Global dynamics of an autoparametric beam structure

• Published in: Nonlinear dynamics Vol. 88; no. 2; pp. 1329 - 1343
• Main Authors: Yu, Tian-Jun, Zhang, Wei, Yang, Xiao-Dong
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.04.2017; Springer Nature B.V
• Subjects: Automotive Engineering; Beams (structural); Bifurcations; Chaos theory; Classical Mechanics; Computer simulation; Control; Dynamic structural analysis; Dynamical Systems; Economic models; Engineering; Galerkin method; Mechanical Engineering; Multiscale analysis; Nonlinear differential equations; Nonlinear equations; Numerical prediction; Ordinary differential equations; Original Paper; Orthogonality; Resonance; Vibration; Šilnikov orbits; Global dynamics; Autoparametric system; Flexible multi-beam structure; Multi-pulse chaotic motions
• ISSN: 0924-090X; 1573-269X
• DOI: 10.1007/s11071-016-3313-0

Global dynamics of an autoparametric beam structure derived from a flexible L-shaped beam subjected to base excitation with one-to-two internal resonance and principal resonance are investigated. Hamilton’s principle is employed to obtain the nonlinear partial differential governing equations of the multi-beam structure. A linear theoretical analysis is implemented to derive the modal functions, and the orthogonality conditions are established. The analytical modal functions obtained are then adopted to truncate the partial differential governing equations into a set of coupled nonlinear ordinary differential equations via the Galerkin’s procedure. The method of multiple scales is applied to yield a set of autonomous equations of the first-order approximations to the response of the dynamical system. The Energy-Phase method is used to study the global bifurcation and multi-pulse chaotic dynamics of such autoparametric system. The present analysis indicates that the chaotic dynamics results from the existence of Šilnikov’s type of homoclinic orbits and the parameter set for which the system may exhibit chaotic motions in the sense of Smale horseshoes are predicted analytically. Numerical simulations are performed to validate the theoretical results.