# 臺灣博碩士論文加值系統

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 The objective of this thesis is to develop a model for studying of nonlinear vibration of bearing-supported rigid shaft systems subjected to collision constraints. The system consists of a rigid shaft and a rigid disk. The shaft is supported by bearings which are modeled as springs and dampers. The kinetic energy and the potentail energy of the system are derived. The work done by supported forces of bearings are also considered. By using the Lagrange equations, the equations of motion of the bearing-supported shaft systems are derived. Next, the geometrical constraints when the collision occurs are found. At last, the relations between the velocities of the shaft before and after impact are determind.Analyses of transient responses of systems with and without collision are carried out, and are compared with those obtained using linearized equations of motion. Then, the individual influence of the parameters such as coefficient of restitution, bearing stiffness, the locations of bearings and collision constraints on the dynamic responses are investigated.
 誌謝 i中文摘要 iiSUMMARY iii章節目錄 iv表目錄 vi圖目錄 vii符號索引 xx第一章 序論 11.1 前言 11.2 文獻回顧 11.3 研究目的與內容 3第二章　基礎理論推導 42.1 運動方程式之推導 42.2 線性化系統之解析解 112.3 系統數值解之模式 122.4 碰撞條件式與碰撞後速度之推導 152.4.1 碰撞幾何條件式之推導 152.4.2 桿碰撞後速度推導 17第三章　系統無碰撞與有碰撞之響應分析 233.1 無碰撞之響應 233.2 單碰撞邊界 323.3 雙碰撞邊界 48第四章 邊界條件對系統響應之影響 554.1恢復係數e改變對系統響應之影響 554.2 邊界位置改變對系統響應之影響 644.2.1 邊界水平位置 改變對系統響應之影響 644.2.2 垂直邊界位置 改變對系統響應之影響 724-3 系統軸承徑向勁度值改變對系統響應之影響 794.3.1 處軸承徑向勁度值 改變對系統響應之影響 804.3.2 處軸承徑向勁度值 改變對系統響應之影響 874-4 系統軸承 方向勁度值 改變對系統響應之影響 964-5 系統支撐軸承 位置改變對系統響應之影響 1034.5.1 支撐軸承 位置 改變對系統響應之影響 1034.5.2 改變支撐軸承 位置 改變對系統響應之影響 110第五章 結論與未來展望 1185.1 結論 1185.2 未來展望 119參考文獻 120
 [1] G. Ioannou, A. Woowat and R. Gohar “Multiple Roller Bearing-Shaft Systems” Wear, Vol. 81, 1982, PP. 47-58.[2] C.W. Lee, Y.G. Jei“Modal Analysis of Continuous Rotor-Bearing Systems” Journal of Sound and Vibration, Vol. 126, Issue 2, 1988, PP. 345-361.[3] N. H. Shabaneh, Jean W. Zu“Dynamic Analysis of Rotor Shaft Systems with Viscoelastically Supported Bearings”Mechanism and Machine Theory, Vol. 35, 2000, PP. 1313-1330.[4] S.P. Harsha“Nonlinear Dynamic Analysis of a High-Speed Rotor Supported by Rolling Element Bearings”Journal of Sound and Vibration, Vol. 290, 2006, PP. 65-100.[5] M.H. Eissa, U.H. Hegazy, Y.A. Amer“Dynamic Behavior of an AMB Supported Rotor Subject to Harmonic Excitation”Applied Mathematical Modelling, Vol. 32, 2008, PP. 1370-1380.[6] Mohammed F. Abdul Azeez, Alexander F. Vakakis,“Numerical and Experimental Analysis of a Continuous Overhung Rotor Undergoing Vibro-Impacts”International Journal of Non-Linear Mechanics, Vol. 34, 1999, PP.415-435.[7] J.Zapomel, C.H.J.Fox, and EMalenovsky, “Numerical Investigation of a Rotor System with Disc-Housing Impact”Journal of Sound and Vibration, Vol. 2, No.243, 2001, PP.215-240.[8] Q. -S. Lu, Q. -H. Li, E. H. Twizell“The Existence of Periodic Motions in Rub-Impact Rotor Systems”Journal of Sound and Vibration, Vol. 264, 2003, PP. 1127-1137.[9] O. Grapis, V. Tamuzsb, N.-G. Ohlson, J. Andersonsb,“Overcritical High-Speed Rotor Systems, Full Annular Rub and Accident” Journal of Sound and Vibration, Vol. 290, 2006, PP. 910-927.[10] S. Popprath, H. Ecker“Nonlinear Dynamics of a Rotor Contacting an Elastically Suspended Stator”Journal of Sound and Vibration, Vol. 308, 2007, PP. 767-784.[11] S. Popprath, H. Ecker“The Effect of Bearing Cage Run-Out on the Nonlinear Dynamics of a Rotating Shaft”Journal of Sound and Vibration, Vol. 308, 2007, PP. 767-784.
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