Dynamical Integrity and Control of Nonlinear Mechanical Oscillators

Abstract

The dynamical integrity of nonlinear mechanical oscillators is analyzed in a systematic way extending a previous authors' work. The definition of the safe basin, which is a crucial point that entails choosing what is dynamically acceptable, is critically reviewed. Two different integrity measures are used to quantify the magnitude of the safe basin. When drawn as functions of a varying parameter, they give the so-called erosion profiles, which are the key tool for studying the variation of dynamical integrity. The main focus is on the practically most interesting cases in which the parameter is the excitation amplitude and the integrity reduces as it increases. With the aim of reducing erosion, namely of shifting the erosion profiles toward larger excitation amplitudes, a control method is then applied. It is based on eliminating the homo/heteroclinic bifurcation of the hilltop saddle, which is the triggering event for the considered erosions, by optimally choosing the shape of the periodic excitation. The erosion curves of four different mechanical oscillators, chosen with the aim of covering some main mechanical, dynamical and control features, are numerically constructed and systematically compared with each other. It is found that the control is always able to shift the erosion profiles, although to different extents. Furthermore, its effectiveness may extend above, sometimes well above, the theoretical predictions. Several supplementary specific issues of dynamics and control interest are discussed in detail.