Numerical continuation algorithms rely on derivative calculations to detect and track bifurcations in parameter space. In an experiment, the presence of noise will, in general, prevent any attempt to directly measure a bifurcation point and can significantly deteriorate derivative estimations. To improve CBC robustness to noise and enable (limit-point) bifurcation tracking, a novel approach markedly different from the approaches typically used in a numerical context was taken. The proposed method collects suitably positioned data points and estimates the actual position of the bifurcation using a (polynomial) regression. This approach has the advantage of being simple and robust to noise as the estimated location of the bifurcation point is based on a series of measurements instead of a single derivative. The proposed method was demonstrated on a single-degree-of-freedom oscillator for two different configurations of the nonlinearity. The results were shown to agree very well with reference bifurcation curves calculated from detailed data sets capturing the complete response surface and Gaussian process regression. Compared to this latter approach, the proposed method was also shown to considerably reduce the overall testing time.
The developed method can also be extended to detect the presence of isolated response curves, which are typically very challenging to detect using classical experimental techniques and can lead to a dramatic underestimation of the resonance amplitude of the system. More generally, the work performed in this project paves the way for the use of more-general data regression techniques to build local surrogate models of the tested system and perform continuation in noisy experiments. Surrogate models have the advantage to be evaluated cheaply to numerical accuracy, enabling the use of established numerical methods.
The challenging environment of a wind tunnel was considered to further demonstrate the developed algorithm (and, more generally, CBC) on new experiments. A collaboration with Prof. Lowenberg (Bristol) was established to characterise the flight dynamics of an HAWK aircraft model using CBC. The aircraft exhibits complicated behaviours, including bistability and limit cycle oscillations. Preliminary experiments have shown that the aircraft limit cycle oscillations could be controlled with the aircraft control surfaces. However, important time delays in the feedback control loop prevented a rapid attenuation of the turbulences generated in the open-jet wind tunnel. As such, no clear steady-state response of the aircraft could be reached and CBC could not be applied directly. The improvement of both the controller hardware and software to reduce delays are currently under investigation. Another structure comprising a rigid aerofoil with three degrees of freedom (one in pitch, one in plunge, and one flap) will be tested using CBC. This new rig was designed by the research group of Dr. Djamel Rezgui to reach stall flutter before the classical “linear” flutter and exhibit complicated nonlinear dynamic behaviours such as subcritical limit-cycle oscillations.
The work performed in this project was published in one international, peer-reviewed journal and presented at several conferences, including the ISMA 2016, the IMAC 2017, and the ENOC 2017. The results of my research were also presented in several seminars and are regularly updated on my research website. The creation of an open repository containing CBC algorithms is in preparation. Experimental data collected during this project are also freely available on the Research Data Repository of the University of Bristol.
