Numerical Continuation of Limit Cycle Oscillations and Bifurcations in High-Aspect-Ratio Wings

Abstract

This paper applies numerical continuation techniques to a nonlinear aeroelastic model of a highly flexible, high-aspect-ratio wing. Using continuation, it is shown that subcritical limit cycle oscillations, which are highly undesirable phenomena previously observed in numerical and experimental studies, can exist due to geometric nonlinearity alone, without need for nonlinear or even unsteady aerodynamics. A fully nonlinear, reduced-order beam model is combined with strip theory and one-parameter continuation is used to directly obtain equilibria and periodic solutions for varying airspeeds. The two-parameter continuation of specific bifurcations (i.e., Hopf points and periodic folds) reveals the sensitivity of these complex dynamics to variations in out-of-plane, in-plane and torsional stiffness and a ‘wash out’ stiffness coupling parameter. Overall, this paper demonstrates the applicability of continuation to nonlinear aeroelastic analysis and shows that complex dynamical phenomena, which cannot be obtained by linear methods or numerical integration, readily exist in this type of system due to geometric nonlinearity.

Efficient aeroelastic beam modelling and the selection of a structural shape basis

“…Each C ν component value is written above the relevant panel; overall a scalar sum of c ν = 1.332 × 10 −3 is achieved. Panels (d)-(f) repeat this test using seventeen shape functions (7,5,5). The better convergence can be directly observed from these latter three plots and an improved convergence measure c ν = 1.262 × 10 −6 is achieved.…”

“…al consider the large amplitude motion of cylindrical shells in supersonic flow, using numerical continuation to highlight the nonlinear characteristics of flutter beyond the linear stability boundary. In [7] Eaton et. al.…”

An extension of the structured singular value to nonlinear systems with application to robust flutter analysis

“…Viceversa, if the bifurcation is subcritical, then for V < V H an unstable LCO exists and this often transitions into a stable one featured by higher amplitudes. This is a far more dangerous scenario since the system will suddenly jump to this LCO branch for V slightly larger than V H , and the absence of oscillations cannot be recovered by simply decreasing V, because of hysteresis [12,36]. Figure 4 shows the corresponding bifurcation diagrams with V on the x-axis and the normalized plunge DOF h b on the y-axis (in case of branches of LCO, this is the maximum value over a period).…”

“…The main idea is to use bifurcation theory [23] to define the conditions by which stability is lost. This technique has been amply used in the aerospace community [1,7,12] and its choice in this particular context is motivated by the fact that equilibria of nonlinear aeroelastic systems typically exhibit loss of stability in the form of limit cycle oscillations (LCO), which can be seen as a limited amplitude flutter. In fact, the onset of LCOs corresponds to a Hopf bifurcation point in the system [8], since the stable branch of equilibria (corresponding to the stable configuration of the system at low speeds) loses stability and meet a branch of periodic solutions.…”

Special Issue: Aeroelasticity

“…The resulting set of closed-form linear ordinary differential equations are solved analytically or by using a Runge-Kutta-Fehlberg algorithm. Eaton et al [3] applied a numerical continuation method to an aeroelastic plant with geometric nonlinearity to predict subcritical limit cycle oscillations. Vio et al [4] investigated transient temperature effects that dominate the aerothermoelastic behavior of hypersonic vehicles.…”