Quasi-stability and exponential attractors for a non-gradient system---applications to piston-theoretic plates with internal damping

Abstract

We consider a nonlinear (Berger or Von Karman) clamped plate model with a piston-theoretic right hand side-which include non-dissipative, non-conservative lower order terms. The model arises in aeroelasticity when a panel is immersed in a high velocity linear potential flow; in this case the effect of the flow can be captured by a dynamic pressure term written in terms of the material derivative of the plate's displacement. The effect of fully-supported internal damping is studied for both Berger and von Karman dynamics. The non-dissipative nature of the dynamics preclude the use of strong tools such as backward-in-time smallness of velocities and finiteness of the dissipation integral. Modern quasi-stability techniques are utilized to show the existence of compact global attractors and generalized fractal exponential attractors. Specific results depending on the size of the damping parameter and the nonlinearity in force. For the Berger plate, in the presence of large damping, the existence of a proper global attractor (whose fractal dimension is finite in the state space) is shown via a decomposition of the nonlinear dynamics. This leads to the construction of a compact set upon which quasi-stability theory can be implemented. Numerical investigations for appropriate 1-D models are presented which explore and support the abstract results presented herein.2 these two terms is the key driver of interest in this plate problem (and in the general flutter phenomenon). Such a non-dissipative, piston-theoretic model has been intensively studied in for different boundary conditions and resistive damping forces, see [5,9,10].In line with the aeroelasticity literature [16,17] (and references therein), we primarily focus on the theory of large deflections for the plate. Taking the boundary to be clamped (as we do) or hinged, von Karman theory is considered accurate [15,16,24]. Moreover, as we consider thin plates, we do not consider rotational inertia effects [24], nor in plane accelerations [25]. In addition, by making an ad hoc simplification to von Karman's scalar plate model, one arrives at a "friendlier" plate model, known as the Berger equation [3, 33, 39] 2 . Other work on flutter dynamics have considered Berger-like beam and plate models (see [9] and references therein), as well as the scalar von Karman equations [10,13,14,38]. Physically, when the plate has no portion of its boundary free, the Berger approximation of von Karman's dynamics is taken to be valid [20,34,37].Remark 1.1. In 1-D, the appropriate model (reflecting an appropriate singular limit of the full von Karman dynamics [25] as in-plane accelerations vanish) is the 1-D analog of the Berger model presented here (sometimes also referred to as a Timoshenko beam [32], or Krieger beam [30]). In Section 7 we consider some numerical results for this 1-D Berger beam model.Many of the results herein are simultaneously valid for both the von Karman and Berger plate dynamics. However, the principal results presented below in Section 1.4.1 on the existence and ...

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.…”

“…Other mathematical references addressing piston-theoretic models include [25,11,6,12]. The article [25]: (i) performs numerical simulations (which guide the simulations herein); (ii) theoretically addresses various classes of attractors arising in the C beam/plate configuration. 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.…”

“…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.…”

“…Beam (or plate) flutter is a topic of great interest in the engineering literature [7,16,1,34,40,45,41,42], as well as, more recently, the mathematical literature [10,25,31,32] (and many references therein). Though there is a vast (mostly engineering-oriented) literature on flow-structure instabilities, we provide a straight-forward analysis utilizing a 1-D structural model in multiple configurations of interest.…”

“…However, to study the qualitative properties of the dynamics in the post-flutter regime, analysis will require some physical nonlinear restoring force to keep trajectories bounded in time [16,14,10]. Indeed, for the post-flutter dynamics of a given system, the theory of nonlinear dynamical systems is often used, e.g., compact global attractors [12,10] as well as exponential attractors [25,9,12].…”