本文探討高速列車在不同車速下與土壤之影響。經由理論推導利用數值分析求得其數值解，再透過三維有限元素法模擬高速列車行車振動比較之。
在理論解部份，乃模擬一移動載重在半無限土層中所產生的振動。本研究所探討的是軟弱土壤。車輛行駛在軟弱土層的時，較容易達到發生共振之臨界波速。有限元素法分析全區域之三維模型，分為三個步驟：第一、建立整體系統模型，包含橋樑、列車以及土壤模型。在此土壤方面設定為一均質(homogenous)土層。第二、利用有限元素法分析列車行經之振動反應，包括橋樑、地面之震動。第三、將歷時分析之結果轉至頻率域並以DB (decibel)表示。
由數值分析可知，當車行速度接近土壤雷利波速時，必會產生一極大的瞬間變位。同理我們比較有限元素分析，觀察此一現象是否存在。然而在有限元素分析中，當列車速接近土壤雷利波速時，列車跟土壤並未產生一明顯的共振行為。當列車運行在橋樑上的時候，其所造成的振動，皆會被橋樑系統所分散，對土層影響反而較少。因此對於高鐵振動分析來說，列車-橋樑共振的危害性遠大於列車-土壤共振。垂直變位大部份由列車-橋樑共振所影響，對結構物來說列車-土壤共振幾乎可以忽略。所以在考慮高鐵系統安全性時，設計車輛跟橋樑的自然振動頻率不同，能有效的避免結構共振的發生。
關鍵字: 橋樑、三維有限元素法、1/3倍頻法、高速鐵路、共振、雷利波、振動

When high-speed trains pass a bridge, there are normally accompanied with increased transient movements of the ground. How to avoid resonance and reduce the vibrations of ground surface induced by a moving load with different speeds has become an important issue.
The numerical analysis and the three-dimensional finite element analysis were used in this research to investigate surface ground vibration due to a moving train. This 3D finite element mesh includes the soil and bridge meshes modeled by using the 3D 8-node element. After FEM analysis, 1/3 octave band method was used in frequency domain to compute the vibration from the ground velocity in time domain. The resonant vibration in this study can be clearly seen from the finite element result. As an illustration of the numerical analysis, the train speed v approaches the Rayleigh wave velocity in supporting soil; a large peak has been occurred in the figure approaching track-soil critical velocities. However, it does not have a large increase in generated ground vibrations in the bridge-train-soil model simulated by the finite element method. The vehicle-bridge resonance triggers the greater part of vibrations than the vehicle-soil resonance for the three-dimensional vehicle-bridge-soil system. To be valid for avoid bridge and train resonance, the first dominated train frequency and the first bridge natural frequency in each direction should be as different as possible.
Keywords: 1/3 octave band, 3D FEM, Bridge, High-speed train, Resonance, Rayleigh-wave, Vibration.

CONTENTS
摘要 I
ABSTRACT Ⅱ
誌謝 III
CONTENTS IV
LIST OF TABLES Ⅶ
LIST OF FIGURES Ⅷ
CHAPTER 1. INTRODUCTION 1
1.1 Background and purpose 1
1.2 Brief account of the research 2
1.3 Literature review 3
1.4 Illustrate of the finite element program 5
CHAPTER 2. THE THEORIES USED IN FEM ANALYSIS 10
2.1 Absorbing boundary 10
2.2 Wheel element 14
2.31/3 octave band method 16
2.4 Newmark direct integration method 17
CHAPTER 3. THEORETICAL BACKGROUND 21
3.1 The description of moving-load problem on an elastic half-space21
3.2 Solution of the basic equations 21
3.3 Subsonic speed 26
3.4 transonic speed 29
CHAPTER 4. ANALYSIS ILLUSTRATIONS AND RESULTS 33
4.1 Numerical study and observations 33
4.2 Compare the displacement with the different locations 36
4.3 Discussion 40
CHAPTER 5. MESH GENERATION PROGRAM OF THE BRIDGE SYSTEM AND FINITE ELEMENT MODEL41
5.1 Establish Ground Mesh by VASJAPAN 41
5.2 The required files 42
5.3 Execute program VASJAPAN 47
5.4 The vehicle-bridge-soil system 47
5.5 Illustration of finite element model 49
CHAPTER 6. FINITE ELEMENT ANALYSIS 57
6.1 Illustrate the generation of bridge foundation mesh 57
6.2 Calculate vibration velocity as program G. 62
6.3 Execute program G 63
6.4 Compare the displacement with the different velocity 64
6.5 Discussion 84
CHAPTER 7. DISCUSSION AND CONCLUSION 86
7.1 DISCUSSION AND CONCLUSION 86
REFERENCES 89
APPENDIX A. Trapezoidal rules for double integrals 92

REFERENCES
1. Alabi B., A model for the problem of ground vibration induced by the wheels of a moving train, Applied Mathematical Modelling 13, 710-715, 1989.
2. Alabi B., A parameter study on some aspects of groundborne vibration due to rail traffic, Journal of Sound and Vibration 153, 77-87, 1992.
3. Chua K. H., Balendra T. and Lo K. W., Groundborne vibration due to trains in tunnels, Earthquake Engineering and Structure Dynamics 21, 445-460, 1992.
4. Cardona S., Fernandez-Diaz E., De Los Santos M. A. and Tejedo J. L., Simple Model for the Time History of the Ground Vibrations Generated by Underground Trains as the Means of Monitoring the Riding Quality of the Wheels of a Bogie, Vehicle System Dynamics 33, 421-434, 2000.
5. Eason G., The stresses produced in a semi-infinite solid by a moving surface force, Int. J. Engng. Sci. 2, 581-609, 1965.
6. Erlingsson S., Bodare A., Live load induced vibrations in Ullevi Stadium - Dynamic soil analysis, Soil Dynamics and Earthquake Engineering 15(3), 171-188, 1996.
7. Erlingsson S., Three-dimensional dynamic soil analysis of a live load in Ullevi Stadium, Soil Dynamics and Earthquake Engineering 18(5), 373-386, JUL 1999.
8. Fryba L., Vibration of Solid and Structures Under Moving Loads, Groningen, The Nethrlands: Noordhoff International Publishing, 1972.
9. Gordon C. G., Generic criteria for vibration sensitive equipment. SPIE 1619, 71-75, 1991.
10. Hood R. A., Green R. J., Breslin M. and Williams P. R., The calculation and assessment of ground-borne noise and perceptible vibration from trains in tunnels, Journal of Sound and Vibration 193(1), 215-255, 1996.
11. Hunt H. E. M., Modelling of rail vehicles and track for calculation of ground-vibration transmission into buildings, Journal of Sound and Vibration 193 (1), 185-194, 1996.
12. Jones C. J. C., Sheng X. and Petyt M., Simulations of ground vibration from a moving harmonic load on a railway track, Journal of Sound and Vibration 231(3), 739-751, MAR 30 2000.
13. Jones C. J. C., Thompson D. J. and Petyt M., A model for ground vibration from railway tunnels, Proceedings of the Institution of Civil
Engineers Transports 153(2), 121-129, May 2002.
14. Jones D. V., Laghrouche O. and LeHouedec D., Petyt M., round vibration in the vicinity of a rectangular load acting on a viscoelastic layer over a rigid foundation, Journal of Sound and Vibration 203(2), 307-319, JUN 5 1997.
15. Jones D. V., Houedec D. Le., Peplow A. T. and Petyt M., Ground vibration in the vicinity of a moving harmonic rectangular load on a half space, European Journal of Mechanics A/Solid 17, 153-166, 1998.
16. Ju S.H., 剛性聯結及介面元素應用於關結應力分折之研究, 國科會報告, (NSC-84-2213-E-006-119), 1995.
17. Ju S.H. and Kung K.S., Mass types, element orders and solving schemes for the Richards equation, Computers & Geosciences, 23:175-187, 1997.
18. Ju S.H., Finite element analysis of wave propagations due to high-speed train across bridges, 國科會報告, 2000.
19. Ju S.H., and Wang Y.M., Time-dependent absorbing boundary conditions for elastic wave propagation, International Journal for Numerical Method in Engineering, 50, 2159-2174, 2001.
20. Ju S.H., Modeling flexible pile foundations using finite elements, Proceedings of the 3rd Japan-Korean-Taiwan Joint Seminar on Earthquake Engineering for Building Structures, Taipei, Nov, 2001, 127-134, 2001.
21. Ju S.H., Finite element analyses of wave propagations due to high-speed train across bridges Accepted by International Journal for Numerical Methods in Engineering, 2001.
22. Ju S.H., Evaluating foundation mass, damping and stiffness by least-squares method, Revised in Earthquake Engineering and structural Dynamics.
23. Ju S.H., 3D finite element analyses of wave barriers for reduction of train-induced vibrations, Accepted by Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 2002.
24. Ju S.H., Development a nonlinear finite element program with rigid link and contact element, Report of NSC in R.O.C., NSC-86-2213-E-006-063, 66-81. This report and source codes in an executable compressed file nfemnew.exe of MS-DOS can be obtained from the FTP server 140.116.36.4
25. Krylov V. V., Dawson A. R., Heelis M. E. and Collop A. C., Rail movement and ground waves caused by high-speed trains approaching track-soil critical velocities, Proc Instn Mech Engrs Vol 214 Part F, (2000).
26. Lefeuve-Mesgouez G., Le Houedec D., and Peplow A. T., Ground vibration in the vicinity of a high-speed moving harmonic strip load, Journal of Sound and Vibration 231(5), 1289-1309, APR 13 2000.
27. Lefeuve-Mesgouez G., Peplow A.T., Le Houedec D., Surface vibration due to a sequence of high speed moving harmonic rectangular loads, Soil Dynamics and Earthquake Engineering, 22(6), 459-473, AUG 2002.
28. Metrikine A. V. and Vrouwenvelder A. C. W. M., Surface Ground vibration due to moving train in tunnel : Two-dimensional model , Journal of Sound and Vibration 234(1), 43-66, 2000.
29. Madshus C. and Kaynia A. M., High-speed railway lines on soft ground : Dynamic behaviour at critical train speed, Journal of Sound and Vibration 23(3), 689-701, 2000.
30. Nelson J. T., Recent developments in ground-borne noise and vibration control , Journal of Sound and Vibration 193(1), 367-376, 1996.
31. Higdon R.L., Absorbing boundary conditions for acoustic and elastic waves in stratified media, Journal of Computational Physics, 101, 386-418, 1992.
32. Sheng X. and Jones C. J. C., Petyt M., Ground vibration generated by a harmonic load acting on a railway track, Journal of Sound and Vibration 225(1), 3-28, AUG 5 1999.
33. Sheng X. and Jones C. J. C., Petyt M., Ground vibration generation by load moving along a railway track, Journal of Sound and Vibration 288(1), 129-156, 1999.
34. Wu T. X. and Thompson D. J., The effects of local preload on the foundation stiffness and vertical vibration of railway , Journal of Sound and Vibration 219(5), 881-904, 1999.
35. Yoshida D. M. and Weaver W., Finite element analysis of bemas and plates with moving loads, Publication of International Association for Bridge and Structural Engineering 31, 179-195, 1971.