度的動態分析，這些參數在以前的文獻通常視為常數。系統之暫態響應將利用四階阮奇-庫達(Runge-Kutta)數值積分方法求解。最後探討使用黏彈支承與最佳化方法來降低旋轉軸產生永久變形後所導致的振動響應。
由數值結果可知，在分析齒輪轉子軸承系統的動態特性時，具時變嚙合勁度之齒輪轉子軸承系統比具常數嚙合勁度的系統更能精確的呈現系統之動態行為。在探討具時變嚙合勁度之系統響應與具常數嚙合勁度之系統響應時，兩者系統響應間的差異可藉由使用較高的齒輪嚙合勁度來降低。旋轉軸產生殘留軸曲後，將增加齒輪轉子軸承系統的振動響應，但是對於系統的自然頻率幾乎沒影響。在齒輪轉子軸承系統振動響應分析中，加裝黏彈支承能有效降低系統之振動響應。藉由使用最佳化來調整軸承支承旋轉軸之位置能降低由旋轉軸產生殘留軸曲後所導致之振動響應。

In this dissertation, the dynamic analysis of a geared rotor-bearing system is investigated. Shafts in the system are modeled as Timoshenko beam by taking into account the effects of rotary inertia and gyroscopic moment. The coupling effect between lateral and torsional motions due to gears are considered. A finite element model of the system is developed by using Lagrangian approach for determination of the natural frequencies of the system and the steady-state responses due to disk unbalance. The pressure angle, contact ratio and gear-mesh stiffness of the gear pair in the system are treated as time-varying variables in the proposed model while they were considered as constant in previous models. Two gear-mesh stiffness models, step gear-mesh and nonlinear gear-mesh stiffnesses, are investigated. Direct time numerical integration, based on a fourth-order Runge-Kutta algorithm, is used to perform the dynamic analysis of transient responses. Viscoelastic bearing supports are used to reduce the vibration of geared rotor-bearing systems. Optimization algorithms are employed to lower the vibration caused by residual shaft bow and transmitted forces in viscoelastic bearing supports.
The present results show that increasing the bearing stiffnesses may suppress the pressure angle and raise the contact ratio of gear pair. The difference between gear-mesh deformations of the geared rotor-bearing system with a constant gear-mesh stiffness and with a time-varying gear-mesh stiffness can be reduced by using higher gear-mesh stiffness. Residual shaft bows influence significantly lateral responses of the system, but hardly affect the natural frequencies of the system. The influences of residual shaft bow on vibration responses of the geared rotor-bearing system and on the transmitted forces in viscoelastic bearing supports can be reduced through optimization.

[1] Lalanne, M., and Ferrairs, G., 1991, Rotor Dynamics Prediction in Engineering, John Wiley and Sons, New York.
[2] Rao, J. S., 1992, Rotor Dynamics, 2nd edition, John Wiley and Sons, New York.
[3] Lund, J. W., and Sternlicht, B., 1962, “Rotor-Bearing Dynamics with Emphasis on Attenuation,” Journal of Basic Engineering, ASME, 84 (4), pp. 491-498.
[4] Dworski, J., 1964, “High-Speed Rotor Suspension Formed by Fully Floating Hydrodynamic Radial and Thrust Bearings,” Journal of Engineering for Power, ASME, 86 (2), pp. 149-160.
[5] Gunter, E. J., 1970, “Influence of Flexibility Mounted Rolling Element Bearings on Rotor Response, Part 1: Linear Analysis,” Journal of Lubrication Technology, ASME, 92 (1), pp. 59-69.
[6] Lund, J. W., 1980, “Sensitivity of the Critical Speeds of a Rotor to Changes in the Design,” Journal of Mechanical Design, ASME, 102 (1), pp. 115-121.
[7] Buccuiarell, L. L., 1982, “On the Instability of Rotating Shafts Due to Internal Damping,” Journal of Applied Mechanics, ASME, 49 (2), pp. 425-428.
[8] Saito, S., and Someya, T., 1984, “Study of Damped Critical Speeds and Damping Ratios of Flexible Rotors,” Journal of Vibration, Acoustics, Stress and Reliability in Design, ASME, 106 (1), pp. 62-71.
[9] Ruhl, R. L., and Booker, J. F., 1972, “A Finite Element Model for Distributed Parameter Turborotor Systems,” Journal of Manufacturing Science and Engineering, ASME, 94, pp. 126-132.
[10] Nelson, H. D., and McVaugh, J. M., 1976, “The Dynamics of Rotor-Bearing Systems Using Finite Elements,” Journal of Manufacturing Science and Engineering, ASME, 98 (2), pp. 593-600.
[11] Nelson, H. D., 1980, “A Finite Rotating Shaft Element Using Timoshenko Beam Theory,” Journal of Mechanical Design, ASME, 102, pp. 793-803.
[12] Childs, D. W., and Graviss, K., 1982, “A Note on Critical-Speed Solutions for Finite-Element-Based Rotor Models,” Journal of Mechanical Design, ASME, 104 (2), pp. 412-415.
[13] Ozguven, H. N., and Ozkan, Z. L., 1984, “Whirl Speeds and Unbalance Response of Multibearing Rotors Using Finite Elements,” Journal of Vibration, Acoustics, Stress and Reliability in Design, ASME, 106, pp. 72-79.
[14] Rajan, M., Nelson, H. D., and Chen, W. J., 1986, “Parameter Sensitivity in the Dynamics of Rotor-Bearing Systems,” Journal of Vibration, Acoustics, Stress and Reliability in Design, ASME, 108 (2), pp. 197-206.
[15] Chen, L. W., and Ku, D. M., 1992, “Dynamic Stability of Cantilever Shaft-Disk System,” Journal of Vibration, Acoustics, Stress and Reliability in Design, ASME, 114 (3), pp. 326-329.
[16] Shiau, T. N., and Hwang, J. L., 1993, “Generalized Polynomial Expansion Method for the Dynamic Analysis of Rotor-Bearing Systems,” Journal of Engineering for Gas turbines and Power, ASME, 115, pp. 209-217.
[17] Lund, J., 1978, “Critical Speed, Stability and Response of a Geared Train of Rotors,” Journal of Mechanical Design, ASME, 100, pp. 535-538.
[18] Iida, H., Tamura, A., Kikuchi, K., and Agata, H., 1980, “Coupled Torsional -Flexural Vibration of a Shaft in a Geared System of Rotors,” Bulletin of the JSME, 23, pp. 2111-2117.
[19] Neriya, S. V., Bhat, R. B., and Sankar, T. S., 1984, “Effect of Coupled Torsional-Flexural Vibration of a Geared Shaft System on Dynamic Tooth Load,” The Shock and Vibration Bulletin, 354, pp. 67-75.
[20] Choy, F. K., Tu, Y. K., Zakrajsek, J. J., and Townsend, D. P., 1991, “Effects of Gear Box Vibration and Mass Imbalance on the Dynamics of Multistage Gear Transmission,” Journal of Vibration, Acoustics, Stress and Reliability in Design, ASME, 113, pp. 333-344.
[21] Neriya, S. V., Bhat, R. B., and Sankar, T. S., 1985, “Coupled Torsional-Flexural Vibration of a Geared Shaft System Using Finite Element Method,” The Shock and Vibration Bulletin, 355, pp. 13-25.
[22] Kahraman, A., Ozguven, H. N., Houser, D. R., and Zakrajsek, J. J., 1992, “Dynamics Analysis of Geared Rotors by Finite Elements,” Journal of Mechanical Design, ASME, 114 (3), pp. 507-514.
[23] Shiau, T. N., Choi, S. T., and Chang, J. R., 1998, “Theoretical Analysis of Lateral Response Due to Torsional Excitation of Geared Rotors,” Mechanism and Machine Theory, 33 (6), pp. 761-783.
[24] Rao, J. S., Shiau, T. N., and Chang, J. R., 1995, “Coupled Bending-Torsional Vibration of Geared Rotors,” Design Engineering Technical Conference, Boston, DE-Vol. 84-2, 3, Part B, pp. 977-989.
[25] Shiau, T. N., Rao, J. S., Chang, J. R., and Choi, S. T., 1999, “Dynamic Behavior of Geared Rotors,” Journal of Engineering for Gas turbines and Power, ASME, 121, pp. 494-503.
[26] Choi, S. T., and Mau, S. Y., 2001, “Dynamic Analysis of Geared Rotor-Bearing Systems by the Transfer Matrix Method,” Journal of Mechanical Design, ASME, 123, pp. 562-568.
[27] Lee, A. S., Ha, J. W., and Choi, D. H., 2003, “Coupled Lateral and Torsional Vibration Characteristics of a Speed Increasing Geared Rotor-Bearing System,” Journal of Sound and Vibration, 263 (4), pp. 725-742.
[28] Khabou, M. T., Bouchaala, N., Chaari, F., Fakhfakh, T., and Haddar, M., 2011, “Study of a Spur Gear Dynamic Behavior in Transient Regime,” Mechanical Systems and Signal Processing, 25, pp. 3089-3101.
[29] Han, Q., Zhao, J., and Chu, F., 2012, “Dynamic Analysis of a Geared Rotor System Considering a Slant Crack on the Shaft,” Journal of Sound and Vibration, 331, pp. 5803-5823.
[30] Kasuba, R., and Evans, J. W., 1981, “An Extended Model for Determining Dynamic Loads in Spur Gearing,” Journal of Mechanical Design, ASME, 103, pp. 398-409.
[31] Umezawa, K., Sato, T., and Ishikawa, J., 1984, “Simulation on Rotational Vibration of Spur Gears,” Bulletin of the JSME, 27, pp. 102-109.
[32] David, J. W., and Park, N. G., 1986, “The Vibration Problem in Gear Coupled Rotor Systems,” Bulletin of the JSME, 29, pp. 297-303.
[33] Nevzat, O. H., and Houser, D. R., 1988, “Dynamic Analysis of High Speed Gears by Using Loaded Static Transmission Error,” Journal of Sound and Vibration, 125 (1), pp. 71-83.
[34] Lin, J., and Parker, R. G., 2002, “Mesh Stiffness Vibration Instabilities in Two-Stage Gear System,” Journal of Vibration, Acoustics, Stress and Reliability in Design, ASME, 124, pp. 68-76.
[35] Cornell, R. W., 1981, “Compliance and Stress Sensitivity of Spur Gear Teeth,” Journal of Mechanical Design, ASME, 103, pp. 447-459.
[36] Yoon, K. Y., and Rao, S. S., 1996, “Dynamic Load Analysis of Spur Gears Using a New Tooth Profile,” Journal of Mechanical Design, ASME, 118, pp. 1-6.
[37] Kahraman, A., and Blankenship, G. W., 1999, “Effect of Involute Contact Ratio on Spur Gear Dynamics,” Journal of Mechanical Design, ASME, 121, pp. 112-118.
[38] Huang, K. J., and Liu, T. S., 2000, “Dynamic Analysis of a Spur Gear By the Dynamic Stiffness Method,” Journal of Sound and Vibration, 234 (2), pp. 311-329.
[39] Sainsot P., Vwlex P., and Duverger, O., 2004, “Contribution of Gear Body to Tooth Deflections－A New Bidimensional Analytical Formula,” Journal of Mechanical Design, ASME, 126, pp. 748-752.
[40] Chaari, F., Baccar, W., Abbes, M., and Haddar, M., 2008, “Effect of Spalling or Tooth Breakage on Gearmesh Stiffness and Dynamic Response of a One-Stage Spur Gear Transmission,” European Journal of Mechanics A/Solids, 27, pp. 691-705.
[41] Chaari, F., Fakhfakh, T., and Haddar, M., 2009, “Analytical Modeling of Spur Gear Tooth Crack and Influence on Gear mesh Stiffness,” European Journal of Mechanics A/Solids, 28, pp. 461-468.
[42] Li, M., Lim, T. C., and Shepard, W. S., 2004, “Modeling Active Vibration Control of a Geared Rotor System,” Smart Materials and Structures, 13, pp. 449-458.
[43] Kahraman, A., and Singh, R., 1990, “Non-linear Dynamics of a Spur Gear Pair,” Journal of Sound and Vibration, 142 (1), pp. 49-75.
[44] Kahraman, A., and Singh, R., 1991, “Interactions Between Time-Varying Mesh Stiffness and Clearance Non-Linearities in a Geared System,” Journal of Sound and Vibration, 146 (1), pp. 135-156.
[45] Gao, Q., Tanabe, M., and Nishhara, K., 2009, “Contact-Impact Analysis of Geared Rotor Systems,” Journal of Sound and Vibration, 319 (1-2), pp. 463-475.
[46] Baguet, S., and Jacquenot, G., 2010, “Nonlinear Couplings in a Gear-Shaft-Bearing system,” Mechanism and Machine Theory, 45, pp. 1777-1796.
[47] Kim, W., Yoo, H. H., and Hung, J. C., 2010, “Dynamic Analysis for a Pair of Spur Gears with Translational Motion Due to Bearing Deformation,” Journal of Sound and Vibration, 329 (21), pp. 4409-4421.
[48] Nicholas, J. C., Gunter, E. J., and Allaire, P. E., “1976, Effect of Residual Shaft Bow on Unbalance Response and Balancing of a Single Mass Flexible Rotor,” Journal of Engineering for Power, ASME, 98, pp. 171-181.
[49] Flack, R. D., and Rooke, J. H., 1980, “A Theoretical–Experimental Comparison of the Synchronous Response of a Bowed Rotor in Five Different Sets of Fluid Film Bearings,” Journal of Sound and Vibration, 73 (4), pp. 505-517.
[50] Shiau, T. N., and Lee, E. K., 1989, “The Residual Shaft Bow Effect on Dynamic Response of a Simply Supported Rotor with Disk Skew and Mass Unbalances,” Journal of Vibration, Acoustics, Stress and Reliability in Design, ASME, 111 (2), pp. 170-178.
[51] Rao, J. S., 2001, “A Note on Jeffcott Warped Rotor,” Mechanism and Machine Theory, 36, pp. 563-575.
[52] Shiau, T. N., and Lee, E. K., 2003, “Dynamic Response of Rotor Bearing System under the Effect of Residual Shaft Bow Mounted on Viscoelastic Suspensions,” The International Symposium on Stability Control of Rotating Machinery-2, Gdansk, Poland.
[53] Song, G. F., Yang, Z. J., Ji, C., and Wang, F. P., 2013, “Theoretical-Experimental Study on a Rotor with a Residual Shaft Bow,” Mechanism and Machine Theory, 63, pp. 50-58.
[54] Kang, C. H., 2012, “The Correction of Geared Rotor-Bearing System under Residual Bow Effect by Using the Optimization Algorithms with Behavior Constraints,” Advanced Science Letters, 5, pp. 914-919.
[55] Bland, D. R., and Lee, E. H., 1956, “On the Determination of a Viscoelastic Model for Stress Analysis of Plastics,” Journal of Applied Mechanics, ASME, 23, pp. 416-420.
[56] Kulkarni, P. S., and Nakra, B. C., 1993, “Unbalance Response and Stability of a Rotating System with Viscoelastically Supported Bearings,” Mechanism and Machine Theory, 28, pp. 427-436.
[57] Bhattacharyya, K., and Dutt, J. K., 1997, “Unbalance Response and Stability Analysis of Horizontal Rotor Systems Mounted on Nonlinear Rolling Element Bearings with Viscoelastic Supports,” Journal of Vibration, Acoustics, Stress and Reliability in Design, ASME, 119, pp. 539-544.
[58] Shabaneh, N. H., and Zu, J. W., 2000, “Dynamic Analysis of Rotor-Shaft Systems with Viscoelastically Supported Bearings,” Mechanism and Machine Theory, 35, pp. 1313-1330.
[59] Jones, D. G., 2001, Handbook of Viscoelastic Vibration Damping, John Wiley and Sons, New York.
[60] Kang, C. H., Hsu, W. C., Lee, E. K., and Shiau, T. N., 2011, “Dynamic Analysis of Gear-Rotor System with Viscoelastic Supports under Residual Shaft Bow Effect,” Mechanism and Machine Theory, 46 (3), pp. 264-275.
[61] Davis, L., 1991, Handbook of Genetic Algorithms, John Wiley and Sons, New York.
[62] Kennedy, J., and Eberhart, R., 1995, “Particle Swarm Optimization,” In Proceedings of IEEE International Conference on Neural Networks, Washington, DC, 4, pp. 1942-1948.