The Mapping Dynamics of Periodic Motions for a Harmonic Excited Negative Stiffness Vibro-Impact Oscillator with Narrow Gap of Rigid Constraints

Abstract In this paper, a new vibro-impact oscillator incorporating negative stiffness and dual rigidly constrained narrow gaps is proposed. An analytical approach for determining periodic orbits in such a non-smooth dynamical system is developed, enabling systematic investigation of bifurcation phenomena in mapping dynamics through parametric variations of external harmonic excitation and vibro-impact. Firstly, a linear approximation of the irrational restoring force due to geometric nonlinearity is carried out, and the original nonlinear non-smooth dynamical model can be closely approximated by a linear non-smooth dynamical system. The theoretical framework of mapping dynamics is proposed by defining switching planes, basic mappings and their composite mapping structures for unilateral, bilateral and mixed subharmonic vibro-impact motions. Secondly, the definitions and geometric interpretations of all types of periodic orbits for such a non-smooth system are given in combination with the mapping dynamics, and the existence theorem of the unilateral impacting three times periodic orbits is provided. Then the initial conditions of periodic motions can be obtained analytically. In addition, the effects of the collision coefficient and excitation amplitude on stability and bifurcation are studied, and the critical parameter values for period-doubling and saddle-node bifurcations are analytically determined. Finally, numerical results show good agreement with theoretical analysis, confirming the validity of the stability and bifurcation analysis and revealing the system's complex bifurcation characteristics.