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

Abstract

We consider a beam model representing the transverse deflections of a one dimensional elastic structure immersed in an axial fluid flow. The model includes a nonlinear elastic restoring force, with damping and non-conservative terms provided through the flow effects. Three different configurations are considered: a clamped panel, a hinged panel, and a flag (a cantilever clamped at the leading edge, free at the trailing edge). After providing the functional framework for the dynamics, recent results on well-posedness and long-time behavior of the associated dynamical system for solutions are presented. Having provided this theoretical context, in-depth numerical stability analyses are provided, focusing both at the onset of flow-induced instability (flutter), and qualitative properties of the post-flutter dynamics across configurations. Modal approximations are utilized, as well as finite difference schemes.

Large deflections of inextensible cantilevers: modeling, theory, and simulation

“…Via this approach, a fully nonlinear (implicit) system of ODEs is obtained by expanding the solution in in-vacuo mode shapes and implementing a Galerkin procedure to determine time-dependent Fourier coefficients. Owing to the complex nature of the nonlinearities for the inextensible beam model, finite difference methods are not developed here, as are used, for instance, in the beam study [25], which compares modal methods and spatially discretized methods.…”

“…As with all flutter problems, the onset of instability can be studied from the point of view of a linear structural theory [13,27] -typically as an eigenvalue problem (see [25,50,56] for recent discussions). However, if one wishes to study dynamics in the post-flutter regime, the analysis will require some nonlinear restoring force that will keep solutions bounded in time [8,26].…”

“…Mathematically, the linear theory of Euler-Bernoulli, Rayleigh, shear, and Timoshenko beams -across all boundary configurations -has been established for some time (see, for instance, the nice survey [22]). Nonlinear beam models, such as Kirchhoff or Krieger-Woinowsky beams, have been considered, typically based on the property of extensibility -see [5,12,24,57] for some older references, as well as the more recent [25,26,38]. Extensible beams are characterized by a nonlinear restoring force that accounts for the effects of stretching on bending; these are often cubic-type semilinear models.…”

“…Based on the model in [32], the paper [26] studied the well-posedness and long-time behavior of a reduced, scalar version in the cantilevered configuration with a nonconservative loading. Further numerical work appearing in [25] addressed unstable extensible beams across all physical configurations.…”

“…Follower forces -maintained tangential forces at the free end -have also recently studied numerically [33,40]. These non-conservative terms can lead to structural bifurcation and associated large deflections through limit cycles oscillations (LCOs) or even chaos [24,25]. More specifically, cantilevers in an axial 1 airflow can experience an instability known as flutter, even at low flow velocities.…”