Safe Basins of Escape of a Particle from an Asymmetrically Truncated, Quadratic Potential Well under Harmonic Excitation

Escape from a potential well is a classic problem arising in various fields of natural science [1] and engineering [2]. Escape can be caused by different types of excitation, starting with the appropriate initial conditions via harmonic excitation [1] to stochastic noise [2] and impact loading. In the case of nonlinear force fields and harmonic excitation, two basic escape mechanisms were observed, the so-called maximum mechanism and the saddle point mechanism [3]. However, the methods applied in [3] only allow the analysis in the nearby 1:1 resonance manifold. The analytic description of the safe basins of escape from selected benchmark potentials under harmonic excitation was performed in [4]. The current work investigates a more general case where the analysis is not limited to the vicinity of the 1:1 resonance anymore; instead, the excitation frequency can take arbitrary values. The safe basins of escape in the plane of the initial conditions (IC) take the form of an ellipse for the irrational ratio of the eigenfrequency of the potential well and the excitation frequency. However, if the above ratio is rational and thus the resulting motion, consisting of the homogeneous and particular solution, has some periodicity, the safe basins can take completely different shapes, yet their form can be estimated analytically with very high accuracy. ... mehrOverall, a rational ratio of the frequencies leads to larger safe basins or the existence of safe basins even then when there is no safe basin at all for an irrational ratio of the frequencies. Furthermore, the introduced method allows the extension of the excitation by additional harmonic terms. The trade-off one has to pay to make this generalization is that the applicability of the process might be limited to the case of an asymmetrically truncated quadratic potential well.

References
[1] Virgin L. N. et al. (1992) Prediction of escape from a potential well under harmonic excitation. Int. J. Non. Linear. Mech., 27(3): 357–365.
[2] Orlando D. et al. (2017) Influence of the mechanics of escape on the instability of von Mises truss and its control. Procedia Eng., 199: 778–783.
[3] Gendelman, O.V., Karmi, G. Basic mechanisms of escape of a harmonically forced classical particle from a potential well. Nonlinear Dyn 98, 2775–2792 (2019). https://doi.org/10.1007/s11071-019-04985-9
[4] Karmi, G., Kravetc, P. & Gendelman, O. Analytic exploration of safe basins in a benchmark problem of forced escape. Nonlinear Dyn 106, 1573–1589 (2021). https://doi.org/10.1007/s11071-021-06942-x