Abstract
One critical case for the motion of a periodically excited oscillator with continuous and piecewise-linear restoring force is that the motion happens to graze a switching plane between two linear regions of the restoring force. This article presents a numerical scheme for locating the periodic grazing orbit first. Then, through a brief analysis, the article shows that the grazing phenomenon turns the stability trend of the periodic orbit so abruptly that it may be impossible to predict an incident local bifurcation with the variation of a control parameter from the concept of smooth dynamic systems. The numerical simulation in the article well supports the scheme and the analysis, and shows an abundance of grazing phenomena in an engineering range of the excitation frequency.
Highlights
The dynamics of a periodically excited oscillator with piecewise-linear restoring force has drawn great attention in recent years, because the oscillator can model a large number of mechanical systems with clearances and elastic stops
If a periodic orbit of the oscillator starting from a certain initial state grazes a switching plane between two linear regions, the neighboring periodic orbit, due to a small variation of the initial state or a control parameter, either penetrates the switching plane or not
The purpose of this article is to study grazing phenomena of an oscillator with continuous and piecewise-linear restoring force, especially to determine the periodic grazing orbits, and to discuss their effect on local bifurcations from the viewpoint of numerical analysis based on the geometrical concepts of dynamic systems
Summary
The dynamics of a periodically excited oscillator with piecewise-linear restoring force has drawn great attention in recent years, because the oscillator can model a large number of mechanical systems with clearances and elastic stops. The oscillator may behave topologically different near a periodic grazing orbit Such an orbit highlights the nature of an oscillator with piecewise-linear restoring force. To the author's knowledge, the study on grazing phenomena of an oscillator with continuous and piecewise-linear restoring force remains rather untouched, many elastic stops in engineering are not rigid enough to fit the instantaneous restitution law. The purpose of this article is to study grazing phenomena of an oscillator with continuous and piecewise-linear restoring force, especially to determine the periodic grazing orbits, and to discuss their effect on local bifurcations from the viewpoint of numerical analysis based on the geometrical concepts of dynamic systems. I(x, y, t, A), which consists of the periodic excitation and a dissipative restoring force, is continuous and piecewise-linear with respect to state vector v, sufficiently smooth with respect to time t and a control parameter A, and of period To as the excitation. It is easy to prove that the closed curve is an ellipse if the oscillator is linear and harmonically excited, and that the curve passes through point Vs at
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