In this paper the following equation for the parametric excitation of a nonlinear aeroelastic oscillator of seesaw type is considered: θ¨+1−εa0cos(ωt)θ=εF(θ,θ˙,μ). In this equation εF represents the aeroelastic force, μ the wind velocity and ε denotes a small parameter. To study the dynamics of the oscillator we use the method of averaging. In absence of parametric excitation one typically finds that above a critical wind velocity the oscillators rest position becomes unstable and stable oscillations with finite amplitude result. Addition of the parametric excitation changes this simple picture. On changing the wind velocity local bifurcations like pitchfork, saddle-node and Hopf bifurcations lead to new nontrivial critical points and limit cycles in the averaged equations. In addition, a global saddle-connection bifurcation is found which either creates or destroys a limit cycle. Note that critical points and limit cycles in the averaged system correspond to periodic solutions and periodically modulated solutions of the original system. An analysis for the possible stability diagrams of the trivial solution and the location of bifurcations in the parameter space is presented. Finally, the numerical calculations performed match with the obtained analytical results and provide phaseportraits for some especial cases.
Limit cycles appearing from the perturbation of a system with a multiple line of critical points
