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1. Large Deflections of Inextensible Cantilevers: Modeling, Theory, and Simulation

   https://doi.org/10.1051/mmnp/2020033  

   Deliyianni, Maria; Gudibanda, Varun; Howell, Jason; Thomas Webster, Justin (July 2020, Mathematical Modelling of Natural Phenomena)

   A recent large deflection cantilever model is considered. The principal nonlinear effects come through the beam’s  inextensibility —local arc length preservation—rather than traditional extensible effects attributed to fully restricted boundary conditions. Enforcing inextensibility leads to:  nonlinear stiff- ness  terms, which appear as quasilinear and semilinear effects, as well as  nonlinear inertia  effects, appearing as nonlocal terms that make the beam implicit in the acceleration.In this paper we discuss the derivation of the equations of motion via Hamilton’s principle with a Lagrange multiplier to enforce the  effective inextensibility constraint . We then provide the functional framework for weak and strong solutions before presenting novel results on the existence and uniqueness of strong solutions. A distinguishing feature is that the two types of nonlinear terms prevent independent challenges: the quasilinear nature of the stiffness forces higher topologies for solutions, while the nonlocal inertia requires the consideration of Kelvin-Voigt type damping to close estimates. Finally, a modal approach is used to produce mathematically-oriented numerical simulations that provide insight to the features and limitations of the inextensible model. 

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2. Nonlinear Response of an Inextensible, Cantilevered Beam Subjected to a Nonconservative Follower Force

   https://doi.org/10.1115/1.4042324  

   McHugh, Kevin A.; Dowell, Earl H. (March 2019, Journal of Computational and Nonlinear Dynamics)

   The dynamic stability of a cantilevered beam actuated by a nonconservative follower force has previously been studied for its interesting dynamical properties and its applications to engineering designs such as thrusters. However, most of the literature considers a linear model. A modest number of papers consider a nonlinear model. Here, a system of nonlinear equations is derived from a new energy approach for an inextensible cantilevered beam with a follower force acting upon it. The equations are solved in time, and the agreement is shown with published results for the critical force including the effects of damping (as determined by a linear model). This model readily allows the determination of both in-plane and out-of-plane deflections as well as the constraint force. With this novel transparency into the system dynamics, the nonlinear postcritical limit cycle oscillations (LCO) are studied including a concentration on the force which enforces the inextensibility constraint. 

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3. Vortex Filament Solutions of the Navier‐Stokes Equations

   https://doi.org/10.1002/cpa.22091  

   Bedrossian, Jacob; Germain, Pierre; Harrop‐Griffiths, Benjamin (April 2023, Communications on Pure and Applied Mathematics)

   Abstract We consider solutions of the Navier‐Stokes equations in 3d with vortex filament initial data of arbitrary circulation, that is, initial vorticity given by a divergence‐free vector‐valued measure of arbitrary mass supported on a smooth curve. First, we prove global well‐posedness for perturbations of the Oseen vortex column in scaling‐critical spaces. Second, we prove local well‐posedness (in a sense to be made precise) when the filament is a smooth, closed, non‐self‐intersecting curve. Besides their physical interest, these results are the first to give well‐posedness in a neighborhood of large self‐similar solutions of 3d Navier‐Stokes, as well as solutions that are locally approximately self‐similar. © 2023 Wiley Periodicals LLC. 

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4. Nonlinear inviscid damping near monotonic shear flows

   https://doi.org/10.4310/ACTA.2023.v230.n2.a2  

   Ionescu, Alexandru D.; Jia, Hao (July 2023, Acta Mathematica)

   Tobias Ekholm (Ed.)

   We prove nonlinear asymptotic stability of a large class of monotonic shear flows among solutions of the 2D Euler equations in the channel $$\mathbb{T}\times[0,1]$$. More precisely, we consider shear flows $(b(y),0)$ given by a function $$b$$ which is Gevrey smooth, strictly increasing, and linear outside a compact subset of the interval $(0,1)$ (to avoid boundary contributions which are incompatible with inviscid damping). We also assume that the associated linearized operator satisfies a suitable spectral condition, which is needed to prove linear inviscid damping. Under these assumptions, we show that if $$u$$ is a solution which is a small and Gevrey smooth perturbation of such a shear flow $(b(y),0)$ at time $t=0$, then the velocity field $$u$$ converges strongly to a nearby shear flow as the time goes to infinity. This is the first nonlinear asymptotic stability result for Euler equations around general steady solutions for which the linearized flow cannot be explicitly solved. 

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5. Large deflections of a structurally damped panel in a subsonic flow

   https://doi.org/10.1007/s11071-020-05805-1  

   Balakrishna, Abhishek; Webster, Justin T. (July 2020, Nonlinear Dynamics)

   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 ℝ3 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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