Analysis of periodic delayed vibratory systems using the liapunov-floquet technique

Delay differential equations (DDEs) are used to model a number of vibratory processes. Specifically, time-periodic DDEs describe the tool oscillations in machining operations such as milling and honing. Stability analysis of DDEs has received much attention in the literature. Chebyshev collocation method, semi-discretization, and temporal finite-element analysis allow converting the periodic DDEs into maps for stability analysis. The Chebyshev spectral continuous time approximation (ChSCTA) technique extends the method of approximating the infinitesimal generator of a solution operator of a constant DDE based on pseudospectral differencing and allows one to represent a single periodic DDE as a large-order periodic ordinary differential equation (ODE), which can now be analysed by the techniques developed for periodic ODEs. By application of Liapunov-Floquet Transformation (LFT), the analysis of periodic ODEs can be confined to that of ODEs with constant linear parts which was successfully implemented for various applications in the linear and nonlinear periodic systems analysis. In this work, LFT is applied to a periodic DDE discretized by ChSCTA. The proposed combined approach allows for the stability and time-response analysis of a large-order constant ODE analogue of the periodic DDE by applying LFT. For the discretized version of delayed Mathieu equation used as an illustrative example, the periodic LFT matrix is computed by fitting a Fourier series through every entry evaluated at the equispaced points on the period. However, some implementation issues arise which can be resolved via an order reduction technique in which only the modes corresponding to accurate multipliers are retained in the state transition and LFT matrices.