Shooting-Based Complete Bifurcation Prediction for Aeroelastic Systems with Freeplay

In recent years there have been several applications of numerical continuation approaches to aeroelastic systems with freeplay. While some of these have been successful, the general application of the method to such systems remains problematic. Numerical continuation can fail in the presence of complex bifurcations, numerous nearby periodic solution branches and other factors. In this paper, a three-part procedure for applying numerical continuation to aeroelastic systems with freeplay is proposed, designed to ensure that the complete periodic behavior is identified, even for systems with very complex bifurcation diagrams. First, the equivalent linearization approach is used to determine approximations to the periodic solutions of the nonlinear system. Then, a shooting-based technique is applied separately to each linearized approximation in order to pinpoint the nearest exact periodic solution. This process results in a cloud of periodic solutions, representing points on all the solution branches and sub-branches. Finally, a branch-following shooting procedure is applied to this cloud of points in order to obtain a complete description of every branch of periodic solutions. The methodology is applied to a simple aeroelastic system with three degrees of freedom and freeplay in the control surface. This system has been often studied but never fully characterised. It is shown that the proposed method succeeds in describing the complete bifurcation behaviour of the system and explaining its limit cycle response.