Challenges in nonlinear structural dynamics: New optimisation perspectives

nalysis of structural dynamics is of fundamental importance to countless engineering applications. Analyses in both research and industrial settings have traditionally relied on linear or close to linear approximations of the underlying physics. Perhaps the most pervasive framework, modal analysis, has become the default framework for consideration of linear dynamics. Modern hardware and software solutions have placed linear analysis of structural dynamics squarely in the mainstream. However, as demands for stronger and lighter structures increase, and as advanced manufacturing enables more and more intricate geometries, the assumption of linearity is becoming less and less realistic. This thesis envisages three grand challenges for the treatment of nonlinearity in structural dynamics. These are: nonlinear system identification, exact solutions to nonlinear differential equations, and a nonlinear extension to linear modal analysis. Of these challenges, this thesis presents results pertaining to the latter two.

The first component of this thesis is the consideration of methods that may yield exact solutions to nonlinear differential equations. Here, the task of finding an exact solution is cast as a heuristic search problem. The structure of the search problem is analysed with a view to motivate methods that are predisposed to finding exact solutions. To this end, a novel methodology, the affine regression tree, is proposed. The novel approach is compared against alternatives from the literature in an expansive benchmark study.

Also considered, are nonlinear extensions to linear modal analysis. Historically, several frameworks have been proposed, each of which is able to retain only a subset of the properties of the linear case. It is argued here that retention of the utilities of linear modal analysis should be viewed as the criteria for a practical nonlinear modal decomposition. A promising direction is seen to be the recently-proposed framework of Worden and Green. The approach takes a machine-learning viewpoint that requires statistical independence between the modal coordinates. In this thesis, a robust consideration of the method from several directions is attempted. Results from several analyses demonstrate that statistical-independence and other inductive biases can be sufficient for a meaningful nonlinear modal decomposition, opening the door to a practical, nonlinear extension to modal analysis.

The results in this thesis take small but positive steps towards two pressing challenges facing nonlinear structural dynamics. It is hoped that further work will be able to build upon the results presented here to develop a greater understanding and treatment of nonlinearity in structural dynamics and elsewhere.