YI Zhuang-peng, ZHANG Yong, WANG Lian-hua. Nonlinear Dynamic Response and Bifurcation Analysis of the Elastically Constrained Shallow Arch[J]. Applied Mathematics and Mechanics, 2013, 34(11): 1182-1196. doi: 10.3879/j.issn.1000-0887.2013.11.008
 Citation: YI Zhuang-peng, ZHANG Yong, WANG Lian-hua. Nonlinear Dynamic Response and Bifurcation Analysis of the Elastically Constrained Shallow Arch[J]. Applied Mathematics and Mechanics, 2013, 34(11): 1182-1196.

Nonlinear Dynamic Response and Bifurcation Analysis of the Elastically Constrained Shallow Arch

doi: 10.3879/j.issn.1000-0887.2013.11.008
Funds:  The National Natural Science Foundation of China(11002030; 11032004); The Program for New Century Excellent Talents in University of China(NCET-09-0335)
• Received Date: 2013-07-05
• Publish Date: 2013-11-15
• When the ends are elastically constrained in vertical and rotation directions for the shallow arch, the natural frequencies and modes are quite different from those of the case of ideal hinged or fixed boundary condition, and the different constraint stiffness will change the nonlinear responses and the parameter fields of various bifurcations under external excitation. The dimensionless dynamic equation was established by introducing the fundamental assumptions of shallow arch, and the method that the effects by the boundary constraint stiffness were considered in the natural frequencies and modes solution was employed, then the full-basis Galerkin discretization and the multi-scale perturbation methods were used to obtain the polar- and Cartesian-type averaging equations, of which the coefficients have one-to-one correspondence with the values of constraint stiffness. With the application of numerical calculation, the dynamic behaviors of the vertical elastically constrained system in the case of one-to-two internal resonance between the lowest two modes under periodic excitation were studied. Both the comparison of calculated results with finite element results and the convergence of the coefficients in averaging equations proved the feasibility of the present method. Also, the numerical results show that there exist several bifurcation points with the variation of the amplitude and frequency of excitation, and the parameter distributions for the occurrence of bifurcations are associated with the values of constraint stiffness. Moreover, there are a series of steady-state solution, periodic solution, quasi-periodic solution and chaotic solution windows in the vicinity of the unstable areas or resonance regions which are connected by the bifurcation points, and the period-doubling bifurcation can be observed with the variation of parameters.
•  [1] Malhotra N, Namachchivaya N S. Chaotic dynamics of shallow arch structures under 1∶2 resonance[J]. Journal of Engineering Mechanics,1997, 123(6): 612-619. [2] Malhotra N, Namachchivaya N S. Chaotic motion of shallow arch structures under 1∶1 internal resonance[J]. Journal of Engineering Mechanics,1997, 123(6): 620-627. [3] Bi Q, Dai H H. Analysis of non-linear dynamics and bifurcations of a shallow arch subjected to periodic excitation with internal resonance[J]. Journal of Sound and Vibration,2000, 233(4): 557-571. [4] 王钟羡, 江波, 孙保昌．周期激励浅拱的全局分岔[J]. 江苏大学学报（自然科学版）, 2004, 25(1): 85-88.(WANG Zhong-xian, JIANG Bo, SUN Bao-chang. Global bifurcation of shallow arch with periodic excitation[J]. Journal of Jiangsu University (Natural Science Edition), 2004, 25(1): 85-88.(in Chinese)) [5] Lacarbonara W, Arafat H N, Nayfeh A H. Non-linear interactions in imperfect beams at veering[J]. International Journal of NonLinear Mechanics,2005, 40(7): 987-1003. [6]Zhou L Q, Chen Y S, Chen F Q. Global bifurcation analysis and chaos of an arch structure with parametric and forced excitation[J]. Mechanics Research Communications,2010, 37(1): 67-71. [6] 刘习军, 陈予恕, 侯书军. 拱型结构在参、强激励下的非线性振动分析[J]. 力学学报, 2000, 32(1): 99-102.(LIU Xi-jun, CHEN Yu-shu, HOU Shu-jun. Analysis of nonlinear vibration of the arch structures under the parametric and forced exciting[J].Acta Mechanica Sinica,2000, 32(1): 99-102.(in Chinese)) [7] Chen J S, Li Y T. Effects of elastic foundation on the snapthrough buckling of a shallow arch under a moving point load[J]. International Journal of Solids and Structures,2006, 43(14/15): 4220-4237. [8] Xu J X, Huang H, Zhang P Z, Zhou J Q. Dynamic stability of shallow arch with elastic supports-application in the dynamic stability analysis of inner winding of transformer during short circuit[J].International Journal of NonLinear Mechanics,2002, 37(4/5): 909-920. [9] Nayfeh A H, Mook D T. Nonlinear Oscillations [M]. New York: John Wiley & Sons Inc, 1979: 320-385. [10] Nayfeh A H, Balachandran B. Applied Nonlinear Dynamics [M]. New York: WileyInterscience, 1995: 423-460. [11] Ermentrout B. Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students [M]. Philadelphia: Siam, 2002: 161-193. [12] Lacarbonara W, Rega G, Nayfeh A H. Resonant non-linear normal modes—part I: analytical treatment for structural one-dimenensional systems[J]. International Journal of nNon-Linear Mechanics,2003, 38(6): 851-872. [13] Lacarbonara W, Rega G. Resonant nonlinear normal modes—part II: activation orthogonality conditions for shallow structural systems[J]. International Journal of Non-Linear Mechanics,2003, 38(6): 873-887.

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