---

Journal of Applied Nonlinear Dynamics 2(4) (2013) 343--354 | DOI:10.5890/JAND.2013.11.003

Jiazhong Zhang; Liying Chen; Sheng Ren

School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an 710049, P.R. China

Download Full Text PDF

 

Abstract

n comparison with Traditional Galerkin’s Method (TGM), an Approx- imate Inertial Manifolds with Time Delay (AIMTDs) is presented for the model reduction of shallow arch with large deformation, a kind of nonlinear continuous dynamic system governed by partial differ- ential equation with second order in time, and the numerical simulation is also presented for the dynamic buckling analysis of shallow arch under impact. By this method, the nonlinear governing equations, which is a kind of infinite dimensional dynamic system, are studied in the phase space, and the solutions of the governing equations are projected onto the complete space spanned by the eigenfunctions of the linear operator of the governing equations. With the introduction of AIMTDs, it decomposes the infinite-dimensional phase space into two complementary subspaces, namely, a finite-dimensional one and its infinite-dimensional complement, and the relation between these two subspaces is the one with a time delay, that is, the evolution of the high modes is not only relevant to the instantaneous low modes, but also to the past high modes. Hence, the system can be projected onto the subspace with finite-dimension, in combination with the relation between the two subspaces, implying the model is reduced. Then, the nonlinear Galerkin’s procedure is adapted to approach the solution on AIMTDs in which the final equilibrium positions are included. Finally, the method is applied to the numerical simulation of dynamic buck- ling of the shallow arch under impact, and some comparisons between Traditional Galerkin’s procedure and Approximate Inertial Manifolds with Time Delay are given. It can be concluded that the method pre- sented give a guarantee for the truncation of buckling modes as mode expansion is used, and are generally feasible for the model reduction of the nonlinear continuous dynamic systems with second order in time, and it is an efficient numerical method requiring less computing time and high accuracy. More, the AIMTDs can be easily approached by finite element method.

Acknowledgments

This work was supported by the National Fundamental Research Program of China (973 Program), No. 2012CB026002, and the National High Technology Research Program of China (863 Program), No. 2012AA052303.

References

1. [1]	Zhang, J.Z., Campen, D.H., and Zhang, G.Q. (2001), Dynamic stability of doubly curved orthotropic shallow shells under impact, AIAA Journal, 39, 956-961.

2. [2]	Romeo Giulio and Frulla Giaomo (1997), Post-buckling behaviour of graphite/epoxy stiffened panels with initial imperfections subjected to eccentric biaxial compression loading, International Journal of Nonlinear Mechanics, 32, 1017-1033.

3. [3]	Ette, A.M. (1997), Dynamic buckling of an imperfect spherical shell under an axial impulse, International Journal of Nonlinear Mechanics, 32, 201-209.

4. [4]	Ikeda, K. and Murota, K. (1999), Systematic description of imperfect bifurcation behavior of symmetric systems, International Journal of Solid and Structures, 36, 1561-1596.

5. [5]	Tabiei, A. and Jiang, Y. (1998), Instability of laminated cylindrical shells with materials non-linear effect, International Journal of Nonlinear Mechanics, 33, 407-415.

6. [6]	Eduard Riks, Charles Rankin C., and Francis Brogan, A. (1996), On the solution of mode jumping phenomena in thin-Walled shell structures, Computer Methods in Applied Mechanics and Engineering, 136, 59-92.

7. [7]	Cheng, C.J., and Shang, X.C. (1997), Mode jumping of simply supported rectangular plates on nonlinear elastic foundation, International Journal of Nonlinear Mechanics, 32, 161-172.

8. [8]	Varadan, T.K., Prathap G., and Ramani H.V. (1989), Nonlinear free flexural vibration of thin circular cylindrical shells, AIAA Journal, 27, 1303-1304.

9. [9]	Raouf Raouf A., and Palazotto Anthony N. (1994), On the non-linear free vibrations of curved orthotropic panels, International Journal of Nonlinear Mechanics, 29, 507-514.

10. [10]	Eduard Riks, Charles Rankin C., and Francis Brogan, A. (1996), On the solution of mode jumping phenomena in thin-walled shell structures, Computer Methods in Applied Mechanics and Engineering, 136, 59-92.

11. [11]	Hilburger, M.W., Waas, A.M., and Jr. Starnes, J.H. (1997), Modeling the dynamic response and establishing post-buckling/post snap-thru equilibrium of discrete structures via a transient analysis, ASME Journal of Applied Mechanics, 64, 590-595.

12. [12]	Zhang, J.Z. and Campen, D.H. (2003), Stability and bifurcation of doubly curved shallow panels under quasi-static uniform load, International Journal of Non-linear Mechanics, 38, 457-466.

13. [13]	Xu, J.X., Huang, H., Zhang, P.Z., and Zhou, J.Q. (2002), Dynamic stability of shallow arch with elastic supportsapplication in the dynamic stability analysis of inner winding of transformer during short circuit, International Journal of Non-linear Mechanics, 37, 909-920.

14. [14]	Temam, R. (1997), Infinite-Dimensional Dynamical System in Mechanics and Physics, Spinger-Verlag, New York.

15. [15]	Titi, E.S. (1990), On approximate inertial manifolds to the Navier-Stokes equations, Journal of Mathematical Analysis and Applications, 149, 540-557.

16. [16]	Jauberteau, F., Rosier, C., and Temam, R. (1990), A nonlinear Galerkin method for the Navier-Stokes equations, Computer Methods in Applied Mechanics and Engineering, 80, 245-260.

17. [17]	Debussche, A. and Temam, R. (1995), Inertial manifolds with delay, Applied mathematics Letter, 8, 21-24.

18. [18]	Zhang, J.Z., Liu, Y., and Chen, D. M. (2005), Error estimate for influence of model reduction of nonlinear dissipative autonomous dynamical system on long-term behaviors, Applied Mathematics and Mechanics, 26, 938-943.

19. [19]	He, Y.N. and Li, K.T. (1999), Optimum finite element nonlinear Galerkin algorithm for the Navier-Stokes equations, Mathematica Numerica Sinica, 21, 29-38 (in Chinese).

20. [20]	Zhang, J.Z., Ren, S., and Mei, G.H. (2011), Model reduction on inertial manifolds for N-S equations approached by multilevel finite element method, Communications in Nonlinear Science and Numerical Simulation, 16, 195- 205.