Spatially localized vibrations in a rotor subjected to flutter

Abstract

The current push toward lightweight structures in aerospace and aeronautical engineering is leading to slender design airfoils, which are more likely to undergo large deformation, hence experiencing geometrical nonlinearities. The problem of vibration localization in a rotor constituted by N coupled airfoils with plunge and pitch degrees of freedom subjected to flutter instability is considered. For a single airfoil, it is shown that depending on the system parameters, multiple static and dynamic equilibria coexist which may be a fixed point, a limit cycle, or irregular motion. By elastically coupling N airfoils, a simplified rotor model is obtained. The nonlinear dynamical response of the rotor is studied via time integration with particular attention to the emergence of localized vibrating solutions, which have been classified introducing a localization coefficient. Finally, the concept of basin stability is exploited to ascertain the likelihood of the system to converge to a certain localized state as a function of the airstream velocity. We found that homogeneous and slightly localized states are more likely to appear with respect to strongly localized states.

Complex nonlinear dynamics and vibration suppression of conceptual airfoil models: A state-of-the-art overview

“…Bisplinghoff and Ashley 1 pointed out that the dynamic behaviors of a wing in a real flight process can be approximately described by selecting a binary section at 70%-75% of the focal line from the root to the tip of the wing structure. It is important to emphasize that the simplified airfoil model does not completely reflect the real situation, but it is commonly used for the studies of aeroelasticity problems in different areas, such as aircraft wing, [1][2][3][4][5][6][7][8][9] aeroengine turbine blades, 10 and others. A series of research results on conceptual airfoil models have been reported, including bifurcation and stability analysis, flutter control, [35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54] airfoil-based energy harvesting, [55][56][57][58][59][60] etc.…”

“…10 recently investigated the problem of vibration localization in a rotor constituted by multiple coupled airfoils with two degrees of freedom. At the same time, the concept of basin stability introduced by Menck et al,…”

Dynamical integrity assessment of stable equilibria: a new rapid iterative procedure

“…Furthermore, it is practically unusable experimentally. Probabilistic approaches, based on Monte Carlo sampling, are an alternative method for reducing computational cost [39,50,51,63]; however, they do not provide any insight about the system dynamics, and their outcome is not comparable with integrity measures [32]. The cell mapping method, first developed by Hsu [20,21,52], is probably the most efficient numerical method for BOA estimation; its basic idea is to consider the state space not as a continuum, but rather as a collection of a large number of state cells, with each cell taken as a state entity; this method is computationally very efficient, having the advantage of being perfectly suited for parallel computation [1].…”

bSTAB: an open-source software for computing the basin stability of multi-stable dynamical systems

“…Depending on the initial condition, multi-stable systems exhibit different time-asymptotic behavior. Recently, multistability has obtained attention in mechanical systems which are not necessarily cyclic, nor of network form: fluttering airfoils [19], few-degree-offreedom friction oscillators [20][21][22], friction oscillator chains [23], and (hyper-chaotic) self-excited oscillators [24] amongst others. Experimental observations of multi-stable mechanical systems range from bi-stable automotive friction brake vibrations [25] , bi-stable responses of helicopter blades [26], windtunnel airfoil tests [27,28], to small cyclic mechanical structures with gap-induced nonlinear vibration localization [29].…”