New Ways Forward for Nonlinear Structural Dynamics

Lead Research Organisation: University of Sheffield
Department Name: Mechanical Engineering


Structural Dynamics (or the Theory of Vibrations), is one of the most important fields of Engineering. Understanding vibrations is vital for new design standards and technologies; it is a key enabler in the design of lighter, greener and safer future-generation structures. A 'Grand Challenge' faced by dynamics is the property of nonlinearity. Unfortunately, almost all real structures are nonlinear to some extent, and highly resistant to mathematical analysis, because mathematics has been built on linear foundations. Although engineers have made progress by using approximations and computer power, they have been denied the insight that comes from exact solutions of structural equations of motion, because those equations have been impossible to solve using traditional methods. The same issue means that it is often impossible to prove that exact solutions even exist, or are unique. The first aim of the programme of research here is to find exact results by non-traditional methods; using state-of-the-art machine learning/evolutionary search methods, based on the PI's 30+ years of experience in nonlinear dynamics and modern machine learning.

Because approximation and computation partly removed the need for exact solutions, engineers turned to a more immediately pressing problem - that of finding equations of motion in the first place. This is often impossible from first principles because the unknown physics of joining processes (e.g. welding), obscures the analysis of all but the simplest built-up structures. The problem was solved by developing 'system identification' (SI) methods, where the required equations were inferred from measured data. Again, linear systems were 'solved' first. Although linear SI proved to have technical difficulties, after fifty years of development, it is now established in working theory and practice which engineers can exploit. Arguably the most powerful technology for linear systems is that of 'modal analysis'; this method has the seemingly miraculous property that problems involving many coupled dynamical systems can be reduced to a set of uncoupled problems, each involving a single mass oscillating on its own spring. Unfortunately - as in the case of exact solutions - modal analysis does not generalise to nonlinear systems.

Lacking an underpinning general technology, engineers have been forced to develop a 'toolbox' philosophy, whereby different types of nonlinear systems require different nonlinear SI (NLSI) methods. Although there have been hints at general approaches, no one technology has emerged as 'the one ring to rule them all'. Some versions of nonlinear modal analysis have been developed, but none exhibit all the desirable properties of the linear theory. The second aim of this programme will be to create a completely general framework for NLSI, which can derive equations of motion together with statistical confidences in their predictions. The programme will also consider new approaches to decoupling nonlinear systems - new ways of looking at nonlinear modal analysis.

The research here will provide very new ways forward in nonlinear dynamics. New and general ways of finding equations of motion will be developed. Given the equations, the programme will provide new ways to solve them; exact solutions to problems which have never been solved before and do not have the prospect of solution using analytical methods. Problems will include: exact solution of nonlinear differential equations; exact and approximate transformation of nonlinear systems into linear ones, and the exact and approximate decoupling of multivariate systems (nonlinear modal analysis).

Creating a research culture with an expectation of finding exact solutions is a truly new way of thinking about nonlinear dynamics. In some ways, new exact solutions will be as important as the discovery of new species in zoology; by dissecting them, one can advance knowledge in the whole subject.


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