Stability and Limit Cycle Oscillation Amplitude Prediction for Multi-DOF Aeroelastic Systems with Piecewise Linear Non-Linearities

Abstract

Discontinuous non-linearities such as freeplay and bilinear stiffness are often encountered in aeroelastic systems, sometimes as a result of wear and tear. It is important to predict the effect of such non-linearities on the dynamic behaviour of a system, so that adequate safety guidelines can be drafted. As a consequence, the prediction of the bifurcation behaviour of a system featuring a discontinuous nonlinearity is crucial. Additionally, the post-bifurcation behaviour of the system is also of interest since it may consist of relatively harmless Limit Cycle Oscillations (LCO) of low amplitude or of unexpected catastrophic high amplitude LCOs. In this paper the bifurcation and post-bifurcation behaviour of a simulated Multi-DOF aeroelastic system with bilinear and freeplay nonlinearities are investigated using the Harmonic Balance method and a novel method for the prediction of the bifurcation conditions and LCO amplitudes. The method is based on the fact that the nonlinearities investigated are piecewise linear. The ratios of the real parts of the system eigenvalues in the various ranges of the bilinear spring are used in order to infer LCO amplitude information. By means of a demonstration on a simulated aeroelastic system with piece-wise linear stiffness, it is shown that the proposed approach is successful in yielding the full bifurcation and post-bifurcation behaviour of the system. Comparison of the amplitude predictions obtained from the Harmonic Balance technique and the Piecewise Linearisation proposed approach show that the latter are more consistent and closer to the true amplitudes throughout the airspeed range. The bifurcation analysis is extended to the special case where the inner stiffness of the bilinear spring is equal to zero, i.e. freeplay stiffness. It is shown that the Piecewise Linear analysis fails to capture the bifurcation behaviour for this case, while the Harmonic Balance method still produces some accurate predictions