Title:

The development of solvers for Symbolic Computational Dynamics

The research in this thesis deals primarily with the solution of the differential equation based models that are encountered in the study of engineering dynamics, and concentrates exclusively on enhancing processes of approximate analytical solution. This enhancement centres initially on the premise that asymptotic methods offer a powerful and adaptable group of formalisms that can be used, with care, to solve a very wide range of nonlinear dynamics problems, represented either as sets of nonlinear ordinary differential equations or as sets of nonlinear partial differential equations and boundary conditions. A further premise is that asymptotic methods can be structured in such a way that the user can apply them algorithmically, and an excellent example of this, which is used in this thesis, is the perturbation method of multiple scales. The final premise is that the algorithmic structure of the solution method lends itself to symbolic computation, and that this then offers the user an extremely powerful baseline tool for investigation, on the assumption that the tool is tested and reasonably validated. It has been shown in this thesis that the perturbation method of multiple scales can be fully automated and that it is then capable of analysing very largescale systems with many degrees of freedom. The usual expectation would be that the symbolic computations required to solve the problem would then give way to a numerical phase during which suitable data is substituted into the equations and then some form of high impact visualisation tool would be implemented for data output. However, numerical methods are highly dependent on the boundary conditions and the data used and they cannot necessarily generate a truly generalised solution to the problem in hand. They act like a 'black box' and as such can sometimes be seen to fail to present clearly the general physics of the systems being examined. The alternative approach in applied dynamics is to use numerical integration directly to solve either the reducedorder, yet physically representative nonlinear model of the system, or even a fuller higher dimensional differential equation model. But it is still the case that numerical output does not necessarily give a general picture, nor does it offer a full understanding of the relative significance of terms within the governing equations In order to enhance the generality offered by approximate analytical modelling a new generation of solver has been proposed in the form of a wider computational study which has been termed Symbolic Computational Dynamics. The application process of Symbolic Computational Dynamics, as it is considered in this thesis, comprises two strands of symbolic computation. The first strand leads to the analytical solution to the differential equation model using a method such as multiple scales, noting that the code for this is termed the core solver. The second strand interrogates that solution continually, as it evolves in a structured manner, so that the underlying mathematicalphysical links are established between terms, quantities, and operators. This process starts from the initial statement of the governing equations and continues right through to the final solution, and the strand of computation that does this is termed the termtracker. Therefore, the requirement of the solver part is to generate a general solution which gives an insight into the physics of the system, and the highly nonlinear interactions that frequently occur. Currently, practical application of these methods tends to be limited to models comprising a few degrees of freedom (or generalised coordinates), as they are mostly applied manually. This restriction is lifted when using a computer to do the analytical calculations and one can envisage moving from a few degrees of freedom to hundreds or even thousands. Therefore the core solver comprises a symbolic approximate analytical method, based here on the perturbation method of multiple scales, and running as a process that is computationally transparent, rather like that which emerges when using a pen and a paper. The termtracker generates all the extra information during the solution procedure. The fully symbolic solution procedure provides detailed information about each step of the analysis, while the termtracker can highlight the connection between the assumptions and decisions that underpin the physical model, the resulting equations of motions, the solution procedure and then the final result in specific equation form. In this study, the concept of a Symbolic Computational Dynamics solver has been advanced in several ways; firstly, an implementable version of early proposals for the Source and Evolution Encoding Method (SEEM) for comprehensive termtracking is introduced, and a computerised basis for this is established. A generalised algorithm has been developed to apply the SEEM potentially to any symbolic solution procedure, so it is not at all limited to use with the perturbation method of multiple scales. The SEEM makes it possible for the analyst to track each individual physical parameter and all physicalmathematical assumptions and simplifications through an approximate analytical solution process. Secondly, the Blueprint visualisation method has been proposed, as a first attempt at visualisation of the results of a Symbolic Computational Dynamic solver. This method can interactively illustrate the connections between equations and the generated SEEM encodings information in a 3D graphical structure. This visualisation provides the analyst with new information that was previously hidden within the structure of the adopted solution procedure. Finally, a combination of the number of quantities in each term and their encoding information is used to define a new normalised parameter, called the Strength Factor (SF). The SF value is implemented into the Blueprint visualisation method. The SF value can assist the user to estimate the strength of each term in an equation. Under a series of predefined conditions some terms can be considered generically negligible and then removed from the analysis; this being a somewhat new result in the application of asymptotic procedures such as multiple scales to problems of nonlinear dynamics. As a further development the possibility of manufacturing a solid threedimensional printed structure for each solution method has been suggested. Moreover, the outputs of the new developments of Symbolic Computational Dynamic solver are discussed for two well known, and somewhat challenging, nonlinear dynamics case studies; those of the parametrically excited pendulum and the autoparametrically coupled beam problem, respectively.
