含不確定性參數旋轉軸之高階有限元素模型

This study aims at bringing the simulation results closer to those of the actual vibrating rotor-bearing system. There are some commericial software which can analyze deterministic structure efficiently, but these are solutions that cannot be added in other setting of analysis, such as uncertainties. In this thesis, high-order shape functions are used to reduce the sizes of matrices, and reduce the computation time while maintain accuracy. Legendre polynomials are used to build high-order shape functions. The property of polynomials is hierarchic, and makes the low-order entries independent of the high-order modes. Correction term of high-order shape functions are introduced. The number of constraints is the variables. Stiffness matrix, mass matrix, stiffness matrix of axial-loaded effects, and gyroscopic matrix of the rotating structure are established using the high-order finite element. According to these matrices, natural frequencies and mode shapes of the structure can be solved. The computational efficiency is also presented by calculation of numerical errors. With almost the same number of degrees of freedom, the more high-order terms are added, the more elements are reduced. The iteration time of system matrices is reduced, and the efficiency of the computation is improved. Bearing stiffness is affected by uncertainties, such as material properties of balls and manufacturing errors. The computation method of the equivalent bearing stiffness are presented and added into the rotating system for uncertainty analysis. Efficiency and accuracy of simulation is raised by this new method. Finally, the actual shaft will undergoing is modal testing. The experimental values from modal testing is compared with the simulated counterpart.本研究主要是使轉軸-軸承系統的振動模擬更接近實際。某些商業軟體可以有效地分析確定參數之結構，但卻無法增加其他設定上的分析，例如不確定性。 在本文中，高階曲線函數用來減少整個結構的有限元素矩陣大小，並在減少計算時間的前提下，同時提高準確度。高階曲線函數是使用Legendre多項式所建立，多項式的特性為階級式，能使矩陣之低階項獨立於高階曲線函數的數目，以便於選取需要的高階項數目。本研究亦提出高階曲線函數的修正項，用來使約束與固有的數值數量相同。剛性矩陣、質量矩陣、受軸向負載的幾何剛性矩陣和轉動結構的陀螺矩陣皆使用高階曲線函數建立。由這些矩陣，能計算結構的自然頻率與模態。計算效率亦藉由計算誤差數值比對而呈現。增加使用的高次曲線函數，可以使選取的元素數量減少，系統矩陣大小亦會減少，更能改善計算效率。 軸承剛性受不確定性所影響，例如滾珠的材料性質不同或製造誤差。本研究提出一個等效軸承剛性的預估方法，並應用在旋轉系統中加以不確定性分析，藉由這種新方法來提高模擬效率與準確性。最後，實際主軸模態測試得出的實驗值，與模擬值進行比對。Acknowledgements i 摘要 ii Abstract iii Table of Contents iv List of Tables vii List of Figures viii List of Symbols xii Chapter 1. Introduction 1 1.1 Background 1 1.2 Literature Review 2 1.2.1 Timoshenko beam theory in Finite Element Analysis 2 1.2.2 High-order finite elements 3 1.2.3 Bearing analysis 4 1.2.4 Uncertainty analysis 5 1.3 Objectives 6 1.4 Methodology and Structure of Thesis 7 1.4.1 Methodology 7 1.4.2 Structure of Thesis 7 Chapter 2. Shape Functions and System Matrices in High-order Timoshenko Beam Finite Elements 9 2.1 Timoshenko beam finite element 9 2.2 Hierarchic high-order shape functions 12 2.3 Correction term 14 2.4 Mass matrix and stiffness matrix 18 2.5 Axial-loaded effect 19 2.6 Gyroscopic effect 20 Chapter 3. Uncertainties of Bearing Stiffness 22 3.1 Introduction 22 3.2 Equivalent stiffness of bearing 23 3.2.1 Bearing simplification 24 3.2.2 Curve fitting 27 3.3 Analysing uncertainties with Monte Carlo simulation 29 Chapter 4. Validation of High-order Timoshenko Finite Element 31 4.1 Simulations 31 4.2 Experimental validation of high-order Timoshenko beam 35 4.2.1 Experimental method 35 4.2.2 Experiment on cantilever beam 38 Chapter 5. Simulation of Bearing Stiffness with Uncertainties 43 5.1 Equivalent bearing stiffness analysis 43 5.2 Analysis of rotor-bearing system with uncertainties 45 5.3 Effects with different bearing spacing 54 5.4 Uncertainty experiment of spindles 58 Chapter 6. Conclusion and Future Work 67 6.1 Conclusions 67 6.2 Future work 68 References 69 Appendix A : Shape functions for deflection & rotation 74 Appendix B : Matrices 7