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.
Highlights
This work overviews and continues recent investigations by the authors (Rega and Lenci, 2005) on the dynamical integrity of nonlinear mechanical oscillators, an issue which was formerly addressed by Thompson and co-workers (Thompson, 19891 Soliman and Thompson, 19891 Lansbury et al, 1992).It is nowadays realized that, for safe practical applications, dynamic attractors must be paralleled by uncorrupted basins
To better highlight the effect of the increasing amplitude of the controlling superharmonic, we report in Figure 7 the erosion profiles for the fixed excitation amplitude 2 1 6 010025, which corresponds to the sudden fall of the curve of harmonic excitation in Figure 6, Figure 6
The basic observation is that control is always able to shift the erosion profiles corresponding to reference harmonic excitation towards higher excitation amplitudes (Figures 1, 2, 5 and 6), and it is broadly effective, to a different extent, in increasing the range of dynamic reliability of the oscillator, this representing a practically appealing feature
Summary
This work overviews and continues recent investigations by the authors (Rega and Lenci, 2005) on the dynamical integrity of nonlinear mechanical oscillators, an issue which was formerly addressed by Thompson and co-workers (Thompson, 19891 Soliman and Thompson, 19891 Lansbury et al, 1992).It is nowadays realized that, for safe practical applications, dynamic attractors must be paralleled by uncorrupted basins. This work overviews and continues recent investigations by the authors (Rega and Lenci, 2005) on the dynamical integrity of nonlinear mechanical oscillators, an issue which was formerly addressed by Thompson and co-workers (Thompson, 19891 Soliman and Thompson, 19891 Lansbury et al, 1992). Dealing with a system only when the erosion is totally prevented may be too conservative, because it can survive safely well above the relevant threshold, if the erosion is not sharp. These considerations call for a detailed investigation of the issue of dynamical integrity, which is the first objective of this work and schematically consists of various basic steps. Choosing the right definition of “safe basin,” i.e., of what is dynamically acceptable
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