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Developing the Next Generation Framework for Testing Nonlinear Dynamic Structures.

Periodic Reporting for period 1 - TNS (Developing the Next Generation Framework for Testing Nonlinear Dynamic Structures.)

Reporting period: 2016-05-01 to 2018-04-30

The constant drive to improve engineering structure performance is increasingly leading to lighter and more flexible designs where nonlinearity is inherent. Though frequently ignored or overlooked at the design stage, nonlinearity is found to affect the dynamics of a significant number of structures and its effects are commonly observed during test campaigns. Nonlinear systems can exhibit a wide range of complicated phenomena inherently very difficult to predict. However, until now, there has been no general-purpose systematic method that can directly measure and characterise such nonlinear dynamic behaviour during laboratory tests; hence it is extremely challenging to incorporate nonlinear features into the model development and validation process.

Control-based continuation (CBC) is a systematic method designed to fill this void. The method uses sensors and actuators to intelligently probe a physical system using a feedback control system in conjunction with numerical continuation algorithms that track different types of behaviour as the system inputs are changed. In this way, CBC modifies, on-line, the excitation signal in order to directly interrogate the nonlinear dynamical features of interest, thus offering the best conditions to analyse them in detail. There is the potential to apply CBC to dynamic experiments across the breadth of engineering but, while demonstrating great promises, the method lacks robustness and cannot currently be applied to real-world applications.

The aim of this research is to turn CBC into a more general framework for testing nonlinear dynamic structures. In particular, the algorithms currently used within CBC are ideal for mathematical models as they can be cheaply evaluated to high precision. Neither of these benefits is realised for an experiment where measurement noise is present. As a result, experiments could only be performed in ideal environments with low noise levels.

In this project, a novel continuation algorithm that is robust to noise was developed and demonstrated experimentally by tracking the limit-point bifurcations of a nonlinear oscillator with adjustable nonlinearity. The project also pioneered the application of the method to wind tunnel experiments.
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.
The complex phenomena emerging from the presence of nonlinearity often leads to important delays and additional development costs in industry because existing methods cannot fully encompass nonlinear dynamic phenomena. This Marie-Curie Fellowship has contributed to the developments of a rigorous, general and systematic method for testing nonlinear dynamic systems. In particular, CBC was turned into a more robust and versatile tool that can track limit-point bifurcations directly during tests. Experimental data collected using CBC is valuable for the development of mathematical models and brings new perspectives in nonlinear model validation, and nonlinear design.

The academic research community shows an increasingly important interest for CBC. Following this project, I was invited as early-career keynote speaker at the final meeting of the Engineering Nonlinearity EPSRC programme grant (Sheffield, July 2017) and I will deliver a one-hour tutorial talk at the International Modal Analysis Conference in January 2018. The broad applicability of CBC also offers perspectives of new research directions and collaborations. For instance, the application of CBC to wind tunnel experiments was initiated in this project. Industry is also showing increasing interest for CBC. For instance, a company involved in the study of rotating systems has recently funded a PhD student to work on the application of CBC to their specific problem.