Subharmonic Melnikov method of four-dimensional non-autonomous systems and application to a rectangular thin plate
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The existence and bifurcations of the subharmonic orbits for four-dimensional non-autonomous nonlinear systems are investigated in this paper. The improved subharmonic Melnikov method is presented by using the periodic transformations and Poincaré map. The theoretical results and the formulas are obtained, which can be used to analyze the subharmonic dynamic responses of four-dimensional non-autonomous nonlinear systems. The improved subharmonic Melnikov method is used to investigate the subharmonic orbits of a simply supported rectangular thin plate under combined parametric and external excitations for verifying the validity and applicability of the method. The theoretical results indicate that the subharmonic orbits can occur in the rectangular thin plate with 1:1 and 1:2 internal resonances. The results of numerical simulation also indicate the existence of the subharmonic orbits for the rectangular thin plate, which can verify the analytical predictions.
KeywordsSubharmonic Melnikov method Bifurcation Periodic solution Rectangular plate
This research was supported by the National Natural Science Foundation of China (No. 11402165 and 11402170). The authors would like to thank the referees for their valuable suggestions.
- 1.Melnikov, V.K.: On the stability of the center for time periodic perturbation. Trans. Moscow Math. Soc. 12, 1–56 (1963)Google Scholar
- 9.Greenspan, B.D., Holmes, P.J.: Homoclinic orbits, subharmonics and global bifurcations in forced oscillations. In: Barenblatt, G., Iooss, G., Joseph, D.D. (eds.) Nonlinear Dynamics and Turbulence. Pitman, London, pp. 172–214 (1983)Google Scholar
- 24.Awrejcewicz, J., Krysko, V.A., Narkaitis, G.G.: Bifurcations of a thin plate-strip excited transversally and axially. Nonlinear Dyn. 32, 187–209 (2003)Google Scholar
- 25.Zhang, W., Chen, J.E., Cao, D.X., Chen, L.H.: Nonlinear dynamic responses of a truss core sandwich plate. Compos. Struct. 108, 367–386 (2014)Google Scholar
- 26.Amabili, M., Karazis, K., Khorshidi, K.: Nonlinear vibrations of rectangular laminated composite plates with different boundary conditions. Int. J. Struct. Stab. Dyn. 11, 673–695 (2011)Google Scholar