摩擦單擺支承(FPS)為目前使用最廣之滑動式隔震支承，其對一般震波雖有不錯的隔震效果，但在具長週期成分之近斷層震波作用下則易造成低頻似共振現象，因而降低其減震效果。故有學者提出具變曲率曲面之多項式摩擦單擺支承(Polynomial Friction Pendulum Isolation，簡寫PFPI)，此支承滑動曲面為多項式函數之曲面，經適當選取係數可將滑動曲面分為軟化段與硬化段兩種功能區段。其中軟化段以降低結構之加速度反應，硬化段則可於強震下用以抑制隔震支承所可能產生之鉅大位移量。本文擬探討FPS與PFPI在三軸向震波作用下之動力行為。
再者，目前常用之滑動隔震分析模型大多適用於支承回復勁度為常數之情形，同時不考慮滑動曲面效應所引致之高階項及水平雙向耦合運動效應。此種簡化式模型在支承位移較小時或可提供足夠的精度，但在極限地震力或近斷層地震力作用下，支承位移與速度有可能急速增加，此時滑動曲面效應對支承動力行為所引致之高階項則可能對隔震行為產生可觀之影響，並使支承水平及垂直向運動產生複雜之耦合效應。有鑑於此，本文乃針對軸對稱型滑動隔震支承於極限位移下受三軸地震力作用下之隔震行為進行研究。首先，以拉格朗運動方程式推導滑動隔震支承模型於三軸向震波作用下之完整運動方程式。該分析模型可模擬完整曲面效應對支承回復力、軸壓力及摩擦力之影響，同時，亦可分析水平雙向及水平垂直之耦合運動行為。本文並以剛性隔震結構作為初步研究對象，並利用振動台之實驗數據驗證本文模型之正確性。結果顯示，當支承位移、滑動速度與相對加速度變大時，常用簡化模型與本文完整模型隔震行為之差異的確實會增加。不過，簡化模型對支承最大位移及上構最大加速度之分析結果較為保守，但對上構垂直加速度及支承軸壓力之估算則會過於低估，此點於設計應用時應加以留意。

A conventional isolation system with constant isolation frequency is usually a long-period dynamic system, so its seismic response is likely to be amplified in earthquakes with long-period wave components. To solve this problem the sliding isolators with variable curvature (SIVCs), which have the property of variable isolation stiffness and frequency was proposed in literature.
Presently, the analytical models proposed for a SIVC isolator were usually derived by using static equilibrium condition. This kind of models (also called simplified model) is not able to account for dynamic coupling effects between each vibration direction as well as the curvature effect such as centrifugal force. As a result, these models will lose their accuracy when the isolators are subjected to extreme earthquakes that produce large velocity or displacement. In addition, most of the existing models did not consider the effect of vertical ground motions on the isolation performance.
In the light of the above discussions, this thesis aims to establish an accurate general analytical model for an isolation system that consists of SIVC and subjected to tri-axial ground motions, and also to investigate the dynamic coupling effects on the isolation performance. Then, the proposed analytical model was verified by shaking table tests. The results show that the simplified model underestimates the vertical response of the isolation system. The results also show that the difference of the vertical responses increased, when the horizontal responses of the isolation become larger.

1. Almazan J L, Llear J C, and Inandi J A,(1998) “Modelling Aspects Of Structures Isolated With The Frictional Pendulum System.”Earthquake Engineering And Structural Dynamics, 27, 845-867
2. Calio I and Massimo Marletta and Francesco Vinciprova (2003)”Seismic response of multi-storey buildings base-isolated by friction devices with restoring properties.” Computers and Structures ,81(28-29): 2589-2599.
3. Castaldo P,Tubaldi E, (2015) “Influence of FPS bearing properties on the seismic performance of base-isolated structures.” Earthquake Engineering & Structural Dynamics, 44:2817–2836
4. Faramarz K and Montazar R (2010) “Seismic response of double concave friction pendulum base-isolated structures considering vertical component of earthquake.” Advances in Structural Engineering, 13:1-13.
5. FEMA (2011) Reducing the risks of nonstructural earthquake damage - A practical guide. Report no. FEMA E-74, Federal Emergency Management Agency, Washington, D.C, USA.
6. Fenz D M and Constantinou M C (2006) “Behaviour of the double concave Friction Pendulum bearing.” Earthquake Engineering and Structural Dynamics Vol.35 1403-1424
7. Furukawa S. (2011) "Performance of structures and equipment in base-isolated medical facilities subjected to severe earthquake motions." Kyoto University.
8. Hwang J S and Hsu T Y (2000), “Experimentalstudy Ofisolatedbuilding Undertriaxial Groundexcitations.” Journal of Structural Engineering, Vol.126,No.8, 879-886
9. Jangid R.S.(1996),“seismic response of sliding structures to bidirectional earthquake excitation”, Earthquake Engineering and Structural Dynamics, vol. 25, 1301-1306
10. Jangid R.S,(2004), “Optimum friction pendulum system for near-fault motions.” Engineering Structures,Vol.27,No.3,349-359
11. Khoshnudian F, and Montazar R (2010) “Seismic Response of Double Concave Friction Pendulum Base-Isolated Structures Considering Vertical Component of Earthquake”, Advances in Structural Engineering Vol. 13 No. 1
12. Khoshnudian F, and Motamedi D (2013) “Seismic Response of Asymmetric Steel Isolated Structures Considering Vertical Component of Earthquakes.” Journal of Civil Engineering Vol.17,No.6,1333-1347
13. Krishnamoorthy A (2013) “Variable curvature pendulum isolator and viscous fluid damper for seismic isolation of structures.” Journal of Vibration and Control Vol.17 No.12 1779-1790
14. Loghman V ,Khoshnoudian F and Banazadeh M ,(2013) “Effect of vertical component of earthquake on seismic responses of triple concave friction pendulum base-isolated structures.” Journal of Vibration and Control Vol.21 No.11 2099-2113
15. Lu L Y , M. H. Shih, C. S. Chang Chien and W. N. Chang (2002) “Seismic performance of sliding isolated structures in near-fault areas”,Proceedings of the 7 th US National Conference on Earthquake Engineering, Session AT-2, July 21-25, Boston, MA, USA.
16. Lu L.Y., M. H. Shih, S. W. Tzeng and C. S. Chang Chien (2003), “Experiment of a sliding isolated structure subjected to near-fault ground motion”,Proceedings of the 7th Pacific Conference on Earthquake 65 Engineering, February 13-15, Christchurch, New Zealand.
17. Lu,L.Y., M.H.Shih, C.Y.Wu (2004a), “Near-Fault Seismic Isolation Using Sliding Bearings with Variable Curvatures,Proceedings of the 13th World Conference on Earthquake Engineering”, August 1-6, Vancouver, BC, Canada, Paper no. 3264.
18. Lu. L. Y., M. H. Shih, S. W. Tzeng, J. Wang (2004b) “Performance evaluation of sliding isolated structures subjected to near-fault ground motion”,Proceedings of International Conference in Commemoration of the 5th Anniversary of the 1999 Chi-Chi Earthquake, September 8-9, Taipei, Taiwan.
19. Lu, L. Y., T. Y. Lee, S. Y. Juang, S. W. Yeh (2013) “Polynomial friction pendulum isolators (PFPIs) for building floor isolation: an experimental and theoretical study” Engineering Structures, Vol. 56, 970-982.
20. Lu, L. Y., C. C. Hsu (2013a) “Eccentric rocking bearings with a designable friction property for seismic isolation: experiment and analysis”,Earthquake Spectra, Vol. 29, No. 3, 869-895.
21. Lu, L. Y., C. C. Lin, G. L. Lin (2013b) “Experimental evaluation of supplemental viscous damping for a sliding isolation system under pulse-like base excitation”. Journal of Sound and Vibration, Vol. 332, No. 8, 1982-1999.
22. Lu, L. Y., C. C. Hsu (2013b)“ Experimental study of variable-frequency rocking bearings for seismic isolation” Engineering Structures, Vol. 46, No. 1, 116-129.
23. Lu, L. Y., T. Y. Lee, S. W. Yeh (2011) “Theory and experimental study for sliding isolators with variable curvature” Earthquake Engineering and Structural Dynamics, Vol. 40, No. 14, 1609-1627.
24. Lu, L. Y., T. Y. Lee, I. L. Yeh, H. Chang (2010) “ Rocking bearings with variable frequency for near-fault seismic isolation (in Chinese)” Journal of the Chinese Institute of Civil and Hydraulic Engineering, Vol. 22, No. 3, 283-298.
25. Lu, L. Y., T. Y. Lee, S. Y. Juang, S. W. Yeh (2013) “Polynomial friction pendulum isolators (PFPIs) for building floor isolation: an experimental and theoretical study.” Engineering Structures, Vol. 56, 970-982
26. Malekzadeh,M.and Taghikhany,T (2015) “Adaptive Behavior of Double Concave Friction Pendulum Bearing and its Advantages over Friction Pendulum Systems, “Scientia Iranica Transaction A-Civil Engineering,Vol.17,No.2,81-88
27. Mosqueda G,Whittaker A S,Fenves G L (2004) “Characterization and Modeling of Friction Pendulum Bearings Subjected to Multiple Components of Excitation.” Journal of Structural Engineering Vol.130,No.3,433-442
28. Monti D,and Petrone F (2016), “Analytical thermo-mechanics 3D model of friction pendulum bearings.” Earthquake Engineering & Structural Dynamics, 45:957–977
29. Naeim, F. and J. M. Kelly (1999) “Design of Seismic Isolated Structures, Chapter 4.” John Wiley & Sons, Inc., New York.
30. Ou Y C, Song J, Lee G C (2010) “A parametric study of seismic behavior of roller seismic isolation bearings for highway bridge.” Earthquake Engineering And Structural Dynamics Vol.39,541-559
31. Panchal V R and Jangid R S (2009) “Seismic Response of Structures with Variable Friction Pendulum System.” Journal of Earthquake Engineering Vol.13,No.2,193-216
32. Panchal V R ,Jangid R S , Soni D P and Mistry B B (2010), “Response of the Double Variable Frequency Pendulum Isolator under Triaxial Ground Excitations.”Journal of Earthquake Engineering, 14:4, 527-558
33. Papazoglou A. J. and Elnashai A. S. (1996) “Analytical and field evidence of the damaging effect of vertical earthquake ground motion.” Earthquake Engineering and Structural Dynamics, 25:1109-1137.
34. Pranesh, M. and R. Sinha (2000) ”VFPI：an isolation device for a seismic deign.” Earthquake Engineering and Structural Dynamics, 29, 603-627.
35. Pranesh, M. and R. Sinha (2002) ,”Earthquake Resistant Design of Structures using the Variable Frequency Pendulum Isolator.”Journal of Structural Engineering, 128, 870.
36. Pranesh, M. and R. Sinha (2004) “Behavior of Torsionally Coupled Structures with Variable Frequency Pendulum Isolator.”Journal of Structural Engineering, 130, 1041.
37. Pranesh,M. and R. Sinha (2004) ”Aseismic design of structure-equipment systems using variable frequency pendulum isolator”, Nuclear Engineering and Design, v 231, n 2, June, 2004, p 129-139
38. Politopoulos I. and Moussallam N. (2012) “Horizontal floor response spectra of base-isolated buildings due to vertical excitation.” Earthquake Engineering and Structural Dynamics, 41:587-592.
39. Rabiei.M and Khoshnoudian.F, (2011) “Response of multistory friction pendulum baseisolated buildings including the vertical component of earthquakes.” Can. J. Civ. Eng.38: 1045–1059
40. Sharma A,Jangid RS,(2010),“SeismicResponse of Base-IsolatedBenchmark Building with Variable Sliding Isolators.” Journal of Earthquake Engineering, 14:1063–1091
41. Shakib.H, and Fuladgar,A (2003a), “Effect of vertical component of earthquake on the response of pure-friction base-isolated asymmetric buildings”, Engineering Structures 25,1841–1850
42. Shakib.H, and Fuladgar,A (2003b), “ Response of pure-friction sliding structures to three components of earthquake excitation.” Computers and Structures NO.81,189–19
43. Tao LI, Dagen Weng, Yuan Zhou, Yang Zhao(2014) “Shaking Table Testsand Nonlinear Time-history Analyses of Multiple Friction Pendulum Base-isolated Building Subjected to Earthquakes with Vertical Components.” Sixth World Conference on Structural Control and Monitoring, July 15-17, Barcelona, Spain, Paper No. 510
44. Tsopelas P C,Roussis P C,Constantinou M C,Buchanan and Reinhorn A M (2005) “3D-BASIS-ME-MB：Computer Program for Nonlinear Dynamic Analysis of Seismically Isolated Structures.”Technical Report MCEER-05-0009
45. Warn G. P. and Whittaker A. S. (2008) “Vertical earthquake loads on seismic isolation systems in bridges”, Journal of Structural Engineering, 134:1696-1704
46. Yang, Y. B., L.Y.Lu, J. D. Yau (2005) “Chapter 22: Structure and Equipment Isolation.”Vibration and Shock Handbook,edited by C. W. de Silva, CRC Press, Taylor & Francis Group
47. Zayas VA, Low SS, Mahin SA (1990), “A simple pendulum technique for achieving seismic isolation.”Earthquake Spectra;6:317–333.
48. 范揚志(2015) “黏滯型隔震系統之隔震阻尼比最佳化設計公式及雙向變曲率型摩擦隔震系統理論與實驗研究”國立臺灣大學工學院土木工程學系，7月，指導教授：鍾立來
49. 盧煉元、鍾立來(1999) “國內外結構控制技術之進展”，土木技術(防災科技專題)，四月號，第14期，81-95頁。
50. 盧煉元、張婉妮 (2000)，「垂直地震力對滑動式隔震結構之影響」，第五屆結構工程研討會，南投縣溪頭，8月28 ~ 30日，251~257頁。
51. 盧煉元、張婉妮 (2001)，「垂直地震力對滑動式隔震結構之影響」，國家地震工程研究中心報告，報告編號 NCREE-01-025。
52. 盧煉元、施明祥、張婉妮(2003) “近斷層震波對滑動式隔震結構之影響評估”，結構工程，十二月刊
53. 盧煉元、施明祥、吳政彥(2004) “變曲率滑動隔震支承之遲滯行為理論與實驗研究”，第七屆結構工程研討會，桃園大溪，8月22-24日，論文編號：H19。盧煉元、施明祥、曾旭玟、吳政彥(2005a) “滑動隔震支承之研發與其受近斷層震波行為之實驗探討”，結構工程，第二十卷，第三期，29-59頁。
54. 盧煉元、吳政彥、王健(2005b) 變曲面滑動隔震系統於近斷層隔震之應用研究(一)，93年度國科會專題研究計畫成果報告書，計畫編號 NSC93-2625-Z-327-002。
55. 盧煉元、吳政彥、葉弈麟 (2009) “圓錐形摩擦單擺支承之隔震應用研究”，結構工程，二十四卷，第二期，91-116頁。(NSC 94-2625-Z-327-004)
56. 蔡諄昶 (2015), “具摩擦可變特性之滑動隔震系統於垂直震波作用下之實驗與分析” 高雄第一科技大學營建工程系碩士論文，7月，指導教授：盧煉元
57. 蔡諄昶、盧煉元、王亮偉、鍾立來 (2016) , “摩擦單擺支承於水平垂直雙向地震力作用下之曲面效應研究”,第十三屆結構工程研討會暨第三屆地震工程研討會，桃園，8月24-26日
58. 王勝宣 (2015) “隔震系統最佳化設計流程及摩擦消能水平雙向隔震試驗之研究”，國立臺灣大學工學院土木工程學系，7月，指導教授：鍾立來