The Evolution of Analytical Treatments of Vibro-Impact Oscillators

Abstract

Prior to 1980 vibro-impact systems had been studied, primarily in the context of impact dampers, by several researchers. The focus of those efforts was on the existence and stability of simple periodic responses, and their utility in vibration attenuation, generally using models with two degrees of freedom (DOF). The treatments of single DOF systems by Senator and two DOF systems by Masri and Caughey were the most sophisticated and thorough of these works. In the early 1980’s, simulation studies by Thompson and Ghaffari showed that even the simplest single DOF vibro-impact model displayed a rich variety of previously unexpected responses, including chaos, and exhibited several interesting bifurcations relating these responses as system and/or excitation parameters were varied.  Around the same time Holmes showed that a simple model for a ball bouncing on a vibrating table (a vibro-impact system with gravity as the restoring force) possessed Smale horseshoes, that is, the topology required for chaotic dynamics. Those works motivated my Ph.D. research on piecewise linear single DOF systems, including the limiting case of vibro-impact dynamics. We attempted to attack these problems by casting the traditional approaches used for analyzing periodic responses in these systems in the form of a more general Poincare map. This approach allowed for systematic local and bifurcation analyses, and provided the means to show that chaos, in the form of horseshoes were present in the system models. It was after completion of my Ph.D. in 1983 that I learned of the extensive, earlier work of Peterka on these systems, which included detailed observations about bifurcations and chaos using analog computer simulations. Also, around this time a number of people started to work on vibro-impact systems, focusing on the peculiarities that arise due to the non-smooth nature of the system dynamics. Most notable of these was the work by Nordmark, who developed systematic methods for investigating grazing dynamics and their attendant bifurcations, providing results which laid the foundation for many subsequent studies. In this paper I will briefly outline some of this history and show how these early studies of vibro-impact systems paved the way for several ongoing efforts in the area of piece-wise smooth systems. I will also show some very recent results by others on vibro-impact systems in some new application areas, including micro-systems and turbomachinery.