Journal of the Architectural Institute of Korea Structure & Construction (대한건축학회논문집:구조계)

Architectural Institute of Korea (대한건축학회)
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Nonlinear Dynamic Response of Arch Structures with Asymmetric Mode using Multi-interval Taylor Series Method

다분할 테일러급수 해법을 이용한 비대칭 모드를 갖는 아치구조물의 비선형 동적 응답

  • 손수덕 (한국기술교육대학교 건축공학부) ;
  • 하준홍 (한국기술교육대학교 교양학부) ;
  • 이승재 (한국기술교육대학교 건축공학부)
  • Received : 2012.07.05
  • Published : 2012.11.25
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Abstract

This study aims at obtaining an analytical solution of nonlinear dynamic system by using multi-interval Taylor series method, and adopting a arch model to its application. For this purpose, the nonlinear governing equations of the arch are formulated and the examples of symmetric and asymmetric modes are dealt with to obtain the analytical solution within each interval. The results of dynamic analysis using this method is also conducted to evaluate the computed solution for the behavior of the arches subjected to step excitations. As the results, the polynomial series solutions with respect to time are able to be obtained from this method. In this paper, the dynamic snapping of the shallow arches is observed very well from the result of the symmetric mode example, and a change of attractor in phase space is investigated as well. A significant growth of displacement response and the appearance of strange attractor are also manifested in the asymmetric mode model under step excitation. These results from the response and the attractor investigation through Taylor method application reveal that the method is valid in explaining the sensitive behavior of shallow arches. In conclusion, The multi-interval Taylor series method as the result of this work can be applied to deal with the nonlinear differential equations for the sensitive dynamic system like arch structures and the reliability of the analysis results is able to expected.

Keywords

Acknowledgement

Supported by : 한국연구재단

References

  1. 강주원, 김기철, 수동 TMD의 적용을 통한 아치 구조물의 지진응답 제어, 대한건축학회논문집 구조계, 26(7), 37-44, 2010
  2. 강주원, 석근영, 아치구조물의 기둥강성 변화에 따른 지진응답 특성분석을 위한 진동대 실험, 대한건축학회논문집 구조계, 26(1), 87-94, 2010
  3. 김기철, 김광일, 강주원, 면진 트러스 아치 구조물의 지진거동 분석, 한국공간구조학회논문집, 8(2), 73-84, 2008
  4. 김승덕, 윤태영, 손수덕, 스텝하중을 받는 얕은 정현형 아치의 연속 응답 스펙트럼에 의한 동적 좌굴 특성 분석, 대한건축학회논문집 구조계, 20(10), 3-10, 2004
  5. 김연태, 허택녕, 김문겸, 황학주, 비선형 운동해석에 의한 낮은 아치의 동적 임계좌굴하중의 결정, 대한토목학회논문집, 12(2), 43-54, 1992
  6. 박광규, 김문겸, 황학주, 낮은 포물선아치의 동적 안정영역에 관한 연구, 대한토목학회논문집, 6(3), 1-9, 1986
  7. 정찬우, 석근영, 강주원, 갤러킨법을 이용한 아치의 고유진동 해석, 한국공간구조학회논문집, 7(4), 55-61, 2007
  8. Adomian, G., and Rach, R., Generalization of adomian polynomials to functions of several variables, Computers & mathematics with Applications, 24, 11-24, 1992
  9. Adomian, G., and Rach, R., Modified adomian polynomials, Mathematical and Computer Modeling, 24, 39-46, 1996
  10. Ariaratnam, S.T., and Sankar, T.S., Dynamic snap-through of shallow arches under stochastic loads, AIAA Journal, 6(5), 798-802, 1968 https://doi.org/10.2514/3.4601
  11. Barrio, R., Performance of the Taylor series method for ODEs/DAEs, Applied Mathematics and Computation, 163, 525-545, 2005 https://doi.org/10.1016/j.amc.2004.02.015
  12. Barrio, R., Blesa, F., and Lara, M., VSVO Formulation of the Taylor method for the numerical solution of ODEs, Computers & mathematics with Applications, 50, 93-111, 2005 https://doi.org/10.1016/j.camwa.2005.02.010
  13. Bi, Q., and Dai, H.H., Analysis of non-linear dynamics and bifurcations of a shallow arch subjected to periodic excitation with internal resonance, Journal of Sound and Vibration, 233(4), 557-571, 2000
  14. Blair, K.B., Krousgrill, C.M., and Farris, T.N., Non-linear dynamic response of shallow arches to harmonic forcing, Journal of Sound and Vibration, 194(3), 353-367, 1996 https://doi.org/10.1006/jsvi.1996.0363
  15. Budiansky, B., and Roth, R.S. Axisymmetric dynamic buckling of clamped shallow spherical shells; Collected papers on instability of shells structures, NASA TND-1510, 597-606, 1962
  16. Chen, J.S., and Lin, J.S., Stability of a shallow arch with one end moving at constant speed, International Journal of Non-linear Mechanics, 41, 706-715, 2006 https://doi.org/10.1016/j.ijnonlinmec.2006.04.004
  17. Chen, J.S., and Li, Y.T., Effects of elastic foundation on the snap-through buckling of a shallow arch under a moving point load, International Journal of Solids and Structures, 43, 4220-4237, 2006 https://doi.org/10.1016/j.ijsolstr.2005.04.040
  18. Chowdhury, M.S.H., Hashim, I., and Momani, S., The multistage homotopy-perturbation method: A powerful scheme for handling the Lorenz system, Chaos Solutions & Fractal, 40, 1929-1937, 2009 https://doi.org/10.1016/j.chaos.2007.09.073
  19. De Rosa, M.A., and Franciosi, C., Exact and approximate dynamic analysis of circular arches using DQM, International Journal of Solids and Structures, 37, 1103-1117, 2000 https://doi.org/10.1016/S0020-7683(98)00275-3
  20. Donaldson, M.T., and Plaut, R.H., Dynamic stability boundaries for a sinusoidal shallow arch under pulse loads, AIAA Journal, 21(3), 469-471, 1983 https://doi.org/10.2514/3.8097
  21. He, J.H., Homotopy perturbation method: a new nonlinear analytical technique, Applied Mathematics and Computation, 135, 73-79, 2003 https://doi.org/10.1016/S0096-3003(01)00312-5
  22. He, J.H., Application of homotopy perturbation method to nonlinear wave equations, Chaos Solitons & Fractals, 26, 695-700, 2005 https://doi.org/10.1016/j.chaos.2005.03.006
  23. Hoff, N.J., and Bruce, V.G., Dynamic analysis of the buckling of laterally loaded flat arches, J. Math. Phys., 32(4), 276-288, 1954
  24. Hsu, C.S., Equilibrium configurations of a shallow arch of arbitrary shape and their dynamic stability character, International Journal of Nonlinear Mechanics, 3(2), 113-136, 1968 https://doi.org/10.1016/0020-7462(68)90011-5
  25. Humphreys, J.S., On dynamic snap buckling of shallow arches, AIAA Journal, 4(5), 878-886, 1966 https://doi.org/10.2514/3.3561
  26. Kong, X., Wang, B., and Hu, J., Dynamic snap buckling of an elastoplastic shallow arch with elastically supported and clamped ends, Computer and Structures, 55(1), 163-166, 1995 https://doi.org/10.1016/0045-7949(94)00416-Z
  27. Kounadis, A.N., Raftoyiannis, J., and Mallis, J., Dynamic buckling of an arch model under impact loading, Journal of Sound and Vibration, 134(2), 193-202, 1989 https://doi.org/10.1016/0022-460X(89)90648-2
  28. Lacarbonara, W., and Rega, G., Resonant nonlinear normal modes-part2: activation/ orthogonality conditions for shallow structural systems, International Journal of Non-linear Mechanics, 38, 873-887, 2003 https://doi.org/10.1016/S0020-7462(02)00034-3
  29. Levitas, J., Singer, J., and Weller, T., Global dynamic stability of a shallow arch by poincare-like simple cell mapping, International Journal of Non-linear Mechanics, 32(2), 411-424, 1997 https://doi.org/10.1016/S0020-7462(96)00046-7
  30. Lin, J.S., and Chen, J.S., Dynamic snap-through of a laterally loaded arch under prescribed end motion, International Journal of Solids and Structures, 40, 4769-4787, 2003 https://doi.org/10.1016/S0020-7683(03)00181-1
  31. Lock, M.H., Snapping of a shallow sinusoidal arch under a step pressure load, AIAA Journal, 4(7), 1249-1256, 1996
  32. Sadighi, A., Ganji, D.D., and Ganjavi, B., Travelling wave solutions of the sine-gordon and the coupled sine-gordon equations using the homotopy perturbation method, Scientia Iranica Transaction B: Mechanical Engineering, 16, 189-195, 2007
  33. Sundararajan, V., and Kumani, D.S., Dynamic snapbuckling of shallow arches under inclined loads, AIAA Journal, 10(8), 1090-1091, 1972 https://doi.org/10.2514/3.50304
  34. Tseng, Y.P., Huang, C.S., and Lin, C.J., Dynamic stiffness analysis for in-plane vibrations of arches with variable curvature, Journal of Sound and Vibration, 207(1), 15-31, 1997 https://doi.org/10.1006/jsvi.1997.1112