Numerical Analysis of Nonlinear Parameter-Dependent Systems with Continuation Methods

Abstract

During the development of most aerospace systems, much effort is spent on deriving detailed models that describe the system dynamics. Powerful analysis tools are then required to extract a comprehensive understanding of the system's behavior throughout its operational envelope from its mathematical description. This dissertation introduces new numerical analysis methods for this purpose, with focus on studying the effect of parameters on the system dynamics. The research extends the methods of numerical bifurcation analysis to address issues specific to the aerospace sector. A framework for bifurcation analysis of multi-parameter systems in the presence of equality constraints on states and parameters is derived first, allowing analysis of particular parts of the operational envelope as specified by the constraints. The approach is then extended to bifurcation analysis of the zero dynamics for systems with input-output structure. To expose how local dynamical properties change throughout the operating envelope of the system, a method for computation of equilibrium conditions that satisfy constraints involving the eigenmodes of the linearized dynamics is developed next. A modification to the pseudo-arclength continuation algorithm underlying these methods is suggested to enable application to problems that are continuous, but only piecewise differentiable. Finally, a method to verify that the operating equilibrium of a system with parameter uncertainty does not experience bifurcation for any parameter combination is derived.