Static aeroelasticity

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

“…The pressure can then be written in terms of the down-wash of the fluid W = (∂ t + U ∂ x )w, where w(x, t) is the transverse displacement of the beam, and U is the unperturbed axial flow velocity. This results in a nonlinear expression [1] [19], which is then linearized to produce the so called linear piston theory used on the RHS of the beam:…”

“…For more general studies of fluttering beams and plates (with extensive references), we point to [19]. In the discussion of piston theory, we mention the modern engineering references [41,42], classic engineering references [1,7], and the recent mathematical-minded survey [10].…”

“…To simply determine the stability of the problem as a function of the given parameters, we invoke a standard engineering ansatz [42,19] (and references therein): assume simple harmonic motion in time of the beam according to some dominant (perturbed) frequency ω; we allow possible contribution from all functions s n for n = 1, 2, ..., N via coefficients labeled α n : (5.4) where N is a chosen dimensional truncation. We repeat by multiplying by mode functions s n , and then integration produces an eigenvalue problem in the perturbed frequency ω.…”

“…In this treatment we provide a multifaceted numerical study-from the dynamical systems point of view-for a simplified class of 1-D models representing the physical phenomenon known as aeroelastic flutter. Flutter is fluid-structure (feedback) instability that occurs between elastic displacements of a structure and responsive aerodynamic pressure changes at the flow-structure interface [7,19,10]. The onset of flutter, for particular flow parameters, represents a bifurcation in the linearization of the system [23], and the qualitative properties of the post-flutter dynamics can be analyzed from an infinite dimensional/control-theoretic point of view [24,10,14].…”

“…As with all flutter problems [19,34], and as we will demonstrate in the sequel, the onset of instability can be studied via linear theory-typically as an eigenvalue problem (see e.g., [41,42]). Indeed, if one can model the dynamic load across the structure, the issue is to determine under what conditions the non-conservative flow loading destabilizes structural (eigen)modes.…”