Evaluation and uncertainty quantification of bifurcation diagram: Landing gear, a case study

Bifurcations are features of dynamical systems and occur when a small change in a system parameter results in a sudden qualitative change in the system behaviour. Such events can occur in linear and non-linear systems. The analysis of this phenomenon is challenging in the presence of non-linearity even for simple systems since they lack the properties and principles common to linear systems, e.g. the superposition principle. In the last few decades, numerical methodologies have been developed in order to efficiently determine bifurcation diagrams. Such techniques include continuation, normal form analysis, harmonic balance, and more recently branch and bounds methods. However, very little work has been considered for quantification of uncertain non-linear systems. In this paper, a methodology is provided to propagate parametric uncertainties and define bounds for the bifurcation diagrams taking into account the factors that most influence the behaviour of the analysed dynamical system. To this end, the developed methodology includes sensitivity analysis and uncertainty quantification. The methodology exploits numerical continuation methods, (high order) singular value decompositions, interval analysis and Bayesian approach, which is adopted in the Kriging method to develop surrogate models . The proposed method is validated considering a complex non-linear system from the aeronautical field: a landing gear (LG) system. The LG system has been the subject of several deterministic studies predicting the occurrence of shimmy, which is a selfexcited oscillation dangerous for the integrity of the aircraft. This phenomenon arises from Hopf bifurcations. The case study explores the effects of uncertainty on this phenomenon.