Global Chaos in a Periodically Forced, Linear System With a Dead-Zone Restoring Force

Abstract

The Poincare mapping and the corresponding mapping sections for global motions in a linear system possessing a dead-zone restoring force are developed through the switching planes pertaining to the two constraints. The global periodic motions based on the Poincare mapping are determined, and the analysis for the stability and bifurcation of periodic motion is carried out. From the global periodic motions, the global chaos in such a system is investigated numerically. The bifurcation scenario with varying parameters was presented. The mapping structures of periodic and chaotic motions are discussed. The Poincare mapping sections for global chaos are given for illustration. The grazing phenomenon embedded in chaotic motion is observed.

Dynamics of a nearly symmetrical piecewise linear oscillator close to grazing incidence: Modelling and experimental verification

“…The extreme cases where the primary stiffness is zero, the so-called dead zone restoring force, have also been studied, e.g. [8]. Many higher order systems exhibit similar behaviour due to one or more discontinuities, for example rotating machinery with imbalance [9,10] or an impact system with progression [11].…”

“…Piiroinen et al [19] reported period adding for a pendulum grazing with a rigid stop, modelled using a coefficient of restitution approach. The chaotic response of a trilinear piecewise system with dead-zone restoring force was investigated by use of a mapping scheme [8].…”

Homoclinic bifurcations and chaotic dynamics for a piecewise linear system under a periodic excitation and a viscous damping

“…There is no doubt that piecewise smooth linear systems, which can be seen as the approximation of many real nonsmooth systems with some nonlinear characteristics, are good test platforms in which the dynamical behavior can be investigated analytically and numerically at the same time. The dynamical behavior of piecewise linear systems with bilinear or trilinear characteristic has been an active topic of intensive research, see, for example, [21][22][23][24][25][26][27][28][29], in which many kinds of periodic motions, bifurcations and chaotic motions were studied by analytical and numerical techniques. Cao [29] also obtained that three-piecewise approach to an archetypal oscillator is an effective method to study the smooth and discontinuous dynamics.…”

Proportional and derivative control for steady-state vibration mitigation in a piecewise linear beam system

“…Depending on the amount of nonlinearity and damping, the steady-state behavior of controlled and uncontrolled piecewise linear systems may be highly varying [10,11,[17][18][19][20][21][22]. In general, next to harmonic resonances, also super and subharmonic resonances may occur.…”