Non-linear vibrations of free-edge thin spherical shells: modal interaction rules and 1:1:2 internal resonance

Abstract

This paper is devoted to the derivation and the analysis of vibrations of shallow spherical shell subjected to large amplitude transverse displacement. The analog for thin shallow shells of von KármánÕs theory for large deflection of plates is used. The validity range of the approximations is assessed by comparing the analytical modal analysis with a numerical solution. The specific case of a free edge is considered. The governing partial differential equations are expanded onto the natural modes of vibration of the shell. The problem is replaced by an infinite set of coupled second-order differential equations with quadratic and cubic non-linear terms. Analytical expressions of the non-linear coefficients are derived and a number of them are found to vanish, as a consequence of the symmetry of revolution of the structure. Then, for all the possible internal resonances, a number of rules are deduced, thus predicting the activation of the energy exchanges between the involved modes. Finally, a specific mode coupling due to a 1:1:2 internal resonance between two companion modes and an axisymmetric mode is studied.

Transition to chaotic vibrations for harmonically forced perfect and imperfect circular plates

“…At the beginnig, for 0.04≤ F ≤0.65, the two configurations, (2,0,C) and (2,0,S) are simultaneously excited. But this regime does not appear to be very stable: it is followed by a modulation, and from F=0.65, the energy is solely transferred to (2,0,S), which is in agreement with the analytical result obtained in [41] where it was shown that in a 1:1:2 resonance, the energy is transferred to one configuration only. The regime appearing in the response is that of a perioddoubling, according to the internal resonance relationship, so that in the Poincaré section two points are visible for each value of the forcing amplitude.…”

“…The gray region (green with online colors) represents the chaotic state, while the two light-gray tongues (yellow with on-line colors) appearing around 5.26 and 10.52 are all the points where the coupling due to the 1:1:2 internal resonance has been numerically observed. In particular, the shape of the coupling 1:1:2 region around mode (0,1) at 10.52 is completely consistent with the theoretical ones that can be found in [41]. For comparison, the limiting value F cr for the perfect plate is reported in Fig.…”

“…The critical parameters in such Galerkin expansions is the number of retained mode in the numerical analysis. In this study, the number of inplane modes has been fixed to 12, a sufficient value to ensure a five-digits accuracy for the cubic non-linear coefficients up to the 15 th modes, and four-digits accuracy up to the 26 th modes [40,41]. This accuracy is sufficient for the truncations we will study in the remainder of the paper.…”

Nonlinear forced vibrations of thin structures with tuned eigenfrequencies: the cases of 1:2:4 and 1:2:2 internal resonances

“…Once the amplitude a 3 is larger than I a , the sdof solution becomes unstable. One can notice that this first instability is completely equivalent to a simple 1:2 internal resonance where the higher mode is excited; see, e.g., [23,24,33]. The instability region is fully characterized by its minimum value:…”

“…Chin and Nayfeh studied the case of a 1:1:2 resonance in a circular cylindrical shell, where only one of the two lowfrequency modes were excited [8]. Thomas et al studied theoretically and experimentally the 1:1:2 resonance occurring in shallow spherical shells, where the driven mode is the high-frequency one [33,34]. The case of a 1:1:1:2 internal resonance occurring in closed circular cylindrical shells was also tackled by Amabili, Pellicano, and Vakakis [5,28].…”