臺灣博碩士論文加值系統

本論文分成四部分探討，第一章為討論本文撰寫動機、目的和有關本文之相關研究情況和重要文獻之探討，包括非線性轉子動力學動態反應、油膜軸承動態反應、轉動轉子與機殼間隙接觸摩擦非線性模式力學之既有資料研究。
第二章為進行非線性分析軟體，來模擬多芬(Duffing)方程式的動態反應，包括時間領域之位置位移反應和頻率領域之頻譜分析、磐卡門氏(Poincare’)之分叉圖(bifurcation plot)、和相位圖繪製，據以這些圖和既有日本人UEDA氏發表論文所作之分析比對，發現本文之發展軟體非常吻合上述UEDA氏發表之解析解。
第三部份為完成本文題目轉子軸承動力系統之非線性行為及混沌運動之分析的數學理論模式之推導，研究推導包括一般傳統軸承、油膜軸承、轉子與機殼之間隙接觸非線性力學，在此亦考慮經由並聯式加工機得到誤差(理論數學式模擬與實驗誤差模擬)值加入轉子系統之軸系(shaft)製造加工上的間隙接觸非線性力學數學模式上，因此因誤差產生非線性間隙於軸系上亦會造成實際轉子與機殼位移，而影響轉子與機殼動態自激振動接觸力產生，及含有位移考慮狀況之軸承座。而油膜軸承為選用油膜為2 型且未發生有氣洞之數學模式，軸承座支撐是考慮非線性彈性力其關係為與位移成三次方比例。經由軟體分析本論文作出此轉子軸承動力系統當考慮軸承座支撐是非線性彈性力其關係為與位移成三次方比例之非線性行為反應混沌(chaos)解，為當轉子軸承動力系統轉速在主軸之自然頻率三倍及八倍附近最為顯著，這是本文利用數值模擬的重要一大發現。在本章亦完成油膜軸承不同滑油黏度之下之分叉現象分析和轉子與機殼間隙接觸及負載振動之磐卡門氏(Poincare’) 動態反應分叉圖。
第四部份為進行轉子軸承動力系統動態反應量測實驗，轉子軸承系統結合一部Carl Schenck A.G. M470/450之轉子平衡測試儀，利用此實驗裝置(test rig)可用來驅動轉子系統在不同轉速之下，模擬無位移支撐固定式軸承座及非線性彈性有位移支撐軸承座，來檢驗此兩種不同支撐之軸承座受不同的氣動驅動轉速下，它們的軸承座位置動態反應實驗。結果經由實驗顯示，在不同的平衡量激發下，無位移支撐固定式軸承座之位置動態反應呈現單一解，而非線性彈性有位移支撐軸承座實驗時，軸承座之位置動態反應呈現非單一解(複解)，此點與前述之非線性數學理論之動態反應是相當吻合。

In the development of many components of the engineering mechanical process, the nonlinear dynamic response of the rotor bearing system is of essential importance. The multiple and chaotic solutions of the steady-state responses in nonlinear rotor bearing system depend upon different types of bearings such as rolling-element type and fluid-film type as well as the foundation of journal housing. Sometimes the rotor bearing dynamic nonlinear response could induce a lot of vibration, the analysis of multiple and chaos dynamic responses of the rotor-bearing system became the important topic.
An analytical model involving a residual balanced rotor shaft and bearing system on the damped flexible journal housing with Duffing’s stiffness and whirl of fluid film force motion in journal mass is constructed and the appropriate equations of the nonlinear dynamic system are developed. To confirm the multiple response of the rotor-bearing system with/without fluid film force due to the unbalanced force with Duffing stiffness bearing housing are analyzed. First, an algorithm is developed to find out the Poincare points representing the steady state periodic solution from SDOF Duffing’s equation. The results are compared with the published of the solutions of Ueda to confirmed with the exact solution of the Fortran algorithm. Second, the equations of motion are solved for a long two-bearing fluid-film symmetrical flexible rotor on deformable foundation with contact force and contour error model of shaft in manufacturing. Then, the bifurcation plot for parameter v.s. dimensionless spin speed on speed up were then performed in which we take a simple case that focus on the effects of an nonlinear term subjected to the nonlinear foundation in the flexible rotor and responses of rotor bearing system with contact force and contour error model of shaft in manufacturing observing the orbit motion journal center; disk center; and housing center etc. After the mentioned above processes, the nonlinear terms in fluid-film forces and the accessory vibration of shaft (coupled with gear transfer system) were included in the dynamic equations to observe the bifurcation plots of the journal center. Finally, an experiment was performed to investigate the multiple/single dynamic responses of the rotor-bearing system supported on the deformable bearing housing.

CONTENTS

ABSTRACT (ENGLISH)………………………………………………x
ABSTRACT (CHINESE)………………………………………………x
CONTENTS ……………...…………………………………………….i
LIST OF TABLES………………………………………..…………….i
LIST OF FIGURES…………………………………………..……….16
Chapter page
1. Introduction…………………………………………………….. 1
1.1 Motivation and objective…………………………………………………....… 1.
1.2 Literature review……………………………………………………….….……3
1.2.1 Response of the fluid-film journal bearing …………………………………4
1.2.2 Response of a flexible rotor dynamic system………………………………..7
1.3 Dissertation outline……………………………………………………………...9
2. The analysis of Duffing’s oscillator equation…………....12
2.1 Introduction……………………………………………………………….……12
2.2. Chaotic dynamics and multiple periods in Duffing oscillator …………….…..13
2.3 Numerical results and discussions ……………………………………….…….14
2.4 Preliminary conclusion …………………………………………………….…..15
3. Mathematical model ………………………………...……24
3.1 Description of the rotor-bearing dynamic system………………………….…...24
3.2 Equation of motion ……………………………………………………….……27
3.3 Non-dimensionalization of the equation of motion ……………………………31
3.4 Hydrodynamic fluid-film bearing force Preliminary conclusion………………34
4. Nonlinear behavior and chaotic motion of the rotor- bearing dynamic system without fluid film force ………41
4.1 Introduction…………………………………………………………………….41.
4.2. Mathematic for the rotor-bearing dynamic system without fluid film force ….42
4.3 Dynamic response analysis for the rotor-bearing dynamic system
without fluid film force ………………………………………………….…..50
4.3.1 Numerical results and discussion of the rotor-bearing system
without fluid film force …………………………………………………...…53
4.3.2. Preliminary conclusion………………………………………………………56
4.4 Analysis of rotor bearing system with contact force and of contour
error model 1 of rotor shaft…………………………………………………59
4.4.1 Numerical results and discussion of rotor bearing system with contact force and of contour error model 1………………….………..72
4.4.2 Preliminary conclusion of high rotor bearing system with contact force and
of contour error model 1………………………………………….…..75
4.5 Analysis of rotor bearing system with contact force and of contour
error model 2 of rotor shaft due to ITRI ballbar experiment’s data………...76
4.5.1 Numerical results and discussion of rotor bearing system with contact
force and of Contour model 2of rotor shaft………………………..…..93
4.5.2 Preliminary conclusion of rotor bearing system with contact force and
of Contour model 2 of rotor shaft………………………………….…..98
4.6 Chaos measurement method……………………………….…………..……..100
5. Analysis of Dynamic response of the rotor bearing system with squeeze film force ………………………..………..150
5.1 Summary for the equations of the rotor bearing system with squeeze film force…………………………………………………………………………150
5.2 Analysis parameters for the rotor-bearing dynamic system with fluid film force …………………………………………………………………………153
5.2.1 Numerical results and discussion of the rotor-bearing system with fluid film force…………………………………………………………………….…159
5.2.2 Preliminary conclusion…………………………..………………………...162
5.3 Analysis parameters for the rotor-bearing dynamic system with rotor rubbing stator /fluid film force under vibration………………………………......163
5.3.1 Numerical results and discussion for the rotor-bearing dynamic system with rotor rubbing stator /fluid film force under vibration……………………...169
5.3.2 Preliminary conclusion of the rotor-bearing dynamic system with rotor rubbing stator /fluid film force under vibration……………………………………..172
6. Experiments on the dynamic response of the rotor bearing system…………………………....…………….186
6.1 Introduction ……………………………………………………………….…186
6.2 Experimental Facility………………………………………………………...187
6.3 Testing procedure………………………………………………………….…194
6.4 Experiment results and discussion…………………………………………...195
7. Conclusion……………………….…………………….....207
REFERENCE………………………………………………211
Appendix……………………………………………………218

References
[1] F.N. Loreze, “Deterministic Non-period Flow,” J. Atmos. Sci. 20, pp.130-141, 1963.
[2] R. Shaw,. 1981, “Strange Attractors, Chaotic Behaviour, and Information Flow,” Z. Naturforsch, Vol. 36a, pp.80-112.
[3] S. H. Shaw, 1988, “Chaotic Dyanmics of a Slender Beam Rotating about its Longitudinal Axis,” Journal of Sound and Vibration, Vol. 124, No. 2, pp.329-343.
[4] P. R. Trumpler, “Design of film Bearings,” Macmillan, New York, 1966.
[5] R. Holmes, “The Vibration of a Rigid Shaft on Short Sleeve bearings,” Journal of Mechanical Engineering Science 2, pp377-380, 1970
[6] P. Bonello, M. J. Brenan, R. Holmes, “Non-linear Modelling of Rotor Dynamic Systems with Squeeze film Dampers-An Efficient Integrated Approach,” J. of Sound and Vibration., 249(4) 743-773, (2002)
[7] P. BONELLO and M. J. BRENNAN, “Modelling the Dynamic Behaviour of a Supercritical Rotor on a Flexible Foundation Using the Mechanical Impedance Technique,” Journal of Sound and Vibration 239, 445-466, (2001)
[8] J.Y. Zhao, I.W. Linnett, L.J., Mclean, “Unbalance Response of a Flexible Rotor Supported by a Squeeze Film Damper,” J. of Vibration and Acoustics, 120 (1998) 32-38.
[9] E. J. HAHN and P. Y. P. CHEN, “Harmonic Balance Analysis of General Squeeze Film Damped Multi-Degree of-Freedom Rotor Bearing Systems,” American Society of Mechanical Engineers Journal of Tribology 116, 499-507, (1994)
[10] M. T. M. Crooijmans, H. J. H. Brouwers, D. H. van Campen, A.de Kraker, “Limit Cycle Predictions of a Nonlinear Journal-Bearing System,” Engineering for Industry, Vol. 112, pp. 168-171, (1990)
[11] Y. B. Kim, S. T. Noah, “Bifurcation Analysis for a Modified Jefcott Rotor with Bearing Clearances,” Nonlinear Dynamics, Vol. 1, pp.221-241, (1990)
[12] Y. B. Kim, S. T. Noah, “Response and Bifurcation Analysis of MDOF Rotor System with a Strong Nonlinearity,” Nonlinear Dynamics, Vol. 2, pp.215, (1991)
[13] D.L. Taylor, B.R.K. Kumar, Closed-Form, Steady-state Solution for the Unbalance Response of a rigid Rotor in Squeeze Film Damper, J. of Engineering for Power., 105 Jul., (1983) 551-559.
[14] J.S., Guo, Characteristics of the Nonlinear Hysteresis loop for Rotor_Bearing Instability, Phd dissertation, Case western (1996).
[15] F.F. Ehrich, “Sum and Difference Frequencies in Vibration of High Speed Rotating Machinery,” ASME Journal of Engineering for Industry, Feb.,pp.181-184 (1972).
[16] D.E. Bently, “Forced Subrotative Speed Dynamic Action of Rotating Machinery,” ASME Paper No. 74-Pet-16, (1974)
[17] D.W. Child, “Factional Frequency Rotor Motion Due to Nonsymmetric Clearance Effects,” ASME Journal of Engineering for Power, July, pp.533-541 (1982)
[18] F.F. Ehrich, “High Order Subharmonic Response of High Speed Rotor in Bearing Clearance,” ASME Journal of Vibration and Acoustics, Stress and Reliability in Design, Jan.,pp.9-16, (1987)
[19] A. Muszynska, “Partial Lateral Rotor to Stator Rubs,” ImechE Paper No. C281/84, 1984.
[20] Y.S. Choi, S.T. Noah, “Nonlinear Steady-state Response of a Rotor-Support system,” ASME. Journal of Vibration, Acoustics, Stress and Reliability in Design, July, pp.255-261, (1987)
[21] F.F. Ehrich, “Spontaneous Sidebanding in High Speed Rotordynamics,” J. Vib. Acoustics, Vol.114, pp.498-505, (1992).
[22] S. W.,Shaw, P. J., Holmes, “A Periodically Forced Piecewise Linear Oscillator,” Journal of Sound and Vibration. Vol., 90. No.1, pp.129-155(1983).
[23] F.F. Ehrich, “Nonlinear Phenomena in Dynamic Response of Rotors in Anisotrpic Mounting Systems,” Transactions of the ASME , Special 50th Anniversary Design Issue, pp.154-161, Vol. 117,June, (1995)
[24] Y. Hori, “Anti-earthquake considerations in rotordynamics,” Proceedings of the IMechE 4th Intl. Conf. on Vibration in roting Machinery, Edinbur5gh, Sept. C318/88, pp.1-8., (1988)
[25] J. E. H. Sykes, R. Holems, “ The effect of bearing misalignment on the non-linear vibration of aero-engine rotor-damper assemblies,” Proc. Insn Mech. Engrs, Part G, Vol. 204(G2), pp.83-99, (1990)
[26] P. Goldman., A. Muszynska., “Chaotic Vibrations of Rotor/Bearing/Stator Stator Systems with Looseness or Rubs,” ASME Vibration and Noise Conference, nonlinear Vibrations, DE.Vol.54. (1993).
[27] S.A. Spiewak, T. Nickel, Vibration Based preload Estimation in Machine Tool Spindles, Int. J. of Machine tool and Manufact., 41, 567-588, (2001)
[28] S.M. Kim, J.H. Ha, S.H. Jeong, S.K. Lee, “Effect of Joint Condition on the Dynamic Behavior of a Grinding Wheel Spindle,” Int. J. of Machine tool and Manufact., 41,1749-1761, (2001)
[29] C.M. Chang, C.P. Kuo, “The study of nonlinear dynamic response of spindle in the hybrid parallel-link machine tools with contour error,” J. of the Chinese Society of Mechanical Eng., Vol.25, No 1, pp.69-78 (2004)
[30] A.C. Lee, Y. Kang, K. L. Tsai, and K. M. Hsiao,” Transient Analysis of an Asymmetric Rotor-Bearing System During Acceleration,” ASME Journal of Engineering Industry, Vol. 114, November, pp.465-475, (1992)
[31] T. N. Shiau, G. J. Sheu, and C. D. Yang, 1994, “Vibration and Control of a Flexible Rotor in Magnetic Bearing Using Hybrid Method and Control Theory”, ASME Journal of Engineering for Gas Turbine and Power.
[32] S. C. Huang, Y.M, Huang, and S. H. Shieh, “Vibration and Stability of a Rotation Shaft Containing a Transverse Crack.” Journal of Sound and Vibration, (1992)
[33] J. H. Wang, and M. T. Tsai, “Instability Due to Fluid Leakage of a Rotor System with Anisotropic Support,” ASME Journal of Vibration, Acoustics, Stress, and Reliability in Design, Vol. 111, pp. 27-34, (1989)
[34] A.C. Lee, Y. Kang, K. L. Tsai, and K. M. Hsiao,” Transient Analysis of an Asymmetric Rotor-Bearing System During Acceleration,” ASME Journal of Engineering Industry, Vol. 114, November, pp.465-475, (1992)
[35] J.M.T.,Thompson, H.B.,Stewart, “Nonlinear Dynamics and Chaos,” John Wiley. (1987)
[36] Vance, H.B.,Stewart, “Nonlinear Dynamics and Chaos,” John Wiley. (1987)
[37] T.H. Chang, C.P. Kuo, C.M. Chang, The study of geometric error of the hybrid parallel-link machine tools, J. of the Chinese Society of Mechanical Eng., Vol.23, No 2, pp.163-172 (2002)
[38] Y.,Kakino, Y., Ihara, and Y., Nakatsu, The measurement of motion errors of NC machine tools and diagnosis of their origins by using telescoping magnetic ball bar method, Annals of CIRP Vol. 36, 1987, pp.377-383
[39] A. Wolf, J. B. Swift, H. L. Swinney, J. A. Vastano, “Determining Lyapunov Exponents from a Time Series,” Physica, Vol. 160, pp.285－317, (1985)
[40] R. Brown, P. Bryant, H. D. I. Abarbanel, “computing the Lyapunov spectrum of a dynamical system from an observed time series,” Physical Review A, Vol. 43, pp.2787-2806, (1991)
[41] P. Grassberger, “An Optimized box-assisted algorithm for fractal dimensions,” Physics Letter A, Vol. 148, No. 1, 2, pp.63-68, (1990)
[42] F. Takens, “Detecting strange attractors in turbulence,” Springer lecture notes in mathematics, Vol. 898, pp.366-381, (1981)
[43] A. M. Fraser, H. L. Swinney, “Independent coordinates for strange attractors from mutual information, “Physical Review A, Vol.33, No. 2, pp. 1134-1140, (1986)
[44] P. Grassberger, I. Procaccia, “Characterization of Stranger Attractors,” Physical Review Letters, Bol. 50, NO. 5, pp.346-349, (1983)
[45] M. A. H. Nerenberg, C. Essex, “Correlation dimension and Systematic geometric effects,” Physical Review A, Vol. 42, No. 12, pp.7065-7074, (1990)