Extending the functionality of a symbolic computational dynamic solver by using a novel term-tracking method

Symbolic computational dynamic solvers are currently under development in order to provide new and powerful tools for modelling nonlinear dynamical systems. Such solvers consist of two parts; the core solver, which comprises an approximate analytical method based on perturbation, averaging, or harmonic balance, and a specialised term-tracker. A term-tracking approach has been introduced to provide a powerful new feature into computational approximate analytical solutions by highlighting the many mathematical connections that exist, but which are invariably lost through processing, between the physical model of the system, the solution procedure itself, and the final result which is usually expressed in equation form. This is achieved by a highly robust process of term-tracking, recording, and identification of all the symbolic mathematical information within the problem. In this paper, the novel source and evolution encoding method is introduced for the first time and an implementation in Mathematica is described through the development of a specialised algorithm

Constraints Implementation in the Application of Reinforcement Learning to the Reactive Control of a Point Absorber

Here, least-squares policy iteration, a reinforcement learning algorithm, is applied to the reactive control of a wave energy converter for the first time. Simulations of a linear point absorber are used for this analysis. The focus of this study is on the implementation of displacement constraints. The use of a penalty term is effective in teaching the controller to avoid the selection of combinations of the damping and stiffness coefficients that would result in excessive displacements in particular sea states. However, the controller can learn that the actions are bad only after trying them, as shown by the simulations. For this reason, a lower-level control scheme is proposed, which changes the sign of the controller force based on the magnitude of the float displacement and sign of its velocity. Its effectiveness is proven in both regular and irregular waves, although greater care is required for the determination of soft constraints

Multiple scales analyses of the dynamics of weakly nonlinear mechanical systems

This review article starts by addressing the mathematical principles of the perturbation method of multiple scales in the context of mechanical systems which are defined by weakly nonlinear ordinary differential equations. At this stage the paper investigates some different forms of typical nonlinearities which are frequently encountered in machine and structural dynamics. This leads to conclusions relating to the relevance and scope of this popular and versatile method, its strengths, its adaptability and potential for different variant forms, and also its weaknesses. Key examples from the literature are used to develop and consolidate these themes. In addition to this the paper examines the role of term-ordering, the integration of the so-called small (ie, perturbation) parameter within system constants, nondimensionalization and time-scaling, series truncation, inclusion and exclusion of higher order nonlinearities, and typical problems in the handling of secular terms. This general discussion is then applied to models of the dynamics of space tethers given that these systems are nonlinear and necessarily highly susceptible to modelling accuracy, thus offering a rigorous and testing applications case-study area for the multiple scales method. The paper concludes with comments on the use of variants of the multiple scales method, and also on the constraints that the method can bring to expectations of modelling accuracy