Nonlinear dynamics of the quasi-zero-stiffness SD oscillator based upon the local and global bifurcation analyses

Zhifeng Hao, Qingjie Cao* (Corresponding Author), Marian Wiercigroch

*Corresponding author for this work

Research output: Contribution to journalArticle

26 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)987-1014
Number of pages28
JournalNonlinear Dynamics
Volume87
Issue number2
Early online date23 Sep 2016
DOIs
Publication statusPublished - 31 Jan 2017

Fingerprint

Local Bifurcations
Global Bifurcation
Nonlinear Dynamics
Stiffness
Zero
Chaos theory
Chaos
Vibration Isolation
Floquet Theory
Continuation Method
Invariant Manifolds
Direct numerical simulation
Topological Structure
Saddle
Bifurcation Diagram
Complex Dynamics
Parameter Space
Attractor
Periodic Solution
Gravity

Keywords

  • Floquet theory
  • Invariant manifold
  • Local and global bifurcation analyses
  • Numerical continuation
  • QZS/HSLDS
  • SD oscillator

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Aerospace Engineering
  • Ocean Engineering
  • Mechanical Engineering
  • Applied Mathematics
  • Electrical and Electronic Engineering

Cite this

Nonlinear dynamics of the quasi-zero-stiffness SD oscillator based upon the local and global bifurcation analyses. / Hao, Zhifeng; Cao, Qingjie (Corresponding Author); Wiercigroch, Marian.

In: Nonlinear Dynamics, Vol. 87, No. 2, 31.01.2017, p. 987-1014.

Research output: Contribution to journalArticle

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abstract = "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.",
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