A Cantilevered Extensible Beam in Axial Flow: Semigroup Well-posedness and Postflutter Regimes

Abstract

We consider a cantilevered (clamped-free) beam in an axial potential flow. Certain flow velocities may bring about a bounded-response instability in the structure, termed flutter. As a preliminary analysis, we employ the theory of large deflections and utilize a piston-theoretic approximation of the flow for appropriate parameters, yielding a nonlinear (Berger/Woinowsky-Krieger) beam equation with a non-dissipative RHS. As we obtain this structural model via a simplification, we arrive at a nonstandard nonlinear boundary condition that necessitates careful well-posedness analysis. We account for rotational inertia effects in the beam and discuss technical issues that necessitate this feature.We demonstrate nonlinear semigroup well-posedness of the model with the rotational inertia terms. For the case with no rotational inertia, we utilize a Galerkin approach to establish existence of weak, possibly non-unique, solutions. For the former, inertial model, we prove that the associated non-gradient dynamical system has a compact global attractor. Finally, we study stability regimes and post-flutter dynamics (non-stationary end behaviors) using numerical methods for models with, and without, the rotational inertia terms.

A thorough look at the (in)stability of piston-theoretic beams

“…Yet, even this simplified model yields interesting behaviors worthy of rigorous study, complemented by a thorough numerical examination done from a dynamical systems point of view. Our conclusions verify empirical conclusions from the engineering literature, as well as provide validation for recent abstract (infinite dimensional) results for beams and plates [25,26]. Our studies also serve as a baseline for future studies of more complex models, testing conjectured similarities between these dynamics in certain regimes.…”

“…Remark 1.1. At present, all rigorous mathematical analyses of the conservative (1.2) model (or the 2-D analog) require α > 0 [30,26]. However, when considering purely transverse w dynamics of thin beams and plates, α = 0 is almost always taken (see the discussion in [29,Ch.1]).…”

“…It is certainly noteworthy that the inherited boundary condition is nonlinear and nonlocal (but collapses to the standard free boundary condition if b 1 = b 2 = 0). (1.4) is mathematically non-trivial, as well as numerically challenging- [26] addresses this condition, and we elaborate on it below in Section 6.…”

“…The inclusion of structural nonlinearity distinguishes this work from [42,41,39,27], allowing us to discuss qualitative properties of post-flutter behaviors. Our detailed parameter studies are done while bearing in mind the large body of recent theoretical results, in particular those in [25,10], and those in [26] for cantilevers with piston-theoretic terms.…”

“…We follow much of that analysis there, in particular tracking stability/instability of dynamics with respect to piston-theoretic terms, as well as types and size of damping effects. The recent [26] follows the general theory/numerics approach in [25], but focuses specifically on the CF configuration and the effect of rotational inertia in the beam.…”