In this paper, the complicated nonlinear dynamics of the harmonically forced quasi-zero-stiffness SD (smooth and discontinuous) oscillator is investigated via direct numerical simulations. This oscillator considered that the gravity is composed of a lumped mass connected with a vertical spring of positive stiffness and a pair of horizontally compressed springs providing negative stiffness, which can achieve the quasi-zero stiffness widely used in vibration isolation. The local and global bifurcation analyses are implemented to reveal the complex dynamic phenomena of this system. The double-parameter bifurcation diagrams are constructed to demonstrate the overall topological structures for the distribution of various responses in parameter spaces. Using the Floquet theory and parameter continuation method, the local bifurcation patterns of periodic solutions are obtained. Moreover, the global bifurcation mechanisms for the crises of chaos and metamorphoses of basin boundaries are examined by analysing the attractors and attraction basins, exploring the evolutions of invariant manifolds and constructing the basin cells. Meanwhile, additional nonlinear dynamic phenomena and characteristics closely related to the bifurcations are discussed including the resonant tongues, jump phenomena, amplitude–frequency responses, chaotic seas, transient chaos, chaotic saddles, and also their generation mechanisms are presented.
The first two authors would like to acknowledge the financial supports from the NSFC (Grant Nos. 11372082,11572096) and National Basic Research Program (973 Program) of China (Grant No. 2015CB057405). ZF Hao is also indebted to the scholarship for international visiting program of Harbin Institute of Technology and the hospitality of the University of Aberdeen, where professor Yoshisuke Ueda provided many valuable suggestions in improving this work.
- Floquet theory
- Invariant manifold
- Local and global bifurcation analyses
- Numerical continuation
- SD oscillator