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April 4, 2019 | 1:00 p.m. - 2:00 p.m.
Category: Seminar
Location: Faculty/Administration #1140 | Map
656 W. Kirby
Detroit, MI 48202
Cost: Free

Speaker: Dr. Justin Webster (Assistant Professor, Department of Mathematics and Statistics, University of Maryland, Baltimore County)

Title: Large Deflections of A Cantilever Driven by A Flow


Abstract:
Flutter is a self-excitation instability of an elastic structure in a surrounding fluid flow. Much can be said at the qualitative level about panel, flag, and airfoil flutter, as these phenomena are of great interest in engineering. Mathematically, there is a lack of rigorous analysis of the associated partial differential equation (PDE) models. Beyond the obvious applications in aeroscience,  flutter arises in the biomedical realm and in sustainable energies. Motivated by specific interest in {em piezoelectric energy harvesting}, we consider the large deflections of an elastic cantilever driven by non-conservative flow terms.

We begin with recent results for models of panel flutter, a simpler situation involving a clamped, nonlinearly extensible plate. We then discuss the ways in which the modeling and analysis break down if a portion of the structural boundary is free. We must consider nonlinear restoring forces coming from an inextensibility constraint, rather than local stretching. This leads to nonlinear inertia and stiffness terms, introducing nonlocality and spatial quasilinearity.

To obtain existence of solutions, we utilize a Galerkin procedure with cantilever modes. Due to the nonlinearity, there is no natural weak formulation, and identifying weak limits requires additional compactness, forcing higher topologies for smooth data.  Inertial terms require the addition of strong damping or rotary inertia to provide meaning to each PDE term. Local existence of strong solutions is obtained. Uniqueness follows from a novel decomposition of the dynamics, and superlinearity provides low frequency control, opening the door to long-time considerations. Time permitting, we show recent numerical results for flow-cantilever simulations, and present the 2-D system for an inextensible, cantilevered rectangular plate.

For more information about this event, please contact Department of Mathematics at 3135772479 or math@wayne.edu.