Global dynamics and integrity of a two-dof model of a parametrically excited cylindrical shell

Abstract

In this paper the global dynamics and topological integrity of the basins of attraction of a parametrically excited cylindrical shell are investigated through a two-degree-of-freedom reduced order model. This model, as shown in previous authors' works, is capable of describing qualitatively the complex nonlinear static and dynamic buckling behavior of the shell. The discretized model is obtained by employing Donnell shallow shell theory and the Galerkin method. The shell is subjected to an axial static preloading and then to a harmonic axial load. When the static load is between the buckling load and the minimum post-critical load, a three potential well is obtained. Under these circumstances the shell may exhibit pre-and post-buckling solutions confined to each of the potential wells as well as large cross-well motions. The aim of the paper is to analyze in a systematic way the bifurcation sequences arising from each of the three stable static solutions, obtaining in this way the parametric instability and escape boundaries. The global dynamics of the system is analyzed through the evolution of the various basins of attraction in the four-dimensional phase space. The concepts of safe basin and integrity measures quantifying its magnitude are used to obtain the erosion profile of the various solutions. A detailed parametric analysis shows how the basins of the various solutions interfere with each other and how this influences the integrity measures. Special attention is dedicated to the topological integrity of the various solutions confined to the pre-buckling well. This allows one to evaluate the safety and dynamic integrity of the mechanical system. Two characteristic cases, one associated with a sub-critical parametric bifurcation and another with a super-critical parametric bifurcation, are considered in the analysis.Keywords Dynamic integrity · Safe basin erosion · Multi-well system · Escape · Thin-walled shells · Parametric instability 62 P.B. Gonçalves et al.

Influence of Axial Loads on the Nonplanar Vibrations of Cantilever Beams

“…13, along with the reference one in the absence of axial load. Figure 14 shows that for this static load level, subcritical bifurcations are associated with the descending left-side instability boundary, while supercritical bifurcations characterize the ascending right side instability boundary; this is similar to what observed in [31] for a parametrically excited cylindrical shell at principal resonance.…”

Multimode interaction in axially excited cylindrical shells

“…Of course, due to the complexity of the nonlinear shell vibration observed in the present work and the dense frequency spectrum of shell structures, additional research must be conducted to unveil the influence of modal interaction in the nonlinear shell dynamics. This may influence significantly the dynamic integrity of the shell in a dynamic environment [7].…”

Load carrying capacity of systems within a global safety perspective. Part II. Attractor/basin integrity under dynamic excitations

“…Yet, this behavior has no effects in terms of load carrying capacity since it practically ends (in the Koiter sense) at PD low Figure 10 shows that GIM is the most conservative measure of integrity. The inequality GIM<IF is quite uncommon and has not been systematically observed in other works [7][8][9][10], where, however, the driving parameter is usually the dynamic excitation amplitude. A justification is reported in the following Subsection 4.2, based on the different depth of the two potential wells.…”

“…They also highlight some (minor) effect associated with the topological erosion of the reference attractor basin. However, in terms of dynamical integrity, actual erosion profiles are obtained by increasing the excitation amplitude q 1 , at fixed values of the axial load p [7][8][9][10]. In the absence of axial load (p=0) both the GIM(q 1 ) and IF(q 1 ) are reported in Fig.…”