Analytical criterion of a multimodal snap-through flutter of thin-walled panels with initial imperfections
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ORIGINAL PAPER
Analytical criterion of a multimodal snap-through flutter of thin-walled panels with initial imperfections Valery Pilipchuk
Received: 6 May 2020 / Accepted: 15 October 2020 © Springer Nature B.V. 2020
Abstract This work deals with snap-through flutter dynamics of thin-walled shallow panels accompanied by flexural mode transitions assuming cylindrical bending conditions. The problem is therefore multimodal and, in addition, essentially non-local due to the presence of multiple equilibrium positions. The corresponding analysis is based on the asymptotic of a perfectly flexible panel with a continuous manifold of equilibrium configurations. It is assumed that trajectories of the snap-through dynamics are close to such a manifold, which is interpreted as a family of generating solutions. It is shown that the two-mode approximation depicts major physical specifics of the snapthrough process, whereas higher modes can be reasonably treated as a perturbation. As a main result of the analysis, the analytical estimate for the critical speed of airflow leading to the cyclical snap-through flutter is derived. Keywords Panel flutter · Snap-through dynamics · Slow manifolds
1 Introduction The goal of this study is clarification of physical conditions under which the snap-through dynamics of V. Pilipchuk (B) Department of Mechanical Engineering, Wayne State University, Detroit, MI, USA e-mail: [email protected]
shallow elastic structures with multiple equilibrium positions becomes cyclical due to the energy inflow from aerodynamic loads. The related analysis is complicated by the fact that every snap-through cycle is usually accompanied by modal transitions of spatial forms as the elastic structure tends to avoid potential barriers between its remote equilibrium positions. As a result, the problem becomes both severely nonlinear/nonlocal and multidimensional. Such situations are rather insufficiently covered in the literature although the flutter dynamics of thin-walled elastic structures has been investigated from different standpoints for several decades [8,11–14]. Note that the term ‘flutter’ often applies to quite different structural dynamics, such as the flutter of thin plates and shells in supersonic gas flows, the bending torsional wing flutter observed, for instance, in suspension bridges [4–6,19]. Some recent studies in the area consider the flutter of panels made of non-conventional composite materials and subjected to severe environmental conditions in order to target certain industrial applications. For instance, nanocomposite plates with different weight fractions of randomly oriented carbon nanotubes interacting with highly supersonic flows were studied in [6] through both linear and nonlinear dynamic analyses aimed to investigate the effect of the nanotubes weight fraction onto the flutter and post-flutter condition. The multiplescale perturbation analysis was applied to characterize the flutter boundaries and the type of Hopf bifurcation at flutter. The present study belongs rather to the
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