Switching Dynamics of a Periodically Forced Discontinuous System With an Inclined Boundary

Abstract

This paper presents the switching dynamics of flow from one domain into another one in the periodically driven, discontinuous dynamical system. The simple inclined straight line boundary in phase space is considered as a control law for the dynamical system to switch. The normal vector-field product for flow switching on the separation boundary is introduced, and the passability condition of flow to the discontinuous boundary is presented. The sliding and grazing conditions to the separation boundary are presented as well. Using mapping structures, periodic motions of such a discontinuous system are predicted, and the local stability and bifurcation analysis are carried out. Numerical illustrations of periodic motions with grazing to the boundary and/or sliding on the boundary are given, and the normal vector fields are illustrated to show the analytical criteria.

On motions and switchability in a periodically forced, discontinuous system with a parabolic boundary

“…To understand the switching dynamics of discontinuous systems, in 2007 Luo and Rapp [1] used the theory of Luo [2] to study a system with a simple straight line boundary in phase plane. The straight line boundary is a special case in the sliding mode control.…”

“…The mass m is connected with a switchable spring of stiffness k α (α = 1, 2) and a switchable damper of coefficient r α (α = 1, 2) in the α-region. The mass of the oscillator is excited by a periodical force, i.e., P α = Q 0 cos(Ωt + φ) + U α for α = 1, 2 (1) where Q 0 and Ω are excitation amplitude and frequency and U α is constant force. The coordinate system is defined by (x, t) both t and x are time and the displacement of mass, respectively.…”