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Large deflections of a structurally damped panel in a subsonic flow

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Abstract

The large deflections of panels in subsonic flow are considered, specifically a fully clamped von Karman plate accounting for both rotational inertia in plate filaments and (mild) structural damping. The panel is taken to be embedded in the boundary of the positive half-space in \(\mathbb R^3\) containing a linear, subsonic potential flow. Solutions are constructed via a semigroup approach despite the lack of natural dissipativity associated with the generator of the linear dynamics. The flow–plate dynamics are then reduced—via an explicit Neumann-to-Dirichlet (downwash-to-pressure) solver for the flow—to a memory-type dynamical system for the plate. For the non-conservative plate dynamics, a global attractor is explicitly constructed via Lyapunov and recent quasi-stability methods. Finally, it is shown that, via the compactness of the attractor and finiteness of the dissipation integral, all trajectories converge strongly to the set of stationary states.

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Notes

  1. If \(\varOmega \) is a rectangle, no results below are affected.

  2. The perturbation \(\mu >0\) is introduced to dispense with the zero eigenvalue; later that this will be taken as a bounded perturbation on Y and removed to obtain the problem as originally stated.

  3. Without loss of generality, take \(t_0=0\).

  4. Barbalat’s Lemma : Suppose \(f(t) \in C^1(a, \infty )\) and \(\displaystyle \lim _{t\rightarrow \infty } f(t) =\alpha < \infty .\) If \(f'(t)\) is uniformly continuous, then \(\displaystyle \lim _{t\rightarrow \infty } f'(t) = 0.\) In our case, we take \(f(t)=\int _0^{t}\Big |(M_{\alpha }u_t,w)\Big |^2\mathrm{d}\tau \).

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Acknowledgements

The authors would like thank the referees for their critical reading and thoughtful criticisms and suggestions which greatly helped improve the quality of the manuscript. The second author would like to thank the National Science Foundation for its generous support.

Funding

Justin T. Webster would like to thank the National Science Foundation and acknowledge his partial funding from NSF Grant DMS-1907620.

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  1. University of Maryland, Baltimore county, Baltimore, MD, USA

    Abhishek Balakrishna & Justin T. Webster

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  1. Abhishek Balakrishna
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  2. Justin T. Webster
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Correspondence to Abhishek Balakrishna.

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Balakrishna, A., Webster, J.T. Large deflections of a structurally damped panel in a subsonic flow. Nonlinear Dyn 103, 3165–3186 (2021). https://doi.org/10.1007/s11071-020-05805-1

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