The Basin Stability of Bi-Stable Friction-Excited Oscillators

Abstract

Stability considerations play a central role in structural dynamics to determine states that are robust against perturbations during the operation. Linear stability concepts, such as the complex eigenvalue analysis, constitute the core of analysis approaches in engineering reality. However, most stability concepts are limited to local perturbations, i.e. they can only measure a state’s stability against small perturbations. Recently, the concept of basin stability has been proposed as a global stability concept for multi-stable systems. As multi-stability is a well-known property of a range of nonlinear dynamical systems, this work studies the basin stability of bi-stable mechanical oscillators that are affected and self-excited by dry friction. The results indicate how the basin stability complements the classical binary stability concepts for quantifying how stable a state is given a set of permissible perturbations.

Dynamical integrity assessment of stable equilibria: a new rapid iterative procedure

“…This phenomenon exists in many and various dynamical systems, ranging from mechanical ones, such as braking systems (generating brake squeal [22,51]) and aircraft landing gears (causing the generation of shimmy vibrations [2,56]), to very different systems such as traffic flow or power grids, where the escape from the BOA can cause traffic jams [41] and power blackouts [11,37,42]. The implications of limited BOA are well-known to scientists dealing with dynamical systems, and indeed several quantitative measures of system robustness exist (often referred to as dynamical integrity measures [44,48,53,54]).…”

“…Furthermore, it is practically unusable experimentally. Probabilistic approaches, based on Monte Carlo sampling, are an alternative method for reducing computational cost [39,50,51,63]; however, they do not provide any insight about the system dynamics, and their outcome is not comparable with integrity measures [32]. The cell mapping method, first developed by Hsu [20,21,52], is probably the most efficient numerical method for BOA estimation; its basic idea is to consider the state space not as a continuum, but rather as a collection of a large number of state cells, with each cell taken as a state entity; this method is computationally very efficient, having the advantage of being perfectly suited for parallel computation [1].…”