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Modeling and analysis of an inextensible beam with inertial and geometric nonlinearities

Applied Mathematics and Mechanics(English Edition) 2024年第12期45卷 2113-2130页

作者： Zhanhuan YAo Tieding GUo Wanzhi QIAo College of Civil and Architecture Engineering Guangxi University School of Aerospace Engineering and Applied Mechanics Tongji University School of Civil and Environment Engineering Harbin Institute of TechnologyShenzhen

The present study focuses on an inextensible beam and its relevant inertia nonlinearity, which are essentially distinct from the commonly treated extensible beam that is dominated by the geometric nonlinearity. Explicitly, by considering a weakly constrained or free end(in the longitudinal direction), the inextensibility assumption and inertial nonlinearity(with and without an initial curvature) are introduced. For a straight beam, a multi-scale analysis of hardening/softening dynamics reveals the effects of the end stiffness/mass. Extending the straight scenario, a refined inextensible curved beam model is further proposed, accounting for both its inertial nonlinearity and geometric nonlinearity induced by the initial curvature. The numerical results for the frequency responses are also presented to illustrate the dynamic effects of the initial curvature and axial constraint, i.e., the end mass and end stiffness.

关键词： 70K30 geometric non-linearity hardening/softening dynamics inertia nonlinearity inextensible beam initial curvature o322

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Mixed-mode fast-slow oscillations in the frequency switching Duffing system with a 1:n frequency ratio

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Applied Mathematics and Mechanics(English Edition) 2024年第12期45卷 2131-2146页

作者： Shiping JIANG Xiujing HAN Hailong YU School of Physics Nanjing University of Science & Technology Faculty of Civil Engineering and Mechanics Jiangsu University School of Energy and Power Engineering Nanjing University of Science & Technology

Sliding fast-slow oscillations are interesting oscillation patterns discovered recently in the Duffing system with frequency switching. Such oscillations have been obtained with a fixed 1:2 low frequency ratio in the previous work. The present paper aims to explore composite fast-slow dynamics when the frequency ratio is variable. As a result,a novel route to composite fast-slow dynamics is obtained. We find that, when presented with variable frequency ratios in a 1:n fashion, the sliding fast-slow oscillations may turn into the ones characterized by the fact that the clusters of large-amplitude oscillations of relaxational type are exhibited in each period of the oscillations, and hence the mixedmode fast-slow oscillations. Depending on whether the transition of the trajectory is from the upper subsystem via the fold bifurcation or not, these interesting oscillations are divided into two classes, both of which are investigated numerically. our study shows that,when the frequency ratio n is increased from n = 3, newly created boundary equilibrium bifurcation points may appear on the original sliding boundary line, which is divided into smaller parts, showing sliding and downward crossing dynamical characteristics. This is the root cause of the clusters, showing large-amplitude oscillations of relaxational type,resulting in the formation of mixed-mode fast-slow oscillations. Thus, a novel route to composite fast-slow dynamics by frequency switching is explained. Besides, the effects of the forcing on the mixed-mode fast-slow oscillations are explored. The magnitude of the forcing frequency may have some effects on the number of large-amplitude oscillations in the clusters. The magnitude of the forcing amplitude determines whether the fast-slow characteristics can be produced.

关键词： 34C23 34C26 Duffing system fold bifurcation frequency switching mixed-mode fast-slow oscillation o322 sliding fast-slow dynamics

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