Effect of hidden dynamic states on floquet eigenvalues

Despite the relative simplicity and power of computational Floquet theory, in recent years analysts have encountered a stumbling block in its use. In particular, the emphasis of recent work has been on the addition of complicated aerodynamic theories both for the air loads (e.g., dynamic stall) and for the wake (e.g., vortex-lattice methods). Most such aerodynamic theories involve computational algorithms that imply additional dynamic states (associated with the circulation). Unfortunately, these states are often implicit and cannot, therefore, be perturbed directly as required by Floquet theory. Because of this limitation, dynamicists have resorted to alternative means for overcoming the problem of hidden aerodynamic states. One method is simply to constrain the aerodynamic states during the perturbation dynamics. This is equivalent to truncation of the differential equations. A second approach to the hidden state variables is to allow these aerodynamic states to undergo their own dynamics during perturbations of the structure, but to ignore any direct perturbations of the aerodynamic states. This is equivalent to truncation of the Floquet transition matrix. However, no systematic study has been performed that would indicate the accuracy of either of these strategies. The purpose of this technical note is to demonstrate the degree to which these approaches can be used and to outline the conditions for which they cannot be used.