The maximal transient energy level over the plane of forcing parameters˜ξparameters˜ parameters˜ξ(Ω, f ); a) 3D plot, b) 2D projection with iso-energy level lines. Level line˜ξline˜ line˜ξ = 1 corresponds to the escape curve of the potential well.

Contexts in source publication

Context 1

... (Ω) and the latter to f < f s (Ω) and therefore, the lower root is chosen. The latter yields a single positive real root. Recalling that the maximal transient energy level is bounded by the upper bound of the well˜ξwell˜ well˜ξ = 1, an injective mapping between excitation parameters and the resulting maximal energy level is obtained, as shown in Fig. 8. The black curve the divides the plane into two regions corresponds to f s (Ω). The descending and ascending lines correspond to f m,0 (Ω|˜ξΩ|˜ Ω|˜ξ) (f > f s (Ω)) and f m,π (Ω|˜ξΩ|˜ Ω|˜ξ) (f < f s (Ω)), respectively. The yellow region ( ˜ ξ = 1) corresponds to escape from the well (bifurcation of type I). The black level lines are ...

Context 2

... The yellow region ( ˜ ξ = 1) corresponds to escape from the well (bifurcation of type I). The black level lines are iso-energy lines that describe sets of excitation parameters that lead to identical maximal transient energy levels. The escape envelope of the well corresponds to the top iso-energy line of˜ξof˜ of˜ξ = 1. As one can see in Fig. 8, the minimum of the iso-energy lines shifts to the right until the minimum of the escape curve is obtained in the vicinity of Ω = 1. This similarity to weakly-nonlinear potential wells with a linear term is somewhat surprising. Although the current well is purely nonlinear and absent a linear term-the shape of the escape envelope and ...

Context 3

... minimum of the escape curve is obtained in the vicinity of Ω = 1. This similarity to weakly-nonlinear potential wells with a linear term is somewhat surprising. Although the current well is purely nonlinear and absent a linear term-the shape of the escape envelope and the location of its minimum is still preserved. The graphical representation in Fig. 8 gives a full perspective on the predicted response energy levels for any set of excitation parameters. In the perspective of equivalent engineering systems or PEAs, Fig. 8 given the designer a complete understanding of the response of the system or energy absorption capabilities over the parameters space of monochromatic harmonic ...

Context 4

... current well is purely nonlinear and absent a linear term-the shape of the escape envelope and the location of its minimum is still preserved. The graphical representation in Fig. 8 gives a full perspective on the predicted response energy levels for any set of excitation parameters. In the perspective of equivalent engineering systems or PEAs, Fig. 8 given the designer a complete understanding of the response of the system or energy absorption capabilities over the parameters space of monochromatic harmonic ...

Context 5

... The latter separates the plane of excitation parameters into two basins: the escape basin, and the safe basin. The former is characterized by transient energy levels that exceed the critical threshold˜ξthreshold˜ threshold˜ξ = 1, while the latter is associated with lower energy levels˜ξlevels˜ levels˜ξ < 1 and correspond to the yellow basin in Fig. 8. The escape envelope of the well is the perimeter of the safe basin in the excitation parameters plane. The main goal of the current section is to obtain an analytical expression for the escape curve of the particle. This will be performed by leveraging the fact that a type I bifurcation (escape) is a particular case of the type II ...