Analysis of a piecewise linear aeroelastic system with and without tuned vibration absorber
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ORIGINAL PAPER
Analysis of a piecewise linear aeroelastic system with and without tuned vibration absorber János Lelkes
· Tamás Kalmár-Nagy
Received: 25 January 2020 / Accepted: 25 May 2020 © The Author(s) 2020
Abstract The dynamics of a two-degrees-of-freedom (pitch–plunge) aeroelastic system is investigated. The aerodynamic force is modeled as a piecewise linear function of the effective angle of attack. Conditions for admissible (existing) and virtual equilibria are determined. The stability and bifurcations of equilibria are analyzed. We find saddle-node, border collision and rapid bifurcations. The analysis shows that the pitch– plunge model with a simple piecewise linear approximation of the aerodynamic force can reproduce the transition from divergence to the complex aeroelastic phenomenon of stall flutter. A linear tuned vibration absorber is applied to increase stall flutter wind speed and eliminate limit cycle oscillations. The effect of the absorber parameters on the stability of equilibria is investigated using the Liénard–Chipart criterion. We find that with the vibration absorber the onset of the rapid bifurcation can be shifted to higher wind speed or the oscillations can be eliminated altogether. Keywords Piecewise linear system · Aeroelasticity · Bifurcation · Limit cycle oscillations · Linear vibration absorber J. Lelkes (B) · T. Kalmár-Nagy Department of Fluid Mechanics, Faculty of Mechanical Engineering, Budapest University of Technology and Economics, Budapest, Hungary J. Lelkes e-mail: [email protected] T. Kalmár-Nagy e-mail: [email protected]
1 Introduction Nonlinear aeroelastic phenomena affect several types of aeroelastic systems such as flexible wings, helicopter rotor blades and wind turbines. Nonlinear aeroelasticity studies the interactions between inertial, elastic and aerodynamic forces on flexible structures that are exposed to airflow and feature non-negligible nonlinearity [1]. The theory of aeroelasticity is extensively covered in the literature [2–4]. The sources of nonlinearity in aeroelastic systems include geometric nonlinearity, structural nonlinearity, flow separation, friction, free-play in actuators, backlash in gears, nonlinear control laws, oscillating shock waves and other nonlinear phenomena [5,6]. A comprehensive study for such nonlinearities was presented by Lee et al. in [7] together with the derivation of the equations of motion of a 2D airfoil oscillating in pitch and plunge. Dowell et al. [8] summarized the physical basis and the effect of nonlinear aeroelasticity on the flight and its association with limit cycle oscillations (LCO). The effect of structural nonlinearities on the dynamical behavior of the system was investigated in [9,10]. Aerodynamic nonlinearities were studied by Dowell et al. in [11] using the describing function method. A combination of structural nonlinearity and the nonlinear ONERA stall aerodynamic model was investigated in the aeroelastic response of a nonrotating helicopter blade in the work of Tang et al. [12]. The airfoil model
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