Abstract
Considering Peierls-Nabarro (P-N) force and viscous effect of material, the dynamic behavior of one-dimensional infinite metallic thin bar subjected to axially periodic load is investigated. Governing equation, which is sine-Gordon type equation, is derived. By means of collective-coordinates, the partial equation can be reduced to ordinary differential dynamical system to describe motion of breather. Nonlinear dynamic analysis shows that the amplitude and frequency of P-N force would influence positions of hyperbolic saddle points and change subharmonic bifurcation point, while the path to chaos through odd subharmonic bifurcations remains. Several examples are taken to indicate the effects of amplitude and period of P-N force on the dynamical response of the bar. The simulation states that the area of chaos is half-infinite. This area increases along with enhancement of the amplitude of P-N force. And the frequency of P-N force has similar influence on the system.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Zhao Guanghui, Zhang Nianmei, Yang Guitong. Nonlinear complex dynamic phenomena of the perturbed metallic bar considering dissipating effect[J]. Applied Mathematics and Mechanics (English Edition), 2005, 26(2):142–149.
Kivshar Y S, Malomed B A. Dynamics of solitons in nearly integrable systems[J]. Rev Mod Phys, 1989, 61:763–916.
Quintero N R, Sánchez A. DC motion of ac driven SG solitons[J]. Physics Letters A, 1998, 247:161–166.
Forinash K, Willis C R. Nonlinear response of the SG breather to an ac driver[J]. Physica D, 2001, 149:95–106.
Laurent Nana, Timoléon C Kofané, Ernest Kaptouom. Subharmonic and homoclinic bifurcations in the driven and damped SG system[J]. Physica D, 1999, 134:61–74.
Matsuda T. A variational analysis of the collision of solitary solutions[J]. Lett Nuovo Cimento, 1979, 24(7):207–212.
Meyers M A, Chawla K K. Mechanical Metallurgy[M]. Prentice Hall, Inc, New Jersey, 1984.
Shu Xuefeng, Yang Guitong. The influence of material properties on dynamic behavior of structures[C]. In: Senoo M (ed). Proceedings of IMMM’97, Mie University Press, Kamihama, 1997, 279–284.
Bishop A B, Lomdahl P S. Nonlinear dynamics in driven, damped sine-Gordon systems[J]. Physica D, 1986, 18:54–66.
Cicogna G. A theoretical prediction of the threshold for chaos in a Josephson junction[J]. Physics Letters A, 1987, 121(8/9):403–406.
Zhang Nianmei, Yang Guitong. Solitary waves and chaos in nonlinear visco-elastic rod[J]. European Journal of Mechanics A/Solids, 2003, 22(6):917–923.
Author information
Authors and Affiliations
Corresponding author
Additional information
Contributed by YANG Gui-tong, Original Member of Editorial Committee, AMM
Project supported by the National Natural Science Foundation of China (Nos.10172063, 10672112), the Youth Science Foundation of Shanxi Province (No.20051004) and the Youth Academic Leader Foundation of Shanxi Province
Rights and permissions
About this article
Cite this article
Zhao, Gh., Zhang, Nm. & Yang, Gt. Analysis of breather state in thin bar by using collective coordinate. Appl Math Mech 27, 1597–1605 (2006). https://doi.org/10.1007/s10483-006-1202-1
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/s10483-006-1202-1